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THE 



ARCHITECT'S AND BUILDER'S 

POCKET- BOOK 

OF 

MENSURATION, GEOMETRY. GEOMETRICAL PROBLEMS, TRIGG 

NO METRICAL FORMULAS AND TABLES. STRENGTH AND 

STABILITY OF FOUNDATIONS, WALLS. BUTTRESSES, 

PIERS, ARCHES. POSTS, TIES, BEAMS, GIRDERS, 

TRUSSES, FLOORS, ROOFS, ETC. 

IN ADDITION TO WHICH IS 

A GREAT AMOUNT OF CONDENSED INFORMATION: 

STATISTICS AND TABLES RELATING TO CARPENTRV, MASONRY. 
DRAINAGE, PAINTING AND GLAZING, PLUMBING, PLAS- 
TERING, ROOFING, HEATING AND VENTILATION, 
WEIGHTS OF MATERIALS, CAPACITY AND 
DIMENSIONS OF NOTED CHURCHES, 
THEATRES, DOMES, TOWERS, 
SPIRES, ETC., 

WITH A GREAT VARIETY OF MISCELLANEOUS INFORMATION. 

BY 

FRANK EUGENE KIDDER, C.E., Ph.D., 

OONBULTINO ABCHITEOT, DEITVSB, OOLO. 



ILLUSTRATED WITH OVER 500 ENGRAVINGS, MOSTLY FROM ORJGINAL DESIGNS 

TWELFTH EDITION, 

REVISED AND GREATLY ENLARGED. 
INCLUDING A GLOSSARY OF TECHNICAL TERMS — ANCIENT AND MODERN. 

FIRST THOUSAND. 

NEW YORK: 

JOHN WILEY & SONS, 

53 East Tenth Street. 



Engin. Library 

I SI 



coftriqht, 
By F. B. KIDDEB, 



Press of J. T- Little & Co^ 
A.8tor Place, I-iew Y(«k. 



.'/ 









CfliS Booft 



IS RESPECTFULLY DEDICATED TO THOSE WHOSE KINDNESS 
HAS ENABLED ME TO PRODUCE IT. 

TO MY PARENTS, 
WHO GAVE ME THE EDUCATION UPON WHICH IT IS BASED; 

TO MY WIFE, 

FOR HER LOVING SYMPATHY, ENCOURAGEMENT, AND ASSIST- 
ANCE; 

TO ORLANDO W. NORCROSS 

OF WORCESTER, MASS., 

WHOSE SUPERIOR PRACTICAL KNOWLEDGE OF ALL THAT 

PERTAINS TO BUILDING HAS GIVEN ME A MORE 

INTELLIGENT AND PRACTICAL VIEW OF 

THE SCIENCE OF CONSTRUCTION 

THAN I SHOULD OTHERWISE 

HAVE OBTAINED. 



TWELFTH EDITION. 

The following revisions and additions have been made in this 
edition : , 

The chapter on Fireproof Floors has been entirely rewritten and 
ext.ended to conform to present practice, and several pages of re- 
visions and additions have been made in Chapter XXV. 

Several pages of tables relating to iron beams have been omitted, 
and other tables substituted in their place. New tables have been 
added in Chapter XI., giving the strength of H -shaped and rectan- 
gular cast-iron columns, and of the new ** Gray " steel column. A 
special article on the Strength of Cast-iron Bearing Plates has been 
added to Chapter X., and new tables are given in Chapter VI. for 
the Strength of Masonry. 

There are also several changes in Part III., particularly a revision 
of the article on Steam-heating, and several new pages giving the 
cost per square and cubic foot of public and private buildings. 
Altogether there are about one hundred pages of revised and new 
matter in this edition. 

F. E. Kidder. 

Denver, Mcvreh 1, 1895. 



PREFACE TO THE NINTH EDITION. 

Within the past four years the introduction of steel in building 
construction has been so rapid, and the changes thereby occasioned 
in the tables relating to the strength of materials so great, that it 
became necessary to revise all that portion of the book which 
relates to iron and steel coi^struction. After undertaking this 
revision, it was found that the changes would be so groat as to 
necessitate resetting a large portion of the book, and the author 
then decided to improve the opportunity to rearrange Part 111., and 
to make certain additions thereto that he has had in contemplation 
for some time. The present edition, therefore, is largely a new 
book, all of Chapters XXIIl. and XXY., and nearly all of Chapters 
XL, Xlll., and XIV., being rewritten, and one hundred pages of 
new matter added io the second part alone. 

Part 111. has been rearranged and enlarged by about eighty 
pages of miscellaneous information of especial value to architects, 
and a glossary of sixty pages added as an appendix. 

The new matter contained on pages 746-773, it is believed, will 
be of especial interest to architects and draughtsmen, as the data 
there given are not readily accessible elsewhere. It will be noticed 
that in the list of Noted American Architects there are many dates 
wanting; if such readers as may be able to supply them will kindly 
inform the author, he will be greatly obliged. 

The author is always pleased to receive criticism and suggestions, 
and is ever willing to give further explanation of any portion of 
the book that may not be readily understood. 

F. E. KiDDEB. 

Denver, Col., November 3, 1891. 



PEEFAOE. 



In preparing the following pages, it has ever been the aim of 
the author to give to the architects and bnilders of this country 
a r^erenee hook which should be for them what Trautwine's 
** Pocket-Book" is to engineers, — a compendium of practical 
facts, rules, and tables, presented in a form as convenient for 
application as possible, and as reliable as our present knowledge 
will permit. Only so much theory has been given as will render 
the application of the formulas more apparent, and aid the stu- 
dent in understanding, in some measure, the principles upon 
which the formulas are based. It is believed that nothing has 
been given in this book but what has been borne out in practice. 

As this book was not written for engineers^ the more intricate 
problems of building construction, which may fairly be said to 
'iome within the province of the civil engineer, have been omitted. 

Desiring to give as much information as possible likely to be of 
service to architects and builders, the author has borrowed and 
ouoted from many sources, in most cases with the permission of 
the authors. Much practical information has been derived from 
the various handbooks published by the large manufacturers of 
rolled-iron beams, bars, etc. ; and the author has always found the 
publishers willing to aid him whenever requested. 

Although but very little has been taken from Trautwine's 
" Pocket-Book for Engineers," yet this valuable book has served 
the author as a model, which he has tried to imitate as well as the 
difference in the subjects would permit; and if his work shall 
prove of as much value to architects and builders as Mr. Traut- 
wine's has to engineers, he will feel amply rewarded for his 
labor. 



viii PREFACE. 

As it is impossible for the author to verify all of the dimensions 
and miscellaneous information contained in Part III. , he cannot 
speak for their accuracy, except that they were in all cases taken 
from what were considered reliable sources of information. The 
tables in Part II. have been carefully computed, and it is believed 
are free from any large errors. There are so many points of in- 
formation often required by architects and builders, that it is 
difficult for one person to compile them all; and although the 
present volume is by no means a small one, yet the author desires 
to make his work as useful as possible to those for whom it has 
been prepared, and he will therefore be pleased to receive any in- 
formation of a serviceable nature pertaining to architecture or 
building, that it may be inserted in future editions should such 
become necessary, and for the correction of any errors that may 
be found. 

The author, while compiling this volume, has consulted a great 
number of works relating to architecture and building; and as he 
has frequently been asked by students and draughtsmen to refer 
them to books from which they might acquire a better knowledge 
of construction and building, the following list of books is given 
as valuable works on the various subjects indicated by the 
titles: — 

" Notes on Building Construction," compiled for the use of the 
students in the science and art schools. South Kensington, Eng-* 
land. 3 vols. Rivingtons, publishers, London. 

"Building Superintendence," by T. M. Clark, architect and 
professor of architecture, Massachusetts Institute of Technology. 
J. R. Osgood A Co., publishers, Boston. 

" The American House Carpenter" and ** The Theory of Trans- 
verse Strains," both by Mr. R. G. Hatfield, architect, formerly of 
New York. 

** Graphical Analysis of Ro Trusses," by Professor Charles E. 
Green of the University of 3higan. 

"The Fire Protection '*' ' by C. J. H. Wcjpdbury, in- 

spector for the Factory urance Companies. John 

Wiley & Sons, publisl 



PREFACE. ix 

** House Drainage and Water Service,*' by James C. Bayles, 
editor of "The Iron Age" and "The Metal Worker." David 
Williams, publisher, New York. 

"The Builders' Guide and Estimators' Price-Book," and "Plas- 
ter and Plastering, Mortars, and Cements," by Fred. T. Hodgson, 
editor of " The Builder and Wood Worker." Industrial Publica- 
tion Company, New York. 

"Foundations and Concrete Works" and "Art of Building," 
by E. Dobson. Weale's Series, London. 

It would be well if all of the above books might be found in 
every architect's ofl&ce; but if the expense prevents that, the 
ambitious student and draughtsman should at least make himself 
acquainted with their contents. These works will also be found 
of great value to the enterprising builder. 



PREFACE TO THE FOURTH EDITION. 



It is now a little more than two years since " The Architect's 
and Builder's Pocket-Book" was first introduced to the public. 
Daring that time the author has received so many encouraging 
words and suggestions from a large niunber of architects and 
bonders, that he desires to acknowledge their kindness, and to 
express the hope that the book will always merit their com-> 
mendation. 

When preparing the book for publication, especial care and 
tiiooght were given to the second part of the book; trusting 
that, if once well done, it would need but little revision for a 
number of years. The first part, also, it is believed, is quite 
complete in its way. For Part III., however, the author found 
time merely to compile such matter as he believed to be of practi- 
cal value to architects or builders, thinking that, should the book 
prove a success, this part could be easily revised and enlarged; 
and, since the second edition was published, the author has de- 
voted such time as he could command to revising such portions 
as upon investigation seemed to require it, and preparing addi- 
tional matter. 

It is the intention of the author, seconded by the publishers, 
to make each edition of the book more complete and perfect 
than the one preceding, in the hope that it may in time become 
to the architects of the present day what Gwilt's "Encyclopaedia" 
was to those of former days. The great diversity of informa- 
tion, however, required by an architect, or those having to do 



xii PREFACE. 

of time to devote to the work, to make such a book as complete 
as could be desired. 

In the Preface to the first edition it was requested that those 
who might have information or suggestions which would increase 
the value of the book would kindly send them to the author, or 
advise him of any errors that should be discovered. 

Several persons generously replied to this invitation ; and several 
small errors have been corrected, and additional information 
given, as the result. It is believed, however, that there are yet 
many who have thought, at. times, of various ways in which the 
book could be improved, or have in their private note-books 
practical data or suggestions which others in the profession would 
be glad to possess; and it is hoped all such will feel it for the 
interest of. the profession to forward such items to the author. 

Any records or reports of tests of the strength of building 
materials of any kind will be especially appreciated. 

To the list of books given in the former Preface the author 
would add the following, which have been of much assistance 
in the preparation of the pages on steam4ieating, and in his 
professional practice : -^ 

"The Principles of Heating and Ventilation, and their Prac- 
tical Application," by John S. Billings, M.D., LL.D., Sanitary 
Engineer, New York. 

"Steam-Heating for Buildings; or, Hints to Steam-Fitters, by 
William J. Baldwin, M.E. John Wiley & Sons, New York. 

"Steam." Babcock & Wilcox Company, New York and Glas- 



gow. 



CONTENTS. 



PART I. 

PAOV 

AbithmbticaIi Sign? and Characters 3 

Involution  . 3 

Evolution, Scjuark and Cube Root, Rules, and Tables . 4 

Wkiqhts and Measures 25 

Thk Metric System 30 

Scripture and Ancient Measures and Weights .... 33 

Mbnsuration 35 

Geometrical Problems 68 

Table of Chords o . 85 

Hip and Jack Rafters 04 

Trioonombtrv, Formulas and Tables ..•»••«,. 95 



PART II. 

Introdiiction i . , ^ . . 123 

CnAPTEK I. 
Definitions of Terms used in Mechanics 125 

CHAPTER II. 
Foundations •••.. IIM) 

CHAPTER III. 
Masonry Walls 149 

CHAPTER IV. 

Composition and Resolution of Forces. — Centre o 
Gravity ..,,.., 



XIV CONTENTS. 

CHAPTER V. p^^^ 

Bbtainikg Walls • • • . 167 

CHAPTER VL 

StRBNGTH OF MaSONBY 171 

CHAPTER VII. 
Stability of Pibbs and Buttbessbs < 187 

CHAPTER VIIL 
Thb Stability of Abches % 191 

CHAPTER IX. 
Rf^istancb to Tension 206 

CHAPTER X. 

Resistance to Shearing and Strength of Pins . • • • • 238 
Pbopobtions of Cast-Ibon Beabino Plates 242a 

CHAPTER XI. 
Strength of Posts, Struts, and Columns 2ia 

CHAPTER XII. 
Bbnding-Moments 290 

CHAPTER XIII. 

Moments of Inertia and Resistance, and Radius of Gy- 
ration 2&7 

CHAPTER XIV. 

General Principles of the Strength of Beams, and 

Strength of Iron Beams 829 

CHAPTER XV. 

Strength of Cast-Iron, Wooden, and Stone Beams. — 
Solid Built Beams 871 

CHAPTER XVI. 



CONTENTS. XV 

CHAPTER XVII. „,_ 

Stbekoth and Stiffness of Continuous Girders .... S92 

CHAPTER XVIII. 
Flitch Plate Girders 401 

CHAPTER XIX. 
Tr^tssbd Beams 404 

CHAPTER XX. 
Riveted Plate-Iron and Steel Beam Girders 410 

CHAPTER XXI. 
Strength of Cast-Iron -Arch-Girders 422 

CHAPTER XXII. 
Strength and Stiffness of Wooden Floors 425 

CHAPTER XXIII. 
Fire-Proof Floors • . 488 

CHAPTER XXIV. 
Mill Construction 466 

CHAPTER XXV. 

Materials and Methods of Firf^Proof Construction for 

Buildings 467 

CHAPTER XXVI. 
Wooden Roof-Trusses, with Details . 486 

CHAPTER XXVII. 

Iron Roofs and Roof-Trusses, with Details of Construc- 
tion 510 

CHAPTER XXVIII. 
Thbory of Roof-Trusses 521 

CHAPTER XXIX. 
JqIMTS 550 



xvi CONTENTS. 



PART III. 

PA 

Chimneys 5 

Rules for Proportioning Chimneys £ 

Examples of Large Chimneys 5 

Wrought-iron Chimneys 5 

Flow of Gas in Pipes, and Gas Memoranda 5 

Piping a House for Gas 5 

Stairs and Tables of Treads and Risers 5 

Seating Space in Theatres and Schools 5 

Symbols for the Apostles and Saints 5 

Dimensions of the Largest Ringing Bells 5 

Dimensions of the Principal Domes 5 

Dimensions of Clock Faces 5 

Height op Buildings, Columns, Towers; Domes, Spires, etc. 5 
Capacity and Dimensions of Churches, Theatres, Opera 

Houses, etc 5) 

Dimensions of English Cathedrals 5 

Dimensions of Obelisks 5' 

Dimensions of Well-known European and American Build- 
ings 5 

Length and Description of Notable Bridges 6 

Lead Memoranda 6 

Weight of Wrought-iron and Steel (Rules) 6 

Weight of Flat, Square, and Round Iron 6 

Weight of Flat Bar Iron 6 

Weight of Cast-iron Plates 6 

Weight of Lead, Copper, and Brass 6 

Weight of Bolts, Nuts, and Bolt Heads 6 

Weight of Rivets, Nails, and Spikes 6 

Weight of Cast-iron Pipes 6 

Weight of Cast-iron Columns 6 

Weight of Wrought-iron Pipes and Tubes 6 

American and Birmingham Wire Gauges 6 

Galvanized and Black Iron, Plain and Corrugated . . 6 

Memoranda for Excavators and Well Diggers .... 6 

Memoranda for Bricklayers, Tables, etc ^ 6 

Measurement of Stone Work 6 

Description and Capacity of Drain Pipe 6 

Tables of Board Measure of Lumber 6 

'iling Memoranda e 

BANDA FOR PLASTERERS 6 



CJONTENTS. XVU 

PASS 

IXDA FOR Roofers 653 

:lics of Plumbing 6S9 

LXDA FOR PaIXTERS 666 

tSQ COXDCCTORS 667 

[CAL DEFixmoys and Formula ? . . 660 

AND Requirements for Lkcandbscext Lightinu . . 675 

f Glass : Price List, etc .... 687 

TUM 6QS 

lsphalt 6M 

T of Freight Cars 607 

• of Substances per Cubic Foot 697 

OSS AND Weight of Church Bells TOO 

' AN'D Cost of Buildings 701 

LSD Tear of Building Materials 7TO 

T of Cisterns axd Tanks T08 

• AND Composition of Air T06 

isoN of Thermometers 706 

OF Iron caused by Heat 707 

J Point and Expansion of Metals 708 

toPERTiES of Water TOO 

PTioN of Water in Cities 711 

bscence on Brickwork 712 

noN OF Rain-water Conducttors to Roof Surface . 712 

TE Strength of Sulphur, Lead, and Cement . . . 713 

ient of Friction 714 

vE Blue Prints of Tracings 715 

L Wool 716 

TE ILvrdness of Woods 718 

ooD LuMHER Grades 718 

x)wer 719 

' OF Castings (Rules) 719 

)F Drums and Pulleys (Rules for) 7t30 

• of Grindstones 720 

.ANEOUs Memoranda 721 

IONS of Pianos, Wagons, Carriages, etc 722 

' of Sash Weights, Lumber, etc 723 

[VK FoRCK OF Blasting Materials 724 

OF the Wind 725 

iutes 725 

erators 726 

AL MoULiiiNOS 728 

jissicAL Orders ... 729 



XVlll CONTENTS. 

PAGS 

List of Noted Foreign Architects 740 

List of Noted American Architects , » 746 

Architects of Noted Buildinos 753 

Cost of Buildings per Cubic Foot 700 

Cost of Buildings per Square Foot leOg 

Charges and Professional Practice op Architects . . 7607* 

Standard Building Contract 764 

Architectural S(-hools and Classes in the United States 769 

Travelling Fellowships and Scholarships 772 

List of Architectural Books 774 

^fTEAM Heating 776 

Residence Heating 807 

APPENDIX. 

Glossary of Technical Terms, Ancient and Modern, used 

BY Architects, Builders, and Draughtsmen . . . I-53 
Legal Definition of Architectural Terms 54-58 



PART L 



PRACTICAL 



Arithmetic. Geometry, and Trigonometry. 



Rules, Tables, and Problems 



PEACTICAL 



ARITHMETIC AND GEOMETRY. 



SIGNS AND CHARACTERS. 

The following signs an() cliaitictjrt; 3tre generally nsed to denote 
and abbreviate the several mathematical operations : — 

The sign = means equal to, or equality. 

— means minus or less, or subtraction. 
+ means phis, or addition. 
X means midtiplied by, or multiplication, 
-r means divided by, or division. 
2 ( Index or power, meaning that the number to which 
* c they are added is to be squared (^) or cubed {^), 
: is to 1 

:: so is [ Signs of proportion. 
: to J 
J means that the square root of the number before 

which it is placed is required. 
A^ means that the cube root of the number before 

which it is placed is required. 
' the bar indicates that all the numbers under it are 

to be talien together. 
{) the parenthenis means that all the numbers between 

are to be taken as one quantity. 
. means decimal parts; thus, 2.5 means 2^^, 0.46 

means ^^. 
® means degrees, ' minutes, '' seconds. 
•*. means hence. 

INVOLUTION. 

To square a number, multiply the number by itself, and the 
product will be the square; thus, the square ofl8 = 18xl8 = 324. 

The cube of a number is the product obtained by multi-* 
plying the number by itself, and that product by the number 
agftin; thus, the cube of 14 = 14 x 14 x 14 = 2744. 



4 EVOLUTION. 

The fotirtli power of a number is the product obtained 
by multiplyini; tlie number by itself four times; thus, the fourth 
power of 10 = 10 x 10 x 10 X 10 = 10000. 

EVOLUTION. 

Square Boot. — Rule for determining the square root of a 
^umber. 

1st, Divide the given number into periods of two figures each, 
conunencing at the right if it is a whole number, and at the 

• • • « • 

decimal-point if there are decimals; thus, 10286.812(5. 

2d, Find the largest square In the left-hand period, and place its 
root in the quotient; subtract the said square from the left-hand 
period, and to the remainder bring dowu the next period for a new 
dividend. 

3d, Double the root already foiuid, and annex one cipher for a 
trial divisor, see how many times it will go in the dividend, and 
put the number in the quotient; also, in place of the cipher in the 
divisor, multiply this final divisor by the number in the quotient 
just found, and subtract the product from the dividend, and to the 
remainuer bring down the next period for a new dividend, and 
proceed as before. If it should be foiuid that the trial divisor 
cannot be contained in the dividend, bring down the next perio<l 
for a new dividend, and annex another cipher to the trial divisor, 
and put a cipher in the quotient, and proceed as before. 

KxAMPLB. 10236.8126 ( 101.17 square root. 

1 

20l]0236 
201 



2021 ) 3581 
2021 



20227 ) 156026 
141589 



14437 



Cube Root. — To extract the cube root of a number, point off 
the number from right to left into periods of three figm*es each, 
and, if there is a decimal, commence at the decimal-point, and point 
off into periods, going both ways. 

Ascertain the highest root of the first period, and place to right 

of number, as in long division; cube the root thus found, and sub- 

fi-* he first period ; to the remainder annex the next period : 

lae root already found, and multiply by three, and annex 



CUBE ROOT. 



two ciphers for the trial divisor. Find how oftrn this trial divisoi 
is contained in the dividend, and write the result in the root. 

Add together the trial divisor, three times the proiuct of the first 
figure of the root by the second with one cipher annexed, and the 
square of the second figure in the root; multiply the sum by the last 
figure in the root, and subtract from the dividend ; to the remain- 
der annex the next period, and proceed as before. 

When the trial divisor is greater than the dividend, write a cipher 
in the root, annex the next period to the dividend, and proceed as 
before. 

Desired the ^493039. 

493039 ( 79 cube root. 
7 X 7 X 7 = 343 



7x7X3 = 14700 


150039 


7X9X3= 1890 




9X9= 81 




16671 


150039 



Desired the ^4035a3.419. 



403583.419 ( 73.9 cube root. 
7 X 7 X 7 = :343 



7x7x3 = 14700 

7X3X3= 630 

3X3= 9 



15339 



73 X 73 X 3 = 1598700 

7a X 9X3= 19710 

9X9= 81 



1618491 
Desired the ^158252.632929. 



60583 



46017 



14566419 



14566419 



158252.632929 ( 54.09 cube root 
5 X 5 X 5 = 125 



5X5X3 = 7500 

5X4X3= 600 

4X4= 16 



8116 



540 X 540 X 3 = 87480000 

540 X 9X3= 145800 

9X9= 81 



87625881 



33225 



32464 



788632929 



788632929 



TABLE 



OF 



SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND 

RECIPROCALS, 



yroiii 1 to lOS"^* 



The following table, taken from Searle's " Field Engineering,'* 
will be found of great convenience in finding the square, cube, 
square root, cube root, and reciprocal of any number from 1 to 1054. 
The reciprocal of a number is the quotient obtained by dividing 1 
by the number. Thus the recipixxjal of 8 is 1 -r 8 = 0.125, 



SQUARES, CUBES, SQUARE ROOTS, 



No. 



1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
10 
17 
10 
10 

20 

Ll 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
82 
33 
34 
85 
36 
87 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
61 
52 
53 
54 
55 
56 
57 
58 
59 



Squares. 


Cubes. 


Square 
lioots. 


1 


1 


1.0000000 


4 


8 


1.4142136 


9 


27 


1.7020508 


16 


64 


2.0000000 


25 


125 


2.2300680 


36 


216 


2 4494897 


49 


343 


2.6457513 


64 


512 


2.8284271 


81 


T-J9 


. 3.0000000 


100 


1000 


8.1622777 


121 


1331 


8.3166248 


144 


1728 


8.4641016 


169 


2197 


8.6055513 


196 


2744 


8.7a3E74 


225 


8375 


8.8729833 


256 


4096 


4.0000000 


289 


4913 


4.1231056 


324 


5as2 


4.S42C407 


361 


6859 


4.3588989 


400 


8000 


4.4721360 


441 


92G1 


4.1825757 


484 


10348 


4.0904158 


529 


12167 


4.7958315 


576 


13824 


4.8009795 


625 


15025 


6.G0C0000 


676 


17576 


5.C0C0195 


729 


19683 


6.1CG1E24 


784 


21952 


6.2015026 


^1 


24389 


5.3851648 


900 


27000 


6.47T2256 


961 


29791 


5.5677614 


l(m 


82768 


5.C568&42 


1069 


85937 


6.7445626 


1156 


89304 


6.8309519 


1225 


42875 


6.9160798 


1296 


46656 


6.0000000 


1369 


50653 


6.0G27625 


1444 


54872 


6.1644140 


1521 


69319 


6.2449980 


1600 


64000 


6.3245553 


1081 


68921 


6.4001242 


1764 


74088 


6.40)7407 


1849 


79507 


6.5574C85 


1036 


85184 


6.f>l;r496 


2025 


91125 


6.703!D039 


2116 


97236 


6.7G23G00 


2209 


10G823 


6.C55G&46 


2304 


110592 


6.9282032 


2401 


117049 


7.0000000 


2500 


125000 


7.0710G78 


2001 


132651 


7.1414284 


2704 


140608 


7.2111026 


2809 


148877 


7.2801099 


2916 


1574G4 


7.3484692 


3025 


16C375 


7.41G1C85 


8136 


175016 


7.4830148 


3249 


185103 


7.5498344 


8364 


195112 


7.6157731 


8481 


205379 


7.6811457 


8600 


216000 


7.7469667 


3721 


226081 


7.810^197 


3844 


238328 


7.8740079 



Cube Roots. 


Reciprocals. 


1.0000000 


1.000000000 


1.2599210 


.500000000 


1.4422496 


.S333333J^ 


1.5874011 


.250000000 


1.7099759 


.200000000 


1.8171206 


.166006667 


1.9129312 


.142857143 


2.CCC0000 


.125000000 


2.0800637 


.111111111 


2.1544347 


.100000000 


2.2239601 


.090909091 


2.2894286 


.083233333 


8.8513347 


.076923077 


8.4101422 


.0714!C8571 


S.4G62121 


Mimm&t 


2.5198421 


.062500000 


2.5712816 


.05882Rr29 


2.6207414 


.05.5555556 


2.6684016 


.C5i2631579 


2.7144177 


,050000000 


2. -J 589243 


.C4701SC48 


2.8020393 


.04M54545 


2.8438670 


.04C478£G1 


2.C844991 


.041GCe667 


2.C340177 


.c-:ccooooo 


2. £624900 


.l£8461538 


8.CC0G00O 


.0&70C7037 


8.CCG5669 


.035714286 


8.07^168 


.C3448275© 


8.1072325 


.033333333 


8.1413806 


.032.968065 


8.1746021 


.03J250000 


8.207C343 


020303030 


8.23S0118 


.029411765 


8.2710603 


.0:^571429 


8.S019272 


.027777^78 


8.332£218 


.027027027 


B.ZQIOTU 


.0£C315789 


8.3912114 


.025641026 


8.4199519 


.025000000 


8.44021':"2 


.0:4390244 


8.47CC266 


.023809524 


8.C03G981 


.023255814 


8.5C0C483 


.02272727^ 


8.55C8C33 


.022222222 


8.58£Oi79 


.021739130 


8.G088261 


.021276600 


8.G342411 


.fl£G833aS3 


8.6593057 


.020406163 


8.6840314 


.020000000 


8.7G&4298 


.019607843 


8.7325111 


.019280769 


8.7502858 


.018867925 


8.7797031 


.018518519 


8.8029525 


.018181818 


8.8258624 


.017857143 


8.8185011 


.017543860 


8.8708766 


.017241379 


8.8929965 


.016949153 


8.9148678 


.016666667 


8.9364973 


.016393443 


8.9578915 


016129080 



CUBE ROOTS, AND RECIPROCALS. 



9 



No. 


Squares. 


Cubes. 


Square 
icbots. 


Cube Roots. 


Reciprocals. 


C3 


3969 


250047 


7.9372539 


8.9790571 


.015873016 


64 


4096 


262144 


8.0000000 


4.0000000 


.015625000 


65 


4225 


2^46^ 


8.0a225V7 


4.0207256 


.015384015 


60 


4856 


267490 


8.12403^ 


4.011^101 


.015151515 


67 


4489 


800763 


8.1853528 


4.0315480 


.014923373 


68 


4624 


314432 


8.24G2113 


4.0816551 


.014;05882 


69 


4761 


328509 


8.3066239 


4.1015661 


.014492754 


TO 


4900 


813000 


8.3666003 


4.1212S53 


.014285714 


71 


6041 


357911 


8.42G1493 


4.1408178 


.014084307 


73 


51&1 


873248 


8.4852814 


4.1601C76 


.0138888GD 


78 


5329 


889017 


8.5440037 


4.1793390 


.013698630 


74 


5476 


405224 


8.6023253 


4.1988364 


.013513514 


75 


5625 


421875 


8.6602540 


4.2171633 


.013333333 


76 


6776 


438976 


8.7177979 


4.2358236 


.013157895 


77 


5929 


45G533 


8.7749614 


4.2543210 


.0121:87013 


78 


6061 


474552 


8.8317600 


4.2726586 


.012820313 


79 


6241 


493039 


8.8681944 


4.2906404 


.012058228 


80 


6400 


B12000 


8.9442719 


4.3068695 


.012500000 


81 


6561 


631441 


9.0000030 


4.3007487 


.0123450; 9 


82 


6724 


5513C8 


9.0353851 


4.3144815 


.012193122 


83 


6889 


671787 


9.1101836 


4.3320707 


.012016193 


84 


7056 


692704 


9.1G51514 


4.3795191 


011901762 


8S 


7225 


614125 


9.2195445 


4.3968296 


.011761706 


86 


7396 


636056 


9.2730185 


4.4140019 


.011027907 


87 


7569 


658503 


9.3278791 


4.4310176 


.011494253 


88 


T?44 


681472 


9.3806315 


4.4479G02 


.011363636 


89 


7921 


7019G9 


9.4339811 


4.4647451 


.011235955 


90 


8100 


■reoooo 


9.4868330 


4.4814017 


.011111111 


91 


8281 


733571 


9.5393920 


4.4979114 


.010089011 


98 


8164 


778683 


9.5916G30 


4.5143574 


.010369565 


93. 


8649 


801357 


9.6430508 


4.5306519 


.010752688 


94 


8836 


830584 


9.6958597 


4.5468359 


.010638298 


96 


9025 


857375 


9.7467943 


4.5629026 


.010326316 


96 


9216 


884786 


9.7979590 


4.5788570 


.010416667 


97 


9409 


912373 


9.8188578 


4.5917009 


.010309278 


98 


9604 


941192 


9.8994019 


4.6101363 


.010204062 


99 


9601 


970299 


9.9498744 


4.6260650 


.010101010 


100 


10000 


1000000 


10.0000000 


4.6415888 


.010000000 


l(Ml 


10201 


1030301 


10.0498756 


4.657C096 


.OOOJ00990 


lOS 


10404 


1061208 


10.0995019 


4.6723287 


.009803923 


108 


10609 


1092727 


10.1488916 


4.6875482 


.000708738 


104 


10816 


1124864 


10.1980390 


4.702GG94 


.009015385 


106 


11025 


1167625 


10.2469508 


4.7176940 


.009328810 


106 


11236 


1191016 


10.2956301 


4.732G235 


.009133962 


107 


11449 


1225013 


10.3440604 


4.7474594 


.009*45791 


108 


11664 


1259712 


10.3923018 


4.7622032 


.00:259259 


109 


11881 


1295029 


10.4403065 


4.7768562 


.009174312 


110 


12100 


1331000 


10.4880885 


4.7914199 


.009090909 


111 


12321 


1367631 


10.5356538 


4.8058955 


.009009000 


112 


12544 


1404928 


10.5830052 


4.8202815 


.008928571 


118 


12769 


1442897 


10.6801458 


4.8315881 


.003849338 


114 


12996 


1481544 


10.C770783 


4.8188076 


.008771930 


116 


13225 


1520875 


10.7238053 


4.8629442 


.008095652 


116 


13456 


1560696 


10.rr03296 


4.87C9990 


.008C20C90 


117 


13689 


1601613 


10.8166538 


4.8909732 


.008317009 


118 


139^ 


1643032 


10.8627805 


4.9048681 


.008174576 


119 


14161 


1685159 


10.9087121 


4.9186847 


.008403361 


190 


14400 


1728000 


10.9544512 


4.9324242 


.008333333 


m 


14641 


1771561 


11 MJXm 


4.9160674 


.008261463 


Itt 


14884 


1815848 


11.0153610 


4.9596757 


.008196^1 


198 


16189 


1660667 


11.0905365 


4.9731898 


.008130081 


IM 


15376 


1906624 


11.1355287 


4.9866310 


.008064516 



10 



SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square 
Koots. 


Cube Roots. 


Reciprocals. 


123 


15625 


1953125 


11.1303399 


6.0000000 


.006000000 


126 


15876 


20C0376 


11.2^9723 


6.0132979 


.007936508 


127 


16129 


2048383 


11.2094277 


6.0265257 


.007874016 


128 


1G3S4 


2097158 


11.3137085 


6.0396843 


.007818500 


129 


16641 


8146689 


U.8578167 


6.0537748 


.0077S1968 


laa 


16900 


2197000 


11.4017548 


6.0657970 


.007698306 


131 


17161 


2248091 


11.4455231 


- 6.0787531 


.007638588 


1C2 


174S4 


2299968 


11.4891253 


6.0916434 


.007575758 


133 


17G39 


2352637 


11.5325G26 


6.1044687 


.007518797 


VA 


17056 


»106104 


11.5758369 


6.1172299 


.007462687 


1C5 


18225 


8160675 


11.6181^00 


6.1299278 


.007407407 


1:3 


18196 


2515456 


11.6619038 


6.1425638 


.007858041 


1:7 


18769 


2571853 


11.7040099 


6.1551367 


.007S99&nO 


108 


19044 


262807^ 


11.7478401 


6.1676498 


.007^16377 


lo9 


19321 


8685619 


11.7896261 


6.1801015 


. .007194845 


140 


19600 


8744000 


11.8321596 


6.1924941 


.00n48887 


111 


19881 


880S221 


11.874&121 


6.2048279 


.00^02199 


112 


20164 


8863288 


11.9168753 


6.2171034 


.007048854 


143 


20449 


89^1207 


11.9582607 


6.2293215 


.006998007 


144 


20736 


8985984 


12.0000000 


6.3414828 


.006944444 


145 


2102s 


8048625 


12.0415946 


6.2535879 


.006890668 


146 


21316 


8112136 


12.0630460 


6.2656374 


.006848615 


147 


21609 


8170523 


12.1243557 


6.2776321 


.006808781 


148 


21904 


8241793 


12.1055251 


6.2895725 


.0067^6757 


14d 


82201 


8307949 


12.2005556 


6.3014598 


.000711409 


150 


22500 


8375000 


12.»174487 


6.3132928 


•006600067 


151 


22801 


8142951 


12.2882057 


5.3?.>0740 


.0066SS517 


153 


23104 


8611808 


13.3288280 


6.3368088 


.006678047 


153 


23409 


8581577 


12.3693169 


6.3481818 


.006536048 


154 


23716 


8652264 


12.4096736 


6.8601064 


.006498606 


155 


^1025 


8723875 


12.4496996 


6.3710R54 


.006461018 


156 


24336 


8796416 


13.4899060 


6.3632126 


.000410860 


157 


24649 


8869603 


12.5299611 


6 3916907 


.000909187 


158 


24964 


8944813 


12.5096051 


6.4061208 


.000839114 


159 


25281 


4019679 


12.6095203 


6.4175015 


.006880806 


160 


85600 


4096000 


12.6491106 


6.4888368 


.000850000 


IGl 


25921 


4173281 


12.0885775 


6.4101218 


.000311180 


1G2 


26244 


4251528 


13.7279221 


6.4513618 


.000178640 


1G8 


86569 


4330747 


13.7671453 


6.4625550 


.000184800 


164 


26896 


4410944 


13.8062486 


6.4787087 


.000007801 


1G5 


27225 


4492125 


13.^52326 


6.4848065 


•006000600 


166 


87556 


4574296 


12.8840987 


6.4958647 


.006081000 


167 


27889 


4657463 


12.9228480 


6.5068784 


.005866084 


168 


88224 


4741633 


12.9014814 


6.5178484 


.O0G9S8881 


169 


28561 


4826809 


13.0000000 


6.6887748 


.006017100 


170 


88900 


4913000 


13.03«<4048 


6.5396588 


.006008058 


171 


29241 


6000211 


13.07GC968 


6.5501991 


.006847968 


172 


29584 


5068448 


13.1148770 


6.5618978 


.006818868 


173 


29929 


6177717 


18.1529164 


6.5780546 


.006780617 


174 


80276 


6268024 


18.1909060 


6.6827703 


.006747180 


175 


80625 


6359375 


13.2287566 


6.6984447 


.008714866 


176 


80976 


M5irr6 


18.2664992 


6.6040787 


.000661818 


177 


81329 


6545233 


18.3041847 


6.6146734 


.OO6O40n8 


178 


81684 


6639753 


13.3116641 


6.6352268 


.006017998 


179 


82041 


6735339 


13.8790683 


6.685740B 


.006686608 


180 


82400 


6832000 


13.4164079 


6.6468109 


tymmmmmmm 




181 


82761 


6929741 


13.4536240 


6.6566588 


.0068tMB08 


182 


83124 


6028568 


18.4907376 


6.6670511 


.O0O4O4B05 


1{» 


83489 


6128187 


13.5277493 


6.6774114 


.006104481 


184 


83856 


6229504 


18.5646600 


6.6877840 


JXMMTSO 


185 


84225 


6331685 


18.6014705 


6.6060198 


J000lfl040^ 


186 


84596 


6481856 


13.6381817 


6.7088675 


jOQBoni' 



Ct'BE ROOTS, AND RECIPROCALS. 



a 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Roots. ' 

! 


Reciprocals. 


187 


84969 


6639203 


13.6747943 


5.7184791 


.005347594 


188 


85814 


6644672 


13.7113092 


5.7286543 


.005319149 


180 


85721 


6751260 


13.7477271 


5.7387936 


.005291005 


190 


86100 


6R50000 


13.7840488 


6.7488971 


.005263158 


191 


86481 


6067871 


13.8202750 


6.758CGn2 


.Cai235602 


103 


86864 


7077888 


13.8564065 


5.7689982 


.0052C8383 


108 


87249 


7180057 


13.8924440 


5.77899G6 


.005181347 


194 


87636 


7801384 


13.928S883 


5.7889604 


.005154689 


195 


88025 


7414875 


13.9642400 


5.7988900 


.005128205 


106 


88416 


7529536 


14.0000000 


5.8087857 


.Oa5102041 


197 


88809 


7645373 


14.035C088 


5.818&179 


.005076142 


193 


89204 


7762392 


14.0712473 


5.8284767 


.OCr,050505 


199 


89601 


7880599 


14.10673G0 


5.6382?^ 


.005025126 


200 


40000 


6000000 


14.1421356 


5.8460855 


.005000000 


2C1 


40401 


8120601 


14.17744C9 


5.8577660 


.004975124 


203 


40604 


8^42408 


14.2126704 


5.^674643 


.004950495 


208 


41209 


6365427 


14.2478068 


6.8771307 


.004926106 


204 


41016 


^89664 


14.28285G9 


5.8867653 


.0(M901961 


205 


42025 


8615125 


14.3178211 


5.8963685 


.004878049 


2oa 


42436 


8741816 


14.3527001 


5.9050400 


.004854369 


207 


42849 


8869743 


14.8874946 


5 9154817 


.004880918 


203 


43264 


8098912 


14.42J:2C.'31 


5 9^9921 


.004807692 


200 


48681 


0128820 


14.45683i:3 


5.9344721 


.004784689 


210 


44100 


0281000 


14.4918767 


5.9439220 


.004761905 


211 


44521 


0308031 


14.525&90 


5.£CcS4l8 


.004739386 


212 


44C44 


0528128 


14.5GQ2196 


5.9G2';&20 


.004716981 


218 


45369 


0668597 


14.5945195 


5.9720926 


.004694836 


214 


45796 


06C0344 


14.6287288 


5. £814240 


.004672897 


215 


46225 


9988875 


14.6G28783 


5.91:07204 


.004651163 


216 


46656 


100776C0 


14.69CC8e5 


6.CC0C0C0 


.004629630 


217 


47089 


10218313 


14.73C9109 


6.CC£24C0 


.0046C8295 


218 


47524 


108GC2S3 


14.7(Via:£l 


6.C184C17 


.C04587156 


219 


47961 


10606459 


14.798C4£6 


6.G27G5G2 


.004566210 


220 


48400 


10648000 


14.8323070 


6.0868107 


.004545455 


21:1 


48841 


10708801 


14.eCCCGC7 


6.C459435 


.0045248.87 


223 


49284 


10041048 


14.898C&44 


6.Cn50489 


.004504505 


2:^8 


49128 


11060507 


14.9331845 


6.CC41270 


.004484805 


2;:4 


fiOlTB 


11230424 


14.96(K£05 


6.0731779 


.004464286 


225 


60025 


11890C25 


15.C0CCC00 


C.C822020 


.004444444 


226 


51076 


11548176 


15.03S2CG4 


6.0911994 


.004424779 


227 


61529 


11G07(«3 


15.0CC5192 


6.1C01'<02 


.004405286 


298 


61964 


11C52C52 


15.CCCCC89 


6.1C01147 


.004385C65 


229 


62441 


12006969 


15.1E274C0 


6.1180S32 


.0043CC812 


290 


62000 


121G7000 


15.1657509 


6.1269257 


.004347826 


281 


533G1 


12326391 


15.108C&12 


6.1857924 


.004829004 


im 


68824 


124enG8 


15.2315402 


6.144C337 


.004310345 


238 


54289 


12649337 


15.264S375 


6.1{:S4495 


.004291845 


234 


547n6 


12G129M 


15.2970585 


6.1C22401 


.004273504 


2ii5 


55225 


12977875 


15.3297007 


6.171C068 


.004255319 


236 


55096 


18144256 


15.3622915 


6.1707466 


.C04237288 


287 


601C0 


13812053 


15.3948043 


6.1884628 


.C04219409 


238 


£6044 


18481272 


15.4272486 


6.19n544 


.004201681 


2S9 


57121 


18651919 


15.4596248 


6.2058218 


.004184100 


240 


57600 


18824000 


15.4919334 


6.2144C50 


004166667 


241 


58061 


18897521 


15.5241747 


6 2i:ccr43 


.CC414C378 


2i2 


58564 


14170488 


15.. '5503402 


6.2310797 


.C04182231 


243 


59040 


14848907 


15.5HR4573 


6.2402515 


.004115226 


244 


50536 


14526784 


15.G204994 


6.^487008 


.004008861 


243 


60025 


14706125 


15.C524758 


6.2573248 


.004081083 


'VM 


60516 


14886036 


15.C84*)871 


6.2058206 


cc-^ccrx4i 


r 


61G00 


15069223 


15.71C0:>:G 


! 6.274S054 


.C0404r:r3 


^ 


^504 


152S2092 


15.7480157 


1 6.2827613 


.004a32258 



SQUAKES, CUBES, SQUARE HOOTS, 



„. 


ftsr 


Cube Boots. S 




iJlfl 


1S.77W7338 


e.WllDM 


wwimu 


sso 


15.8113883 


<I.W9<)a53 


00400000Q 






t.dcmaas 


oosgstOM 






e.aesaM 






15:b058737 


a.8iM703B 






lS,»3rjr!3 


B.xaoM 




SiU 




e.su^r 




UK 










lO! 031:2] ue 


siaajsoii 


0CO8910S1 




IB.0CS3;»1 


fl.3G009«8 


WB7SM0 


m 








m 


10.13U1KI 


B.ssaaota 


OOSSUIM 








003831418 










xa 


ittiauofos 


ist. 


offiw 




i6.i.T!aa» 


i.mim 


mnw» 










aa; 






0(B74O3ie 


m 


in.aroTOJi 


8,*173037 


OOJiMSU 


aea 


1S.40I3185 


B.4K3148 


oo3;m» 


■m 


IB!4a!»77S 


8^4712736 




f 


l«.4»itiW 


8,4ri«J3« 


«H0T04n 












ie!fi[W).<i 




IKI3UieS85 




10.B831S40 




ODSKsesM 


2Tfl 


la.fliaaiTT 


8,5106300 


taxasm 




10.6433i:0 


B.siama 


oaxioioe 










a;s 






0(Ki58uai 


280 


16.7S3i!0OS 


o.seisss 


ocispwo 




iB.Tastow 






ani 








283 






001533668 


«81 


wisoiiiifao 


oisvaia^a 


oosaiiK 


2d5 


Ht.8319l30 


6.5aCWH3 


ooBosra 


















SSS 








m 






OLtuiijjoe 


m 


17.tO»38M 


6.C10I0CO 


oosMsm 




















iTinrei* 


8;8418Ma 


oojiiaiao 




r,i40iii»i 


e.wBSOM 


xH4oi3ai 


Wi 








£9e 






M33;s3;s 


S97 


Y-asntaxs 


e:oriBio3 


0(B.-)07«B 


SZB 


r.aciWTOs 


e.orowx) 


Oft-;i-->T05 












T..T!CW<1 


€.C3*3KI5 




toi 


-,;yy^iu 


CTOlTMa 




303 


7.3.-|!urii 


8.70B17S9 


003311SS8 


803 


.iom^ 


G.nSfiJOO 


ae3(n330 








C0398M74 




;4e4»t'ia 


oliaiaias 


0U:378(>8S 




,4M8.V.7 


0.7380011 


(X0387W4 






li.74j!ni6T 


(Mao^aa* 


308 


!ftiui»« 




oonsiuras 


3je 






)i3«*i3<B 




.soisius 


e.TBWJifi 1 





CUBE ROOTS, AND RECIPROCALS. 



13 



No. 


Squares. 


811 


96721 


312 


97344 


313 


97969 


314 


98596 


315 


99225 


316 


99656 


317 


100489 


818 


101124 


319 


101701 


830 


102400 


821 


103041 


a22 


103G84 


323 


104329 


324 


104076 


325 


105625 


326 


106276 


327 


106029 


328 


107584 


329 


10B241 


390 


108900 


331 


109561 


332 


110224 


338 


110389 


334 


111556 


335 


112225 


336 


112806 


337 


113569 


338 


114244 


339 


114921 


Z40 


115600 



342 
343 
344 
315 
346 
U7 
»8 
249 

350 
351 
852 
353 
354 
£55 
356 
357 
G58 

SCO 
CCl 
862 
363 
364 
3t3 
366 
367 
8G8 
300 



116964 
117C49 
118336 
119025 
119716 
120109 
121104 

tiimi 

12^j00 
l;3o^01 
123004 
124G09 
125316 
120025 
12ci736 
127449 

1J:J881 

120G00 
l.^Jt-21 
131044 
1317G9 
102136 
133225 
133956 

l.T^l^^l 
13G1C1 

130900 

137641 
138984 



Cubes. 


Square 
llootB. 


80080231 


17.6351921 


80371328 


17.6635217 


80664297 


17.6918000 


80959144 


17.7200451 


81255875 


17.7482398 


81554496 


17.7703888 


81855018 


17.8044938 


82157432 


17.8G25515 


82461759 


17.8605711 


82768000 


17.8885438 


8307G161 


17.9104729 


83386248 


17.9448584 


83698267 


17.9722008 


84012224 


18.0000000 


84328125 


18.0277504 


84645076 


18.0554701 


84965788 


18.0831418 


85287552 


18. 1107703 


85011280 


18.1383571 


85087000 


18.1659021 


862&10i)l 


18.19&4054 


86594G08 


18.2206072 


86UJXC37 


18.2482S76 


87250704 


18.2750009 


87595371> 


18.30G0052 


87983056 


18.33aXK3 


88272753 


18.35755i« 


88614472 


18.3847768 


88958219 


18.4119526 


89304000 


18.4390889 


89051821 


18.46CiJw3 


40001CC3 


18.4932120 


40363007 


18.520e.:92 


40707584 


18.54r;J;V0 


41063025 


18.5741756 


414217S6 


18.60107:2 


41781023 


18.C27C:,00 


42144102 


18.G547L81 


42508549 


18.0815417 


42875000 


18.7062860 


43243551 


18.7340040 


43614206 


18.7G1CC30 


43966977 


18.788^042 


44361864 


18.814^>077 


44788875 


18.8414137 


45118016 


18.8679G23 


4'>499203 


18.894 4- Jo6 


4.>'«2712 


18.0:»>/i9 


46268279 


:0.9472953 



Cube Boots. 



Reciprocals. 



46666000 
4701.':-.M 
474370Ji 
47832117 
482285!! 
48627125 
40027?v90 

5024^109 

50653000 

510G?fll 
6147rj8l8 



t 18.9736660 
. )O.(XX>0C/j0 
I 19.0262076 
I 19.0525.'i89 
10.0787^0 
■i9.l04a7r:2 
19.1311CV, 
ia.l5?^Ml 

10.2093727 

19.2353W1 
19.2Gl.r.08 
19.2fJ7S^15 



I r 



6.7751690 
6.782;;J29 
6.7896013 
6.7968844 
6.8040921 
6.8112847 
6.8184620 
6.8250242 
6.8327714 

6.8899087 
6.8470213 
6.8641240 
6.8612120 
6.8682855 
6.6753443 
6.8y23e88 
6.8694188 
6.690i.'345 
6.9034;;59 

6.C1042S2 
6.91739&4 
6.9213556 
6.0313006 
6.9Ui2S21 
6.9451496 
6.9520583 
6.9589434 
6.0058198 
6.9?26826 

6.9795S81 
0.0^03081 
6.9031006 
7.0(KXX)00 
7.00GrOC3 

7.(mo',vi 

7.C20a400 
7.0»n053 
7.0388197 
7.0405806 

7.0472967 
7.0M0(M1 
7.060C0f;7 
7.0C737C7 
7.0740440 
7.080r/JH8 
7.0H7.'>ni 
7.C039^(J9 

7.ioor/>c> 

7.1071087 

7.11378G6 
7.1200074 
7.1260360 
7.13aiai> 
7.1400370 
1405695 
7.1030901 
7.15950f« 
7.1660X7 
7.1725809 

7 1790544 
7.1tV>K2 
7.1919663 



.003215434 
.003205128 
.003191888 
.003184718 
.003174608 
.003164557 
.0CJJ151574 
.00:il44(i51 
.003134796 

.008125000 
.00311526S 
.003105590 
.0()::C05976 
.0(3X80420 
.00PXJ70923 
.00.'XX;74fc5 
.000058104 
.00.'X>187a) 
.003009514 

.00;»30808 
,co::x;i204« 

.000003003 
.crj:W4012 
.0(U;b5075 
.002976100 
.W)20073i:9 
C029585J:0 
.002949853 

.002941176 
.002932551 
.C02923977 
.00291545a 
.ar290C977 
.002898551 
.002890173 
.002881844 
.002873503 
.002865330 

.002857143 

.C02849003 

.caAioooo 
.cojjC2j;gi 

.C..:<'.iilfJ59 
.002810001 

.C-'.-^/Jlli.'O 

.c/)<j7««j:>i 

.0^f.r785515 

.002777778 
.f.0J7700«? 
.a/;w^e2431 
.(/.ZTTAf-il 
.C;r/47253 
.r. ,2739728 
.C 0/732210 

.r//:r72i706 

.002710027 
.002702708 

.0026.75172 



^.--i 



i\ 



11 














f 

1 

1 


No. 


1 


CtA«. ' 


BqiMra 1 


CUbaBootiL 


Becfprocak 


i 


zn 


13919 


5i^sb::7 


i»si»n9 : 


T.lSBtfGO 


.000680065 


' a-.i 


la*^ 


5iilir.;ii 


19 33Wnt i 


Tjmsn 


AȣUX^i 


i:5 


14i«£S 


&*:*«:i73 


ida6i»i«r E 


7.21UIS9 


JUOBSBUBSi 


V.i 


i4iaf:«  


53lC7o76 


IdSWnM r 


Tjunas 


.00^0674 


£77 i 


142:2> ' 


.Vi.>i>j:d 


19.41M8» 


T.SHMM 




ir-j i 


14A*4 


&;..i';ia 


19.4t2S21 


T.»>ttflB 


.ttEGisecs 


&T5P 


14£k41 : 


5UiabUUA 


i.>.4d:ae8S 


T.Msna 


.000688622 


5SN) 


lUlO) ' 


WtTSWO 


19.4995807 


T.MSISK 


.00001679 


Tvl 


i«:»:i 


5r;>i«;>«i 


19.5lStS>l3 


T.MSOMB 


.0QEBE9l6i8 


5 &"2 i 


UZJ^ 


t5T42«8 


19.M4San 


T.S556I15 


.000617801 


! &<J ! 


\V^j:Q 


Ki?i*r 


19.570RKV) 


7.aa2i6a 


.0QB610066 


fr>4 1 


Uli^A \ 


&>^1«>4 


19.5956119 


7.2;4;;sN 


. .OQB0M167 




3Ki 


\4?-:£& 


67'.»>.':25 


19.6214169 


.00007408 




3^ 1 


14^SW 


6751iM.-.« 


19.646fct27 


T.2B10TM 


.000600674 


Zhl 


I4'y7e9 


6::'>.»><3 


19.C:2?15« 


T.SSTSfilT 


.QOKBOOTO 


&v3 


\:*i:M 


£ :-!i-.-r2 


19.6iK7l5« 


7.;£9d63a0 


.000577380 




doSf 


15132^ 


5cou:ibC9 


l^.T^BXSSi 


T.29980a8 


.QOOBRMOft 




300 


152100 


5050«i«) 


19.7484177 


T.ao6i4as 


.OOOBOdOO 




&01 


i.:j^:.i 


w^r<'^ii 


19.7:37199 


7.3123a» 


.OaBBffi45 




'iC-i 


1.>>X>4 


G^i302S8 


19.7569609 


7.318G114 


OQOB6U0D 




3:>j 


l&i449 


6o«^.^7 


19.8!24iS78 


7.S2I828K 


.00064080 




8!4 


155.^ 


ClKii:^ 


19.8494333 


7.3310900 


.000606071 




a5 


15(Xr25 


61t>2ah75 


19.8746069 


7.3372390 


.000631646 




3ri6 


15CS16 


62U09136 


19.8997487 


7.8431906 


.O0E82BO68 




J>i;7 


15TC09 


6257U773 


19.9^M8588 


7.3195966 


.000618000 




£JH 


15>404 


63.>44;92 


19.9499373 


7.855TK54 


.OOESlOaB 




31/9 


15U201 


68521199 


19.9749t>44 


7.36191:8 


.000500006 




400 


IWOOO 


61000000 


20.0000000 


7.3680680 


.008800000 




Hil 


l(>A\ijl 


64-1 ;ion 


2U.024SI>44 


7.&;41979 


.000498^6 




4(r2 


161tX>4 


&l'J04.-.-08 


20.0499:377 


7.St03227 


.OU018S66O 




40:) 


10:^409 


6545a>27 


20.074i:,':09 


7.Sl^64373 


.000481800 




4r>i 


1C:£216 


C5939204 


20.0997512 


7SU25418 


.060478018 




405 


1GU^25 


60130125 


20.1:^113 


r.3&86363 


.000160186 




4iA 


lOiKiO 


6002:3116 


20.1494417 


7.404?,»6 


.GQ0468O64 




407 


](35m9 


67410143 


20.1742410 


7.4107950 


.000457900 




4<i8 


lGr>lf>4 


G;yi«':il2 


20.1«)aU9 


7.41Ce.:95 


.COMSOKO 




409 


107281 


6&41?d29 


20.2237464 


7.4)^29142 


.000444868 




410 


168100 


68921000 


20.^&15C7 


7.4289589 


.000480001 




411 


1G<''I*;J1 


604^:0531 


20.^531849 


7.4*49938 


.000138000 




412 


10'J744 


60aS4528 


20.2977b31 


7.4410189 


.00242n84 




413 


170509 


70444097 


20.S2iM014 


7.4470342 


.00^21866 




414 


insoo 


70967944 


20.a469e99 


7.45S0S99 


.002415400 




415 


172225 


71473375 


20.8/15488 


7.4590859 


.002400680 




410 


173056 


71991296 


20.8960;81 


7.4050223 


.002408846 




4!7 


1738(<9 


72511713 


20.4205779 


7 4709991 


.CG2S980e2 




413 


174724 


730JM(:32 


20.4450483 


7.47C9G04 


.002392844 




419 


1755G1 


78560059 


20.4694896 


7.4829242 


.002386685 




430 


170400 


74088000 


20.4939015 


7.4888TO1 


.002880962 




4^1 


177-^1 


74M84C1 


20.5182845 


7.4948118 


.002375297 




4:ii 


i:roH4 


75151448 


20.5426386 


7.6007406 


.002369663 




4lKl 


17R{)29 


775G80CG7 


20.5069038 


7.5066607 


.002364066 




4^i 


179776 


70225021 


20 5912603 


7.5125715 


.002358491 




425 


1W)025 


7G7a'3025 


20.6155281 


7.51&4780 


.002352941 




4:(i 


1814:6 


77008776 


20.6897674 


7.5218652 


.002»47418 




4^7 


IP^'X'O 


778M483 


20.6039788 


7.5806M82 


.002*11920 




42\ 


IKJIKI 


7W02753 


20.6881609 


7.5861221 


.002336440 




4^J 


i»-1041 


7)»fia580 


20.7128152 


7.5419667 


.002331002 




4.'X) 


1&1900 


79507000 


20.7364414 


7.6478488 


.002^2558' 




431 


1857C1 


8(X)G2991 


20.7C05395 


7.5536888 






4US 


IBGCa* 


80621568 


20.7846097 


7.5595268 






«» 


1874K9 


81182787 


20.8086520 


7.5038548 






4»l 


188356 


81746501 


20.8826667 


7 5ni743 


-"\ 


 






^^^ 


^SAIIB • 


"-. 


^J 



CUBE ROOTS, AND RECIPROCALS. 



15 



i No. 
4C5 


Squares. 


Cabes. 


Square 
Boocs. 


1 

Cube Root& 


BeciprociUs. 


1302S3 


^312^^75 


i)a.&-'665JK 


7.570»^ 


.0ai£9t^M 


43a 


1'jG0l)6 


83tiK18j6 


20.biM>ldO 


7.&iK>*65 


.OOfciftWSrS 


437 


100009 


83^3453 


20.9O4M5a 


7.5fW57ya 


.v.X]fcSJi^0:5) 


433 


1j1S44 


&ia2;c?2 


d0.9ei844;o 


7.5i^4;>i;53 


.vX122S3UK> 


43d 


iJjt^iSl 


81604519 


20.9ai^(>3 


7.U)01385 


.OOB^fTTOOA 


440 


19^500 


85181000 


20.97617:0 


7.6060019 


.WS73757 


441 


101481 


85766121 


21.0000UOO 


7.6116626 


.a)22G7o74 


412 


ia>15l 


86350333 


21.(123^)00 


7.6174116 


.ae»2443 


443 


193249 


8698S907 


21.047565^ 


7.6231519 


.0^^2257336 


444 


19na6 


87528384 


21.0713075 


7,62888:57 


.0<)225aj52 


445 


19^025 


83121125 


21.0050231 


7.6346007 


.a)224n91 


446 


193916 


887ir»536 


21.1187121 


7.640^13 


.a)iM2152 


447 


199300 


89314623 


21.1423745 


7.6400272 


.a>2237i:50 


448 


203704 


80J15392 


21.1600105 


7.6517217 


.o.>:i52iiJ 


440 


201601 


90518349 


21.1896201 


7.657413J 


.002227171 


450 


20K00 


91125000 


21.?1320f54 

21.£30;0;\i 


7.6630943 


.0^3322222 


451 


203401 


91733151 


7.66870;.5 


.002217205 


45fi2 


204304 


8eJ3454« 


21.2002016 


7.67443LI) 


.0112212389 


453 


2(B209 


92^50377 


21.28370C7 


7.680aCi7 


.rtK2075(K5 


1 454 


206116 


93570304 


21.3072753 


7.6S57a>3 


.002202(H3 


455 


207085 


94i9G;:ro 


21.8307-J:)0 


7.6913717 


.002197802 


456 


207036 


94818310 


21.3541505 


7.6970023 


.0021029ti2 


457 


203349 


95443003 


21.3775583 


7.7036246 


.003188184 


453 


2o:r;64 


9oonni2 


21. 400*5 JO 


7.70t^2583 


.00218:5 UKi 


453 


210681 


9o70257J 


21.4242853 


7.7138443 


.002178649 


460 


211600 


97336000 


21.4476106 


7.n94426 


.002173013 


4'Jl 


212531 


9rJ7;3181 


21.4709106 


7.?250:)25 


.O;)2100107 


4G3 


213444 


080111^:3 


21.4041853 


7.7:500141 


.002101502 


463 


214369 


902a2:U7 


21.5174:543 


7. 7301877 


.ar215082r 


464 


215296 


90307344 


21.5406502 


7.7417582 


.002155172 


465 


216225 


100544025 


21.5fl3a')87 


7.7478109 


.0021505315 


466 


2m56 


101 1940 JO 


21.5870331 


7.7528006 


.00211592:) 


467 


218089 


101847553 


21.6101823 


7.7584023 


.002141328 


468 


210024 


102503232 


21.03.3:3077 


7.7G;59361 


.0021:50752 


469 


219961 


103101703 


21.6564078 


7.7604620 


.002132106 


470 


220900 


103828000 


21.6794834 


7.7749801 


.002127860 


471 


221841 


104487111 


21.702534-4 


7.7804004 


.a)2123142 


473 


222784 


10515404.5 


21.7255010 


7.7850028 


.002118644 


473 


223729 


105823817 


21.748JC32 


7.7014875 


.002114165 


474 


S24676 


103406424 


21.7715411 


7.7060745 


.00210l»705 


475 


225625 


107171875 


21.7044W7 


7.80215:58 


.(X)21 06263 


476 


223576 


107850176 


21.8174212 


7.H070254 


.(X)2100810 


477 


227529 


108531&33 


21.8403207 


7.81:5:5892 


.(X)2()96I30 


473 


223 134 


100215.>52 


21 8632111 


7.8ir4iI56' 


.()>■.■»( )92«.»:»() 


479 


229441 


109002230 


21.8800680 


7.8212042 


.(X»'J()K7(yia 


480 


230400 


110592000 


21.0089021 


7.8297353 


.002083:5.33 


481 


251301 


111234(;U 


21.9517122 


7.8'5510W5 


.u)i^)7m)^ 


482 


23i3:^ 


lli0801O;5 


21.0.5-4-19.U 


7 Hia5910 


.(X)J0710.S'.) 


48;} 


2:iVH9 


112078587 


21.9772010 


7.8100134 


.iA)nniY.m 


4H4 


251258 


113379904 


23.0000000 


7.8514214 


A%innmui 


4a'3 


2]5ii5 


11408412.5 


22.02271.55 


7.850H2;a 


.<);)2(MJ1K50 


480 


2:50196 


114791250 


22 0151077 


7.8022212 


.002057013 


487 


2:571 GO 


115501303 


22.0(V<();05 


7.80701:50 


,(j020.5:j.'i>« 


4S8 


25S144 


1102i:r2 


2,v()^r;r2n 


7. 8720'.) 14 


JHWiiUHf) 


4S'3 


2:50121 


11003010) 


LMUJin 


7.87K5(;sl 


AHMHMM 


490 


210100 


117040000 


22.1350430 


7.8837:552 


.002040816 


1 401 


2110S1 


im3707il 


22.1585103 


7.8800016 


.(XWOJWMKiO 


i 492 


24J0r>t 


110095483 


22.1810730 


7.8044463 


.002^X52.720 


493 


24-'5040 


110823157 


22.203608:) 


7.8097917 


.002028:508 


'94 


241036 


12nfMr784 


22.2261108 


7.9061204 


.002021201 


% 


245025 


121287375 


22.2485055 


7.9104.599 


,002020202 


i^ 


240010 


122028086 


22.2710575 


7.9157832 


.002010129 



16 



SQUARES, CU15ES, SQUARE ROOTS, 



:>\7 
r.H 

51'.) 

n.v) 
:..-,! 

.v»i 
rci5 

r>5rt 



No. 


Squares. 


Cubes. 


Square 
Roots. 


497 


217009 


12276a473 


22.2934908 


4,U 


21<S()0-1 


123505993 


22 315913a 


4J0 


:W9001 


124251499 


2;S.338S079 


500 


2.50000 


125000000 


22.3606793 


r>,;i 


251001 


125751501 


22.3830293 




252)01 


120506008 


22.4033365 


50.) 


25.*J009 


127203527 


22.4276015 


501 


251010 


128024064 


22.449»443 


51)5 


2.55025 


128787623 


22.4722051 


503 


2.5<)0.*JG 


129554216 


22.41M4438 


mr 


2,57049 


130323843 


22.5166003 


5o;j 


25S(X)-4 


131096512 


22.5388553 


oo-j 


259081 


1318?2229 


2J.5GI0283 


510 


200100 


132651000 


22.5831796 


511 


201121 


133432S31 


22.0033001 


512 


2G2144 


134217728 


22.6274170 


513 


2G:U()9 


1.35005097 


22.6495033 


511 


2<H19a 


135790744 


22.0715081 


515 


2(5.5223 


136590f;75 


22 6036114 


516 


2002.50 


13r38S096 


22.715a-331 


617 


207289 


l.*)8188413 


22. 7370.3 JO 


518 


2()H;W4 


1:389918.32 


22.7590134 


519 


209.301 


139798359 


£2.7815713 


620 


270100 


1-40608000 


23 80a50ft5 


521 


271441 


141420701 


22.rr>4244 




272^184 


142236018 


22.8473193 


52.^ 


27.'J529 


113055607 


22.K091933 


524 


271576 


1-43877824 


22.8910403 


525 


275025 


144703125 


22.9128783 


52« 


2^(>076 


14.55.31,576 


22.9:340899 


527 


277?29 


14(>36:3ia3 


22.9504800 


528 


2mS784 


147197972 


22.r.7r.2.5(XJ 


52U 


279811 


148035889 


23.0000000 


5.30 


2S0(N>0 


148877000 


23.«17289 


5;u 


2^1901 


1497212:)1 


£3.0134:372 


5:i2 


2S;5()24 


1.5(V)08708 


23.0051252 


rm 


2S1089 


151419437 


23a^67928 


5:U 


2S.51.50 


1.5227'3;304 


2:5.1084400 


5.15 


2S«;2*i5 


1.5.3130375 


25.1300670 


WW 


2sr2^»<i 


153990650 


2:5.15167:38 


5;J7 


2^s;}«;«) 


1.548341.5.3 


25 1732603 


5.18 


2s:)114 


1,55?20S72 


23 194WrO 


iJiiQ 


2«.*i)521 


156590819 


23.21G3r33 


5!0 


201C,(» 


1.57464000 


23.23T9001 


5U 


2'.fJ.J.si 


1.5S;M0421 


23.25941X57 


512 


2.i:{7i;4 


15'W20088 


23.2S089:]5 


M.i 


2*.) I ' 19 


16.1030(17 


2:3..3t>23<504 


5U 


2'.i.-:i:ii> 


1609891K4 


23.3238076 


5 15 


29 ."'>•.'.> 


161878025 


2:3.34.52:351 


5HJ 


2'.tM0 


1627n:33<J 


23.3606429 



,31 ''4 
3ul . '1 

:i'>:;;',iti 

:3<m;i)4 
::ii.-,s<)'j 

.'iiMi'.llO 

;ii»N)25 
:«)9i:)6 

3HV219 
311:3<>1 



lt>;366732.3 
164.566,592 
16U6»14J 

106375000 
107281151 
l(}.^19(}«i(H 
169112:377 
17(K»146t 
1T0953875 
inK7»)16 
1?2H0K6IM 
173741112 



23.3880:311 
2.3.-l(y:):/'9i 
t:a. 4307490 

23.4520788 
23.473J3892 
23.4tH(5.sift> 
23.51.59.521) 
2:5..53?20I6 
23.55K13H0 
83.6706S23 
S3.600M74 

ss.esssosao 



Cubo Roots. 


Rcclprooala 


7.9210994 


.00201207;3 


7.9264085 


.002008032 


7.9317104 


.0Q20O4OUJ 


7.937005S 


.002000000 


7.9422931 


.001996008 


79475739 


.001992002 


7.0528477 


.001988073 


7.9381144 


.001984isr/ 


7.9633743 


.001980196 


7.9686Sn 


.001976286 


7.9738r31 


.001973387 


7.9791122 


.001968.')(>1 


7.9843444 


.001964637 


7.9895697 


.0019607&1 


7.9947883 


.00105694/ 


8.0000000 


.001953125 


8.C032049 


.0011M9318 


8.0104032 


.001945525 


8.0155946 


.001M1740 


8.0207794 


.001937984 


8.0259574 


.001934236 


8.C3112M7 


.C0193a503 


8.0362935 


.001926783 


8.0414615 


.001933077 


8.0466030 


.001919383 


8.0.517479 


.001915709 


8.0508862 


.001912046 


8.0020180 


.001908397 


8.0671432 


.001904762 


8.0722020 


.0)1901141 


8.0773743 


.001897533 


8.a"24K00 


.0)189.3930 


8.0875794 


.001890359 


8.0920?23 


.001886793 


8.a»77.589 


.0)1883233 


8.1028.390 


.001879699 


8.1079123 


.001876173 


8.1129803 


.a)l«?2659 


8 118a414 


001869153 


8.12309(52 


0)1863073 


8.1281447 


.0)1862197 


8.1331«70 


.0)l858r33 


8.1382230 


.001855283 


8.14.32.529 


.001851853 


8.14t^V03 


.0)1JM8Jl-J 


8 1.532939 


.On.»U5013 


8.1.5S.3(X51 


.0;iH416£l 


8.16.3:3102 


.OM838235 


8.i(5.s:>o::2 


.0)18*48(:3 


8.i7:i::< ^) 


.OUW31503 


8.1782^ 3 


.0M8281.51 


8.18.32;:. 5 


.c :-^:jsn 


8.1882441 


.0U18-n494 


8 1932127 


.001Sl8ir-3 


8.19t317,5.3 


.0»1814«ii 


8.2031319 


.OI181159& 


H. 2080823 


.0)l.s<K3H 


8.2130371 


.INUHO5031 


8.21790.57 


.0)1801803 


8.233R9K5 


.0017118561 


8.237H351 


.001793333 


8.2327463 


.(X)1?J3115 



CUBE KOOTS, AND IIECIPROCALS. 



17 



No. 


^Squares. 


Cubes. 


559 


313481 


174676879 


5G0 


313600 


175616000 


5G1 


314721 


176558481 


562 


315844 


177504328 


6G3 


316969 


178453547 


504 


318096 


179406144 


6C5 


319225 


180362125 


5C6 


320356 


181321496 


EG7 


321489 


182284263 


5G3 


322624 


183250432 


509 


323761 


184220000 


570 


324900 


185103000 


571 


32G041 


186109411 


572 


327184 


187149248 


573 


828329 


188132517 


574 


329476 


189119224 


575 


330625 


190109375 


576 


331776 


191102976 


577 


332929 


192100033 


578 


334084 


193100552 


579 


335241 


194104539 


580 


836400 


195112000 


581 


337561 


196122941 


582 


338724 


19713r368 


583 


330389 


198155287 


684 


341056 


199176704 


585 


342225 


200201625 


586 


343:96 


201230056 


587 


344569 


202262003 


588 


345744 


2032974?2 


589 


346921 


204336469 


590 


348100 


205379000 


591 


349.^1 


206425071 


592 


850464 


207474688 


593 


351649 


208527857 


594 


352836 


209584584 


595 


354025 


210644875 


590 


355316 


211708736 


P97 


356409 


212776173 


t08 


a->7604 


213847192 


C39 


358801 


214921799 


603 


360000 


216000000 


CO I 


SG1201 


217081601 


C02 


SG2404 


218167208 


C33 


CG3G09 


219256227 


604 


SG4S16 


220348864 


605 


3GG025 


221445125 


60G 


CGr236 


222545016 


6'J7 


SG3449 


223648543 


COB 


8C9G64 


224755712 


609 


370881 


225866529 


610 


372100 


226981000 


cn 


373.J21 


228099131 


612 


<:71544 


229220928 


613 


3757G9 


230346397 


614 


37G996 


231475544 


615 


37S225 


232608375 


G16 


379456 


23^3744896 


617 


380G89 


2348a5113 


618 


3«1924 


236029032 


619 


383161 


237176659 


&30 


384400 


238328000 



Square 
Icoots. 


Cube Roots. 


23 6431806 


8.2876614 


23 6643191 


82425706 


23.6854386 


8.2474740 


23.7065392 


8.2523ri5 


23.7276210 


8.2572633 


23.7486842 


8.2621492 


23.7697286 


8.2670294 


23.7907545 


8.2719039 


23.8117018 


8.2';bY/26 


28.8327506 


8.2816355 


2J.853?S09 


8.2864928 


23.8746728 


8.2913444 


23.8956063 


8.2961903 


23.9165215 


8.S010304 


23.9374184 


8.3058651 


23.9582971 


8.3106941 


23.9791676 


8.3155175 


24.0000000 


83203853 


24.0208243 


8.8251475 


24.0416300 


8.3299542 


24.0624188 


8.3347558 


24.0831891 


8.3895509 


»4. 1039416 


8.3443410 


24 1246762 


8.3491256 


24.1453929 


8.3539047 


24.1660919 


8.8586784 


24.1867732 


8.2634466 


24.2074369 


8.3682095 


34.2280829 


8.8729668 


24.248ni3 


8.8777188 


24.2693222 


8.8824653 


24.2899156 


8.8872C65 


24.3104916 


8.3919423 


24.3310501 


8.8966729 


24.3515913 


8.4018981 


24.3721152 


8.<061ie0 


^.3926218 


8.4108S26 


^.4131112 


8.4165419 


24.4335834 


8.4S02460 


24.4540385 


8.4240448 


24.4744765 


8.4296883 


24.4948074 


8.4348267 


24.5153013 


8.4390098 


24.5356883 


8.4486877 


24.5560583 


8.4483605 


24.5704115 


8.4530281 


24.5907478 


8.4576006 


24.6170073 


8.4623479 


34.63r3,00 


8.4670001 


24.a576560 


8.4716471 


24.0779254 


8.4762892 


24.6981';«1 


8.4809261 


24.7184142 


8.4855579 


24.7386338 


8.4901848 


24.7588368 


8.4948065 


24.7790234 


8.4994233 


24.7991935 


8.5040350 


24.8193473 


8.5086417 


24 8394847 


8.5132435 


24.8596058 


8.5178403 


24.8797106 


8.5224321 


24.8997992 


8.6270189 



Reciprocals. 



001788909 

.001785714 
.001782531 
.001779359 
.001776199 
.001773050 
.001769912 
.001766784 
.001768668 
.001760563 
.001757469 

.001754386 
.001751313 
.001748252 
.001745201 
.001742160 
.001789130 
.001736111 
.001783102 
.001730104 
.001727116 

.001724138 
.001721170 
.001718213 
.001715266 
.001712329 
.001709402 
.001706485 
.001703578 
.001700680 
.001697793 

.001694915 
.001692047 
.001689189 
.001686341 
.001683502 
.001680672 
.001677852 
.001675042 
.001672241 
.001669449 

.001666667 
.001668894 
.001661130 
.001658375 
.001655629 
.001652893 
.001650165 
.001647446 
.001644737 
.001642036 

.001639344 
.001636661 
.001633987 
.001631321 
.001628664 
.001626016 
.001623377 
.00162C746 
.001618123 
.001615509 
.001612903 



^.^ 



20 



SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Sqiiaro 
Boots. 


Cube Roots. 


Reciprocals. 


745 


555025 


413493625 


27.2946881 


9.0653677 


.001342282 


V4G 


650510 


415100936 


27.3130006 


9.0o94^20 


.001340483 


T47 


•558009 


416832723 


27.3313007 


9.0734726 


.COl&'WfiRR 


743 


559504 


418508992 


27.3495887 


9.0775197 


.001836898 


749 


561001 


420189749 


27.3678644 


9.0815031 


.001335113 


750 


562500 


421875000 


27.8861279 


9.0856030 


.C01fi38883 


751 


504001 


423564751 


27.4043792 


9.0696392 


.0. 1331658 


752 


565504 


425259008 


27.4226184 


0.0936719 


.G01o20787 


753 


507009 


42695YYVV 


27.4408455 


9.0977010 


.C01828D21 


754 ' 


5G35I6 


428661064 


27.4590604 


9.1017265 


.C01S26260 


.755 


570025 


430368875 


27.47'r263S 


9.1057485 


.1015:24503 


756 


571536 


432081216 


27.4954542 


0.1097669 


.t<;lS22751 


757 


573049 


4337930i>3 


27.5186330 


9.1137818 


.C01821004 


v:>8 


. 574501 


435519512 


27.6317998 


9.1177931 


.001319261 


759 


576081 


43?i45479 


27.5499546 


0.1218010 


.101317623 


730 


B77600 


438976000 


27.5680975 


0.1258053 


.C01315789 


7G1 


579121 


440711081 


27.5862284 


9.1298061 


.C01314060 


732 


580044 


442450728 


27.604*475 


9.1338a34 


.101312886 


703 


532109 


444194047 


27.62.4540 


9.1377971 


.001310616 


7(>4 


583096 


445943744 


27.6405499 


9.1417874 


.001306001 


7J5 


585225 


447697125 


27.6586:334 


9.1457742 


.001807190 


706 


580756 


449455096 


27.6767050 


0.1497576 


.C01805483 


7G7 


583289 


451217063 


27.6947648 


9.1C37375 


.001803781 


708 


63D824 


452984832 


27.7128129 


0.1577189 


.001802068 


709 


591361 


454750009 


27.7808492 


0.1616809 


..C01300890 


770 


592900 


45653;3000 


27.7488?39 


C. 1656505 


.001298701 
.101297^17 


771 


594441 


458314011 


27.7668668 


0.1fc962x;5 


772 


595984 


400099648 


27.7848880 


0.1735852 


.C01296837 


773 


597529 


401889917 


27.8028775 


0.1775445 


.001293661 


774 


599076 


403684824 


27.8208555 


0.1815003 


.001291990 


7.5 


000625 


405484375 


27'. 8388218 


0.1854527 


.C012C0823 


7r6 


002176 


467288576 


27.8507706 


0.1894018 


.C01288660 


777 


603729 


469097433 


27.8747197 


0.1983474 


.001287001 


778 


605284 


470910952 


27.8926514 


0.1972897 


.C012fc5347 


779 


606841 


472729139 


27.9105715 


0.2012286 


.C01288697 


780 


608400 


474552000 


27.9284801 


0.2051641 


.001282061 


781 


609901 


470379541 


27.9463772 


9.2090962 


.001280410 


732 


611524 


478211708 


27.9642029 


9 2180250 


. 001278772 


733 


613039 


480048687 


27.98213?2 


0.2109505 


.001277189 


7.S4 


614056 


481890304 


28.0000000 


0.2206726 


.001275510 


735 


610225 


483730625 


28.0178515 


0.2247914 


.C01273886 


'm 


617796 


485587656 


28.a356915 


9.2287068 


.001272265 


737 


619369 


4874434C3 


28.a535203 


2826189 


.C01270648 


788 


620944 


489303872 


S8.(ynii:i77 


9 2365277 


.C0126C036 


789 


622521 


491109069 


28.0891438 


2404833 


.001267427 


790 


6^4100 


493039000 


28.1069386 


9.2443855 


.0012a')823 


791 


825081 


494913071 


28.1247222 


9 2482344 


.C01C64223 


•592 


02?264 


490793088 


28.1424946 


0.2521300 


.0015:62626 


75)3 


628a49 


49867?257 


28.1602557 


9.25602^ 


.001261034 


794 


630436 


600566184 


28.1780056 


0.2599114 


.C0K.';0446 


795 


632025 


602459875 


28.1957444 


9.2C87973 


.101257862 


796 


633616 


604358336 


28 2134720 


9.207(;7'.;8 


.C012E6281 


797 . 


635209 


600261573 


28.2311884 


9.2715592 


.C0i:c£4705 


798 


636804 


608169592 


28.2488938 


9.2754*52 


.001253133 


799 


638401 


610082399 


28.2665881 


9.2798081 


.001251564 


830 


640000 


512000000 


28.2842712 


0.2831777 


.001250000 


801 


641601 


51392i»01 


28.30194*4 


9 2870440 


.001248439 


orv) 


643204 


615849608 


28.3196045 


9.2909072 


.001246883 




644809 


517^1627 


28.3372546 


0.2W7071 


.001245830 




646416 


CI 0718464 


28.a548938 


9.2980239 


.001243781 




648025 


6210C0125 


28.3725219 


9.8024775 


.001242236 




649636 


623606616 


28.3901391 


9.3063278 


.001240695 



CUBE ROOTS, AND RECIPROCALS. 



21 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Roots. 


Reciprocals. 


S07 


651249 


625557943 


28.4orr4r>4 


9.3101750 


.001239157 


{.03 


652864 


527514112 


28.4253408 


9.3140190 


.0012:37624 


809 


654481 


529475129 


28.4429253 


9.3178599 


.001236094 


010 


656100 


531441000 . 


28.4604989 


9.3216975 


.001234568 


811 


657721 


533411751 


28.4780617 


9.3255320 


.001233046 


812 


659344 


635387328 


28.4956137 


9.3293634 


.001231527 


813 


660969 


537367797 


28.5131549 


9.3331916 


.001230012 


8lt 


662596 


539353144 


28.5306852 


9.3370167 


001228501 


815 


664225 


641343375 


28.5482048 


9.340aS86 


.001226994 


810 


665856 


&43a38496 


28.5657137 


9.3446575 


.001225490 


817 


667489 


545338513 


28.5832119 


9.3484731 


.001223990 


813 


660124 


54734^432 


28.6006993 


9.3522857 


.001222494 


819 


670761 


549353259 


28.6181760 


9.3560952 


.001221001 


830 


672400 


651368000 


28.C)856421 


9.3599016 


.001219512 


621 


674041 


553387661 


28.6530976 


9.3637049 


.001218027 


G22 


675684 


555412248 


28.6705424 


9.3675051 


.001216545 


823 


677329 


557441767 


28.6879766 


9.3713022 


.001215067 


GU 


678976 


559476224 


28.7054002 


9.3750963 


.001213592 


8,35 


680625 


661515625 


28.7228132 


9.3788873 


.001212121 


826 


682276 


563559076 


28.7402157 


9.8826752 


.001210654 


62? 


683929 


565609283 


28.7576077 


9.3864600 


.001209190 


823 


685584 


567663552 


28.7749891 


9.3902419 


.001207729 


829 


687241 


6097^2789 


23.7923601 


9.3940206 


.001206273 


830 


688900 


671787000 


28.8097206 


9.3977964 


.001204819 


8U 


690561 


573a56191 


28.8270706 


9.4015691 


.001203369 


832 


692224 


575930308 


28.8444102 


9.4053387 


.00120192:3 


833 


693889 


678009537 


28.8617:394 


9.4091054 


.001200480 


a3i 


695556 


680093704 


28.8790582 


9.4128690 


.001199041 


a35 


697225 


582182875 


■28.8963066 


9.4166297 


.001197605 


836 


698896 


6&4277056 


28.9136646 


9.4203873 


.001196172 


837 


700569 


686376253 


28.9309523 


9.4iU1420 


.001194743 


833 


702244 


588480472 


28.9482207 


9.4278936 


.001193317 


839 


703921 


590589719 


28.9654967 


9.4316423 


.001191895 


840 


705600 


592704000 


28.9827535 


9.4353880 


.001190476 


841 


707281 


694823321 


29.0000000 


9.4391807 


.001189061 


&i2 


708964 


696947683 


29.0172363 


9.4428704 


.001187648 


&i3 


710649 


699077107 


29.0344023 


9.4466072 


.001186240 


844 


712336 


601211584 


29.0516781 


9.4503410 


.001184834 


845 


714025 


603351125 


29.0688837 


9.4540719 


.001183432 


846 


715718 


605495736 


29.0860791 


9.4577999 


.001182033 


&47 


717405 


607645423 


29.1032644 


9.4615249 


.001180638 


848 


719104 


609800192 


29.1204396 


9.465^70 


.001179245 


849 


720301 


611960049 


29.1376046 


9.4689661 


.001177856 


850 


722500 


614125000 


29.1547595 


9.4726824 


.001176471 


851 


724201 


616295051 


29.1719043 


9.4763957 


.001175088 


'852 


725904 


618470203 


29.1890:390 


9.4801061 


.001173709 


853 


727009 


620650477 


29.2061637 


9.4838136 


.001172333 


854 


729316 


6228358G4 


29.2232784 


9.4875182 


.001170960 


855 


731025 


625026:375 


29.2403830 


9.4912200 


.001169591 


856 


732736 


627222016 


29.2574777 


9.4949188 


.001168224 


857 


734449 


629422793 


29.274502:3 


9.4986147 


.001166861 


858 


7:36164 


631628ri2 


20.2916:370 


9.5023078 


.001165501 


fm • 


737881 


6338397; 9 • 


29.3087018 


9.5059980 


.001164144 


860 


739600 


636056000 


29.3257566 


9.5096854 


.001162791 


861 


741321 


638277:381 


29.3428015 


9.5ia3699 


.001101440 


862 


743044 


64050:3028 


29.359a3a5 


9.5170515 


.001160093 


863 


744709 


042735G47 


29.3768610 


9.5207:303 


.001158749 


864 


746496 


644972544 


29.3938769 


9.52440<;3 


.001157407 


865 


748225 


647214625 


29.410882:3 


9.5280794 


.001156069 


860 


749956 


649461896 


29.4278779 


9.5317497 


.001154734 


867 


751689 


651714:363 


29.4448637 


9.5354172 


.001153403 


868 


7534^4 


0539?2032 


29.4618397 


9.5390818 


.001152074 



22 



SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square 
Hoots. 


Cube Rootq. 


Reciprocalfl. 


8G9 


7551C1 


056234909 


29.4788069 


9.i>127437 


.C01150748 


870 


756900 


C58503000 


29.4957624 


9.5464027 


.001149425 


871 


758641 


G607/()311 


29.5127091 


9.5500089 


.001148106 


873 


760384 


G63054&18 


29.52^^1 


9.5537123 


.001146789 


873 


762129 


6«5;iS8G17 


29.5465734 


9.5578630 


.001145475 


874 


763876 


667627021 


29.5634910 


9.5610106 


.001144165 


875 


V65625 


669921875 


29.580;««» 


9.5646569 


.001142867 


876 


767376 


672221376 


29.5972972 


0.5682982 


.001141553 


877 


769129 


674526i:« 


29.6141858 


0.6719377 


.001140251 


878 


7708&4 


676836158 


29.6:310648 


9.5755745 


.0011380ri2 


879 


7?^641 


679151439 


29.6479342 


0.5792065 


.001137650 


880 


774400 


681472000 


29.6647939 


0.5828397 


.001186864 


881 


776161 


683797841 


29.6816442 


,9.5864682 


.001135074 


882 


7779^^ 


6HJ128968 


29.6984848 


9.5900939 


.001138787 


883 


779689 


68W65387 


29.7163159 


9.5937169 


.001182803 


884 


781456 


690807104 


29.7:321375 


9.5973373 


.001131222 


885 


783225 


693154125 


29.7489196 


9.6009548 


.001129944 


886 


781996 


695506456 


29.7657521 


9.6045C96 


.001128668 


887 


786769 


697864103 


29.7825452 


9.6081817 


.001127896 


888 


788.'>44 


700227072 


29.79932B9 


0.6117911 


.001126126 


889 


790321 


702595369 


29.8161090 


0.01539r/ 


.0011^4650 


890 


792100 


704969000 


29.8328678 


9.6190017 


.001128606 


891 


793881 


707347971 


2:9.8496231 


9.(2i26C30 


.001122834 


S»i 


795661 


709732288 


29.8663690 


9.6262016 


.001121076 


893 


797449 


712121957 


29.8831056 


9.0297975 


.001110621 


894- 


799236 


714516984 


29.8998328 


9.6333007 


.001118668 


895 


801025 


716917:375 


29.9165506 


9.G369812 


.001117818 


896 


802816 


719323136 


29.9332591 


9.C40oC90 


.001116071 


897 


804609 


?217a4273 


29.9499583 


9.044ir>42 


.001114827 


898 


806404 


?241 50792 


29.9666481 


9. 0477^67 


.001113686 


899 


808201 


?-«}5?2699 


29.983328/ 


9.051S1G6 


.001112847 


900 


810000 


729000000 


80.0000000 


9.6.'>48938 


.001111111 


901 


811801 


731432701 


30.0166620 


9.05846^4 


.001109678 


902 


813604 


733870808 


30.0333148 


9.6G2O403 


.001108647 


903 


815409 


736314327 


30.04995&1 


9.0656096 


.001107420 


901 


81?216 


738763264 


30.0665928 


9.6691762 


.0011C6195 


905 


819025 


741217625 


30.0832179 


9.0727403 


.001104072 


906 


820836 


743677416 


30.0998339 


9.0768017 


.001103758 


907 


822649 


746142643 


30.1164407 


9.6798601 


.001102580 


908 


624464 


748613312 


30.1:330383 


9.6834166 


.001101822 


909 


826281 


751089429 


30.1496269 


9.0869701 


.001100110 


910 


828100 


753571000 


30.1662063 


0.G905211 


.001006001 


911 


829921 


756058031 


30.1827765 


0.(5940094 


.001097695 


912 


831744 


758550528 


30.1093377 


9.6976151 


.001096491 


913 


883569 


76104R497 


30.2158899 


9.7011.583 


.001096200 


914 


835396 


763551944 


30.2:3^1329 


9.7046989 


.001094002 


915 


837^25 


766060875 


80.^489669 


9.70S»369 


.001092896 


916 


839056 


768575296 


80.2654919 


9.7117r23 


.001001708 


917 


840889 


771095213 


80.2820079 


9.71.5:30.51 


.001090618 


918 


842724 


773()20632 


30.2985148 


9.71H?C>51 


.001089335 


919 


844561 


776151559 


30.3150128 


9.7223631 


.001088130 


920 


84(M00 


778688000 


30.3315018 


( 9.72588a3 


.001086967 


921 


848241 


781229961 


30.3479818 


9.7294109 


.001085rr6 


922 


850084 


783'/7V448 


30.3614529 


9.7820:309 


.0010^4599 


923 


a51929 


786330467 


30.380915i 


9.7:3(M484 


.G010K3423 


924 


853776 


788889024 


30.3973(W3 


9.7:399634 


.001082251 


925 


855625 


79145:^125 


30.41.38127 


9.7434758 


.001081081 


926 


857476 


794022776 


80.4302481 


9.7469857 


.001079914 


927 


a59329 


796.597983 


80.4466747 


9.7.504930 


.001078749 


928 


861181 


799178752 


30.4630921 


9.7539J)79 


.001077586 


929 


863041 


80176.')0«9 


30.4795013 


t). 7575002 


.001076426 


'WO 


864900 


804357000 


30.4956014 


9.7610001 


001075269 



CUBE ROOTS, AND RECIPROCALS. 



23 



f 

No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Boots. 


Reciprocals. 


031 


866761 


806954491 


30.5122926 


• 0.7644974 


.001074114 


932 


868624 


809557568 


30.5286750 


0.7679922 


.001072961 


933 


870489 


8121l>6237 


30.5450487 


0.7714845 


.001071811 


{m 


872356 


814780504 


30.5614186 


0.7749743 


.C01070664 


935 


874225 


817400375 


30.57r;697 


0.7784616 


.C010C9519 


93G 


87(3096 


820025856 


80 5941171 


0.7819466 


.001068376 


937 


877969 


822656953 


30.6104557 


0.7854288 


.001067236 


938 


879844 


8252936ra 


80.6267857 


9.7689087 


.001066098 


939 


881721 


827936019 


30.6431069 


9.7923861 


.001064963 


940 


883600 


830584000 


80.6594194 


9.7958611 


.001063830 


941 


eii5481 


as;3237621 


30.6757233 


9.7993336 


.001062699 


942 


887364 


835896888 


80.6920185 


9.8028036 


.001061571 


943 


889249 


888561807 


30.7083051 


9.8062711 


.001060445 


944 


891136 


841232384 


}.0 7245830 


9.6097362 


.001059322 


945 


893025 


843906625 


80.7408523 


9.8131989 


.001058201 


946 


894916 


846590536 


80.7571130 


0.8166591 


.001057082 


947 


896809 


849278123 


30.7733651 


0.8201169 


.001055966 


948 


898704 


851971392 


80.7896086 


9.6235723 


.001054852 


949 


900601 


854670349 


30.80J:8436 


9. 62'. 0252 


.001053741 


950 


002c00 


857375000 


£0.8220rOO 


9.6CC4757 


.001052632 


951 


904401 


860085351 


£0.8382879 


9.62o9238 


.001051525 


952 


906304 


862801408 


£0.8544972 


9.&37Se95 


.001050420 


953 


908209 


865523177 


30.870€e81 


9.84C8127 


.CC1049318 


954 


910116 


868250664 


30.8866904 


9.6442526 


.001048218 


955 


912025 


870983875 


80.9020743 


9.8476C20 


.001047120 


956 


913936 


873722816 


£0.9192497 


9.6511260 


.C0104€025 


957 


915849 


876467493 


•£0.9354166 


9.6545017 


.C01044932 


958 


917764 


879217912 


80.9515751 


9. 657 9929 


.001042641 


959 


919681 


881974079 


30. 9677251 


9.6614218 


.CC1C42753 


960 


921600 


884736000 


50.9828668 


9.6648483 


.C01041667 


961 


923521 


C87503681 


31.0C00C0O 


9.6662724 


.CG1C4C563 


962 


925444 


£90277128 


31 .0161248 


9.6716941 


.GClOSSSOl 


963 


927869 


898056347 


31.0322413 


9.8751135 


.C01038422 


064 


929296 


£95841344 


31.0483494 


9.8785305 


.C010S7344 


965 


931225 


898632125 


31.0644491 


9.6819451 


.C01C2G269 


966 


933156 


901428696 


31.0805405 


9.6863574 


.C01C25197 


967 


935080 


904231063 


31.0966236 


9.6687673 


.C01C34126 


968 


937024 


907039232 


31.1126984 


9.6921749 


.001022058 


969 


938961 


909853209 


31.1287648 


9.6955601 


.€01031192 


OTt) 


940900 


912673000 


31.1448230 


9.686C6E0 


.C0103C928 


C71 


942841 


915498611 


31.1608729 


9.{;Ci:£6£5 


.C01C298C6 


972 


944784 


918330048 


31.1769145 


9.CC57817 


.001C26607 


973 


946729 


S21 167317 


31.19294';9 


9. £0917 76 


.C01C27749 


974 


948676 


924010424 


31.2089731 


9.9125712 


.C0102GG94 


975 


950625 


926859375 


31.2249900 


9.915CC24 


.C010i:5G41 


976 


952576 


929714176 


SI. 2409987 


9.9102513 


.C01G24CC0 


977 


954529 


932574883 


31.2569992 


9.9227379 


.C01023541 


978 


956484 


035441352 


SI. 2729915 


0.C2C1222 


.001022495 


979 


958441 


938313739 


£1. £889757 


9 12C5042 


.C01021450 


960 


960400 


941192000 


31.S04D517 


9.S328839 


.001020408 


961 


962361 . 


044076141 


31.£2C91£'5 


9.C2G2013 


.001019268 


082 


964324 


046966168 


S1.33CS7fi2 


9.9SC6263 


.C01016S20 


983 


966289 


949862087 


31.3528SC8 


9.9430C92 


.001017294 


964 


968256 


952763904 


31.3687743 


9.1MG3797 


.001010260 


965 


970225 


955671025 


31.2&47097 


9.9497479 


.00101C228 


966 


972196 


958585256 


31.4CGC3G9 


9.9531128 


.001014199 


967 


974109 


961504803 


31.416.'35G1 


9 9564775 


.001012171 


mo 


976144 


964430272 


31.4324673 


9 9598389 


.001012146 


060 


978121 


967361669 


31 .4483704 


9.9631981 


.001011122 


090 


980100 


970299000 


31.4642654 


9.9665549 


.001010101 


001 


982081 


973242271 


31.4801525 


9.9699095 


.001009062 


092 


984064 


976191488 


81.4960315 


9.9732619 


.001006005 



24 



SQUARES, CUBES, SQUARE ROOTS, ETC. 



No. 


Squares. 


Cubes. 


Square 
Roots. 


Cube Root!. 


Reciprocals. 


903 


930)40 


979140057 


31.5119025 


9.9766120 


.001007049 


901 


9.5303J 


9:?2107784 


31.5277055 


9.9799599 


.00100603G 


905 


900025 


983074875 


31.5436:^ 


9.9833055 


.0010(»Q25 


906 


932010 


9330479:30 


31.5594077 


9.9866488 


.001004016 


937 


904000 


991020973 


31.5753008 


9.9899900 


.001008009 


903 


930004 


994011902 


31 591i;.i80 


9.9933289 


.001002004 


909 


903001 


9070029:)0 


81.C009813 


9.996665G 


.ooiooion 


1000 


1030000 


lOOOOOvWJO 


31.G22r7GG 


10.0000000 


.001000000 


1001 


1002001 


1003003001 


81.6885840 


10.0033322 


.0009990010 


1003 


1004004 


1006012008 


31.6543836 


10.0006022 


.0009960040 


1003 


1006009 


1009027027 


81.6701752 


10.0099899 


.0009970090 


1004 


1008016 


1012.)48064 


31.6859590 


10.0133155 


.0009960159 


1005 


1010025 


1015075125 


81.7017349 


10.0166889 


.0009950249 


1006 


1012036 


1018108216 


81.7175030 


10.0199601 


.0009^)358 


iao7 


1014049 


1021147343 


81.7332033 


10.0232791 


.0009880487 


1003 


1010004 


1024192512 


81.7490157 


10.0265958 


.0009990635 


1009 


1018031 


1027243729 


81.7647603 


10.0299104 


.0009910603 


1010 


1020100 


1033301000 


31.78049r2 


10.0332228 


.0009900990 


1011 


1028121 


1033364331 


81.7962232 


10.0365330 


.0.309891197 


1012 


1024144 


1036433r2.3 


31.8119474 


10.0396410 


.0009681423 


1013 


1026169 


1039503197 


81.8276609 


10.0431469 


.00098n663 


1014 


1028196 


1042593744 


31.8433666 


10.0404506 


.0009861933 


1015 


1030225 


1045673 J75 


81.8590646 


10.0497521 


.00096»2217 


1016 


1032256 


1048772336 


31.8747549 


10.0580514 


.00Og642S») 


1017 


1034289 


1051871913 


81.8904374 


10.0568485 


.0009688843 


1013 


1033324 


1054977.i-32 


81.9061123 


10.0596435 


.0009823188 


1019 


1038361 


1058039850 


31.9217794 


10.0629364 


.0009818643 


1020 


1040400 


1061203000 


81.9374388 


10.0662271 


.0009608982 


1021 


1042441 


1064332201 


31.9530906 


10.0695156 


.0009794319 


10-iZ 


1044484 


1067402643 


31.9837347 


10.0728020 


.0009784796 


1023 


104a529 


1070539167 


31.9843712 


10.076086.3 


.0009775171 


1024 


1048576 


1073741824 


32.(000030 


10.0793884 


.0009766625 


1025 


1050325 


1076390025 


82 0156212 


10.0826484 


.0009766098 


1028 


1052576 


1080045576 


82 0312:313 


10.0859282 


.0009746689 


1027 


1051729 


1083200S33 


32.0483407 


10.0892019 


.0009787098 


1028 


1056734 


10383733/2 


iJ2.0824'331 


10.0924755 


.0009727886 


1029 


1053S11 


10335473 D 


82.0783333 


10.0a57469 


.0009718173 


1030 


1060900 


1092727033 


32.0936131 


10.0990163 


.0009708738 


1031 


1062961 


1095312731 


32.1091837 


10.1022835 


.0009699381 


laiJ 


1085024 


1099104703 


32.1247503 


10.1055487 


.00 9689988 


1033 


1067039 


1102302337 


32.1403173 


10.1088117 


.0009680548 


1034 


1089156 


1105507334 


82.1558704 


10.1120726 


.0009671180 


1035 


1071225 


1103717375 


82.17141.59 


10.1153314 


.0009661886 


1036 


1073296 


1111934053 


82.1869539 


10.11&5882 


.0009658510 


1037 


1075:369 


1115157653 


32.2024314 


10.1218428 


.0009643808 


1038 


1077444 


llia333372 


82.2180374 


10.1250953 


.0009633911 


ia39 


1079521 


1121622319 


82.233.5229 


10.1283457 


.0009624639 


1040 


1031600 


1124364000 


32.2490310 


10.1315941 


.0009616885 


1041 


1083681 


1128111921 


82.2645316 


10.1348403 


.0009606148 


1042 


1035764 


1131:306038 


82.2800248 


10.1380845 


.0009596929 


1043 


1087349 


11*4626507 


82.2955105 


10.1413286 


.0009587738 


1044 


1089936 


11378a31J?4 


82.8109888 


10.1445667 


.0009578544 


1045 


1092025 


1141166125 


82.3264598 


10.1478047 


.0009569378 


1046 


1094116 


1144445336 


82.8419233 


10.1.510106 


.0009.560229 


1047 


1096209 


11477130823 


32 a573794 


10.1.542744 


.0009551096 


1048 


1098304 


1151022592 


32 37289m 


10.1575002 


.0009541985 


1049 


1100401 


1154320649 


32.3882605 


10.1607:359 


.0009532888 


1050 


1102500 


1157625000 


82.40:37035 


10. 1839836 


.0009523810 


1051 


1104601 


11609:35651 


82.4191:301 


10.1671893 


.0009514748 


::o2 


1106704 


1164252(k)8 


a2.4345l?r> 


10.1704129 


.0009505708 


105.3 


1108809 


1167575877 


32.44^)615 


10.1738:344 


.0009490676 


1054 


1110916 


1170905464 


32.4653662 


10.1768639 


.OOO9487n06 















WEIGHTS AND MEASURES. 25 

WBIGHTS AND MEABURBEl 
Measures of Len^b. 
: Inches = 1 foot- 
feet = 1 yard — 38 inches, 
i yards = 1 rod = 188 inches = 18i ft. 
' rods = 1 turlong = 70-20 inches = fiflO ft. = 220 yds, 
furlongs = imile = 63360 inches - 13280 f t. = 1760 yds-, 
yard = 0,0006682 of a mile. [= 320 rods, 
ounteb's chain. 
7.92 Inches = 1 link. 
100 links = 1 cliain = 4 rods = 00 feet. 
80 chains = 1 mile. 



6 feet = 1 fathom. 120 fathoms = 1 cable's length. 

I Deoimals of a 



26 MEASURES OF SURFACE ANT) VOLUME. 

GEOGRAPHICAL AND NAUTICAL. 

1 degree of a great circle of the earth = GO. 77 statute miles. 
1 mile = 2046.58 yards. 

.siio?:makers' measure. 

No. 1 is 4.125 inches in length, and every succeeding number la ' 
-^Mii of an inch. 

'J'here are 28 numbers or divisions, in two series of numbers, vis.,  
iroui 1 to 18, and 1 to 15. 

MISCELLANEOUS. 

1 palm = 3 inches. 1 span = 9 inches. 

1 hand = 4 inches. 1 meter = 3.2800 feet. 

Measures of Surface. 

144 square inches = 1 squanj foot. 

9 square feet - 1 square yard = 1296 square inches. 
100 square feet = 1 square (architects' measure). 

LAND. 

30i square yards ~ 1 stjuare roJ. 

40 square roils = 1 square rood =1210 square yards. 
4 square roods | — 1 acre = 4840 s<^iuare yards. 

10 square chains S = 100 sfiuare rods. 
040 acres ~ 1 scjuare mile = 3007000 square yards = 

208.71 feet square = 1 acre. 1 102400 sq. rods = 25C0 sq. roods. 

A Heciion of land is a square mile, and a quarier-acction is ICO 
acres. 

Measures of Volume. 

1 gallon liquid measure = 231 cubic inches, and contains 8.330 
avoir.liii)o:s pounds of distilled water at 39.8° F. 
1 gallon dr>' measure = 208.S cubic inches. 

1 bushel ( WlncheHicr) contains 2150.42 cubic inches, or TJ.CSft 
],ounils distill«Ml water at 39. ^° F. 

A heape.l bushel contains 2747.715 cubic inches. 

DRY. 

2 pints = 1 quart = 07.2 cubic inches. 
4 quarts = 1 gallon = 8 pints = 20H.8 cubic inches. 
2 gallons = 1 pe<^k = 10 pin Is = 8 quarts = 537.0 cubic inches. 
4 pecks = 1 bushel = 04 pints = 32 quarts = 8 gals. = 2150.42 
1 chaldron = 30 heaped bushels = 57.244 cubic feet. |cu. ia 
1 cord of wood =128 cubic feet. 



MEASURES OF VOLUME AND WEIGHT. 27 

IJQUID. 

4 gills == 1 pint. 

2 pints = 1 quart = 8 gills. 

4 quarts = 1 gallon = .32 gills = 8 pints. 

In the United States and Great Britain I barrel of wine or brand]^ 
= 31i gallons, and contains 4.211 cubic feet. 

A hogshead is 03 gallons, but this term is often applied to casks 
ftf various capacities. 

Cubic Measure. 

/^r^^r 1728 cubic inches = 1 foot. 
27 cubic feet = 1 yard. 

In measuring loood, a pile of wood cut 4 feet long, piled 4 feet 
high, and 8 feet pn the ground, malting 128 cubic feet, is called a 
cord. /^--^ "/ /i'>- -/^. -- S'i^>^ 

16 cubic feet make one cord foot. 

A perch of stone is lOJ feet long, 1 foot high, and li feet thick, 
and contains 242 cubic feet. 

A perch of stone is, however, often computed differently in dif- 
ferent localities; thus, in Philadelphia, 22 cubic feet are called a 
perch, and in some of the New-England States a perch is computed 
at 16i cubic feet. 

A ton^ in computing the tonnage of sliips and other vessels, is 
100 cubic feet of their internal space. 

Fluid Measure, 

60 minims = 1 fluid drachm. 

8 fluid drachms = 1 ounce. 
16 ounces ~ 1 pint. 

8 pints = 1 gallon. 

Miscellaneous. 

Butt of Sherry = 108 gals. Puncheon of Brandy, 110 to 120 gals. 

Pipe of Port = 115 gals. Puncheon of Bum, 100 to 110 gals. 

Butt of Malaga = 105 gals. TTo'?=?hoad of Brandy, 55 to 00 gals. 

Puncheon of Scotch Whis- Hogshead of claret, 4(5 gals. 

key, 110 to 130 gals. 

Measures of Weiglit. 

The standard avoirdupois pound is the weight of 27.7015 cubic 
inches of distilled water weighed in air at 39.83^, the barometer at 
30 inches. 



28 MEASURES OF WEIGHT. 

AvoirdupoiSy or Ordinary Coiumercial Weight. 

16 drachms = 1 ounce, (oz.). 

16 ounces = 1 pound, (lb.). 

100 pounds = 1 himdred weight (cwt. ). 

20 hundred weight = 1 ton. 

Tn collecting duties upon foreign goods at the TJnite<l Sta 
custom-houses, and also in freighting coal, and selling it by who 
«jale, — 

28 poimds = 1 quarter. 

4 quarters, or 112 lbs. = 1 himdred weight. 
20 hundred weight = 1 long ton = 2240 poimds. 
A stone = 14 pounds. 

A quintal = 100 pomids. 

The following measiu*es are sanctioned by custom or law : 

32 poimds of oats = 1 bushel. 

45 poimds of Timothy- seed = 1 bushel. 

48 poimds of barley = 1 bushel. 

50 pounds of rye = 1 bushel. 

56 poimds of Indian corn = 1 bushel. 

50 poimds of Indian meal = 1 bushel. 

60 pounds of wheat = 1 bushel. 

60 pounds of clover-seed = 1 bushel. 

60 pounds of potatoes = 1 bushel. 

56 pounds of butter = 1 firkin. ^ 

100 pounds of meal or flour = I sack. 
100 pounds of grain or flour = 1 cental. 
100 pounds of dr>' fish = 1 quintal. 

100 pounds of nails = 1 cask. 

196 pounds of flour = 1 barrel. 

200 pounds of beef or pork = 1 barrel. 

Troy Weij^ht. 

USED IN WEIGHIXG GOLD OR SILVER. 

24 grains = 1 pennyweight (pwt.). 

20 pennyweights = 1 ounce (oz.). 

12 ounces = 1 pound (lb.). 

A carat of the jewellers, for precious stones, is, in the Uni 
States, 3.2 grains: in London, 3.17 grains, in Paris, 3.18 grains i 
divided into 4 jewellers' grains. In troy, apothecaries', and av( 
dupois weights, the grain is the same. 



MEASURES OF VALUE AND TIMK. 29 



Apothecaries' Weiglit. 

USED IN COMPOUNDING MEDICINES, AND IN PUTTING UP 

MEDICAL PRESCRIPTIONS. 



20 grains (gr.) = 1 scruple ( 3 ). 
;^ scruples = 1 drachm ( 3 ). 



8 drachms = 1 ounce (oz.). 
12 ounces = 1 pound (lb.). 



Measures of Value. 



UNITED STATES STANDARD. 



10 mills = 1 cent. 
10 cents = 1 dime. 



10 dimes = 1 dollar. 
10 dollars = 1 eagle. 



The standard of gold and silver is 900 parts of pure metal and 
100 of alloy in 1000 parts of coin. 

The fineness expresses the quantity of pure metal in 1000 parts. 

The remedy of the mint is the allowance for deviation from the 
exact standard fineness and weight of coins. 



*e»' 



Weigrlit of Coin. 

Double eagle = 516 troy grains. 

Eagle = 258 troy grains. 

Dollar (gold) = 25.8 troy grains. 

Dollar (silver) = 412.5 troy grains. 

Half-dollar = 192 troy grains. 

5-cent piece (nickel) = 77.16 troy grains. 

3-cent piece (nickel) = 30 troy grains. 

Cent (bronze) = 48 troy grains. 

Measure of Time. 



365 days = 1 common year. 

366 days = 1 leap year. 



60 seconds = 1 minute. 
60 minutes = 1 hoiu*. 
24 hours = 1 day. 

A solar day is measured by the rotation of the earth upon its 
ji :1s with respect to the sun. 

in astronomical computation and in nautical time the day com- 
mences at noon, and in the former it is counted throughout the 24 
hours. 

In cixil coinputation the day conunences at midnight, and is 
divided into two portions of 12 hours each. 

A solar year is the time in which the earth makes one revolution 
around the sun; and its average time, called the mean solar year, 
is 305 days, 5 hours, 48 minutes, 49.7 seconds, or nearly 365i days. 

A mean lunar month, or lunation of the moon, is 29 days, 12 
hours, 44 minutes, 2 seconds, and 5.24 thirds. 



30 THE CALENDAR. — ANGULAR MEASURE. 

The Calendar, Old and New Style. 

The Julian Calendar was established by Julius Csesar, 44 B.C., 
and by it one day was inserted in every fourth year. This was the 
same thing as assuming that the length of the solar year was 305 
(lays, 6 hours, instead of the value given above, thus introducin;: 
an accumulative error of 11 minutes, 12 seconds, every year. This 
calendar was adopted by the church in 325 A.I>., at the Council of 
Nice. In tlie year 1582 the annual error of 11 minutes, 12 seconds, 
had amounted to a period of 10 days, which, by order of Pope Greg- 
ory XIII., was suppressed in the calendar, and the 0th of October 
reckomnl as the 15th. To prevent the repetition of this error, it 
was decided to l(^a.ve out three of the inserted days every 400 years, 
and to make this omission in the years which are not exactly divisi- 
ble by 400. Thus, of the years 1700, 1800, 1900, 2000, all of which 
arc leap years according to the Julian Calendar, only the last is a 
leap year according to the licfoinned or Greyorian (/alendar. This 
Ileformed Calendar was not adopted by England until 1752, when 
1 1 days were omitted from the calendar. The two calendars are 
now often called the Old Sft/lc. and the New Style. 

The latter style is now adopted in every Cliristian country except 
liussia. 

Circular and Ang^iilar Measures. 

tSEl) FOK MEASUUINO ANGI^ES AND ARCS, AND FOR DBTSH- 
MININO LATITUDE AND LONGITUDE. 

CO seconds (") = 1 minute ('). 

00 minutes = 1 degree (°). 

360 degrees = 1 circumference (C). 

Herouds are usually subdivided into tenths and hundredths. 

A iiilnute of the circumference of the earth is a geographical 
mile. 

D('(j}'pes of the earth's circumference on a meridian average 69.7.6 
common miles. 



THE METRIC SYSTEM. 

Thf nn'frir. fii/Moni is a system of weiu^lits and measiu'es based 
r.pon a unit called a meter. 

The meter is one ten-millionth part of the distance from the 
equator to either pole, measured on the earth's surface at the level 
jl the sea. 



THE METRIC SYSTEM. 31 

The names of derived metric denominations are formed by pre- 
fixing to the name of the primary unit of a measure — 



Milli (miU'e), a thousandth, 
Centl (sent'e), a hundredth, 
Dec! (des'e), a tenth, 
Deka (dek'a), ten, 



Hecto (hek'to), one hundred, 
Kilo (kil'o), a thousand, 
Myria (mir'ea), ten thousand. 



This system, first adopted by France, has been extensively adopteq 
by other countries, and is much used in the sciences and the arts. 
It was legalized in 1866 by Congress to be used in the United States, 
and is already employed by the Coast Survey, and, to some extent, 
by the Mint and the General Post-Office. 



Linear Measures. 

The meter is the primary unit of lengths. 

Table. 

10 millimeters (mm.) = 1 centimeter (cm.) = 0.393*7 in. 

10 centimeters = 1 decimeter = 3.937 in. 

10 decimeters = 1 meter = 30.37 in. 

10 meters = 1 dekameter = 393.37 in. 

10 dekameters = 1 hectometer = 328 ft. 1 in. 

10 hectometers = 1 kilometer (km.) = 0.62137 mi. 

10 kilometers — 1 myriameter = 6.2137 mi. 

The meter is used in ordinary measurements; the centimeter or 
jnillimeterf in reckoning very small distances; and the kilometer y 
for roads or great distances. 

A centimeter is about ^ of an inch ; a meter is about 3 feet 3 
inches and | ; a kilometer is about 200 rods, or $ of a mile. 

Surface Measures. 

The square meter is the primary unit of ordinary surfaces. 
The are (air), a square, each of whose sides is ten wicie/vs, is 
the unit of land measures. 



Table. 



100 square millimeters (sq. mm.) = 1 square ) _. ^ -^j-- j^^l^ 

centimeter (sq. cm. ) S 

100 square centimeters = 1 square decimeter = 15.5 sq. inches. 
100 square decimeters ^ 1 square I ^ ^55^ .^^ ^^ j jgg y^^ 

IMTEB (sq. ni. I ) 



Axao 
100 centUrea, <x sq. meters, = 1 AR 



A square meter, or one emttari, tl 
Bquare yards, and a hectare Is ftboot St X' 



CnMol 

The cubic meter, or itert (stair), t> the 
Tablk. 
1000 cubic inillimM«ra (en. mm. ) = 1 cut 



The atere is the tuune given to the i 
wood and timber. A t«ittli of & itae Is 

are a JefciMtere. 
A cubic meter, or etere, Is about 11 cub 

feet. 

Liquid and Dry M< 

The liter (leeter} is the primary unit 
and is a cube, each of whose edgee is a t( 

The kectnliter Is the unit In meaanring 
fruits, roota, and liquids. 

Table. 

10 milliliters (ml.) = 1 centiliter (d-) 

lO-centinters = 1 decUlter 

10 deciliters = 1 lttbb (1.) 

10 liters = 1 dekaliter 

10 dekaliters = 1 BECTOLITEB (hi 

10 hectoliters = 1 kiloUter 

A centiliter is abotit i of a flidd oonee; a (Iter Is about liV H 
quarts, or I'.f of a dry quart; mJieetoUter Is about 2) bmheb; a 



The gram it the primary unit of wel^its, and Is tbs 
vacuum of a cubic ceutlmeter of dlaUUed water at Uw 
«f SU.2 degrees FkhrenbdL 



ANCIENT MEASURES AND WEIGHTS. 



SZ 



Table. 
10 milligrams (mg.) = 1 centigram 



10 centigrams 
10 decigrams 
10 grams 
10 dekagrams 
10 hectograms 
10 kilograms 
10 myriagrams 
10 quintals 



0.1543 troy grain. 
1.543 troy grains. 
15.432 troy grains. 
0.3527 avoir, ounce. 
3.5274 avoir, ounces. 
2.2046 avoir, pounds. 
22.046 avoir, pounds. 
220.46 avoir, pounds. 



= 1 decigram = 

= 1 GRAM (g. ) = 

= 1 dekagram = 

= 1 hectogram = 

= 1 KILOGRAM (k.) = 

= 1 myriagram = 
= 1 quintal = 

= 1 TONNE AU (t. ) = 2204.6 avoir, pounds. 

The gram is used in weighing gold, jewels, letters, and small 
quantities of things. The kilotjram, or, for brevity, kiloy is used 
by grocers; and the tonneau (tonno), or metric toji, is used in find- 
ing the weight of very heavy articles. 

A gram is about 15i grains troy; the kilo about 2i pounds avoir- 
dupois; and the metric ton, about 2205 pounds. 

A kilo is the weight of a liter of water at its greatest density; and 
the metric ton, of a cubic meter of water. 

Metric numbers are written with the decimal -point (.) at 
the right of the figures denoting the unit; thus, 15 meters and 3 
centimeters are written, 15.03 m. 

When metric numbers are expressed by figures, the part of tha 
expression at the left of the decimal-point is read as the number 
of the unit, and the part at the right, if any, as a number of the 
lowest denomination indicated, or as a decimal part of the unit; 
thus, 46.525 m. is read 46 meters and 525 millimeters, or 46 and 525 
thousandths meters. 

In writing and reading metric numbers, according as the scale is 
10, 100, or 1000, each denomination should be allowed one, two, op 
three orders of figures. 



SCRIPTURE AND AKCIfilTT MEASURES AKD 

"WEIGHTS. 

Scripture Long: Measures. 





Inches. 




Feet. 


Inches. 


Digit 


= 0.912 


Cubit 


= 1 


9.888 


Palm 


= 3.648 


Fathom 


= 7 


3.552 


Span 


= 10.944 









Egryptian Longr Measures. 

Kahad cubit ^ 1 foot 5.71 Indies. Royal cubit s= 1 foot 8.66 inches. 



-J^.ZOrZ IdJL^'iZ^ AyZv "TTE^aHT^ 









>^. = -'trr. C-ra = 1 a.406 



^-± A!kxiziirUi5 Tiiia* = 11.11912 



'■»•- f 415.1 

l-:r.-^:i^- z:±^ = -.>^ I 431 J! 



^ = ;,-i^.' r>r.w>-. = ije.5 



 : .• :i.*r. -jk  .tr . ■» ^ ": tk .-• -^ .-"■;:;■> i-x. ^•.i\«:. i:-r ibe maw wdghL 

Miscellaneous. 



lii' 7: -r:lir: f-»: = l.I-H Hobn-w o::b!X =1.817 



MENSURATION. - DEFINITIONS. 



85 



Fig.l 

A Curved Line. 



BflllNSTJRATION. 
Definitions. 

A point is that which has only position. 

A plane is a surface in which, any two points heing taken, thfi 
straight line joining them will be wholly in the 
surface. 

A curved line is a line of which no portion is 
sti-aight (Fig. 1). 

Parallel lines are such as are wholly in the same plane, and have 
the same dii-ection (Fig. 2). 

A broken line is a line composed of a 
series of dashes ; thus, . fig- 2 

An angle is the opening between two Parallel Lines, 

lines meeting at a point, and is tenued a riyJit angle when the two 
lines are perpendicular to each other, 
an acute angle when it is less or 
sharper than a right angle, and ob- 
iune when it is greater than a right 
angle. Thus, in Fig. 3, 

A A A A are acute angles, 

O O O O are obtuse angles, 

K K R R are right angles. 

Polygons. 

A polygon Is a portion of a plane bounded by straight lines. 

A triangle is a polygon of three sides. 

A scalene triangle has none of its sides equal; an isosceles tri* 
angle has two of its sides equal; an equi- 
lateral triangle has all three of its sides 
equal. 

A right-angle triangle is one which has a 
right angle. The side opposite the right Fig. 4. 

angle is called the hypothenuse; the side on Right-angle Triangle. 
which the trian^e is supposed to stand is called its bane, and the 
other side, its altitude. 








FI9.6. 

Triangle. 



Fig. 6. Fig. 7. 

lso8C«les Triangle. Bquilateral Triangl« 



GEOMETRICAL TERMS. 



.1. 



quadrilateral is a polygon of four sides. 
Quadrilaterals are divided into classes, as follows, — the irape' 
zium (Fig. 8), which has no two of its sides parallel; the trapezoid 
(Fig. 9), which has two of its sides parallel; and the paralleloyram 
(Fig. 10), which is bounded by two pairs of parallel sides. 




\ 






/ 



Fig. 8. 



Fig. 9. 



Fig. 10. 



A parallelogram whose sides are not equal, and its angles not 
right angles, is called a rhomboid (Fig. 11); when the sides are all 
equal, but the angles are not right angles, it is called a rhombvtt 
)Fig. 12) ; and, when the angles are right angles, it is called a rectan- 
gle ( Fig. 13). A rectangle whose sides are all equal is called a square 
(Fig. 14). Polygoils whose sides are all equal are called regular. 



L 



I 




Fig. 11. 



Fig. 12. 



Fig. 13. 



Fig. 14. 



Besides the square and equilateral triangles, there are 

The i)entaf(ow (Fig. 15), which has five sides; 
The hexagon (Fig. 16), which has six sides; 
The heptagon (Fig. 17), which has seven sides; 
The octagon (Fig. 18), which has eight sides. 





Fig. 15. 



Fig. 16. 





Rg. 17. 



\ / 

Rg. 18. 



The enneagon has nine sides. 
The decagon has ten sides. 
The dodecagon has twelve sides. 

For all polygons, the side upon which it is supposed Co stand h 
called its base ; the pei*pendlcular distance from the highest side oi 



GEOMETRICAL TERMS. 



37 



angle to the base (prolonged, if necessary) is called the altitude ; and 
a linf. joining any two angles not adjacent is called a diagonal, 

A perimeter is the boundary line of a plane figure. 

A circle is a portion of a plane bounded by a curve, all the pointi 
of which are equally distant from a point witliin called the centre 
(Fig. 19). 

The clrcurnference is the curve which bounds the circle. 

A radius is any straight line drawn from the centre to the cir- 
cumference. 

Any straight line drawn through the centre to the circumference 
on each side is called a diameter. 

An arc of a circle is any part of its circumference. 

A chord is any straight line joining two points of the circumfer- 
ence, as bd. 

A segment is a portion of the circle 
included between the arc and its 
chord, as A in Fig. 19. 

A sector is the space included be- 
tween an arc and two radii drawn to 
its extremities, as B, Fig. 19. In the 
figure, (U) is a radius, cd a diameter, 
and db is a chord subtending the arc 
bed, A tangent is a right hne which / 
in passing a curve touches without 
cutting it, as fg, Fig. 19. 




Fig. 19. 



Volumes. 

A prism is a volume whose ends are equal and parallel polygons, 
and whose sides are parallelograms. 

A prism is triangular f rectangular, etc., according as its ends 
are triangles, rectangles, etc. 

A cube is a rectangular prism all of whose sides are squares. 

A cylinder is a volume of uniform diameter, bounded by a cm \o 1 
surface and two equal and parallel circles. 

A pyramid is a volume whose base is a polygon, 
and whose sides are triangles meeting in a point 
called the vertex, 

A p3rramid is triangular, quadrangular, etc., ac- 
cording as its base is a triangle, quadrilateral, etc. 

A cone is a volume whose base is a circle, from 
which the remaining surface tapera uniformly to 
a point or vertex (Fig. 20). P»fl- 20. 

Conic ucUona are the figures made by a plane cutting a cone. 




38 



MENSURATION. 



An ellipse is the section of a cone wlien cut by a plane passing 
obliquely through both sides, as at «6, Fig. 21. 

A paxcthola is a section of a cone cut by a plane parallel to its 
side, as at cd. 

A hyjnrhola is a section of a cone cut by a plane at a greater 
angle through the base than is made by the side of the cone, as 
at i'lu 

In the ellipse, the tranarerse axis, or loju/ 
diameter f is the longest line that can be drawn 
through it. The conjugate axis, or short di- 
ametery is a line drawn through the centre, 
at right angles to the long diameter. 

A frustum of a jyyramid or cone is tliat 
which remains after cutting off the upper part 
of it by a plane parallel to the base. 

A sphere is a volume boimded by a curved 
surface, all points of which are equally dis- 
tant from a point within, called the centre. 
Mensuration treats of the meas:urement of llnesy surfaces, 
and volumes. 




^Flg.21. ^ 



To compute the area of a square, a rectangle, a rhombus^ or a 
rhomboid. 

Rule. — Multiply the length by the breadth or height; thus, in 
either of Figs. 22, 23, 24, the area = ab X be. 





Fig.23 




To coinpiite the area of a triangle. 

c Rule. —Multiply the base by the alti- 

tude, and divide by 2; thus, in Fig. 25, 

ab X cd 
area of abc = 2 

'^ To find the length of the hypothenuse qfa 
right-angle triangle when both <idef 
are knoion. . .. 




MENSURATION. - POLYGONS. 



39 



Fig.26 




KuLE. — Square the length of each of the sides making the right 
angle, add their squares together, and take the j^ 

square root of their suiu. Thus (Fig. 2(3), the 
length of at* = 3, and of 6c = 4; then 

a6 = 3 X 3 = 9 + (4 X 4) = + 10 = 25. 

^25 = 5, or a6 = 5. a 

To find the length of the base or altitude of a right-angle triangle, 
when the length of the hypothenuse and one side is known. 

Rule. — From the square of the length of the hypothenuse 
subtract the square of the length of the 
other side, and take the square root of 
the remainder. 

To find the area of a trapezium. 

Rule. — Multiply the diagonal by the 
sum of the two perpendiculars falling 
upon it from the opposite angles, and 
divide the product by 2. Or, 



ah X (cfi-f (70 
2 




= area (Fig. 27). 



To find the area of a trapezoid (Fig. 28). 

Rule. — Multiply the sum of the two par- 
allel sides by the perpendicular distance between 
them, and divide the product by 2. 

To compute the area of an irregidar polygon. 

Rule. — Divide the polygon into triangles 
by means of diagonal lines, and then add to- 
jrether the areas of all the triangles, as A, B, 
and C (Fig. 29). 

To find the area of a regular polygon. 

Rule. —Multiply the length of a side by 
Jie i>eri)endicular distance to the centre (as 
.((>, Fig. 30), and that product by the nunibcH- 
of sides, and divide the result by 2. 

To compute the area of a regular polygon 
tohen the length of a side only is given. 

Rule. — Multiply the square of the side by 
the luoltipUer opposite to the name of the 
polygon in column A of the following table: — 





a 



Fig.30 



40 



MENSURATION. -POLYGONS AND CIRCLES. 







A. 


B. 


C. 


D. 


Name of Polygon. 


No. of 
BldeB. 


Area. 


liadius of 
circum- 
scribing 
circle. 


Length of 
the side. 


Radius of 
inscrilxKl 
circle. 


Triangle . . . 


3 


0.43d013 


0.5773 


1.732 


0.2887 


Tetragon . . 


4 


1 


0.7071 


1.4142 


0.5 


Pentagon . . . 


5 


1.720477 


0.8506 


1.1756 


0.0S82 


Hexagon . . . 


6 


2.598076 


1 


1 


O.SOti 


Heptagon . . . 


7 


8.633912 


1.1524 


0.8677 


1.0:J8:j 


Octagon . . . 


8 


4.828427 


1.3066 


0.7653 


1.2071 


Nouagon . . . 


9 


6.181824 


1.4619 


0.684 


1.3737 


Decagon . . . 


10 


7.094209 


1.618 


0.618 


1.5383 


UudecagOD . . 


11 


9.36564 


1.7747 


0.5634 


1.7028 


I>odecagon . . 


12 


11.196152 


1.9319 


0.5176 


1.86(5 



To compute the radius of a circumscribing circle when the length 
of a side only is given. 

Rule. — Multiply the length of a side of the polygon by the 
number in column B, 

Example. — Wliat is the radius of a circle that will contain a 
hexagon, the length of one side being 5 inches ? 

Ans, 5X1=5 inches. 

To compute the length of a side of a polygon that is contained in 
a given circle, when the radius of the circle is given* 

Rule. — Multiply the radius of the circle by the number opposite 
the name of the polygon in column C 

Example. — What is the length of the side of a pentagon con- 
tained in a ch'cle 8 feet in diameter ? 

Ans. 8 ft. diameter -^ 2 = 4 ft. radius, 4 X 1.1756 = 4.7024 ft. 

To compute the radius of a circle that can be inscribed in a given 
polygon, when the length of a side is given. 

Rule. — Multiply the length of a side of the polygon by tl>« 
number opposite the name of the polygon in column D, 

Example. — What is the radius of the circle that can be inscribed 
in an octagon, the length of one side being 6 inches. 

Ans. G X 1.2071 = 7.2420 inches. 



Circles. 

To compute the circiunference of a circle. 

Rule. — Multiply the diameter by 3.1416; or, for most purposes, 
by 3 j is sufficiently accurate. 

Example. — What is the circumference of a circle 7 inches in 
diameter ? 

A\is. 7 X 3.1410 = 21.9912 inches, or 7 X 3} = 22 inches^ tht 
error in this last being 0.0088 of an inch. 




MENSURATION. — CIRCLES. 41 

To find the diameter of a circle when the circumference is given. 

Rule. — Divide the circumference by 3.1416, or for a very neai 
approximate result multiply by 7 and divide by 22. 
To find the radius, of an arc, lohen the chord and rise or versed 
sine are given. 

Rule. — Square one-half the chord, also square the rise; divide 
their sum by twice the rise; the result will 
be the radius. 

Example. — The length of the chord ac. 
Fig. 30J, is 48 inches, and the rise, ho, is 6 
inches. What is the radius of the arc ? 

Ans, Rad = '-^i±J^ = ?^±^ = 51 ins. "«• 304- 

2bo 12 

To find the rise or versed sine of a circular arc, when the chord 

and radius are given. 

Rule. — Square the radius; also square one-half the chord; sulx 
tract the latter from the former, and take the square root of the 
remainder. Subtract the result from the radius, and the remainder 
will be the rise. 

Example. — A given chord has a radius of 51 inches, and a 
chord of 48 inches. What is the rise ? 

Ans, Rise = rad — ^md^ — ichord2 = 51 — v^2601 - 576 

= 51 — 45 = 6 inches = rise. 
To compute the area of a circle. 

Rule. — Multiply the square of the diameter by 0.7854, or mul 
tiply the square of the radius by 3. 1416. 

Example. — What is the area of a circle 10 inches in diameter V 

Ans. 10 X 10 X 0.7854 = 78.54 square inches, or 5 X 5 X 8.1410 
= 78.54 square inches. 

The following tables will be found very convenient for finding 
the circumference and area of circles. 



44 



MENSURATION. — CIRCLES. 



ABEAS AND CIRCUMFERENCES OF CIRCLES 

{Advancing by TentJis, ) 



Diam. 

.JO.O 


Area. 


Cirenm. 


Diam. 

35.0 


Area. 


Cireom. 


Dian. 

40.0 


Area. 


CireiiB. 


706.8583 


94.2478 


962.1128 


109.9557 


1256.6371 


125.6637 


.1 


711.5786 


94.5619 


.1 


967.6184 


110.2699 


.1 


1262.9281 


125.9779 


.2 


716.3145 


94.8761 


.2 


973.1397 


110.5841 


.2 


1269.2848 


126.2920 


.3 


721.0662 


95.1903 


.3 


978.6768 


110.8982 


.3 


1275.5573 


126.6062 


.4 


725.8336 


95.5044 


.4 


984.2296 


111.2124 


.4 


1281.8955 


126.9203 


.5 


730.6167 


95.8186 


.5 


989.7980 


111.5265 


.5 


1288.2493 


127.2345 


.6 


735.4154 


96.1327 


.6 


995.3822 


111.8407 


.6 


1294.6189 


127.6487 


.7 


740.2299 


96.4469 


.7 


1000.9821 


112.1549 


.7 


1301.0042 


127.8628 


.8 


745.0601 


96.7611 


.8 


1006.5977 


112.4690 


.8 


1307.4052 


128.1770 


.9 


749.9060 


97.0752 


.9 


1012.2290 


112.7832 


.9 


1313.8219 


128.4911 


31.0 


754.7676 


97.3894 


.36.0 


1017.8760 


113.0973 


41.0 


1320.2543 


128.8053 


.1 


759.6450 


97.7035 


.1 


10-23.5381 


113.4115 


.1 


1326.70-24 


129.1195 


.2 


764.5380 


98.0177 


.2 


1029.2172 


113.7267 


.2 


1333.1663 


129.4336 


.3 


769.4467 


98.3319 


.3 


1034.9113 


114.0398 


.8 


1339.6458 


129.7478 


.4 


774.3712 


98.6460 


.4 


1040.6212 


114.3540 


.4 


1346.1410 


180.0610 


.5 


779.3113 


98.9602 


.5 


1046.3467 


114.6681 


.5 


1352.6520 


180.3761 


.6 


784.2672 


99.2743 


.6 


1052.0880 


114.9823 


.6 


1359.1786 


130.6903 


.7 


789.2388 


99.5885 


.7 


1057.8449 


115.2965 


.7 


1365.7210 


ISl.OOU 


.8 


794.2260 


99.9026 


.8 


1063.6176 


115.6106 


.8 


1372.2791 


131.3186 


.9 


799.2290 


100.2168 


.9 


1069.4060 


115.9248 


.9 


1378.8529 


131.6827 


32.0 


804.2477 


100.5310 


37.0 


1075.2101 


116.2389 


42.0 


1385.4424 


131.9469 


.1 


809.2821 


100.8451 


.1 


1081.0-299 


116.5531 


.1 


1392.0476 


132.2611 


.2 


814.3322 


101.1593 


.2 


1086.8654 


116.8672 


.2 


1398.6685 


132.6752 


.3 


819.3980 


101.4734 


.3 


1092.7166 


117.1814 


.3 


1405.30.)1 


132.8894 


.4 


824.4796 


101.7876 


.4 


1098.5835 


117.4956 


.4 


1411.9574 


133.2036 


.5 


829.5768 


102.1018 


.5 


1104.4662 


117.8097 


.5 


1418.0254 


133.5177 


.6 


834.6898 


102.4159 


.6 


1110.3645 


118.1239 


.6 


1425.3092 


188.8318 


.7 


839.8185 


102.7301 


.7 


1116 2786 


118.4380 


.7 


1432.0086 


184.1460 


.8 


844.9628 


103.0442 


.8 


1122.2033 


118.7522 


.8 


1438.7238 


1^.4602 


.9 


850.1229 


103.3584 


.9 


1128.1538 


119.0664 


.9 


1445.4546 


1»4.7743 


33.0 


855.2986 


103.6726 


38.0 


1134.1149 


119.3805 


43.0 


1452.2012 


186.0886 


.1 


860.4902 


103.9867 


.1 


1140.0018 


119.6947 


.1 


1458.9635 


136.4026 


.2 


86).6973 


104.3009 


.2 


1146.0844 


120.C088 


.2 


1465.7415 


186.7168 


.3 


870.9202 


104.6150 


.3 


1152.0927 


120..3230 


.3 


1472.5352 


186.0310 


.4 


8^0.1588 


104.9292 


.4 


1158.1167 


120.6372 


.4 


1479.3446 


186.3461 


.5 


88/. 4131 


105.2434 


.5 


1164.1564 


120.9513 


.5 


1486.1697 


136.6598 


.() 


886.6831 


105.5575 


.6 


1170.2118 


121.2655 


.6 


1493.0105 


186.0734 


.7 


891.9688 


10>.8717 


.7 


1176.2830 


121.5796 


.7 


1499.8670 


187.2876 


.« 


85)7.2703 


106.18.58 


.8 


1182.3698 


121.8938 


.8 


1506.7393 


187.6018 


.9, 


902.5874 


100.5000 


.9 


1188.4724 


122.2080 


.9 


1513.6272 


187.0150 


34.0 


907.9203 


106.8142 


39.0 


1194..5900 


122.5221 


44.0 


1520.5308 


188.2301 


.1 


9i:{.2688 


107.1283 


.1 


1200.7246 


122.8363 


.1 


1527.4502 


188.5443 


.2 


918.63:31 


107.4425 


.2 


1206.S742 


1-23.1.'>04 


.2 


1534.3853 


188.8584 


.3 


924.0131 


107.7566 


aJ 


1213.0396 


123.4646 


.3 


1541.3360 


130.1726 


.4 


929.4088 


108.0708 


.4 


1219.2207 


123.7788 


.4 


1548.3025 


199.4867 


.5 


934.8202 


108.3849 


.5 


1225.4175 


124.0929 


.5 


1555.2847 


iao.8000 


.6 


940.2473 


108.6991 


.6 


1231.6300 


1-24.4071 


.6 


1562.2896 


140.U63 


.7 


945.6901 


109.0133 


.7 


12:J7.8582 


1-24.7212 


.7 


1569.2962 


1404202 


.8 


951.1486 


109..3274 


.8 


1244.1021 


125.0354 


.8 


1576.33ft6 


140.7484 


.9 


956.6228 


109.6416 


.9 


1250.:i617 


125.3495 


.9 


1583U)700 


141.0575 



MENSl "BATION. - CIRCLBS. 



AREAS AND CIBCnMFBBENOBS OF CIBCLBS. 

{Aii»aHcing by Tenthi.) 



MENSURATION. - CIRCLES. 



AREAS AND CIBCUMFERBNCES OP CIRCLBa 

{Adtnncia'j On Tenlli».) 



s'si«7.4S27  

7 M8T,74T4 











■21«,0W,'. 


■■' 


4UT.4S18 
43Se.lM> 


»a.TSM 






43T0.seu 


warn 


ii8!ti6eo 






aujan 




> 


MH.sau 


mmi 


"'■'■'*" 









MENSURATION. - C'lRCLBS. 



&BEAB AND CIRCUMFEBENCES OF CIRCLB8. 

(Adoancing by Tenths.) 



48 



MENSURATION. — CIRCLES. 



AREAS AND CIRGUMFEBENGES OF CIRCLES. 

{Advancing by Tenths, ) 



Diam 

90.0 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


Diam. 

97.0 


Area. 


CirCBD. 


6361.7251 


282.7433 


93.5 


6866.1471 


293.7389 


7389.8113 


304.7345 


.1 


6375.8701 


283.0575 


.6 


6880.8419 


294.0531 


.1 


7405.0559 


305.0486 


.2 


6390.0309 


283.3717 


.7 


6895.5524 


294.3672 


.2 


7420.3162 


305.3628 


J3 


6404.2073 


283.6858 


.8 


6910.2786 


294.6814 


.3 


7435.5922 


305.6770 


.4 


6418.3995 


284.0000 


.9 


6925.0205 


294.9956 


.4 


7450.8839 


305.9911 


.5 


6432.6073 


284.3141 


94.0 


6039.7782 


295.3097 


.5 


7466.1913 


306.3053 


.6 


6146.8309 


284.6283 


.1 


6954.5515 


295.6239 


.6 


7481.5144 


306.6194 


.7 


6461.0701 


284.9425 


.2 


6969.3106 


295.9380 


.7 


7496.8532 


306.9336 


.8 


6475.3251 


285.2566 


.3 


6984.1453 


296.2522 


.8 


7512.2078 


307.2478 


.9 


6489.5958 


285.5708 


.4 


6998.9658 


296.5663 


.9 


7527.5780 


307.5619 


91.0 


6503.8822 


285.8849 


.5 


7013.8019 


296.8805 


98.0 


7542.9640 


307.8761 


.1 


6518.1843 


286.1991 


.6 


7028.6538 


297.1947 


.1 


7558.3656 


308.1902 


.2 


6532.5021 


286.5133 


.7 


7043.5214 


297.5088 


.2 


7673.7830 


308.5044 


.3 


6.546.8356 


286.8274 


.8 


7058.4047 


297.8230 


.3 


7589.2161 


308.8186 


.4 


6561.1848 


287.1416 


.9 


7073.3033 


298.1371 


.4 


7604.6648 


300.1327 


.5 


6575.5498 


287.4657 


95.0 


7088.2184 


298.4513 


.51 7620.1293 


309.4400 


.6 


6589.9304 


287.7699 


.1 


7103.1488 


298.7655 


.er 


7635.6095 


309.7610 


.7 


6604.3268 


288.0840 


.2 


7118.1950 


299.0796 


.7 


7651. lOM 


310.0752 


.8 


6618.7388 


288.3982 


.3 


7133.0568 


299.3938 


.8 


7666.6170 


310.3894 


.9 


6633.1666 


288.7124 


.4 


7148.0343 


299.7079 


.9 


7682.1444 


310.7085 


92.0 


6647.6101 


289.0265 


.5 


7163.0276 


300.0221 


99.0 


7697.6893 


311.0177 


.1 


6662.0692 


289.3407 


.6 


7178.0366 


300.3363 


.1 


7713.-2461 


311.3318 


.2 


6676..5441 


289.6548 


.7 


7193.0612 


300.6504 


.2 


7728.8206 


311.6460 


.3 


6691.0347 


289.9690 


.8 


7208.1016 


300.9646 


.3 


7744.4107 


311.9602 


.4 


6705.5410 


290.2832 


.9 


7223.1577 


301.2787 


.4 


7760.0166 


312.2743 


.5 


6720.0630 


290.5973 


96.0 


7238.2295 


301.5929 


.5 


7775.6382 


812.5885 


.6 


6734.6008 


290.9115 


.1 


7253.3170 


301.9071 


.6 


7791.2754 


812.9026 


.7 


6749.1542 


291.2256 


.2 


7268.4202 


302.2212 


.7 


7806.9284 


818.2168 


.8 


6763.7233 


291.5398 


.3 


7283.5391 


302.5354 


.8 


7822.6971 


818.6309 


.9 


6778.3082 


291.8540 


.4 


7298.6737 


302.8405 


.9 


7838.2815 


818.8451 


93.0 


6792.9087 


292.1681 


.5 


7313.8240 


303.1637 


100.0 


7853.9816 


314.1503 


.1 


6807.5250 


292.4823 


.6 


7328.9901 


303.4779 








.2 


6822.1.'>69 


292.7964 


.7 


7344.1718 


303.7920 








.3 


6836.8046 


293.1106 


.8 


7359.3693 


304.1062 








.4 


6851.4680 


293.4248 


.9 


7374.5824 


304.4203 









MENSURATION. — CIRCLES. 



49 



AREAS OF CIRCLES. 

^ADVANCING BY EIGHTHS.) 
AREAS. 



Diam. 




0.0 


0.1 


0-1 


o.| 


H 


o-# 


O.f 


O.J 


0.0 


0.0122 


0.0490 


0.1104 


0.1963 


0.3068 


0.4417 


0.6013 


1 


0.7854 


0.9940 


1.227 


1.484 


1.767 


2.073 


2.405 


*i.761 


2 


3.1416 


3.546 


3.976 


4.430 


4.908 


5.411 


5.9.39 


6.491 


3 


7.068 


7.669 


8.295 


8.946 


9.621 


10.32 


11.04 


11.79 


4 


12.56 


13.36 


14.18 


15.03 


15.90 


16.80 


17.72 


18.66 


5 


19.63 


20.62 


21.64 


22.69 


23.75 


24.85 


25.96 


27.10 


6 


28.27 


29.46 


30.67 


31.91 


33.18 


34.47 


35.78 


37.12 


7 


38.48 


39.87 


41.28 


42.71 


44.17 


45.66 


47.17 


48.70 


8 


50.26 


51.^ 


53.45 


55.08 


56.74 


58.42 


60.13 


61.86 


9 


63.61 


65.39 


67.20 


69.02 


70.88 


72.75 


74.66 


76.58 


10 


78.54 


80.51 


82.51 


84.54 


86.50 


88.66 


90.76 


92.88 


11 


95.03 


97.20 


99.40 


101.6 


103.8 


106.1 


108.4 


110.7 


12 


113.0 


115.4 


117.8 


120^ 


122.7 


125.1 


127.6 


130.1 


13 


132.7 


135.2 


137.8 


140.5 


143.1 


145.8 


148.4 


151.2 


14 


153.9 


156.6 


159.4 


162.2 


165.1 


167.9 


170.8 


173.7 


15 


176.7 


179.6 


182.6 


185.6 


188.6 


191.7 


194.8 


197.9 


16 


-201.0 


204.2 


207.3 


210.5 


213.8 


217.0 


220.3 


223.6 


17 


226.9 


230.3 


233.7 


237.1 


240.5 


243.9 


247.4 


250.9 


18 


254.4 


258.0 


261.5 


265.1 


268.8 


272.4 


276.1 


279.8 


19 


283.5 


287.2 


291.0 


294.8 


298.6 


.302.4 


306.3 


310.2 


20 


814.1 


318.1 


322.0 


326.0 


330.0 


334.1 


338.1 


342.2 


21 


346.3 


850.4 


854.6 


358.8 


363.0 


367.2 


371.5 


375.8 


22 


380.1 


384.4 


388.8 


393.2 


397.6 


402.0 


406.4 


410.9 


23 


415.4 


420.0 


424.5 


429.1 


433.7 


438.3 


443.0 


447.6 


24 


452.3 


457.1 


461.8 


466.6 


471.4 


476.2 


481.1 


485.9 


25 


490.8 


495.7 


600.7 


605.7 


510.7 


515.7 


520.7 


525.8 


26 


630.9 


636.0 


641.1 


646.3 


551.5 


656.7 


562.0 


567.2 


27 


572.5 


677.8 


683.2 


688.5 


693.9 


599.3 


604.8 


610.2 


28 


615.7 


621.2 . 


626.7 


632.3 


637.9 


643.5 


649.1 


6.54.8 


29 


660.5 


666.2 


671.9 


677.7 


683.4 


689.2 


695.1 


700.9 


30 

1 


706.8 


712.7 


718.6 


724.6 


730.6 


736.6 


742.6 


748.6 


1 
31 


754.8 


760.9 


767.0 


773.1 


779.3 


785.5 


791.7 


798.0 


32 


804.3 


810.6 


816.9 


823.2 


829.6 


836.0 


842.4 


848.8 


33 


855.3 


861.8 


868.3 


874.9 


881.4 


888.0 


894.6 


901.3 


34 


907.9 


914.7 


921.3 


928.1 


934.8 


941.6 


948.4 


955.3 


35 


962.1 


969.0 


975.9 


982.8 


989.8 


996.8 


1003.8 


1010.8 


36 


1017.9 


1025.0 


1032.1 


1039.2 


10i6.3 


1053.5 


1060.7 


1068.0 


37 


1075.2 


1082.5 


1089.8 


1097.1 


1104.5 


1111.8 


1119.2 


1126.7 


38 


1134.1 


1141.6 


1149.1 


1156.6 


1164.2 


1171.7 


1179.3 


1186.9 


39 


1194.6 


1202.3 


1210.0 


1217.7 


1225.4 


1233.2 


1241.0 


1248.8 


40 


1256.6 


1261.5 


1272.4 


1280.3 


1288.2 


1296.2 


1304.2 


1312.2 


41 


1320.3 


1328.3 


1336.4 


1344.5 


1352.7 


1360.8 


1369.0 


1377.2 


42 


1385.4 


1393.7 


1402.0 


1410.3 


1418.6 


1427.0 


1435.4 


1443.8 


43 


1452.2 


1460.7 


1469.1 


1477.6 


1486.2 


1494.7 


1503.3 


1511.9 


44 


1520.5 


1629.2 


1537.9 


1546.6 


1655.3 


1564.0 


1572.8 


1581.6 


45 


1590.4 


1699 J) 


1608.2 


1617.0 


1626.0 


1634.9 


1643.9 


1652.9 



50 



MENSURATION. - CIRCUMFERENCES. 



CIRCUMPBRBNCBS OP CIRCLES. 

(advancing by eighths.) 

CIRCUMFERENCES. 



Diam. 




0.0 


04 


o-i 


0.| 


O.J 


0-1 


Of 


O.J 


0.0 


0.3927 


0.7854 


1.178 


1.570 


1.963 


2.356 


2.748 


1 


3.141 


3.534 


3.927 


4.319 


4.712 


5.105 


5.497 


5.890 


2 


6.283 


6.675 


7.068 


7.461 


7.854 


8.246 


8.639 


9.032 


3 


9.424 


9.817 


10.21 


10.60 


10.99 


11.38 


11.78 


12 17 


4 


12.56 


12.95 


13.35 


13.74 


14.13 


14.52 


14.92 


15.31 


5 


15.70 


16.10 


16.49 


16.88 


17.27 


17.67 


18.06 


18.45 


6 


18.84 


19.24 


19.63 


20.02 


20.42 


20.81 


21.20 


21.60 


7 


21.99 


22.38 


22.77 


23.16 


23.56 


23.95 


24.34 


24.74 


8 


25.13 


25.52 


25.91 


26.31 


26.70 


27.09* 


27.48 


27.88 


9 


28.27 


28.66 


29.05 


29.45 


29.84 


30.23 


30.63 


31.02 


10 


31.41 


31.80 


32.20 


32.59 


32.98 


33.37 


33.77 


34.10 


11 


34.55 


34.95 


35.34 


35.73 


36.12 


36.52 


36.91 


37.30 


12 


37.69 


38.09 


38.48 


38.87 


39.27 


39.66 


40.05 


40.44 


13 


40.84 


41.23 


41.62 


42.01 


42.41 


42.80 


43.10 


48.68 


14 


43.98 


44.37 


44.76 


45.16 


45.55 


45.94 


46.33 


46.73 


15 


47.12 


47.51 


47.90 


48.30 


48.69 


49.08 


49.48 


49.87 


16 


50.26 


50.65 


51.05 


51.44 


51.83 


52.22 


52.62 


63.01 


17 


53.40 


53.79 


54.19 


54.58 


54.97 


55.37 


66.76 


66.15 


18 


56.54 


56.94 


57.33 


57.72 


58.11 


58.51 


58.90 


60.29 


19 


59.69 


60.08 


60.47 


60.86 


61.26 


61.65 


62.04 


62.48 


20 


62.83 


63.22 


63.61 


64.01 


64.40 


64.79 


66.18 


66.58 


21 


65.97 


66.36 


66.75 


67.15 


67.54 


67.93 


68.32 


68.72 


22 


69.11 


69.50 


69.90 


70.29 


70.68 


71.07 


n.47 


71.88 


23 


72.25 


72.64 


73.01 


73.43 


73.82 


74.22 


74.61 


76.00 


24 


75.39 


75.79 


76.18 


76.57 


76.96 


77.36 


77.75 


78.14 


25 


78.54 


78.93 


79.32 


79.71 


80.10 


80.50 


80.89 


81.28 


26 


81.68 


82.07 


82.46 


82.85 


83.25 


83.64 


84.03 


84.48 


27 


84.82 


85.21 


85.60 


86.00 


86.39 


86.78 


87.17 


87.57 


28 


87.96 


88.35 


88.75 


89.14 


89.53 


89.92 


90.32 


00.71 


29 


91.10 


91.49 


91.89 


92.28 


92.67 


93.06 


93.46 


03.85 


30 


94.24 


94.64 


95.03 


95.42 


95.81 


06.21 


06.60 


96.90 


31 


97.39 


97.78 


98.17 


98.57 


98.96 


99.35 


99.76 


100.14 


32 


100.53 


100.92 


101.32 


101.71 


102.10 


102.49 


102.89 


103.20 


33 


103.07 


104.07 


104.46 


104.85 


105.24 


105.64 


106.03 


106.42 


34 


106.81 


107.21 


107.60 


107.99 


108.39 


108.78 


109.17 


109.56 


36 


109.96 


110.35 


110.74 


111.13 


111.53 


111.92 


112.81 


112.71 


36 


113.10 


113.49 


113.88 


114.28 


114.67 


115.06 


116.46 


115.85 


37 


116.24 


116.63 


117.02 


117.42 


117.81 


118.20 


118.60 


118.90 


38 


119.38 


119.77 


120.17 


120.56 


120.95 


121.34 


121.74 


122.13 


39 


122.52 


122.92 


123.31 


12:J.70 


124.09 


124.49 


124.88 


125.27 


40 


125.66 


126.06 


126.45 


126.84 


127.24 


127.63 


128.02 


128.41 


41 


128.81 


129.20 


127.59 


129.98 


130.38 


130.77 


131.16 


181 J5 


42 


131.95 


132.34 


132.73 


133.13 


133.52 


133.91 


134.30 


184.70 


43 


135.09 


135.48 


135.87 


136.27 


136.66 


137.05 


187.4ft 


187.84 


44 


138.23 


138.62 


139.02 


139.41 


139.80 


140.19 


140.60 


l¥iM 


45 


141.37 


141.76 


142.16 


142.55 


142.94 


143.34 


148.78 


tuja 



MENSURATION. - CIECLES. 



51 



AREAS AND CIRCUMPBRBNCES OP CIRCLES. 



From I to 50 Feet. 



(advancing by one inch.) 



IHam. 


Area. 


Cimim. 


Diaffl. 


Area. 


Circnm. 


Diam. 


Area. 


Circum. 


Ft. 


Feei. 


Ft, In. 


Ft. 


Feet. 


Ft. In. 


Ft. 


Feet. 


Ft. In. 


1 


0.7854 


ill 


5 


19.635 


15 8t 
15 llg 


9 


63.6174 


28 3^ 


1 


0.9217 


1 


20.2947 


1 


64.8006 


28 6} 


2 


1.069 


3 8 


2 


20.9656 


16 21 


2 


65.9951 


28 9 


3 


1.2271 


3 11 


3 


21.6475 


16 5j 


3 


67.2007 


29 f 


4 


1.3062 


4 2| 


4 


22.34 


16 9 


4 


68.4166 


29 3 


5 


1.5761 


4 5 


5 


23.0437 


17 


5 


69.644 


29 7 


6 


1.7671 


4 8 


6 


23.7583 


17 3 


6 


70.8823 


29 10 


7 


1.9689 


4 11 


7 


24.4835 


17 6 


7 


72.1309 


30 1 
30 4 


8 


2.1816 


5 2 


8 


25.2199 


17 9| 


8 


73.391 


9 


2.4052 


5 ^ 


9 


25.9672 


18 3 


9 


74.662 


30 7i 


10 


2.6398 


5 9 


10 


26.7251 


18 3 


10 


75.94^ 


30 lift 


11 


2.8852 


6 2| 


11 


27.4943 


18 7 


11 


77.2362 


31 Ijj 


20 


3.1416 


6 Si 


60 


28.2744 


18 104 


10 


78.54 


31 5 


1 


3.4087 


6 6 


1 


29.0649 


19 1 


1 


79.854 


31 83 


2 


3.6869 


6 9 


2 


29.8668 


19 43 


2 


81.1795 


31 in 


3 


3.976 


7 


3 


30.6796 


19 7* 
19 10 


3 


82.516 


32 2i 


4 


4.276 


7 31 


4 


31.5029 


4 


83.8627 


32 5| 


5 


4.5869 


7 7 


5 


32.3376 


20 1| 


6 


85.2211. 


32 8j 


6 


4.9087 


7 10^ 


6 


33.1831 


20 ^ 
20 8| 


6 


86.5903 


32 111 


7 


5.2413 


8 1 


7 


34.0391 


7 


87.9697- 


33 2| 


8 


5.585 


8 4l 


8 


34.9065 


20 111 


8 


89.3608 


33 65 


9 


5.9395 


8 7jr 


9 


35.7847 


21 2} 


9 


90.7627 


33 91 


10 


6.3049 


8 10| 


10 


36.6735 


21 5, 


10 


92.1749 


34 f 


11 


6.6813 


9 l| 


11 


37.5736 


21 8i{ 


11 


93.5986 


34 3| 


30 


7.0686 


9 5 


7 


38.4846 


21 llj 


11 


95.0334 


34 6| 


1 


7.4666 


9 8^ 


1 


39.406 


22 3 


1 


96.4783 


34 9.^ 


2 


7.8757 


9 11 


2 


40.3388 


22 61 


2 


97.9347 


35 1 

35 43 


• 3 


8.2957 


10 2 


3 


41.2825 


22 91 


3 


99.4021 


4 


8.7265 


10 5 


4 


42.2367 


23 1 


4 


100.8797 


35 7. 


5 


9.1683 


10 8} 


5 


43.2022 


23 2i 


5 


102.3689 


35 lOJ 


6 


9.6211 


10 llj 


C 


44.1787 


23 ^ 


6 


103.8691 


36 l| 


1 


10.0346 


11 3 


7 


45.1656 


23 9| 

24 1} 


7 


105.3794 


36 45 


s 


10.5591 


11 6 


i; 


46.1638 


8 


106.9013 


36 71} 


9 


n.0W6 


11 9 


ft 


47.173 


24 4J 


9 


108.4342 


36 10| 


10 


r.5403 


12 k 
12 3 


10 


48.1962 


24 7} 
24 lOf 


10 


109.9772 


37 25 


11 


ij.om 


11 


49.22,36 


11 


111.5319 


37 51 


4 


12.5664 


12 6J 


8 


50.2656 


25 li 
25 4| 


12 


113.0976 


37 83 


1 


13.0952 


12 9| 


1 


51.3178 


1 


114.6732 


37 111 


•; 


13.63>3 


13 1 


2 


52.3816 


25 7j 


2 


116.2607 


38 ^ 


3 


14.1862 


13 4 


n 

i> 


63.4562 


25 11 


3 


117.859 


38 5i 


4 


14.7479 


13 7 


4 


54.5412 


26 2 
26 5 


4 


119.4674 


38 8| 


5 


15.3206 


13 10 


5 


55.6377 


5 


121.0876 


39 


6 


15.9043 


14 1 


6 


56.7451 


26 8 


6 


122.7187 


39 3a 


7 


16.4986 


14 4 


7 


57.8628 


26 llJ 


7 


124.3598 


39 ^ 


8 


17.1041 
17.7205 


14 7 


8 


58.992 


27 2i 


8 


126.0127 


39 9, 


9 


14 11 


9 


60.1321 


27 51 


9 


127.6765 


40 1 


10 


1S.3476 


15 2| 

16 6j 


10 


61.2826 


27 9 


10 


129.3504 


40 33 


11 


18.8858 


.11 


62.4445 


28 i 


11 


131.036 


40 6| 



52 



MENSURATION. - CIRCLES. 



Areas and Circumferences of Circles (Feet and Inches) 


. 


1 
Diam. 


Aw. 


Cireiim. 


Dbni. 


Area. 


Cirenra. 


Diam. 


Area. 


Cirenn. 


Ft. 


Feet. 


Ft. III. 


/Y. 


Feet. 


Ft. 


In. 


Ft, 


Feet. 


Ft. III. 


13 


132.73-26 


40 10 


18 


254.4696 


56 


tii 


23 


415.4766 


1> 3 


1 


134.4:391 


41 U 


1 


256 8303 


56 


93 


1 


418.4915 


7; 6jj 


2 


136.1574 


41 4^ 


2 


259.2033 


57 


2 


421 518J 


IL 9 


3 


137.8867 


41 Ih 


3 


261.5872 


57 


4 


3 


424..')577 




4 


139.626 


41 10.' 

42 n 


4 


263.9807 


57 


■^1 


4 


427.6055 


7:3 31 


5 


141.3771 


5 


266.3864 


57 


10 


5 


430.6658 


7:3 6^ 


6 


143.1391 


42 4i 


6 


268.8031 


58 


1 


6 


433.7371 


7;J 9^ 


i 


144.9111 


42 8 


7 


271.2293 


58 


4 


7 


436.8175 


74 I 


8 


146.6049 


42 IJi 


8 


273.6678 


68 


7 


8 


439.9106 


74 4i 


9 


148.4896 


43 2\ 


9 


276.1171 


58 lOi 


9 


443.0146 


74 7> 


10 


150.2943 


43 5ft 
43 8| 


10 


278.5761 


58 


2 


10 


446.1278 


74 10 

75 1 


11 


152.1109 


11 


281.0472 


69 


H 


11 


449.2536 


UO 


153.9384 


43 Hi 


19 


283.5294 


69 


81 


24 


452.3904 


75 4| 


1 


155.7758 


44 2j 


1 


286.021 


59 lU 


1 


455.5362 


75 71 


2 


157.625 


44 6 


2 


288.5249 


60 


2 


2 


458.6948 


75 11 


3 


159.4852 


44 9} 

45 J 


3 


291.0307 


60 


5 


3 


461.8642 


76 2| 


4 


161.3553 


4 


293.5641 


60 


sl 


4 


465.0428 


76 5l 


5 


163.2373 


45 Sl 


5 


296.1107 


60 


Hi 


5 


468.2341 


76 8 


6 


165.1303 


45 6{ 


6 


298.6483 


60 


H 


6 


471.4363 


76 11 


7 


167.0331 


45 9i 


7 


301.2054 


61 


el 


7 


474.6476 


77 24 


8 


168.9479 


46 i 


8 


303.7747 


61 


9ft 


8 


477.8716 


77 6i 





170.8735 


46 4 


9 


306.365 


61 


 ; 


9 


481.1065 


77 9 


10 


172.8091 


46 71 
46 111 


10 


308.9448 


61 


31 


10 


484.3506 


78 1 
78 3} 


11 


174.7565 


11 


311.5469 


62 


6j 


11 


487.6073 


15 


176.715 


47 1ft 


20 


314.16 


62 


9| 


25 


490.875 


78 6ft 
78 9i 


1 


178.6832 


47 4 


1 


316.7824 


62 


n 


1 


494.1516 


2 


180.6634 


47 73 


2 


319.4173 


63 


4 

••• 


2 


497.4411 


79 1 

70 8| 


3 


182.6545 


47 10- 


3 


322.063 


63 


3 


600.7415 


4 


184.6555 


48 2ft 


4 


324.7182 


63 


lU 


4 


504.051 


70 7 


 


6 


1S6;6684 


48 5 


5 


327.3858 


63 


If 


6 


507.3732 


79 n 




6 


18^6923 


48 8, 





330.0643 


64 


n 


6 


510.7063 


80 1 




7 


19;X726 


48 11 


7 


332.7522 


64 


7j 


7 


514.0484 


80 4 




8 


192^7716 


49 2 


8 


:j35.4525 


64 


11 


8 


517.4034 


80 7 




9 


194:8282 


49 5 


9 


338.1637 


65 


2i 


9 


520.7692 


80 10 




10 


190.8946 


49 8| 


10 


340.8844 


66 


H 


10 


524.1441 


81 1 




11 


198.973 


50 


11 


343.6174 


66 


8} 


11 


527.5318 


81 5 


16 


201.0024 


50 3| 

50 (U 


21 


346.3614 


«5 


lll 


26 


530.9304 


81 81 
81 11} 


1 


203.161'. 


I 


349.1147 


66 


'A 


1 


534.3.'379 


2 


205.2726 


Th) 9 


2 


351.8804 


66 


H 


2 


537.7583 


82 2| 


3 


207.S94li 


51 


3 


3.)4.657l 


66 


9 


3 


641.18JKJ 


82 5 i 


4 


209.5264 


51 31 


4 


357.4432 


66 


4 


4 


544.6209 


82 k} 


5 


211.6703 


51 4 


5 


360.2417 


67 


6 


648.083 


82 112 


6 


213.82.51 


51 10 


(i 


363.0511 


67 


6A 


6 


551. .5471 


83 3 


7 


215.9896 


52 n 


7 


365.8698 


67 


of 


7 


655.0201 


88 Oft 


8 


218.1662 


52 4i 


8 


368.7011 


68 


1 


. 8 


558.5069 


83 OJ 




9 


220.3537 


52 n 


9 


371.5432 


6S 


3| 


9 


662.0027 


84 i 




10 


222.551 


52 10ft 


10 


374.3947 


68 


1 


10 


665.5084 


84 3 




11 


224.76J3 


53 if 


11 


377.2587 


68 


10", 


11 


569.027 


84 0| 




17 


226.9806 


53 41 


22 


380.13:36 


69 


1| 


27 


572.5560 


84 OZ 


1 


229.2105 


5:1 8 


1 


:i8'3.0177 


69 


41 


1 


576.0940 


86 1 


2 


231.4525 


53 in 


2 


385.9144 


69 


7I 


2 


679.6463 


85 4 




3 


233.7055 


54 2 


8 


383.822 


69 lOj 1 


8 


683.2066 


85 8 




4 


235.9682 


.54 5 


4 


391.7389 


70 


n 


4 


686.7796 


85 11 




5 


238.2m 


54 85 


5 


394.6683 


70 


5 


5 


600.3637 


80 1 




6 


240.5287 


.54 llg 


6 


.397.6087 


70 


8] 


6 


603.0587 


80 4 




7 


242.8241 


55 21 


7 


400.558!J 


70 111 1 


7 


607.5026 


80 7 




8 


245.1316 


55 6 


8 


403.5204 


71 


A 


8 


601.1793 


80 11 





247.45 


55 9 
66 


9 


406.49:35 


71 


^i 


9 


004.807 


87 4 


10 


249.7781 


10 


409.4759 


71 


85 


10 


608.4436 


87 U 


11 


252.1184 


56 3^ 


11 


412.4707 


71 


ni 


11 


612.0(R)1 


87 ^ 



MENSUIIATION. - CIRCLES. 



58 



Areas and Circumferences of Circles (Feet and Inches). 



Dim. 



Ft. 

28 
1 
2 
3 
4 
5 
6 

I 

8 

9 

10 

11 

29 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 

30 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 



31 




1 
2 
3 
4 
5 
6 
< 

8 

9 

10 

11 

32 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 



Area. 



Feft. 
610.7536 
619.4228 
623.105 
626.7982 
630.5002 
634.2152 
6:J7.9411 
641.6758 
645.4235 
649.1821 
652.9495 
656.73 

660.5214 
664.3214 
668.1:346 
671.9587 
675.7915 
679.6375 
6H3.4943 
687.3.598 
691.2385 
695.1028 
699.0263 
702.9377 

706.86 

710.791 

714.735 

718.69 

722.654 

726.631 

730.618 

734.615 

738.624 

742.645 

746.674 

750.716 

754.769 
758.831 
762.906 
766.992 
771.086 
775.191 
779.313 
783.440 
787.581 
791.732 
795.892 
800.065 

804.25 

808.442 

812.648 

816.865 

821.090 

825.3-29 

829.579 

833.837 

838.103 

»42.:«)1 

846.681 

8o0.»85 



€irr«in. 


Ihim. 

Ft. 


Ft. III. 


87 \\\ 


'&\ 


88 21 


1 


88 5^ 


2 


88 9 


3 


89 1 
89 3j 


4 


5 


89 6j 


6 


89 9} 


7 


90 


8 


90> 3| 


9 


90 6^ 


10 


90 11 


11 


91 n 


34 


91 H 


1 


91 n 

91 lOf 


2 


3 


92 r 


4 


92 4 


5 


92 »! 


6 


92 \\\ 


< 


93 2| 


8 


93 ol 


9 


9:J S| 


10 


93 11^ 


11 


94 •>» 


:l) 


94 (5 


1 


94 9i 

95 i 


2 


3 


95 3A 


4 


95 6| 


5 


95 9j 


6 


96 2 


i 


96 4 


8 


96 7| 
96 lOj 


9 


10 


97 U 


11 


97 4| 


36 


97 7| 


1 


97 10| 


2 


9H 2 


3 


98 h\ 


4 


98 Sji 


5 


98 in 

99 2| 


6 


7 


99 52 
99 8| 


8 


9 


100 


10 


100 3| 


11 


100 6j{ 


37 


100 9^ 


1 


101 \ 


•> 

mm 


101 3^ 


3 


101 6} 


4 


101 10 


5 


102 U 


6 


102 4 


7 


102 1\ 
102 10{ 


8 


9 


103 1 
103 4| 


10 


11 



Area. 



FeH. 

855.301 

859.624 

863.961 

868..'J09 

872.665 

877.035 

881.415 

885.804 

890.206 

894.619 

899.041 

903.476 

907.922 
912.377 
916.844 
921.323 
925.810 
930.311 
934.822 
939.342 
943.875 
948.419 
952.972 
957.538 

962.115 

966.770 

971.299 

975.908 

980.526 

985.158 

989.803 

994.451 

999.115 

1003.79 

1008.473 

1013.170 

1017.878 
1022.594 
1027.324 
1032.064 
1036.813 
1041. .576 
1046..349 
10.)1.130 
1055.926 
1060.731 
10f>5.546 
1070.374 

1075.2126 

10SO.059 

1084.920 

10S0.791 

1094.671 

109:>.564 

1104.469 

1109.3S1 

1114.307 

1119.244 

1124.1M9 

1129.148 



Oirron. 



109 %\ 

109 11^ 

110 28 
110 h\ 

110 8^ 

111 
111 3J 
111 6 
111 9 
112 
112 



117 
117 
117 
117 
118 
118 
118 



\ 
3ii 



112 6| 

112 10 

113 1| 
113 4| 
113 78 

113 lOf 

114 15 
114 4^ 

114 8 

114 in 

115 2\ 
115 5# 
115 9 

115 11 

116 2| 
116 6 
116 9 J 



■t 

^\ 

6i 

9| 

4 
74 



118 10J 

119 \i 



Ihan 



Ft. 

38 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 

39 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 

40 
1 
2 
3 
4 

• 

5 
6 
7 
8 
9 
10 
11 

41 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 

42 
1 
2 
3 
4 
5 
6 
I 

8 
9 

10 
11 



Area. 



Feet. 
1134.118 
1139.095 
1144.087 
1149.089 
1154.110 
1159.124 
1164.159 
1169.202 
1174.259 
1179.327 
1184.403 
1189.493 

1194.593 
1199.719 
1204.824 
1-209.958 
1-215.099 
1-2-20.254 
1-225.420 
12.30.594 
1-235.782 
1*240.981 
1-246.188 
1-251 .408 

1-256.64 

1-261.879 

1267.i:i3 

1-272.397 

1277.669 

1282.955 

1288.252 

1293.557 

1298.876 

1.304.206 

1309.543 

1314.895 

1320.267 

1.325.6-28 

1331.012 

1.3.36.407 

1341.810 

1347.-2-27 

1352.6.)5 

1358.001 

130.3.541 

1369.001 

1374.47 

1379.952 

13S5.446 

1.390.-247 

1396.462 

1401.988 

1407.522 

1413.07 

1418.6-29 

1424.195 

14-29.776 

1435.367 

1440.967 

1446.580 



Cirfnm. 



Ft. In. I 
119 4i 

119 7j : 

119 105 

120 2 I 
120 5| i 
1-20 Hi 1 

120 Ui 
1-21 2A 

121 5J 

121 8^ 

121 in 

122 31 

122 61 
1-22 9^ 

123 i 
123 3ji 

123 6J 
1*23 9| 

124 IJ 
1-24 4i 1 
124 7H i 
1-24 KU \ 
1-25 if 



1-2.) 



^ 



125 7| 
1-25 11 
1*26 2\ 

126 bi 
1-26 S4 
1-26 1l| 
1*27 25 

127 5,' 
1*27 9 
1*28 i 
1-28 3g 
1-28 6j 

1-28 9| 

1*29 I 

1*29 3j 

1*29 7 

1*29 101 

130 U 

130 4i 

130 7| 

130 lOS 

131 n 1 

131 5 i 

131 8^ I 

1.31 Hi 

132 -2^ 
132 51 
132 HI 

132 111 

133 3 
133 OH 

133 91 

134 i 
134 ^ 
134 63 
134 9| 



54 



MENSURATION. -CIRCULAR ARCS. 



Areas and Circumferences of Circles (Feet etnd Inches). 



Diam. 



Ft. 

43 

1 

2 

I 

5 
6 
7 
8 
9 
10 
11 

44 
1 
2 
3 
4 

' 5 
6 
7 
8 
9 
10 
11 

45 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 



Area. 



Feet. 
1452.205 
1457.836 
1463.483 
1469.14 
1474.804 
1480.48.} 
1486.173 
1491.870 
1497.532 
1503..^) 
1509.035 
1514.779 

1520.534 
1526.297 
1532.074 
1537.862 
1543.058 
1549.478 
1555.288 
1561.116 
1566.959 
1572.812 
1578.673 
1584.549 

1590.435 
1596.:V29 
1602.237 
160S.155 
1614.0S2 
1620.023 
1625.974 
1631.9.33 
1637.907 
1643.891 
1649.883 
1655.889 



CircBm. 


Diam. 


Ft. In. 


Ft. 


135 1 


46 


135 4 
135 1, 


1 


2 


135 10 


3 


136 1 


4 


136 4i 


5 


136 7| 


6 


136 11 


7 


137 2i 
137 5j 
137 83 


8 


9 


10 


137 lit 


11 


138 2^ 


47 


138 5| 


1 


138 9 


2 


139 


3 


139 31 


4 


139 6 


5 


139 9 


6 


140 


7 


140 3 


8 


140 7 


9 


141 10, 


10 


141 l| 


11 


141 43 


48 


141 74 


1 


141 105 

142 l| 


2 


3 


142 5 


4 


142 8i 


5 


142 11 


6 


143 21 


7 


143 5 


8 


143 8^ 


9 


143 11 


10 


144 3 


11 



Area. 



Feet. 
1661.906 
1667.931 
1673 97 
1680.02 
1686.077 
1692.148 
1698.231 
1704.321 
1710.425 
1716.641 
1722.663 
1728.801 

1734.947 

1741.104 

1747.274 

1753.455 

1759.643 

1765.845 

1772.059 

1778.28 

1784.515 

1790.761 

1797.015 

1803.283 

1809..562 
1815.848 
1822.149 
1828.460 
1834.779 
1841.173 
1847.457 
1853.809 
1860.175 
1866.552 
1872.937 
1879.335 



Cireum. 


Diam. 


Ft. In. 


Ft. 


144 6 


49 


144 9, 


1 


145 


2 


145 3i 


3 


145 6: 


4 


145 9| 


5 


146 1 


6 


146 4 


7 


146 7 


8 


146 10 


9 


147 U 


10 


147 4 


11 


147 73 


50 


147 11 




148 2 




148 5 




148 8 




148 11 




149 2 




149 5 




149 82 




150 




150 3 




150 6 




150 9i 




151 




151 3| 




151 6 




151 10 




152 1 




152 4: 




152 Ik 
152 10 






153 13 




153 4i 




153 8| 





Area. 



Feet. 
1885.745 
1892.172 
1898.504 
1905.037 
1911.497 
1917.961 
19-24.426 
1930.919 
1937.316 
1943.914 
1950.439 
1956.969 

1963.5 



Cireia. 



Ft. In. 

153 llj 

154 2| 
154 5) 
154 8j 

154 llj 

155 2| 
155 6 

155 9J 
156 
1.56 

156 61 

156 9^ 

157 I 



Circular Arcs. 

To find the length of a circular arc when its chord and height, or 
versed sine is given; by the following table. 

Rule. — Divide the height by the chord; find in the column of 
heights the number equal to tills quotient. Take out the corre- 
sponding number from the colunm of lengths. Multiply this 
number by the given chord. 

Example. — The chord of an arc is 80 and Its versed 6ine is 30, 
what is the length of the arc ? 

Ans. 30 -r 80 = 0.875. The lenglh of an arc for a height of 0.375 
we find from table to be 1.840t«. 80 X 1.34063 = 107.2504 =? 
length of arc. 



MENSUKATION. — CIHCULAR ARCS. 



55 



TABLE OP CIRCULAR ARCS. 



Hght8. 


Lengths. 


Hghts. 


Lengths. 


Hghts. 


Lengths. 


Hghts. 


Lengths. 


Hghts. 


Lengths. 


.001 


1.00001 


.062 


1.01021 


.1-23 


1.03987 


.184 


1.08797 


.245 


1.15.308 


.002 


1.00001 


.063 


1.01054 


.124 


1.04051 


.185 


1.08890 


.-246 


1.154-28 


.00:J 


1.00002 


.064 


1.01088 


.125 


1.04116 


.186 


1.08984 


.247 


1.15,U9 


.001 


1.00004 


.065 


1.01123 


.126 


1.04181 


.187 


1.09079 


.248 


1.15670 


.005 


1.00007 


.066 


1.01158 


.127 


1.04247 


.188 


1.09174 


.249 


1.15791 


.oo-^ 


1.00010 


.067 


1.01193 


.128 


1.04313 


.189 


1.09269 


.250 


1.15912 


. .0J7 


1.00013 


.068 


1.01228 


.129 


1.04380 


.190 


1.09365 


.251 


1.16034 


.OOS 


1.00017 


.069 


1.01264 


.130 


1.04447 


.191 


1.09461 


.252 


1.16156 


.OO.J 


1.00022 


.070 


1.01301 


.131 


1.04515 


.192 


1.09557 


.253 


1.16279 


.010 


1.00027 


.071 


1.01338 


.132 


1.04584 


.193 


1.09654 


.254 


1.16402 


.Oil 


1.00032 


.072 


1.01376 


.133 


1.04662 


.194 


1.09752 


.255 


1.16526 


.012 


1.00038 


.073 


1.01414 


.134 


1.047-22 


.196 


1.09850 


.256 


1.16650 


.013 


1.00045 


.074 


1. 01453 


.135 


1.04792 


.196 


1.09949 


.-257 


1.16774 


.014 


1.00053 


.075 


1.01493 


.136 


1.04862 


.197 


1.10048 


.258 


1.16899 


.01.5 


1.00061 


.076 


1.01533 


.137 


1.04932 


.198 


1.10147 


.259 


1.170-24 


.016 


1.00060 


.077 


1.01673 


.138 


1.05003 


.199 


1.10247 


.260 


1.17150 


.017 


1.00078 


.078 


1.01614 


.1:39 


1.05075 


.200 


l.ia347 


.261 


1.17-276 


.018 


1.00087 


.079 


1.01656 


.140 


1.05147 


.-201 


1.10447 


.262 


1.17403 


.019 


1.00097 


.080 


1.01698 


.141 


1.05-2-20 


.'202 


1.10548 


.26:3 


1.17530 


.020 


1.00107 


.081 


1.01741 


.142 


1.05293 


.203 


1.10650 


.264 


1.176.57 


.021 


1.00117 


.082 


1.01784 


.143 


1.05367 


.204 


1.10752 


.265 


1.17784 


.022 


1.00128 


.083 


1.01828 


.144 


1.05441 


.-205 


1.10855 


.266 


1.17912 


.023 


1.00140 


.084 


1.01872 


.145 


1.05516 


.206 


1.10958 


.267 


1.18040 


.024 


1.00153 


.085 


1.01916 


.146 


1.05591 


.207 


1.11062 


.268 


1.18169 


.025 


1.00167 


.086 


1.01961 


.147 


1.0566? 


.208 


1.11165 


.269 


1.18-299 


.026 


1.00182 


.087 


1.02006 


.148 


1.05743 


.209 


1.11-269 


.270 


1.184-29 


.027 


1.00196 


.088 


1.02052 


.149 


1.05819 


.210 


1.11374 


.271 


1.18559 


.028 


1.00210 


.089 


1.02098 


.150 


1.0.")S96 


.211 


1.11479 


.272 


1.18689 


.029 


1.00-225 


.090 


1.02145 


.151 


1.0)973 


.212 


1.11584 


.273 


1.188-20 


.030 


1.00240 


.091 


1.02192 


.152 


1.06051 


.213 


1.11690 


.274 


1.18951 


.031 


1.00256 


.092 


1.02240 


.153 


1.06130 


.214 


1.11796 


.275 


1.19082 


.032 


1.00272 


.093 


1.02289 


.154 


1.06-209 


.215 


1.11904 


.276 


1.19-214 


.033 


1.00289 


.094 


1.02339 


.155 


1.06-288 


.216 


1.1-2011 


.277 


1.19346 


.034 


1.00307 


.095 


1.02389 


.156 


1.06:368 


.217 


1.12118 


.278 


1.19479 


.035 


1.00327 


.096 


1.02440 


.157 


1.06449 


.218 


1.1-2-225 


.279 


1.19612 


.036 


1.00345 


.097 


1.02491 


.158 


1.06530 


.219 


1.1-23:34 


.280 


1.19746 


.037 


1.00361 


.098 


1.02542 


.159 


1.06611 


.2-20 


1.1*2444 


.-281 


1.198S0 


.038 


1.0a384 


.039 


1.02593 


.160 


1.06693 


.221 


1.12554 


.282 


1. -20014 


.039 


1.00405 


.10) 


1.02645 


.161 


1.06775 


.-2-22 


1.1-2664 


.283 


1.20149 


.040 


1. 00426 


.101 


1.0289S 


.162 


1.068.58 


.2-23 


1.1-2774 


.284 


1.20284 


.(Wtl 


1.00447 


.102 


1.02752 


.163 


1.06941 


.2-24 


1.1-2885 


.285 


1.20419 


.042 


1.00469 


.10 J 


1.02S06 


.164 


1.070-25 


.-2-25 


1.12997 


.286 


1.20555 


.043 


1.00492 


.104 


1.02860 


.165 


1.07109 


.-2*26 


1.13108 


.287 


1.20691 


.044 


1.00->15 


.105 


1.02914 


.166 


1.07194 


.227 


1.1.3-219 


.288 


1. -20827 1 


.0*'> 


1.00 ).39 


.103 


1.0-2970 


.167 


1.07279 


.228 


1.13331 


.289 


1.-20^)64 1 


.046 


1.0056:3 


.107 


1.03026 


.168 


1.07365 


.2-29 


1.13444 


.290 


1.21102 


.047 


1.00587 


.108 


1.03082 


.169 


1.07451 


.-230 


1.13557 


.291 


1.21-2.'39 


.048 


1.00612 


.103 


1.03139 


.170 


1.07537 


.'231 


1.13671 


.292 


1.21377 


.049 


1.0033S 


.110 


1.03198 


.171 


1.076-24 


.232 


1.13785 


.293 


1.-21015 


.050 


1.00665 


.111 


1.03254 


.172 


1.07711 


.233 


1.13900 


.294 


1.-21654 


.051 


1.00692 


.112 


1.03312 


.173 


1.07799 


.'234 


1.14015 


.295 


1.21794 


.(►.-.2 


1.00720 


.113 


1.0:3371 


.174 


1.07888 


.235 


1.14131 


.-296 


1.219:33 


.05:J 


1.00748 


.114 


1.03430 


.175 


1.07977 


.236 


1.14247 


.297 


1.2-2073 


.0.>4 


1.00776 


.115 


1.03 J90 


.176 


1.08066 


.-237 


1.14363 


.298 


1.2-2-213 


.055 


1.00805 


.116 


1.03551 


.177 


1. OS 1 56 


.-238 


1.14480 


.299 


1.22:354 


.050 


1.00834 


.117 


1.03611 


.178 


1.0S246 


.-230 


1.14597 


.300 


1 .-2-2495 


.057 


1.00864 


.118 


1.03672 


.179 


1.083;J7 


.-240 


1.14714 


.301 


1. -226:36 


.058 


1.00895 


.119 


1.03734 


.180 


1.0S42S 


.2 41 


1.148.32 


.302 


1.-2-2778 


.059 


1.00928 


.120 


1.03797 


.181 


1.08519 


.242 


1.14951 


.30:j 


1.-2-2920 


.060 


1.00957 


.121 


1.03860 


.182 


1.08611 


.243 


1.15070 


.304 


1.'2:]063 


.061 


1.00989 


.122 


1.03923 


.183 


1.08704 


.-244 


1.15189 


.:305 


1.23206 



56 



MENSURATION. — CIRCULAR ARCS. 



Table of Circular Aros {conciuded)], 



Hghts. 

.306 


lengths. 


Hghts. 


Ungths. 


lights. 


lengths. 


Ughts. 


lengths. 


Hghts. 


Leigths. 


1.23349 


.345 


1.29*209 


.384 


1.35575 


.423 


1.42402 


.462 


1.49651 


.307 


1.23492 


.346 


1.29.366 


.385 


1.3.5744 


.424 


1.42583 


.463 


1.49842 


.308 


1.23636 


.347 


1. -29523 


.386 


1.. 35014 


.425 


1.42764 


.464 


1.50033 


..309 


1.2.3781 


.348 


1.29681 


.387 


1.. 36084 


.426 


1.42945 


.465 


1.:iO-224 , 


.310 


1.23926 


.349 


1.29839 


.388 


1.36254 


.427 


1.43127 


.466 


1.50416 


.311 


1.24070 


.350 


1.29997 


.389 


1.36425 


.428 


1.4.3309 


.467 


1.50608 I 


.312 


1.24216 


.351 


1.30156 


.390 


1.. 36596 


.429 


1.4.3491 


.468 


1.50800 ; 


.313 


1.24361 


.352 


1.30315 


.391 


1.36767 


.430 


1.43673 


.469 


1.50992 1 


.314 


1.24507 


.353 


1.30474 


.392 


1.30939 


.431 


1.43856 


.470 


1.51185 ! 


.315 


1.24654 


.354 


1.306.34 


.393 


1.37111 


.432 


1.44039 


.471 


1.51378 


.316 


1.24801 


.355 


1.30794 


.394 


1.37283 


.433 


1.44222 


.472 


1.51571 


.317 


1.24948 


.356 


l.:50954 


.395 


1.. 37455 


.434 


1.44405 


.473 


1.51764 


.318 


1.25095 


.357 


1.31115 


.398 


l.:37628 


.435 


1.44589 


.474 


1.51958 


.319 


1.25243 


.358 


1.31276 


.397 


1.37801 


.436 


1.44773 


.475 


1.52152 


.320 


1.25391 


.359 


1.314:37 


.398 


1.37974 


.437 


1.44957 


.476 


1.52346 


.321 


1.25540 


.360 


1.31599 


.399 


1.38148 


.438 


1.45142 


.477 


1.52541 


.322 


1.25689 


..361 


l.:31761 


.400 


1.38.3-22 


.439 


1.45327 


.478 


1.527:36 


.323 


1.25838 


.362 


1.31923 


.401 


l.:38496 


.440 


1.45512 


.479 


1.52931 


.324 


1.25988 


.363 


1.. 3-2086 


.402 


1.38671 


.441 


1.45697 


.480 


1.53126 


.325 


1.26138 


.364 


l.:j-2-249 


.403 


l.:38846 


.442 


1.45883 


.481 


1.53322 


.326 


1.26288 


.365 


1.32413 


.404 


1.39021 


.443 


1.46069 


.482 


1.53518 


.327 


1.26437 


..366 


1.32577 


.405 


1.. 391 96 


.444 


1.46255 


.483 


1.53714 


.328 


1.2(5)88 


.367 


1.32741 


.406 


l.:i9372 


.445 


1.46441 


.484 


1.53910 


.329 


1.23740 


..36S 


1.32905 


.407 


l.:39548 


.446 


1.46628 


.485 


1.54106 


.330 


1.26892 


..383 


1.33069 


.408 


1.397-24 


.447 


1.46815 


.486 


1.54302 


:.m 


1.270 U 


.3'<0 


1.33-2:34 


.409 


1.39900 


.448 


1.47002 


.487 


1.54499 


.332 


1.2719 J 


.371 


1.. 33399 


.410 


1.40077 


.449 


1.47189 


.488 


1.54696 


.3:i3 


1.27349 


.372 


1.. 3:3564 


.411 


1.40254 


.450 


1.47377 


.489 


1.54893 


.:«4 


1.27502 


.373 


l.a3730 


.412 


1.404:J2 


.461 


1.47565 


.490 


1.55091 


.33> 


1.27656 


.374 


l.:3:3896 


.413 


1.40610 


.452 


1.47753 


.491 


1.55289 


.336 


1.27810 


.375 


1.34063 


.414 


1.40788 


.453 


1.47942 


.492 


1.55487 


.337 


1.27964 


..376 


1.342-29 


.415 


1.40966 


.454 


1.48131 


.493 


1.55685 


.338 


1.28118 


.377 


1.34:396 


.416 


1.4H45 


.455 


1.48320 


.494 


1.55884 


.339 


1.28273 


.378 


1.34583 


.417 


1.413-24 


.456 


1.48509 


.495 


1.56063 


.340 


1.28428 


.379 


1.. 34731 


.418 


1.41503 


.457 


1.48699 


.496 


1.56292 


.341 


1.28583 


.380 


1.:J4899 


.419 


1.41682 


.458 


1.48889 


.497 


1.56481 


.342 


1.28739 


..381 


1.3506S 


.4-20 


1.41861 


.459 


1.49079 


.498 


1.56681 


.343 


1.28895 


.382 


1.35-237 


.421 


1.4-2041 


.460 


1.49269 


.499 


1.56881 


.344 


1.29052 


.38:3 


1.35406 


.422 


1.42-221 


.461 


1.49460 


.500 


1.57<M0 



Table of Leiig^ths of Circular Arcs whose Radius 

is 1. 

Rule. — Knowing the measure of the circle and the measure of 
the arc in degrees, minutes, and seconds; take from the table the 
lengths opposite the number of degrees, minutes, and seconds in 
the arc, and multiply their sum by the radius of the circle. 

Example. — What is the length of an arc subtending an angle 
of 13° 27' 8", with a radius of 8 fe<»t. 

Ana. Length for 13° = 0.2268928 

27'= 0.0078540 

8"= 0.0000388 

1.30 27' 8"= 0.2:J47850 

8 

Length of arc = 1.8782848 feeL \l 



MENSURATION. —CIRCULAU ARCS. 
IjeugthB of Circular Arcs ; BadiuB = 1. 



I 



ziT^T^ri^rj.:* — -.i:sr:a> -if chords. ' 



• .:f Vir ■■••i-f ,T I,, f- r7»-M '.'§•* rft'tr^i '^f k'llf th€ arf^ and 

." '*'r^9t"L it/i-i C7"i [jirtfji. (The vprswl 
^^.J-,^ "ri^L* - ■Lit; ^itr^eniii.'alakT N/. Fig. 31.) 

, ^- A •-..'.. — J-'im :Iie siTau** of tli«* clionl of " 

B-^ -, ;;i-r -ir lt: ?;iijcru!!: "I^e itjo^ire of the versed 

iini-. lali ~jL£ii T3¥'jx zjifi si^iujre root of the ; 

i 



I 

-.:... i:' .J. — T'lr -iiirr. .t u-r' ^Iie ir: is •5i>, and the versed '■ 



^ i.t. •'I- — J'- = =;X4, and \'J3kM = 48, ; 
iii -Ifr < z = \ft5^ the chord. 

••i"j-( ir f:i f:-: r//*;;! :,itf diiMtter and versed »ine ! 



I. 



ji ■.-:." I" jt* '-rse-i sill' v- i. xai sobtncc the product from | 
.:•• li.i^it*. -r -j.-;i -jaijui-ur^ "-i- iiiiarn ot the reiuaiuder from 
.^' -r. ;:i.~- .r -.Air. lia.nrL-^r. i:ii ::4Ju uhe ft^aare root of that re- 



>  ■•. 1-1- 

3:. 4 ' ..:. — Titt riiOiHC-r }f a '."jrcfe is !♦». and the vereed 
?»j.r .1 ui i.^- .i\ vjiL :r :L.e iIikc i»ik the iTC ? 

-i ... .._*_^ = -;;. : i: - Ti = 2S. lOUfS — 28* = 0216. 

\ -ii: f = f**. :iif ■-•xiodi oif the are. 

r- .1. ' I- I- ••■'■ r ■/ ' - 'I ■«••: jc&ea tAe cAord of the arc 

.::. — TLi^ vre <»; iir^ r-x-c or the sum of the squares of the 
■■.>«■•: ?..:•' i:: . ;r '-« ' -ir .'iLOri of the arc. 

Y;> • \ • ..-.. — 7iH .-li.ri :r iji an: b ^ and the versed shie 30, 

V '..i^ > . It- :i:i.ri .c '■'■«^"*' "lit; at:? 



'"' " - >" • /■ •■: ■"' V.:""* T.1 'JT}: irA«i the diameter and rerW 

•. . '  . '-.1. 

"..•.— Xi ::i~ :j:h i^^i^iiecer by the versed sine, and take tlie 

"■. .■■-'■ '■.■«. C ."C l^*z.? V'"^.\1".JJ-. 

 — V" ■'"■ •-■: *->.*f 5i;ujLre oi the chonl of half the arc by 

Kv.Y i. — A:.: :Jir? <i;ujLrv of half the chord of the arc to the 
o. ar*. . : :L*n: -■ rs •: <!-: . i:l<.I divide this sum by the versed sine. 



MENSURATION. —ARCS AND VERSED SINES. 59 

Example. — What is the radius of an arc whose chord is 96, and 
whose versed sine is 36 ? 

Ans. 482 + 362 = o^qqq^ 3(500 -^ 36 = lOO, the diameter, 
and radius = 50. 

To compute the versed sine. 

Rule. — Divide the square of tlie chord of half the arc by the 
diameter. 

To compute the versed sine ivhen the chord of the arc and the 
diameter are given, ' 

Rule. — From the square of the diameter subtract the square 
of the chord, and extract the square root of tlie remainder; sul>- 
tract this root from the diameter, and halve the remainder. 

To compute the length of an arc of a circle when the number of 
degrees and the radius are given. 

Rule 1. — Multiply the number of degrees in the arc by 3.1416 
multiplied by the radius, and divide by 180. The result will be the 
length of the arc in the same unit as the radius. 

Rule 2. — Multiply the radius of the circle by 0.01745, and the 
product by the degrees in the arc. 

Example. — The number of degrees in an arc is 60, and the 
radius is 10 inches, what is the length of the arc in inches ? 

Ans. 10 X 3.1416 X 00 = 1884.96 -f 180 = 10.47 inches; 
or, 10 X 0.01745 X 60 = 10.47 inches. 

To compute the length of the arc of a circle when the length is 
given in degreesj minutes, and seconds. 

Rule 1.^ Multiply the number of degrees by 0.01745329, and 
the product by the radius. 

Rule 2. — Multiply the number of minutes by 0.00029, and that 
product by the radius. 

Rule 3. — Multiply the number of seconds by 0.00000448 times 
the radius. Add together these three results for the length of the 
arc. 

See also table, p. 57. 

Example. —What is the length of an arc of 60° 10' 5", the 
radius being 4 feet ? 

Ans. 1. 60° X 0.01745329 X 4 = 4.188789 feet. 

2. 10' X 0.00029 X 4 = 0.0116 feet. 

3. 5" X 0.0000048 X 4 = 0.000W)6 feet. 



4.200485 feet. 



MFNSVRATION.--C1RCULAR SEGMENTS, ETC. 

7 -:'.:•:-:■: ■:/" -i a^rtf^r of circle ichen the degreea of the 

^ ^ .:'•: 'An'! th^ rodvis are given (Fig. 82). 

F 5-32 

^,.— -^=^-— ^^^ RvLii. — Multiply the number of degrees in 

J ' _____r:?i.i ...^ _^ ; ^. .j^^ area of the whole circle, anddi- 

Ex-OiPLE. — Wliat is the area of a sector of 
A :.r.'-. ^* Lose radius is 5, and the length of the 



\ 



.y . 



■--.■» 



.: >. A:rA •:: c'.role = 10 X 10 X 0.7854 = 78.54. 

78.0 X 00 _ ^^^ 
TLrr. infa of sector = — ^^ — — 13.09. 

" .  .•.-•■-.-,. .„ •-.-. :•;- .j.-ijrf'fs and mini(tes, reduce it 
>. v.  :v.v..:.v V  \ :':.v Arxra of the whole circle, and divide 



« 



I 



 - i. ,--• .: rir^le irhen the length of the 

' m * - 

— '^. .. ■•:'-• T-r:":. •"»: the arc by half the length of the 



X 



«' -.«h  aw «*,«« ■»« ^« 



X % 



I . 



\ 



 - ■*  --ir.'/^ irhen the chord and 
. :\: -.' 'r'.'j* or diatui'ter of the circle 



"» -VN :'•. tw a xcmicircle). — Ascer 
>v.:,7 "..A* '.^ :"::o same arc as the segment, 
. .-\.. .' \ :-..%".-'-: fonuea l>v the chord of the 
■."..-: s-:v:^'r. and late the difference of 



■V V 



 < ■•--••-— rV:n a ^mirirrle). — As- 

^ -.. . : . : .ir-.,* i*: :he le>5er iH)rtion of the 

, Ar.A . : :"..t ul-.ole «.-ii\-le, and the remain- 



\ vv 



\ 



.:' . • :*-.; ::n.ujnfen»nce, and the 

. V   . \ >:::■:* *^v of a sphere of 10 inches 

•- > V.:4li^= S1.416 inches,* 
- > : . . : :.t: >U7ftic« of sphere 



MENSURATION. — SPHERES AND SPHEROIDS. 



61 




To compute the surface of a segment of a sphei'e. 

Rule. — Multiply the height (be, Fig. 38) 
by the circumference of the sphere, and add 
the product to the area of the hase. 

To find the area of the base, we have the 
diameter of the sphere and the length of the 
versed sine of the arc abdy and we can find 
the length of the chord ad by the nde on 
p. 56. Having, then, the length of the chord 
ad for the diameter of the base, we can easily Fig. 33 

find the area. 

Example. — The height, be, of a segment abd, is 36 inches, and 
the diameter of the sphere is 100 inches. What is the convex sur- 
face, and what the whole surface? 

Ans. 100 X 8.1416 = 314.16 inches, the circumference of sphere. 
36 X 814.16 = 11309.76, th e conv ex surface. 
The length of ad = 100 — 30 x 2 = 28. 

V1OO2 — 28-^ = 96, the chord cwi. 

962 X 0.7854 = 7238.2464, the area of base. 



11309.76 + 7238.2464 = 18548.0064, 
the total area. 

To compute the surface of a spherical 
zone. 

Rule. — Multiply the height (cd, Fig. 34) ^ 
by the circumference of the sphere for the 
convex surface, and add to it the area of 
the two ends for the whole area. 



Fig.34 




Spheroids, or Ellipsoids. 

Definition. — Spheroids, or ellipsoids, are figures generated by 
the revolution of a semi-ellipse about one of its diameters. 

When the revolution is about the short diameter, they are pro- 
late ; and, when it is about the long diameter, they are oblate. 

To compute the surface of a spheroid when the apheroid is prolate. 

Rule. — Square the diameters, ami nmltiply the square root of 
half their sum by 3.1416, anil this procluct by the short diamettn*. 

Example. — A prolate spheroid has diameters of 10 and 14 
Uiches, what is its surface ? 

Ans. 10=2 = 100, and 142 = 19n._ 

Tlieirsum = 296, andi/-^ = 12.1655. 
12.1655 x 3.1416 X 10 = 382.191 square inches. 




62 MENSURATION.- CONES AND PYRAMIDS. 

To compute the mirface of n ipheroid when the fipheroid is obVite. 

KuLK. — Square the diameters, aud multiply the square root of 
lulf their smu by :i.l4ie, and tlila product by the long diameter. 
To tumipute thf mir/uM iff n ryllndm: 
liiiLK. — Multiply tlie ittngth by tiie circumference for the cod 
:X sarface, and add to the product the ares o> 
e two ends for the whole sm-face. 
I compute the HeetiontU urea of a circwtoi 

ring (Kg. 35). 
Ri'Mf. —Find the area of liotli circles, and 
subtract the area of t1ie sinaller from the area 
of tlie larger: the remainder will be the area of 
Fig.3S the ring. 

To i:im\\mti: the Hurfare of a eone. 

}{<T^E■ — Multiply the perimeter or circumference of the base by 
one-lialf the slant height, or side of the cone, for the convex area. 
Add (o this tlie ai'ea of the base, for ilie whole area. 

Example. —The diameter of the base of a cone ie 3 inches, and 
the slant height 15 inches, what Is the area of the cone f 

Ans. 3 X 3,141(i = 8.4248 = circumference of tmte. 

6.4*248 X 7i ~ Hi.mi squai-e inches, the convex stu^ace. 
3 X 3 X 0.TS54 = T.CI68 3(|iiare inches, the ares of baae. 
Area of cone = 77.7.J4 square Inches. 
PI jg To enmpute the itiea nf the surfneeof thefiru*- 

RULii. — MiUtipty tlie sum of the perinietets 
of the two cnils by the sinjit height of tlie fois- 
tnm, and <iivide l>y '2, fur the convex surface. 
Add tlie area of the lop and bottom surfaces. 
To rompiile the nurface ufa pyramid. 

Rule, — Multiply the perimeter of llie base 
by one-half the slant height, aud add to Uie 
product the area of the base. 
To i^nmpiite the nvrface of the fruttum <tf It 
pyrcmi.l. 
lti:i.K. — Multiply the sum of the perimeters of the two ends by 
the slant height of the frustiuu, lialve the pnxluct, aud add lo Uie 
result the area of the two euds. 



MENSURATION. - PIUSMS. 



63 



BfENSURATION OF SOLIDa 

To compute the volvme of a prism, 

RiJi.K. — Multiply tlie area of tli^ base by the height. 
This rule applies to any prism of any shape on the base, as long 
as the top and bottom surfaces are parallel. 

To compute the volume qf a prismoid. 

Definition. — A prismoid is 
a solid having parallel ends or 
bases dissimilar in sliape with 
qiuidri lateral sides. 

KuLK. — To the sum of the 
are^s of the two ends add four 
times the area of the middle 
section pai*allel to them, and ^ 
nmltiply this sum by one-sixth 
of the perpendicular height. 

Example. — What is the vol- 
ume of a quadrangular prismoid, as in Fig. 37, in which ah = 0", 
C(i = 4", ac = he = 10", ce = 8", ^ = 8", and //* = 6" ? 




Ans. Area of top 



Area of bottom 



Area of middle section 



6jfJ 

2 
8 + 6 

2 
« + ($ 



X JO = 50. 



X 10 = 70. 



X 10 = 60. 



|50 + 70 + (4 X 60)1 X J^ = 600 cubic inches. 
Note. — The length of the end of the middle section, as mn in Fig. 37 = 

To find the volume of a prism 
truncated obliquely. 

Rule. — Multiply the area of 
the base by the average height 
of the edges. 

Example. — What is the 
volume of a truncated prism, 
as in Fig. 38, where (f = 6 
inches, y7i = 10 inches, ea = 10, 
ft = 12, (?// = 8, an(l/^ = 8? 

Ans, Area of base = 6X10 =60 square inches. 

10+12 + 8 + 8 




Fig. 38 



Average height of edges = 



= 9i inches. 



60 X 9i = 970 cubic inches. 



66 MEiNSUUATlON. — SPHEROIDS, PAUAB0L0ID3, ETC. 





the square of the radius of the base phis the square of the lieight 

10:3 X 4 X 0.5236 = 341.3872 cubic inches vol- 
ume. 

Second Solution. — By the rule for fin«l- 
ing the diameter of a circle when a chord 
and its versed sine are given, we find that 
the diameter of tlie sphere in this case is 16.2o 
inches; then, by Rule 2, (3 X 16.25) — (2 X 4) 
= 40.75, and '!0.75 x 4^ X 0.5236 = 341.3872 
Fig. 41. cubic inclies, the volume of the segment. 

To cowpiite the volume of a spherical zone. 

Definition. — The part of a sphere in- 
cluded between two parallel planes (Fig. 

42). 

Rule. — To the sum of the squares of 

the radii of the two ends add one-third 

of the square of the height of the zone; 

nndtiply this sum by the height, and that 

Fig. 42. pi*oduct by 1.5708. 

To compute the volume of a nphei'ohh 

Rule. — Multiply the square of the revolv- 
ing axis by the fixed axis, and this product by 
0.5236. 

To compute the volume of a parafjolold of revo- 
lution (Fig. 43). 

Rule. —Multiply the area of the base by half 
rifl.43 the altitude. 

To compute the volume of a hjperholoid of revolution (Fig. 44). 

Rule. — To the s(|Uare of the I'adius of the 
base add the square of the middle diameter; 
nmltiply tliis sum by the height, and the pix>tl- 
uct by 0.5236. 

To compute the volume of any Jiyure ^f revo- 
^'^'^ lution. 

Rule. — Multiply the area of the generating surface by the clr- 
cuniference described by its centre of gi-avity. 

To compute the volume of an excavation, where the ground uf irrey- 
ular, and the bottom of the excavation is level (Fig. 45). 

Rule. — Divide the surface of the ground to be excavated Into 
equal squares of about 10 feet on a side, and ascertain by ineuu 





MENSURATION. — EXCAVATIONS. 



67 



a 



a 



A 



d 



a 



d 



d 



b 
Fig.45 



d 



a 



a 



a 



of a level the height of each comer, a, a, a, ft, 6, &, etc., abo\e the 

level to which the ground is to be excavated. Then add togcllier 

the heights of all the corneis that only come into one scjuare. 

Next take twice the sum of the heights of all the corners that come 

in two squares, as 6, h, b ; 

next three times the sum 

of the lieiglits of all the 

corners that come in three 

squares, as r, c, c ; and 

then four timies the sum 

of the heights of all the j^ 

corners that belong to foiu* 

squares, as d, r2, d, etc. 

Add togetlwr all these ^ 

quantities, and multiply 

their sum by one-foiuth 

the ai-ea of one of the squares. The result will be the volume of 

the excavation. 

Example. — Let the plan of the excavation for a cellar be as in 
the figure, and the heights of each corner above the proposed bot- 
tom of the cellar be as given by the numbers in the figure, then the 
volume of the cellar would he as follows, the area of each square 
being 10 X 10 = 100 sqHai*e feet: — 

Volume = i of 100 (a's + 2 b's 4- 8 c's + 4 tZ's). 

The a's in this case = 4 + « + :J + 2+1 + 7 + 4 = 27 

2 X the siun of the 6's = 2 X (3 + ($ + 1 + 4 + :{ + 4 )= 42 

3 X the sum of the c's = 3 x ( 1 + ;^ + 4) =24 

4 X the siuu of the *rs = 4 X (2 + 3 + + 2) =52 

145 

Volume = 25 X 145 = 3625 cubic feet, tlui <iUiintity of eailh to be 
exjavatetL 



68 



GEOMETKICAL PROBLEMS. 



OEOMETRICAL PROBLEMS. 

Problem 1 . — To bisect j or flimde into equal partSy a (/hen 

Ihu'.ah (Fig. 46). 

^ From a and ft, with any radius greater 

tlian half of aft, describe ares intersecting 

in c and d. The line cd, connecting these 

intersections, will bisect a)), and be perpen- 



Fig.46 



X^ 



l> diciilar to it. 

Pkoblkm 2. — To draw a perpendicular 
to a given straiyht line from a point witJf 
out it, 

1st Method (Fig. 47). —From the point a describe an arc with 

sufficient radius that it will cut the line he 
« V X in two places, as e and /. From e and / 

describe two arcs, with the same radius, 
intersecting in g; then a line drawn from 
a to fj will be pei-peudicular to the line ftc. 

2d Method (Fig. 

48). — From any two 
» ^ • •s.-w —r^ points, d and c, at some 

distance apart in the 

given line, and with 
radii da and c« respectively, describe arcs cut- 
ting at a and e. Di-aw ae, and it wili l)c the 
I)erpendicular required. This method is useful 
where the given point is opposite the end of 
the line, or nearly so. 

Problem 3. — To draw a perpendicular to 
a straiyht line from a given point, a, in that 
line. 





>.i 



a 
Fig.49 



1st Method (Fig. 49).— With any 
radius, from the given point a in the 
line, describe arcs cutting the line in 
the points ft and c. Then with b and 
c as centres, and with any radius 
greater than ab or ac, describe arcs 
cutting each other at d. The line Ja 
will be the perpendicular desireiL 



GEOMETRICAL PROBLEMS. 



69 




2d Method (Fig. 50, when the given point is at the end of 
the line). — From any point, 6, outside of the 
line, and with a radius ba, describe a semi- 
circle passing through a, and cutting the 
given line at rL Through b and d draw a 
straight line intersecting the semicircle at 6. 
The line ea will then be perpendicular to the 
line uc at the point a, 

3d Method (Fig. 51) or the 3, 4, and 5 
Method. — From the point a on the given line measure off 4 
inches, ot-4 feet, or 4 of any other unit, and with the same unit of 
measure describe an arc, with a as a centre 
and 3 units as a radius. Then from b describe 
an arc, with a radius of 5 units, cutting the 
first arc in c. Then ca will be the perpen- 
dicular. This method is particularly useful 
in laying out a right angle on the ground, or 
framing a house where the foot is used as 
the unit, and the lines laid off by straight edges. 

In laying out a right angle on the ground, the proportions of the 
triangle may be 30, 40, and 50, or any other multiple of 3, 4, and 5; 
and it can best be laid out with the tape. Thus, first measure off, 
say 40 feet from (c on the given line, then let one person hold the 
end of the tape at b, another hold the tape at the 80-foot mark at 
a, and a third person take hold of the tape at the 50-foot mark, 
with his thumb and finger, and pull the tape taut. The 50-foot 
mark will then be at the point c in the line of the pei*pendicular. 

Problem 4. — To draw a strali/ht line parallel to a given line 
at a given distance apart (Fig. 52). 




i 


B 


d 








( 


» 


Fig.52 I 


> 



From any two points near the ends of the given line describe 
two arcs about opposite the line. Draw the line cd tangent to 
these arcs, and it will be parallel to ab. 



70 



GEOMETRICAL PROBLEMS. 



Problem 5. — To eonstriici an (vngle equal to a given angie. 

With the point ^4, at the apex of the given 
angle, as a centre, and any radius, describe the 
arc BC, Then witli the point <r, at the vertex of 
tlie new angle, as a centre, and with the same 
radius as before, describe an arc like BC, Then 
with JiC as a radius, and h as a centre, describe 
an arc cutting the other at c. Then will cab b*» 
equal to the given angle CAB. 

Problem 6. — From a point on a given line 
to draw a line making an angle qf 6(P with. tJie 
(jiven line (Fig. 54). 

Take any distance, as ab, as a radius, and, with a as a centre, de- 
^crilie the arc 6c. Then with 6 as a centre, and the same radius, 
describe an arc cutting the first one at c. Draw from a a line 
through (', and it will luake with ab an angle of 60^. 





Fig.54 




Fig.55 



Problem 7. — From a given point, A, on a given line, AE, to 
draw a line making an angle of 4^^ with the given line (Fig. 55). 

Measure off from A, on AE, any distance, -46, and at 6 draw a 
line perpendicular to AE. Measure off on this perpendicular be 
equal to Ab, and draw a line from A through c, and it will make 
an angle with AE of 45^. 

Problem 8. — From any point, A, on a given line, to draw a line 
which shall make any desired angle with the given line (Fig. 56). 

To perform this problem we must have a 
table of chords at hand (such as is found on 
pp. 85-'.)3), which we use as follows. Find 
in the table the length of chord to a radius 
1, for the given angle. Then take any ra- 
^ dius, as large as convenient, describe an 
arc of a circle be with A as a centre. Mul- 
tiply the chord of the angle, found in the table, by the length of the 
radius Ab^ and with the product as a new radius, and 6 as a centre, 
describe a short arc cutting be in d. Draw a line from A throngl: 
&, and it will make the desired aaglc with DE, 




Fig.56 



GEOMETRICAL PROBLEMS 



71 



Example. — Draw a line from A on DE^ making an angle of 
440 40' with DE. 

Solution. — We find that tlie largest convenient radius for our 
arc is 8 inches: so with ^ as a centre, and 8 inches as a radius, we 
describe the arc be. Then, looking in the table of chords, we find 
the chord for an angle or arc of 44° 40' to a radius 1 is 0.76. Mul- 
tiplying this by 8 inches, we have, for the length of our new radius, 
6.08 inches, and with this as a radius, and 6 as a centre, we describe 
an arc cutting be in d. Ad will then be the line desired. 

Problem 9. — To biseet a given 
angle, as BAG (Fig. 57). 

With ^ as a centre, and any radius, 
descrl an arc, as eb. With c and b as 
centres, and any radius greater than 
one-half of eb, describe two arcs inter- 
secting in d. Draw from A a line 
through d, and it will bisect the angle BAC, 

Problem 10. — To biseet the anyle contained between two linen^ 
(IS A B and CI), when the vertex of the angle is not on the drawing 
(Fig. 58) 





Draw fe parallel to AB, and cd parallel to CD, so that the two 
lines will intersect each other, as at i. Bisect the angle cidy as in 
the preceding problem, and draw a line through i and o which will 
bisect the angle between the two given lines. 

Problem 11. — Through two given points, 
B and C, to describe an arc of a circle with 
a given radius (Fig. 59). 

With B and C as centres, and a radius 
equal to the given radius, describe two arcs 
intersecting at A» With ^ as a centre, and 
the same radius/ describe the ait; be, which 




Fig. 59 



will be found to pass through the given points, B and C 



72 



GEOMETRICAL PROBLEMS. 



Problem 12. — To find the centre of a given circle (Pig: W)). 

Draw any chord in the circle, as ah, and bisect this chord by 
the perpendicu/ar cd. This line will pass through the centre 
of the circle, and ef will be a diameter of the circle. Bisect ^, and 
the centre o will be the centre of the circle. 





T*R0BLEM 13. — To draw a circular arc through three gii>en 
pointH, as A, B, and C (Fig. 61). 

Draw a line from ^ to J5 and from B to C. Bisect AB and BC 
by the lines aa and cc, and prolong these lines until they intersect 
at 0, which will be the centre for the arc sought. With o as a 
centre, and Ao as a radius, describe the arc ABC, 

Problem 14. — To describe a circular arc parsing through three 
given points^ when the centre is not availaJjle, by means of a tri- 
angle (Fig. 62). 

B^ Let il, JB, and C 

be the given points. 
Insert two stiff pins 
or nails at A and C. 
Place two strips of 
wood, SS, as shown 
in the figure; one 
against A, the other 
against C, and in 
clined so that tlieir 
intersection shall 
come at the third 
point, B. Fasten the strips together at their intersection, and nail 
a third strip, T, to their other ends, so as to make a firm trian^e. 
Place the pencil-point at B, and, keeping the edges of the trian^ 
against A and B, move the triangle to the left and right, and tbv 
l>eneil will describe the arc sought. 




OEOMETHICAL PROBLEMS. 



73 







X 




%/ 






/ 




4 


ra 

Fig. 63 



V 



When the points A and C are at the same distance from B^ if a 
strip of wood be nailed to the triangle, so tliat its edge de shall be 
at right angles to a line joining A and C as the triangle is moved 
one way or the other, the edge de will always point to the centre of 
the circle. This principle is used in the perspective linear cZ. 

PuoBLEM 15. .— To find a circular arc which shall be ianfjent to 
a f/iven point, A, on a straiyht lincj and ^ 

pass through a given point, C, ouUnde the 
line (Fig. 63). 

Draw from A a line perpendicular to 
the given line. Connect A and C by a 
straight line, and bisect it by the perpen- 
dicular ac. The point whei-e these two 
perpendiculars intersect will be the centre 
of the circle. 

Pkoblbm 16. — To connect two parallel linen by a reversed curve 
composed qf two circular arcs of equal radius, and tangent to the 
lines at given points, a« A and B (Fig. 64). 

Join A and B, and di- 
vide the line into two 
equal parts at C. Bisect 
CA and CB by perpen- 
diculars. At A and B 
erect i)erpendicu]ars to 
the given lines, and the 
intersections a and b 
will be the centres of the 
arcs composing the required curve. 

Pboblbm 17. ^On a given line, as AB, to construct a com- 
pound curve qf three arcs of circles, the radii of the two siue. ones 
being equal and qf a 
given length, atid their 
centres in the given 
line; the central arc 
to pans through a given 
point, C, on the perpen- 
dicular bisecting the^ 
given line, and tangent 
to Uie other two arcs 
(Fig. 66). 

Draw tlie pttpendlc- 
nlar CIX Lftj off Aa^ 
Bbf aud CCf eiudi equal to the given radius of the side arcs; join 




Fig.64 




\:: 






/ 



/ 



I / 




Fig. 65 



74 



GEOMETRICAL PROBLEMS. 



ac; bisect ac by a perpendicular. The intersection of this line with 
tlie perpendicular CD will be the required centre of the central 
arc. Through n and h draw the lines De and De' ; from a and b, 
with the given radius, equal to Aa, Bby describe the arcs Ae'sind 
lie; from D as a centre, and CD as a radius, describe the arc eCef 
which completes the ciu^e required. 

Phoblem 18. — To conairuct a triangle upon a given straight 
line or bane, the length of the two tildes being given (Fig. 66). 

First (an equilateral triangle. Fig. 66a). — With the extremities 
A and B of the given line as centres, and AB sasi radius, descril)e 
arcs cutting each other at C Joiu AC and BC, 




Fig.GGa 




Fig. 60 b 



Second (when the sides are unequal, Fig. 66b). — Let ADh^ tt.e 
given base, and the other two sides be equal to C and B. With /) 
as a centre, and a radius equal to C, describe an indefinite arc 
With ^ as a centre, and B as a radius, describe an arc cutting the 
first at E. Join E ^dth A and 2>, and it will give the required 
triangle. 

Problem 19. — To describe a circle about a triangle (Fig. 67). 

Bisect two of the sides, us AC and CB, of the triangle, and at 
their centres erect perpendicular lines, as ae and />e, intersecting at 
e. With e as a centre, and eC as a radius, descril)e a circle, aud U 
will be found to pass through A and B. 





Fig. 67 



Problem 20. — To inscribe a circle in a triangle (Fig. tSB), 
Bisect two of the angles, A and B, of the triangle by lines cntting 
each other at o. With o as a centre, aud oe as a radius, dMeribe » 
circle, which will be found to just touch the other two sideiu 



GEOMETRICAL rUOBLKMS. 



40 



PnoBLEM 21. — To inscribe a square in a circle^ and to describe 
a circle about a square (Fig. 69). 

To inscribe the square. Draw two diameters. AB and CDy r.t 
rigliL angles to each other. Johi the points A, 1), B, C, and we 
liavc the inscribed square. 

To describe the circle. Draw the diagonals as before, intersecting 
at E, and, with ^ as a centre and AE as a radius, describe the 
circle. 





PROBLKM 22. — To inscribe a circle in a fiqvare, and to deticrihe 
a square about a circle (Fig. 70). 

To inscribe the circle. Draw the diagonals AB and C7>,. inter- 
secting at E. Draw the i)eipendicular EG to one of the sides. 
Tlien with J^ as a centre, and EG as a radius, describe a circle, 
which will be found to touch all four sides of the square. 

To describe the square. Draw two diameters, AB and CD, at 
right angles to each other, and prolonged beyond the circumference. 
Draw the diameter GF, bisecting the angle CEA or BED. Drnw 
lines through G and JF* perpendicular to GF, and terminating in 
the diagonals. Draw AD and CB to complete the square. 

Pkoulem 23. — To inscribe a penta- 
gon in a circle (Fig. 71). 

Draw two diametei*s, AB and CD, at ^ 
right angles to each other. Bisect AG ,^ 
at E. AVith ^ as a centre, and EC as a A| 
« radius, cut OB at F. AVith C as a centre, 
and CF as a radius, cut the circle at G 
and U. With these points as centres, and 
the same radius, cut the circle at I and 
J. Join /, J, ff, G, and C, and we then 
have inscribed in tlie circle a regular pentagon. 




pROBt.KM 24. — 7b i»w(!rl6« B rflffntei 
SoLUTioH. — Lay off on tiie dm 
circle six times, and connoct the p 




Prohi.em 25. — To coniitmct a re^ 
KlriiiuUt line, AB (Fig. TA). 

From A and II, wiih a nullus equal 
at O. With aa a centre, aiid a rM 
circle, and from A and JS lay o'J tU 
fci'ciice of tlie circle, and join tbe 
result will be a regular liexagon. 

Pkoblem 26, — To coimtnict n re, 
atraUjkt line, AB (Fig. 74). 

Produce the line AB both vaya, an 
and Bb, of iudeflnite lei^^' Kaect 
B, and niake the length of the Itiiea e 
draw lines parallel to Aa, and eijual in 
centres G and I) describe arcs, witli i 
peiidiculara Aa and Bb In Fand E. 



? 




E- 


Y 


' 


\ 






A 



Flg.74 

I'noni.KM 27.—Toniakeurrj/ultirct 

Dian- the diagonals .<1D and BC, 

C, and D, with a nuUin cqnal to A 



GEOMETRICAL PROBLEMS. 



77 



sides of the square in a, ft, c, d, c, /, h, and L Join these points 
to complete the octa^gon. 

Problem 28. — To inscribe a regular octagon in a circle (Fig. 
76). 

Draw two diameters, AB and CD, at right angles to each other. 
Bisect the angles AOB and AOC by the diameters EF and Gfl. 
Join Af Ey I), 11 J B, etc., for the inscribed figure. 

a 





Fig.ZS /ig.77 

PiiODLEM 29. — To inHcrihe a circle within a regular poh/f/on. 

Fimt (when the polygon has an even number of sides, as in Fig. 
T7). — Bisect two opposite sides at -4 and /?, and drawylZ?, and 
bisect it at C by a diagonal, DE, drawn between two opposite 
angles. With the radius CA describe the circle as required. 

Second (when the number of sides is odd, as in Fig. 78). — Bisect 
two of the sides at A and By and draw 
lines, AE and BD^ to the opposite angles, 
intersecting at C With C as a centre, 
and (J A as a radius, describe the circle as 
required. 

Pkoblem 30. — To deacribe a circle 
without a regular polygon. 

When the mmil)er of the sides is even, 
draw two diagonals from opposite angles, 
as ED and 67/ (Fig. 77), intersecting at 
C; and from C\ with CD as a radius, 
describe the circle required. 

When the number of sides is odd, find the centre, C, as in last 
pi-oblem; and with C as a centre, and CD (Fig. 78) as a radius, 
describe the cii'cle required. 




Fi8.78 



GEOMETRICAL PKUBLEH8. 



Plioni.EM 31. —To describe an ellipse, the lengtli and hrei 
the Uro iiiex, behiji iiieeii. 







iH-hlg gh 






On All a 






as aiamcte 






from the 






centre, 0, ri 


A 




the circles. 




and CLDK 






nZberTf 






on the cir. 






eiicc of th. 






circle, aa 






6", etc, an 






tlieiii dra» 




G 


to tlie cen 




FiB.79 


cutting th, 

circle at tht 


a, a', a", 


elc, respectively. 


l,„.ji the points h, b', eU'., dra 


parAllel 


to the shorter axi 


3; ami from the points n, a*, etc 


t^ 


. 1 


lines parallel to tht 

— .,,,.^^ axis, and inlerset^ 

\, first set of lines i 

\ c", etc These last 

\ will be points in 

lipse, anil, h; obta 


K 


v\ ^ 


I the ellipse can ea 

y^ 2n Method (P 
^^^ — Take the stmigl 




c 


of a stiff piece of 




FiB.eo 


canlboani, or woe 


Smm sor 


lie point, as ii, lu.ii 


rk off „b eqwal to half the sharle) 



/ 



GEOMETRICAL PROBLEMS. 



79 



eter, and ac equal to half the longer diameter. Place the straight 
eilge so that tlie point h shall l>e on the longer diameter, and tlie 
point c on the shorter: then will the point a be over a point in 
the ellipse. Make on the paper a dot at a, and move the slip 
around, always keeping the points b and c over the major and 
minor axes. In this way any number of points in the ellipse may 
be obtained, which may be connected by a curve drawn freehand. 

3d Method (Fig. 81, given the two axes AB and CD.) — FroTM 
the point Z> as a centre, 

and a radius A O, equal to D 

one-half of AB, describe 
an arc cutting AB at F 
andF'. These two points 
are called the foci of the 
ellipse. jOne property of 
the ellipse is, that the 
sum of the distances of 
any two points on the 
circumference from the 
foci is the same. Thus 
F'D + DF= F'E -f EF 
or F'G + GF.] Fix a 
couple of pins into the axis A B at F and F\ and loop a thread 
or cord upon them equal in length, when fastened to the pins, to 
AB, so as, when stretched as per dotted line FDF\ just to reach 
the extremity D of the short axis. Place a pencil-point inside 
the chord, as at E, and move the pencil along, always keeping the 
cord stretched tight. In this way the pencil will trace the outline 
of the ellipse. 

Problem 32. — To draw a tangent to an ellipse at a given point 
on the curve (Fig. 
82). 

Let it be re- 
quired to draw a 
tangent at the 
point E on the 
ellipse shown in 
Fig. 82, First 
find the foci F 
and F'j as in the 
third method for 
describing an el- 
lipse, Hnuk from 







1*^' 




80 



GKOMETRICAL PUOBl.EMS. 



E (li-aw lines EF and EF\ Prolong EF' to a, so that Ea shall 
equal EF. Bisect the angle uEF as iii 6, and through 6 draw a 
line touching the ciu-ve at E. Tliis line will be the tangent 
required. If It were tlt?sii*ed to draw a line normal to the ciu've 
at E, as, for instance, the joint of an elliptical arch, bisect the 
angle FEF\ and draw the bisecting line through E, and it will be 
the normal to the curve, and the proper line for the joint of an 
elliptical arch at that point. 

Problem 33. — To dmto a tarty ent to an ellipse from a yiven 
point without the curve (Fig. 83). 




Fig.83 



From the point T as a centre, and a radius equal to Uio distance 
to the nearer focus F, describe a circle. From F' as a centre, and 
a radius equal to the length of the longer axis, describe arcs cutting 
the circle just described at a and b. Draw lines from F' to a and 
/;, cutting the circumference of the ellipse at E and G, Draw lines 
from T through E and G, and they will be the tangents reqiiired. 

PitOBLEM 34. — To describe an ellipse approximately, by means 
of circular arcs. 

First (with arcs of two radii, Fig. 84). —Take half the difference 
of the two axes AH and CD, and set it off fiom the centre O to (f 
and c on OA and OC ; draw ac, and set off half ac tx) d; draw dl 
parallel to ac; set off Oc equal to Od; join c /, and draw em and dm 
parallels to di and ic. On nt as a centre, with a i-adlus mC, describe 
an arc through C, terminating in 1 and 2; and with i as a'oentre, 
and id as a radius, describe an arc tlu'ough X>, terminating in points 
3 and 4. On d and e as centres describe arcs through A and JS, 
connecting the points 1 and 4, 2 and 3. The four arcs' Urns de- 



GEOMETRICAL PROBLEMS. 



81 



smhod form approxiuiately an ellipse. Tliis methotl does not apply 
satisfaciov^^ when the conjugate axis is less than two-thirds of the 
li-ansvei-se axfs; 

Rg.04 
C 




Second (willi arcs of three radii, Fig. 85). — On the ti-ansverse 
r.xis AB draw the rectangle AGEB, equal in height to 0C\ half 




the conjitgatc axis. Di-aw GD perpendicular to AC. Set off OK 
eqnal to OC^ and on AK as a diameter describe the semicircle 



82 



GEOMETlllCAL PROBLEMS. 



ANK, Draw a radiiis parallel to OC, intersecting the semicircle 
at N, and the line GE at P. Extend OC to L and to D. Set off 
OM equal to PJV, and on D as a centre, with a radius DM, descrilKj 
an arc. From A and B as centres, with a radius OX, intersect 
this arc at a and h. The i^oints //, a, 2), 6, //', are the centres of 
the arcs required. Produce tlie lines a/T, Da, Dh, hW, and the 
spaces enclosed determine the lengths of each arc. This process 
works well for nearly all ellipses. It is employed in striking: out 
vaults, stone arches, and bridges. 

Note. — In this example the point IT happens to coincide with the point K^ 
but this need not nccesuariiy be the case. 



The Parabola* 

PjtoiJT.KM 35. — To construct a parabola token the vertex A, the 
axis AB, and a jjoint, 21, of the curve, are given (Fig. 86). 




Construct the rectangle ABMC, Divide MC into any nmnbor 
of equal parts, four for instance. Divide ^C in like manner. Con- 
nect Al, A2, and ^13. Through 1', 2', 3', draw parallels to the axis. 
The intersections I, II, and III, of these lines, are i)olnt8 in the 
required ciure. 

Pkoblem 36. — To draw a tangent to a given points II, €f Hie 

parabola (Fig. 86). 

From the given point II let fall a perpendicular on the axis at 6. 
JCxteml the axis to the left of A, Make Aa equal to Ah, Draw 
(dl, and it is the tangent required. 

The lines perpendicular to the tangent are called normals. To 
find the, normal to any point 1, harhif/ the tangent to any oUier 
point, 11. Draw the normal lie. From I let fall a perpendicular 
Id, on the axis AB, Lay off de equal to be. Connect Ic, and we 
have the nonnal required. The tangent may be drawn at I bf 
iaying off a perpendicular to the uonnal le at L 



OKOMKTUICAL PIIOBLKMS. 83 

Hie Hyperbola. 

The hyi>erbola possesses the characteristic that if, from any point, 
P, two sti-aiglit lines be drawn to two fixed points, F and jF", the 
foci, their difference shall always be the same. 

Phobi.em 37. — To ddHcrihe an hyperbola throvffh a </iven vertex, 
a, icith the (jwcu difference ahy and one of the foci, F (Fig 87). 




Draw the axis of the hyperbola AB, with the giveji distance ah 
and the focus F marked on it. From b lay off bFx equal to aF 
for the other focus. Take any point, as 1 on AB, and with a\ as 
a radius, and F as a centre, describe two short arcs above and 
below the axis. With 61 as a radius, and F' as a centre, describe 
arcs cutting those just described at P and P'. Take several points, 
as 2, :^, and 4, and obtain the corresponding points P.^, P;,, and P4 
in the same way. Join these points with a curved line, am) it will 
be an hyperbola. 

To draw a tant/ent to any point of an hyperbola, draw linos from 
the givi'Ji point to each of the foci, and bisect the angle thus 
formed. The bisecting line will be the tangent recpiircd. 



84 



GEOMETlllCAL PROBLEMS. 



The Cycloid. 



__^^^_^QQ 




The cycloid Is the curve descrribed 
by a x>oint hi the circumference of a 
circle rolling in a straight line. 

Problem ;^. — To deacrihc a cy- 
cloid {Fi^. m. 

Draw the straight line AB slz the 
base. Describe the generating circle 
tangent to this line at the centre, and 
through the centre of Uic circle, C, 
draw the line EE parallel t<: the base. 
Let fall a perpendicular from C upon 
the base. Divide the semi-circumfer- 
ence into any number of equal parts, 
for instance, six. Lay off on A B and 
. CE distances 0*1', J '2', etc., equal to 
Q« the divisions of the circiunferencc. 
5» Draw the chords Dl, D2, etc. From 
the points 1', 2', 3', on the line CE, with 
radii equal to the generating circle, 
describe arcs. From the points 1', 2^, 
3', 4', 5', on the line BA, and with 
radii equal respectively to the chords 
2)1, 7)2, D3, D4y 2)5, describe arcs 
cutting the preceding, and the inter- 
sections will be points of the curve 
required. 



GROMF-TRICAl. ritOBLEMS. 8B 

TABLE OF CHORDS ; Badios = 1.0000. 



KG 



GEOMETRICAL PROBLEMS. 



Table of Chords; Radius = l.OCXX) {continued). 



M. 

1 


IV 


1J8- 


13- 


14- 


1 


.1917 


.2091 


.2264 


.2437 


I 


.l'>20 


.2093 


.22*57 


.•2440 


'2 


.1923 


.2096 


.2270 


.'2443 


3 


.1926 


.2099 


.2273 


.2446 


4 


.192S 


.2102 


.2276 


.•2449 


f» 


.19:J1 


.2105 


.2279 


.•2452 


1 


.1931 


.2108 


.2281 


.'2465 


1 7 


.1937 


.2111 


.2284 


.•2458 


! 8 


.1^0 


.2114 


.2287 


.•24t>0 


J « 


.1943 


.2117 


.2290 


.•24<5:i 


■10 


.1946 


.2119 


.2293 


.•2466 


11 


.1949 


.2122 


.22JK> 


.•2469 


12 


.1952 


.2125 


.22i>9 


.•2472 


13 


.I9.'i5 


.2128 


.2:102 


.•2475 


14 


.1M7 


.2131 


.2305 


.•2478 


15 


.1960 


.2134 


.2307 


.•2481 


16 


.1W>3 


.2137 


.2310 


.2*84 


17 


.1^)66 


.2140 


.2313 


.•2486 


18 


.1969 


.214.-1 


.2316 


••2489 


19 


.1972 


.2146 


.2319 


.•2492 


2U 


.1975 


.2148 


.2322 


.*^95 


21 


.197S 


.2151 


.2:J25 


.•2498 


22 


.1981 


.2154 


.2328 


.•2501 


23 


.198.1 


.2157 


.•2331 


.•2504 


24 


.198«> 


.2UiO 


.2:133 


.•2507 


2.') 


.1989 


.216:$ 


.2:1:16 


••2.510 


21'. 


.IW2 


.2106 


.2339 


.2512 


27 


.llW:') 


.2169 


.2342 


.•2515 


2S 


.1998 


.2172 


.2:146 


.•2518 


2V 


.2001 


.2174 


^148 


.2521 


30 


.2004 


.2177 


.2:151 


.•2524 


31 


.2007 


.2180 


.2354 


.2527 


'.)■> 


.2010 


.2183 


.2367 


.25:10 


3-i 


.2012 


.2186 


.2:159 


.253:1 


34 


.2015 


.2189 


.2:162 


.2636 


3.-) 


.2018 


.2192 


.2365 


.2.v:i8 


36 


.2021 


.2195 


.2368 


.2.>H 


37 


.2024 


.2198 


.•2371 


.2544 


3K 


.2t)27 


.2200 


.2:174 


.2547 


3H 


.20:iO 


.220:1 


.2377 


.2660 


40 


.20: UJ 


.2206 


.2380 


.2653 


41 


.20;i6 


.2209 


.•2383 


.2656 


42 


.2038 


.2212 


.2385 


.•2559 


4ii 


.2041 


.2215 


.'2388 


.'2561 


44 


.2044 


.2218 


.•2:191 


.2664 


4i> 


.2047 


.2221 


.2:194 


.2567 


46 


.2t).'>0 


.2224 


.2397 


.•2570 


J 47 


.2a'»;) 


.222ti 


.2400 


.•2573 


.2a'HJ 


.2229 


.2401 


.2f>76 


4H 


.20.')9 


.22;i2 


.240«) 


.•2679 


(K) 


.2WJ2 


.22:15 


.2409 


.•2^.82 


Til 


.20t)5 


.2238 


.2411 


.2585 


."•2 


.2067 


.2241 


.2414 


.•2587 


61 


.2070 


.2244 


.SM17 


.2590 


fi4 


.2073 


.2247 


.2420 


.25it:i 


r.5 


•>076 


.2260 


.2423 


.2596 


f)6 


.2079 


.2253 


.2426 


.2599 


•u 


.2082 


.2256 


.24'29 


.2t)02 


r>s 


.2085 


.2258 


.•24:12 


.2605 


M 


.2088 


.2261 


.24:14 


.2»M)8 


6tl 


.2091 


.2264 


.'2437 


.•2611 



16' 



.2611 
.2613 
.•2616 
.2619 

.2<)'25 
.•2ti28 
.2631 
.•2»>W 
.'2636 
.'26:19 

.♦2642 
.'2645 
.'2648 
.2651 
.•2654 
.'2657 
.'2660 
.•2662 
.'2605 
.•2668 

.2671 
.'2674 
.2677 
.'2680 
.2683 
.'2685 
.2688 
.2691 
.2694 
.2697 

.2700 
.270:1 
.2706 
.2709 
.2711 
.'2714 
.'2717 
.'27'20 
.•27'2:j 
.'2726 

.27^29 
.27:12 
.27:14 
.2737 
.2740 
.274:1 
.2746 
.•2749 
.2752 
.2755 

.2758 
.•27(50 
.•276:i 
.2766 
.27459 
.2772 
.'2775 
.•2778 
.•2781 
.•2783 



16* 



.2783 
.2786 
.2789 
.2792 
.'2795 
.2798 
.•2801 
.'2804 
.'2807 
.'2809 
.•2812 

.'2815 
.•2818 
.•2821 
.•28-24 
.'2827 
.2830 
.'2832 
.'2835 
.'28:18 
.2841 

.•2844 
.'2847 
.'2850 
.'2853 
.2855 
.2858 
.'2861 
.2864 
.2867 
.•2870 

.'2873 
.•2876 i 
.'2878 . 
.•2vS81 , 
.•2S84 ' 
.•2887 ' 
.2890 I 
.•2893 ; 
.'2896 
.'2899 j 

.•2902! 

.2904 1 

.2907 ! 

.•2910 

.•2D13 

.•2916 

.•2919 ! 

.29^22 

.•2925 

.'2927 

.'2930 
.'29:i:i 

.•29:m 

.-29:19 
.2942 
.-2945 
.•2948 
.•2950 
.295:1 
.'2956 



17' 



.2966 
.'2959 
.'2962 
.'2966 
.'2968 
.2971 
.2973 
.2976 
.'2979 
.2982 
.2986 

.2988 
.2991 
.2994 
.2996 
.'2999 
.3002 
.3005 
.3008 
.3011 
.3014 

.3017 
.3019 
.3022 
.3026 
.30'28 
.30:11 
.3034 
.3037 
.3040 
.3042 

.3046 
.:1048 
.3051 
.:1054 
.:1057 
.3060 
.3063 
.:1065 
.3068 
.3071 

.3074 
.:J077 
.:1080 
.3083 
.3086 
.:i088 
.3091 
.3094 
.3097 
.3100 

.310:1 
.3106 
.3109 
.3111 
.3114 
.3117 
.3120 
.312:1 
.31-26 
.31-29 




.3129 
.3132 
.31^4 
.3137 
.3140 
.3143 
.3146 
.3149 
.3162 
.3155 
.3167 

.3160 
.3163 
.3166 
.3169 
.3172 
.3176 
.3178 
.3180 
.3183 
.3186 

.3189 
.3192 
.3195 
.3198 
.3*200 
.3-203 
.3'206 
.3'209 
.3212 
.3216 

.3218 
.3221 
.3*223 
.3*226 
.3-229 
.3'232 
.3-235 
.3238 
.3241 
.3'244 

.3-246 j 

.3-249 

.3-252 

.3255 

.3-258 

.3261 

.3264 

.3267 

.3269 

.3272 

.3275 
.3278 
.3*281 
.3-284 
.3287 
.3-289 
.3-292 
.3-295 
.3-298 
.3301 



.3801 
.3304 
.3307 
.3310 
.3312 
.3316 
.3318 
.3321 
.3324 
.33*27 
.3330 

.3333 
.3335 
.3338 
.3541 

.3au 

.3347 
.3350 
.3363 
.3356 
.3358 

.3361 
.2i\M 
.3367 
.3370 
.3373 
.3376 
.3378 
.3381 
.3384 
.3387 

.3390 
.3393 
.3306 
.3398 
.^1401 
.3404 
.3407 
.3410 
.3413 
.3416 

.:1419 
.:1421 
.3424 
.34-27 
:M80 
.a433 
.3436 
.:1439 
.:i441 

oil 1 

.Ol'l 1 

.:1447 
.3450 
.3463 
.3466 
.3469 
.:14<J2 

.3467 
.3470 
.3473 



«©• 


ai* 


.3473 


..1645 


.3476 


.3(U8 


.3479 


.3660 


.3482 


.3663 


.8484 


.3656 


.3487 


.3659 


.3490 


^1662 


.3493 


.3665 


.3406 


.3668 


.S409 


.3670 


.8502 


.3673 


.3504 


.3676 


.3507 


.3679 


.3510 


.3682 


.3513 


.3686 


.3516 


.3688 


M^'klO 


.3690 


.3522 


.3603 


.3626 


.3606 


.3527 


.3600 


.3530 


.3702 


.3533 


JNO& 


.3636 


.3708 


.3530 


.3710 


.3542 


.3713 


.3645 


.3716 


.3547 


.3719 


.3650 


.3722 


.3663 


.3726 


.3666 


.3728 


.3660 


.3730 


..3662 


.3733 


.3665 


.3736 


.3567 


.3730 


.3670 


.3742 


.3573 


.3745 


.3676 


.3748 


.3670 


.3750 


.3682 


.3753 


.3685 


.3766 


.3687 


.3760 


.3600 


.3762 


.3603 


.3765 


.3606 


.3768 


mIuOO 


.3770 


.8602 


.3773 


.3605 


.8776 


.3608 


.3770 


.3610 


.8782 


.3613 


.3786 


.3616 


.3788 


.3619 


.3700 


.36*22 


.3703 


.3626 


.8706 


.3628 


.3709 


.3030 


jsaoi' 


.3633 


.3805 


.36:16 


.3808 


.3630 


U»10 


.3642 


.3813 


.8046 


.3816 



6 
7 
8 

10 

11 
12 
13 
U 
15 
16 
17 
18 
10 
20 

21 
22 
23 
24 
25 
26 
27 
28 
20 
30 

81 
32 
33 
34 
35 
30 
37 
38 
30 
40 

41 
42 
43 
44 
45 
40 
47 
48 
40 
50 

51 
52 
53 
&4 
55 
56 
57 
58 
50 
«0 



GEOMETRICAL PROBLEMS. 



87 



Table of Chords; 


Radius 


= 1.0000 (continued). 




M. 


aa* 


88' 


«4- 


»5' 


«6' 


«?• 


28* 


»9' 


30* 


3V 


32' 


M. 

0' 


(K 


.3816 


.3987 


.4158 


.4329 


.4499 


.4669 


.48.38 


.5008 


.5176 


.5345 


.5513 


1 


.3819 


.3990 


.4161 


.4332 


.4502 


.4672 


.4841 


.5010 


.5179 


.5348 


.5516 


1 


2 


.3822 


.3993 


.4164 


.4334 


.4505 


.4675 


.4844 


.5013 


.5182 


.5350 


.5518 


2 


S 


.3825 


•oVvD 


.4167 


.4337 


.4508 


.4677 


.4847 


.5016 


.5185 


.5353 


.5521 


3 


4 


..3828 


•«J«I<T(I 


.4170 


.4340 


.4510 


.4680 


.4850 


.5019 


.5188 


.5356 


.5524 


4 


5 


.3830 


.4002 


.4172 


.4343 


.4513 


.4683 


.4853 


.5022 


.5190 


.5359 


.5527 


5 


6 


.3833 


.4004 


.4175 


.4346 


.4516 


.4686 


.4855 


.5024 


.5193 


.5362 


.5530 


6 


m 
I 


.3836 


.4007 


.4178 


.4349 


.4519 


.4689 


.4858 


.5027 


.5196 


.5364 


.5532 


7 


1 8 


.3839 


.4010 


.4181 


.4352 


.4522 


.4692 


.4861 


.5030 


.5199 


.5367 


.5535 


8 


9 


.3842 


.4013 


.4184 


.4354 


.4525 


.4694 


.4864 


.5033 


.5202 


.5370 


.5538 


9 


lu 


.3845 


.4016 


.4187 


.4357 


.4527 


.4697 


.4867 


.5036 


.5204 


.5373 


.5541 


10 


11 


.3848 


.4019 


.4190 


.4360 


.4530 


.4700 


.4869 


.5039 


.5207 


.5376 


.5543 


11 


12 


.3850 


.4022 


.4192 


.4363 


.4533 


.4703 


.4872 


.5041 


.5210 


.5378 


.5546 


12 


13 


.3853 


.4024 


.4195 


.4366 


.4536 


.4706 


.4875 


.5044 


.5213 


.5381 


.5549 


13 


14 


.:3856 


.4027 


.4198 


.4369 


.4539 


.4708 


.4878 


.5047 


.5216 


.5384 


.5552 


14 


15 


.3859 


.4030 


.4201 


.4371 


.4542 


.4711 


.4881 


.5050 


.5219 


.5387 


.5555 


15 


16 


.3862 


.4033 


.4204 


.4374 


.4544 


.4714 


.4884 


.5053 


.5221 


.5390 


.6557 


16 


17 


.3865 


.4036 


.4207 


.4377 


.4547 


.4717 


.4886 


.5055 


.5224 


.5392 


.5560 


17 


18 


.3868 


.4039 


.4209 


.4380 


.4550 


.4720 


.4889 


.5058 


.5227 


.5395 


.5563 


18 


19 


.3870 


.4042 


.4212 


.4383 


.4553 


.4723 


.4892 


.5061 


.5230 


.5398 


.5566 


19 


20 


.3873 


.4044 


.4215 


.4:i86 


.4556 


.4725 


.4895 


.5064 


,5233 


.5401 


.5569 


20 


21 


.3876 


.4047 


.4218 


.4388 


.4559 


.4728 


.4898 


.5067 


.5235 


.5404 


.5571 


21 


22 


.3879 


.4050 


.4221 


.4391 


.4561 


.4731 


.4901 


.5070 


.5238 


.5406 


.5574 


22 


23 


.3882 


.4053 


.4224 


.4394 


.4564 


.4734 


.4903 


.5072 


.5241 


.5409 


.5577 


23 


24 


.3885 


.4056 


.4226 


.4397 


.4567 


.4737 


.4906 


.5075 


.5244 


.5412 


.5580 


24 


25 


.3888 


.4059 


.4229 


.4400 


.4570 


.4740 


.4909 


.5078 


.5247 


.5415 


.5583 


25 


26 


.3890 


.4061 


.4232 


.4403 


.4573 


.4742 


.4912 


.5081 


.5249 


.5418 


.5585 


26 


27 


.3893 


.4064 


.4235 


.4405 


.4576 


.4745 


.4915 


.5084 


.5252 


.5420 


.5588 


27 


28 


.3896 


.4067 


.42.38 


.4408 


.4578 


.4748 


.4917 


.5086 


.5255 


.5423 


.5591 


28 


29 


.3899 


.4070 


.4241 


.4411 


.4581 


.4751 


.4920 


.5089 


.5258 


.5426 


.5594 


29 


30 


.3902 


.4073 


.4244 


.4414 


.4584 


.4754 


.4923 


.5092 


.5261 


.5429 


.5597 


30 


31 


.3905 


.4076 


.4246 


.4417 


.4587 


.4757 


.4926 


.5095 


.5263 


.5432 


.5599 


31 


32 


.3908 


.4079 


.4249 


.4420 


.4590 


.4759 


.4929 


.5098 


.5266 


.5434 


.5602 


32 


33 


.3910 


.4081 


.4252 


.4422 


.4593 


.4762 


.4932 


.5100 


.5269 


.5437 


.5605 


33 


34 


.3913 


.4084 


.4255 


.4425 


.4595 


.4765 


.4934 


.5103 


.5272 


.5440 


.5608 


34 


35 


.3916 


.4087 


.4258 


.4428 


.4598 


.4768 


.4937 


.5106 


.5275 


.5443 


.5611 


35 


36 


.3919 


.4090 


.4261 


.4431 


.4601 


.4771 


.4940 


.5109 


.5277 


.5446 


.5613 


36 


37 


.3922 


.4093 


.4263 


.4434 


.4604 


.4773 


.4943 


.5112 


.5280 


.5448 


.5616 


37 


38 


.3925 


.4096 


.4266 


.4437 


.4607 


.4776 


.4946 


.5115 


.5283 


.5451 


.5619 


38 


39 


.3927 


.4098 


.4269 


.4439 


.4609 


.4779 


.4948 


.5117 


.5286 


.5454 


.5622 


39 


40 


.3930 


.4101 


.4272 


.4442 


.4612 


.4782 


.4951 


.5120 


.5289 


.5457 


.5625 


40 


'.41 


.3933 


.4104 


.4275 


.4445 


.4615 


.4785 


.4954 


.5123 


,5291 


.5460 


.5627 


41 


142 


.3936 


.4107 


.4278 


.4448 


.4618 


.4788 


.4957 


.5126 


.5294 


.5462 


.5630 


42 


43 


.3939 


.4110 


.4280 


.4451 


.4621 


.4790 


.4960 


.5129 


.5297 


.5465 


.5633 


43 


44 


.3942 


.4113 


.4283 


.4454 


.4624 


.4793 


.4963 


.5131 


.5300 


.5468 


.5636 


44 


45 


.3945 


.4116 


.4288 


.4456 


.4626 


.4796 


.4965 


.5134 


.5303 


.5471 


.5638 


45 


46 


.3947 


.4118 


.4289 


.4459 


.4629 


.4799 


.4968 


.5137 


.5306 


.5474 


.5641 


46 


47 


.3950 


.4121 


.4292 


.4462 


.4632 


.4802 


.4971 


.5140 


.5308 


.5476 


.5644 


47, 


48 


.3953 


.4124 


.4295 


.4465 


.4635 


.4805 


.4974 


.5143 


.5311 


.5479 


.5647 


48; 


49 


.3956 


.4127 


.4298 


.4468 


.4638 


.4807 


.4977 


.5145 


.5314 


.5482 


.5650 


49 


50 


.3959 


.41.30 


.4300 


.4471 


.4641 


.4810 


.4979 


.5148 


.5317 


.5485 


.5652 


50 


bl 


.3962 


.4133 


.4303 


.4474 


.4643 


.4813 


.4982 


.5151 


.5320 


.5488 


.5655 


51 


i 52 


.3965 


.4135 


.4306 


.4476 


.4646 


.4816 


.4985 


.5154 


.5322 


.5490 


.5658 


52 


53 


.3967 


.4138 


.4309 


.4479 


.4649 


.4819 


.4988 


.5157 


.5325 


.5493 


.5661 


53 


54 


.3970 


.4141 


.4312 


.4482 


.4652 


.4822 


.4991 


.5160 


.5328 


.5496 


.5664 


54 


55 


.3973 


.4144 


.4315 


.4485 


.4655 


.4824 


.4994 


.5162 


.5331 


.5499 


.5666 


55 


56 


.3076 


.4147 


.4317 


.4488 


.4658 


.4827 


.4996 


.5165 


.5334 


.5502 


.5669 


56 


57 


.3979 


AlbO 


.4320 


.4491 


.4660 


.4830 


.4999 


.5168 


.5336 


.5504 


.5672 


57 


58 


.3982 


.4153 


.4323 


.4493 


.4663 


.4S:J3 


.5002 


.5171 


.5339 


.5507 


.5075 


58 


59 


.3085 


.4155 


.4326 


.4496 


.4666 


.48:j6 


.5005 


.5174 


.5342 


.5510 


.5678 


59 


60 


.3987 


.4158 


.4329 


.4499 


.4669 


.4838 


.5008 


.5176 


.5345 


.5513 


.5080 


60 



88 



GEOMETRICAL PROBLEMS. 



Table of Chords ; Radius = 1.0000 {continued) , 




0' 

1 

2 
3 

4 
5 
6 

8 

9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
21 
25 
26 
27 
2S 
29 
30 

31 
32 
•M 
34 
35 
.'it; 
37 

:is 

39 
40 

I 41 
42 
43 
44 

j45 
'46 
i47 
'48 
I 49 
■50 

51 
,52 
;53 

I 54 
:6:. 
' 5'> 
I 57 
: 5S 
I 59 
I 30 



5680 
5683 
56S6 
5689 
5691 
5694 
5697 
5700 
5703 
57a5 
5708 

5711 
5714 
5717 
5719 
5722 
5725 
5728 
57:U) 

57;j:j 

5 



'TM 



5739 
5742 
5744 
5747 
5750 
5753 
5756 
575S 
5701 
5704 

5767 
5769 
5772 

:> M O 

577S 
5781 
5783 
5786 
5789 
5792 

5795 
5797 
5800 
5803 
5806 
5S0S 
5811 
58 14 
5817 
5820 

5SJJ 

5h28 
58:;i 
:.s:j4 
5s:;r» 
.'i.s:j9 
.■•s4-.; 

58 J.') 
5S47 



.5847 
.5850 
.5853 
.5866 
.5859 
.5861 
.5864 
.5867 
.5870 
.5872 
.5875 

.5878 
.5881 
.5884 
.5886 
.5889 
.58;»2 
.5895 
.5897 
.5900 
.5'.Ktt 

..5903 
.5909 
.5911 
.5914 
.5917 
.5920 
.5922 
.5925 
.5928 
.5931 

.5931 
.5936 
.59:19 
.5942 
.5945 
.5947 
.5950 
.5953 
.5956 
.5959 

.5961 
.5964 
.5967 
.5970 
..5972 
.5975 
.5978 
.5981 
.5984 
..5980 

.59S0 
.5992 
.5995 
.59'.)7 
.iUMH) 

.004 »6 

.0011 
.0014 



.6014 
.6017 
.6(r20 
.6022 
.6025 
.6028 

.6a-a 

.6034 

.etm 

.6039 
.6042 

.6045 
.mil 
.605i) 

.605:^ 

.60r»6 
.6058 
.60«)1 

jMm 

.6067 
.6»)70 

.0072 
.6075 
.6;»7S 
.6t)Sl 
.608:) 
.6;)8:'. 
.6089 
.6(K»2 
.60{»5 
.0097 

.61 on 

.6103 
.6106 
.6108 
.6111 
.6114 
.6117 
.6119 
.6122 
.6125 

.6128 
.6130 

.61:^:) 

.6i:Mi 
.6139 
.6142 
.6144 
.6147 
.6150 
.6153 

.6155 
.0158 
.0101 
.61('>4 
.010<\ 
.010!) 
.0172 
.0175 
.iil7S 
.01 SO 



I 



.6180 

.6183; 

.6186 

.6189 

.6191 

.6194 

.6197 

.6200 

.6202 

.6205 

.6208 

.6211 
.6214 
.6216 
.6219 
.6222 
.6225 
.6227 
.6230 
.6233 
.6-236 

.6238 
.6241 
.6244 
.6247 
.6249 
.6252 
.6255 
.6258 
.6260 
.6263 

.6266 
.6269 
.6272 
.6274 
.6277 
.6280 
.6283 
.6285 
.6288 
.6291 

.Ui94 
.62<H5 
.6299 
.6302 
.6305 
.6307 
.6,310 
.6313 
.6316 
.(»18 

.6321  
.6:J24 
.r>327 I 
.O:i:io I 



.<^i:>2 
.»):j;j5 

.(Viil 

.o:n:i 

.0:> l«i 



r I 



37' 


38' 

.6511 


39» 


40* 


4V 


4«' 


43* 


M. 

1 


.6346 


.6676 


.6840 


.7004 


.7167 


.7330 


0'. 


.6349 


.6514 


.6679 


.6843 


.7007 


.7170 


.7333 


i; 


.6.%>2 


.6517 


.6682 


.6846 


.7010 


.n73 


.7335 


2 


.63.54 


.6520 


.6684 


.6849 


.7012 


.n76 


.7338 


3 


.6:j57 


.6522 


.6687 


.6851 


.7015 


.n78 


.7341 


4 


.62)60 


6.525 


.6«J90 


.68.54 


.7018 


.nsi 


.7344 


5 


XuViH .6528 


.6693 


.68.57 


.7020 


.7181 


.7346 


«■ 


.631^5 i .6531 


.6695 


.6860 


.7023 


.7186 


.7340 


4 


.6:J6S , .65:w 


.6098 


.6862 


.7026 


.n89 


.7352 


8i 


.6371 1 .6536 


.6701 


.6865 


.7029 


.7192 


.7354 


»i 


.6374 .6.5.39 


.6704 


.6868 


.7081 


.7195 


.7357 


101 


.a376 


.6542 


.6706 


.6870 


.7034 


.7197 


.7360 


11 


.6379  .6544 


.6709 


.6873 


.7Cte7 


.7200 


.73G2 


12 


.6:182 


.6547 


.6712 


.6876 


.7040 


.7203 


.73«5 


13 


.0385 


.6550 


.6715 


.6879 


.7042 


.7205 


.7368 


14 


.6387 


.6553 


.6717 


.6881 


.7045 


.7208 


.7371 


15 


.6390 


.6555 


.6720 


.6884 


.7048 


.7211 


.7373 


16 


.6393 


.6558 


.6723 


.6887 


.7050 


.7214 


.7376 


17 


.0:»6 


.6561 


.6?25 


.6890 


.7053 


.7216 


.7370 


18 


.6398 


.6564 


.6728 


.6892 


.7096 


.7210 


.7381 


19 


.«U01 


.6566 


.6731 


.6895 


.7059 


.7222 


.7384 


20 


.6404 


.6569 


.6734 


.6898 


.7061 


.7224 


.7387 


21 


.6407 


.6572 


.6736 


.6901 


.7064 


.7227 


.7300 


22 


.MIO 


.6575 


.67.39 


.6903 


.7067 


.7230 


.7302 


23 


.6412 


.6577 


.6742 


.6906 


.7069 


.7232 


.7395 


24 


.6415 


.6580 


.6715 


.6909 


.7072 


.7235 


.7308 


25 


.6418 


.6583 


.6747 


.6911 


.7075 


.7238 


.7400 


26 


.6421 


.6586 


.6750 


.0914 


.7078 


.7241 


.7408 


27 


.6423 


.6588 


.675:$ 


.6917 


.7080 


.7213 


.7406 


28 


.6426 


.6591 


.6756 


.692«) 


.7083 


.7246 


.7408 


SO 


.0)429 


.6594 


.6758 


.6922 


.7086 


.7240 


.7411 


30 


.6432 


.6597 


.6761 


.6925 


.7089 


.7251 


.7414 


31 


.6434 


.6599 


.6764 


.6928 


.7091 


.7254 


.7417 


32 


.6437 


.6602 


.6767 


.6931 


.7094 


.7257 


.7410 


38 


.6440 


.6605 


.6769 


.6933 


.7097 


.7260 


.7^2 


84 


.6443 


.6608 


.6772 


.6936 


.7099 


.7202 


.7425 


35 


.6445 


.6610 


.6775 


.6039 


.7102 


.7285 


.7427 


ao 


.6448 


.6613 


.6777 


.6941 


.7105 


.7268 


.7480 


37 


.6451 


.6616 


.6780 


.6944 


.7108 


.7270 


.7433 


38 


.6454 


.6619 


.67X3 


.6947 


.7110 


.7278 


.7433 


39 


.♦U56 


.6621 


.6786 


.6950 


.7118 


.7276 


.7438 


40 


.6459 


.6624 


.(.788 


.6952 


.7116 


.7270 


.7441 


41! 


.0462 


.6<)27 


.6791 


.6955 


.7118 


.7281 


.7443 


42- 


.6465 


.60;M) 


.0794 


.6958 


.7121 


.7284 


.7446 


43: 


.0407 


.66:J2 


.6797 


.6961 


.7124 


.7287 


.7440 


44 


.6470 


.66:)5 


.6799 


.6963 


.7127 


.7280 


.7468 


45 


.6473 


.663S 


.6802 


.6966 


.7129 


.7292 


.74U 


46 


.6470 


.6640 


.6805 


.61HJ9 


.7132 


.7205 


.7467 


47 


.W78 


.6<U3 


.0S08 


.6971 


.7135 


.7298 


.7400 


48 


.6481 


.(•)640 


.6810 


.6974 


.7137 


.7900 


.7402 


49> 


.6484 Am\) 


.0813 


.0977 


.7140 


.7308 


.7405 


50> 


.6487 ' .0051 


.0810 


.0981 


.714:5 


.7306 


.7468 


51 


.6tSt| .60 "i 4 


.r.8l9 


.0982 


.7146 


.7308 


.7471 


52' 


.fi4l»2 ; .0<r)7 


.«>X21 


.0985 


.7148 


.7311 


.7478 


53, 


.r>4y5 1 .ooiKJ 


.0824 


.6!>88 


.7151 


.7814 


.7476 


54 


.o49*< 1 sm-i 


.r>.S27 


.0901 


.7154 


.7316 


.7470 


55 


.»>;'i<M) .fiOiV) 


AW1\) 


.6'.>1»3 


.7150 


.7310 


.7481 


50; 


,0511:; .Oi'is 


.0k:12 


.0LKH5 


.7159 


.7322 


.7484 


57 i 


.O'ltMi .OiTl 


.ov;5 


.6i>.t'.i 


.7Hi2 


.7325 


.7487 


68 


.0V»'.» .007:') 


.ris;;^ 


.7001 


.7165 


.7527 


.7480 


M 


.o-.ii 


.(•►«>7r» 


.0840 


, .7(M»4 


.7107 


.7330 


.7402 


00 



.ble of Chords 


; Radius 


= 1.0000 (continued) 


• 


4' 


46* 


46' 


47' 


48' 


49' 


SO" 


51- 


68- 


1 


54* 


M. 

0' 


192 


.7654 


.7815 


.7975 


.8135 


.8-294 


.8452 


.8610 


.8767 


.8924 


.9080 


m 


.7656- 


.7817 


.797:^ 


.8137 


.8297 


.8455 


.8613 


.8770 


.8927 


.9082 


1 


m 


.7659 


.7820 


.7981) 


.8140 


.8299 


.8458 


.8615 


.8773 


.8929 


.9085 


2 


wo 


.7662 


.7823 


.7983 


.8143 


.8302 


.8460 


.8618 


.8775 


.8932 


.9088 


3 


i03 


.7664 


.7825 


.7986 


.8145 


.8304 


.8463 


.8621 


, .8778 


.8934 


.9090 


4 


m 


.7667 


.7828 


.7988 


.8148 


.8307 


.8466 


.8623 


.8780 


.8937 


.9093 


5 


m 


.7670 


.7831 


.7991 


.8151 


.8310 


.8468 


.8626 


.8783 


.8940 


.9096 


6 


»ii 


.7672 


.7833 


.7994 


.8153 


.8312 


.8471 


.8629 


.8786 


.8942 


.9098 


7 


il4 


.7675 


.7836 


.7996 


.8156 


.8315 


.8473 


.8631 


.8788 


.8945 


.9101 


8 


>16 


.7678 


.7839 


.7999 


.8159 


.8318 


.8476 


.8634 


.8791 


.8947 


.9103 


9 


»19 


.7681 


.7841 


.8002 


.8161 


.8320 


.8479 


.8636 


.8794 


.8950 


.9106 


10 


>22 


.7683 


.7844 


.8004 


.8164 


.8323 


.8481 


.8639 


.8796 


.8953 


.9108 


11 


»24 


.7686 


.7847 


.8007 


.8167 


.8826 


.8484 


.8642 


.8799 


.8955 


.9111 


12 


»27 


.7689 


.7849 


.8010 


.8169 


.8328 


.8487 


.8644 


.8801 


.8958 


.9113 


13 


30 


.7691 


.7852 


.8012 


.8172 


.8331 


.8489 


.8647 


.8804 


.8960 


.9116 


14 


33 


.7694 


.7855 


.8015 


.8175 


.8334 


.8492 


.8650 


.8807 


.8963 


.9119 


15 


35 


.7697 


.7857 


.8018 


.8177 


.8336 


.8495 


.8652 


.8809 


.8966 


.9121 


16 


38 


.7699 


.7860 


.8020 


.8180 


.8339 


.8497 


.8655 


.8812 


.8968 


.9124 


17 


41 


.7702 


.7863 


.8023 


.8183 


.8341 


.8500 


.8657 


.8814 


.8971 


.9126 


18 


•43 


.7705 


.7865 


.8026 


.8185 


.8344 


.8502 


.8660 


.8817 


.8973 


.9129 


19 


•46 


.7707 


.7868 


.8028 


.8188 


.8347 


.8505 


.8663 


.8820 


.8976 


.9132 


20 


49 


.7710 


.7871 


.8031 


.8190 


.8349 


.8508 


.8665 


.8822 


.8979 


.9134 


21 


•51 


.7713 


.7873 


.8034 


.8193 


.8:J52 


.8510 


.8668 


.8825 


.8981 


.9187 


22 


•54 


.7715 


.7876 


.8036 


.8196 


.8355 


.8513 


.8671 


.8828 


.8984 


.9139 


23 


.57 


.7718 


.7879 


.8039 


.8198 


.8357 


.8516 


.8673 


.8830 


.8986 


.9142 


24 


•60 


.7721 


.7882 


.8042 


.8201 


.8360 


.8518 


.8676 


.8833 


.8989 


.9145 


25 


•62 


.7723 


.7884 


.8044 


.8204 


.8363 


.8521 


.8678 


.8835 


.8992 


.9147 


26 


•65 


.7726 


.7887 


.8047 


.8206 


.8365 


.8523 


.8681 


.8838 


.8994 


.9150 


27 


•68 


.7729 


.7890 


.80 JO 


.8209 


.8368 


.8526 


.8684 


.8841 


.8997 


.9152 


28 


.70 


.7731 


.7892 


.8052 


.8212 


.8371 


.8529 


.8686 


.8843 


.8999 


.9155 


29 


•73 


.7734 


.7895 


.8055 


.8214 


.8373 


.8531 


.8689 


.8846 


.9002 


.9157 


30 


•76 


.7737 


.7898 


.8058 


.8217 


.8376 


.8534 


.8692 


.8848 


.9005 


.9160 


31 


.78 


.7740 


.7900 


.8060 


.8220 


.8378 


.8537 


.8694 


.8851 


.9007 


.9163 


32 


•81 


.7742 


.7903 


.8063 


.8222 


.8381 


.8539 


.8697 .8854 


.9010 


.9165 


33 


•84 


.7745 


.7906 


.8066 


.8225 


.8384 


.8542 


.8699 


.8856 


.9012 


.9168 


34 


86 


.7748 


.7908 


.8068 


.8228 


.8386 


.8545 


.8702 


.8859 


.9015 


.9170 


35 


.89 


.7750 


.7911 


.8071 


.8230 


.8389 


.8547 


.8705 


.8861 


.9018 


.9173 


36 


.92 


.7753 


.7914 


.8074 


.8233 


.8392 


.8550 


.8707 


.8864 


.9020 


.9176 


37 


.9.) 


.7756 


.7916 


.8076 


.8236 


.8394 


.8552 


.8710 


.8867 


.9023 


.9178 


38 


.97 


.7758 


.7919 


.8079 


.8238 


.8397 


.8555 


.8712 


.8869 


.9025 


.9181 


39 


iOO 


.7761 


.7922 


.8082 


.8241 


.8400 


.8558 


.8715 


.8872 


.9028 


.9183 


40 


m 


.7764 


.7924 


.8084 


.8244 


.8402 


.8560 


.8718 


.8874 


.9031 


.9186 


41 


105 


.7766 


.7927 


.8087 


.8246 


.8405 


.8563 


.8720 


.8877 


.9033 


.9188 


42 


i08 


.7769 


.7930 


.8090 


.8249 


.8408 


.8566 


.8723 


.8880 


.9036 


.9191 


43 


ill 


.7772 


.7932 


.8092 


.8251 


.8410 


.8568 


.8726 


.8882 


.9038 


.9194 


44 


il3 


.7774 


.7935 


.8095 


.8254 


.8413 


.8571 


.8728 


.8885 


.9041 


.9190 


45 1 


•16 


.7777 


.7938 


.8098 


.8257 


.8415 


.8573 


.8731 


.8887 


.9044 


.9199 


40. 


•19 


.7780 


.7940 


.8100 


.8259 


.8418 


.8576 


.8734 


.8890 


.9046 


.9201 


47, 


121 


.7782 


.7943 


.8103 


.8262 


.8421 


.8579 


.8736 


.8893 


.9049 


.9204 


48 , 


124 


.7785 


.7946 


.sio,-) 


.826) 


.8423 


.8581 


.8739 


.8895 


.9051 


.9207 


49, 


127 


.7788 


.7948 


.8108 


.8267 


.8426 


.8584 


.8741 


.8898 


.9054 


.9209 


50' 


i-29 


.7791 


.7951 


.8111 


.8270 


.8429 


.8.587 


.8744 


.8900 .9056 


.9212 


51 


•32 


.7793 


.7954 


.S113 


.8273 


.8431 


.8589 


.8747 


.8903 


.9059 


.9214 


52 


35 


.7796 


7956 


.8110 


.8275 


.S434 


.8)92 


.8749 


.8906 


.9002 


.9217 


53 1 


i38 


.7799 


.7959 


.8119 


.S27S 


.8437 


.8594 


.8752 


.8908 


.9064 


.9219 


54; 


m 


.7801 


.7962 


.8121 


.8281 


.8439 


.8597 


.8754 


.8911 


.9067 


.9222 


55 


43 


.7804 


.7964 


.8124 


.S283 


.8442 


.8000 


.8757 


.8914 


.9069 


.9225 


56 


46 


.7807 


.7967 


.8127 


.8286 


.8444 


.8602 


.8760 


.8916 


.9072 


.9227 


57 


48 


.7809 


.7970 


.8129 


.8289 


.8447 


.8605 


.8762 


.8919 


.9075 


.9230 


5S 


>51 


.7812 


.7972 


.8132 


.8291 


.8450 


.8608 


.8765 


.8921 


.9077 


.92:32 


59 1 


154 


.7815 


.7975 


.8135 


.8294 


.8452 


.8610 


.8767 


.8924 


.9080 


.9235 60 



90 



GEOMETllICAL PROBLEMS. 



Table of Chords 


; Radius = 


= 1.0000 1 


[continued) 


• 


M. 


55" 


66' 


67* 


68" 


50" 


eo* 


or 


62' 


«8' 


64* 


M. 


.9235 


.9389 


.9543 


.9696 


.9848 


1.0000 


1.0161 


1.0301 


1.0460 


1.0698 


1 


.9238 


.9392 


.9546 


.9699 


.9861 


1.0003 


1.0163 


1.0303 


li)462 


1.0601 


1 


2 


.9240 


.9395 


.9548 


.9701 


.9854 


1.0005 


1.0166 


1.0306 


1.0466 


1.0603 


2 


3 


.9243 


.9397 


.9551 


.9704 


.9856 


1.0008 


1.0168 


1.0308 


1.0467 


1.0606 


3 


4 


.9245 


.9400 


.9553 


.9706 


.9859 


1.0010 


1.0161 


1.0311 


1.0460 


1.0608 


4 


T) 


.9248 


.9402 


.9566 


.9709 


.9861 


1.0013 


1.0163 


1.0313 


1.0462 


1.0611 


5 


6 


.9250 


.9405 


.9559 


.9711 


.9864 


1.0015 


1.0166 


1.0316 


1.0466 


1.0613 


6 


7 


.9253 


.9407 


.9561 


.9714 


.9866 


1.0018 


1.0168 


1.0318 


1.0467 


1.0616 


7 


8 


.9256 


.9410 


.9564 


.9717 


.9869 


1.0020 


1.0171 


1.0321 


1.0470 


1.0618 


8 


9 


.9258 


.9413 


,9566 


.9719 


.9871 


1.0023 


1.0173 


1.0323 


1.0472 


1.0621 


9 


10 


.9261 


.9415 


.9569 


.9722 


.9874 


1.00-26 


1.0176 


1.0326 


1.0476 


1.0623 


10 


11 


.926:$ 


.9418 


.9571 


.9724 


.9876 


1.0028 


1.0178 


1.0328 


1.0477 


1.0626 


11 


12 


.9266 


.9420 


.9574 


.9727 


.9879 


1.0030 


1.0181 


1.0331 


1.0480 


1.0028 


12 


13 


.9268 


.94-23 


.9576 


.9729 


.9881 


i.oa33 


1.0183 


1.0333 


1.0482 


1.0690 


13 


14 


.9271 


.9425 


.9579 


.9732 


.9884 


1.0035 


1.0186 


1.0336 


1.0486 


1.0633 


14 


15 


.9274 


.9428 


.9581 


.9734 


.9886 


1.0038 


1.0188 


1.0338 


1.0487 


1.0636 


15 


16 


.9276 


.9430 


.9584 


.9737 


.9889 


1.0040 


1.0191 


1.0341 


1.0490 


1.0638 


16 


17 


.9279 


.9433 


.9587 


.9739 


.9891 


1.0043 


1.0193 


1.0343 


1.0492 


1.0640 


17 


18 


.9281 


.9436 


.9589 


.9742 


.9894 


1.0045 


1.0196 


1.0346 


1.0406 


1.0043 


18 


19 


.9284 


.9438 


.9592 


.9744 


.9897 


1.0048 


1.0198 


1.0348 


1.0407 


1.0646 


19 


20 


.9287 


.9441 


.9594 


.9747 


.9899 


1.0050 


1.0201 


1.0361 


1.0600 


1.0648 


20 


21 


.9289 


.9443 


.9597 


.9750 


.9902 


1.0053 


1.0203 


1.0363 


1.0602 


1.0660 


21 


22 


.9292 


.9446 


.9699 


.9752 


.9904 


1.0055 


1.0206 


1.0366 


1.0604 


1.0668 


22 


23 


.9294 


.9448 


.9602 


.9755 


.9907 


1.0058 


1.0208 


1.0368 


1.0607 


1.0666 


23 


24 


.9297 


.9451 


.9604 


.9757 


.9909 


1.0060 


1.0-211 


1.0361 


1.0600 


1.0668 


24 


25 


.9299 


.9464 


.9607 


.9760 


.9912 


1.006;i 


1.0213 


1.0363 


1.0612 


1.0660 


2ft 


26 


.9302 


.9456 


.9610 


.9762 


.9914 


1.0060 


1.0-216 


1.0366 


1.0514 


1.0602 


26 


27 


.9305 


.9459 


.9612 


.9765 


.9917 


1.00(58 


1.0218 


1.0368 


1.0517 


1.0005 


27 


28 


.9307 


.9461 


.9615 


.9767 


.9919 


1.0070 


1.0-221 


1.0370 


1.0519 


1.0007 


28 


29 


.9310 


• J7^rO*T 


.9617 


.9770 


.9922 


1.0073 


1.0-2-23 


1.0373 


1.0622 


1.0070 


29 


30 


.9312 


.9466 


.9620 


.9772 


.9924 


1.0075 


1.0226 


1.0376 


1.0624 


1U)072 


30 


31 


.9315 


.9469 


.96-22 


.9775 


.9927 


1.0078 


1.0228 


1.0378 


1.0627 


1.0076 


31 


32 


.9317 


.9472 


.9625 


.9778 


.9929 


1.0080 


1.0231 


1.0380 


1.0629 


1.0077 


32 


33 


.9320 


.9474 


.9627 


.9780 


.9932 


1.0083 


1.0233 


1.0.383 


1.0632 


1.0080 


33 


34 


.9323 


.9477 


.9630 


.9783 


.9934 


1.0086 


1.0-236 


1.0386 


1.0534 


1.0082 


34 


35 


.9325 


.9479 


.9633 


.9785 


.9937 


1.0088 


1.0-238 


1.0388 


1.0537 


1.0086 


36 


36 


.9328 


.9482 


.96;}5 


.9788 


.9939 


1.0091 


1.0-241 


1.0390 


1.0539 


1.0087 


36 


37 


.9330 


.9484 


.96:JS 


.9790 


.9942 


1.0093 


1.0-243 


1.0393 


1.0W2 


1.0090 


37 


38 


.9333 


.9487 


.9640 


.9793 


.9945 


1.0096 


1.0-246 


1.0396 


1.0544 


1.0092 


38 


39 


.93;t5 


.9489 


.9643 


.9795 


.9947 


1.0098 


1.0248 


1.0398 


1.0547 


1.0004 


39 


40 


.9338 


.9492 


.9645 


.9798 


.9950 


1.0101 


1.0251 


1.0400 


1.0649 


1.0097 


40 


41 


.9341 


.9495 


.9648 


.9800 


.9952 


1.0103 


1.0253 


1.0403 


1.0661 


1.0099 


41 


42 


.934:i 


.9497 


.9650 


.9803 


.9955 


1.0106 


1.0-256 


1.0406 


1.0664 


1.0702 


42 


43 


.9346 .9500 


.9653 


.9805 


.9957 


1.0108 


1.0258 


1.0408 


1.0566 


1.0704 


43 


44 


.9348 .9502 


.9665 


.9808 


.9960 


1.0111 


1.0261 


1.0410 


1.0669 


1.0707 


44 


45 


.93.')! 


.9505 


.9658 


.9810 


.9962 


1.0113 


1.026:} 


1.0413 


1.0661 


1J0700 


46 


4t3 


.9353 


.9507 


.9661 


.9813 


.9965 


1.0116 


1.0-266 


1.0415 


1.0564 


1.0712 


40 


47 


.9351) .9510 


.9663 


.9816 


.99<>7 


1.0118 


1.0-268 


1.0418 


1.0566 


1.0714 


47 


48 


.9359 j .9512 


.9660 


.9818 


.9970 


1.01-21 


1.0-271 


1.0420 


1.0560 


1.0717 


48 


49 


.93<)1 i .9515 


.9668 


.9821 


.9972 


1.01-23 


1.0273 


1.04-23 


1.0571 


1.0719 


49 


50 


.9364 1 .9518 


.9671 


.9823 


.9975 


1.01-26 


1.0-276 


1.0425 


1.0674 


1.0721 


50 


51 


.93(56  .9520 


.9673 


.0«2») 


.9977 


1.01-28 


1.0278 


1.0428 


1.0576 


1.0724 


61 


52 


.93()J 


.9523 


Mid 


.9828 


.9980 


1.0131 


1.0281 


1.04:50 


1.0^79 


1.0726 


52 


53 


.9371 


.9525 


.9678 


.9831 


.9982 


1.0133 


1.0-283 


1.0433 


1.0681 


1.0729 


63 


54 


.9374 


.9528 


.9681 


.08:5^$ 


.99S5 


1.0136 


1.0286 


1.04:15 


1.0684 


1.0731 


64 


55 


.9377 


.9530 


.9(^3 


.9836 


.99S7 1 1.013S 


1.0-2S8 


1.0438 


1.0686 


1.0784 


65 


56 


.9379 


.9533 


.9086 


.9S3S 


.9990 j 1.U141 


1.0-291 


1.0440 


1.0589 


1.0730 


66 


57 


.9382 


.95.36 


.9689 


.9841 


.9992; 1.0143 


1.0-293 


1.0443 


1.0591 


1.0730 


57 


58 


.9384 


.9538 


.9691 


.9843 


.9i>95 


1.0146 


1.0-296 


1.0446 


1.0603 


1.0741 


68 


59 


.9387 


.9541 


.9694 


.9846 


.9998 


1.0148 


1.0-298 


1.0447 


1.0696 


1.0744 


50 


60 


.9389 


.9543 


.9696 


.9848 


1.0000 


1.0151 


1.0301 


1.0460 


1.0508 


iun4o 


00 



GEOMETRICAL PROBLEMS. 



91 



Table of Chords ; Radius 


= 1.00O0 


(continued] 


1. 


M. 


65* 


66" 


67' 


es" 


69* 


70- 


w 


78* 


73' 


M. 

0' 


1.0746 


1.0893 


1.1039 


1.1184 


1.1328 


1.1472 


1.1614 


1.1766 


1.1896 


1 


1.0748 


1.0895 


1.1041 


1.1186 


1.1331 


1.1474 


1.1616 


1.1758 


1.1899 


1 


2 


1.0751 


1.0898 


I.IOU 


1.1189 


1.1333 


1.1476 


1.1619 


1.1760 


1.1901 


2 


8 


1.0753 


1.0900 


1.1046 


1.1191 


1.1335 


1.1479 


1.16-21 


1.1763 


1.1903 


3 


4 


1.0756 


1.0903 


1.1048 


1.1194 


1.1338 


1.1481 


1.1624 


1.1765 


1.1906 


4 


5 


1.0758 


1.0905 


1.1051 


1.1196 


1.1340 


1.1483 


1.1626 


1.1767 


1.1908 


5 


6 


1.0761 


1.0907 


1.1053 


1.1198 


1.1342 


1.1486 


1.1628 


1.1770 


1.1910 


6 


7 


1.0763 


1.0910 


1.1056 


1.1201 


1.1345 


1.1488 


1.1631 


1.1772 


1.1913 


7 


8 


1.0766 


1.0912 


1.1058 


1.1203 


1.1347 


1.1491 


1.1633 


1.1775 


1.1915 


8 


9 


1.0768 


1.0915 


1.1061 


1.1206 


1.1350 


1.1493 


1.1635 


1.1777 


1.1917 


9 


10 


1.0771 


1.0917 


1.1063 


1.1208 


1.1352 


1.1495 


1.1638 


1.1779 


1.1920 


10 


11 


1.0773 


1.0920 


1.1065 


1.1210 


1.1354 


1.1498 


1.1640 


1.1782 


1.1922 


11 


12 


1.0775 


1.0922 


1.1068 


1.1213 


1.1357 


1.1500 


1.1642 


1.1784 


1.1924 


12 


18 


1.0778 


1.0924 


1.1070 


1.1215 


1.1359 


1.1502 


1.1645 


1.1786 


1.1927 


13 


14 


1.0780 


1.0927 


1.1073 


1.1218 


1.1362 


1.1505 


1.1647 


1.1789 


1.1929 


14 


15 


1.0783 


1.0929 


1.1075 


1.1220 


1.1364 


1.1507 


1.1650 


1.1791 


1.1931 


15 


16 


1.0785 


1.0932 


1.1078 


1.1222 


1.1366 


1.1510 


1.1652 


1.1793 


1.1934 


16 


17 


1.0788 


1.0934 


1.1080 


1.12-25 


1.1369 


1.1512 


1.1654 


1.1796 


1.1936 


17 


18 


1.0790 


1.0937 


1.1082 


1.1227 


1.1371 


1.1514 


1.1657 


1.1798 


1.1938 


18 


19 


1.0793 


1.0939 


1.1085 


1.1230 


1.1374 


1.1517 


1.1659 


1.1800 


1.1941 


19 


20 


1.0795 


1.0942 


1.1087 


1.1232 


1.1376 


1.1519 


1.1661 


1.1803 


1.1943 


20 


21 


1.0797 


1.0944 


1.1090 


1.123+ 


1.1378 


1.1522 


1.1664 


1.1805 


1.1946 


21 


22 


1.0800 


1.0946 


1.1092 


1.1237 


1.1381 


1.1524 


1.1666 


1.1807 


1.1948 


22 


28 


1.0802 


1.0949 


1.1094 


1.1239 


1.1383 


1.1526 


1.1668 


1.1810 


1.1950 


23 


24 


1.0805 


1.0951 


1.1097 


1.1242 


1.1386 


1.1529 


1.1671 


1.1812 


1.1952 


24 


25 


1.0807 


1.0954 


1.1099 


1.1244 


1.1388 


1.1531 


1.1673 


1.1814 


1.1955 


25 


26 


1.0810 


1.0956 


1.1102 


1.1246 


1.1390 


1.1533 


1.1676 


1.1817 


1.1957 


26 


27 


1.0812 


1.0959 


1.1104 


1.1249 


1.1393 


1.1536 


1.1678 


1.1819 


1.1959 


27 


28 


1.0815 


1.0961 


1.1107 


1.1251 


1.1395 


1.1538 


1.1680 


1.1821 


1.1962 


28 


29 


1.0817 


1.0963 


1.1109 


1.1254 


1.1398 


1.1541 


1.1683 


1.1824 


1.1964 


29 


80 


1.0820 


1.0966 


1.1111 


1.1256 


1.1400 


1.1543 


1.1685 


1.1826 


1.1966 


30 


31 


1.0822 


1.0968 


1.1114 


1.1258 


1.1402 


1.1545 


1.1687 


1.1829 


1.1969 


31 


32 


1.0824 


1.0971 


1.1116 


1.1261 


1.1405 


1.1548 


1.1690 


1.1831 


1.1971 


32 


38 


1.0827 


1.0973 


1.1119 


1.1263 


1.1407 


1.1550 


1.1692 


1.1833 


1.1973 


33 


34 


1.0829 


1.0976 


1.1121 


1.1266 


1.1409 


1.1552 


1.1694 


1.1836 


1.1976 


34 


35 


1.0832 


1.0978 


1.1123 


1.1268 


1.1412 


1.1555 


1.1697 


1.1838 


1.1978 


35 


36 


1.0834 


1.0980 


1.1126 


1.1271 


1.1414 


1.1557 


1.1699 


1.1840 


1.1980 


36 


37 


1.0837 


1.0983 


1.1128 


1.1273 


1.1417 


1.1560 


1.1702 


1.1843 


1.1983 


37 


38 


1.0839 


1.0985 


1.1131 


1.1275 


1.1419 


1.1562 


1.1704 


1.1845 


1.1985 


38 


39 


1.0841 


1.0988 


1.1133 


1.1278 


1.1421 


1.1564 


1.1706 


1.1847 


1.1987 


39 


40 


1.0844 


1.0990 


1.1136 


1.1280 


1.1424 


1.1567 


1.1709 


1.1850 


1.1990 


40 


41 


1.0846 


1.0993 


1.1138 


1.1283 


1.1426 


1.1569 


1.1711 


1.1852 


1.1992 


41 


42 


1.0S49 


1.0995 


1.1140 


1.1285 


1.1429 


1.1571 


1.1713 


1.1854 


1.1994 


42 


43 


i.oajvi 


1.0997 


1.1143 


1.1287 


1.1431 


1.1574 


1.1716 


1.1857 


1.1997 


43 


44 


1.0854 


1.1000 


1.1145 


1.1290 


1.1433 


1.1576 


1.1718 


1.1859 


1.1999 


44 


45 


1.0856 


1.1002 


1.1148 


1.1292 


1.1436 


1.1579 


1.1720 


1.1861 


1.2001 


45 


46 


1.0859 


1.1005 


1.1150 


1.1295 


1.1438 


1.1581 


1.1723 


1.1864 


1.2004 


46 


47 


1.0861 


1.1007 


1.1152 


1.1297 


1.1441 


1.1583 


1.1725 


1.1866 


1.2006 


47 


48 


1.0863 


1.1010 


1.1155 


1.1299 


1.1443 


1.1 58() 


1.1727 


1.186S 


1.-2008 


48 


49 


1.0866 


1.1012 


1.1157 


1.1302 


1.1 44r) 


1.1 5S8 


1.1730 


1.1871 


1.-2011 


49 


50 


1.0868 


1.1014 


1.1160 


1.1304 


1.1448 


l.ir)90 


1.1732 


1.1 S73 


1.2013 


50 


51 


1.0871 


1.1017 


1.1162 


1.1307 


1.1450 


1.1.51)3 


1.1735 


1.1875 


1.2015 


51 


52 


1.0873 


1.1019 


1.1105 


1.13()« 


1.14r)2 


1.1.505 


1.1737 


1.1878 


1.2018 


52 


I 53 


1.0876 


1.1022 


1.1167 


1.1311 


i.i4r)r) 


1.150S 


1.1730 


1.1880 


1.2020 


53 


! .-4 


1.0S78 


1.1024 


1.1109 


1.1314 


1.1457 


1.1000 


1.1742 


1.1882 


1.20-22 


54 


1 55 


1.0881 


1.1027 


1.1172 


1.1316 


1.1460 


1.1002 


1.1744 


1.1885 


1.2025 


55 


1 56 


1.0883 


1.1029 


1.1174 


1.1319 


1.1402 


1.1005 


1.1740 


1.1887 


1.20*27 


56 


, 57 


1.0885 


1.1031 


1.1177 


1.1321 


1.1404 


1.1607 


1.1749 


1.1889 


1.2029 


57 


58 


1.0888 


1.1084 


1.1179 


1.13-23 


1.1467 


1.1609 


1.1751 


1.1892 


1.2032 


58 


50 


1.0890 


1.1036 


1.1181 


1.1326 


1.1469 


1.1612 


1.1753 


1.1894 


1.2034 


59 


60 


iMm 


1.1089 


1.1184 


1.1328 


1.1472 


1.1614 


1.1756 


1.1896 


1.2036 


60 



92 



GEOMETRICAL PROBLEMS. 



Table of Chords; Radius = 1,0000 (continued). 



M. 


740 


76- 


76* 


77* 


78- 


70' 


80' 


81* 


S^" 


M. 


1.2036 


1.2175 


1.2313 


1.2450 


1.2586 


1.2722 


1.2866 


1.2989 


1.3121 


O' 


1 


1.2039 


1.2178 


1.2316 


1.2453 


1.2689 


1.2724 


1.2868 


1.2991 


1.3123 


1 


2 


1.2041 


1.2180 


1.2318 


1.2455 


1.2591 


1.2726 


1.2860 


1.2993 


1.3126 


2 


3 


1.2043 


1.2182 


1.2320 


1.2457 


1.2593 


1.2728 


1.2862 


1.2996 


1.3128 


3 


4 


1.2046 


1.2184 


1.2322 


1.2459 


1.2595 


1.2731 


1.2866 


1.2998 


1^130 


4 


5 


1.2048 


1.2187 


1.2325 


1.2462 


1.2598 


1.2733 


1.2867 


1.8000 


1.3132 


5 


6 


1.2050 


1.2189 


1.2327 


1.2464 


1.2600 


1.2735 


1.2869 


1.3002 


1.3134 


6 


7 


1.2053 


1.2191 


1.23-29 


1.2466 


1.2602 


1.2737 


1.2871 


1.3004 


1.3137 


7 


8 


1.2055 


1.2194 


1.2332 


1.2468 


1.2604 


1.2740 


1.2874 


1.3007 


1.3130 


8 


9 


1.2057 


1.2196 


1.2334 


1.2471 


1.2607 


1.2742 


1.2876 


1.3009 


1^41 





10 


1.2060 


1.2198 


1.2336 


1.2473 


1.2609 


1.2744 


1.2878 


1.8011 


1.3143 


10 


11 


1.2062 


1.2201 


1.2338 


1.2475 


1.2611 


1.2746 


1.2880 


1.3013 


1.3146 


11 


12 


1.2064 


1.2203 


1.2341 


1.2478 


1.2614 


1.2748 


1.2882 


1.3016 


1.3147 


.12 


13 


1.2066 


1.2205 


1.2343 


1.2480 


1.2616 


1.2751 


1.2886 


1.3018 


1.3150 


13 


14 


1.2069 


1.2208 


1.2345 


1.2482 


1.2618 


1.2763 


1.2887 


1.3020 


1.8152 


li 


16 


1.2071 


1.2210 


1.2348 


1.2484 


1.2020 


1.2755 


1.2889 


1.8022 


1.3154 


15 


16 


1.2073 


1.2212 


1.2350 


1.2487 


1.2623 


1.2767 


1.2891 


1.3024 


1.3156 


16 


17 


1.2076 


1.2214 


1.2352 


1.2489 


1.2625 


1.2760 


1.2894 


1.3027 


1.3158 


17 


18 


1.2078 


1.2217 


1.2354 


1.2491 


1.2627 


1.2762 


1.2896 


1.3029 


1.3161 


18 


19 


1.2080 


1.2219 


1.2357 


1.2493 


1.2629 


1.2764 


1.2898 


1J»81 


1.3163 


10 


20 


1.2083 


1.2221 


1.2359 


1.2496 


1.2632 


1.2766 


1.2900 


1.8038 


1.3165 


20 


21 


1.2085 


1.2224 


1.2361 


1.2498 


1.2634 


1.2769 


1.2903 


1.3085 


1.3167 


21 


22 


1.2087 


1.2226 


1.2364 


1.2500 


1.2636 


1.2771 


1.2905 


1.3088 


1.3169 


22 


23 


1.2090 


1.2228 


1.2366 


1.2503 


1.2638 


1.2773 


1.2907 


1.3040 


1.3172 


28 


24 


1.2092 


1.2231 


1.2368 


1.2505 


1.2641 


1.2776 


1.2909 


1.8042 


1.3174 


24 


25 


1.2094 


1.2233 


1.2370 


1.2507 


1.^2643 


1.2778 


1.2911 


1.3044 


1.3176 


25 


26 


1.2097 


1.2235 


1.2373 


1.2509 


1.2646 


1.2780 


1.2914 


1.8046 


1.8178 


26 


27 


1.2099 


1.2237 


1.2375 


1.2512 


1.2648 


1.2782 


1.2916 


1.8040 


1.3180 


27 


28 


1.2101 


1.2240 


1.2377 


1.2514 


1.2650 


1.2784 


1.2918 


1.3061 


1.3183 


28 


29 


1.2104 


1.2242 


1'.2380 


1.2516 


1.2652 


1.2787 


1.2920 


1.8068 


1.3185 


20 


30 


1.2106 


1.2244 


1.2382 


1.2518 


1.2664 


1.2789 


1.2922 


1.9056 


1.8187 


30 


31 


1.2108 


1.2247 


1.2384 


1.2521 


1.2656 


1.2791 


1.2925 


1.8057 


1.3180 


31 


32 


1.2111 


1.2249 


1.2386 


1.2523 


1.2659 


1.2793 


1.2927 


1.8060 


IJSlOl 


82 


33 


1.2113 


1.2251 


1.2389 


1.2525 


1.2661 


1.2796 


1.2929 


1.3062 


1^08 


38 


34 


1.2115 


1.2254 


1.2391 


1.2528 


1.2663 


1.2798 


1.2931 


1.8064 


1.8196 


84 


35 


1.2117 


1.22.56 


1.2393 


1.2530 


1.2665 


1.2800 


1.2934 


1.8066 


IJSIW 


85 


36 


1.2120 


1.2258 


1.2396 


1.2532 


1.2668 


1.2802 


1.2936 


1.8068 


1.82U0 


36 


37 


1.2122 


1.2260 


1.2398 


1.2534 


1.2670 


1.2804 


1.2938 


1.8071 


1.8902 


37 


38 


1.2124 


1.2263 


1.2400 


1.2537 


1.2672 


1.2807 


1.2940 


1.8073 


1.8204 


38 


39 


1.2127 


1.2265 


1.2402 


1.2539 


1.2674 


1.2809 


1.2942 


1.8075 


1.3207 


30 


40 


1.2129 


1.2267 


1.2405 


1.2541 


1.2677 


1.2811 


1.2946 


1.3077 


1.32U0 


40 


41 


1.2131 


1.2270 


1.2407 


1.2543 


1.2679 


1.2813 


1.2947 


1.8070 


1.3211 


41 


42 


1.2134 


1.2272 


1.2409 


1.2546 


1.2681 


1.2816 


1.2949 


1.3082 


1.8218 


42 


43 


1.2136 


1.2274 


1.2412 


1.2548 


1.2683 


1.2818 


1.2961 


1.8084 


1.8315 


43 


44 


1.2138 


1.2277 


1.2414 


1.2550 


1.2686 


1.2820 


1.2954 


l.'UKUt 


1.8318 


44 


45 


1.2141 


1.2279 


1.2416 


1.2552 


1.2688 


1.2822 


1.2956 


1.8088 


1.3220 


45 


46 


1.2143 


1.2281 


1.2418 


1.2555 


1.2690 


1.2825 


1.2958 


1.3090 


1.3222 


46 


47 


1.2145 


1.2283 


1.2421 


1.2557 


1.2692 


1.2827 


1.2960 


1.8003 


1JI224 


47 


4S 


1.2148 


1.2286 


1.2423 


1.2559 


1.2695 


1.2829 


1.2962 


1.3005 


1.8226 


48 


49 


1.2150 


1.2288 


1.2425 


1.2562 


1.2697 


1.2831 


1.2965 


1.3007 


1.8228 


40 


50 


1.2152 


1.2290 


1.2428 


1.2564 


1.2699 


1.2833 


1.2967 


1.8000 


1.8231 


SO 


51 


1.21.54 


1.2293 


1.2430 


1.2566 


1.2701 


1.2836 


1.2969 


1.3101 


1.3288 


51 


52 


1.2157 


1.2295 


1.2432 


1.2568 


1.2704 


1.28.18 


1.2971 


1.3104 


1.8285 


52 


53 


1.2159 


1.2297 


1.2434 


1.2.571 


1.2706 


1.2840 


1.2973 


1.3106 


1.3237 


53 


54 


1.2161 


1.2299 


1.2437 


1.2573 


1.2708 


1.2842 


1.2976 


1.3108 


1.3280 


54 


55 


1.2164 


1.2302 


1.2439 


1.2575 


1.2710 


1.2845 


1.2978 


1.3110 


1.3242 


55 


50 


1.2166 


1.2304 


1.2441 


1.2577 


1.2713 


1.2847 


1.2980 


1.3112 


1.3244 


56 


57 


1.2168 


1.2306 


1.2443 


1.2580 


1.2715 


1.2849 


1.2982 


1.8115 


1.8246 


67 


58 


1.2171 


1.2309 


1.2446 


1.2582 


1.2717 


1.2851 


1.2985 


1.8117 


1.8M8 


68 


59 


1.2173 


1.2311 


1.2448 


1.2584 


1.2719 


1.2864 


1.2987 


1.3110 


1.8860 


50 


60 


1.2175 


1.2313 


1.2450 


1.2586 


1.27-22 


1.2856 


1.2989 


1.3121 


1.8259 


00 



OBOHBTRICAI. PROBLEMS. 93 



94 



HIP AND JACK RAFTERS. 



Lengrtlis and Bevels of Hip and Jack Rafters. 

The lines ab and be in Fig. 89 represent the walls at the angle 
of a building; be is the seat of the hip-rafter, and (jf of a jack-rafter. 
Draw eh at right angles to be, and make it equal to the rise of the 
roof; join b and 7^, and hb will be the length of the hip-rafter. 
Through e draw di at right angles to be. Upon b, with the radius 
bh, describe the arc hiy cutting di in L Join b and i, and extend nf 




to meet bi in.; ; then r/j will be the length of the jack-rafter. The 
length of each jack-rafter is found in the same manner, — by ex- 
tending its seat to cut the line ht. From/ draw yik at right angles 
to /r/, also fl at right angles to be. Makefk equal to fl by the arc 
Ik, or make u^' equal to (ij by the arc./A-V then the angle at J will be 
the top bevel of the jack-rafters, and the one at h- the down bevel. 

Backhu/ of the hip-rnftoy. At any conv(Miient place in be (Fig. 
8i)), as o, draw mn at right angles to be. From o describe a circle, 
tangent to bh, cutting be in s. Join m and h and n and b ; then 
these lines will form at s the proper angle for bevelling the top of 
the hip-rafter. 



TRIGONOMETRY. 95 



TRIGONOMETR7. 

ot the purpose of the author to teach the use of trigonom- 
^hat it is; but, for the benefit of those readers who have 
icquired a knowledge of this science, the following con- 
formulas, and tables of natural sines and tangents, have 
erted. To those who know how to apply these trigono- 
iinctions, they will often be found of great convenience 

ty. 

tables are taken from Searle's "Field Engineering," John 
Sons, publishers, by permission. 



96 



T&IGONUMETRIC i'X>UMUlJLS. 



Tkioosomktbic FtTscnonL 
ljetA(Fig. lOT) = BJoglo BAC = mre Br^ajid let the radius Af— AB = 
We then hATe 



dii.f 


= DC 


eos^ 


= AC 


tan^ 


= DF 


txAA 


^HO 


wocA 


^AD 


eosee A 


= AG 


Tenia -4 


= CF= BE 


covers^ 


= i;;.: = i.x 


exsec A 


= i?Z> 


cuerstx! -4 


= BG 


chord -1 


^BF 


^kOx^^A 


z=Zl=2LC 




FicKK. 



Ie tbe liglitnan.eltxl triangle ABC iTi|:. 107) 
' L-et AB = r, -4C? = ft, end ^C = o 
j We then have : 



L sin.4 

2. eo8.4 

S. tan.f 

4. col .4 

\ 8ec.4 
6t 



= — =cosi? 

c 



c 
a 
b 
b 
u 

f 
li 



= cotB 
= IoxlB 



.4 = - = p«*o B 
a 



c - h 
7. Ters -4 = = ch^v^ts B 

c 

c - h 
R. cxaeo .4 =t js ('MeTStv B 

^ 



0, coT««r«^ 



r - rt 



:- vorsin B 



10. omtxftHi.l .. - «»\mhW? 



u 



«U iiriM^ 



It a =-.ctinA = hUnA 

li. b =: ccosA = acot^ 

ah 
a.a .4 c*XiA 

H, o =ccos^ = 6cc>t^ 

i:v 6 =3 c sin J7 r= a tan ^ 

-- ah 

cvuj B t^u B 

17. a =3 «' ^c -r 6» kc — ~6r 



TRIGONOMETRIC FORMULAS. 



9: 



Boixmov OF Oiiu^uB Trumo: 




Fio. 10& 



GIVEN. 



23 



A,B,a 



23 



84 

as 

2G 



tit 



28 
29 

81 



as 



-A. a, 6 



C,a,6 



a,b,c 



souoar. 



C, 6, c 



-B, C, c 



-<<,;&, O.a 






Foiann.«. 



' Bin ^ 



c = -r - - sin (^ 4- B) 



Rill ^ = -  . 6, 
a 



O=180«»-U4-P), 



-T . sin C. 



area 



area 



sin A 
tanHU-J3)="-^^tanHU + P) 

K^y^abelnC. 



cos 



be 



«^=/n7^'«-H^yvs 



.mA = . — ; 

be 



vers A = 



2 Cf - fc) (a —^c) 
6c 



J: = ♦'a (« - a> (a — b) (j — c) 
a* sin B.tdn C 



K:^ 



»B1U ^ 



98 TRIGONOMETRIC FORMULAS. 



GENERAL PORHULA. 



34 sin ^ = = 4/ 1 — cos^ A = tan A cos A 
comic A 

35 sin ^ = 2 sin J^ A cos l^A = vers ^ cot J4 -4 



36 sin^ = |/ levers 2 -4 = f/j^d'— co8'2\4) 

1 



37 cos ^ = = V 1 — sina A = cot ^ sin A 

BOO ^ 

as cos ^ = 1 - vers ^ = 2 cos^ Y^A — l = 1—2 sin« ^ ^ 
cos^ = cos» 14 ^ — Bina 14 ^ = i^ 34"+>.i co8"2^ 

40 t;in^l = -;- ^ ?^" ^ = ^"i^c^'A—l 
cot ^ cos A 

y cos-* ^ cos^ l+cos2^ 

^ - . . 1 — cos 2 ^ vers 2 A ^ .. w ^ 

42 tan -4 = - . = — ^ — -— - = exsec -4. cot JiS jl 

sin 2 ^ sm 2 -4 '^ 

^« i. J 1 cos A , r— : T 

« '^'^ = tSn = Bn-3 = ♦'c«»ec'^-l 

44 cot u4 = - -as ss ' - . 

1 — COS 2 A vers 2 ^ sin 2 ^ 

45 cot ^ = —  , 

40 vers -4 = 1— cos A t= sin -4 tan ^ ^4 = 2 sin* ^ j1 
47 vers A — e::r*c A cos A 



48 



40 



exsec A = sec -4 — 1 = tan A tan X^A — — — .- 

^* cos A 



. .. . /l — cos A / 

smH^ = i/ 2 = i/- 



vers -4 



2 

BO Kin 2 A — 2r.Iny(cos-4 

kt 1 y ^ /l + COS -<^ 

Bl cosj.^^ =1 i/' 2 ' 

53 cos 2 ^ = 2 cos« A — 1 = ccs'^ A — Bin* >i m 1 ^tMn*^ 



TRIGONOMETRIC FORMULAS. 09 

1 



General Fobmula. 



tan A J u A 1 — cos^_^ /l — cos ^2 



2^ = 



2 tan A 
1 — tan»-4 



. _ sin A l_-f coSj4 1 

^ ~ vers A ~ sin -4 "~ cosec A — cot ^ 



« ^ = — :; 



lH^ = 



2cot^ 

J<^ vers ^ 1 — cos A 



1+*^1 — ^vers^ 2+ V2(l4-cos']4) 
i2A=:2 Bin* -4 

,, . 1 — cos^ 

(1 + cos ^) + V;si (1 4- cos ^) 

2 tana ^ 



3C2 A = 



1 - tan« A 
iA ± B) = ^nA. cos P ± sin P . cos ji 
(-4 ± P) = cos A . cos J? 7 sin ^ . sin ^ 

4 4- sin P = 2sin J^(4 + P)cos^(^ — B) 
4 — sin B = 2 cos ^ M + B) sin ^ (^ — S) 
^ -f cos B = 2 cos Ji^ (-4 H- 5) cos JiS (^ — -B) 
jB — cos ^ = 2 sin H (^ + J?) sin Ji^ U — B) 

A — sin« P = cos» B — cos« A = sin (^ + B) sin (^ — B) 
' ^ — 8in« J5 = cos (^ 4- B) cos (-4 — B) 

' COS ^ . COS B 



COS^.COSB 



J 



NATURAL SINES AND COSINES. 



101 



m 


6 


!. 1 


6«» 


7 





8* 1 


9 


1 


9 


9 


Sine Cosin 


Sine Cosin 
71045?'. 99462 


Sine 


Cosin 


Sine 


Cosin 


Sine Cosin 


"o ToKTior.owioi 


.12187 


.99255 


7l3J)17 


.99027 


715643 ".iW760 60 


1 !.0874'> .996171!. 104831 


.99440 


.12216 


.99251 


.13946 


.99023 


.15(572 .98764' 59 


2J.0H774 


.99014 l.ia511 


.99446! 


.12245 


.90218 


.13075 


.90019 


'.15701 .987(50: 5M 


8 ! .0H80.-) 


.99C12 1.10540 


.9944:3 


.12274 


.99214; 


.14004 


.99015 


1.157:301.98755; 57 


4'.0«831 


.99609 1.10560 


.99410 


.12302 


.99210 


.14033 


.99011 


i.l57.':8 .98751: 50 


5;.0K8C0 


.99607 


.10597 


.99437 


.12331 


.992371 


.14001 


.99000 


.15787 


.98740. So 


6 


.08889 


.99604 


.10626 


.99434 1. 12360 1.90283 


.14000 


.99002 


; .15816 


.98741 1 64 


7 


.08918 


99602 


.10055 


.99431 .123891.992:30 


.14119 


.08908 


1.15845 


.987371 53 


8 .0K&17 


.99599 


.10G84 


.99428 .121181.99220 


.14143 


.98991 


1.15873 


.98732: 52 


9 ; .0»^C .99596 


.10718 


.90124 .12447' 


.992221 


.14177 


.989CU 


;. 15002 


.987281 51 


10 - .09005 .99594 

1 


.10742 


.99421 


.12176 


.99219 


.14205 


.98980 


 .15931 


.98?23; 60 


11 .09034 .90591 


.10771 


.99418' 


.12501 


.90215 


.14234 


.98982 


1 .15959 


.98718! 49 


lSi.0(K)&3 .99588' 


.10800 


.99415; 


.125331.99211! 


.14i>63 


.98078 


.is'jHy 


.98714' 48 


18 ; .00092 .99586 


.10829 


.994121 


.12662 


.902081 


.14292 


.98973 


.16017 


.98700, 47 


14  .09121 .99583 1.10858 


.994091 


.12591 


.90204! 


.14320 


.98900 


. .16040 


.98701140 


15 .09150 .99580 1.10887 


.994061 


.12620 .99200 


.14349 


.98965 


.16074 


.98700) 45 


16 .09179 .99578 


1.10916 


.994021 


.12649 .99107 


.14378 


.08961 


.16103 


.98695! 44 


17  .09308 .99575 


1.10945 


.09309. 


.12(}78 .90103 


.14407 


.98957 


: .16132 


.96690 43 


16 ' .09287 .905?2 


1.10973 


.99306: 


.12706 


.99180! 


.144:36 


.98953 


' .16160 


.986801 42 


19 : .09306 .99570 


;. 11003 .993JW 1 


.12735 


.99186 


.144(54 


.96948 


• .10189 


.98681 1 41 


20 .09295 


.99567 


.11031 


.99390 


.12764 


.99182 


.14493 


.98911 


j .16218 


.98676 40 


tl .09821 


.99564 


.11060 


.99386 


.12TJ)3 


.99178 


.14522 


.08940 


'.16246 


.96671 80 


22!. 09353 


.<K)5G2 


.110S9 


.99ai3i 


.12822 


.90175 


.14551 


.989:iG 


.16275 


.98607: 38 


28 .09382 .99559 


.11118 


.993801 


.12C)1 


.99171 


.145801.98931 


.16304 


.98602 87 


24 .09411 .99556 


.11117 


.99:^771 


.12880 


.90167 


.14(508 


.98927 


.16333 


.98657' 30 


261.09140 


.99553 


.11176 


.90374, 


.12008 


.99163 


.14637 


.98923 


.16361 


.986521 ») 


29 .09409 


.99551 


.11205 


.99370, 


.12037 


.99160 


.14006 


.98910 


.16390 


.98048: 84 


27 .09496 


.99518 


.11234 


.99:W7 


.12066 


.99156 


.14C05 


.98914 


.16419 


.98(543 8:3 


26 .09027 


.99545 


.11263 


.993:{1 


.12005 


.99152 


,14723 


.98910 


.16447 


.98038 82 


29 .00556 


.99542 


.11201 


.99»iO 


.1:3024 


.99148 


.14': 52 .98000 


1 .16476 


.986:3:3: 81 


80 


.09566 


.99540 


.11320 


.99357; 


.13053 


.99144 


.14781 .98902 


.16505 


.98629 80 


81 


.09614 


.99537 


.11349 


.99354' 


.13081 


.99141 


.14810 '.98897 


.16533 


.98024 20 


82 


.09612 


.995*4 


.11378 


.99a>l ; 


.13110 


.99187 


.14KJH .98803 


.165(52 


.98610, 28 


88 


.09671 


.99531 


.11407 


.99:347 


.18i:J0 


.9013:r 


.14807!. 98880 


1 .ia591 


.98(514 27 


84 


.09700 


.99528 


.11436 


.99:M41 


.1:31(58 


.90120 , 


.14800 


.98881 


1 .16020 


.98600, 20 


85 .09729 


.99526 


.11465 


.993111 


.13107 


.90125 


.14025 


.98880 


1.16048 


.98004 25 


86 


.09758 


.90523 


.11491 


.993:J7 


.13226 


.9912«> 


.1405.1 


.98876 


 .10677 


.98000,24 


87 


.09787 


.9'J520 


.11523 


.99*i4 


.13254 


.901181 


.14082 


.988n 


.16706 


.98595: 2:3 


88 ! .09816 


.99517 


.11552 


.993:31 


.1328:3 


.99114; 


.15011 


.98867 


.167:i4 .985001 22 | 


88 .09845 


.99514 


.11580 


.99327 


.1^312 


.991101 


.15010 


.98863 


.107(53 


.98585; 21 


40 


.00674 


.99511 


.11609 


.99324! 


.18341 


.99100 


.16069 


.98858 


.16792 


.98580,20 


41 


.00006 


.09508 


.116^ 


.99390' 


.ia370 


.99102 


.16097 


.96854 


.16820 


.98575 10 


42 


.00932 


.99506 


.11667 


.99:317 


.13:jy0 


.99008 


.15120 


.98W0 


;.16W0 


.98570 18 


48 


.09961 


.99503 


.11606 


.09314 


.1^427 


.99091 


.15155 


.98845 


.16878 


.98565 17 


44 


.00990 


.99500 


.11725 


.99310' 


.l&4.-iC 


.990911 


.16184 


.96841 


.16006 .98501, 10 


45 


.10019 


.99497 


.11751 


•993071 


.18485 


.99087 


.15212 


.988:20 


.160351.98556, 15 


46 


.10048 


.99494 


.11783 


.99333' 


.ia514 


.90083 


.15241 


.988:^2 


;. 16964 !.9&'}5ll 14 


47 


.10077 


.9(M91 


.11812 


.09300 


.135:3 


.O'joro 


.15270 


.98827 


.10002 :.98,>161 13 


46 


.10106 


.99488 


.11840 


.99207; 


.ia')72 


.90075 


.152001.98823 


.170211.985-111 1:J 


49 


.iai33 


.99485 


.11860 


.992a3 


.I3c;)0 


.90071 


.15327 .98818 


i.i7a"'.0j.985;:(> 11 


50 


.li)lti4 


.99482 


.11898 


.09290. 


.13029 


.91H)G7 


.15350 


.98814 


.17078 .98531 10 


51 


.10192 


.9MTD 


.11927 


.99286' 


.136^9 


.900^3 


.in3R5 


.98809 


.17107 .98526 9 


52 


.loe-Ji 


.9^170 


.HOW 


.99283' 


.130 17 


.oixno; 


.ir>iii 


.9W'()5 


 .171 :m .98521 S 


53 


.lieso 


.99473 


.li985 


.992791 


.1:3710 


.OO(XV) 


.15412 


.98800 


; .17101 


.98510 


r* 


54 


.i.WTO 


.99470 


!. 12014 


.99276' 


.13711 


.9<K)51 


.15-171 


.98706 


: .17103 


.9a')11 


6 


55 


.10908 


.99467 


1.12013 


.992^2 


.137?:^ 


.90017 


.15.'i<-)0'. 98701 


.17222 


.98500: 5 


66 


.10337 


.994^^1 


..12071 


.99209: 


.13802 


.00013: 


.15.V>0 .98787 


.172.-)0 


.98501 


4 


57 


1086C 


.9!»W1 


.12100 


.99265 


.i:38:u 


.9;)i):iol 


.15:.57 .98782 


. 17279 i. 9849(5 


3 


56 


.10695 .99158 


; .12129 


.992(J2 


.138<X) 


.aKlT)! 


.\X,:m .98778 


:. 17308'. 98101 


2 


50 


.10121 .99455 


.12158 


.99258 


.13KS0 


.95K):Jl 1 


.l.':(515'.9877;J 


.17a-«5!. 98480 


1 


60 


.10168 


.9M52 


i .12187 

;Co6ln 

1   


.99255 
Bine ; 


.13017 
Cosin 


.90027 
"Sine" 


.1501):. 98709 


,.17305 1.98481 

1 1 — . . . 


JO 


/ 


Oorin 


Bine 


Cosin 

a: 


Sine 


1 Cosin 


Sino 




84* 


1 88* 


8 


2» 


V* 


8( 



NATURAL SINES AND COSINES. 



103 



— 1 



I 



16« 





1 

2 
3 

4 
5 
G 
7 
8 
U 
10 

11 
12 
i:) 
14 
15 
10 
17 
18 
19 
20 

21 
22 
23 
21 
25 
20 
27 
28 
29 
30 

31 

a2 

33 
31 

35 
I 36 
I 37 

3^ 
• :iJ 

- 40 

 

: '» 

. 42 
. 43 

- 41 

! -15 

' 47 
48 
t.) 
Tit) I 



ie« 



J7« 



18< 



Sine 



SlnQ j Cmnn Slne^ i Cosln 

.ai882^ 9659:3 '."27l)« .9612fi ; .29237 

.2W10.98585^ .27592 .96118 .29^*fi5 

.25938.90578 .27620 .9(5110 .29293 

. 25966 . W r)70 . 27648 . JWl 02 ' . 29321 

.25991 .f^'j«'^  .27676 .960!)4 .298-18 
1-26022 .SMK65 .2770t .96086 .at)376 

.26050 .96547: .27731 .96078; .29404 

.26079 .96510 .27759 •96070. .29432 
' .26107 .96532, .27787 .96062' .29460 

.aJ135 .{MW24 '1.27815 .96054:' .29487 
;. 26163 .96517 1.. 27813 .96016j .29515 

' .26191 .96.509 ' .27871 .96037 1 .29543 

26219 .96502 .27899 .96029' .29571 

: .26247 .9(^94 .27937' .96021 1 .29599 .9r)519; .31261 

.26275 .96186 .27955 .96013 1 . 29620;. 93511 j .31289 

.26303 ■.96179, .27983.96005 .29651 .95502 .81316 

.29682.9.>493i .31344 

.29710 .95485 1 .81372 

.29737 '.95476; .31399 



Cosin ' Sine 

.95(»j' ^30902 

.95623 .301)29 

.95613 .80957 

.956(X; .:^^)^H5 

.9559(> .81013 

.(»5588 .31040 

.95579 .310(58 

.95571, .31095 

.9.")562! .81123 

.95554 .81151 



.95545 
.95536 



.31178 
.31206 



. 9-3528 ! .81233 



. .26331 ; .96471 1 .28011 ' .95997 
.26359. 96163 1.280391.95989 
.263871. 9(hl56 i .28067 1.95981 



i .26415 .96148 
.26443 1.9(»40 

.26471 1. 96188 
.26500 .9<U25 
.26528 '.9(>117 
.26556' 96410 
.265&tl.96402 

.266681.96:379 
.26696,. 96371 
.26?^:. 96363 

.26752'. 96353 
.20780 1.96*47 
.26808 >. 96340, 
.2G836;.06332i 
.26861 .96321' 
.20892 1. 96:316 
.26990,. 96308 
.2G948'.9'J:301 
.»J976 .9<5293 
.27001;'. 96283 



.2T03S'.9fl27? 
.2ro(K)'.iMW69, 
.270681.06261' 
.2nl6i. 96253; 
.271441.96216; 
.271721.962381 
.25200. 9C333: 
.27228 .9(5222, 
.2723(5 .1K;->14 
.2<)»lj. 96206 

•M 1.27812'. 96198 

-■> 



. 28095 ' . 95973 . 297(5.-) ' . 95-107 



.28123,. 95964 

.28I50L959.56 
.28178 .95948 
.28206 -.95940 
.282:31 .959:31 
.28263 .95923 
.28390 .95915 
.28:318;. 95907 
.283461.95898 
.28374 .95890 
.28402 05882 

.28429 '.05874 



.29793 ,.95459 

.29821 '.95450 

.29849 .95441 

.2J)876 .0.5433 

.29904 .95421 
; .209:33 .95415 
. .29960; .95107 ' .31630 
1.2993/1.953031;. 31 648 



.31437 
.81454 



' .81482 
'■ .31510 
I .81537 
I .31505 
i .81593 



.30015 
I .80048 
; .30071 



.9.');i89i.. 31675 
.0.->:380; .8170) 
.95872, .81730 



.28485;. 95857 
.28513 ,.95849 
.28511 .ft5841 
.28569 !.9.'y5S3 
.285971.95821 
.28625 '.95816 
.28652 ■.95807' .SO.*^ 
.80348 



ii 



.28457 '.95805, .80120 
1.80164 
:. 80182 
1.80209 
I.80C37 
! .80305 
■.8C292 



.80098 .95363,;. 81758 



.28680,. 95799 

.287081.95791 
.28786 .95783 
.287Wi. 95774 



.95766 
.96757 
.95749 
.95740 
.95782 



1.80376 
i. 30403 
,'.30181 
1 .80159 
.80486 



.28792 

■7 
.28875 
.28903 
.28931 .95724; .80597 



.9535^4 ; .31760 
.9o345 .81813 
.95337 1 1.31841 
.95328; '.31808 
.95319 1;. 31890 
.95310'!. 31 033 
.95301 1 '.81951 
.9539:3 ; .81079 
.05281 Ij. 83006 

.9.'5275' .82034 

.95.3^3 1 .83001 
.95357 ■.33089 
.95218 ;,. 33116 
.952:0 ;.83144 



.80514 .OrrSA ' .33171 



.8a'>12 
.80570 



.28959 .95n5 
.289871.95707 



.80025 



.95333,;. £3100 
.95313 N. 33337 
.95304'i.833.>4 
.95195, ,'.33282 



I, 



.306r)3 .ailRO .32809 

2r310 .Wil90 .3:*0iu;.9r)0{)8l .:3(M>«{)i 95177 i.S.':]:)7 

.290i2 .950W| .80708 .95:08 ' .338(>4 

. 29070 . it'yCm ' . 3or;J0 1 . 951 r.O . ;33:J0:5 

.21)098 .9.757:3' . 30708 ;.Ori 150 .;33t19 

. 2912(5 . 9500 4 . 3( )791 , . iul 13 . ;334 17 

.29154.95650, .3(W19l.9'"31.8:5 .;33t74 



.V3 .'SilV:^ .9»51K3 

54 .373JK5 .9(5174 

55  .27424 .Wl&i 

56 , .27452 .9(5158 

57 ' .27480 .96160 
.'58 . .27508 .96142 
.VJ .27580 .00181 
tiU j .27561 .96126 

j'Cosin; Sine i 



.2SJ182 .9rm7\ .30p«10.95l3i .:33.";03 



.2lr309 .95639 
. 20287 !. 95(530 

Cosia|sine 
78^ 



.30874,. 951 15 .3-i.->3i) 



.3(KXW 
Osin 



9510(5 .33557 
Sine : Cosin 



r2o 



Oosin 

.9.5106 
.95(X»7 
.9.')0^«8 
.9507!» 
.9r,070 
.9s)<)(51 
.95053 
.95043 
.950:3:5 
.9.')0::4 
.95015 

.95006 
.94997 
.94988 
.949r9 
.94970 
.94961 
.91953 
.94943 
.949:3:3 
.04924 

.04915 
.94906 
.94897 
.94888 
.94878 
.948(59 
.94800 
.9-1851 
.948l.'3 
.04832 

.94823 
.94814 
.94805 
.9-1795 
.94780 
.94777 
.94708 
.94758 
.94749 
.94740 

.94730 
.94731 
.94713 
.04703 
.94093 
.04084 
.94074 
.94005 

.940-ly 

.94037 

.94037 

.94018 

.9-40t')0| 

.94509' 

.9I.7M);' 

.9:3.5801 

.9-1.571 

.91501 

.94553 

Siup 



W 



IV 



Sine 

.82557 
.33.584 
.33(513 
.83089 
.83f5<57 
..83()<M 
.83733 
.83749 
.33777 
.83Hi)4 
.338.33 



60 



Cosin 

.945.53 

.94543, 59 

.945:3:31 .58 

.91.53:3 

.94511 

.*.).4504 

.94195 

.94-1K5 

.91470 



57 
56 
55 
54 
.5:3 
53 



.91400! 51 



.9145^ 

.83H.59;. 94-4.47 
.33HS7l.944:3;< 
.82914;. 94 438 
.33{)43 .94418 
.339(59 .94409 



.94:399 
.94300 
.948S0 
.94370 
.94301 

.94351 



.33097 
.38031 
.83051 
.3:3079 
.83106 

.831.%1 
.83101 
.3:3189 
.33316 
.83244 
.83371 
.83308 
.3a']3(5 
.3:3:55:3 
.33381 

.88408 
.38130 
.a8108 
.83400 
.a*35l8 
.8:3545 
.a3573 
' .83000 
.3:3037 
.33055 

.33083 
.3:3710 
.a8787 
.3:3704 
.83793 
.88J!19 
.33816 
.3:3874 
.3:}:k)1 
.S3J39 

.330.50 

.3:3:i-v;,.91()jo 
.aio 11!. 01050 
.8ii»;ii.ni(h>o 

.84005.. 91010 
.Siir.i;j;.94(K)0 
.8413<) .93009 
.84117;.980H0 
.3 1175!. 93970 
.3 1303 j. 93000 

Cosin Siuo 



50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 



.94251 
.04245 
.94235 
.94335 
.94215 
.9430(5 
.94190 
.9418(5 
.9417(5 
.94107 



.94313' 38 
.94:3:331 37 
.043331 86 
.948131 85 
.94.8a8l 34 
.943o:r 3:3 
.943^1; 33 
.913741 31 
.94304 30 

29 
28 
27 
26 
25 
24 
2:3 
23 
21 
20 

10 

1!5 

37 
10 
15 
14 
38 
13 
11 
10 

9 
8 
7 
6 
5 
4 
3 
2 
1 




.94157 
.94147 
.94187 
.94137 
.94118 
.94108 

.oioor. 

.9108H 
.91078 
.91008 

.04058 



70^ 



NATURAL SINES AND COSINBS, 





0° 1 




Sine ICoain Sine 


-5 


OOdOO Ono. .01741> 


» 


a 






s 








Mild Ono. 1 .01B03 








Ono. .01801 








One, 








O03W 


Ono. 


•"igiS 






IXe38 


One. 








wim 


One. 






10 


aaai 


One. 


!0!0S6 


a 




00890 


.99901 


.OlOSS 


a 




.oau 








IS 








I 








'.osaa 










.oaiai 


IS 








.di9ii 


H 


17 


'.OMt 


!eoo9e 


.0224C 


IS 


IS 








B 




ioOUK 




:02M 






.00683 




.(e^ 


10 


21 








n 




ioooii 






» 


23 


.OOKIi! 






n 


*4 


.006ft 






K 


K 




'.mg! 




B 


M 


'.wai 


.00907 




U 




.OKK 


.09901 




n 




.OOftll 






n 


SO 




'.max 




n 


» 


:008T! 


.99996 


:<H618 


» 






,99096 




» 


3S 




.00906 


:o£(i7( 


IB 


33 


.OOOO 


.99e-5 


.0370; 


tr 




.OOM 


.990% 




K 


as 


.UlOll 




llBTOI 


IB 


86 




Inoos 


jOBnt 


H 


ST 


'.oimi 


.gODH 


.taw 


>S 


38 


.Olios 


.ooou 


.QBSX 


a 




.OllSl .OOQH 


.0^71 


n 


40 


.01101, OOBW 


.0SW8 


ID 




,01103' .99B9S 


.OM38 


» 






.OJOO 


IS 


43 


:oiiaii:TO3aa 


.om 


17 


44 


.01880.90903 


.0303 


It 


49 


.01300 .wool 


.030^ 


U 


46 


.01S33'. 90091 


.03033 




47 


.01837.90001 


^11! 




48 


.01396.99090 


.0314: 


It 


4« 


.tU4ra .00000 






BO 


-OMM;. 90089 


^OUO! 




51 


.m4H3.9S9l« 


.eua 


9 


m 


.01313 .ou:mo 




8 




.oi(na.9onm 


'.mi 


9 


u 


.oian.oooiw 


.03310 


9 


cm 


.oiodoI.eooK!' 


.OSMl 


8 


s» 






4 




Mivii '.mm 








.OICRT .OOOfiB 


:0S4.'!; 


 




.01719 .Booes 




1 






]0340( 





~ 


Cosln"|l5r 


Codn 




BJf 


ft 



X^ %.^>^AA« .A^ftk^S 



ri6 

174 
)03 

m 

)18 
M7 

we 



.mi9 

.09617 
.99614 
.99612 
.99600 
.99607 
.99604 
99602 
.99399 
.99596 
)05 .99594 

)84'. 90591 
)63 .99588 
)92. .09586 
121 '.99583 
l50 .99580 
179 .99578 
J08 .00575 
S87i.9g3?2 
!66 .99570: 
295 .99567 



&4 
)53 



.99564 
.90562 



82 .99359 



Ul 
MO 
109 
106 

)86 

n4 

)42 

571 

roo 

789 

r87 

?16 
374 

333 
361 
990 
)19 
>18 
377 
106 
133 
l(» 

192 
221 
230 
279 
308 
W7 
366 
)95 

m 



.90556 
.99553 
.90551 
.00548 
.99546 
.00542 
.99i^ 

.99537 
tVuiyn 
.99531 
.99528 
.99526 
.99523 
.99520 
.99317 
.90514 
.90511 

.99506 
.90503 
.00500 
.99497 
.99494 
.99401 
.09488 
.00485 
.09482 

.90479 
.09-170 
.99473 
.90170 
.90107 
.9n4<J4 
.00101 
.90458 
.99455 



168..9S452 
iin|siiie 



^6« 

Sine 

71045? 
.10482 
.10511 
.10540 
.10360 
.10597 
.10626 
.10655 
.10084 
.10713 
.10742 

.10771 

.10800 

.10829 

.10a38 

.10387 

.10916 

.10045 

.10973 

.110021 

.11031 

.11060 
.110S9 
.11118 
.11147 
.11176 
.11205 
.11234 
.11283 
.11291 
.11320 

.11349 
.11378 
.11407 
.11436 
.11465 
.11494 
.11523 
.11652 
.11580 
.11609 

.116® 
.11667 
.11696 
.11725 
.11734 
.11783 
.11812 
.11840 
.11869 
.11898 

.11027 
: .11030 
.li985 
1.12014 
.12013 
.12071 
.12100 
.121'J9 
.12138 
.12187 

Cosin 



Cosin, 

.90452: 

.99440 

.99446' 

.99443 

.99410: 

.99437 

.99434 

.99431 i 

.99428 

.99424 

.99421 

.99418 

.994151 

. 99412 i 

.99409, 

.99406 

.99402, 

.09a')9i 

.99306; 

.90303! 

.993001 

.99386; 

.99333^ 

.993H0 

.993771 

.99374 

.99370 

.99367 

.99334 

.99360 

.99357, 

.99334 

.99331! 

.90347 

.90:»4| 

.99311 

.993:^7 

.99334 

.99:i31 

.99J27 

.99324 

.99320 

.99:317 

.99314, 

.993101 

•99307 1 

.99333' 

.99300 

.99297- 

.99203 

.99290 

.99286' 

.90283 

.992791 

.99276 

.99272 

.99209 

.90205 

.00262 

.00238' 

.00255, 



Sine 

2187 
2216 
2245 
2274 
2302 

23;n| 

2360: 
2380! 
2418' 
JW47 
12476 

2504 

2533 

2562 

2501 

2620 

2&49i 

2(578 

2706 i 

2735J 

2764 

2703 

2822 
2S51 
2880 
2006 
2037 
2966 
2005 
3024 
3053 

3081 
3110 
31.'}9 
31(i8 
3107 
3226 
32.34 
328:3 
3312 
3341 

8370 
a309 
3427 
34.36 
3485 
3314 
35 J3 
a372 

3G29 

3638 
30 7 
3710 
371 1 
3773 
3802 
;38:U 
3860 
3KS9 
8017 



Sine Cosin Sino 



88< 



i| 



• I 

Cosin 

T9^i35 

.992511 

.9JfcM8 

.99-^Ui 

.99:^ 

.99237. 

.99233 

.99230 

.99226 

.992221 

.09219: 

.90215 

.00211 

.90208 

.90204 

.99200 

.99197 

.90103 

.00180' 

.00186 

.99182, 

.99178' 

.99175 

.99171 

.99167 

.99163 

.99160 

.99156 

.99162 

.99148 

.991^ 

.99141 

.99187 

.9913:3, 

.99120 ; 

.99123 

.99122 

.90118; 

.99114! 

.99110 

.99106 

.99102 
.99008 
.99004 
.00091 
.99087 
.90083 
.0:X)70 
.99075 
.90071 
.00007 

.90063 
.000.50 
.00(V)3 
.00031 
.90047 
.00013 
.00030 
.900.^3 
.90031 
.00027 



82< 



8^^ 

Sine Cosin 

.99027 
.99023 
.90019 
.99015 
.99011 
.99000 
.09002 
.98908 
.98994 
.98900 
.98086 



13017 
13046 
13073 
14004 
14033 
140G1 
14aJ0 
14110 
14143 
11177 
14205 

14234 
14263 
14292 
14320 
14349 
14378 
14407 
14436 
14464 
14493 

14622 

14351 

14580: 

14608 

14637 

14666 

14005 

14723 

14^32 

14781 

14810 
14fc3« 
14807 
14806 
14923 
14034 
14082 
16011 
15040 
16069 

15097 
16126 
16155 

16184 
16212 
15JM1 
16270 
15290 
15327 
15336 

15385 
13-n I 
13113 
16171 
15,'M)0 
13.'):>0 
15."i7 
in.")«6 

inoi5 

15613 1 
Cosin j 



?* 

^ine j Cosin 

15672} 
15701 1 
157:301 

i57r.y 

157871 

15816; 

158451 

15873 

15002 

15931 



.98962 
.98978 
.98973 
.98969 
.98965 
.98961 
.98957 
.08953 
.96948 
.98044 

.98940 

.98936!! 

.989311 

.98927ii 

.98923 

.98919 I 

.98914 

.98910 

.98906 

.989002 

.98897 
.98803 
.98880 
.98884 
.98880 
.98876 
.988n 
.96867 
.98863 
.98868 

.96854 
.98840 
.98845 
.96841 
.98836 
.98832 
.98827 
.96823 
.98818 
.98814 

.98809 
.98803 
.98800 
.98706 
.98701 
.08787 
.08782 
.98778 
.98773 
.98769 

Sine 



01* 



15959 
15C88 
16017 
16046 
16074 
16103 
16182 
16160 
16180 
16218 

16246 
16275 
16304 
16333 
16361 
16300 
16410 
1&447 
16476 
16605 

16538 
16362 
16301 
16620 
16&18 
16677 
16706 
16734 
10763 
16792 

16820 

16649 

16878 

16006 

16033 

160&1 

16002 

17021- 

17030' 

17078 

17107 
17i:36, 

1710-11 
171031 

1 r'.).>k> I 

17230! 
172701 
17308! 
173:36 i 
17863 



.98769 60 
.987&4:50 
.06700 5^.1 
.08755, 57 
.08751 ! 60 
.98746 5.3 
.96741164 
.98737,63 
.98732 62 
.98728 51 
.987^160 



45 
41 
43 
42 



.98718: 49 
.98714" 48 
.98709, 47 
.987011 46 
.98700 
.98695 
.96690 
.96686 
.96681:41 
.06676| 40 

.9667l'89 
.98667 38 
.96662! 87 
.98657 86 
.96662185 
.98648' 84 
.986431 83 
.98638! 82 



.98633 
.98629 



81 
80 



.96624 29 
.98619 28 
.986141 27 
.98609 26 
.98604^25 
.98600 24 
.06695 
.98590 
.98585 
.98580 



23 
22 

21 
20 



19 
18 
17 
16 
15 
14 



Cosin 



.98576 
.98570 
.98565 
.96561 
.96556! 
.96551 
.98546! 13 
.96541: V2 
.miZa 11 
.98531 1 10 

.98526' 9 

.985211 8 

.985161 7 

.98311 

.9a500 

.9^301 

.98496 

.98401 

.96486 

^^3481 

Sine 



80* 



NATURAL SINES AND COSINES. 



, 1 10° 


IP i| 12 




l; 


• II K- 




line CMin 


SlDS 


COBlni Kna 


CoBln! 


Bii» 


Ccnbtl' OMtn 




■ol.]786!i:«»81 


.1008: 








-K^i' Tmrao 


«> 


I ;.IT393 .<)H4T( 












91023 




2 .lT4aa.lWI71 


llMBt 






:ai553 


97431 




0701B 




3:.lTJ51.0K16a 


.loie; 




9T7W, 


.aasHO 


.97417 




97l» 




^|. 17419, .OSlOl 






97701 


.aaous 






07001 


5« i 




!llM3i 


iosissj 










96091 




0l:i7B87|:«il50 


.19253 


.88189 




:B3a85 


«7S08 




OflOW 




7 .iraw.twm 


.l!«a 


.981« 




.83008 


97801 




98980 




s!.]<391 


.lkU4l) 


.1030 


.wtiia 




.(awa 


97384 




00073 as 






.1033 






.SSTW 






90906 Gl i 


10 1 1™: 


:iKH3CI 


.lOSM 


;»8I07 






07871 




00950. » 




,0«425 




.fflSIM 


077*8 


.a»»7 












!lft4S3 




07743 








oooS ^ 


IS iiTTS; 


!S8414 


.loite 












08037:47 


Hi.rra 


.W109 






07720 




oraiB 




oooaoiM 




.BSHM 






97733, 


.33930 


07888 






lii': ?fB 




!]»S8 






.32948 






B^lti 






.IBSOO 




9mi 


.83977 






96000 43 






.liBOS 


osoai 


9770S 


.asocB 


S7B18 




oeoce 42 






.19683 


OSMfl 


97008 


.23038 


9731 




SosmUi 


so : Tsa? 




.isesa 




07003 




OTKH 




9«a87|« 


M 


.17« 






B9M4I 


OTBSe' 


.83090 


07296 




00880 30 










SHOW 


SiWO 


.23118 


97-JO 




90878:18 


2a 








03038 


ITS73 


JU14D 




















JW17S 






9affi8 


>0 














sttsa 


eran 




08861 


s 














.2U31 


9TS64 




06814 




» .lain 










.sssoo 






9683; 


S3 


W .1B1« 




Jseao 


ftWU 




.23388 








IB 


!» .1B1« 


!o9isi 


Ams 


Bra» 


)ro3fl 




071M4 




B68aS 




SOl.ltKM 


.OBsa 




97003 




.^ 


975187 




06815 


30 




.98330 




07^!! 




23Brm 


BTSBO 




06807 


(0 






.MOM 








07.a8 




BG80O 


a 


















0670E 


a 




















a> 






!i!aa7D 




)7j93 




97:3)3 






19 


30 ilKSM 


:OSiM 






oraoa: 












37 .IMM 


,98a« 






975KI 




mw 




06704 




38 .lUSI 


.uttss 






iTsn 










» 




.vserr 




07010 




'naim 








« 


■wiiiaaot 


.D&tTS 




0WS4I 


iraao 


33827 


OTia 


;»sw 






41 .18SB8 


,98OT 


.2HS0 


oT%ie 


07500 




cmos 






» 


a'.iesm 


.IM»1 


a 


07933, 


OIMS 


.S308J 


onis 






IS 


4a!.185M 




07918 


17547 


28718 


07148 


iS«i 




IT 


wi.isea 


.iwaa 


.0X00 


BTOIOI 






97141 




10 


45l.lHll3li 


.ma 


.tanu 




jmt. 












40 .U<UBI 


.vsau 






msa 














.WBM 




''^' H 


Ksi: 














.fti-UO 




9mr ffiR 
















.U«i3 


.aun .ma-A^m 












1 


a) iionc 


.turn 


.imb«».9W!b:!Sb' 












10 


M .IIBU 


.<H«13' 


.aosi!i'.inn»'!.ii»fl)' 














U .IBSSJ 


.ftt«7> 


.xa» .omn .aiWK 




230Rt 












.iiH«n; 


.aunwiLKtw .awj? 


)7WJ 


23095 










M Iwrni .wiwi 


.axtanl.BjMBi : .aaaa 


JT47B 


24033 










(B .IUBSHHW 


.3MIB..DTK1S ;.3:!Xn 














0«..I«H7 .WIMJ 


.ann? .07«k)' .a^m: 




34079 










S7 ,11WBS .BMT9; 


.90?» .imwt .awiii 


07437 


«10S 










W..lilUSi .(B174 


.»!nt..m>ST .fom 


J7450; 


91180 




.raBj 






Rt -i!io5a .mm 


.annSj.nHBi. .»H7 




M104 




:«mi 






<W 1 .IDMn .361A3 




37487' 


21103 












CoSS|-Bb.e- 


Ocali 


-ffiaiciSSi 


5^ 


Oosta 


aST 


<MB(ak. 


~ 




79- 


 "l 


.- 11' 77 




7fl 




73- 



wl 


WTW 


.aSSST 


irmr,^ 






"mbstI 
































































































































itai Stool 


JoBlDJ Sine 1 Oslul 61na iCoalnl Sio' [;< 


















• ll 



106 



NATURAL SINES AND COSINES. 




1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 



80» 



Sine Cosin 



.50000 
.50025 
.50050 
.5007G 
.50101 
.50126 
.5()151 
.50176 
.50201 
..^)0>37 
.50252 

.50277 

.G0a>7 
.50352 
.50377 
.50403 
.5042S 
.60453 
.6W78 
20 .60503 



21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

11 
12 
43 

41 
45 
46 
47 
48 
49 
50 

51 
ryi 
5:J 
54 
55 
56 
57 
58 
59 
60 



sr 



Sine ' Cosin 



82« 



88« 



.60528 
.50563 
.60578 
.60603 
.60628 
.50651 
.60679 
.60704 
.50729 
.50754 

.50779 
.60804 
.50629 
.50864 
.60879 
.50904 
.50929 
.50954 
.50979 
.51004 

.51029 

.51054 

.510?J 

.51104 

.51129 

.51164 

.61179 

.61204' 

.61229 

.61;354i 



.86588 
.80573 
.80559 
.86544 
.80530 
.80515 
.86501 
.86486 
.P6471 

.m^7 

.80442 
.80-127 
.8(J413 
.86393 
.86384 
.8(>369 
.86354 
.86340 
.8(J325 
.86810 

86205 
.86281 
.86266 
.8(5251 
.86237 
.80222 
.86207 
.86192 
.80178 
.86163 

.86148 

.80133 

.86119 

.86104 

.860891 

.86074' 

.86059 

.8(5045 

.8(5030 

.80015, 

.86000' 

.K-)1)S5 

.K->970 

.K5956 

.K>D41 

.8")926 

.85911 

.K'>896 

.F.")S81 

.83806 



.61279 

.5l;)04 

.51829 

.51854 

.51879 

.51404 

.51429 

.514r>4 

.61479, 

.51504 



.85851 
.r...iW 

.a">so6 

.R'>792 

)l M . 

.a")762 
.K-)747 
.K">732 

.K->n7 



51504 
51529 
51554 
51579 
51604 
51628 
51653 
51678 
51703 
6172.i 
61753 

61778 
51803 
5182.i 
5185^ 
51877 
51902 
61927 
61952 

6i9rr 

62002 

82020 
52051 
52076 
52101 
621C6 
62151 
52175 
52200 
52225 

52275 
522LJ 
52324 
62349 
62374 
52399 
62423 
52448 
52473 
62498 

82522 
52547 
625?^ 
62597 
52621 
52646 
5:^71 
52096 
52720 
52745 

52770 
52794 
52819 
528'14 
62869 
52893 
62J)18 
62i)43 
52967 
62992 i 



.85717 
.85702 
.85687 
.85672 
.8r^(5.-)7 
.85<>42 
.a->027 
.a")612 
.^55^7 
.KV)S2 
.85567 

.85551 
.a5536 
.85521 
.85506 
.85491 
.85476 
.85461 

.85431 
.85416 

.85401 

.avivS 

.85;j70 
.85355 

.avjio 

.a-)325 

.a"),jio 

.a">294 
.K-)279 
.85264 

.ft^>249 

.a-i218' 
.85203! 
.a5188, 
.85173, 
.851571 
.851421 
.a-)1271 
.85112 1 

.85006' 
.85081 
.a-iOOO 
.a-)051 : 
.aj035 

.a^oos 

.81<)89i 

.81974 

.a4U59 



.84948 

.8Ji)28 

.8J913 

.81897 

.81882 

.84866 

.a4851| 

.81836 

.84820 

.84805 



Sine 

529!)2 

63017 

53041: 

63066 

63091 

53115 

63140 

63104 

531i:9; 

5^14 

63263 



63312 
53337 
53.361 
&3:»6 
6.3411 
5^435 
531(W 
63484 

ssm 

63.>34 
53558 
63583 
63607 
63032 

63081 
63705 
53730 

63754 
53779 
53*^)4 
63f^ 

istm 

63877 
63{)02 
63926 
63a')l 
53975 

64000 

54024 

540191 

W()73 

64097 

64122 

64146 

64171 

541 a'} 

64220 

64244 
642: ;9 
5421W 
5l:J17 
64:J42 
54306 
&1391 
M415 
&4410 
&44I54 



Cosin 

.&4805 
.a4789 
.&4774 
.&4759 
.&4743 
.&4?28 
.a4712 
.84697 
.84681 
.a4606 
.84650 

.o4635'! 

.&1019,' 

.&4604l| 

.84588 I 

.84573 ' 

.&4557 

.84542 

.84526 

.84511; 

.84495 

.84480' 

.84464 

.84448 

.84433 

.8W17 

.84 402 

.84386 

.84370, 

.84:J55 

.84339 

.84324 

.84;yiw 

.84292 

.84277 

.84J61: 

.81245 

.84^J0 

.81214 

.84198 

.8U82 

.84167 

.84151 

.84135 

.84120 

.841(V4: 

.84088 

.84072 

.84057 

.84041 

.84025 

.84009 

.a-5:^>4 

.88978,1 

.83%2;! 

.839461 

.&ia*JO! 

.83915  

.83899 

.888K;J 

.88K67 



.83676 
.83660 
.83045 
.83629 
.83613 
.83397 
.83581 
.83.365 
.88549 



Sine 'Oosin 

544<^ .83867 
64488 .88851 
54513 .83835 
545371.88819 
645611.88804 
64586 = .83788 
fr4610 .83772 
64635 .83756 
546,39 .83740 
546835.83724 
54708 .83708 

54732 
54756 
&4781 
M805 
54829 
54a>4 
64878 
64902 
M927 
54951 

64975 

54999! 

6501^4! 

53048 

53072 

53097 

55121 

65145 

65109 

55194 

56218 
55242 
55206 
652J1 
53:}15 
65:};39 
65:}03 
65388 
66-412 
55436 

65460 
56484 
65509 
6(>533 
65657 
65581 
65605 
65630 
53654 
55678 

65702 
65?^5 
66750 
65775 
65799 
66823 
65847 
558n 
53895 
65919 



84« 



Sine ICk>sln' 



— / 



' .65919 
.66013 
.559(53 
.53992 
.66016 

! .66040 
.56064 
.66088 
.66112 
.56186 
.60160 

.66184 
.66308 
.66282 
.66256 
.66280 
.66805 
.66329 
.66853 
.56377 
.66401 



.88583 
.83517 ; 
.88501 ! 
.83485 •' 
.83469 , 
.88458 
.88437. 
.83421! 
.834051 
.83389! 

.88873' 

.88356 

.88;^ 

.83324 

.83806, 

.882921 

.832761 

.83260' 

.8:5«4 

.83;iS8. 



Cosin I Sine ; Cosin bine 



fiS** 



53« 



Cosin I Slue Cosiu Sine 



.88318 

.83105 

.83179 

.88103: 

.88147, 

.88181 ! 

.88115 

.88098 

.880^2 

.83066 

.88050 

.880171! 
.88001! i 

.82{)85; 
.82960'! 
.821K38 • 
.82936 I 
.82920 I 
.82901 ; 



.60125 
.66449 
.66473 
.66497 
.66521 
.66546 
.66600 
.66593 
.66617 
.66041 

.66665 
.66689 
.66713 
.66736 
.66760 
.66784 
.66608 
.66882 
.66856 
.66880 

'.66004 
.668S8 
.66052 
.66976 

.vrooo 

.670S4 
.67047 
.6TO71 
.57003 
.67119 



67« 



6e» 



.6T148 

.5no7 

.W191 
.67i:i5 
.57388 
.67862 
.57^280 
.C?7810 
.573U 
0)7858 

Cosin Bino 



.83904 60 
.82887 60 
.82871 68 
.82855 57 
.82889 66 
.83823 65 
.83806 51 
.83790 68 
.837r3 63 
.K757i 61 
.83741 60 

.837^' 49 
.83708 48 
.83093 47 
.82075 46 
.83659 45 
.82043 44 
.82026 48 
.83610 43 
.82593 41 
.83677 40 

.82S6l'fl9 

.83544 88 

.63538 37 

.83511 86 

.63495 86 

.83478 81 

.83463 88 

.83446 as 

.8^439 81 

.8(9413 86 

.88806' SO 
.83880 38 
.88868 37 
.88847 30 
.68880 89 
.88814' 84 
.83897; 38 
.8838r83 
.88304 31 
.81^48 30 

.88881' 10 
.88814; 18 
.68198 17 
.68181 16 
.68165 15 
.68148. 14 
.63183 18 
.83116 13 
.88098 11 
.88063 10 

.880651 
.63048- 
.83033 
.68015, 
.819991 
.81983 
.61965. 8 
.61949 $ 
.81983: 1 
.81915 



NATURAL SINES AND COSINES. 



107 



85< 



Sine Cosin 



86< 



.673811 
.67405! 

.67453 

.57477; 
.67501; 
.575iJii 
8 .67548. 




1 
2 

3 
4 
6 
G 



9 



.57^-'>" 



572' 



10 . .5759G 

11 . .57619 

12 .57043 

13 .576C7 

14  .67091 

15 . .57715 
10 .57738 

17 .577(52 

18 .57786 

19 .57810 

20 .57833^ 

21 

22 

23 

24 

25' 

26 

27' 

28' 

29 

80 



biJ)i5 

81899 
81882 
81805 
81848 
81832 
81815 
81798 
81782 
817G5 
81748 



I 

81 
S2 
83 

ai 

85 

86 = 

87 J 
88 
89 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 



.578571 

.578811 

.679041 

.5TD28' 

.57052 

.57976 

.57999 

.58023 

.58047 

.58070 

.5S094! 
.C3118: 
.CJ141 
.58165; 
.CS189! 
.582121 
.58236 
.58260 
5«383 
.58307 

.58830' 

.58354 

.58378. 

.58401 

.584^ 

.5f^9 

.58472 

.68496: 

.68519) 

.58643 



51 .5656;' 

62 .6a'i00 

63 .58014 
M .68037 
55 .68661 
6«; .686&4 

57 .68708 

58 .68731 

59 .58755- 
W .68779' 

,C0Bin| 



.81731 I 

.8m4 

.81098 

.81081  

.81604'; 

.81647,' 

.81031 

.81614 

.01597 

.81580 

.81563= 

.81546 

.815::0 

.81513 

.81406 

.81479. 

.814C3 

.81443 

.r.l4C3 

.81413 

.81395 

.81.078 

.81361 

.81&44 

.81327; 

.81310 

.81293 

.81276 

.81C59 

.81242 

.81225- 

.81208 

.81191 

.81174 

.81157 

.81140 

.81123 

.81106 

.810H9 

.81072 



Sine 'CJosin 
r5sr;9 

.6SSa2 

.68826 

n. 



.a.^19 

.5c^^r3 

.668961 

.63920! 

.58943 

.58907! 

.68990 

.69014 

.59037 
.69061 
.59084 
.50108 
.50131 
.59154 
.59178 
.59201 
' .59225 
j .59248 

.59272 
'.59295 
' .59318 
.50342 
.59305 
■.59389 
.694121 
.594361 
.59159' 
.594831 



.8iWS5 
.80SC7 
.WS'jO 
.8i>S33 
.r<fcjil6 
.fX)?J9 
.80782 
.80765 
.60748 
.80730 

.80718' 

.80096' 

.80679 

.80002 

.800^4 

.80027 

.80010 



87* 



88< 



'■1 



.59506' 
' .5a"'>29. 
.69552 
.69576 
.59599 
.59022' 
.690461 

.no(;93 

.69716 

.59739 
 .6JTG3' 

.wm 

.5I,'**09 

.6JS32, 

.6'.>S56 

.5:)S79 

.6iHK>2 

..'):i!»26 

.69949 



.81055 


.59072 


.81(»38 


.6;»(K>5 


.81021 


.0«i!l9 


.81004 


.&x>i-« 


.8IKI87 


.600«J5 


.80;»70 


.eoosG 


.80953 


.60112 


.809'36 


.001:35 


.80919 


.601.58 


.809(» 


.60182 


Sine 


Cosin 



.sa^ro 

.8055S 

.8a>ii 

.805:w4 
.805U7 
.SO4H0 
.80472 
.8<>455 
.804.-i3 
.804::X) 
.80403 
.80386 

.8^ 

.803.:4 
.80316 
.80203 
.80Z-2 
.80204 
.80247 
.80230 
.80212 

.80195 

.80178 
.80160 
.8.)143 
.&.)125 
.80108 

.8*K)r3 

.8iwi5(; 
.800;id 

.80021 
.TlO-0 

.7;mk« 

,.7IKi.")l 

.70910 
.79J-:»9 
.7fKsl 

Bine 



Sine Cosin 

7G0182 /40.-'O4 
.GtKN)5 
.(i0228 
.00251 
.aK>74 .7 
.G(VJii6 .7 



Sine 'Cosin 



89< 



7' 



r9846 

rosu 

07ii3 
51776 



I . 



.6(ui44 , 
.00307, 
.0a390 
.60414: 

.60437 

.60400 

.60483 

.00506' 

.C(K529 

.60553 

.C057C 

.cav.:9 

.G0G22 
.60645 

.60008 
.t<>>91 
.C0714 

.ror:>M 

.CUTOl 
.607H4 
.00807 

00.^)3 1 
.60876 

.60809 
.60028 
.60945 
.C09C8' 

.ccooi i 

C1015I 
OIOCS' 
ClOCl ' 
6iaS4' 
011C7 



.79758 
,79741 
.79723 
.70700 
.79688 

.7%n 

.79053 
.70035 
.70018 
.70000 
.70583 
.70505 
.70.>I7 

.70512 

.79i04 

"•f • — 

.7W.')0 
.70441 
.70iC4 

.70-:c;o 

. <  " »yO 

.79^35. 
.70318' 

.7w:a) 

.7120.1 

.7r.~Mr 

.7f211 
.7010.'} 

.7oi';o 

.70156 



.oiyi4 

.6l:«7 
.613001 

.01 nw; 

.ou-,>o 

.01 t.>i 
.01 574  
.CI 107 : 
.01520 
.6l.'>43" 



.70105 

.70(:-7 

.70<.»f;o 

.79a"i! 
.700:W 
.70010 
.7M('. S 
.TbOJS) 

.78962 
.7'-0-!l 
.7vr:(j 

.7^^:>i 

.7.'^^73 
.7>*-.')5 
.7»>:i7 
.7^19 
.7*W01 



.615<;0 

.61580. 

.61612" 

.616:i5 

.61058 

.61681 

.61704 

.61720  

.01740 

.617?J 

.61705 

.61818 
.61&41 
.01804 
.61887 
.61900 
.61052 
.01055 
.01078 
.020CJ1 
.02024 

.02040 
.02(X;0 
.G2(H0 
.02115 
.62138 
.02100 
.0218:3 
.02200 

.02251 

.62274 
.02207 

. CCS 05 
.62.':88 
.02411 
.624:« 
.02-150 
.62479 



.61130 '.70140 .62502 

.6115:3 .7I.1C2 

.61170 

.61 ICO 

.01-^22 

.61 '.'-45 1 

.C1C(;8! 

.01201 



.02524 
.62547 
.62570 
.62502 
.62015 
.6vaS 
.C20<X) 



.78^■;ll 
.78783 
.7870o 
.78747 
.78720 
.78711 
.78004 
.78070 
.78058 
.780i0 
.78622 

.786^-' 
.78580 '■■ 
.78508 
.78550 
.7K532 - 
.78514 
.78490 
.7^478 : 
.7t>400 
.■:^>442  

. 784^4 
.784a") 
.;-8.387 
.78:>;0 
.78351 
.7833:3 
.78315 
.78297 
.78279 
.78261 

.78243 
.7^5225 
.78C06 
.78188 
.78170 
.78152 
.7«134 
.78110 
.78008 
.78U79 

.78061 
.7h04:3 
.7H)25 
.78007' 
.77988 
'"""070 . 



1 ( 

.77052 
.779:34 

G20S:J .7701'; 

62700 .77697 



64" 



63- 



Cosin ; Sine 
62« 



.65728 

. J.. . I I 
.02?'J«j 
.C2«10 
.02S-I-2 

.o-.i^;i 

.62KS7 

.020.32 
Cobiu 



.77879 
.77h;i 
.'.-7X13 
.7:.'<01 

. < < I T 
. < I <0.l 
. t ^ i.^1 

.(it •/•! 

.III 10 

Sine 



Sine Cosin 

.62055 
.62077 

.om'O 
.6:3(n:2 

.0C0-i5 

.63000 
.63113 
.0:3125 
.63158 



.77715, eo 
.7701.0 £0 

. 770781 £8 
57 
50 



.63180 
.0:3203 
.C3225 
.C3248 
.03271 
.63203 
.63310 
, .63338 
.63001 
.63383 

.63400 
.634-8 
.63451 
.0*473 
.03400 
.03518 
.035-10 
.0350:i 
.63585 
.03a'8 

.63630' 

.0:3053 

.03075 

.G30i!8 

.0:3720 

.63742 

.63705 

.C3787 

.CO.'^IO 

.63832 

.essw' 

.0:3677 
.C3809 
.03022 
.0::044 
, .0:3900 
.C:3G60 
.(■4011 
.(.10:33 
.04050 

.64078 
.04100 
.61123 
.041*5 
.64167 
. 6419*3 
.64212 
.642:31 
.642.V) 
.61270 

Cosin 



.'iTOOO' 
.77041 ' 
. 77023 i 
.77005' 
.77586 
.775i'»8 
.77550 
.77531 

.77513 
.77494 
.77476 
.77458 
.77439 
.77421144 , 
.77402 1 43 ' 
.77384' 42 
.77306 41 
.77347: 40 



65 
54 

.•il 

60 

40 
48 
47 
40 
45 



.77329 
.77310 
.77202 
.77273 
.77255 



.77230 
.77218 
.77100 
.77181 
.77162 

.7714-4 
.77125 
.77107 

.77070 
.77051 
.77a0:3 
.77014 
.7CO-0 
.70077 



.7CO,'50 
.70040 
.7-0021; 
.7000:3 

.':os>4 

.70^:1} 

.70.^1:. 

.7'<>e8 

.r-.^i.j 

.70702 

I -, S^ I 

. ((:i <•; 

r . ■""• i 

. I Jt •' I 

.7()735 
.7()?17 
.7(''50.S 

.■;';<;:o 
.70001; 

.70042' 

.7002:J 

.70(Xi4 

Sine 



39 

33  
37 i 
30 
3-3 i 
34 
33 

O.) 

31 i 

30 ' 

29 I 

28 : 

20 ! 
25 : 

24 
23 

21 . 

SiO : 

19 

IS  

17 

10 

!.-> 

11 

1:3 

li 

1! 








5 

4: 



1 




51' 



60° 



NATURAL TANGENTS AND COTANGENTS. 





4- 


8- 


8° 


7- 


Ta^ 


rang 


Cotang 


Tflng_ 


Cotang 


Tang lOo 


J 




B749 


iIImis 


:iS 


IS 


:!SI 


s* 




.07051 










.123B8 


b! 


s 






iiisiea 




a'.issa 




8.1 




:o7iio ; 


)S8«« 


11.3789' 




0.40904 


;]£G97 


B.I 












0.88307 


.IZ^ 










iiiaois 






.is;:8 




7 








; 0718 








8 




06983 


ii!iaifl 






:i3515 


t': 


fl 


.0^56 


K»18 




; 0J76 






?.■ 










.10606 


oiassw 


;i267* 




11 




own 


11. 0837 




9.S80I6 


.13808 


7. 


IS 


,07314 


WlOl 


10.0882 




B.OAiie 


.12i:C3 




13 


.07273 




10.CM9 




9.1S028 
















9.16651 


! 12692 


7" 




!o74ai 


09189 






B.18093 


.isras 


7. 


IG 


.07 Ul 


KSIS 


10.6183 


.0981 


9,IOW0 


.12^1 




17 


.07 30 




10.8139 


. 1011 


























!07.>J8 








0!03370 


!i^o 


7. 


sa 


.07 a 1 


09336 


10:711B 


: 1090 


0.00063 


.12889 




ai 


.07(107 ; 


(^65 


J0.6T8S 


.11138 


.98698 


.12809 
















. 2as9 






:07effi 1 




lOloilB 


:ilJ8T 




.2908 


7- 


SI 


.07005 




I0.6;S9 


.11217 




.20t8 


7. 


25 


.OTTi* 




10.6462 






. 8017 


T. 




















:o77aa 




I0:4U13 


illEfflS 


;81551 


.'SOTO 


7.- 


sa 


.07818 


0057U 


]0,4i3I 


.11335 


.83^53 


. 3106 


7. 


so 


.OTSII 


ocaoo 








. 8136 


7.' 






09620 


lOisSM 


;ilB94 


:77«W 


.ISWB 




SI 


.07800 


09668 


'*'-^ 


.11423 


.nta 


. 8196 


7. 


CJ 




X!038 








. 8324 












J1482 




:a 






ioToaz 


K)T-I6 


loiaooa 




isSTOl 




E5 


.OS017 


M7r« 


10.2294 




.86483 


. SS13 


7. 


K6 


.OSOlfl 




10.1963 






. 8848 


7. 


















Kl 




05S04 


loiinai 


:il829 








m 


.osiai 


T.^ 


IO.108O 


.iie.'-.o 


.67718 


. W33 






.08103 






.nm 




.3461 


7. 






19952 


lo.was 


.117I8 








43 


.08321 








.tiacB 








.08251 








.49188 


:i8560 


7. 






10010 


OioOOOT 


:iIS06 




.18680 








lOOCO 


9.03101 


.IIBSO 


:44Foa 






40 










.4£;95 


.')8«3a 




47 










.40705 


.18689 


T.. 


IS 




101 ns 


0!k14S3 




.SEcas 


.13808 


7.. 


4» 


!o81CT 


ioii;7 


B.t.in4i 


;iir54 


.SKM 


.ISiSB 




00 


.08406 


10218 


B.;8S17 


.11963 


8.34496 


.1K58 






.08485 


I0SJ8 




.laoj!. 


B.aE446 


.I3JB7 






.OKU 




9:rji7 


.13)42 


B.B&:06 






63 






9-71_H41 




S.atffiTB 








losses 












7.: 






10363 




! 13131 






T. 




'.mm 


03!a 


oioL-aw 


.laico 


BisSftM 


.'l808S 






.aim 


0422 


B.RllOO 




s.soura 


.18008 




68 










8.18370 


.18906 


7. 


69 










e.I«3B8 


.14004 


7.' 


60 


!oer40 
Cotang TmiE | C 


0310 
otang 


B:ai-i38 


:i8S78 


8.14486 


.14064 


7. 


TanB 


c«»¥ 


Tang 


Cotang 


~T 


8S' 1. 


84° 


88' 


.W 



NATURAL TANGENTS AND COTANGENTS. 



ni 



8» 


90 


10*» 


11"» 


60 


'smg 


Cotanff 


Tang 
.158:38 


Cotang 


Tang 
.17633 


Cotang 


Tang 
.19438 


Cotang 


14034 


7.11537 


0.31375 


5.67128 


5.14455 


14061 


7.10038 


.15803 


6.30189 


.17003 


5.66165 


.19468 


5.13658 


59 


14118 


7.08540 


.1589.J 


6.29007 


.17093 


6.65205 


.19493 


5.12862 


58 


:414d 7.07059 


.15928 


6.27829 


.17723 


5.64248 


.19329 


5.12069 


57 


4173 7.03379 


.15950 


6.26655 


.17753 


6.63295 


.19559 


5.11279 


56 


43(Xb 


7.04105 


.15938 


6.25486 


.irr83 


5.62344 


.19589 


5.10490 


55 


4232 


7.02a'J7 , 


.16C17 


6.24321 


.17813 


5.61397 


.19619 


5.09704 


54 


4262 


0.91174 


.16047 


0.23160 


.17843 


5.60452 


.19649 


5.08921 


53 


4291 


6.99718 


i .icor7 


0.22003 


.17873 


5.59511 


.19080 


5.08139 


52 


4321 


6.98368 


.10107 


C.20&51 


.17903 


5.58573 


.19710 


5.07360 


51 


4351 


0.96823 


.16137 


6.19703 


.17933 


5.57638 


.19740 


5.06584 


50 


4881 


6.93385 


.161C7 


6.18559 


.17963 


5.56706 


.19770 


5.05809 


49 


4-110 


6.93953 


•IClCj 6.17419 


.17993 


5.55r<7 


.l9;;oi 


5.03037 148 


4440 


C.925?o 


.16220 6.1G283 


.18023 


5.51831 


.19831 


5.042G7 


47 


4170 


6.91104 


.10250 


0.13151 1 


.18053 


5.53927 


.19vS01 


5.03499 


46 


4499 


6.89688 


.16230 


0.14023 


.18033 


5.53007 


.19891 


5.02731 


45 


4529 


6.83278 


.16316 


6.12309 


.18113 


5.52090 


.19921 


5.01971 


44 


4539 6.8G3T4 


.ICOiG 


6.11779 


.18143 


5.51176 


.19932 


5.01210 


43 


1588 


6.83475 1 


.iGc;o 


6.10004 


.18173 


5.60264 


.199^13 


5.00151 


42 


1618 


0.84083 


.16435 


6.095:2 


.18203 


5.49356 


.20012 


4.99695 


41 


1648 


6.82694 


.10435 


6.08114 


.18233 


5.4W51 


.20042 


4.98940 


40 


1678 


0.81312 


.16465 


6.07340 ' 


.182G3 


5.47548 


.20073 


4.98188 


39 


1707 


6.7JK)36 


.16495 


6.0GC10 


.18233 


5.400:8 


.20103 


4.97438 


38 


1787 


6.re5G4 


.16523 


6.C5143 


.18323 


5.45751 


.20133 


4.96G90 


37 


1767 


6.77199 


.10355 


6.04031 1 


.18333 


5.44837 


.20104 


4.95045 


36 


1796 


6.73838 


.16585 


6.02332 


.18384 


5.439G6 


.20194 


4.95201 


35 


1826 


6.74403 


.16615 


6.01878 


.18414 


5.43077 


.20224 


4.94460 


34 


1856 


6.73183 


.16645 


6.00797 1 


.18444 


5.42192 


.20254 


4.93rai 


33 


1886 


6.71789 


.10074 


5.99720 ! 


.18474 


5.41309 


.20285 


4.92084 


32 


1915 


6.70450 


.1G7J4 


5.98G40 


.18504 


5.40429 


.20315 


4.92219 


31 


1945 


6.69116 


.16731 


5.97576 : 

1 


.18334 


5.89552 


.20345 


4.91516 


30 


1975 


6.67787 


.16764 


5.9C310 


.18564 


5.38677 


.20376 


4.90785 


29 


Kxe 


6.60463 


.10794 


5.95413 


.18504 


5.37805 


.20406 


4.90056 


28 


Km 


6.65144 


.16824 


5.94000 


.18021 


5.36936 


.20436 


4.89330 


27 


3064 


6.6,ia31 


.16834 


5.93355 


.18a51 


5.36070 


.20406 


4.88005 


26 


S094 


6.62523 


.16834 


5.92283 


.180iil 


5.35206 


.20497 


4.87882 


25 


S124 


6.61219 


.10914 


5.91236 


.18714 


5.34315 


.20527 


4.87162 


24 


S153 


6.59921 


.16944 


5.90101 


.18743 


5.33487 


.20557 


4.86444 


23 


3183 


6.53627 


.16974 


5.89151 


. 18775 


5.32G31 


.20588 


4.85727 


22 


3218 


6.57339 


.170D4 


5.83111 


.18805 


5.31778 


.20018 


4.85013 


21 


3243 


6.56055 


.17033 


5.87080 


.18835 


5.30928 


.20648 


4.81300 


20 


3272 


6.54777 


.17063 


5.86051 


.18865 


5.30080 


.20679 


4.83590 


19 


3802 


6.53503 


.17033 


5.85021 


.1«895 


5.2oe:5 


.20709 


4.82882 


18 


5332 


6.52234 


.17123 


5.84001 


.18925 


5.28CJ3 


.20739 


4.82175 


17 


5362 


6.50970 


.17153 


5.82982 


.18935 


5.27553 


1 .20770 


4.81471 


16 


5391 


6.49710 


.17183 


5.819G6- 


.18930 


5.20715 


.20800 


4.80769 


15 


>121 


6.48456 


.17213 


5.80953 


.19016 


5.25880 


.208':0 


4.80068 


14 


5451 


6.47208 


.17243 


5.79944 


.19013 


5.25048 


.208G1 


4.79870 


13 


5481 


6.45961 1 


.17273 


5.7f;938 


.19070 


5.24218 


.20891 


4.78673 


12 


5511 


6.44720 


.17303 


5.779:36 


.19106 


5.23301 


.20921 


4.77978 


11 


j&lO 


6.43484 


.17333 


5.7G937 


.19130 


5.22566 


.20932 


4.ri28C 


10 


5570 


0.42253 


.17363 


5.75941 


.19166 


5.21744 


 .20982 


4.76595 


9 


5600 


6.11023 


.17393 


5.71019 


.101'.i7 


5.200-35 


.21013 


4.75900 


8 


TAW 


G.:}0y04 


.1742:3 


5.7:W0O 1 


.19227 


5.20107 


.21013 


4.75210 


7 


5060 


G.3S3H7 


.17453 


5.70074 


.19257 


5.19203 


.21073 


4. 745:) 4 


6 


3r>.S9 


6.37374 


.17433 


5.71992 


.19287 


5.18480 


.21104 


4.73851 




5719 


6.3G165 


.17513 


5.71013 


.19317 


5.17671 


.21134 


4.73170 


4 


5749 


6.34961 


.17343 


5.700:37 ! 


.19347 


5.16863 


.21104 


4.72490 


31 


3779 


6.83761 


.17573 


5.69004 


.19378 


5.16058 


.21195 


4.71813 


2| 


5:^09 


6.32566 


i .17603 


5.68094 


.19408 


5.15256 


.21225 


4.71137 


1 ' 


5838 


6.81373 


.17033 
Cotang 

1 


5.G7128 j 


.10138 
Cotang 


5.141.55 


.21236_ 
Cotang 


4.70463 


1\ 


tang 


Tang 


Tang 


Tang 


Tang 


1 
/ 


sv i 


1 SO** 1 


79° 1 


7 


B° 



NATURAL TANOEMTS AND COTANGENTS. 



NATURAL TANGENTS AND COTANOKNTe. 



113 



le- 


l7» 1 


18» 


19« 


1 

/ 

60 


Tang 
.30578 


Cotang 


Tang '• Cotang 
.83493 1 8.07768 


Tang 
.84433 


Cotang 


nb 


3.48741 


3.27085 


2.90421 


w 


3.^8850 ' 


.30606 


8.26745 


.32524 


8.07464 


.84466 


2.90147 


59 


■as 


3.4'«Tn7 ! 


.30037 


8.26406 


.82556 


3.07160 


.84496 


2.89678 


58 


-m i 3.47596 


.80GC9 


3.20067 


.32588 


3.06857 


.84530 


2.89600 


57 


100 8.47216 


.30700 


3.25729 


.82621 


3.06554 


.84563 


2.89827 


56 


138 3.46837 


.80783 


3.253P3 


.82653 


3.06253 


.84596 


2.89055 


55 


(64 


3.46458 


.30764 


3.25065 


.82G85 


3.05950 


.84628 


2.88788 


54 


m 


3.46060 


.80796 


3.24719 


.32717 


3.05649 


.84601 


2.88511 


68 


w 


3.45703 


.80828 


3.24383 


.82749 


8.05^49 


.84093 


2.88240 


63 


158 


3.45327 


.80800 


3.24049 


.82782 


3.05049 


.84726 


3.87970 


51 


190 


3.44951 


.80891 


3.23714 


.82814 


3.04749 


.84758 


2.87700 


50 


61 


3.44578 


.80038 


3.23881 


.82846 


3.04450 


.84791 


2.87480 


49 


33 


3.44202 


.80055 


3.2:J048 


.82878 


3.04152 


.84824 


2.87161 


48 


84 


3.43829 


.30987 


3.22715 


.32911 


3.03854 


.84856 


2.86893 


47 


16 


3.43456 


.31019 


3.22384 


.32043 


3.03566 


.84889 


3.80624 


40 


47 


3.43084 


.81051 


3.22053 


.32973 


3.03260 


.84922 


2.80366 


45 


79 


3.42713 


.81083 


3 21723 


.83007 


3.02903 


.84a>4 


2.86069 


44 


10 


8.42313 


.31115 


3.21393 


.83040 


3.02007 


.84987 


3.85823 


43 


42 


3.41073 


.31147 


3.21063 


.83072 


3.02372 


.35020 


3.85555 


42 


74 


3.41604 


.81178 


3.20734 


.83104 1 3.02077 


.35052 


3.85289 


41 


OS 


3.41236 


.81210 


3.20406 


.83180 


3.01783 i 


.35085 


2.85028 


40 


37 


8.40R69 


.81243 


3.20079 


.88169 


3.01489 ' 


.35118 


3.84758 


39 


68 


3.4Uj02 


.31274 


3.10752 


.83201 


3.01196 


.85150 


2.81494 


38 


00 


3.4J136 


.81300 


3.10426 


.83233 


3.00CC3 


.35183 


2.81229 137 


32 


3.39771 


.31338 


3.10100 


.832(30 


3.00011 


: .35216 


3.83965 i33 


08 


8.30406 


.81370 


3.18775 


.33298 


3.00319 


.85218 


2.83702 


33 


95 


8.39012 


.31402 


3.ia461 


.33330 


8.00028 


.2&:in 


2.83139 


34 


S6 


8.38679 


.81434 


3.10127 


.33303 


2.09738 


.35314 


2.83176 


33 


68 


3.3:]317 


.81466 


3.17804. 


.38395 


2.99447 


.35,^6 


2.82914 


32 


90 


3.C7Ga3 


.81493 


3.17181 


.83427 


2.99158 


.85379 


2.82653 


31 


fn 


3.S7594 


.31530 


3.17159 


.83400 


2.98868 


.85412 


2.82301 


30 


58 


8.371M4 


.31663 


3.10838 


.83492 


2.98580 


.85445 


2.82130 !20| 


86 


8.CJj75 


.31594 


3 10517 


.3S5C4 


2.98292 


.35477 


2.81870 


28 


16 


8.3G516 


.31026 


3.10197 


.Zr^iii 


2.98004 ' 


.35510 


2.81610 


27 


48 


8.33158 


.81058 


3.15877 


.33569 


2.9m7 1 


.35548 


2.81360 '26 


80 


3.r,-;800 


.01090 


3.ir;,-,58 


.33621 


2.97430 


.85576 


2.81091 25 


11 


8.avi4d 


.31722 


3.15240 


.330,>4 


2.97144 


.35008 


2.60638 24 


43 


8.33087 


.31754 


3.11923 


.3;^cso 


2.96858 


.35041 


2.80574 23 


75 


8.ai733 


.31786 


3.14605 


.83718 


2.CG5?3 


.35074 


2.80316 22 


06 


3.31377 


.31818 


8.14288 


.33751 


2.90288 


.857V7 


2.80050 121 


38 


8.34028 


.31860 


. 3.18973 


.33783 


2.96004 

1 


.85740 


3.79802 


20 


70 


8.33670 


.31883 


3.18656 


.83810 


2.95721 


.85772 


3.79545 


10 


01 3.33317 


.31014 


3.13341 


.33^48 


2.95437 


.35805 


3.79289 


18 


33 8.8::365 


.31046 


3.13027 


.33881 


2.95155 


.35838 


2.79033 


17 


65 


8.3r:614 


.31978 


3.12718 


.33913 


2.94872  


.35871 


2.78r;8 


10 


97 


8.3^264 


.32010 


8.12400 


.33045 


2.94591 


.35004 


2.78623 


15 


38 


3.31914 


.32043 


8.12087 


.33978 


2.94309 


.35a37 


2.78269 


14 


60 


3.31565 


.32074 


3.11775 


.34010 


2.94028 


.35009 


2.78014 


13 


IM 


3.31216 


.32106 


8.11104 


.3-1013 


2.93748 


.30002 


2.VY761 


12 


34 


.S.CIXSS 


.32139 


3.11153 


.34(m'5 


2.0.'W(W 


.300:« 


2.77507 


11 


56 


8.U)o81 


.32171 


3.10843 


.34108 


2.93189 


.30008 


2.77254 


10 


87 


8.30174 


.325^3 3.10533 


.34140 


2.92910 


.30101 


2.77002 


9 


19 


3.:J.Ki29 


.32235 3.10223 


.34173 


2.92a*J2 


.miu 


2.707;.0 


8 


51 


3.;::) 183 


.3':.:07 3.n:)014 


.3 12: '5 


2.92.%! 


.30107 


2.701J,8 


7 


83 


3.'JJ139 


.3i,VJ9 


a. mm ' 


.312;]^ 


2.92070 


.3<;i09 


2.70217 


6 


14 


3.2S795 


,ii::m 


3.()ik>98 


.34270 


2.«17J)9 


.;i02:J2 


2.75{«)0 


5 


46 


8.38453 


.32303 


3.CS991 , 


.34:^3 


2.91.':C:J 


.3<;2(r) 


2.75740 


4 


78 


8.2:^100 


.32396 


3.0S685 . 


.343:« 


2.012-10 


.30208 


2.75400 


3 


00 


8.27767 


.3'>l28 


3.OS.379 


,34Ci.8 


^:.{:c:.7i 


.303:31 


2.76246 


2 


41 


8.27436 


.awx) 


3.0S073 1 


.3+400 


2.90C96 


.30304 


2.74997 


1 


.78 


S.2T0R6 


.32493 
Cotang 


3.07708 


.344.'i3 
Cotang 


2.00421 1 


.3o;m>7 

Cotang 


2.74718 


_0 


ng 


Tiuig 


Tang 


Ta:ig 


Tang 


/ 


7 


y 


TS'* 


71° ,1 


70« ! 





114 



NATURAL TANGENTS AND COTANGENTS. 



20<= 



2V 





Ji 

3! 
4. 
5 
6' 

7; 
8 
9 
10 

11 
la 
13 
14 
15 
IG 
17 
18 
19 
20 

21 
22 
23 
24 
23 
26 
27 
2^ 
23 
80 

81 
82 
83 
34 
85 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 

5:»| 

53, 

51 

55 

56 

57 

5ii 

59 

GO 



Tanff 

36430 
86-163 
36106 
3G529 
36562 
86593 
36628 
36661 
36694 
36727 

36760 
:56793 
3r>823 
30859 
30'M)2 
3G925 
36958 
30991 
37024 
37057 

870?0 
37123 
37157 
S7190 
3?^^ 
3?256 
87289 
37:522 

3ra55 

87388 

87422 
37453 
37483 
37521 
37554 
37588 
37621 
37654 
37687 
37720 

37754 
37787 
37820 
37a53 
37{}37 
37920 
37953 
3r9«6 
3 '1.30 
38053 



3«»0W] 
3S120 
'JS153 
3^1NJ 
3S220 
38253 
382:46 

3rvj20 

38*r>3 
383S8_ 

Cotarr: 



Cotang 

~2.74r48 
2.74499 
2.74251 
2.74004 
2.73756 
2.73509 
2.73i263 
2.?3017 
2.73771 
2.7.1526 
2.7^2281 

2.7.'3036 

2.71792 

2.71548 

2.71305 

2.71003 I 

2 70.^19 i 

2.70577 

2.70333 

2.7't034 

2.03858 

2.00612 
2.0J371 
2.03131 
2.0:\392 
2.CJ053 
2.CM14 
2.0:175 
2.0V.:'57 

2.orroo 

2.67462 



67225 

069S9 
GG752 
66516 
66281 
2.66046 
2.a5811 
2.0.5376 
2.0.5342 
2.05109 

261875 

2.01 ; 13 
2 01410 
2.01177 
2 &3315 
2.03714 
2.C3483 
2.01-152 
2.0^21 
2.02791 

2.62361 
2.(;ii32 
2.02103 
2.01M74 
2.01046 
2.G1418 
2.01190 
2.00063 
2.(50736 
2.0)509 



G0« 



Tan?r 
.38386 


C->tanfr 


2.a;o09 


.88420 


2.60283 


.8&453 


2.60057 


.88487 


2.53331 


.88520 


2.53006 


.38553 


2.59381 


.38587 


2.59156 


.38020 


2.C3933 


.3"^54 


2.58708 


.38687 


2.58484 


.88721 


2.68261 


.88754 


2.68088 


.38^7 


2.57815 


.83821 


2.57593 


.33854 


2.57871 


.SS888 


2.5n50 


.889:1 


2.56928 


.33955 


2.56707 


.83983 


2.66487 


.39023 


2.50206 


.89056 


2.56046 


.89089 


2.55827 


.33123 


2.53CL8 


.89158 


2.55389 


.39190 


2.r5170 


.33?-?3 


2.W052 


.S3C57 


2.51734 


.83200 


2.5-;516 


.333:4 


2.54209 


.33357 


2.51003 


.89391 


2.53805 


.imvi 


2.53^48 


.33453 


2.';3433 


.33192 


2.:i3217 


.33523 


2.53001 


.33539 


2.5:;:8 


.33393 


2.;"::571 


.39023 


2.r,:357 


.80000 


2.fJ142 


.30094 


2.:;19i:9 


.39^7 


2.51715 


.89761 


2.51502 


.::3795 


2.512S9 


.::3H29 


2.51076 


.83803 


2.50304 


..^'3896 


2.50r;3 


.339::o 


2.5a«0 


.33963 


2.:;0e->9 


.33997 


2.rHWH 


..rx)3i 


2.10807 


.40005 


2. :3597 



.40008 2.193'-<0 

.40133 I 2.l.;i77 

.4«)106 , 2.-I ■:);)7 

..;;)■»(« ! 2.! wrvS 

.4tU;n ' 2..J043 

. 40:^07 2 'l".04O 

.40301 2.18183 

.40335 2.-17924 

.40309 2.47716 

.40403 2. 17500 

Cotanf? '1 -.r.i,' Cotaiicj Tang 



22» 


Tangr 


CotanfjT 


.40408 


2.47509 


.40436 


8.47303 


.40470 


2.47095 


.40504 


2.46888 


.40538 


2.4668S 


.40573 


2.46476 


.40606 


2.46270 


.40640 


2.46065 


.40074 


8.45860 


.40707 


2.45655 


.40741 


8.46461 


.40775 


2.4S946 


.40809 


2.45048 


.40843 


8.44839 


.403?r 


8.44686 


.40911 


8.44488 


.40945 


8.44280 


.40379 


8.44027 


.<1018 


2.43::25 


.41017 


2.43023 


.11081 


8.43428 


.41115 


8.48220 


.41149 


8.43019 


.41183 


2.42319 


.41217 


2.42618 


.41251 


8.43418 


.4121^5 


8.42218 


.41319 


2.42019 


.4ia53 


2.41819 


.413;?7 


2.41020 


.41421 


8.41421 


.41455 


8.41223 


.414'J0 


2.410J3 


.41524 


2.4ai27 


.4m58 


8.406C9 


.41693 


8.40133 


.416C6 


8.40233 


.41660 


2.40033 


.41694 


2.89841 


.41T-:3 


2.890-13 


.41763 


2.89449 


.41797 


2.89258 


.'118:U 


2.29053 


 ..!lC<i5 


2.88863 


41099 


2.88668 


i .41933 


2.88173 


i ..-inr.s 


2.88279 


.4J003 


2.38084 


.42086 


2.87891 


.'!'2070 


2.37097 


..42105 


2.37J04 


.42189 


r:.37811 


.12173 


2.37118 


, .'422t)7 


2.86925 


' .42-213 


2.86733 


.422'. 6 


2.86541 


.42310 


2.86349 


.42845 


2.86158 


.42379 


2.86967 


i .42418 


2.86776 


! .42447 


2.a'S5W 



23< 



Tang 



«M47 
48488 
42516 
42551 
42585 
42619 
42651 
42033 
42723 
42757 
42791 

42r28 
42800 
42694 
420S9 
48968 
42996 
48088 
48067 
48101 
4S186 

4S170 
48205 
48383 
48274 
43308 
43a43 
43378 
46418 
43447 
46181 

48516 
43550 
48585 

48634 
48689 
43734 
48758 
48703 
48888 

48868 

48897 
48038 
48006 
44001 
44033 
44071 
44103 
44140 
44173 

44210 
44214 
442rr9 
44314 
44349 
44884 
44418 
44468 
4(488 
44388 



Cotang 

8.85686 
8.85335 
8.83205 
8.85016 
8.84825 
8.64686 
8.84147 
8.81858 
8.84069 
8 83881 
8.83608 

8.83606 
8.83817 
8.88180 
8.8891S 
8.88756 
8.S35TO 
8.88888 
8.88197 
8.83018 
8.81886 

8.81641 
8.81466 
8.81871 
8.81086 
8.80008 
8.80718 
8.80631 
8.80861 
2.80167 
8.89981 

8.80601 
8.89619 
8.89487 
8.89851 
8.80078 
8.88891 
8.88710 
8.88588 
8.83318 
8.88107 

8.8T067 
887806 
8.87686 
8.87447 
8.87867 
8.87068 
8.80009 
8.86780 
8.866R8 
8.36374 

8.86106 
8.86018 
8.80&1O 
8.85063 
8.25186 
8.85800 
8.86188 
8.81056 

8 8mo 

8.81001 



GV 



QV 



Cotang Tung 







NATURAL TANGENTS ANE 


> COTANGENTS. 1 


15 




24«» 


25° 


' £8° 


27^ 


60 


Tangr 
.44523 


Cotang 


Tang 
.40631 


C'otang 


1 Tang 
.484 73 


Cotang 


Tang 

.50953 


Cotang 


2.24004 


2.14^^31 


2.05C30 


1.96;t01 


1 


.44558 


2.24428 


.46666 


2.14288 


.48809 


2.04879 


.509ii9 


1.90120 


59 


2 


.44593 


2.24252 


.40702 


2.14125 


.4^345 


2.04728 


.61020 


1.95979 


58 


8 


.44627 


2.24077 


.46737 


2.13963 


.4LS881 


2.04577 


.51003 


1.95838 


57 


4 


.44662 


2.23902 


.46778 


2.1SS01 


.48017 


2.04426 


.510CD 


1.95698 


56 


6 


.44097 


2.23727 


.46808 


2.18639 


.48953 


2.04276 


.51136 


1.95557 


55 


6 


.44738 


2.23553 


.46843 


2.13477 


.48989 ; 2.04125 i 


.51173 


1.95417 :54 


7 


.41767 


2.23378 


.46879 


2.1.3G16 


.49026 


2.0C975 


.51209 


1.05277 i53 


8 


.44802 


2.23204 


.4C014 


2.181M 


.49CG2 


2.03P25 


.51246 


1.95137 ;52 


9 


.41837 


2.23030 


.40950 


2.12093 


.49098 


2.0CC:a 


.51283 


1.94C97 


51 


10 


.44872 


2.22857 


.40985 


2.12832 


.49134 


2.a3526 


.51319 


1.94858 


50 


11 


.44907 


2.23683 


.47021 


2.12671 


.49170 


2.03376 


.51356 


1.94718 


49 


12 


.44942 


2.22510 


.47056 


2.12511 


.49^06 


2.0S227 


.51393 


1.94579 ,'48 


13 


.44977 


2.22337 


.47092 


2.12350 


.492:2 


2.03078 


.61430 


1.94440 


47 


14 


.45012 


2.^164 


.47128 


2.12190 


.492^8 


2.02023 


..01467 


1.94301 


46 


15 


.45047 


2.21992 


.47163 


2.12030 


.49315 


2.02780 i 


.51503 


1.94162 


45 


IG 


.45082 


2.21819 


.47199 


2.11871 


.49351 ! 2.02031 


.51540 


1.94023 


44 


17 


.45117 


2.21047 


i A?23i 


2.11711 


.49387 


2.02483 


.51577 


1.93885 


43 


18 


.45152 


2.21475 


.47270 


2.11552 


.494::3 


2.02335 


.51614 


1.93746 


42 


19 


.45187 


2.21804 


.47005 


2.11C92 


.494:9 


2.021B7 


.C1651 


1.03608 '41 


20 


.45222 


2.21132 


.47341 


2.11233 


.49495 


2.02039 


.51688 


1.93470 


40 


21 


.45257 


2.20961 


.47377 


2.11075 


.49532 


2.01891 


.51724 


1.03332 


39 


22 


.45292 


2.20790 


.47412 


2.1GJI6 


.40.:l8 


2.01743 


.61761 


1.93195 


38 


23 


.45327 


2.20019 


.47448 


2.10758 


.49604 


2.01596 


.51798 


1.93057 


37 


24 


.45S6J 


2.204i9 


.47483 


2.10600 


.49640 


2.01449 


.£1835 


1.92920 


36 


25 


.45897 


2.20278 


.47519 


2.10442 


.49077 


2.01302 


.51872 


1.92782 


35 


26 


.45432 


2.20108 


.47555 


2.10284 


.49713 


2.01155 


.51909 


1.92645 


34 


27 


.45407 


2.100C8 


.47590 


2.10126 


.49749 


2.01008 


.51946 


1.92508 


33 


28 


.45502 


2.197G9 


.47626 


2.09969 


.49786 


2.00862 


.51983 


1.92371 


32 


29 


.45538 


2.195C9 


.47062 


2.09011 


.49822 


2.00715 


.52020 


1.92235 


31 


80 


.45573 


2.19430 


.47698 


2.09654 


.49858 


2.00569 


.52057 


1.92098 


30 


81 


.45608 


2.19261 


.47733 


2.09498 


.49894 


2.00423 


.52094 


1.91962 


29 


82 


.45643 


2.10092 


.47769 


2.09341 


.49931 


2.00277 


.52131 


1.91826 


28 


83 


.45078 


2.18923 


.47805 


2.03184 


.49967 


2.00131 


.52168 


1.91690 


27 


84 


.45718 


2.18755 


.47840 


2.09028 


.50004 


1.99986 


.52205 


1.91554 


26 


86 


.45748 


2.18587 


.47876 


2.08872 


.SOOiO 


1.99841 


.52242 


1.91418 


25 


86 


.45784 


2.18419 


.47912 


2.08716 


.50076 


1.99695 


.52279 


1.91282 


24 


87 


.45819 


2.18251 


.47948 


2.08500 


.50113 


1.C3550 


.52316 


1.91147 


23 


88 


.45854 


2.18064 


.47984 


2.08405 


.50149 


1.C3406 


.52353 


1.91012 


22 


89 


.45889 


2.17916 


.43019 


2.03250 


.60185 


1.C9261 


.52390 


1.90876 


21 


40 


.45024 


2.17749 


.43055 


2.08094 


.50222 


1.99116 


.52487 


1.90741 


20 


41 


.45960 


2.17582 


i48091 


2.07939 


.50258 


1.98972 


.52464 


1.90607 


19 


42 


.45995 


2.17416 


.43127 


2.07785 


.50295 


1.98828 


.52501 


1.90472 


18 


43 


.46080 


2.17249 


.48163 


2.07630 


.50331 


1.98684 


.52538 


1.90337 


17 


4( 


.40065 


2.17083 


.48198 


2.07476 


.50308 


1.98540 


.52575 


1.90203 


10 


45 


.40101 


2.10917 


.48234 


2.07321 


.50404 


1.98396 


.62613 


1.90009 


15 


46 


.46186 2.10751 


.48270 


2.07167 


.50141 


1.98253 


.52650 


1.89935 


14 


47 


.46171 1 


2.10585 


.48306 


2.07014 


.50477 


1.93110 


.52687 


1.89801 13 


48 .40208 


2.10^120 


.48342 


2.06860 


.50514 


1.97966 


.62724 


1.89607 12 


49! .4G242 


2.1C55 ' 


.48.378 


2.06706 


.50550 


1.97823 


.52761 


J. 895;^ 11 


£0 


.46277 


2.16090 i 


.48414 


2.06553 


.50587 


1.97681 


.52798 


1.89400 


10 


51 


.46312 


2.15925 


.48450 


2.06400 


.50623 


1.97538 


.52836 


1.8926b 


9 


1.2 


.40348 


2.157C0 


.4-3486 


2.00247 


.50060 


1.97395 


.52873 


1.89133 


8 


53 


.46383 


2.15596 


.43521 


2.00094 


.50096 


1.97253 


.52910 


1.89O00 


7 


54 


.46418 


2.15482 


.48557 


2.05942 


.50733 


1.97111 


.52947 


1.88r.G7 


6 


55 


.40454 


2.15268 


.48593 


2.05790 


.50769 


1.96969 


.62985 


1.88734 


5 


56 


.40489 


2.15101 


.48629 


2.05037 


.50806 


1.96827 


.53022 


1.88602 


4 


57 


.40525 


2.14940 


.48665 


2.05485 


.50843 


1.96685 


.53059 


1.884C9 


3 


58 


.46560 


2.147r7 


.48701 


2.05383 


.50879 


1.96544 


.58096 


1.88337 


2 


59 


.46595 


8.14614 


.48787 


2.05182 


i .50916 


1.96402 


.53134 


1.88205 


1 


1 

/ 


.46681 
Cotang 


2.14451 


.48773 


2.05030 


.50953 


1.962G1 


.53171 


1.88073 


_0 


Tang 


Cotangi 


'I'ang 


Cotang 


Tang 


Cotang 


Tang 


f 


66» 


e4« 


6 


30 


6! 


2» 





NATURAL TANGENTS AND COTANGENTS. 









NATURAL TANGENTS AND 


COTANGENTS. 1 


17 


* 


82* ': 


88<» 


84» 1! 


85» 


(io 


m 


Tangf I Ck>taiig: '  
.62487 1.600:43 , 


Tang 
.04941 


Cotong 


Tang j Cotang 1 
.67451 . 1.4H2r)6 , 


Tang 

.70021 


Cotang 







1.539S6 


1.42815 




ll 


.62527 


1.50980 


.61982 


1.53888; .67493 


1.48103 


.70UG4 


1.42720 50 




2| .625U8 


1.59826 . 


.65024 


1.53791 .67536 


1.48070 


.70107 


1.42(:g:} \:a 




3 .6:%0d 


1.537e:j 


.65005 


1.53093 .67578 


1.47JJ77 


.70151 


1.42550 57 




4 .62649 


1.59da0 1 


.65106 


1.53595 : .67C20 


1.47R85 


.70194 


1.42462 i56 




5 .62689 


1.59517 


.^148 


1.53197 . .07ca 


1.47792 


.70238 


1.42^^74 ;55 




6 .G2r^ 


1.59414 


.65189 


1.53100 : 


.677l'5 


1.47G99 


.70281 


1.42286 54 
1.42198 53 
1.42110 52 




7 .68770 


1.59311 


.652:31 


1.63302 


.Ci i <8 


1.47007 


.70325 




8 .62811 


1.5020^3 


.65272 


1.5305 


.6771:0 


1.47514 


.70:jf;.8 




9 


.62852 


1.59105 


.6>314 


1.53107 


.67S:i2 


1.47422 


.70412 1.42022 ;51 1 




10 


.628S» 


1.50002 


.05355 


1.53010 


.67875 


1.47330 


.70455 1.41931 


50 




11 


.62933 


1.53900 


.6.5307 


1.B2913 


.67917 


1.47238 . 


.70199 1.41817 


49 




12 


.C;^r3 


1.5J7J7 


.0>4:38 


1.52816 


.679f» 


1.47146 
1.47053 ' 


.70:>42 1.41759 


48 




13 


.63014 


1.5S695 


.05480 


1.52719 


.08002 


.705S6 1.41C72 


47 




14 


.CSOxS 


1.53593 


.05521 


1.52C22 


.68045 


1.4C002 =: 


.70029 1.415K1 


40 




15 


.030;» 


1.58190 


.C5563 


1.51S25 


.68088 


1.40S70 :.' 


.70073 


1.41497 


45 




16 .63136 


1.583^38 


.65004 1.52129 1 


.68130 1.40778 .; 


.70717 


1.41409 


44 




17! .63177 


1.58286 


.65646 


1.62332 : 


.68173 1.4GC86 :■ 


.70700 


1.41322 


^^ 




\^\ .fSSXl 


1.581W 


.6"5C'« 


1.52235 


.68215 1.4C595  


.70804 


1.41£:« :42 




19- .63258 


1.58«3 


1 .or^^ 


\XIV^ 


.6<*2.:S . 1.40503 .1 


.70«4S 


1.411 LS 


41 




2U .C32D9 


1.579bl 


.OjiVI 


1.52013 


.68301 


1.40411 ;| 


.70801 


1.41001 


40 




21 


.63810 ' 


1.57879 


.R-seia 1.5194C 


.68313 


1.40320 '' 


.70035 


1.40974 


39 




22 


.63S30 


1.577:1 


.C^oCl ' 1.51850 


.CK->6 


1.4UJJ9 1 


.70.,r9 


1.40'-S7 


3:^ 




23 


.63121 


1.57G.0 


.C^S>5 1.51754 .6S4:w;» 


1.40137 


.710a 


I.41.UK) 


37 




24 


.63162 


1.57675 


.65033 1.51C.-i8 . 


.68471 


1.40046 ; 


.7Ki<:6 


1.40714 


30 




25 


.635C3 


1.574T4 


: .65SS0 1.515C2 


.0S511 


1.450.'i5 ; 


.71110 


1.40»;27 


35 




25 


.63514 


1.573T2 


.66021 


1.6l4r» . 


.08557 


1.45:01 ! 


.71151 


1.4U"10 


•:4 




27 .63581 


\.h"i:i\ 


.660C3 


1.51870 


.68000 


1.45ii'3 ' 


.7ir.Pi 


1.40:.-4 


3:1 




2) .63625 


1.5n70 


.66105 


1.51275 


.68012 


1.450-^2 


.7ic:}2 


1.40:.:;7 


32 




29 


.63006 


1.670C9 


.66147 


1.51179 


.m>h 


1.45.v:r2 


.712-5 


1.4^2^1 '31 


ao| 


.63707 


1.56969 


.661S9 


1.510c4 ; 


.^I'A 


1.455U1 


.71329 


1.40105 "30 

1 




81 


.63718 


1.56868 


.66230 


i.soaas 


.68771 ' 1.4*^110 


\ .71373 


1.40109 20 




32 


.63789 


1.5STC7 


.00272 


1.50SI*3 


.f-^U : 1.45:,.'0 


! .71417 


1.4'i:r« 'O'i 




83 


.63830 


1.5o007 


.00314 


l.D£r797 : 


.OS857 ; 1.4.-; J ;9 . 


.71401 


1.80036 27 




81 .63071 


1.56566 


.CC3::6 


1.50702 


.CS900 . 1.45!:;3 : 


.7151)5 


l.S0!OC :2*^. 




85 .63912 


1.56406 


.60308 


1.5«3C-/7 


.0=^342 1 1.45049 


.715 J9 1.80701 25 




86 .63L53 


1.5C3G6 


.66140  1.50:^12 ; .CS0S5 ' 1.4;'.:.S 1 


.71503 1.89C79 .21 




87 .C39'J1 


1.56265 


.661S2 


1.50417! .C9028 1.4'l=';S ; 


.71637 1.89:03 23 




8rt .61085 


1.56105 


.60524 


1.50322 .«'>j71 1.44773! 


.71081 ' 1.805"7 22 




89 .61076 


1.500G5 


.60506 


1.60223 ; .rolU 1.44C^S 


.n725 1.89 ni 21 




40 .6ai7 


1.559C6 


.66608 


1.50133 i .(^157 1.44598 : 


.71769 1.393:;6 :-^ 




41 .61158 


1.55866 


.666.')0 


1.50038 ' .09200' 1.44508 


.71P13 ; 1.39250 'l9 


\Ai\ .e4JiH> 


1.557W 


.60092 


1.4JJ44 .«.j0;3 i 1.44413 


.71?.-.7 


1.8..: ;5 1 ; 




43 


.61:^0 1.55CW 


. Xf-jt '-4 


1.4aS49 .Uj2o6 1.44:^:9 
1.49^55 .Cj3i9 1.4i2:^ 
1.49601 .69372 1.4!Ii9 ' 


.710 :l 


1.30 :9 17 




44 


.6J281 1.55567 


.66776 


.':i:.46 


1.3. 04 10 




45 


.6i:jii 1.SM67 


.66818 


.71000 


1.3S0U0 1.- 




46 


.OldGS 1.5986^ 


. .Oi-CO 


1.49566 = .09116 \AV/A 


.7-a:.M  1.3->.^4 11 




47 


.64404 1.5SSG3 


.6akl2 


1.4:^72 . .ei^;59 i.4p%ro 


.7JJ78 l..'i:7:;S IC 




43 


.64446 1 1.55irj 


.6C:^ i 1.4&G73 «! .005.-2 1.43& 1 


.72122 l.?J^.-J 1: 




49 


.61487 


1.55071 


.er;:;j5 ! 1.49171 : .6X-i5 . 1.4.'?n'2 


.72107 1.3'.-. S 11 




50 


.04538 


1.54972 


.6r0t23 1.49190 '• .695>)8 X.A^r,^ 


.72211 1. ;:>;>! u 




51 


.61509 1 1.54873 


.67071 1.49097 | .69^31 1.4CC11 


.7225.5 1.3S.r9 9 




52 


.64010 


1.&4774 


.67113 1.4>yj3 


.€:..; 5 1.4:i:.:5 


.72i:/J 1.3SJ11 f= 




53 


.64052 


1 5JC75 


.671.5 1.4:i;9 


.607:8 i.4;;;-5 


.72311 1.3---. J 7 




;>i 


.61003 


l.r^J576 


.671 W 1.48.3:6 


.60; 01 i.4.>i;7 


.72;--J 1.3^145 6 




ns 


.647^ 


1.514:3 


.67;,i9 1.4-722 - .Cf^- 4 1.4oi.S 


. .721':2 i 1.3SC-» 5 




56 


• .647^5 


i.5ij:9 


.67::r3 \A:^?J!i 1 .c:.;7 1.4;: .3 


.721:7 ; l.:i7:.:'3 4 




57 


' .64817 : 1.51iSl 


.6r.J4 1.4S536 ' .e: 1 1.43 ■) 


.72: .1 ' 1.37fe:i 3 




.^e 


\ .61858 


1.541S3 


.67J36 1.4S«2 


.f/y^i i.4;<>2 


.?i: 5 i.37¥«>r 2 

.72C:0 : 1.37722 1 
.72?;-.! 1 .37cr;s 

Cotacg] Ting 




ES 


» .64SII0 


1.51035 


.67416 1.4«'^9 


.Uw 1 7 1 .-x^-- -3 


'6C 


» .64941 


1.589B6 


.67451 1.4^j6 


.7aei 1.4715 


1 f 


Cotaogt Tang 


1 Cotcn^ 1 Tacg 


C^L-a.;: T^EiJ 




w 


Vf 


: 




65' 


\ 


34- 


_J 



118 



N.-iTURAl, TAXOKNTS AND COTANGENTS. 



36« 



Tang I C()t:irijf 




1 
2 
3 

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120 NATURAL TANGENTS AND COTANGENTS. 



PART II. 



Strength of Materials, and Stability of 

Structures. 



UTTRODUCnON. 

the chapters constituting this part of the book, the author 
ideavored to present to architects and builders handy and 
e rules and tables for determining the strength or stability of 
ece of work they may have in hand. Every pains has been 

to present the rules in the simplest form consistent with 
accuracy; and it is believed that all constants and theories 
ced are fully up to the knowledge of the present day, some 
! constants on transverse strength having but recently been 
lined. The rules for wrought-iron columns have lately been 
y changed by some engineers; but as the question of the 
th of wrought-iron columns has not yet been satisfactorily 
I, and as the formulas herein given undoubtedly err on the 
ide if at all, we have thought best not to change them, espe- 
as they are still used by many bridge engineers. 

question of the wind-pressure on roofs has not been taken 
as thorough manner as would be needed for pitch roofs of 
Teat span ; but for ordinary wooden roofs, and iron roofs not 
ling one hundred feet span, the method given in Chap. 
I. is sufficiently accurate. 

r one wishing to study the most accurate method of obtaining 
feet of the wind-pressure on roofs will find it in Professor 
's excellent work on " Graphical Analysis of Roof Trusses." 
©nclusion, the author recommends these chapters as present- 
icurate and modern rules, especially adapted to the require- 

of American practice. 



EXPLANATION OF SIGNS AND TERMS USED IN 
THE FOLLOTVING FORMULAS. 

Besides the usual arithmetical signs and characters in general 
use, the following characters and abbreviations will frequently be 

used : — 

The sign y^ means square root of number behind. 

^ means cube root of number beliind. 
( ) means that all the numbers between are to be 
taken as one quantity, 
means decimal parts; 2.5 = 2t^, or .46 = iVo. 
The letter A denotes the co-efficient of strength for beams one 

inch square, and one foot between the supports. 

C denotes resistance, in pounds, of a block of any 

material to crushing, per square inch of section. 

E denotes the modulus of elasticity of any material, 

in pounds per square inch, 
e denotes constant for stiffness of beams. 
F denotes resistance of any material to shearing, per 

square inch. 
B denotes the modulus of rupture of any material. 
aS denotes a factor of safety. 

T denotes resistance of any material to being pulled 
apart, in pounds, per square inch of cross-section. 

Breadth is used to denote the least side of a rectangular piece, 
and is always measured in inches. 

Depth denotes the vertical height of a beam or girder, and is 
always to be taken in inches, unless expressly stated otherwise. 

LetKjth denotes the distance between supports in feetf unless 

otlu*rwis(» specified. 

Abbreviations. — In order to shorten the formulas, it has 
()ft(Mi been found necessary to use cerUin abbreviations; such as 
bet. Tor Ix'twiMjn, hot. for bottom, dist. for distance, diam. for 
diaimtcr, lior. tor horizontal, scj. for square, etc., which, however, 
can in no cast' Wiul to uncertainty as to their meaning. 

Wli( IV tlie word "ton" is used in this volume, it always means 

2(M)0 pounds. 



CHAPTER T. 
DEFINITIONS OF TERMS USED IN MECHANICS. 

The following terms frequently occur in treating of mechanical 
construction, and it is essential that their meaning be well under- 
stood. 

Mechanics is the science which treats of the action of forces. 

Applied Mechanics treats of the laws of mechanics which 
relate to works of human art ; such as beams, trusses, arches, etc. 

Rest is the relation between two points, when the straight line 
joining them does not change in length or direction. 

A body is at rest relatively to a point, when any point in the 
body is at rest relatively to the first-mentioned point. 

Motion is the relation between two points, when the straight 
line Joining them changes in length or direction, or in both. 

A body moves relatively to a point, when any point in the body 
moves relatively to the point first mentioned. 

Force is that which changes, or tends to change, the state of a 
body in reference to rest or motion. It is a cause regarding the 
essential nature of which we are ignorant. We cannot deal with 
forces properly, but only with the laws of their action. 

Kqiiilibrium is that condition of a body in which the forces 
acting upon it balance or neutralize each other. 

Statics is that part of Applied Mechanics which treats of the 
conditions of equilibrium, and is divided into: — 

a. Statics of rigid bodies. 

6. Hydrostatics. 

In building we have to deal only with the former. 

Structures are artificial constnictions in which all the parts 
are intended to be in«equilibrium and at rest, as in the case of a 
bridge or roof-truss. 

They consist of two or more solid bodies, called pieces, which 
are connected at portions of their surfaces called joints. 

There are three conditions of equilibrium in a structure; viz. : — 

I. The forces exerted on the whole structure must balance each 
other. These forces are: — 

a. The weight of the structure. 

h. The load it carries. 



126 DEFINITIONS OF TERMS 

c. The supporting pressures, or resistance of the foundation?, 
called external forces. 

II. The forces exerted on each piece must balance each other. 
These forces are: — 

rt. The weight of the piece. 

b. The load it carries. 

c. The resistance of its joints. 

III. The forces exerted on each of the parts into which any 
piece may be supposed to be divided must balance each other. 

Stability consists in the fulfilment of conditions I. and II., 
that is, the ability of the structure to resist displacement of its 
parts. 

Streng'th consists in the fulfilment of condition III., that is, 
the ability of a piece to resist breaking. 

Stiffness consists in the ability of a piece to resist bending. 

The theory of structures is divided into two parts; viz. : — 

I. That which treats of strength and stiffness, dealing only with 
single pieces, and generally known as strength of liiaterialH* 

II. That which treats of stability, dealing with structures. 
Stress. — The load or system of forces acting on any piece of 

material is often denoted by the term " stress,'* and the word will 
be so used in the following pages. 

The i)} tensity of the stress per square inch on any normal sur- 
face of a solid is the total stress divided by the area of the section 
in square inches. Thus, if we had a bar ten feet long and two 
inches square, with a load of 8000 poimds pulling in the direction 
of its length, the stress on any normal section of the rod would be 
8000 pounds ; and the intensity of the stress per square inch would 
be 80{K) -f 4, or 2000 pounds. 

Strain. — When a solid body is subjected to any kind of stress, 
an alteration is produced in the volume and figure of the body, and 
this alteration is called the ** strain." In the case of the bar given 
al)ovo, the strain would be the amount that the bar would stretch 
under its load. 

The Ultimate Stronprth, or Breaking: Load, of a body 
is the load riHiuircd to prothK-e fracture in some specified way. 

The Safe Load is the load that a piece can support without 
impairing: its strciii^tii. 

Factors of Safety. — When not otherwise specified, & factor 
of safety means the ratio in which the breaking load exceeds the 
safe load. In designing a i)i{^ce of material to sustain a certain 
load, it is required that it shall be perfectly safe under all circum- 
stances; and henc(^ ii. is necessary to make an allowance for any 
defects in the material, workmanship, etc. It is obviona, that, for 



USED IN MECHANICS. 127 

Is of different composition, different factors of safety will 
ired. Thus, iron being more homogeneous than wood, and 
»le to defects, it does not require so great a factor of safety, 
^in, different kinds of strains require difiPerent factors of 

Thus, a long wooden column or strut requires a greater 
»f safety than a wooden beam. As the factors thus vary 
irent kinds of strains and materials, we will give the proper 
of safety for the different strains when considering the 
ce of the material to those strains. 
iiiction between Dead and Live liOad. — The 
dead load," as used in mechanics, means a load that is ap- 
j imperceptible degrees, and that remains steady; such as 
3;ht of the structure itself. 

ive load '' is one that is applied suddenly, or accompanied 
.brations; such as swift trains travelling over a railway- 
or a force exerted in a moving machine. 
\ been found by experience, that the effect of a live load on 
or other piece of material is twice as severe as that of a 
id of the same weight: hence a piece of material designed 
r a live load should have a factor of safety twice as large 
lesigned to carry a dead load. 

load produced by a crowd of people walking on a floor is 
considered to produce an effect which is a mean between 

a dead and live load, and a factor of safety is adopted 

modulus of Rupture is a constant quantity found in 
aulas for strength of iron beams, and is eighteen times the 
: the constant " A." 

ulus of Elasticity. — If we take a bar of any elastic 
1, one inch square, and of any length, secured at one end, 
he other apply a force pulling in the direction of its length, 
i find by careful measurement that the bar has been stretched 
;ated by the action of the force. 

if we divide the total elongation in inches by the original 
)f the bar in inches, we shall have the elongation of the bar 
b of length; and, if we divide the pulling-forre per square 
this latter quantity, we shall have what is known as the 
s of elasticity. 

e we may define the hkkIhIiis of fUintirUij an the pullinfj or 
uiing force per unit of .'section divided by the elongation 
iresnion i)er unit of Unfjth. 

\ example of the method of determining the modulus of 
y of any, material, we v^ill take the following: — 
)8e we have a bar of wroiight-iron, two inches square and 



ten feet long, securely fastened at one end, and to the other end 
we apply a pulling-force of 40,000 pounds. This force causes the 
bar to stretch, and by careful measurement we find the elongation 
to be 0.0414 of an inch. Now, as the bar is ten feet, or 120 inches, 
long, if we divide 0.0414 by 120, we shall have the elongation of the 
bar per unit of length. 

Perfonning this operation, we have as the result 0.00034 of an 
inch. As the bar is two inches square, the area of cross-section 
is four s(iuare inches, and hence the pulling-force per square inch 
is 10,000 pounds. Then, dividing 10,000 by 0.00084, we have as the 
modulus of elasticity of the bar 29,400,000 pounds. 

This is the method generally employed to determine the modulus 
of elasticity of iron ties; but it can also be obtained from the 
deflection of beams, and it is in that way that the values of the 
modulus for most woods have been foiuid. 

Another definition of the modulus of elasticity, and which is a 
natural consequence of the one just given, is the number of 
pounds that would be required to stretch or shorten a bar one inch 
square by an amount equal to its length, provided that the law of 
pei-fect elasticity would hold good for so great a range. The mod- 
uhis of elasticity is generally denoted by E, and is used in the 
detomiination of the stiffness of beams. 

Moment. — If we take any solid body, and pivot it at any 
point, and apply a force to the body, acting in any direction 
except in a line with the pivot, we shall produce rotation of the 
body, provided the force is sufficiently strong. This rotation is 
produced by what is called the moment of the force; and the 
moment of a force about any given point or pivot is the product 
of the force into the perpendicular distance from the pivot to the 
lin(i of action of the force, or,an common phrase, the product cf 
the force into the arm with which it acta. 

The Centre of Gravity of a body is the point through 
which tlie resultant of the weight of the body always acts, no mat- 
ter in what, position the body be. If a body be suspended at its 
centre of tjjravity, and revolved In any direction, it will always be 
in e<iuilihriinn. 

(For centre of gravity of surfaces, lines, and soliils, see Chap. IV.) 



CLASSIFJCATION OF STRAINS. 120 

CI.A88IFICATION OF STRAINS WHICH MAT BE 
PRODUCED IN A SOLID BOD7. 

The dififerent strains to which building-materials may be exposed 
are: — 

I. Tension, as in the case of a weight suspended from one end 
of a rod, rope, tie-bar, eta; the other end being fixed, tending to 
stretch or lengthen the fibres. 

II. Shearing Strain^ as in the case of treenails, pins in 
bridges, etc., where equal forces are applied on opposite sides in 
such a manner as to tend to force one part over the adjacent one. 

III. Conipressiony as in the case of a weight resting on top 
of a column or post, tending to compress the fibres. 

IV. Transversa or Cross Strain, as in the case of a load 
on a beam, tending to bend it. 

V. Torsion, a twisting strain, which seldom occurs in build- 
ing-construction, though quite frequently in machinery. 



130 FOUNDATIONS. 



CHAPTER n. 
FOUNDATIONS. 

The following chapter on Foundations is intended to furnish 
the reader with only a general knowledge of the subject, and to 
enable him to be sure that he is within the limits of safety if he 
follows what is here given. For foundations of large works, or 
buildings upon soil of questionable firmness, the compressibility of 
the soil should be determined by experiments. 

The term ^'foundation" is used to designate all that portion of 
any structure which serves only as a basis on which to erect the 
superstructure. 

This term is sometimes applied to that portion of the solid mate- 
rial of the earth upon which the structure rests, and also to the 
artificial arrangements which may be made to support the base. 

In the following pages these will be designated by the term 
" foundation-bed." 

Object of Foundations.— The object to be obtained in the 
construction of any foundation is to form such a solid base for the 
superstructure that no movement shall take place after its erection. 
But all structures built of coarse masonry, whether of stone, or 
brick, will settle to a certain extent; and, with a few exceptions, 
all soils will become compressed under the weight of almost any 
building. 

Our main object, therefore, is not to prevent settlement entirely, 
but to insure that it shall be uniform ; so that, after the structure is 
finished, it will have no cnacks or flaws, however irregularly it may 
be disposed over the aroa of its site. 

Foundations Classed. — Foundations maybe divided into 
two classes : — 

Class I. — Foundations constructed in situations where the 
natural soil is sufficienthj flnn to bear the weight of the intended 
structure. 

Class II. — Foundations in situations where an artyicicU bear^ 
ing-stratum must be formed, in consequence of the 9rftne89 or 
looseness of the soil. 



FOUNDATIONS. 131 

Each of these two great classes may be subdivided into two 
divisions: — 

a. Foundations in situations wliere water offers no impediment 
to the execution of the work. 

6. Foundations under water. 

It is seldom that architects design buildings whose foundations 
are under water; and, as this division of the subject enters rather 
deeply into the science of engineering, we shall not discuss it here. 

Boringf. — Before we can decide wliat kind of foundation it 
will be necessary to build, we must know the nature of the subsoil. 
If not already known, this is deterininetl,* ordinarily, by digging a 
trench, or making a pit, close to the site of the proposed works, to 
a depth sufficient to allow the different strata to be seen. 

For important structures, the nature of the subsoil is often de- 
termined by boring with the tools usually employed for this pur- 
pose. When this method is employed, the different kinds and 
thickness of the strata are determined by examining the speci- 
mens brought up by the auger used in boring. 

Foundations of tlie First Class.— -The foundations in- 
cluded under this class may be divided into two cases, according to 
the different kinds of soil on which the foundation is to be built : —% 

Case I. — Foundations on soil composed of mateiHals whose 
stability is not aff^cteA by saturation with water, and which are 
firm enough to support the weight of the structure. 

Under this case belong, — 

Foundations on Rock. — To prepare a rock foundation for being 
bfuilt upon, all that is generally required is to cut away the loose 
and decayed portions of the rock, and to dress the rock to a plane 
surfsice as nearly perpendicular to the direction of the pressure as 
is practicable; or, if the rock forms an inclined plane, to cut a 
series of plane surfaces, like those of steps, for the wall to rest on. 
If there are any fissures in the rock, they should be filled with con- 
crete or rubble masonry. Concrete is better for this purpose, as, 
when once set, it is nearly incompressible under any thing short of 
a crushing-force; so that it forms a base almost as solid as the 
rock itself, while the compression of the mortar joints of the 
masonry is certain to cause some irregular settlement. 

If it is unavoidably necessary that some parts of the foundation 
shall start from a lower level than others, care should be taken to 
keep the mortar Joints as close as possible, or to execute the lower 
portions of the work in cement, or some hard-setting mortar: other- 
wise the foundations will settle unequally, and thus cause much 
injury to the superstructure. The load placed on the rock should 
at no time exceed one-eighth of that necessary to crush it. Pro' 



132 FOUNDATIONS. 

fessor Rankine gives the following examples of the actual intensity 
of the pressure per square foot on some existing rock founda- 
tions: — 

Average of ordinary cases, the rock being at least as strong 

as the strongest reil bricks 2000(; 

Pressures at tlie base of St. KoIIox chimney (450 feet below 
the summit) 

On a layer of strong concrete or beton, 6 feet deep .... 0070 

On sandstone below the beton, so soft that it crumbles in the 

hand 4000 

The last example sliows the pressure which is safely borne in 
practice by one of the weakest substances to which the name of 
rock can be applied. 

M. Jules Graudard, C.E., states, that, on a rocky ground, the 
Roquefavour aqueduct exerts a pressure of 26,800 pounds to the 
square foot. A bed of solid rock is unyielding, and appears at first 
sight to offer all the advantages of a secure foundation. It is gen- 
erally found in practice, however, that, in lai^ge buildings^ part of 
the fowidations will not rest on the rock, but on the adjacent soil; 
and as the soil, of whatever material it may be composed, is sure to 
be compressed somewhat, irregular settlement will almost invariably 
take place, and give much trouble. The only remedy in such a case 
is to make the bed for the foundation resting on the soil as firm as 
possible, and lay the wall, to the level of the rock, in cement or 
hard-setting mortar. 

Foundation on Compact Stony Earths, such as Graieel or Sand. 
— Strong gravel may be considered as one of the best soils to build 
upon ; as it is almost incompressible, is not affected by exposure to 
the atmosphere, and is easily levelled. 

Sand is also almost incompressible, and forms an excellent foun- 
dation as long as it can be kept from escaping; but as it has no 
cohesion, and acts like a fluid when exposed to running water, it 
should be treated with great caution. 

The foundation bed in soils of this kind is prepared by digging a 
trench from four to six feet deep, so that the foundation may be 
started below the reacli of the disintegrating effects of frost. 

The bottom of the trench is levelled ; and, if parts of it are required 
to be at different levels, it is broken into steps. 

Care shoulil l)e taken to keep the surface-water from running into 
the trench; and, if necessary, drains should be made at the bottom 
to carry away the water. 

The weight resting on the bottom of the trench should be pro* 
portional to the resistance of the material forming the bed. 



FOUNDATIONS. 133 

Mr. Gaudard says that a load of 10,500 to 18,300 pounds per 
square foot has been put upon close sand in tlie foundations of 
Gorai Bridge, and on gravel in the Lock Ken Viaduct at Bordeaux. 

In the bridge at Nantes, there is a load of 15,200 pounds to the 
square foot on sand; but some settlement has already taken place. 

Ilankine gives the greatest intensity of pressure on foundations 
in firm earth at from 2500 to 3500 pounds per square foot 

In order to distribute the pressure arising from the weight of the 
structure over a greater surface, it is usual to give additional breadth 
to the foundation courses: this increase of breadth is called the 
spread. In compact, strong earth, the spread is made one and a 
half times the thickness of the wall, and, in ordinary earth or sand, 
twice that thickness. 

Case II. — Foundations on soils firm enough to support the 
weight of the strtictiire, but whose fttaMUty l8 affected by water. 

The principal soil imder this class, with which we have to do, is 
a clay soil. 

In this soil the bed is prepared by digging a trench, as in rocky 
soils; and the foundation must be sure to start below the frost-line, 
for the effect of frost in clay soils is very great. 

The soil is also much affected by the action of water; and hence 
the ground should be well drained before the work is begun, and 
the trenches so arranged that the water shall not remain in them. 
And, in general, the less a soil of this kind is exposed to the air and 
weather, and the sooner it is protected from exposure, the better for 
the work. In building on a clay bank, great caution should be used 
to secure thorough drainage, that the clay may not have a tendency 
to slide daring wet weather. 

The safe load for stiff clay and marl is given by Mr. Gaudard at 
from 5500 to 11,000 pounds per square foot. Under the cylindrical 
piers of the Szegedin Bridge in Hungary, the soil, consisting of 
clay intermixed with fine sand, bears a load of 13,300 pounds to 
the square foot; but it was deemed expedient to increase its sup- 
porting power by driving some piles in the interior of the cylinder, 
and also to protect the cylinder by sheeting outside. 

Mr. McAlpine, M. Inst. C.E., in building a high wall at Albany, 
N.Y., succeeded in safely loading a wet clay soil with two tons to 
the square foot, but with a settlement depending on the depth of 
the excavation. In order to prevent a great influx of water, and 
consequent softening of the soil, he surrounded the excavation 
with a puddle trench ten feet high and four feet wide, and he also 
spread a layer of coarse gravel on the bottom. 

Foundations in Soft Eurths. — There are three materials in gen- 
eral use for forming an aitificial bearing-stratum in soft soils. 



134 FOUNDATIONS. 

Whichever material is employed, the bed is first prepared by ezca^ 
vating a trench sufficiently deep to place the foundation-courses 
below the action of frost and rain. Great caution should be used 
in cases of this kind to prevent unequal settling. 

The bottom of the trench is made level, and covered with a bed 
of stones, sand, or concrete. 

Stones. — When stone is used, the bottom of the trench should 
In; paved with rubble or cobble stones, well settled in place by 
ramming. On this paving, a bed of concrete is then laid. 

Sand. — In all situations where the ground, although soft, is of 
sufficient consistency to confine the sand, this material may be used 
with many advantages as regards both the cost and the stability of 
the work. The quality which sand possesses, of distributing the 
pressure put upon it, in both a horizontal and vertical direction, 
makes it especially valuable for a foundation bed in this kind of 
soil; as the lateral pressure exerted against the sides of the founda- 
tion pit greatly relieves the bottom. 

There are two methods of using sand; viz., in layers and as piles. 
In fonning a stratum of sand, it is spread in layers of about nine 
inches in thickness, and each layer well rammed before the next 
one is spread. The total depth of sand used should be sufficient 
to admit of the pressiu^ on the upper surface of the sand being 
distributed over the entire bottom of the trench. 

Sand-piling is a very economical and efficient method of forming 
a foundation under some circumstances. It would not, however, 
be effective in very loose, wet soils; as the sand would work into 
the surrounding ground. 

Sand-piling is executed by making holes in the soil, or in the 
bottom of the trench, about six or seven inches in diameter, and 
about six feet deep, and filling them with damp sand, well rammed 
so as to force it into every cavity. 

In situations where the stability of piles arises from the pressure 
of the ground around them, sand-piles are found of more service 
than timber ones, for the reason that the timber-pile transmits 
pressure only in a vertical direction, while the sand-pile transmits it 
over the whole surface of the hole it fills, thus acting on a large 
area of bearing-surface. The ground above the piles should be 
covered with planking, concrete, or masonry, to prevent its being 
forced up by the lateral pressure exerted by the piles: and, on the 
stratum thus formed, the fomidation walls may be built in the usual 
manner. 

Fouiidatious on Piles. — Where the soil upon which we 
wish to build is not firm enough to support the foundation, one 
of the most common metliods of fonnhig a solid foundation bed is 



FOUNDATIONS. 136 

by driving wooden piles into tlie soil, ami placing the foundation 
wails upon these. 

The piles are generally round, and have a length of ahout twenty 
times iheir mean diameter of cross-section. The diameter of the 
hcjid varies from nine to eighteen inches. The piles should be 
straight grained, and free from knots and ring strokes. Fir, beach, 
oak, anil Florida yellow-pine are the best woods for piles; though 
spruce and hemlock are very commonly used. 

Where piles are exposed to tide-water, they are generally driven 
with their bark on. In other cases, it is not essential. 

Piles which are driven through hard ground, generally require to 
have an iron hoop fixed tightly on their heads to prevent them from 
splitting, and also to be shod with iron shoes, either of cast or 
wrought iron. 

Long piles may be divided into two classes, — those which trans- 
mit the load to a firm soil, thus acting as pillars; and those where 
the pile and its load are wholly supported by the friction of the 
earth on the sides of the pile. 

In order to ascertain the safe load which it will do to put upon 
a pile of the first class, it is only necessary to calculate the safe 
crushing-strength of the wood; but, for piles of the second and 
more common class, it is not so easy to determine the maximum 
load which they will safely support. 

Many writers have endeavored to give rules for calculating the 
effect of a given blow in sinking a pile; but investigations of this 
kind are of little practical value, because we can never be in pos- 
session of sufficient data to obtain even an approximate result. 
The effect of each blow on the pile will depend on the momentum 
of the blow, the velocity of the ram, the relative weights of the 
ram and the pile, the elasticity of the pile-head, and the resistance 
offered by the ground through which the pile is passing; and, as 
the last-named conditions cannot well be ascertained, any calcula- 
tions in which they are only assumed must of necessity Ikj mere 
guesses. 

I^ad on Piles. — Professor Rankine gives the limits of the 
safe load on pilesy based upon practical examples, as follows : — 

For piles driveil till they reach the firm ground, 1000 pomids per 
square inch of ar^ of head. 

For piles standing in soft ground by friction, 200 pounds per 
square inch of area of head. 

But as, in the latter case, so much depends upon the character of 
the soil in which, the piles are driven, such a gcneml rule as the 
above is hardly to be reconunended. 

Several rules for the bearing-load on piles have been given, 



Perhaps tho nile most commonly given is that of Major Sanders, 
United-States En«jint;er. He experimented largely at Fort Dela' 
ware, and in 1851 gave the following rule as reliable for ordinary 
pikMlriving. 

Sanders's Rule for determining the load for a common 
wooden pile, driven until it sinks through only small and nearly 
equal distances under successive blows : — 

,, , , , . „ weight of liammer in lbs. X fall in inches 
Safe load m lbs. = SXslnkin^t iS^blo^v^ 

Mr. John C. Trautwine, C.E., in his pocket-book for engineers, 
gives a rul(i which appears to agree very well with actual results. 
His rule is expressed as follows: — 

cube root of weight of x O^.*! 

Extreme load in _ fall in feet ha m mer in Ib:^. "'"^^ 
tons of 2240 lbs. ~ Last sinking in inches -h 1 

For the safe load he recommends that one-half the extreme load 
should be taken for i)iles thoroughly driven in firm soils, and one- 
fourth when driven in river-mud or marsh. 

According to Mr. Trautwine, the French engineers consider a 
pile safe for a load of 25 tons when it refuses to sink under a liam- 
mer of 1344 pounds falling 4 feet. 

The test of a pile having been sufficiently driven, acconling to 
the best authorities, is that it shall not sink more than one-fifth of 
an incli under thirty blows of a ram weighing 800 pounds, falling 
5 f(H>t at each blow. 

A more common rule is to consider the pile fully driven wlien it 
does not sink more than one-fourth of an inch at the last blow of a 
ram weighing 2500 pounds, falling 80 feet. 

In ordinary pile-driving for buildings, however, the piles often 
sink more than this at the last blow; but, as the piles are seldom 
loaded to their full capacity, it is not necessary to be so i)articular as 
in tlie foundations of engineering structures. A common practice 
witli :ircliitects is to specify the lengih of the piles to be usi»d, and 
the ])iles ;in» driven imtil their heads are juat al)Ove ground, and 
then left to he levelled off afterwards. 

Kxamplo of I^ile Foundation. — As an example of the 
ni"thf/!l of di'termining the necessary numl>er of plh»8 to 8up]K)rt 
a i:iv«'n building, we will determine tho numlKT of piles nM|ulr«Ml 
to MUi)port the sidivwalls of a warehouse (of which a vertical sec- 



tlon is shown in Fig. 1). The walls aro of brick, and the weight 
may be taken at 110 pounds per cubic foot of masonry. 

The piles are to be driven in two rows, two feet on centres; and 
It is found that a pile 20 feet long and 10 inches at the top will sink 




Fig. 1. 

one inch under a 1200-pound hammer falling 20 feet after the pile 
has been entirely driven into the soil. What distance should the 
piles ba on centres lengthwise of the wall ? 



 4 



138 FOUNDATIONS. 

Hy calculation wc find tliat the wall contains 157i^ cubic feet of 
masonry per running foot, and hence weiglis 17,306 pounds. 
The load from the floors which comes upon the wall is: — 

From the first floor 1500 lbs. 

From the second floor loSO ll>s. 

From the third floor 1380 lbs. 

From the fourth floor 790 lbs. 

From the fifth floor 720 lbs. 

From the sixth floor 720 lbs. 

From the roof 240 lbs. 

Total 6730 lbs. 

Hence the total Ayeight of the wall and its load per running foot is 

24,0:56 pounds. 

Tlie load which one of the piles will support is, by Sanders's rule, 

1200 X 240 

— ^^"^7~f — — 36000 pounds. 

By Trautwine's rule, using a factor of safety of 2.5, the safe load 

would be 

(^20 X 1200 X 0.023 

— *j 5 X (14-1 ) ~ ^'^ ^^^^ ^^^ ^^'^ pounds), or 33600 pounds. 

Then one pair of piles would support 72,000, or 67,200 pounds, 
according to which rule we take. 

Dividing these numbers by the weight of one foot of the wall 
and its load, we find, that, by Sandei-s's rule, one pair of piles will 
support 3 feet of the wall, and, by Trautwine's rule, 2.8 feet of wall: 
hence the piles should be placed 2 feet 9 inches or 3 feet on centres. 

In very heavy buildings, heavy timbers are sometimes bolted to 
the tops of the piles, and the foundation walls built on these. 

In Boston, Mass., a large part of the city is built upon made 
land, and hence the buildings have to be supported by pile founda- 
tions. The Building Laws of the city require that all buildings 
"exceeding thirty-five feet in height (with pile foundation) shAll 
have not less than two rows of piles under all external and party 
walls, and the piles shall be spaced not over three feet on centres 
in the direction of the length of the wall." 

^l.s (m example of the load which ordinary piles in the made 
land of Boston will support, it may be stated that the piles under 
Trinity ('hurch in Boston support two tons each, approKimately. 

For engineering works, various kinds of iron piles are used; baft 
they are too rarely used for foundations of buildingB to come 
within the scope of this chapter. For a description of these 



FOUNDATIONS. 139 

le reader should consult some standard work on engineering, 
good description of iron piles is given in "Wheeler's Civil 
jering," and also in " Trautwine's Handbook." 
icrete Foundation Beds. — Concrete is largely used 
ndation beds in soft soil, and is a very valuable material for 
rpose; as it affords a firm solid bed, and can be spread out 

> distribute the pressure over a large area. 

;rete is an artificial compound, generally made by mixing 
cement with sand, water, and some hard material, as bi*oken 
slag, bits of brick, earthenware, burnt clay, shingle, etc. 
e is any choice of the materials forming the base of the 
:e, the preference should be given to fragments of a some- 
K>rous nature, such as pieces of brick or limestone, rather 

> those with smooth surfaces. {See page liSa.) 

broken material used in the concrete is sometimes, for con- 
2e, called the agrjregate, and the mortar in which it is incased, 
sitrix. The aggregate is generally broken so as to pass 
b a li or 2 inch mesh. 

imp ground or imder water, hydraulic lime should of course 
I in mixing the concrete. 

ingr Concrete. — A very common practice in laying con- 
1 to tip the concrete, after mixing, from a height of six or 
3et into the trench where it is to be deposited. This process 
;ted to by the best authorities, on the ground that the heavy 
:ht portions separate while falling, and that the concrete is 
•re not uniform throughout its mass. 

best method is to wheel the concrete in barrows, immedi- 
fter mixing, to the place where it is to be laid, gently tipping 
position, and carefully ramming into layers about twelve 
thick. After each layer has been allowed to set, it should 
pt clean, wetted, and made rough, by means of a pick, for the 
yer. 

; contractors make the concrete courses the exact width 
d, keeping up the sides with boards, if the trench is too 
This is a bad practice; for when the sides of the founcla- 
;s are carefully trimmed, and tlie concrete rammed up solidly 
them, the concrete is less liable to ha crushed and broken 
it has entirely consolidated. It is therefore desirable that 
K:ifications for concrete work should require that the whole 
of the excavation be filled, and that, if the trenches are 
ted too wide, the extra amount of concrete be furnished at 
itractor's expense. 
Tete made with hydraulic lime is sometimes designated as 



140 FOUNDATIONS. 

The pressure allowed on a concrete bed should not exceed one> 
tenth part of its resistance to crushing. Trautwine gives as the 
average crushing-strength of concrete forty tons per square foot. 

Foiiudations in Compressible SoiL— The great diffi- 
cully mot with in fonuing a iinu bed in compressible soils arises 
from the nature of the soil, and its yielding in all directions under 
pressure. (See page 144.) 

There are several methods which have been successfully em- 
ployed in soils of this kind. 

I. When the compressible material is of a moderate depth, the 
excavation is made to extend to the firm soil beneath, and the 
fomulation put in, as in firm soils. 

The principal objection to this method is the expense, which 
would often be very grea.t. 

II. A second method is to drive piles through the soft soil into 
the tlrm soil beneath. The piles are then cut oif at a given level 
and a timber platform laid upon the top of the piles, which serves 
as a support for the foundation, and also ties the tops of the piles 
together. 

III. A modification of the latter method is to use shorter piles^ 
which are only driven in the compressible soil. The platform is 
made to extend over so large an area that the intensity of the press- 
ure per square foot is within the safe limits for this particular 
soil. 

lY. Another modification of the second method consists in 
using piles of only five or six inches in diameter, and only five or 
six feet long, and placing them as near together as they can be 
driven. A platform of timber is tlien placed on the piles, as in the 
second metho<l. 

Tht^ object of the short piles is to compress the soil, and make it 
tirmor. ''This practice is one not to be recommended; its effect 
bein<i^ usually to pound up the soil, and to bring it into a state 
which can best Xh', described by comparing it to batter-pudding." * 

V. Still another method is to surround the site of the work with 
shccL-piling (flat piles driven close together, so as to fonn a sheet), 
to prcvi>nt the esca^Mi of the soil, which is then consolidated by 
driving ]>iles into it at short distunires from each otlier. The piles 
are then sawn oft' level, and the ground excavated between them 
for two or three feet, and filled up with concrete: the whole is tlien 
planked ovt;r to re(!eive the superstructure. 

The great point to be attended to in building foundations in soils 
of this kind is to distribute the weight of the structui'e equally 



1 Dobeon on Fouiidatloiirt. 



FOUNDATIONS, 141 

over the foundation, wtilcfa will then seLlle In a vertical direction, 
and cause little Injui-y; wh^'eas any irregular aettlement would 
rend the work from top to bottom. 

Planking for Poiinaation Beds.— In erecting buildings 
□n soft groimd. where a large briiring-siirface ia required, planking 
may be resorted to with great advantage, provided tbo timber can 
lie kept from decay. If the ground is wet ami the timber good, 
there ia little to fear in thia respect; but in a dry aituatlon, or one 
expoaed to alternations of wet and dry, no dependence can be 
placed on unprepared timber. There are several methods cm- 
ployed for the presei-vation of timber, such as kyanlzing oi' creo- 
Mting: and the timber used for fouiidatlona should be trcaleil by 
one of these methods. 

The advantage of timber Is, tliat it will resist a great cross-strain 
with very triOing flexure; and therefore a wide fooling may l>e ob- 
tained without any excessive spreailing of the bottom courses of 
tbe masonry. The best method of employing planking under walls 
is to cut the stuff into short lengths, which should be placctd 
acroKS Uie foundation, and tied longitudinally by planking laid to 
the width of the bottom course of masonry in tlie direction of the 
length of the wail, and firmly spiked to the bottom planking. 
Another good method of using planking ia to lay down sleepers 
on the ground, and fill to their top with cement, and then place tlie 
planking on the level surface thus formed. For the cross'timbers, 
four-Inch by six-inch timber, laid flatwise, will answer in ordinary 

FouiKlations for Cliimiteys. — As examples of tlie foun- 
dations i'ci|uired for very high chimneys, we quote the following 
front a treatise on foundations, in the latter part of a work on 
"Foundations and Foundation Walls," i»y George T. PowelL 



Fig. 2 represents the l>ase of a cliimiiey erected in IfS 
Chicago Refining Company, 1.51 feet high, and 12 feet aqm 



142 FOUNDATIONS. 

SooL Tlic bnse, merely two courses of lieavy dlmeiuloD stone, lu 
shown, is bedded upon the aurface^ravel near the mouth of the 
rivet, there recently deposited by the lake. The inorUr employeil 
In the joint between thu stone Is rooting-gr&vel in cement. The 
an'a of the base is '£>!; square feet, the woight of chimney, iDcluslvu 
of bnse, 025 tons, giving a pressure of 34 pounds to the square 
inirh. This foundation provei! to \x; perfecL 

Fig. 3 represents the base of a chimney ereelcil in 1872 for tliii 
Hcl'orniick Iteaper Works, Clilcago, which is 160 feet liigh, 14 feet 
square at the foot, with a round flue of (t feet 8 inches diameter. 



FiB- 3. 

The base covers 025 square feet; the weight of the chimney and 
base is approximately 1100 tons; the pressure upon the ground 
(liry liard clay) ia therefore 24^ |>ouniis to the square inch. This 
foundation also proved to be perfect in every respect. 

Bftitrinif Power of Soils. 

{Added to A'Mli JtlditioH.j 
In u imjier publislied in tJiu Ameritmn Arehiteet and BuHdinf 
JVVjuw, November 3. 188«, hy J'rof. Ira O. Baker, C.R.. on the 
Hearing I'owcr of Soils, iio sums up the resulta of his discussion in 
tho following liibli', which t;ivus values which seom to the writer to 
be both praclieal anil I'vliablu. The remiirks ((blowing the tBl>lc 
should al.so bo cart^fully cnnsidifred. 



FOUNDATIONS. 



r4b 



Kind of Matbbial. 



Rock— the hardest— in thick lay^ers, in native bed 

Hock equal to best ashlar masonry 

Kock equal to best brick masonrj' 

Kock equal to poor brick masonry 

Clay on thick beds, always dry 

Clay on thick beds, moderately dry m 

Clay, soft 

Gravel and course tiand, well cemented 

Sand, compact and well cctmeuted 

Sand, clean, dry 

Quicksund, alluvial noils, etc 



Bearing power in tons 
per square foot. 



Min. 


Max. 


200 




25 


30 


15 


20 


5 


10 


4 


6 


2 


4 


1 


2 


8 


10 


4 


6 


2 1 


4 


0.5 


1 



" Conclusion. — It is well to notice that there are some practical 
considerations which modjiy the pressure which may safely be put 
upon the soil. For example, the pressure on the foundation of a 
tall chimney should be considerably less than that of the low mas- 
sive foundation of a fireproof vault. In the former case a slight 
inequality of bearing power, and consequent unequal settling, 
might endanger the stability of the structure; while in the latter 
no serious harm would result. The pressure per unit of area 
should be less for a light structure subject to the passage of heavy 
loads— as, for example, a railroad viaduct — than for a heavy struct- 
ure, subject only to a quiescent load, since the shock and jar of 
the moving load are far more serious than the heavier quiescent 
load." 

The following list of actual known weight on different soils will 
give a very good idea of what has been done in actual practice. 

Rock. — St. Rollox chimney, poorest kind of sandstone, 2 tons 
per square foot. 

Clay. — Chimney, McCormick Reaper Works, Chicago, 1^ tons 
per square foot on dry, hard clay. 

Capitol at Albany, N. Y., rests on blue clay containing from GO 
to 90 per cent, of alumina, the remainder being fino sand, and con- 
taining 40 per cent, of water on an average. The safe load was 
taken at 2 tons per square foot. 

In the case of the Congressional Library at Washington, which 
rests on "yellow clay mixed with sand," 2^ tons per square foot 
was taken Tor the safe load, ** Experience in Central Illinois shows 
that if the foundation is carried down below the action of the frost 
the clay subsoil will bear 1^ to 2 tons per square foot without ap- 
preciable settling. " * 



* In O. Baker, Amerkan Architect, November 8, 1888. 



144 FOUNDATIONS. 

Sand and Gravel. — "In an experiment in Finance, eiean 
river sand, compacted in a trench, supported 100 tons per sqaare 
foot. 

** The p'.ers of the Cincinnati suspension bridge are founded on a 
bed of coarse gravel 12 feet below water; the maximum pressure on 
the gravel is 4 tons per square foot. 

*'Thc piers of the Brooklyn suspension bridge are founded 44 
feet below the bed of the river, upon a layer of sand 2 feet thick 
resting upon bed-rock ; the maximum pressure is about 5^ tons 
per square foot. 

** At Chicago, sand and gra,vel about 15 feet below the sarfaoe 
are successfully loaded with 2 to 2.V tons per square foot. 

'* At Berlin the safe load for sandy soil is generally taken at 2 
to 2^ tons per square foot. 

" The Washington Monument, Washington, D. C, rests upon a 
bed of very fine sand 2 feet thick. The ordinary pressure on cer- 
tain parts of the foundation i^eing not far from 11 tons per square 
foot, which the wind may increase to nearly 14 tons per square 
foot."* 

Foundations on Soft, Yielding Soil, BuUt of Steel 
Seams and Concrete. — On page 141 is described the method 
of planking for foundations, wliich does very well where the timber 
is sure to bo always wot, but, if there is any chance of its ever 
becoming dry, iron or steel beams should be used instead. Steel 
rails were first used embedded in concrete, but they oflfer, however, 
comparatively little resistance to deflection, and for this reason, if 
allowed to project beyond the masonry to any considerable length, 
the concrete filling is liable to crack, and thus the strength of the 
foundation become impaired. 

Steel I-beams, more recently used for this purpose, are found 
to be superior in every respect. A greater depth can be adopted, 
the deflection thus reduced to a minimum and a sufficient saving 
effected to more than compensate for their additional cost per 
pound. 

The foundation should be prepart-d (see illustration, p. 146) by 
first laying ji bed of concrete to a depth of from 4 to 1*3 inches and 
then placing upon this a row of I-beams at right angles to the face 
of the wall. In the case of heavy ])iei's, the beams may be crossed in 
two directions. Their distances apart, from centre to centre, may 
vary from 9 to 24 inches according to circumstances, i.e,, length 
of their projection beyond the masonry, thickness of concrete, esti- 
mated pressure per square foot, etc. They should be plaoed at 
least far enough apart to permit the introduction of the oonczeto 

* Ira O. Baker, American Architect, Novonber 8, 18B8. 



FOUNDATIONa 145 

filling and its proper tamping between the beams. Unless the 
concrete is of unusual thickness, it will not be adyisable to exceed 
20 inches spacing, since otherwise the concrete may not be of suffi- 
cient strength to properly transmit the upward pressure to the 
beams. The most useful application of this method of founding 
is in localities where a thin and comparatively compact stratum 
overlies another of a more yielding nature. By using steel beams 
in such cases, the requisite spread at the base may be obtained 
without either penetrating the firm upper stratum or carrying the 
footing-courses to such a height as to encroach unduly upon the 
basement-room . 

MBTHOD OF OALCULATINa THB 8IZI3 AND 
LENGTH OF THE BEAMS.^ 

Let L — Weight of wall per lineal foot, in tons. 
and h = Assumed bearing capacity of ground, per square 
foot (usually from 1 to 3 tons). 

Thei;i -r = IF =? Required width of foundation, in feet. 

w = Width of lowest course of footing stones. 
p = Projection of beams beyond masonry, in feet. 
8 = Spacing of beams centre to centre, in feet. 

Evidently the size of beams required will depend upon their 
strength as cantilevers of a lengthy, sustaining the upward reaction, 
which may be regarded as a uniformly distributed load. 

Thus ^ & = uniformly distributed load (in tons) on cantilevers, 

per lineal foot of wall, 

and ph8 = uniform load in tons, on each beam. 

The table on the following page gives the safe lengths p for the 
various sizes and weights of steel beams, for s--l foot and 6 rang- 
ing from 1 to 5 tons per square foot. For other values of 8 say 15 
inches, i. «., 1| i^^t, the table may be used by simply considering b 
increased in the same ratio as 8 (see example below). As regards 
the weight of beams, it is advantageous to assign to 8 as great a 
value as is warranted by the other considerations which obtain. 

EXAMPLE SHOWING APPLICATION OF TABLE. 

The weight of a brick wall, together with the load it must sup- 
port, is 40 tons per lineal foot. The width of the lowest footing- 
course of masonry is 6 feet. Allowing a pressure of 2 tops per 

* This and the next page are taken by permiBsion from Carnegie, Phipps & 
Co.*8 Pocket-book. 



FOUNDATIONS. 



Bquare foot od tho foundation, what dse ftnd length of steet I-bemu 

18 inches dcnCre to centre will be required ? 

Am : L -40 ;6 = 2;w = C;a = U. 

Therefore ir = 40 -^ 3 = 20 feet, the required lei^h ol beams. 
The projection jj = HSl* - 8) = 7 feet. 

In order to apply tho table (calculated for « = 1 fool) wc must 
consider 6 increased in tho same ratio as «, t'.e., 6 = 3 x 1^ =S 

In the eolumn for 3 tons, we find the length 7 feet to agree with 
30 inches I-lieams G4.0 pounds per foot. 




TABLE OIVINQ SAFE LENGTHS OF FROJECTIOKS p IN FEBT (BSB 

ILLl'STRATION). FOK I - 1 FOOT AND VALUES OF ft BAITQING 

FROM 1 TO 5 TONS. 



Depth lw«tght 






>. 


Tos 


7i 


Foot). 


u 


11 


7i 


SO SO 


\i 


%\f,i 


!o:o 


a 


15 7S 
15 flO 

IS 1 « 


11 


5 , 10.5 


S,B 




■m 


Wt 


li 40 

10 ! ai. 


6 


V li 








""a 


g ' « 




1 ! h.o 






m 





FOUNDATIONS. 147 

The foregoing table applies to sied beams. Values given leased 
on extreme fibre strains of 16,000 pounds per square inch. 

Chicago Foundations*'" — The architects and builders of 
Chicago probabijT have to deal with the most unfavorable condi- 
tions for securing a good ^foundation for their heavy buildings of 
any people in the world. 

1 he soil under the central part of the city consists of a black 
loamy clay, which is tolerably firm at the surface, and will sustain 
a load of from one to three tons per foot, depending upon locality. 
A few feet below the natural surface of the ground the soil becomes 
quite soft, growing more and more so the deeper the excavation is 
carried, and at a depth of from 12 to 18 feet it is so yielding that 
nothing can be placed upon it with any reliance. Nor is this all. 
It has been discovered, by many failures in buildings, that there is a 
broad subterranean layer of soft mud which lies directly across the 
most heavily built portion of the city, extending under the Post- 
office, and reaching from the lake to the river, a distance of three- 
quarters of a mile. 

The first of the larger structures were built with continuous 
foundation walls, with wide footings, the width being proportioned 
tx) the loads bearing upon them. This method, however, did not 
prove successful, as it was foimd that the wall will settle more than 
a pier, and the comers of the wall will settle less than the centre. 

After experiments of one kind and another, it has come to be the 
accepted practice in Chicago of dividing the foundation into iso- 
lated piers, the footing of each pier being carefully proportioned 
according to the load upon it, its position in the building, char- 
acter of the superstructure, etc., so that all shall settle at exactly 
the same rate without any crackings or detriment to the super 
structure. 

The footings of the piers are built of steel beams and concrete, 
as described on page 145, except that the beams are often crossed 
three and four times ; in this way a great spreading is obtained in 
a small height. 

In determining the area of the footings, the ground is assumed to 
be capable of sustaining a safe load of from 1| to 2^ tons per 
square foot. The loads on the piers of the Board of Trade building 
vary from 2| to S^ tons per square foot. The size of the footings 
under the piers and the corners is made less than under the walls, 
to offset the difference in settlement of the different portions of the 
building. 
•^-.^    

* 0. H. BlMkall, in American Architect, p. 147, Vol. XXUI. 



148 FOUNDATIONS. 

It is found that a heavy pier will sink proportionally more than a 
light one, so that the area under the larger piers is made relatively 
greater than under the smaller ones. 

Again, it is necessary to take into account the material of which 
the superstructure is to be built. Thu?, a footing under a brick 
wall i^ made larger than a footing under a line of iron columns, so 
that if both footings aro loaded with the same weight, thiit under 
the columns will settle the most, to allow for the compression in 
the joints of the mason- work. 

It is impossible to build heavy buildings on the Chicago w)il 
without settlement, and the architect must therefore plan his build- 
ing so that all parts shall settle equally, and this has been success- 
fully done in many of the largest buildings. 

In a building where the footings aro proportioned to give a bear- 
ing weight on the ground of 2+ tons per square foot, it is esti- 
mated that the building will settle about 4 inches altogether. 

Piling has been successfully used under several buildings in 
Chicago, and there seems to bo no reason why it should not be more 
extensively resorted to. 

In the construction of the large grain elevators which are seat> 
tercd through the city the loads are so excessive, reaching as high 
as six tons per foot, that it would be impracticable to support them 
on ordinary footings, and piling has been resorted to. The piles 
are driven a distance of twenty to forty feet down to hard-pan, 
cap[)ed by wooden sleepers, with heavy wooden cross-beams and 
solid planking to receive the masonry. 



CONCBETE FOOTING FOB FOUNDATIONS. 148a 



OONORSTB FOQ!nNQ> FOR FOUNDATIONS. 

For the footings of foundations in nearly all kinds of soil where 
piles are not used, the writer believes a good concrete to be prefer- 
able to even the best dimension stone, for the reason that it acts as 
one piece of masonry and not as individual blocks of stone, and if 
made of sufficient thickness it will possess sufficient transverse 
strength to span any weak place in the soil beneath, if not of large 
area. 

When the best brands of Portland cement are used, the propor- 
tions may be as follows : 

One part Portland cement ; 3 parts clean sharp sand ; 5 parts chip 
stone, in sizes not exceeding 2 x 1^ x 3 inches. Using these pro- 
portions, one barrel of cement will make from 22 to 26 cubic feet 
of concrete. 

The above proportions were used in the concrete for the founda- 
tions of the Mutual Life Insurance Company's Building, New York 
City 

When the cement is not of the best quality, or other cement than 
Portland cement is used, more cement should be used with the 
other material. Using a cement made in the West, the author 
specifies that one part of cement to two of sand and four of broken 
stone should be used, and the result has been very satisfactory. 

It will generally be found wise to keep an inspector constantly 
on the ground while the concrete is being put in, as the temptation 
to the contractor to economize on the cement is very great. 

In mixing the concrete, the stone, sand, and cement should be 
thrown into the mortar box in the order named , and while one man 
turns on the water two or more men should rapidly and thoroughly 
work the material back and forth with shovels, when it should be 
imiuediatelv carried to the trenches. The concrete should be 
deposited in layers not over six inches thick, and each layer \\ell 
rammed. If one layer dries before the next is deposited it should 
be well wet on top, just before depositing the next layer. 

Care should be exercised to see that the trenches are not dug 
wider than the desired width of the footings ; and also in mixing 
the concrete, not to use more water than is necessary to bring the 
mass to a pudding-like consistency, as otherwise the cement may 
be washed away. 



148^ COST OF CONCRETE. 



COST OF OONORSTB. 

The cost of labor in mixing concrete, when the proper facilities 
are provided, need not exceed three cents a cubic foot, and four 
cents is a liberal allowance, with wages at two dollars a day. The 
vunount of materials required to make 100 cubic feet of concrete 
may be taken as follows : proportion of 1 to 6, 5 bbls. cement 
(original package) and 4.4 yards of stone and sand ; proportion of 
1 to 8, 3.9 bbls. of cement and 4i yards of aggregates. 

The cost of concrete at the present time in Denver is about thirty 
cents per cubic foot. 

The weight of concrete varies from 130 to 140 lbs. per cubic foot, 
according to the material used, granite aggregates making nat- 
urally the heaviest concrete. 



MASONRY WALLS. 149 



CHAPTER III. 
MASONRT TV ALLS. 

Footingr Courses. — In commencing the foundation walls 
of a building, it is customary to spread the bottom courses or the 
masonry considerably beyond the face of the wall, whatever be the 
character of the foundation bed, unless, perhaps, it be a solid rock 
bed, in which case the spreading of the walls would be useless. 
These spread courses are technically known as " footing courses." 
They answer two important purposes : — 

:ist, By distributing the weight of the structure over a larger 
area of bearing-surface, tlie Uability to vertical settlement from 
the compression of the ground is greatly diminished. 

2d, By increasing the area of the base of the wall, they add to 
its stability, and form a protection against the danger of the work 
being thrown out of "plumb" by any forces that may act against 
it. ... 

Footings, to have any useful effect, must be securely bonded into 
the body of the work, and have sufficient strength to resist the 
violent cross-strains to which they are exposed. 

Footings of Stone Foundations. — As, the lower any 
stone is placed in a building, the greater the weight it has to sup- 
port and the risk arising from any defects in the laying and dress- 
ing of the stone, the footing courses should be of strong stone 
laid on bed^ with the upper and lower faces dressed true. By laying 
on. bed is me^nt laying the stone the same way that it lay before 
quarryin{]j. 

In la3^ng the footing courses, no back joints should be allowed 
beyond the face of the upper work, except where the footings are 
in double courses; and every stone should bond into the body of 
the work several inches at least. Unless this is attended to, the 
footings will not receive the weight of the superstructure, and will 
be useless, as is shown in Fig. 1. 

In proportion to the weight of the superstructure, the projection 
of each footing course beyond the one above it must be reduced, or 
the cross-strain thrown on the projecting portion of the masonry 
will rend ft from top to bottom^ as shown in Fig. 2. 

la- boildllig 1st)9e mlMses of work, such as the abutments of 



150 



MASONRY WALLS. 



bridges and the like, the proportionate increase of bearing-surface 
obtained by the footings is very slight, and there is generally great 
risk of the latter being broken off by the settlement of the body 



f 



A' 



^ 



P 



] 



^,/ 



EEL 



^IL 



1 



Fig. 1. 



Fig. 2. 



of the work, as in Fig. 2. It is therefore usual in these cases to 
give very little projection to the footing courses, and to bring up 
the work with a battering-face, or with a succession of very slight 
offsets, as in Fig. 3. 



A 



'r* 



/^''//x' 



Hl"l/ -^ 



Fig. 3. 

Footings of undressed rubble built in common mortar should 
never be used for buildings of any importance, as the compression 
of the mortar is sure to cause movements in the superstructun*. 
Jf rubble must be used, it should be laid with cement mortar, £o 
that the whole will form a solid mass; in which case the size aiul 
shape of the stone are of little consequence. 

In general, footing stones should be at least two by three feet on 
the bottom, and eight inches thick. 

The Building Laws of the city of New York require that ttie 
footing under all foundation walls, and under all plejs, columns, 
posts, or pillars resting on the earth, shall be of stone or concrete. 
Under a foundation wall the footing must be at least twelve inches 
wider 1 aan the bottom width of the wall, and under pler% wrtnmnUi 



MASONRY WALLS. 



151 



its, or pillars, at least twelve inches wider on all sides than the 
;tom width of the piers, columns, posts, or pillars, and not less 
m eighteen inches in thickness; and, if huilt of stone, the stones 
ill not he less than two by three feet, and at least eight inches 
ck. 

Vll base-stones shall be well bedded, and laid edge to edge; and, 
Lhe walls are built of isolated piers, then there must be inverted 
hes, at least twelve inches thick, turned under and between the 
rs, or two footing courses of large stone, at least ten inches 
ck in each course. 

The Boston Building Laws require that the bottom course for all 
indation walls resting upon the ground shall be at least twelve 
hes wider than the thickness given for the foundation walls. 
footings of Brick Foundations. — In building with 
ck, the special point to be attended to in the footing courses is 



1 BRICK 



^^. 



T^^ 



<5s. -^=i 



A 



M BRICK 



<>^, 



'y'yy^' ^^ 



H%v. %\ 



E3. 



y>- 'ssr 



''^^ 



Fig. 4. Fig. 5. 

keep the back joints as far as possible from the face of tht: 
rk; and, in ordinary cases, the best plan is to lay the footings in 



2 BRICKS 



'W^Tm 



'////"/" =^ 



y/yy 



'''/// 



\\\ll'^ 



^^-*-^ 



'// 



v/. 



\\\\ 



''//// 



;ll///^'^ I //////,, =g'.' 



y/ 



'^^///„ 



Fee. 

;le courses; the outside of the work being laid all headers, and 
course pix>jecting more than one-fourth brick beyond the one 
>?e ity exo^ in. the case of an eight-inch wail 



154 MASONRY WALLS. 

inches thick below the top floor, and stone walls not less than six- 
teen inches. 

The thickness of the walls required by the laws of the cities of 
Boston, New York, and Denver, Colo., are shown by the tables on 
pp. 155-157. 

The Boston Law also contains the following provisions, which 
form an excellent guide to architects in other localities : 

Section 38. Vaulted walls shall contain, exclusive of withes, 
the same amount of material v.a is required for solid walls, and the 
walls on cither side of the air-space shall be not less than eight 
inches thick, and shall be securely tied together with ties not more 
than two feet apart. 

Section? 39. In reckoning the thickness of walls, no allowance 
shall be made for ashlar, unless it is eight inches or more thick, 
in which case the excess over four inches shall be reckoned as part 
of the thickness of the wall. Ashlar shall be at least four inches 
thick, and properly held by metal clamps to the backing, or prop- 
erly bonded to the same. 

Section 40. External walls may be built in part of iron or steel, 
and when so built may be of less thickness than is above required 
for external walls, provided such walls meet the requirements of 
this act as to strength, and provided that all constructional parts 
are wholly protected from heat by brick or terra-cotta, or by 
plastering three-quarters of an inch thick, with iron furring and 
wiring. 

First and Second Class Buildings. 

Section 45. First and second class buildings hereafter bnilt 
shall have floor bearing supports not over thirty feet apart. These 
supports may be brick walls, trusses or columns and girders. Such 
brick walls may be four inches less in thickness than is required 
by this act for external and party walls of the same height, pro- 
vided they comply with the provisions of this act as to the strength 
of materials, but in no case less than twelve inches thick. When 
trusses are used, the walls upon which they rest shall be at least 
four inches thicker than is otherwise required by sections thirty-siz 
and thirty-seven, for every addition of twenty-five feet or part 
thereof to the length of the truss over thirty feet. 

Section 46. Second class buildings hereafter buHt shall be so 
divided by brick partition walls of (ho thickness prescribed for 
bearing partition walls and carried twelve inches above the roof, 
that no space inside any such building shall exceed in area tea 
thousand square feet, and no existing wall in any aeoond 



MASONRY WALLS. 



165 



building shall be removed so as to leave an area not so enclosed, of 
more than ten thousand square feet. 

Section 47. All walls of a first or second class building meet- 
ing at an angle shall be united every ten feet of their height, by- 
anchors made of at least two inches by half an inch wrought iron 
securely built in to the side or partition walls not less than thirty- 
six inches, and into the front and rear walls at least one-half the 
thickness of such walls. 

The New York Law also provides that the bearing walls of all 
buildings exceeding one hundred and five feet in depth without a 
cross wall, or piers or buttresses, shall be increased four inches in 
thickness for each additional one hundred and five feet in depth 
or part thereof; also, in case the walls of any building are less 
than twenty feet apart and less than forty feet in depth, or there 
are cross walls, or piers or buttresses, which serve to strengthen 
the walls, the thickness of the interior walls may be reduced in 
thickness at the judgment of the superintendent of buildings. In 
comparing the thickness of brick walls in the eastern and western 
portions of the country, it should be taken into consideration that 
the eastern brick arc much harder and stronger than those in the 
west, and that an eight-inch wall in Boston is probably as strong 
(to resist crushing) as a thirteen-inch wall in Denver, Colo. 



THIOKNBS8 OF WAIX8 REQUIRZSD IN DENVER, 

OOI.O. 

FOR DWELLINGS, TENEMENTS, OR LODGING HOUSES. 



Outside and Party Walls. 



Onestorjr, ,. 
Two stories.. 
Three stories. 
Four stories. . 
Five stories.. 
Six stories . . . 







*» 


, 


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One story. .. 
Two atones.. 
Three stories. 
Foor stories. . 
Five stories.. 
Six stories... 



BUILDINGS OTHER THAN THE ABOVE. 



13 


8 










17 


13 


13 








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158 COMPOSITION OF FORCES. ETC. 



CHAPTER IV. 

COMPOSITION AND RESOLUTION OF FORCZSa^ 

CENTRE OP GRAVITY. 

Let us imagine a round ball placed on a plane surface at A (Fig. 
1), the surface being perfectly level, so that the ball will have no 
tendency to move until some force is imparted to it. If, now, we 
impart a force, P, to the ball in the direction indicated by the 
arrow, the ball will move off in the same direction. If, instead of 
imparting only one force, we impart two forces, P and Pi, to the 

ball, it will not move in the direction of 
either of the forces, but will move off in 
the direction of the resultant of these 
>B forces, or in the direction Ab in the figure. 
If the magnitude of the forces P and Pi 
is indicated by the length of the arrows, 
then, if we complete the parallelogram 
ABCDy the diagonal DA will represent the 
direction and magnitude of a force which 
will have the same effect on the ball as the 
two forces Pi and P. If, in addition to the two forces P^ and P, 
we now apply a third force, Pg, the ball will move in the direction 
of the resultant of all three forces, which can be obtained by com- 
pleting the parallelogram ADEF, formed by the resultant!)^ and 

the third force Pg. The diagonal R of this second 
parallelogram will be the resultant of all three of 
the forces, and the ball will move in the direction 
Ae, In the same way we could find the resultant 
of any number of forces. 

Again : suppose we have a ball suspended in the 
air, whose weight is indicated by the line W (Fig. 
2). Now, we do not wish to suspend this ball by a 
vertical line above it, but by two inclined lines or 
Fig. 2. forces, P and Pi. What shall be the magnitude 

of these two forces to keep the ball suspended in just this position ? 
We have here just the opposite of our last case; and, instead of 
finding the diagonal of the resultant, we have the diagonal, which 
is the line IF, and wish to find the sides of the parallelogram. To 
do this, prolong P and Pi , and from B draw lines panUel to thfl^ 





COMPOSITION OF FORCES. 



159 




Fig. 3. 



to complete the parallelogram. Then will CA be the required 
magnitude for P, and CB for Pi. 

Thus we see how one force can be made to have the same effect 
as many, or manv can be made to do the work of one. Bearing 
the above in miad, we are now prepared to study the following 
propositions: — 

I. A force may be represented fry a straight line. 

In considering the action of forces, either in relation to struc- 
tures or by themselves, it is very convenient to represent the force 
gi'aphically, which can easily be done by a straight line having an 
arrow-head, as in Fig. 3. The length of the 
line, if drawn to a scale of pounds, shows 
the value of the force in pounds; the direc- 
tion of the line indicates the direction of the 
force; the arrow-head shows which way it 
acts; and the point A denotes the point of 
application. Thus we have the direction, magnitude, and point 
of application of the force represented, which is all that we need 
lo know. 

Parallelog^ram of Forces, — II. Jf two forces applied at 
one point, and actiny in the same plane, be represented by two 
straight lines inclined to each other, their resultant loill be equal 
to the diagonal qf tlie parallelogram formed on these lines. 

Thus, if the Hues AB and AC (Fig. 4) represent two forces act- 
ing on one point. A, and in the same plane, 
then, to obtain the force which would have the 
same effect as the two forces, we complete the 
parallelogram ABDC, and draw the diagonal 
AD» This line will then represent the result- 
ant of the two forces. 

When the two given forces are at right angles to each other, the 
resultant will, by geometry, be equal to the square root of the sum 
of the squares of the other two forces. 

The Triaui^le of Forces. — III. If 
three forces acting on a point be repre- 
sented in magnitude and direction by the 
aides of a triangle taken in order, they 
icill keep the point in equilibrium. 

Thus let P, Q, and R (Fig. 5) represent 
thi"ee forces acting on the point O. Now, 
if we can draw a triangle like that shown 
at the right of Fig. 5, whose sides shall be 
respectively {Murallel to the forces, and shall 
have thfl^ same relation to each other as do the forces, then the 




Fig. 4. 




160 



COMPOSITION OF FORCES. 



forces will keep the point in equilibrium. If such a triangle 
cannot be drawn, the forces will be unbalanced, and the point will 
not be in equilibrium. 

The Polygon of Forces. — IV. If any nwnher qf forcen 
actiny at a point can be represented in magnitude xmd direction by 
the aides of a polygon taken in order, they will be in equilibriwn. 

This proposition is only the preceding one carried to a greatei 
extent. 

Moments* — In considering the stability of structures and the 
strength of materials, we are often obligexl to take into considera- 
tion the moments of the forces acting on the structure or piece; and 
a knowledge of what a moment is, and the properties of moments, 
is essential to the praper understanding of these subjects. 

When we speak of the moment of a force, we must have in mind 
some fixed point about which the moment is taken. 

The moment of a force about any given point may be defined as 
the product of the force into the perpendicular distance from the 
point to the line of action of the force; or, in other words, the 
moment of a force is the product of the force by the arm with lohich 
it acVi. 
Thus if we have a force F (Fig. G), and wish to determine its 
moment about a point P, we determine the perpen- 
dicular distance Pa, between the point and the line 
of action of the force, and multiply it by the force 
in pounds. For example, if the force F were equal 
to a weight of 500 pounds, and the distance Pa 
were 2 inches, then the moment of the force about 
the point P would be 1000 inch-pounds. 

The following important propositions relating to forces and 
moments should be borne in mind in calculating the strength or 
stability of structures. 
V. — If any number of parallel forces act on a 1>ody, that the 

body shall be in eqvilihrimn, the nmn 
P^ of the forces acting in one direction 




Fig. 6 



P' 



Pi 



4 4 



Fig.7 
Pi,Pj, and P3. 



must equal the sum of the forces actr 
D lug in the opposite direction. 

Thus if we have tlie parallel 
forces P\ P*, P®, and P*, acting on 
the rod AB (Fig. 7), in the opposite 
direction to the forces Pi, P„ P„ 
then, if the rod is in equilibrium, the 
sum of the forces P' , P*, P«, and P», 
must equal the sum of the loroet 



COMPOSITION OF FORCES. 



161 





Fa 


1 


Fs 


4^ 


1^ .. . . 1 ' *■ 


-2 — ^ 


^ 


A ^ 


S ^ 


w 


.n 


^. ^ « _J O^ 


\ 


Fig. 8 


Fi 






. i. 





VI. If any nwnber of parallel forces act on a body in opposite 
directions, then, for the body to he in equilibrium, the sum of the 
moments tending to turn the body in one direction must equal 
the sum of the moments tending to turn the body in the opposite 
direction about any given point. 

Thus let Fig. 8 represent three parallel 
forces acting on a rod AB. Then, for the 
rod to be in equilibrium, the sum of the 
forces Ft and F3 must be equal to Ft. 
Also, if we take the end of the rod, A, 
for our axis, then must the moment of Fj 
be equal to the siun of the moments of 
F2 and Fi about that point, because the 
moment of Fi tends to turn the rod down 
to the right, and the moments of F^ and F^ tend to turn the rod 
up to the left, and there should be no more tendency to turn the 
rod one way than the other. For example, let the forces F^, F^, 
each be represented by 5, and let the distance ^a be represented 
by 2, and the distance Ac by 4. The force F, must equal the sum 
of the forces F3 and Ff, or 10; and its moment must equal the 
sum of the moments of F^ and Fs. If we take the moments around 
A, then the moment of F3 = 5 X 2 = 10, and of Fg = 5 X 4 = 20. 
Their simi equals 30: hence the moment of F| nmst be 30. Divid- 
ing the moment 30 by the force 10, we have for the arm 3; or 
the force Fi must act at a distance 3 from A to keep the rod in 
equilibrium. 

If we took our moments around b, then the force Fi would have 
no moment, not having any arm, and so the moment of F2 about 
5 must equal the moment of F3 about the same point; or, as in this 
case the forces are equal, they must both be applied at the same 
distance from b, showing that b must be halfway between a and c, 
as was proved before. 

Tlie Principle of the Lever.— 
Tills principle is based upon the two pre- 
ceding prox>osltions, and Is of great im- 
portance and convenience. 

VII. Xf three parallel forces acting in 
one place balance each other, then each ^ 
force must Ije proportionaX to tJie distance jq 
between the other two. 

Thus, if we have a rod AB (Figs. 9a, 
Ob, and 9c), with three forces, P|, P^, 
F9, acting QU it» that the rod shall be balanced, we must have the 



15 



12 



Fig. 9 a 



B 



Pi 



162 



COMPOSITION OF FORCES. 



following relation between the forces and their points of applica- 
tion; viz., — 

P, P2 P, 



or 



vn ' An ' AC 

Pi :P^ :Ps ::BC :AB : AC, 



This is the case of the common lever, anil gives the means of 
detennining how much a given lever will raise. 



p Pig.9 b 



B 



h 



Ftg.9o 



The proportion is also true for any arrangement of the forces 
(as shown in Figs, a, b, and c), provided, of course, the forces are 
lettered in the order sho^^Ti in the figures. 

Example. — Let the distance AC be 6 inches, and the distance 
CB be 12 inches. If a weight of 500 pounds is applied at the point 
B, how much will it raise at the other end, and what support will 
be required at C (Fig. 9b)? 

Ans, Applying the rule just given, we have the proportion: ^ 

P:, : P, :: AC : CB, or 500 : (P,) :: 6 : 12. 

Hence P, = 1000 poiuids; or 500 pounds applied at B will lift 1000 
suspended at A. The supporting force at C must, by proposition 
v., be equal to the sum of the forces Pi and Pj, or 1500 ponnds 
in this case. 

Centre of Gravity. — The lines of action of the force of 
gravity converge towards the centre of the earth; but the distance 
of the centre of the earth from the bodies which we have occasion 
to consider, compared with the size of those bodies, is so great, that 
we may consider the lines of action of the forces as parallel. The 
number of tin? forces of gravity acting upon a body may be consicU 
ered as equal to the numbei' of particles composing the body. 

The centre of (jratlty of a body may be defined *a8 the point 
through which the resultant of the parallel forces of graTlty, actiiif 
upon the body, passes in eveiy position of the body. 



CENTRES OF GRAVITY. 163 

If a iKxly be supported at its centre of gravity, and be turned 
about tliat point, it will remain in equilibrium in all positions. 
The resultant of the parallel forces of gravity acting upon a body 
is obviously equal to the weight of the body, and if an equal force 
be applied, acting in a line passing through the centre of gravity of 
the body, the body will be in equilibrium. 

Examples of Centres of Gravity. — Centre of Gravity of 
Lliies. StraiyfU Lines. — By a line is here meant a material line 
whose transverse section is veiy small, such as a very fine wire. 

The centre of gravity of a uniform straight line is at its middle 
point. This proposition is too evident to require demonstration. 

The centre of gravity of the perimeter of a triangle is at the 
centre of the circle inscribed in the lines joining the centres of 
the sides of the given triangle. 

Thus, let ABC (Fig. 10) be the given 
triangle. To find the centre of gravity of 
its perimeter, find the middle points, D, 
E, and F, and connect them by straight 
lines. The centre of the circle inscribed 
in the triangle formed by these lines will g- 
be the centre of gravity sought. 

Symmetrical Lines, — The centre of 
gravity of lines which are sjrmmetrical with reference to a point will 
be at that point. Thus the centre of gravity of the circumference 
of a circle or an ellipse is at the geometrical centre of those figures. 

The centre of gravity of the perimeter of an equilateral triangle, 
or of a regular polygon, is at the centre of the inscribed circle. 

The centre of gravity of the perimeter of a square, rectangle, or 
parallelogram, is at the intersection of the diagonals of those 
figures. 

Centre of Gravity of Surfaces, Definition. — A surface here 
means a very thin plate or shell. 

Symmetrical Surfaces, — If a surface can be divided into two 
symmetrical halves by a line, the centre of gravity will be on that 
line: if it can be divided by two lines, the centre of gravity will be 
at their intersection. 

The centre of gravity of the surface of a circle or an ellipse is 
at the geometrical centre of the figm-e ; of an equilateral triangle 
or a regular polygon, it is at the centre of the inscribed circle; of a 
parallelogram, at the intersection of the diagonals ; of the surface 
of a sphere, or an ellipsoid of revolution, at the geometrical centre 
of the body; of the convex surface of a right cylinder at the 
middle point of the axis of the cylinder. 

Irregular Figures, — 4^ny figure may be divided into rectangles 




164 



CENTRES OF GRAVITY. 



and triangles, and, the centre of gravity of each being found, the 
centre of gravity of the whole may be determined by treating the 
centres of gravity of the separate parts as particles whose weights 
are proportional to the areas of the parts they represent. 

Triangle, — To find the centre of gravity of a triangle, draw a 
line from each of two angles to the middle of the side opposite: the 
intersection of the two lines will give the centre of gravity. 

QuadrilateraL — To find the centre of gravity of any quadrilat- 
eral, draw diagonals, and, from the end of each farthest from their 
intersection, lay ofif, toward the intersection, its shorter segment: 
the two points thus formed with the point of intersection will form 
a triangle whose centre of gravity is that of the quadrilatenl. 

Thus, let Fig. 11 be a quadrilateral 
whose centre of gravity is sought. 
Draw the diagonals AD and BC, and 
from A lay ofif AF= ED, and from 
B lay off BH = EC. From E draw 
, P a line to the middle of FH, and from 
Fa line to the middle of EH. The 
point of intersection of these two lines 
is the centre of gravity of the quadri- 
lateral. This is a method commonly 
used for finding the centre of gravity of the voussoirs of an arch. 

Table qf Centres of Gravity. — Let a denote a line 
drawn f "om the vertex of a figure to the middle point of 
the base^ and D the distance from the vertex to the cen- 
tre of gravity. Then 

In an isosceles triangle D = fa 

chord* 
In a segment of a circle 2) = 12 X area 

2 X chord 





m 





/ 



\ 



V 

Segment. 




In a sector of a circle, the ver- ) 7^ _ « ^^ _ 
tex being at the centre J ' ^ 

In a semicircle, vertex being at ) 

r • * 



X arc 
D = 0.4S6R 



Sector. 



the centre 

In a quadrant of a circle D = IB 

In a semi-ellipse, vertex being ) /) = 426a 

at the centre ) * ' 

In a pai-^bola, vertex at intersection of I D=^hi. 

axis wi* \i curve) ' ' 

In a cone or pyramid D = }a 

In a frustum of a cone or pyramid, let h = hei^t of complete 
cone or pyramid, Ji' = height of f rustiun, and the vertex be at apei 

of complete cone or pyi*amid; then 1> = a/ku^jJ \ * 




GBNTRES OF GRAVITY. 165 

The oommon centre of gi'avity of two figures or bodies external 
to esLob. other is found by the following rule: — 

Multiply the smaller ai'ea or weight by the distance between 
centres of gravity, and divide the product by the sum of the areas 
or weights: the quotient will be the distance of the common centre 
of gravity from the centre of gravity of the larger area. 

Example. — As an example under the above 
rule and tables, let us find the common centre of 
gravity of the semicircle and triangle shown in 
Fig, 12. 

We must first find the centres of gravity of the 
two parts. 

The centre of gravity of the semicircle is 0.425 R Fig. 12 

from A, or 2.975. The centre of gravity of the 

triangle is i of 8", or 2.666^' from A ; and hence the distance 

between the centre of gravity is 2.975" + 2.666", or 5.641". 

3|X49 
The area of the semicircle is approximately — 5 — > ^^^*^ square 

inches. The area of the triangle is 7 X 8, or 56 square inches. 

The sum of the areas is 133 square inches. Then, by the above 

rule, the distance of the common centre of gravity from the centre 

66 X 5.641 
or* gravity of the semicircle is Too — = 2.37 , 

or 

2.975 — 2.37 = 0.605 inches from A, 

Centre of Gravity of Heavy Particles. — Centre of 

Gravity of Two Particles. — Let P be the p^^ 

weight of a particle at A (Fig. 13), and W | 

that at C 

The centre of gravity will be at some 
point, B, on the line joining A and 



^;0 



e 



The point B must be so situated, that if p^ Flo, 13 W 

the two particles were held together by a 

stiflf wire, and were supported at 5 by a force equal to the sum 
of P and W, the two particles would be in equilibrium. 

The problem then comes under the principle of the lever, and 
hence we must have the proportion, 

P+W :P :: AC :BC, 
or 

PX^ 

^^■" P + W 
If TT = P, then BC = AB, or the centre of gravity will be half- 



166 



CENTRES OF GRAVITY. 



way between the two particles. This problem is of great impor- 
tance, for it presents itself in many practical examples. 

Centre of Gravity of Several Heavy Particles. — Let Wj , We, TF3, 
W4 and Ws (Fig. 14) be the weights of the particles. 

Join W] and W2 by a straight line, and find 
their centre of gravity ^ , as in the preceding 
'Ws problem. Join A with W3, and find the cen- 
tre of gravity By which will be the centre of 
gravity of the three weights W^ , Wfy and W^. 
Proceed in the same way with each weight, 
and the last centre of gravity found will be 
the centre of gravity of all the particles. 
In both of these cases the Unes joining the 
particles are supposed to be horizontal lines, or else the horizontal 
projection of the real straight line which would join the points. 




Ws Fig. 14 



RETAINING WALLS. 1^'^ 



CHAPTER V. 
RETAINING VSTALLS. 

A Retaining^ Wall is a wall for sustaining a pressure of 
earth, sand, or other filling or backing deposited behind it after it 
is built, in distinction to a brest or face wall, which is a similar 
structure for preventing the fall of earth which is in its undis- 
turbed natural position, but in which a vertical or inclined face 
has been excavated. 

Fig. 1 gives an illustration of the two kinds of wall. 




Retaining* Walls. — A great deal has been written upon the 
theory of retaining walls, and many theories have been given for 
computing the thrust which a bank of earth exerts against a re- 
taining wall, and for determining the form of wall which affords 
the greatest resistance with the least amount of material. 

There are so many conditions, however, upon which the thrust 
exerted by the backing depends, — such as the cohesion of the 
earth, the dryness of the material, the mode of backing up tlic 
wall, etc., — that in practice it is impossible to determine tli(» exact 
thrust which will be exerted against a wall of a given heiji:ht. 

It is therefore necessary, in designing retaining walls, to be guided 
by experience rather than by theory. As the theory of retaining 
walls is so vague and unsatisfactory, wc shall not offer any in this 
article, but rather give such rules and cautions as have been estab- 
lished by practice and experience. 

In designing a retaining wall there are two things to be consid- 
ered, — the backing and the wall. 

The tendency <^ tAe hacking to slip is very much less when it is 



^^^ BETAINING WALLS. 

in a dry state tlian when it is filled with wnter, and hence eve 
pi'M^aution shouliJ be taken to secure good drainage. Besides bi 
face drainage, tiiere should be openings left iii tlie waJI for Ike | 
water which may accumulate l:>e1iind it to escape aud run off. 

The manner in which the material is HUed agaiust the wftll also 
affects the stability of the baclcings. ff the ground be made irregu- 
lar, as in Fig. 1 , and the earth weil rammed in layers inclined jVom 
tlie uatl, tliit pressure will be very trifling, provided that attention 
be paid to drainage. If, on the other hand, the earth tie tipped, in 
ttie usual manner, in layers sloping toteardu the wall, the full pi 
urc of the earth will be exerted against II, and It must be made of 
correaponding strength. 




Fig.3 



FiB.4 



Fig.! 

The Wall.— lietainingWAlls are generally built with a batter- 
ing (sloping! face, as this Is the strongest wall tor a given amonnt 
of material ; and, if the courses are inclined towards the back. Ilia 
tendency to slide on each other will be overcome, and it will not bs 
necessary Ut depend upon the adhesion of the mortar. 





Fig-I 



FIg.a 



The importance of making tlie resistance independent of tiw 
ailhesion of the mortar Is obviously very great; as It WonU other- 
wise be necessary to delay backing up a n^l until tba iDortar WH 
'horoughly set, which might require several uonllni 



RETAINING WALLS. it™ 

e Back of tlie Wall shonld bo left Roagli.— In 

ivork It would be well to let every third or fourth course 
^t an inch or two. This increases the frietion of the earth 
9t the back, and thus causes tlie resultant of the forces acting 
d the wall to become nion? nearly vertical, and to fall farther 
n the base, giving increased stability. Jt also conduces to 
;tli not to make each course of uniform lielglit throughout the 
less of the wall, but to have some of the stones, especially near 
ick, sufiiciently high to reach up through two or liiree courses, 
is means the wliole masonry becomes more effectually inter- 
1 or bonded tc^etlier as one mass, and less liable to bulge. 
ere deep freezing occurs, the back of the wall should be sloped 
rds for threeor four feet belowitstop, aa at OC (Fig. 2), which 
1 be quite smooth, so aa to lessen the hold of the frost, and 
at displacement. 

i. 3, 4, 5, and 6 show the relative sectional areas of walls of 
snt shapes that would be required to resist the pressure of a 
of earth twelve feet high ("Art of Building," E. Dobson, 
The first three examples are calculated to resist the maxi- 
thnist of wet earth, while the last shows the modified form 
y adopted in practice. 

il's for tbe Tlilckness of tlie Wnll.— As has been 
. the only practical rules for retaining walls which we have 
nplrlcal rules based iiixin experience and practice- 
John C. Trautwiue, C.E., who is considered authority on 
?ering subjects, gives the following table in his " Pocket-Book 
igineers," for the thickness at the base of vertical retaining 
with a sand-backing deposited In the usual manner. 



• first cohmm coulains the verLiea) Iieight CD (Pig. 7) of tht^ 
as compared willi the vertical lieiglil of the wall ; which lal fn' 



170 



KETAINING WALLS. 



is assumed to be 1, so that tlie table begins with backing of the 
same height as the wall. These vertical wails may be battered to 
any extent not exceeding an inch and a half to a foot, or 1 In 8, 
without affecting their stability, and without increasing the base. 

Proportion of Retaining: Walls. 



f 

Total height of the earth com- 


Wall of 


Good mortar, 


Wall of 


pared with the height of the 


cut Btone 


rubble, 


good, dry 


wall above grouud. 


in mortar. 


or brick. 


rubble. 


1 


0.35 


0.40 


0.50 


1.1 


0.42 


0.47 


0.57 


1.2 


0.46 


0.51 


0.61 


1.3 


0.40 


0.54 


0.64 


1.4 


0.51 


0.56 


0.66 


1.5 


0.52 


0.67 


0.67 


1.6 


0.54 


0.59 


0.68 


1.7 


0.55 


0.60 


0.70 


1.8 


0.56 


0.61 


0.71 


2 


0.58 


0.63 


0.78 


2.5 


0.60 


0.65 


0.75 


3 


0.62 


0.67 


o.n 


4 


0.63 


0.68 


0.78 


6 


0.64 


0.69 


0.79 



Brest Walls (from Dobson's "Art of Building").— Where 
che ground to be supported is firm, and the strata are honzontal, 
the office of a brest wall is more to protect tlian to sustain the earth. 
[t should be borne in mind that a trifling force skilfully applied to 
onbroken ground will keep in its place a mass of material, which, 
if once allowed to move, would crush a heavy wall ; and therefore 
great care should be taken not to expose the newly opened ground 
to the influence of air and wet for a moment longer than is requisite 
for sound work, and to avoid leaving the smallest space for motion 
between the back of the wall and the ground. 

The strength of a brest wall nuist be projiortionately increase<1 
when the strata to be supported inclines towards the wall: where 
they incline from it, the wall need be little more than a thin facing 
to protect the ground from disintegration. 

The preservation of the natural drainage is one of the most im- 
portant points to be attended to in the erection of brest walls, as 
upon this their stability in a groat measure depends. Xo rule can 
be given for the best manner of doing this: it must be a matter for 
attentive consideration in each particular case. 



STBEKGTH OF MASOKBY. 171 



CHAPTER VI. 

STRBNGTH OF MASONRY. 

By the term "strength of masonry " we mean its resistance to a 
crushing-force, as that is the only force to which masonry should 
bo subjected. The strength of the different stones and materials 
used in masonry, as determined by experiment, is given in the 
following table. (For Architectural Terra-Cotta, see page 186a.) 

Crushing Resistance of Bricks Stone, and Concretes, {Pressure at 

right angles to bed.) 

Pounds 
per sq. inch. 

Brick : Common, Maspachnsetts. 1U,000 

Common, St. Louis -. 6,417 

Common, Wtibhington, D. C 7,870 

Paving, Illinois .... 6,000 to 13,000 

Granites : Bine, Fox Island, Me 14,875 

Gray, Vinal Haven, Me 18,000 to 18,000 

Westerly, R. I 15,000 

Rockport and Quincy, Mass 17.750 

Milford, Conn 22,600 

Staten Island, N. Y 22,250 

East St. Cloud, Minn 28,000 

Gannison, Colo 18,000 

Red, Platte Caflon. Colo 14,600 

Limestones: Glens Falls, N. Y 11,475 

Joliet,Ill •. 12,775 

Bedford, Ind 6,000 to 10,000 

Salem, Ind 8,625 

Red Wing, Minn 23,000 

Stillwater, Minn *. 10,750 

Sandttones : T)OTche»ter^N.B. {hrovfii) 9,150 

Mary's Point, N. B. (fine grain, dark brown) 7,700 

Connecticut Brown Stone  on lied '. 7,000 to 18,000 

LoDgmeadow, Mass. (reddish brown) 7,000 to 14,000 

'* " average, for good quality 12,000 

Little Falls, N. Y 9,850 

Medi na, N. Y 17,000 

Potsdam. N. Y. (red) 18,000 to 42,000 

Cleveland, Ohio 6,800 

North Amherst, Ohio 6,212 

Beren, Ohio 8,000 to 10,000 

Ilnmmcltitown. I*a 12,810 

Fond du Lac, Minn 8,750 

Fond du Lac, Wis 6,237 

Manitou, Colo, (light red) 6,000 to 11,000 

St. Vrain, Colo, (hard laminated). 11,505 

3Iarble8 : Lee, Mass 22.900 

Rutland, Vt 10,746 

Montgomery Co., Pa .' 10,000 

Colton.Cal 17,783 

Italy 12,156 

Flagging : North River, N. Y 13,425 

Concrete : Rosendale cement 1, pand and stone 7A, 46 months old 1,.544 

Portland cement 1, sand and stone 9, 6 months 2,000 

* This stone should not be set on edge. 



173 STRENGTH OF MASONRY. 

The stones in this table are supposed to be on bed, and the height 
» to be not more than four times the least side. Of the strength of 
rubble masonry, Professor Rankine states that "the resistance 
of fjood coursed rubble masonry to crushing is about four-tenths of 
that of single blocks of the stone it is built with. The resistance 
of common rubble to crushing is not much greater than that of the 
mortar which it contains." 

Stones generally commence to crack or split under about one-half 
of their crushing-load. 

Crushing-Height of Brick and Stone. — If we assume 
the weight of brickwork to be 112 pounds per cubic foot, and that 
it would crush under 450 pounds per square inch, then a vertical 
unifonn column 580 feet high would crush at its base under its own 
weight. 

Average sandstones at 145 pounds per cubic foot would require 
a column 5950 feet high to crush it; and average granite at 165 
pounds per cubic foot would require a column 10,470 feet high. 
The Merchants' shot-tower at Baltimore is 246 feet high, and its 
base sustams a pressure of six tons and a half (of 2240 pounds) 
per square foot. The base of the granite pier of Saltash Bridge (by 
Biiinel) of solid masonry to the height of 96 feet, and supporting 
the ends of two iron spans of 455 feet each, sustains nino tons 
and a half per squai-e f oot . The highest pier of Rocquef avonr stone 
aqueduct, Marseilles, is 305 feet, and sustains a pressure at the base 
of thirteen tons and a half jyar square foot. 

Worldng-Strengtli of Masonry.— The worlring-streiigth 
of masonry is generally taken at from one-sixth to one-tenth of the 
crushing-load for piei's, colunms, etc., and in the case of arches a 
factor of safety of twenty is often recommended for computing tbe 
resistance of the voiissoirs to crushing. 

Mr. Trautwine states that it cannot be considered safe to expose 
even first-class pressed brickwork in cement to more tlian thirteen 
or sixteen tons' pressure per square foot, or good hand-moulded 
brick to more than two-tliirds as nmch. {Seepage 181.) 

Sheet lead is sometinH^s plac(ul at the joints of stone columns 
with a view to equalize the pressure, and thus increase the strength 
of the cohnun. Exi)oriments, however, seem to show that the 
effect is directly the reverse, and that the column is materiaHy 
weakened thereby. ' 

Piers. — Masonry thai is so heavily loaded tliat it Is necessary 
to proporlion it in regard to its strength to resist crushing, is, as a 
general rule, in the form of piers, either of brick or Btoue. As 

1 Trautwine's Pocket-book, p. 176. 



STRENGTH OF MASONRY. 1*^3 

these pien are often in places where it is desirable tliat they should 
occupy as little space as possible, they are oflen loaded to the full 
limit of safety. 

The material generally used for building piers is brick: block or 
cut stone is sometimes used, for the sake of appearance; but rubble- 
work should never be used for piers which are to sustain posts, 
pillara, or columns. Brick piers more than six feet in height 
should never bo less than twelve inches square, and should have 
properly proportioned footing courses of stone not less than a foot 
thick. 

The brick in piers should be hard and well burned, and should 
be laid in cement, or cement mortar at least, and be well wet before 
being laid, as the strength of a pier depends very much upon the 
mortar or cement with which it is laid: those piei*s which have to 
sustain very heavy loads should be built up with the best of the 
Rosendale cements. The size of the pier should be determined by 
calculating the greatest lead which it may ever have to sustain, and 
dividing the load by the safe resistance of one square inch or foot 
of that kind of masonry to crushing. 

Example. — In a large storehouse the floors are supported by a 
girder running lengthwise through the centre of the building. The 
girders are supported every twelve feet by columns, and the lowest 
row of columns is supported on brick piers in the basement. The 
load which may possibly come upon one pier is found to be 65,000 
pounds. What should be the size of the pier ? 

^iM. The masonry being of good quality, and laid in cement 
mortar, we will a^ume 12 tons per square foot, or 166 lbs. per 
square inch (see p. 181), for the safe working load. Dividing 
65,000 lbs. by 166, we have 891 square inches for the size of the 
pier. This would require a pier 20 x 20 inches. 

It is the custom with many architects to specify bond stones in 
brick piers (the full size of the section of the pier) every three or 
four feet in the height of the pier. These bond stones are gener- 
ally alx)ut foiu" inches thick. The object in using them is to 
distribute the pressure on the pier equally through the whole mass. 
Many first-class builders, however, consider that the piers are 
stronger without the bond stone; and it is the opinion of the 
writer that a pier will be just as strong if they are not used. 

Section 3 of the Building Laws of the city of New York requires 
that every isolated pier less tlian ten superficial feet at the base, 
and all piers supporting a wall built of rubble-stone or brick, or 
under any iron beam or arch-girder, or arch on which a wall rests, 
or lintel supporting a wall, shall, at intervals of not less than thirty 
inches in height, have built into it a bond stone not less than 



174 STRENGTH OF MASONRY. 

four inches thick, of a diameter each way equal to the diametei 
of the pier, except that in piers on the street front, above the 
curb, the bond stone may be four inches less than the pier in 
diameter. 

Piers which support colmnns, posts, or pillars, shonld have the 
top covered by a plate of stone or iron, to distribute the pressure 
over the whole cross-section of the pier. 

In Boston, it is required that '*all piera shall be built of good, 
hard, well-burned brick, and laid in clear cement, and all bricks 
used in piers shall be of the hardest quality, and be well wet when 
laid. 

'* Isolated brick piers under all lintels, girders, iron or other col- 
umns, shall have a cap-iron at least two inches thick, or a granite 
cap-stone at least twelve inches thick, the full size of the pier. 

^* Piers or columns supporting walls of masonry shall have for a 
footing course a broad leveller, or levellers, of block stone not less 
than sixteen inches thick, and with a bearing surface equal in area 
to the square of the width of the footing course pluB one foot 
required for a wall of the same thickness and extent as that borne 
by the pier or colunm." 

For the Strength of Manonry WallSj see Chap. UL 

The following tables give the results of some tests on bclckf 
brick piers, and stoue, made under the direction of the 
author, in behalf of the Massachusetts Charitable Mechanics Ajbso- 
ciation. 

The specimens were tested in the government testingi-macliliie 
at Watertown, Mass., and great care was exercised to make tlie 
te~sts as perfect as possible. As the parallel plates between which 
the brick and stone were crushed are fixed in one position, it is 
necessary that the specimen tested should have perfectly parallel 
faces. 

The bricks which were tested were rubbed on a reyolTing bed 
until the top and bottom faces were perfectly true and parallel. 

The preparation of the bricks in this way required a great deal 
of time and expense; and it was so difficult to prepare some of the 
hanler bi'ick, that they had to be broken, and only one-half if 
:he brick prepared at a time. 



STRENGTH OF MASONRY. 



175 



TABLE 

f^howing the UUimaJte and Cracking Strength of the Brick, the 

Size and Area of Face, 



Name of Bbiok. 



Philadelphia Face Brick . . . 



• • • 



41 U 

Average . 



(« 



Cambridge Btiok (Eastern) . 
«< *( (( 









Average 



Boflton Terra-Ck>tU Co.'s Brick, 

l( CI (I (( 



(( 



« It 



Average 



New -England Pressed Brick . 
i( <t «( 



«« 
«i 



<i (« 

11 (« 



Average 



Size. 



Whole brick 
Whole brick 
Whole brick 



Half brick . 
Whole brick 
Half brick . 
Half brick . 



Half brick . 
Whole brick 
Whole brick 



Half brick 
Half brick 
Half brick 
Half brick 



Area of 
face in 
Bq. ins. 



33.7 
32.2 
34.03 



10.89 
25.77 
12.67 
13.43 



11.46 
25.60 
28.88 



12.95 
13.2 
13.30 
13.45 



Commenced 

to crack 

under Iba. 

per sq. inch. 


Net 
strength 
lbs. per 
sq. inch. 


4,303 
3,400 
2,870 


6,062 
5,831 
5,862 


3,527 


5,918 


3,670 
7,760 
3,398 
3,797 


9,825 
12,941 
11,681 
14,296 


4,655 


12,186 


11,518 
8,593 
3,530 


13,839 

11,406 

9,766 


7,880 


11,670 


3,862 
8,180 
2,480 
4,535 


10,270 
13,530 
13,082 
13,085 


4,764 


12,490 



The Philadelphia Brick used in these tests were obtained from a 
Boston dealer, and were fair samples of what is known in Boston 
as Philadelphia Face Brick. They were a very soft brick. 

The Cambridge Brick were the common brick, such as is made 
around Boston. They are about the same as the Eastern Brick. 

The Boston Terra-Cotta Company^ a Brick were manufactured of 
a rather fine clay, and were such as are often used for face brick. 

The New-England Pressed Brick were hydraulic pressed brick, 
and were almost as hard as iron. 

From tests made on the same machine by the United States Gov- 
ernment in 1884, the average strength of three (M. W. Sands) Cam- 
bridge, Mass., face brick was 13,925 pounds, and of his common 
brick, 18,337 pounds per square inch, one brick developing the enor- 
mous strength of 22,351 pounds per square inch. This was a very 
bard-burnt brick. 

Three brick of the Bay State (Mass.) manufacture showed an 
average strength of 11,400 pounds per square inch. 

The New England brick are among the hardest and strongest 
brick in the oonntry, those in many parts of the West not having 
one-fourth of the strength given above, so that in heavy buildings, 



176 STRENGTH OF MASONRY. 

where the strength of the brick to be used is not known by actaal 
tests, it is advisable to have the brick tested. 

Prof. Ira 0. Baker, of the University of Illinois, reported some 
tests on Illinois brick, made on the 100,000 pounds testing machine 
at the university, in 1888-89, which gives the crushing strength of 
soft brick at <574 pounds per square inch, average of three face 
brick, 3,070 pounds ; and of four paving brick, 9,775 pounds. 

In nearly all makes of brick it will be found that the face brick 
are not as strong as the common brick. 

Tests of the Streni^li of Brick Piers laid with 
Various Mortars/ — These tests were made for the purpose of 
testing the strength of brick piers laid up with different cement 
mortars, as compared with those laid up with ordinary mortar. 
The brick used in the piers were procured at M W. Sands's brick- 
yard, Cambridge, Mass., and were good ordinary brick. They were 
from the same lot as the samples of common brick tested as 
described. 

The piers were 8" by 12", and nine coui-ses, or about 224'' high, 
excepting the first, which was but eight courses high. They were 
built Nov. 29, 1881, in one of the storehouses at the United-States 
Arsenal in Watertown, Mass. In order to have the two ends of 
the piers perfectly parallel surfaces, a coat of about half an inch 
thick of pure I'ortland cement was put on the top of each pier, 
and the foot was grouted in the same cement. 

March 8, 1882, three months and five days later, the tops of the 
piers were dressed to plane surfaces at right angles to the sides of 
the piers. On attempting to dress the lower ends of the piers, the 
cement grout peeled off, and it was necessary to remove it entirely, 
and put on a layer of cement similar to that on the top of the piers. 
This was allowed to harden for one month and sixteen days, when 
the piers were tested. At that time the piers were four months and 
twenty-six days old. As the piers were built in cold weather, the 
bricks were not wet. 

The piers were built by a skilled brick-layer, and the mortars 
were mixed under his superintendence. ITie tests were made with 
the government testing-machine at the Arsenal. 

The following table is arranged so as to sbow the resalfc of these 
tests, and to afford a ready means of comparison of the strength of 
brickwork with different mortars. The piers generally failed by 
cracking longitudinally, and some of the brick were crushed. The 



1 The report of these tests was first pablished in the AmBrican Aidiileel^ 
June 8, 1882. 



STRENGTH OF MASONBY. 17'( 

Portland cement used in these tests was known as Brooks, Shoo- 
bridge ft Co. 'a cement. 



As the aetaal strength of brick piers is a very important coneid- 
eration in bnildiog constmetion, the following tests, made by the 
United States Government at Watertown, Mass.. and contained in 
tbe rrport of the tests mode on the (iovcrnment testing machine 
for the year 18B4. are given, as being of much value. 

Three kinds of brick were reprasent«d in the conatruction of the 
piers, and mortars of different composition — ranging in strength 
from lime mortar to neat Portland cement. The piers ranged in 
cross-section dimensions from H' x 8" to 16" x IS", and in 
height from 16" to 10 '. 

The piers were tested at the age of from 18 to 24 months 

The following table gives the reaiUts obtained, and memoranda 
regarding the size and character of the piers. 






« 


SSS223SSS2 


■3 

11 


Jiiiiiiiiii 

,IIWJI|I.I.I. 
1 



lii 




iiilll^ 



5 
J 



1 1 



nrl r;::il 







• il i I 

I 



I 



180 



STRENGTH OF MASONRY. 



Tests of Mortar Cubes. — The following tests of 6" oabesof 

mortar were made by the United States Gk)veniment at Watertown, 
Mass., in the year 1884. 

Ttie mortar cubes were allowed to season in the open air, a 
period of fourteen and a half months, whpn they were tested. 

The age of tlic plaster cube was four months. It should be 
noticed that, while the cube? of Rosendalc cement and Hme-mortar 
showed a greater strength than when sand alone was mixed with 
the cement, with the cubes of Portland cement and lim^-mortar 
the reverse was the case, differing from the result obtained by the 
author. This shows the necessity of a number and variety of tests. 



TABULATED RESULTS, 6" MORTAR CUBES. 
Crubhino Stbbngth. 



No. of 
test. 


Composition. 


First 
crack. 


Ultimate 

Btrength 

persq. in. 


Weight 

per 
CO. ft. 


3a 
Sb 
Zc 


1 part lime, 8 parts sand, 

ti 4( H 


lbs. 


Ibe. 
185 
119 
118 


lbs. 
118 
111 
106 


4a 
4b 
4c 


1 part Portland cement, 2 parts sand, 


• • « • 

11,600 


660 
606 
888 


116 
180 
115 


6a 
bb 
5c 


1 part Rofiendale cement, 2 parts sand, 

(t It It tt 

tt it tt it 


4,600 


166 
186 
148 


•111 
100 
107 


6a 
6b 

ec 


Neat Portland cement, 
kt it 

it ti 


• • • • • • • 

96,000 


2,678 
8,548 

4,887 


196 
189 
185 


7a 
lb 
7c 


Neat Ro^endale cement, 

it it 

it it 


11,000 
19,000 
19,900 


481 
615 
686 


94 
90 

vr 


8a 
8b 
8c 


1 part Portland cement, 2 parte lime-mortar, ^ 

it ti it i( 

It it ii it 


• • • • • • • 


804 
196 
175 


100 
110 
lOi 


9a 
9b 
9c 


1 part Rosendole cement, 2 parts lime-mortar,^ 

ti it ti ti 

it ti it it 
PlasttT-of-Paris. 


• • • • • • 


194 

198 

16-2 

1,981 


105 

1(« 

106 

74 



Workings Stren^h of Masonry.— The faUowing table 

has been compiled as representing the practice of leading engineen, 
and the average requirements of recent building laws. The author 
believes that the values may be relied upon with eafetf , ftod with- 

1 Lime-monar, 1 part lime, 8 parui 



STRENGTH OF MASOKBT. 181 

out andae waste of materials. For the size of oast-iron bearing 
plates on masonry, see page 342&. For strength of architeotnral 
terra-cotta, see page 186a. 

SAFE WORKING LOADS FOR MASONRY. 
Briektoork in isalls or pier», 

TONS FBB SqUABS VOOT. 

Bastem. Western. 

Bed brick in lime mortar 7 6 

** hydraulic lime mortar 6 

*' natural cement mortar, 1 to 3 10 8 

Arch or pressed brick in lime mortar 8 6 

** •* " natural cement 13 9 

** ** ** Portland cement 15 12^ 

Piers exceeding in height six times their least dimensions should 
be increased 4 inches in size for each additional 6 feet. 

Stonework, 

(Tons per square foot.) 

Bubble walls, irregular stones 8 

** coursed, soft stone %^ 

** hard stone 5 to 16 

Dimension stone, squared in cement : 

Sandstone and limestone 10 to 20 

Granite 20 to 40 

Dressed stone, with |-inch dressed joints in cement : 

Granite 60 

Marble or limestone, best 40 

Sandstone 30 

Height of columns not to exceed eight times least diameter. 

CoTicrete. 

Portland cement, 1 to 8 8 to 15 

Rosendale cement. 1 to 6 6 to 10 

Hydraulic lime, best, 1 to 6 5 

HdUow Tile* 

(Safe loads per square inch of effective bearing parts.) 

Hard fire-clay tiles 80 lbs. 

*• ordinary clay tiles 60 ** 

Porous terra-cotta tiles 40 ** 

Mortars. 

(In 4-inch joints, 8 months old, tons per square foot.) 

Portland oement, 1 to 4 40 

Rosendale cement» 1 to 8 18 

Lime mort r, beet. . : 8 to 10 

Best Portl d cement, 1 to 2. in 4-inch joints for bedding 

ixonp tea 70 



182 8TBENGTH OF MASONRY. 

Actual Tests of the Crushingr-Stren^h of Sand- 
stones (made under the direction of the author for the Massachu- 
setts Charitable Mechanics' Association). — These tests were made 
with the Government testing mac^hine at the United States Arsenal, 
Watertown, Mass., and every precaution was taken to secure accu- 
rate results. 

Wood's Point (X.B.) Sandstone. — This stone is of about the 
same color as the Mary's Point stone, but it has a much coarser 
gmin, and is not very hard. 

Block No. 1 measured 4.03" x 4.03" X 8". Sectional area 16.2 
square inches. 

Commenced to crack at 50,000 pounds, on the comers, and con- 
tinued cracking, along the edge^ and at the comers, until it was 
crushed under 80,000 lbs.' pressure, or 4932 lbs. per square inch. 

Block No, a measured 4" x 3.«8" X 7.25". SecUonal area 15.02 
square inches. 

This stone commenced to crack under a pressure of 44,000 
pounds, and failed under a pressure of 62,500 pounds, or 3976 
pounds per square inch. 

Long MEADOW Stone. — The Bay of Fundy Qiiarryhig Com- 
pany also quarry a variety of the Longmeadow (Mass.) sandstone, 
which is a reddish-brown in color. 

Block No. 1 measured 3.S0" x 3.87" X 7.42". Sectional area 
14.71 square inches. 

This stone showed no cracks whatever until the pressure bad 
reached 152,000 pounds, when it conmienced to crack at the cor- 
ners. When the pressure reachetl 200,000 pounds, Uie stone sud- 
denly flew from the machine in fragments, giving an ultluiato 
strength of 13,506 pounds per square inch. 

This stone did not fit into the machine vei7 perfectly. 

lilock No. f measured 3.30" x 3.07" X 7.5". Sectional area 15.6 
square inches. 

The stone commenced to crack along the edges under a pressure 
of 47,000 pounds. Under 78,(KX) pouuils, a large piece of the stone 
split off from the bottom of the block, and under 142,300 pounds* 
pressure, the stone failed, cracking very badly. UUimale lUmngUi 
per aqiuirc inch 0121 jjtmnilfi. 

Bkown Sandstone fhom East Lon«meaj>ow, MAsa. — Quap- 
ried by Norcross Brothers tfe Taylor of East Longmeadow. This finii 
works several (juarries, the stone differing in the degree of hard- 
ness, and a little in color, which is a reddish brown. The different 
varieties take the name of the quarry from which they oome. 

Soft Saulsbubt Bbownstone. — This stone is one of the 



STRENGTH OF MASONRY. 183 

softest varieties quarried by this firm, althougli it is about as liard 
as the ordinary brownstones. The specimens tested were selected 
by the foreman of the stone-yard without knowing tlie purpose for 
wliich they were to be used, and were ratlier below the average of 
this stone in quality. 

Block No. 1 measured 4" X 4" X 7.58". Area of cross-section 16 
square inches. Ultimate strength 141,000 pouuci*, or 8812 j>oi/hc/« 
per square inch. 

Stone did not commence to crack until the pressure had reached 
130,000 pounds. 

Block No. t measured 4" X 4" X 7.85". Area of cross-section 10 
square inches. Ultimate strength 129,000 pounds, or 8062 pounds 
per square inch. 

There were no cracks in the specimen when it was under 100,000 
pounds' pressure. 

Hard Saulsbury Brownstone. — This is one of the hardest 
and finest of the Longmeadow sandstones. 

Block No. 1 measured 4.16" x4.1(')" x 8". Sectional area 17.3 
square inches. Ultimate strength 233,iKK) pounds, or 13,520 pounds 
per square inch. 

Stone did not commence to crack until the T?^*^sure had reachecl 
220,000 pounds, almost the crushing-stronjrth. 

Block No. 2 measured 4.15" X 4.i:>" x S". Sectional area 17.5: 
square inches. Ultimate strength 2,b2,{M) pounds, or 14,650 i^ownd* 
per square inch. 

This specimen did not commence to crack until the pressure had 
reached 240,000 pounds, or 13,953 pounds to the square inch. 

The following table is ari-anged to show the sectional area and 
strength of each specimen, and the average for each variety of 
^tone. The cracking-strength, so to speak, of the stone, is of con- 
sideitible unportance, for, after a stone has commenced to crack, its 
permanent strength is probably reached ; for, if the load which caused 
it to crack were allowed to remain on the stone, it would probably 
in time crush the stone. In testing the blocks, however, some in- 
equality in the faces of the block might cause one corner to ciack 
when the stone itself had not commenced to weaken. 



STKENGTH OF MASONRY. 



Cell. Q. A. Gillinore, a few yeura ago, tested tbe strength  
Uiauy vai'ielies of saii<latoiie by (.'I'lisliing Lwo-liiuli cubes. The r 
suits obtalnetl by bliii laiigtvl fi-otii 4:t50 pounds to 9830 poanda pi 
square inch. Coniparicig the Btrengtli of the stones lealed by tli 
author with these values, we find that tlie specimens of liar 
Sa»lsbiU7 sanilstone had a strengtli far aluve tlie average for smk 
stones, anil tlie oilier specimens have about the same value* i 
tliose obtained by Gen. Gllliuore. 

We should expect, liowever, smaller values from block) 4" X 4 
X n" than fioni two-inch cubes; for, as a rule, small spednMnu ( 
almost any material show a greater strength than large speclmeiu 

It is interesting to note the mode of fractare of the btocki i 
browiistone, which was the same for each spechnen. The block 
fractui'ed by the sides bursting off; and, when takca fram tin ■! 



STRENGTH OF MASONRY. 185 

shine, the specimens had the form of two pyramids, with their 
aj>exes meeting at the centre, and having for their bases the com- 
pressed ends of the block. The pyramids were more clearly shown 
in some specimens than in others, but it was evident that the mode 
of fracture was the same for all specimens. 

KruBK Sandstone. — In 1883 the writer superintended the 
testing of two six-inch cubes of the Kibbe variety of Longmeadow 
sandstone, quarried by Norcross Brothers. One block withstood a 
pressure of 12,590 pounds to the square inch before cracking, and 
the other did not commence to crack until the pressure had reached 
12,185 pounds to the square inch. The ultimate strength of the 
first block was 12,619 pounds, and of the second 12,874 pounds, per 
square inch. 

Strength and H^eight of Colorado BalldiniT 

Stones. 

The following are the most reliable data obtainable of the strength 
and weight of the stones most extensively used for building in 
Colorado. 

* Med Ghranite from Platte Cafton, Crushing strength per square 
inch, 14,600 pounds. Weight per cubic foot, 164 pounds. 

Bed Sandstone from Pike's Peak Quarry, Manitou. Crushing 
strength, 6.000 pounds per square inch. 

** Red Sandstone from Greenlee & Son's quarries, Manitou 
(adjacent to the Pike's Peak quarries). Crushing weight, 11,000 
pounds per square inch on bed. Weight, 140 pounds per cubic foot. 

* Oray Sandstone from Trinidad, Crushing weight, 10,000 
pounds per square inch. Weight, 145 pounds per cubic foot. 

t Ldva Stone, Curry's Quarry, Douglas County, Crushing 
{trength, 10,675 pounds per square inch. Weight, 119 pounds per 
;abic foot. (Experience has shown that this stone is not suitable 
for piers, or where any great strength is required, as it cracks very 
saslly.) 

* Fort Collins, gray sandstone (laminated), much used for foun- 
dations. 

Crushing strength, bed 11,700 pounds, edge 10,700 pounds per 
square inch Weight, 140 pounds per cubic foot. (One ton of 
this stone measures just a perch in the wall.) 

* SI. Vrains, light red sandstone (laminated), excellent stone for 
foundations. Very hard. 



 From tests made for the Board of Capitol Managers (of Colorado) by State 
BnglDeer E. 8. Nettleton, in 1885, on two-inch cnbes. 

t Floiii tests made by Denver Society of Civil Engineers, in 1884, also on two- 
ndi eobes. ♦• Tested at V. S. Arsenal, Watertown, Mass. 



186 STBKNGTH OF MASONRY. 

Crushing strength, bed 11,505 pounds, edge 17,181 pounds per 
square inch. Weight, 150 pounds per cubic loot. 

Eft'ects of Freezing on Mortar.— Both cement and lime- 
mortar, mixed with salt, have been used in freezing weather with- 
out any bad clfcjts. (See American Architect. v«)l. xxi., p. 2>G.) 

Kule for the proportion of salt said to have been used in the works 
at Woolwich Arsenal: *' Dissolve one pound of rock-salt in eighteen 
gallons of water when the temperature is at 32 degrees Fahr., and 
add three ounces of salt for every three degrees of lower tempera- 
ture." 

durability of Hoop Iron Bond.— I believe that, embed- 
ded in liine-mortar at such depth &s to protect it from the air, 
hoop iron bond is indestructible*. In cement mortar containing 
salts of potash and soda, I doubt its lasting 1,500 years iinooRoded. 
— M. C. Meios, May 17, 1887. 

Grouting.* 

It is contended by persons having large experience In building 
that masonry carefully grouted, when the temperature is not lower 
than 40' Fahr., will give the most efficient result. 

The following buildings in New York City have grouted walls : 

Metropolitan Opera House. 

Produce and Cotton Exchanges. 

Mortimer and Mills Buildings. 

Equitable and Mutual Life Insurance Buildings. 

Standard Oil Building. 

Astor Building. 

The Eden Musee. 

The Navarro Buildings. 

Manhattan Bank Building. 

Tho Presbyterian, Gorman, St. Vincent, and Woman's Hospitals. 
etc ; also, the Mersey Docks and Warehouses at Liverpool, £ng: 
land, one of the greatest pieces of masonry in the world, have been 
grouted throughout. It should b(} stated, however, that there arj 
niiiny engineers and others who do not believe in grouting, claim- 
ing that there is a tendency of the materials to separate and fona 
lavers. 



* See American Architect, July 21, 1S87, p. 11. 



STRENGTH OF MASONRY. 186a 



Architectural Terra- Cotta— Weight and Strength. 

The lightness of terra-cotta, combined with its enormous resist- 
ing strength, and taken in connection also with its durability and 
absolute indestructibility by fire, water, frost, etc., renders it 
specially desirable for use in the construction of all large edifices. 

Terra-cotta for building purposes, whether plain or ornamental, 
is generally made of hollow blocks formed with webs inside, so as 
to give extra strength and keep the work true while drying. This 
is necessitated because good, well-burned terra-cotta cannot safely 
be made of more than about 1^ inches in thickness, whereas, when 
required to bond with brick-work, it must be at least four inches 
thick. When extra strength is needed, these hollow spaces are filled 
with concrete or brick- work, which greatly increases the crushing 
strength of terra-cotta, although alone it is able to bear a very heavy 
weight. *• A i'Olid block of terra-cotta of one foot cube has borne a 
crushing strain of 500 tons and over." 

Some exhaustive experiments, made by the Royal Institute of 
British Architects, give the following results as the crushing 
strength of terra-cotta blocks : 

Crushing wt. 
per en. ft. 

1. Solid block of terra-cotta 523 tons. 

2. Hollow block of terra-cotta, unfilled 186 *' 

8. Hollow block of terra-cotta, slightly made and unfilled. 80 " 

Tests of terra-cotta manufactured by the New York Company, 
which were made at the Stevens Institute of Technology in April, 
1888, gave the following results : 

Crushing wt. Crushing wt. 
per cu. in. per cu. ft. 

Terra-cotta block, 2-inch square, red 6,840 lbs. or 492 tons. 

Terra-cotta block, 2-inch square, buff 6,236 *' '* 449 

Terra-cotta block, 2-inch square, gray 5,126 " " 369 



( ( 
(( 



Prom these results, the writer would i)lace the safe working 
strength of terra-cotta blocks in the wall at 5 tons per square foot 
when unfilled, and 10 tons per square foot when filled solid with 
brick-work or concrete. 

The weight of tem-ootta in solid blocks is 122 pounds. When 



186* STBENGTH OP MASONRY. 

made in hollow blocks 1^ inches thick, the weight varies from 6f 
to 85 pounds per cubic foot, the smaller pieces weighing the most. 
For pieces 12" x 18" or larger on the face, 70 pounds per cubic fool 
will probably be a fair average. 

For the exterior facing of fire-proof buildings, terra-cotta is non 
considered as the most suitable material available. 



STABILITY OF PIERS ANP BUTTBESSES. 187 



CHAPTER VIT. 
8TABII1ITT OF PIERS AND BUTTRESSBS. 

A PI Kit or buttress may be cousMered stable when the forces 
acting upon it <lo not cause it to rotate or "tip over," or any 
course of stones or brick to slide on its bed. When a pier has to 
sustain only a vertical load, it is evident that the pier must be 
stable, although it may not liave sufficient strength. 

It is only when the pier receives a thrust such as tliat from a 
rafter or an arch, that its stability must be considered. 

In order to resist rotation, we must have the condition that the 
moment of the tluiist of the pier about any point in the outside of 
the pier shall not exceed the moment of the weight of the pier 
about the same point. 

To illustrate, let us take the pier shown in Fig. 1. 

Let us suppose that this pier receives the foot of a rafter, 
which exerts a thrust T in the direction AB» The tendency of 
this thrust will be to cause the pier to rotate about the outer 
edge b 1 ; and the moment of the thrust about this point will be 
T X a lb I, a lb i being the arm. Now, that the pier shall be just 
in equilibrium, the moment of the weight of the pier about the 
same edge must just equal T X a, 6,. The weight of the pier 
will, of course, act through the centre of gravity of the pier (which 
in this case is at the centre), and in a vertical direction; and its arm 
will be 6|<r, or one-half the thickness of the pier. 

Ilcncc, to liave equilibrium, we must have the equation, 

T X ttibi = W X bic. 

Ihit under this condition the least additional thrust, or the crush- 
ing off of the outer edge, would cause the pier to i-otate: hence, 
to have the pier in safe equilibrium, we must use some factor of 
safety. 

This is generally done by making the moment of the weight c(iual 
to that of the thiiist when referred to a point in the bottom of the 
pier, a certain distance in from the outer tnlge. 

This distance for piers or buttresses should not be less than one- 
fourtb of tbe thlcknesa of the pier. 



18R 



STABILITY OF PIERS AND BUITBESSEI^. 



Rcpresontiiig this point in the figui*e by h, we have the neceasuj 
e(i nation for the safe stability of the pier, 

TX ab= W X it, 

t denoting the width of the pier. 

We cannot from this e<iuation detenuine the dimensions of a 
pier to resist a given thmst; becanse we have the distance ah, /, 
and W, all unknown quantities. Hence, we must first guess at i\w 
size of the pier, then find the length of the line a6, and sec if 
the moment of the pier is equal to that of the thrust. If it is not, 
we must guess again. 





Graphic Method of determining: the Stability of a 
Pier or Buttress. — When it is desii-ed to determine if a givon 
pier or buttress is capable of resisting a given thrust, the probleiu 
can easily be solved graphically in the following manner. 

TiCt ABCD (Fig. 2) represent a pier which sustains a given 
thnist T at B. 

To detennine whether the pier will safely sustain tliis thrust, we 
pi-oceed as follows. 

Draw the indefinite line liX in the direction of the thnisL 
Through the centre of gravity of the pier (which in this case Is at 
the centre of the pier) (h-aw a vortical line until it intersects tint 
line of the thrust at c. As a force may be considered to act any- 
where in its line of direction, we may consider the tlinut and Ih* 
weiixht to act at the point c: and the resultant of these two forces 
can l)e obtained by laying off the ihnist T from e on eX, and Ui« 
wcijrlit of the pier IT, from c on the line cY, lx)th to the same 
scale (pounds to the inch), completing the parallelogram, and dimw- 
ing the diagonal, if this diagonal prolonged cats the base of the 
pier at less than one-fourth of the width of the liase from the outer 
eilge, the pier will l>e unstable, and its dhneusious must beduuigiad. 

The stability of a pie7' may be increased by adding 10 U* ira%|l 



STABILITY OF PIERS AND BUTTRESSES. 186 

(by placing some heavy mnterial on top), or by Increasing Its width 
at the base, by means of " set-offs," as in Fig. 3. 

Figs. 3 (A and B| show the method of determinit^ the stability 
^f a buttress with offsets. 

The flrst step Is to find the vertical line paaslng throngli the 
centre of gravity of tlie whole pier. This is best done by dividing 
the bmtresa up into quadrilaterals, as ABCD, DEFG, and GIIIK 
(Fig. 3A), finding the centre of gruvity of each quadrilateral by 
the method of diagonals, anil then measuring the perpendicular 
distances A'g, A'„ X^, from the diSei^ent centres of gravity to the 
line KI. 

Multiply the area of each qitadrilateral by the distance of its 
centre of gravity from the line KT, and add together the areas 
and the products. Divide the sum of the latter by the sum of the 
former, and the result will be the distance of the centre of gravity 
of the whole buttress from KI. This distance we denote by X^. 




Example I. — Let the buttress shown in Fig. 3A have Ilia 
dimensions given l)etween llie cross-marks. Then the arv& of 
the quadrilaterals and the distances from their centres of gravity to 
KI would be as follows; 

1st area = 35 sq. ft X, = (V.ft-> 1st area x X, - M.2.5 
2d area = 23 sq. ft. X, = t'M 2d area x A\ = 67.85 
3d ai-ea = 11 sq. ft. X-i = i^.OS 3d area X A',, = 54.45 



Total a 



L, mi s<|. ft. 



Total 



t, 155.55 



Tlie sum of the moments is IS.^..^; and, dividing this by the total 
area, we have 2.25 as the distance Xu- Measuiing tliis to the scale 
of the drawing froqj KI, we have a point through which the 
Tertic«l line fMlng through the centre of gravity moat pass. 



190 



STABILITY OF PIERS AND BUTTRESSES. 



After this line is found, the metho<l of dctemiining the stability of 
the pier is the same as that given for the pier in Fig. 2. Fig. 3B 
also illustrates the method. If tlie buttress is more than one foot 
thick (at right angles to the piano of the paper), the cubic contents 
of the buttress must be obtained to find the weight. It is easier. 
howeviT, to divide tlie real thrust by the thickness of the buttress, 
which i^ivi's the thrust per foot of buttress. 

J^ine of lleniiitsince, — Dcjinition, The line of resistance 
or of i>nvs.sures, of a pier or buttress, is a line drawn througli the 
centre of pressure of each joint. 

The centre of prenftitre of any joint is the point where the 
resultant of the forces acting on that portion of the pier above 
the joint cuts it. 

The line of pressures, or of resistance, when drawn in a pier, 
shows liow near the greatest stress on any joint comes to the edges 
of tliat joint. 

It can be drawn by tlie following method. 

Let AIU'I) (Fig. 4) be a pier 
whose line of I'esistance we wish 
to draw. First divide the pier in 
height, into portions two or three 
feet high, by drawing horizontal 
lines. It is more convenient to 
make the i)ortions all of the same 
size. 

Proloiii: the line of the thrust, 
and dr.'iw a verti<'al line through 
th(» centre of giiivity of the pier, 
intersertiiig tlu* line of thrust at 
tin' i)oint (I. From a lay off to a 
scah' the thrust T and the weights 
of the different ]M)rtions of the pier, 
eonnnencing with the w«Mght of the 
upper portion. Thus, ir, r(*pn»s<*nt8 
the wi'ight of the porti(m alM)ve the 
 ir-*t jiiinl : z'*^ represents the w«»ight 
of tin* .s.M'<)n:l iH>rtion; and so on. 
Tin* sum (if the /r's will <M|nal the 
whole \\«'ii:iil of the pier. 

Ilaviiii: iti-oeeeded thus far, etmipMi* a |)Hralleloffraiii, with 7* and 
w^ tor it> two sides. Dniw the diagonal, and prolong U. When- 
it eius iiii> first, joint will Im' a |N>iiil hi the line of mlitAnoe. 
Draw another parallehtgram, with 7' and Wi + lOg for lU iwotklML 
Draw the di;igonal intenMH^ting the second Joint at 8. rromud !■ 




Fi|.4. 



.Ji. ^.«kX:>- - ai2 -«■*■ •■rill.— :_. u- v' •■ ij- :i: 




*• uLk.*5i*-- ■•:««*•■■  »^- .I'-'i^ ii-.*««** — ;!. '.■;• — Lij;"'- li 'i ..:: 

r TIIMMT- f 5^»^ *^*» ^ iiftUii**-— *':i;; T*U* v _i'_ v v. -.y. 

hulki: V "U^-- uUTtt' »•:. i - ti 1 at. a:T '-• f «•'■" v::. 

^ *^^ *■— wfcr- t« jsrjs^tf- * _*•■ *-L ill ;.'zr.?«' lui'-v- ll- i, ... 
;^i»*»' - . ^■**^ nil - - * t^ — — - 



J/ 


= ^ 


-*- 


Z'. = *' 


^  


ZZ J< 


w 


: ji = ^..,: 




^i* 




• » •>...■ 







190 



STABILITY OF PIERS AND BUTTEES8ES. 



After this line is found, the method of determining the stability 
the pier is tlie same as that given for the pier in Fig. 2. Fig. i 
also illustrates the method. If the buttress is more than one fc 
thick (at right angles to the plane of the paper), the cubic contei 
of the buttress must be obtained to find the weight. It is easii 
however, to divide the real thrust by the thickness of the buttre: 
which gives the thrust per foot of buttress. 

J^iiie of Kesistaiice. — Definition, The line of resistan 
or of pressures, of a pier or buttress, is a line drawn through t 
centre of pressure of each joint. 

The centre of pressure of any joint is the point where t 
resultant of the forces actmg on that portio.n of the pier abo 
the joint cuts it. 

The line of pressures, or of resistance, when drawn in a pi< 
shows how near the greatest stress on any joint comes to the edg 
of that joint. 

It can be drawn by the following method. 

Let ABCB (Fig. 4) be a pier 
whose line of resistance we wish 
to draw. First divide the pier in 
height, into portions two or three 
feet high, by drawing horizontal 
lines. It is more convenient to 
make the portions all of the same 
size. 

Prolong the line of the thnist, 
and draw a vertical line through 
the centre of gravity of the pier, 
intersecting the line of thrust at 
the point a. From a lay off to a 
scale the thrust T and the weights 
of the different portions of the pier, 
coiuniencing with the weight of the 
upper portion. Thus, to i represents 
the weight of the portion above the 
lirst joint; i02 represents the weight 
of the second portion; and so on. 
The sum of the to's will equal the 
whole weight of the pier. 

Having proceeded thus far, complete a parallelogram, with T u 
w I for its two sides. Draw the diagonal, and prolong it. Whfi 
it cuts the first joint will be a point in the line of resistanc 
Draw another parallelogram, with T and Wi+Wt for iU two aldf 
Draw the diagonal intersecting the second Joint at % 




Fit.4. 



STABILITY OF PIERS AND BUTTRESSES. 191 

this way, when the last diagonal will intersect the base in 4. Join 
the points 1, 2, 3, and 4, and the resulting line will be the line of 
resistance. 

We have taken the simplest case as an example; but the same 
principle is true for any case. 

Should the line of resistance of a pier at any point approach 
the outside edge of the joint 'neai-er than one-quarter the width 
of the joint, the pier should be considered unsafe. 

As an example embracing all the principles given above, we will 
take the following case. 

Example II. — Let Fig. 5 represent the section of a side wall 
of a church, with a buttress against it. Opposite the buttress, on 
the inside of the. wall, is a hammer-beam truss, which we will sup- 
pose exerts an outward thrust on the walls of the church amount- 
ing to about 9600 pounds. We will further consider that the 
resultant of the thrust acts at P, and at an angle of 60° with a 
horizontal. The dimensions of the wall and buttress are given in 
Fig. 5 A, and the buttress is two feet thick. 

Question. — Is the buttress sufficient to enable the wall to 
withstand the thrust of the truss ? 

The first point to decide is if the line of resistance cuts the 
joint CD at a safe distance in from C To ascertain this, we must 
find the centre of gravity of the wall and buttress above the joint 
CD. We can find this easiest by the method of moments around 
KM (Fig. 5A), as already explained. 

The distance Xi is, of course, half the thickness of the wall, 
or one foot. We next find the centre of gravity of the portion 
CEFG (Fig. 5A), by the method of diagonals, and, scaling the 
distance X«, we find it to be 2.95 feet. 

The area of CEFG = ^g = 10 square feet; and of GIKL = Ax 
= 26 square feet. 

Then we have, 

X,-\ ^, =26 ^, X X, = 26 

Xt = 2.95 ^2 = 10 A^X Xi- 29.5 

36 36 ) 55.5 

Xo = 1.5 

Or the centre of gravity is at a distance 1.5 foot from the line 
ED (Fig. 5). Then on Fig. 5 measure the distance Xn = 1.5 foot, 
and through the point a dmw a vertical line intersecting the line 
of the thrust prolongisd at O. Now, if the thrust is 9600 pounds 
for a buttress two feet thick, it would be half that, or 4800 pounds, 
lor a buttrass one loot thick. We will call the weight of the 



IBS STABILITY OK PIEKS AND BUTTHB88BS. 

masonry of whicb the buttreea itDd wall la built IiJO ponnila pef 
ciibie foot. Then tbe Ihiiist is equivalenl lo 4800 -^ 150, or Hi 
cubic fctt of masonry. Laying tbls off lo a scale from O, in the 
illreotion of the Ihnist ami the area of the masonry, :tl> square feel 
from on tbe vertical line, completing the rectangle, anil (Irawjug 
ilin iliaguiial, we find it cnts ibe joint CD al ti, within tbe Uinlls 
of safety. 
We must next Qud where theliueof resistaoce cuts tlie base ^fi. 



First Hml the centre of gravity of tbe wtiole Ognre, wbUib I* 
fuiMiit by ascertaining the distances X,', X3', in fig. 6A, and 
making the following computation: 



2'.98 A^< 


= 24 A,' ■K J,' = 11.62 


4'.e5 A, 


'= 12 ^,'X ,lV = i»-« 




TO 70 1 imw 



T„' = 2.35 

Then from the line EJi (Fig. 0| lay off the disUncv Xt' = 
2'.2.'i, and ilraw through il a vi'rtlcal line iutcraeuliug tbe line of tlie 
tiirust at V. Un this vertical fi-uni O'jucasurc down the whole 
area 76, and from its extremity lay off tbe thniit T^ U at tl» 



STABILITY OF I'lKRS AND BUTTRESSES. 193 

proper angle. Di*aw the line O'e intersecting the base at c. Tliis 
is the point where the line of resistance cuts the base; and, as it is 
at a safe distance in from A, the buttress has sufficient stability. 

If there were more offsets, we sliould i^roceed in the same way, 
finding where the line of resistance cuts the joint at the top of 
each offset. The reason for doing thisis because the line of resist- 
ance might cut the base at a safe distance from the outer edge, 
while higher up it might come outside of the buttress, so that the 
buttiess would be unstable. 

The method given iu these examples is applicable to piei's of any 
sliape or material. 

Should the line of resistance make an angle less tliau 30^ with 
any joiut, it might cause the stones above Uie joint to slide on 
their bed. This can be prevented either by dowelliug, or by incliu- 
lug the joint. 

It is very seldom in architectural coustruction that such a case 
would occur, however. 



194 THE STABILITY OF ARCHES. 




CHAPrER vin. 

THE STABILITT OF ARCHB8. 

The arch is an arrangeimmt for spanning large openings by 
means of small blocks of stone, or other material, arranged in a par- 
ticular way. As a rule, the arch answers the same purpose as tbe 
beam, but it is widely different in its action and in tbe effect that 
it has upon tlie appearance of an edifice. A beam exerts merely a 
vertical force upon its supports, i>ut the arch exerts both a vertical 
load and an outward thrust. It is this thrust which requires that 
tho arch sliould be used with caution wliere the abutments are not 
abundantly large. 

Before taking up the principles of the • 
arch, we will define the many terms relating 
to It. The distance ec (Fig. 1) is called 
the ftpan of the arch; ai, its rise; b, its 
crown; its lower boundary Hue, eac, its 
9(^t or intrados ; the outer boundary line, pi^l 

its back or extrados. The terms "soffit" 

and "back'' are also applied to the entire lower and upper curved 
surfaces of the whole arch. The ends of the arch, or the sides 
which are seen, are called its faces. The blocks of which the arch 
itself is composed are called voussoh'-s : the centre one, K, is called 
the keystone ; and the lowest ones, .S.S, the tfprintfei'H, In nf*/- 
weiital arches, or those whose intrados is not a complete semicircle, 
the springers generally rest upon two stones, as RR, which luive 
their upper surface cut to receive them: these stones are called 
skewhdcks. The line connecting the lower edges of the springers 
is called the sprinyhKj-Une ; the sides of the arcli are called the 
haunches ; and the load in the triangular space, between the 
haunches and a horizontal line drawn from the crown, is called 
the spandrel. 

The blocks of masonry, or other material, which support two 
sucrcssive arches, are called piers : the extreme blocks, which, in 
the Cease of stone bridges, generally support on one side emlMuak- 
ments of earth, arc calle<l ((hutments. 

A pier strong enough to withstand the thrust of ^ther areh, 
should the other fall down, is sometimes called an nhnUneni pier. 
Resides their own weight, arches usually support a pemnneiit kiad 
or surcharge of masonry or of earth. 

In using arches in architectural constructions! thit flom of fki 



THE STABILITY OF ARCHES. 195 

arch is generally governed by the style of the edifice, or by a limited 
amount of space. The semicircular and segmental forms of arches 
are the best as regards stability, and ai-e the simplest to construct. 
Klliptical and three-centred arches are not as strong as circular 
arches, and should only be used where they can be given all the 
strength desirable. 

The strenytJi of an arch depends very much upon the care with 
which it is built and the quality of the work. 

In stone arches, special care should be taken to cut and lay the 
beds of the stones accurately, and to make the bed-joints thin and 
close, in order that the arch may be strained as little as possible in 
settling. 

To insure this, arches are sometimes built dry, grout or liquid 
mortar being aftei*wards nm into the joints; but the advantage of 
this method is doubtful. 

!Brick Arches may be built either of wedge-shaped bricks, 
moulded or rubbed so as to fit to the radius of the soffit, or of 
bricks of common shape. The former method is imdoubtedly the 
l>est, as it enables the bricks to be thoroughly bonded, as in a wall ; 
but, as it involves considerable expense to make the bricks of the 
proper shape, this method is very seldom employed. Where bricks 
of the ordinary shape are used, they are accommodated to the 
curved figiu-e of the arch by making the bed-joints thinner towards 
the intrados than towards the extrados; or, if the curvature is 
sharp, by driving thin pieces of slate into the outer edges of those 
joints; and different methods are followed for bonding them. The 
most common way is to build the arch in concentric rings, each 
lialf a brick thick; that is, to lay the bricks all stretchers, and to 
depend upon the tenacity of the mortar or cement for the connec- 
tion of the several rings. This method is deficient in strength, 
unless the bricks are laid in cement at least as tenacious as them- 
selves. Another way is to introduce courses of headers at intervals, 
so as to connect pairs of half-brick rings together. 

This may be done either by thickening the joints of the outer of 
a pair of half-brick rings with pieces of slate, so that there shall bo 
the same number of courses of stretchers in each ring between two 
courses of headers, or by placing the courses of headers at such 
distances apart, that between each pair of them there shall be one 
course of stretchers more in the outer than in the inner ring. 

The former method is best suited to arches of long radius ; the 
latter, to those of short radius. Hoop iron laid round the arch, 
between half-brick rings, as well as longitudinally and radially, is 
very useful for strengthening brick arches. The bands of hoop iron 
which traverse the arch radially may also be bent, and prolonged 
In tbe bed-Joints of the backing and spandrels. 



196 



THE STAlilLlTY OF ARCHES. 



By the aid of hoop-iron bond. Sir Marc-lsanibard Brunei 
half-arcli of bricks laid in strong cemtint, which stood, pr< 
from its abutment like a bracket, to tlie distance of sixty fe< 
it was destroyed by its foundation being undermined. 

The New- York City Building Laws make the following i 
ments regarding brick arches: — 

" All arches shall be at least four inches thick. Arches o"\ 
foot span shall be increased in thickness toward the hauu 
additions of four inches in thickness of brick. The first ad< 
thickness shall commence at two and a half feet from the c< 
tli(^ span ; the second addition, at six and one-lialf feet from I 
tre of the span ; and the thickness shall be increased then 
inches for every additional four feet of span towards the liai 

" The said brick arches shall be laid to a line on the centr 
a close joint, and the bricks shall be well wet, and the join 
with cement mortar in proyoitions of not more than two < 
to one of cement by measure. The arches shall be well | 
and pinned, or chinked with slate, and keyed." 

Hide for RadUis of Brick Archett. — A good nUe for the 
of segmental brick arches over windows, doors, and othe 
openings, is to make the radius equal to the width of the Oj 
This gives a good rise to 
the arch, and makes a pleas- 
ing proportion to the eye. 

It is often desirable to 
span openings in a wall by 
means of an arch, when 
there is not sufficient abut- 
ments to withstand the 
thrust or kick of the arch. 
In such a case, the arch can 
be formed on two cast-iron 
skewbacks, which are held 
in place by iron rods, as is 
shown in Fig. 2. 

AVhen this is done, it is necessai^ to proportion the size 
rods to the thrust of the arch. The horizontal thrust of the 
very nearly represented by the following formula: — 

load on arch x span 
Horizontal thrust = y x rise of arch in feet' 

If two tension rods are used, as is generally the case, the 
ter of each rod can be detennined by the following mie: — 




^. . . , / total load on arch X span 

Diameter lu iiicl.es = y/ ^ x rise of aich in fee»^ 



THE STABILITY OF ARCIIES.y 107 

If only one rod is used, 8 should be substituted in the place of 
16, in the denominator of the above rule; and, if three rods are 
used, 24 should be used instead of 1(5. 

Centres for Arches. — A centre is a temporary stnicture, 
generally of timber, by which the voussoirs of an arch are sup- 
ported while the arch is being builU It consists of parallel frames 
or ribs, placed at convenient distances apart, cui'ved on the outside 
to a line parallel to that of the soffit of the arch, and supporting 
a series of ti-ansverse planks, upon which the arch stones rest. 

The most common kind of centre is one which can be lowered, or 
struck all in one piece, by driving out wedges from below it, so as 
to remove the support from every point of the arch at once. 

The centre of an arch should not be struck until the solid part of 
the backing has been built, and the moi*tar has had time to set and 
haixlen ; and, when an arch forms one of a series of arches with 
piers between them, no centre should be struck so as to leave a pier 
with an arch abutting against one side of it only, imless the pier has 
sufficient stability to act as an abutment. 

When possible, the centre of a large brick arch should not be 
struck for two or three months after the arch is built. 

Mechanical Principles of the Arch, — In designing an 
arch, the fii-st question to be settled is the form of the arch; and in 
regard to this there is generally but little choice. Where the abut- 
ments are abundantly large, the segmental arch is the strongest 
fonn ; but, where it is desired to make the abutments of the arch 
as light as possible, a pointed or semicircular arch should be used. 

Depth of Keystone. — Having decided upon the form of the arch, 
the depth of the arch-ring must next be decided. This is generally 
determined by computing the required depth of keystone, and 
making the whole ring of the same or a little larger depth. 

In considering the strength of an arch, the depth of the keystone 
is considered to be only the distance from the exti-ados to the intra- 
dos of the arch; and if the keystone projects above the arch-ring, 
as in Fig. 1, the projection is considered as a part of the load on 
the arch. 

There are several rules for determining the depth of the key- 
r.tone, but all are empirical; and they differ so greatly that it is 
<lifficidt to recommend any particular one. Professor Rankine's 
Itule is often quoted, and is probably true enough for most arches. 
It applies to both circular and elliptical arches, and is as follows: — 

Rankine's Rule. — For the depth of the keystone, take a 
mean proportional between the inside radius at the crown, and 
0.12 of a foot for a single arch, and 0.17 of a foot for an arch form- 
ing one of a series. Or, if represented by a formula, 



•Mi) THE STABILITY OP AECHE8. 

Bnt, if we sliouM compute the stability of a •eraidreular ardi of 
20 foot span, and 1.3 foot depth of keystone, we should find thai 
the arch was vei^ unstablp; hen^e, in this case, we must throw tlw 
rule aside, and go by our own judgment. In the opinion of the 
autlior, such an arcli should have at least 2i feet depth of ucb- 
Ttng, and we wiil try the stability of the arch with that thickness. 

In ali calculations on tlie arch, it is customary to conaltler tlie 
an'U to be one foot thick at rightangles toltsface; for it is evident, 
thai, if an arch one foot thick is stable, any utmiberof arches of the 
same fliiiiensioiis built alongside of it would be stable. 

Graplilc Solution of tlie Stalilllty of tlie Arcli.— 
Tlie most convenient luctbod of detennlning the stability of the 
arch is by the graphic mutliod, as it is called. 

1st Stbi'. — Draw one-half the arch to as large a scale as con- 
venient, and divide it up Into voussoirs of i!qual size. In this 
exaniiile, shown In Fig. '-i. we have divided the arch-ring into ten 
equal voussolrs. (It is not necessary that these should be the 
actttal voussolrs of which the arch is built. ) The next step Is to 
And the area of each voussolr. Where the arch-rfi^ Is divided into 
voussoirs of equal size, this Is easiest done tiy computing th« ana 
of the arch-ring, and dividing by the number of voussoira. 



Fls.3 

Ridi' for 'W'li of •iiif-hiiif vf urdi-rim; is as follows: — 

Area in square feet = 0.7854 X (outside radius squared — itaW.c 
radius squared). 

In this example the wholi' area equals 0.78-Vl X J12.5* — Id*) = 
44.2 s<|iiare feet. As tiiere are ten equal voussoira, the area of «*ch 
vonssilir is 4.4 square feet. 

Having drawn out one-half of the arch-ring, we divide eack Joint 
into tliree equal parts; and from the point A (Fig. 8] we lay off to 
a scale the area of each voussoir, one below the ot' 



THE STABILITY OF ARCHES. 201 

with the top voussoir. The whole length of the line AE will equal 
the whole area drawn to same scale. 

The next step is to find the yertical line passing through the 
centre of gravity of the whole arch-ring. To do this, it is first 
necessary to draw vertical lines through the centre of gravity of 
each voussoir. The centre of gravity of one voussoir may be found 
by the method of diagonals, as in the second voussoir from the top 
(Fig. 3). Having the centre of gravity of one voussoir, the centres 
of gravity of the others can easily be obtained from it. 

Next, from A and E (Fig. 3) draw lines at 4b^ with AE, inter- 
secting at O. Draw 01, 02, 03, etc. Then, where AO intersects 
the first vertical line at a, draw a line parallel to 01, intersecting 
the second vertical at b. Draw 6c parallel to 02, cd parallel to 03, 
and so on to kn parallel to OlO: prolong this line downward until 
it intersects AO, prolonged at D. Then a vertical line drawn 
through 1) will pass through the centre of gravity of the arch-ring. 

2i) Step. — Draw a horizontal line through A (the upper part of 
the middle third), and a vertical line through D; the two lines 
intersecting at C (Fig. 3). 

Now, that the arch shall be stable, it is considered necessary that 
it shall be possible to draw a line of resistance of the arch within 
the middle third. We will, then, first assume that the line of 
resistance shall act at A, and come out at B'. 

Then draw the line CB, and a horizontal line opposite the point 
10, between Q and P. This horizontal line represents the hori- 
zontal thrust at the crown. 

Draw AP equal to QP, and the lines PI, P2, P3, etc. 

Then, from the point where AC prolonged intersects the first 
vertical, draw a line to the second vertical, parallel to PI ; from 
this point a line to the third vertical, parallel to P2 ; and so on. 
The last line should pass through B. If these lines, which we will 
call the line of resistance, all lie within the middle third, the arch 
may bo considered to be stable. Should the line of resistance pass 
outside of the arch-ring, the arch should be considered unstable. 
In Fig. 3 this line does not all lie in the middle third, and we nuist 
see if a line of resistance can yet be drawn within that limit. 

2i) Triai.. — The line of resistance in Fig. 3 passes farthest from 
the middle third at the seventh joint from the top; and we will next 
pass a line of resistance through A and where the lower line of the 
middle third cuts the seventh joint, or at B (Fig. 4). 

To do this, we must prolong the line <jh, parallel to 07 (Fig. 4), 
until it intersects AO. In this case it intersects it at O; but this 
18 merely a coincidence; it would not always do so. Through O 
draw a vertical intersecting PA prolonged at C. Draw a line 



303 THE STABILITY OF ARCHES. 

through C &nd D, and the horizontal line p^, oppoalte the point 7: 
this line represents the new horizonUkl thrust H,. Disw AP = 
pQ, and the lines PI, P2, etc.; then draw the line of resistaniK 
)is before. It should pass through D if drawn correctly. This 
lime we aee that the line of reslatance Ilea within the middle third, 
except jiist a short distance at the springing; and hence we nw} 
consider the arch stable. If it had gone outside the middle third 
this time, to any great extent, we should have considered the anHi 
unatable. 

The above Is the method of determining the stability of M 
unloaded semicircular arch. Such a case very seldom occurs In 
practice; but it is a good example to Illustrate the method, whidi 
applies to all other cases, with a little difference in the method of 
determining the centre of gravity of loadod arches. 



FiB.4 

Example II. — Loaded or awcharf/ed semicircular areh. 

We will take the same arch as in Example L, and snppoM It to 
l>e loaded with a wall of masonry of the same thickness and welgbt 
per square foot as tliat of the arch-ring ; the horizontal snrtece of 
rhc wall being 3 feet C inches above the arch-ring at the crown. 

1st Stei-. — Find centre qfgraHty, 

Commencing at Ibe crown, divide the load and aFch-rlng Into 
strips two feet wide, making the last strip the width of the areb- 
ring at tlie springing. Then draw the joints as shown In Hg, G. 
Measure with the scale the length of each vertical line, Aa, Bb, 
etc. ; then the area of Aalili Is equal to llie length of An + Bb, M 
the distance between them is Just two feet. The area of ffKk li, 
of course, FfX width of areh-ring. 

In this case, the areas of the slices are as shown by the Ognnt on 
their faces (Fig. 5}. 

Now <]lvlde the areh-ring into thirds, and from the top of tba 
middle thin<, at It, lay oS in succession, to a iHmla, tbe ntut td 



THE STABILITY OF ARCHB8. SOB 

iKcefl, commencing with the first slice (ram the crown, AaBb. 
m areas, when measured off, wilt be represented by the line 
2, $ ... B (Pig. 5). From the extremities of this liile, if and 6, 
V lines at 45° with a vertical, intersecting at O. B>om O draw 
t to 1, 2, 3, 4, 5, and 6. Next, draw a vertical line through the 
re of each slice (these lines, in Fig. 5, are nnmlKred 1, 2, 3, 
I. From the point in which the line RO intersects vertical 1 , 
t a line paraJle) to 01, lo the line 2. From this point draw a 
to vertical 3, parallel to 02, and so on. The line parallel to 
will intersect vertical 6 at F. Then through F draw a line 
owards at 4^°, iniersecting OB at X. A vertical Hue drawn 
ngb X will pass through the ceutre of gravity of the arch-rlog 
its load. 



I Step. — To find the thnat at thecrojnnand at the i>pringing. 
) find the thrust at the crown, draw a vertical line through .V, 
a horizontal line through B, intersecting at V, Now, the weight 
■ch and load, and the resultant thrust of arch, must act throi^h 
point. We will also make the condition that the thrust shall 
through Q, the outer edge of the middle third. Then the 
at of the arch must act in the line VQ. Opposite 6, on the 
ical line throi^h B, draw a horizontal line IT, between KA' 
V<i. This horizontal tine represents a horizontal thrust at B, 
•h would cause the resultant thrust of the arch to pass through 
Now draw the horizontal line BP, equal in length to H, and 
I P draw lines 1, 2, 3 ... U. The line P6 represents the thrust 
be Mcb at Uie springing. lie amouut In cubic feet of masonry 
be detennined by measuring its length to the proper scale. 



204 THE STABILITY OF ARCHES. 

3d Step. — To draw the line of resistance. 

The lines PI, P2, P3, etc., represent the magnitude and dirae- 
tion of the thrust at each joint of the arch. Thus PI represents 
the thrust of the first voussoir and its load ; P2, that of the flret 
two voussoirs and their loads; and so on. Then from the point a', 
where the line BP, prolonged, intersects the vertical line 1, draw 
a line a7/ parallel to PI; from 6', on 2, draw a line 6V parallel 
to P2, and so on. The last line should pass through Q, and be 
parallel to P6. 

Now, if we connect the points where the lines a'6', 6V, etc., cnk 
the joints of the arch, we shall have a broken line, which is known 
as the line of resistance of the arch. If this line lies within the 
middle third of the arch, then we conclude that the arch is stable. 
If the line of resistance goes far outside of the middle, we must see 
if it be possible to draw another line' of resistance within the mid- 
dle third; and if, after a trial, we find that it is not possible, we 
must conclude that the arch is not safe, or unstable. 

In the example which we have just been discussing, the line of 
resistance goes a little outside of the middle third; but it is very 
probable that on a second trial we should find that a line of resist- 
ance passed through R and Q' would lie almost entirely within the 
middle third. . 

The method of drawing the second line of resistance was 
explaineil under Example I. ; and, as the same method applies to 
all cases, we will not repeat it. 

The method given for Example II. would apply equally well for 
a semi-elliptical arch. 

Example 111. — Segmental archy with load (Fi^ 6). 

1st Step. — To determine the centre ofgravify. 

In this case we proceed, the same as in the latter, to divide the 
arch-ring and its load into vertical slices two feet wide, and compute 
the area of the slices by measuring the length of the vertical lines 
An, Bh, etc. Having computed the areas of the slices^ we lay them 
off in order from R, to a convenient scale, and then proceed 
exactly as in Example II., the remaining steps detenAinlng the 
tlirust; and the lines of resistance are also the same as given under 
Example 11. 

In a flat segmental arch, there is practically no need of dividing 
the arch-ring into voussoirs by joints radiating from a centre, but 
to consider the joints to be vertical. Of course, when built, they 
must be made to radiate. 

Fig. 6 shows the computation for an arch of 40-loot flpan, and 
with a load 13i feet high at the centre. The depth of the arch- 
ring is 2 feet inches. 

It will be seen, that the curve of pres as lies a iralj irlllifai 

- 



TIiE STABILITY OF ARCHES. 305 

iddle third; uid hence the arch is abundaatlj safe, or stable, 
■tild be remarked, that the line of resIstaDce in a segmental 
should be drawn through the toteer edge of the middle third 
springing. 



lii be noticed that the horizontal thrust, and ttie thrust T, 
springing, are very great as compared wiih those in a seml- 
Lr arch; and hence, aJthough the segmental arch Is the 
er of the two, it requires much heavier abutments, 
se three examples serve to show tlic method of determining 
tUlity and thrust of any arch sucli as is nseA In building. 



20(1 RESISTANCE TO TENSION. 



CHAPTER IX. 



RESISTANCE TO TENSION. 

OR THE STRENGTH OF TIE-BOD8, BARS, ROPES, AND CHAINS. 

The resistance which any material offers to being pulled apart 
is due to the tenacity of its fibres, or the cohesion of the particles 
of which it is composed. 

It is evident that the amount of resistance to tension which any 
cross-section of a body will exert depends only upon the tenacity 
of its fibres, or the cohesion of its particles, and upon the number 
of fibres, or particles, in the cross-section. 

As the number of the fibres, or particles, in the section, is pro- 
portional to the area, the strength of any piece of material must be 
as the area of its cross-section; and hence, if we know the tenacity 
of the material per square inch of cross-section, we can obtain the 
total strength by multiplying it by the area of the section in 
inches. 

The tenacity of different building-materials per square inch hM 
been found by pulling apart a bar of the material of known dimen- 
sions, and dividing the breaking-force by the area of the croti- 
section of the bar. 

Table I. gives the average values for the tenacity of building- 
materials, as determined by the most reliable experiments. 

Knowing the tenacity of one square inch of the material, all 
that is necessary to determine the tenacity of a piece of any uniform 
size is to multiply the area of its cross-section, in square inches, by 
the number in the table opposite the name of the material. Tliii 
would give the weight that would just break the piece; but, as what 
we wish is the safe load, we must divide the result by a factor of 
safety. Most engineers advise using a factor of safety of five f6r 
a (lead load, although the New-York City and also the Boston 
Building Laws require a factor of six. 

Denoting the factor of safety by Sf and the tenacity by T, we 
iKive as a rule. 

For a rectangular bar, 

breadth x depth XT 
Safe load = ^;7-^^^ (1) 



RESISTANCE TO TENSIOlf. 



For a round bar, 

„ , , , 0.1854 X diameter squared x T 

Safe load = ~ -g — — (2J 

ExAMPLBl. — Wliat is the safe load for a tie-bar of wUite pine 
B b; 6 inches ? 

Ans. Here the breadtb and depth both equal G inches, T — 7000, 
and we will let tf = 5; then. 



20tf X RESISTANCE TO TENSION. 







y 



e size of the bar is desired, we have, 

iS X load 
The breadth = g^^j^^^ (3) 

For a round bar, 

_. -S X load 

Diameter squared = q '^054 v T ^^' 

Example II. — It is desired to suspend 20,000 pounds from a 
round rod of wrought-iron : what shall be the diameter of the rod 
to carry the weight in safety ? 

Ans, In this case T = 50,000; and taking 8 at 5, we have 

5X20000 
Diameter squared = 0.7854 x 50000 = ^-M. 

The square root of this is 1.6 or 1§ inches nearly: therefore 
the diameter of the rod should be If inches. 



Tensile Strength and Qualities of SteeL 

The elastic limit of steel should not be less than 40,000 poonds 
per square inch for high grade steel, 36,000 pounds for medium 
steel, and 30,000 pounds for solt steel. 

The ultimate tensile strength of high grade steel should range 
between 70,000 and 80,000 pounds per square inch ; of mediom, 
between 00,000 and 70,000 ; and of soft steel, between 52.000 and 
60,000 pounds per square inch. 

The elongation in a length of 8 inches should be not less than 18 
per cent, for liigh grade steel, 23 per cent, for medium, and 25 per 
cent, for soft stcol. 

The reduction of area at point of fracture should be not less than 
35 per cent, of tho original area. 

Jligh grade steel i85 per cent, carbon) should be used for com- 
pression, bolsters, bearing-plates, pins, and rollers. 

Medium steel (1j per cent, carbon) should be osed for tension 
members, floor system, laterals, bracing, and, unless high gnde 
steel is specified, should be used for all steel members except rivets. 

Soft steel (11 or 12 j)er cent, carbon) should be nsed in rivets only, 
and should bo tested by actually making up into rivets, riveting 
two plates together, and upon being nicked and cut cmt should 
show a good, tough, silky structure, with no crystalline appeannoe. 
Rivet steel should not have over 0.15 per cent, oaifoon. 

Steel made by the Bessemer process shonld not re over 0.06 
per cent, of phosphorus, and open hearth steel e ow -^ cf 1 



RESISTANCE TO TENSION. 209 

per cent. The amount of phosphorus allowable should always be 
stated in the specitications, as this determines the price of the pig 
iron required to make the steel. About 0.04 per cent, of sulphur 
is allowable, and sometimes more.* 

The Working Streiig^h of steel in bi-idges is generally taken 
at 12,000 pounds per square inch, and in roof trusses, and struct- 
ures sustaining a steady load, at 15,000 pounds per square inch ; 
or, in a general way, the strength of steel is generally taken at 
20 per cent, over that allowable for wrought iron under the same 
conditions. 

standard spxsoifioation, adopted by bridgb- 
buhiDErs, for material and workmanship 
of iron and steel structures. 

quality of materials. 

Wn OUGHT Ibon. 

Character and Finisli. — I. All wrought iron must be 
tough, ductile, fibrous, and of uniform quality for each class, 
straight, smooth, free from cinder pockets or injurious flaws, 
buckles, blisters, or gracks. As the thickness of bars approaches 
the maximum that the rolls will produce, the same perfection of 
finish will not be required as in thinner ones. 

2. No specific process or provision of manufacture will be de- 
manded, provided the materia] fulfils the requirements of this 
specification. 

Standard Test Piece. — 3. The tensile strength, limit of 
elasticity and ductility, shall be determined from a standard test 
piece, not less than one quarter inch in thickness, cut from the full- 
size bar, and planed or turned parallel ; if the cross-section is 
reduced, the tangent between shoulders shall be at least twelve 
times its shortest dimension, and the area of minimum cross-sec- 
tion in either case shall be not less than one-quarter of a square 
inch and not more than one square inch. Whenever practicable, 
two opposite sides of the piece are to be left as they come from the 
roils, but the finish of opposite sides must be the same in this 
respect. A full-size bar, when not exceeding the above limitations, 
may be used as its own test piece. In determining the ductility 
the elongation shall be measured, after breaking, on an original 
length the nearest multiple of a qinirttT inch to ten times the 
shortest dimension of the test piece, in which length must occur the 

* JTaioeB BUobo, before the Civil Engineers' Club of Cleveland. 



210 RESISTANCE TO TENSION. 

curve of reduction from stretch on both sides of the point o< frut- 
ure, but in no case on a shorter length than five inches. 

Tension Iron for Open Trusses. — 1. Ail iron to be used 
in the tensile members of open trusses, laterals, pins and bolts, ex- 
cept plate iron over eight inches wide and shaped iron, must show 
by the standard test piece a tensile strength in pounds per square 
inch of : 

f-rt rxr^n. 7,000 X arca of original bar , ,, . . , . 

52,000 r^ — , i- ^.—r-a- (a^ ^ inches), 

circumference of onginal bar 

with an elastic limit not less ttian one-half the strength given by 
this formula, and an elongation of twenty per cent. 

Plate Iron. — .*). Plate iron 24 inches wide and under, and 
more than 8 inches wide, must show by the standard test pieces a 
tensile strength of 4d,C00 pounds per square inch, with an elastic 
limit not less than 26,000 pounds per square inch, and an elonga- 
tion of not less than 1 2 [)er cent. All plates over 24 inches in width 
must have a tensile strength not less than 46,0CO pounds per sqoue 
inch with an elastic limit not less than 26,000 pounds per sqoue 
inch. Plates from 24 inches to 86 inches in width must have An 
elongation of not less than 10 per cent. ; those from 86 inches to 4B 
inches in width, 8 per cent. ; over 48 inches in width, 5 per cent. 

Shaped Iron. — 6. All shaped iron and other iron not herein- 
before . specified must show by the standard test pieces a tensile 
strength in pounds per square inch of : 

7.000 X area of original bar 



50,000- 



circumference of original bar* 



with an elastic limit of not less than one-half the strength given 
by this formula, and an elongation of 15 per cent, for bars fifo- 
eighths of an inch and less in thickness, and of 12 per cent, lor 
bars of greater thickness. 

Hot Bending. — 7. All plates, angles, etc., which are to be 
bent hot, in th(> manufacture must, in addition to the above rs- 
quirements, be capable of bending sharply to a right angle at a 
working heat without sign of fracture. 

Rivet Iron.— 8. All rivet iron must be tough and soft^ and 
pieces of the full diameter of the rivet must be capable of bending 
cold until the sides are in close contact without sign of fracture on 
the convoif side of the curve. 

Bending Tests. — 9. All iron specified in claase 4 most bend 
cold, 180 degrees, without sign of fracture, to a oorve the innflr 
radius of which equals the thickness of the pieoa tested- - 



RESISTANCE TO TENSION. 211 

10. Specimens of full thickness cut from plate iron, or from the 
flanges or webs of shaped iron, must stand bending cold, through 
90 degrees, to a curve the inner radius of which is one and a half 
times its thickness, without sign of fracture. 

Niiiiiber of Test Pieces.— 1 1 . For each contract four stand- 
ard test pieces and one additional for each 50,000 pounds of wrought 
iron will, if required, be furnished and tested by the contractor 
without charge, and if any additional tests arc required by the pur- 
chas'ir, they will be made for him at the rate of $5J!0 each ; or, if 
the contractor desires additional tests, they shall be made at his 
own expense, under the supervision of tlie purchaser, the quality of 
the material to be determined by the result of all the tests in the 
manner set forth in the following clause. 

12. The respective requirements stated are for an average of the 
tests for each, and the lot of bars or plates from which samples 
were selected shall be accepted if the tests give such average results ; 
but, if any test piece gives results more than 4 per cent, below said 
requirements, the particular bar from which it was taken may be 
rejected, but such tests shall be included in making the average. 
If any test piece has a manifest flaw, its test shall not be considered. 
For each bar thus giving results more than 4 per cent, belov/ the re- 
quirements, tests from two additional bars shall be fumishe<l by 
the contractor without charge, and if in a total of not more than 
ten tests, two bars (or, for a larger number of tests, a proportion- 
ately greater number of bars) show results more than 4 pier cent, 
below the requirements, it shall be cause for rejecting the lot from 
which the sample bars were taken. Such lots shall not exceed 20 
tons in weight, and bars of a single pattern, plates rolled in univer- 
sal mill or in grooves, and sheared plates shall each constitute a 
separate lot. 

Time of Inspection. — 13. The inspection and tests of the 
material will be made promptly on its being rolled, and the quality 
determined before it leaves the rolling-mill. All necessary facili- 
ties for this purpose shall be afforded by the manufacturer ; but, if 
the inspector is not present to make the necessary tests, after due 
notice given him, then the contractor shall proceed to make such 
number of tests on the iron then being rolled as may have been 
agreed upon ; or, in the absence of any special agreement, the num- 
ber provided for in clause 11, and the quality of such material shall 
be determined thereby. 

Variation of Weiglit. — 14. A variation in cross-section or 
weight of rolled material of more than 2^ per cent, from that speci- 
fl€d nwy be catiBe for rejection. 



212 liESISTAKOK TO TENSION. 



Steel. 



15. No specific process or provision of manufMetare will be de- 
manded, ])rovided tl^c material fullils the regniremg^ts of this 
specitication. 

Test Bars.— IG. From three seiiarate ingots of each casta 
round sample bar, not less than three-quarters of ivn inch in diame- 
ter, and having a length not less than twelyo diameters between 
jaws of testing machine, shall be furnished and tested by the manu- 
facturer without charge. These bars are to be truly round, and 
shall be linished at a uniform heat, and arranged to cool onifonnljf, 
and fro:n these test pieces alone, the quality of the material ahaU be 
determined as follows : 

Tensile Tests.— 17. All the above described test baramut 
have a tensile strength within 4,000 pounds per square inch of that 
specified, an elastic limit not less than one-half of the tenaile 
strength of the test bar, a percentage of elongation not leas than 
1,200,000 -f- the tensile strength in pounds per square inch, and a 
percentage of reduction of area not less than2,40O,O0O -f- thetensQe 
strength in pounds per square inch. In determining tbe ductUitj 
the elongation shall bo measured after breaking on an original 
length of ten times the shortest dimension of the test piece, ia wliicli 
lengt h must occur the curve of reduction from stretch on both sidn 
of the point of fracture. 

Finish and Reduction of Area on Finished Ban.^ 
IS. Finished bars must be free from injurious flaws or cracks and 
must have c workmanlike finii^h, and round or square test pieoee 
cut therefrom when pulled asunder shall have reduction of area at 
the point of fracture as above specified. 

[Number of Test Pieces.— 19. For each contract foor.snch 
tests respectively for reduction of area and for bending, and one 
additional of each for eax;h 5 J,()()0 pounds of steel will, if zeqoired, 
be made by the contractor witliout charge ; and if the porohaaeris 
not satisfied that the I'eduction of area test correctly indicates the 
effect of the heating and rolling, such additional tests for tenaik) 
strength, limit of elasticity, and ductility, as ho may desire, will bo 
made for him on test pieces confomiing to the provisions of daoso 
8, at the rate of $5.00 each, or, if the contractor desires additional 
tests, he may make them at his own expense, under tho saperviuon 
of the purchaser, the quality of the material to be determined bj 
the result of all the tests in the manner set forth in the fbUowing 
clause. 

20. Except for tensile strength, the respective : ijpiinaiinli 



BESLSTANCE TO TENSION. 213 

stated &re for an average of the tests for each, and the lot of bars 
or plates from which samples were selected shall be accepted if the 
tests give such average results ; but, if any test piece gives results 
more than 4 per cent, below said requirements, the particuhir bar 
from which it was taken maybe rejected, but such tests shall be in- 
cluded in making the average. If any test piece has a manifest 
flaw, its U^st shall not be considered. For each bar thus giving 
results more than 4 per cent, below the requirements, tests from two 
additional bars chall be furnished by the contractor without charge, 
and if in a total of not more than ten tests, two bars (or. for a 
larger number of tests, a proportionately greater number of bars) 
show results more than 4 per cent, below the requirements, it shall 
bo cause for rejecting the lot from which the sample bars were 
taken. Such lot shall not exceed 20 tons in weight, and bars of a 
single pattern, plates rolled in universal mill or in grooves, and 
sheared plates shall each constitute a separate lot. 

Rivet Steel. — 2l. Rivet steel shall have a specified tensile 
strength of 60,000 pounds per square inch, nnd test bars must have 
a tensile strength within 4, 03 pounds per square inch of that spe- 
cified, and an elastic limit, elongation, and reduction o ' area at the 
point of fracture, as stated in clause 17, and be capable of bending 
double, flat, without sign of fracture on the convex surface of the 
bend. 

Time of Inspection. — 22. The inspection and tests of the 
material will be made promptly on its being rolled, and the quality 
determined before it leaves the rolling-mill. All necessary facili- 
ties for this purpose shall bo afforded by the manufacturer ; but, if 
the inspector is not present to mak(^ the necessary tests, alter due 
notice given him, then the contractor shall proceed to make such 
namber of tests on the steel then being rolled as may have been 
agreed upon, or, in the absence of any special agreement, the 
number provided for in clause IG or 10, and the (luality of such 
materia] shall be determined thereby. 

Variation of Weigrhts. — 23. A variation in cross-section 
or weight of rolled material of more than 2^ per cent, from that 
specified may be cause for rejection. 

CAhT Iron. 

24. Except where chilled iron is specifie 1, all c;astings shall be 
of tough gray iron free from injurious cold ^huts or blow holes, true 
to pattern, and of a workmanlike finish. Sample pieces 1 inch 
aqiuune oast from the same heat of metal in sand moulds shall be 



214 RESISTANCE TO TENSION. 

capable of sustaining on a clear span of 4 feet 6 inches a centnl 
load of 500 pounds when tested in the rough bar. 

Workiiiansjiip. 

Inspection. — 25. Inspection of the work shall be made as it 
progresses, and at as early a period as the nature of the work 
permits. 

26. All workmanship must be first-class. All abutting surfaces 
of compression members, except flanges of plate girders where the 
joints are fully spliced, must be planed or turned to even bearings 
so that they shall bo in such contact throughout as may be obtained 
by such means. All finished surfaces must be protected by white 
lead and tallow. 

27. The rivet-holes for splice plates of abutting members shall 
be so accurately spaced that when the members are brought into 
position the holes shall be truly opposite before the rivets are 
driven. 

28. When members are connected by bolts whioh transmit 
shearing strains the holes must be reamed parallel, and the bolts 
turned to a driving fit. 

29. Hollers must be finished perfectly round and roller-beds 
planed. 

Rivets. — 80. Rivets must completely fill the holes, have foil 
heads concentric with the rivet, of a height not less than ,0 the 
diameter of the rivet, and in full contact with the surface^ or be 
countersunk when so requiretl, and machine-driven wherever prM- 
ticabie. 

31. Built members must, when finished, bo true and free from 
twists, kinks, buckles, or open joints between the component pieces. 

Eye Burs and Pin-hole, and Pilot Nuts.— 82. All 
pin-holes must be accurately bored at right angles to the axis of 
the members, unless otherwise shown in the drawings^ and in 
piec^es not adjustable for len^.th no variation of more than one- 
thirty-se3oncl of an inch will be allowed in the length between 
centres of pin-holes ; tlio diameter of the pin-holes shall not exceed 
that of the pins by more than one- thirty-second inch, nor by more 
than one-fiftietli inch for pins under three and one-half inobes 
diameter. Eye bars must Ic strai^^ht before boring; the holes 
must be in the centre of the heads, and on the centre line of 
the bars. Whenever links arc to be packe;! more tiian onemi^tli 
of an inch to the foot of their length out of parallel with the 
axis of the structure, they must bo bent with a gentle 



RESISTANCE TO TENSION. 215 

the head stands at right angles to the pin in their intended position 
before being bored. All links belonging to the same panel, when 
placed in a pile, must allow the pin at each end to pass through at 
the same time without forcing. No welds will be allowed in the 
body of the bar of eye bars, laterals, or counters, except to form 
the loops of laterals, counters, and sway rods ; eyes of laterals, 
stirrups, sway rods, and counters, must be bored ; pins and lateral 
bolts must be finished perfectly round and straight, and the party 
contracting to erect the work must provide pilot nuts where neces- 
sary to preserve the threads while the pins are being driven. 
Thimbles or washers must be used whenever required to fill the 
vacant spaces on pins or bolts. 

Tests of Eyes on Full Size Bars.— 33. To determine the 
strength of the eyes, full size eye bars or rods with eyes may be 
tested to destruction, provided notice is given in advance of the 
number and size required for this purpose, so that the material can 
be rolled at the same time as that required for the structure, and 
any lot of iron bars from which full size samples are tested shall be 
accepted — 

1st, if not more than one-third the bai*s tested break in the eye ; 
or, 

2d, if more than one- third do break in the eye and the average of 
the tests of those which so break shows a tensile strength in pounds 
per square inch of original bar, given by the formula — 

g3 pQQ_7,000 X area of origjnal bar _ ,^^ ^ ^.^^^^ ^^ ^^^ ^j -^ 
Circumference of original bar 

inches), and not more than one-half of those which break in the eye 
fail at more than 5 percent, below the strength given by the formula. 
Any lot of steel bars from which full size samples are tested shall be 
accepted if the average of the tests shows a strength per square inch 
of original bar, in those which break in the eye, within 4,000 
pounds of that specified, as in clause 17 ; but if one- half the full size 
samples break in the eye, it shall be cause for rejecting the lot from 
which the sample bars were taken. All full size sample bars which 
break in the eye at less than the strength here specified shall be at 
the expense of the contractor, unless he shall have made objection 
in writing to the form or dimension of the heads before making the 
eye bars. All others shall be at the expense of the purchaser. If 
the contractor desires additional tests thev shall be made at his own 
expense, under the supervision of the purchaser, the acceptance of 
the bars to be determined by the result of all the tests in the 
manner above set forth. A variation from the specified dimensions 



216 RESISTANCE TO TENSION. 

of the heads will be allowed, in thickness of one-tblrty.second inch 
below and one-sixteenth above that specified, and in diameter of 
OD^ourth inch in either direction. 

Piincbiug' and Reaming. — 34. In iron work, the diameter 
of the punch shall not exceed by more than one-sixteenth inch the 
diameter of the livcts to be used. Rivet-holes must be accurately 
spaced ; the use of drift-pins will be allowed only for bringing 
together the several parts forming a member, and they must not be 
driven with such force as to disturb the metal about the holes ; if 
the hole must be enlarged to adnut the rivet, it must be remade; 
all rivet-holes in steel work, if punched, shall be made with a punch 
one-eighth inch in diameter less than the diameter of the rivet in- 
tended to be used, and shall be reamed to a dluneter one-sixteenth 
inch greater than the rivet. 

Annealing. — 35. In all cases where a steel piece iu which the 
Full strength is required has been partially heated, the wlM^e piece 
must be subsequently annealed. All bends in steel most be nade 
cold, or if the degree of curvature is so great as to require heatings 
the whole piece must be subsequently annealed. 

Painting. — 86. All surfaces inaccessible after assembling 
must be well painted or oiled before the parts are assembled. 

37. The decision of the engineer shall control as to the interpre- 
tation of drawings and specifications during the execution of work 
thereunder, but this shall not deprive the contractor of his right to 
redress, after the completion of the work, for an improper decision. 



BESISTANOE TO TENSION. 



217 



TABLE II. 

Tables showing the Strength given by tJie Form'tUce of Sections 4, 
6, and 88, for Iron Bars of Various Dimensions. 

7,000 X area of original bar 
For Standard Te«t Pi, ce of Bars, 88,000 - i,- j-^SSiS^e ^^tTri^iiiA bif- 

For ey*i8 of Full Size Eye Bars, 

7,000 X area of original bar ^^ . ,. . u * ia^u 

62,000 ,- —i- c . _i 11- - 5 .0 lbs. per inch of width. 

' circumference of original bar 

7,000 X area of original bar 
For Standard Te^t Piece of Angles. 50,000 - ^jrcuiSference of original h^' 



Size of bar. 


1 


X 1 


u 


xU 


u 


xli 


2 


X 2 


2 


X \ 


2 


X I 


2 


X 1 


8 


X i 


8 


X J 


8 


X 1 


4 


X \ 


4 


X 1 


4 


xli 


5 


X } 


5 


xl 


5 


xli 


5 


X li 


5 


x2 


6 


X ? 


6 


x 1 


6 


xll 


6 


X U 


G 


X 2 


7 


X 1 


7 


xli 


7 


X 2 



Standard 
test piece. 



50,250 
49,8>0 
49,380 

48,500 
50, GOO 
50,090 
49,670 

,50,510 
49,91)0 
49,:i80 

49,790 
49,200 
48,070 

49,720 
49,090 
48,500 
47,9G0 
47,010 

49,670 
49,000 
48,390 
47.800 
46,750 

48,940 
47,680 
46,560 



Eyes of full 

size eye 

bars. 



40,150 
49,195 

48.G:J0 

47,500 
49,600 
49,090 
48,670 

49,010 
48,400 
47,880 

47,790 

-47,200 
46,070 

47,220 
46,590 
46.00* » 
45,460 
44,510 

46.070 
40,000 
45,390 
44,800 
43,750 

45.440 
44,180 
48,060 



Size of angle. 



6 X 6 X i 
6 X 6 X J 



4 X 
4 X 



x^ 



2 



f 



3.x 3 X i 
3 X 8 X li 



2 X 2 X i 



ii 



Standard 
test piece. 



48,320 
47.165 



48,750 
47,620 
49,160 

47,870 



49,180 
48,810 



BESISTANCE TO TENSION. 



TABIjB UL 

Strength of Iron Bode.' 

Bxra Tehiilb STBBHaTsn or Round WBODsOT-lHoir Roia i to 4 Ik< 

IK DllKETBB, AND TH* WkioHTS PBn FOOT, TBI SaPK BrBBISTH B 

lAEEH AT 10.000 PoDNDe PIR SqUABE IhCE. 



Tensile Strength and Quality of Wrou^ht-Imm. 

The best American rolled iron has a. breaking tenatle itTength of 
from fifty thousand to sixty thousand pounds per sqaitre Inch tar 
epecimens not exceeding one square inch in section. Ordlnar7bM>- 
iron should not brealt under a less strain than fifty thouNnd 
pounds per square inch, and sliould not take a set under a Knu 
less than twenty-five thousand pounds per square inch. A bar one 
inch square and one foot long should stretch fifteen per cent of Ui 
length before breaking, and should be capable of being bent, coH, 
00° over the edge of an anvil without sign of fracture, ud should 
show a fibrous lestnre when broken. 

Iron IliJit will not meet these re<[airements fs not suitable for 
structures; Imt notliinii is gained by speclfyii^ more severe tMts, 
because, in bars of the sizes and shapes usually required for tneb 
work, nothinp mon? can be atlaineil with certainty, and coniden- 
Eiuus milkers will be unwilling to agree to furnish that which ll la 
not practicable to produce. 

The aorking-iarerirjtb of wrought^iron ties hi trUM 



RESISTANCE TO TENSION. 219 

taken at ten thousand pounds per square inch. In places where 
the load is perfectly steady and constant, twelve thousand pounds 
mav be used. 

The extension ofir&n, for all practical purposes, is as follows : — 

Wrought-iron, ru^no of its length per ton per square inch. 
Cast-iron, ^,,^01) of its length per ton per square inch. 

Appearance of the Fractured Surface of Wrouglit- 

Iron. 

At one time it was thought that a fibrous fracture was a sign of 
good tough wrought-iron, and that a crystalline fracture showed 
that the iron was bad, hard, and brittle. Mr. Kirkaldy's experi- 
ments, however, show conclusively, that, whenever wrought-iron 
breaks suddenly, it invariably presents a crystalline appearance; 
and, when it breaks gradually, it invariably presents a fibrous ap- 
pearance. From the same experiments it was also shown, that the 
appearance of the fractured surface of wrought-iron is, to a certain 
extent, an indication of its quality, provided it is known liow the 
stress was applied which produced I he fracture. 

Small, uniform crystals, of a uniform size and color, or fine, 
close, silky fibres, indicate a good iron. 

Coarse crystals, blotches of color caused by impurities, loose and 
open fibres, are signs of bad iron; and flaws in the fractured surface 
indicate that the piling and welding processes have been imper- 
fectly carried out. 

Kirkaldy^s Conclusious.^ 

Mr. David Kirkaldy of England, who made some of the most 
valuable experiments on record, on the strength of wrought-iron, 
came to some conclusions, many of which differed from what had 
previously been supposed to be true. 

The following are of special importance to the student of build- 
ing construction, and should be carefully studied : — 

" The breaking-strain does not indi(uite the quality, as hitlK'ito 

assumed. 

** A hUjh breaking-strain may be due to the iron being of superior 
quality, dense, fine, and moderately soft, or simply to its being 
very hard and unyielding. 

** A ioKJ breaking-strain may be due to looseness and coarsenc^ss in 
the texture; or to extreme softness, although very close and fine 
in quality. 

1 Kirkaldy *B Ezperiraents on Wrought-iron iind Steel. 



220 RESISTANCE TO TENSION. 

** The contraction of area at fracture, previously oyerlooked, fo 
an essential element in estimating the quality of specimens. 

** The respective merits of various specimens can be correctly as 
tained by comparing the breaking-strain y(9t/i% with the contraci 
of area. 

" Inferior qualities show a much greater variation in the breakj 
strain than superior. 

^* Greater differences exist between small and lai*ge bars inooi 
than in fine varieties. 

''The prevailing opinion of a rough bar being stronger thai 
turned one is erroneous. 

" Rolled bars are slightly hardened by being forged doii'n. 

'' The breaking-strain and contraction of area of iron plates 
greater in the direction in which they are rolled than in a tn 
verse direction. 

*^ Iron is less liable to snap, the more it is worked and rolled. 

'* The ratio of ultimate elongation may be greater in short tl 
in long bars, in some descriptions of iron; whilst in others then 
is not affected by difference in the length. 

'* Iron, like steel, is softened, and the breaking-strain reduced, 
being heated, and allowed to cool slowly. 

'* A great variation exists in the strength of iron bars which hi 
been cut and welded. Whilst some bear almost as much as 
uncut bar, the strength of others is reduced fully a third. 

" The welding of steel bars, owing to their being so easily bun 
by slightly overheating, is a difficult and uncertain operation. 

'^ Iron is injured by being brought to a white or welding heat 
not at the same time hanmiered or rolled. 

'^ The breaking-strain is considerably less when the strain is ai^ 
suddenly instead of gradually, though some have imagined that ' 
reverse is the case. 

'* The specific gravity is found generally to indicate pr^ty correc 
th<* quality of spiH'inieus. 

"' Till' doiisity of iron is decreased by the process of wire-draw 
and by the similar ])rocess of cold rolling,^ instead of increwted, 
previously imagined. 

*' The density of iron is decreased by being drawn out nude 
tensile strain, instead of increased, as believed by some. 

"" It must be abundantly evident, from the facts which have b 



* The couclusioii of Mr. Kirkaidy in renpect to cold rolllDg ia undoubtedly t 
when the rolling amonntrt to wirc-dniwini;: but, when tbe oomprenkm of 
Hurface by rolliiiK diminidheH the MH:tional area in greiUer proportion thtt 
cxtcndd the bar, the result, accordinfc to the experience of tho PlttsbnTj^ mi 
facturerH, ia a slight iucreaise in the density of the Iron. 



1 [STANCE TO TENSION. 221 

produced, that the breahing-strain, when taken alone, gives a false 
impression of, instead of indicating, the real quality of the iron, as 
the experiments which have been instituted reveal the somewhat 
tiarthng fact, that frequently the inferior kinds of iron actually 
yield & higher result than the superior. The reason of this diHer. 
enoe was shown to be due to the fact, that, whilst the one quality 
retained its original area only very slightly decreased by the 
strain, the other was reduced to less than one-half. Now, surely 
this variation, hitherto unaccountably completely overlookedj is of 
importance as indicating the relative hardness or softness of the 
material, and thus, it is submitted, forms an essential element in 
considering the safe load that can be practically applied in various 
structures. It must be borne in mind, that, although the softness of 
the material has the e£fect of lessening the amount of the breaking- 
strain, it has the very opposite effect as regards the workimj-Htrain. 
This holds good for two reasons: first, the softer the iron, the less 
liable it is to snap; and, second, fine or soft iron, being more uni- 
form in quality, can be more depended upon in practice. Hence 
the load which this description of iron can suspend with safety may 
approach much more nearly the limit of its breaking-strain than 
can be attempted with the liarder or coarser sorts, where a greater 
margin must necessarily be left. 

'* As a necessary corollary to what we have just endeavored to 
establish, the writer now submits, in addition, that the working- 
strain should be in proportion to the breaking-strain per square 
inch of fractured area, and not to the breaking-strain per square 
inch of original area, as heretofore. Some kinds of iron experi- 
mented on by the writer will sustain with safety more than double 
the load that others can cuspend, especially in circumstances where 
the load is unsteady, and the structure exposed to concussions, aa 
in a ship or railway bridge." 



Eye-Bars and Screw-Ends* 

Iron ties are generally of flat or round bars attached by eyes 
And pins, or by screw-ends. In either case, it is essential that the 
proportion of the eyes or screw-ends shall be such that the tie will 
not break at the end sooner than in the middle. In importaiit 
work, eyes are forged on the ends of flat or round bars, by hydraulic 
pressure, in suitably shaped dies; and, while the risk of a welded 
eye is thus avoided, a solid and well-formed eye is made from the 
iron of the bar itself. 

A similar process is adopted for enlarging the screw-ends of long 



222 RESISTANCE TO TENSION. 

rods ; so that, when the screw is cut, the diameter of the screw il 
the root of the thread is left a little larger than the body of the rod. 
Frequent trials with saeh rods has proven that they will pull apart 
in tension anywhere else but in the screw ; the threads remaining 
perfect, and the nut turning freely after having been subjected 
to such a severe test. By this means the net section required in 
tension is made available with the least excess of material, and no 
more dead weight is put upon the structure than is actually needed 
to carry the loads imposed. 

T/ie diameter of the eye in flat bars, having the same thiokneBB 
throughout, should be 0.8 the width of the bar. The width of the 
metal on each side of the eye should be \ the width of the bar, and 
in front of the eye should be equal to the width of the bar. Wlien 
it becomes necessary to use a larger pin than here described (as 
when a bar takes hold of the same pin with bars of larger size), the 
amount of metal around the eye should be still further increased. 
The weight of an eye-bar, proportioned as here described, will be 
about equal to that of a plain bar of a length equal to the distaDce 
from centre to centre of the pins, plus twice the diameter of the 
pin multiplied by the width of bar, both in inches. 

The thickness of flat hara should be at least one-fourth of the 
width in order to secure a good bearing surface on the pin, and the 
metal at the eyes should be as thick as the bars on which they are 
upset. 

Table IV. gives the proportion for eye bars, sleeve nuts, and 
clevises, as manufactured by ttie ^ew Jersey Steel & Iron Co. 

Table VI. gives the proportion for upset screw-ends for dif- 
ferent sizes of rods, as adopted by the keystone Bridge Com' 
pany. 

Cast-iron has only about cno-thirJ the tensile strength of 
wroujj:! It-iron ; and as it is liabk* to air-holes, internal strains from 
uiH'ipial contraction in cooling, and other concealed defects, redu- 
cing its effective area for tension, it should never be used where it 
is subject to any great tensile stress. 

Tables. 

The following tables give the strength of iron rods, bars, steel 
and iron wire roi)es, nianila ropes, and dimensions of upset screw* 
ends. 

The diameter in Table III. is the least diameter of the rod; and, 
if the screw is cut into the rod without enlarging the end, the 
effective diameter between the tlu^ads of the icrew dumld be 
ised in calculating the strength of the rod. ' 



BBS:8TANCK TO TEN8IOH. 
TABLE IV. 



Aa 


WE1.DLES3, DIE-FOEGSD EYE BARS, 


1* .SSKISSSSTSSSSSSTJSStESsS 


p 


is3=sai!==Sf=s2"S2"»'s""-*»— a- 


11 


ii.i,,.i,i,.„i,,,%„„i.f. 



I' The snulleM diameter iif i>[n given for each width <ir tiuris the xiandunl i 
11m larger fliea given are Ih« iBivwt that ai-c nJlowatile with each head. 

SThe thlckneaa of the ban ahonld not he more than ) nor lesa than t their wi 
l]n>-ban an hored J, Inch larger than the diameecr of the pin. Other eizes 
befamlMhed- 



224 RESISTANCE TO TENSIOK 

Table YIl. was compiled from data furnished by the John A. 
Roebling's Sons Company of New York. 

The ropes with nineteen wires to the strand are the most pliable, 
and are generally used for hoisting and running rope. The ropes 
with seven wires to the strand are stiffer, and are better adapted 
for standing rope, guys, and rigging. 

Table IX. is taken from Trautwine's " Pocket-Book for Engi- 
neers.*' 

Table X. gives the weight and proof, or safe strength, of ofaains 
manufactured by the New Jersey Steel and Iron Compuiy. 



RESISTANCE TO TENSION. 

TABLE V, 
Safe Strength of Plat Rolled Iron Bar». 



e. per gquare toob. 



226 



RESISTANCE TO TENSION. 



TABLE V. (concluded). 
Safe Strength of Flat Rolled Iron Bars, 



s ^ 


Width iu iucbeB. 


Thicknei 
in incbef 




3J" 


3 J" 


4" 


^" 


H" 


^" 


5" 


H" 


6" 


6i" 




IbB. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


Ibe. 


A- 


2,190 


2,340 


2,500 


2,660 


2,810 


2,970 


3,130 


3,440 


3,750 


4,060 


i 


4,380 


4,690 


5,000 


5,310 


5,630 


5,940 


6,250 


6,880 


7,500 


8,130 


A 


6,560 


7,030 


7,500 


7,970 


8,440 


8,910 


9,380 


10,300 


11,300 


12,200 


i 


8,750 


9,380 


10,000 


10,600 


11,300 


11,900 


12,500 


13,800 


15,000 


16,300 


■h 


10,900 


11,700 


12,500 


13,300 


14,100 


14,800 


15,600 


17,200 


18,800 


20,300 


i 


13,100 


14,100 


15,000 


15,900 


16,900 


17,800 


18,800 


20,600 


22,500 


24,400 


iV 


15,300 


16,400 


17,500 


18,600 


19,700 


20,800 


21,900 


24,100 


26,300 


28,400 


i 


17,500 


18,800 


20,000 


21,300 


22,500 


23,800 


25,000 


27,500 


30,000 


32,500 


A 


19,700 


21,100 


22,500 


23,900 


25,300 


26,700 


28,100 


30,900 


33,800 


36,600 


f 


21,900 


23,400 


25,000 


26,600 


28,100 


29,700 


31,300 


34,400 


37,500 


40,600 


\i 


24,100 


25,800 


27,500 


29,200 


30,900 


32,700 


34,400 


37,800 


41,300 


44,700 


i 


26,300 


28,100 


30,000 


31,900 


33,800 


35,600 


37,500 


41,300 


45,000 


48,800 


+1 


28,400 


30,500 


32,500 


34,500 


36,600 


38,600 


40,600 


44,700 


48,800 


52,800 


1 

8 


30,600 


32,800 


35,000 


37,200 


39,400 


41,600 


43,800 


48,100 


52,500 


56,900 


+? 


32,800 


35,200 


37,500 


39,800 


42,200 


44,500 


46,900 


51,600 


56,300 


60,900 


1 


35,000 


37,500 


40,000 


42,500 


45,000 


47,500 


50,000 


55,000 


60,000 


65,000 


We 


37,200 


39,800 


42,*b00 


45,200 


47,800 


50,500 


53,100 


58,400 


63,800 


69,100 


n 


39,400 


42,200 


45,000 


47,800 


50,600 


53,400 


56,300 


61,900 


67,500 


73,100 


lA 


41,600 


44,500 


47,500 


50,500 


53,400 


56,400 


59,400 


65,300 


71,300 


77,200 


U 


43,800 


46,900 


50,000 


53,100 


56,300 


59,400 


62,500 


68,800 


75,000 


81,300 


n 


48,100 


51,600 


55,000 


58,400 


61,900 


65,300 


68,800 


75,600 


82,500 


89,400 


H 


52,500 


56,300 


60,000 


63,800 


67,500 


71,300 


75,000 


82,500 


90,000 


97,500 


is- 


56,900 


60,900 




65,000 


69,100 


73,100 


77,200 


81.300 


89,400 


97,500 


105,600 


1} 


61,300 


65,600 


70,000 


74,400 


78,800 


83,100 


87,500 


96,300 


ia->,ooo 


113,800 


15 


65,600 


70,300 


75,000 


79,700 


84,400 


89,100 


93,800 


103,100 


112,500 


121,900 


2 


70,000 


75,000 


80,000 


85,000 


90,000 


95,000 


100,000 


110,000 


120,000 


130,000 



RESISTANCE TO TENSION. 2 

TABLE Vi. 

Upset ScretB-End» fm- Round and Square Bars. 

StINDAHD PKOFORTIOm OP THE KETBTOKK BRIDGE COUPAKr. 



RESISTANCE TO TENSION. 



TABLE VI. (concluded). 
Upset Srrew-Enda. 



RE81STANCB TO TKN8I0M. 



TABLE Vn. 
Strength <tf Irott and Steel Wire Bopen, 

Mahutictdbed by thk Jobs A. Koeblikh'b Sons Co., New Tobk. 



In IIh. uf roio 






•i'-'liX. 



CastSte 



230 RESISTANCE TO TENSION. 

Ropes, Hawsers, and Cables. 

(HASWKLL.) 

Ropes of hemp fibres are laid with three or four strands of 
twisted fibres, and run up to a circumference of twelve inches. 

Hawsers are laid with three strands of rope, or with four rope 
strands. 

C<(hles are laid with three strands of rope only. 

Tarred ropes, hawsers, etc., have twenty-five per cent less 
strength than white ropes: this is in consequence of the injury 
the fibres receive from the high temperature of the tar, — 290°. 

Tarred hemp and manila ropes are of about equal strength. 
Manila ropes have from twenty-five to thirty per cent less strength 
than white ropes. Hawsers and cables, from having a less pro- 
portionate number of fibres, and from the increased irregularity 
of the resistance of the fibres, have less strength than ropes; th^ 
diflference varying from thirty-five to forty-five per cent, being 
greatest with the least circumference. 

Ropes of four strands, up to eight inches, are fully sixteen i^er 
cent stronger than those having but three strands. 

Hawsers and cables of three strands, up to twelve inches, are 
fully ten per cent stronger than those having four strands. 

The absorption of tar in weight by the several ropes is as fol- 
lows : — 



Bolt-rope . . . .18 per cent 
Shrouding . . 15 to 18 per cent 



Cables 21 per cent 

Spun-yarn . . 25 to 30 per cent 



White ropes are more durable than tarred. 

The greater the degree of twisting given to the fibres of a rope, 
etc., the less its strength, as the exterior alone resists the greater 
portion of the strain. 

To compute the Strain that can be borne with 
Safety by New Ropes, Hawsers, and Cables, 
deduced from tlie Experiments of tlie Russian 
Government upon tlie Relative Strengtli of 
Different Circumferences of Ropes, Hawsers, 
etc. 

The United-States navy test is 4^00 pounds for a white rope, of 
three strands of best Ri(/a hemp, of one and three-fourths inches in 
cArcvmference (i.e., 17 ^000 pounds per sqxiare inch); but in thefol- 
lowing table 14^000 pounds is taken as the unit of strain that can 
be boime with safety. 

Rule. — Square the circumference of the rope, hawser, etc., and 
multiply it by the following units for ordinary ropes, etc 



EESI6TANCE TO TEN8I0W. 331 

TABLE VIIL 

Showing the Unltx for compiitiny the Safe Strain that may be 

home by Eo/ipk, Ilftienem, nnd Cablea. 



WTien it is required to uncertain the vjeiylit or strain that can 

be borne by ropes, etc., in yeneral use, the above units sliould be 

redut^ed one-third, in order to meet tlie reduction of tlieir atrength 

by chafing, and exposure to ilie weather. 

TABLE IX. 

Streniilb and irpi(/At 0/ Manila Hope. 



m 



RESISTANCE TO TENSION. 



TABLE X. 
Weight and Proof Strength of Chain. 
HE KewJebbet Steel ahd Iroh ( 



StrCDl^rth of Old Iron. — A square link 12 inches broad, 1 
incli (hick, and about 12 feet long was taken from the Kieff Bridge, 
then i ) years old. and tesl-od in comparison with a similar link 
which hiid been preserved in the slock-housc since the bridge was 
built. The following is a record iif a mean of four longitudinal 
test pieces, 1 >i IJ n 8 inehes, taken from each link. 





Old link 
from bridge. 


"•ss^ 




21.8 

n'.a 





















(TlH Hwhaoiul Worid, London.) 



JtSSlBTASCS TO SUEAKINO, 



CHAPTER X. 

RESISTANCX! TO SHBAKINO. 

Bt shearing is meant the pushing of one part of a piece by the 
Other. Thos in Fig. 1, let abed be a, beam resting upon the sup- 
ports 8S, which are very near logclher. If a sufflcientl; heavf 



load were placed upon tlie beam, it nould cause the beam to break, 
not by. bending, but by pushing the whole central part of the beam 
thrai^b between tlie ends, as represented in the figure. This mode 
of fracture is called " shearing." 

The resistance of a body to shearing is, like its resistance lo 
tension, directly proportional to tbe area to lie sheared. Hence, if 
we denote the resistance of one square inch of tlie material to 
shearing by F, we shall have as ihe safe resistance to shearing, 
Safe shearing > _ area to be sheared X 
strength fc S 

ft denoting factor of safety, as before. 

A piece of timber may be sheared either longitudinally or trans- 
versely; and, as the resistance is not the same in both cases, the 
value of F will be different In the two cases. Hence, in substi- 
tuting values for F, we must distinguish whether the force tends 
to shear the piece longituilinally (lengthwise), or Iransyersely 
(across). 

Table I. gives the values of F, as determined by experiment, tor 
) materials employed in architectural con uo- 



(1) 



JtEBlSTANCE TO SHEARING. 



Showini/ the Reninlnn'-.f of Materials to Shearing, hoUi Longtta- 
dlualljf and Traii^terxelf/, or the Values of f. 



MATsnr^tLs. 


VaiuMofr. 








It«. 

MO'l 

470 d 

640. 

732* 


lb*. 

K.7(»i> 

as: 

si: 

4!«)0c 

a,«uc 

6.700 « 
^000. 


























!J3;i:^°« 









Tliere are but few cases in ai-fliEtectural construction in vrbicb 
tbe resistance to siiearing tms to lie provided for. The one moat 
frequently met witii is at the end of a tie-beam, as in Pig. S. 



Fifl. 2. 

Tlie I'afier U e\pits a iluiisl ivliicli teiKls to push or shear off the 
pifice A HVD, ami tli« area of the section at CD slioiild offer enough 
resiatanci' to kei^p tliu rafter In place. This area is eqnal to CD 

• Ranklnt^. bKlrkaldy. c Tcuulwtm. >1 Hntfield. o Uu)Ied.SUt« iSomtB- 



RESISTANCE TO SHEARING. 235 

times the breadth of the tie-beam; and, as the breadth is fixed, we 
have to determine the length, CD. If we let // denote the hori- 
zontal thrust of the rafter, then, by ,a simple deduction from 
formula 1, we have the rule: — 

Length of CD in inches = b.^th o^beam x r <2) 

F, in this case, being the resistance to shearing longitudinally. 

Example I. — The horizontal thrust of a rafter is 20,000 pounds, 
the tie-beam is of Oregon pine, and is ten inches wide: how far 
should the beam extend beyond the point D f 

Ana. In this case H = 20,000 pounds, and from Table X. we find 
that jP = 840; aS we will take at 5. Then 

5 X 20000 
= 10 X 840 * ^^ nearly 12 inches. 

Practically a large part of the thrust is generally taken up by an 
iron bolt or strap passed through or over the foot of the rafter and 
tie-beam, as at A (Fig. 2). When this is done, the rod or strap 
should be as obliquely inclined to the beam as is possible; and, 
whenever it can be done, a sti-ap should be used in preference to 
a rod, as the rod cuts into the wood, and thus weakens it. 

The two principal cases in building construction where the 
shearing strength must be computed, are pins and rivets; for the 
latter see pages o57-565. 

Strength of Pins in Iron Bridge and Roof Trusses. 
— Iron and steel trusses are now so generally used that it is neces- 
sary for the architect who is at all advanced in his profession to 
know how to determine the strength of the joints, and especially of 
pin joints ; and to facilitate the calculation of the necessary size of 
pins, we give Table II , which shows the single shearing strength 
and bearing value of pins, and Table III., showing the maximum 
bending moment allowed in pins. 

Pins must be calculated for shearing, bending, and bearing 
strains, but one ol" the latter two only (in almost every case) deter- 
mines the size to be used. 

By bearing s( rain is meant the force required to crush the edges of 
the iron plales against, which the pin bears. 

The several strains usually allowed per square inch on pin con- 
nections in bridges are : shearing, 7,500 pounds; crushing, 12,000 
pounds ; and bending, 15,000 pounds for iron, and 20,000 pounds 
for steel. 

The shearing strain is measured on the area of cross-section ; the 



236 



STRENGTH OF PINS. 



crushing strain, on the area measured by the product of the diame- 
ter of the pin, by the thickness of the plate or web on which it bears. 

The bending moment is determined by the same rules as given 
for determining the bendiug moment of beams. 

When gi'oupsof bars are connected to the same pin, as in the 
lower chords of trusses, the sizes of bars must be so chosen, and the 
bars so placed, that at no point on the pin will there be an exces- 
sive bending strain, on the presumption that all the bars are 
strained equally per square inch. 

The following example will show the method of determining the 
size of pin in a simple joint. 

Example.— Fig. 3. Determine the size of pin for the joint in 
the lower chord of a truss, shown in Fig. 3, the middle bar being a 
vertical suspension rod, merely to hold the chord in place. 



40,000 



IX 4' 



I -. ^ IX 4'^40,000 



* IX 4'-40.000 ^ 



40,000 



1X4' 



4- 



i 



Fig. 8. 



Ans, The shearing and crushing strain in this case is 40,000 
pounds. The bending moment will be 40,0(iO x 1"; the distance 
between the centres of the two outer bars = 40,000 pounds. Prom 
Table III. , we find that to sustain a bending moment of 40,000 lbs., 
with a fibre strain of 15,000 lbs., will require a 3" or 3^" pin. 
From Table II., we find that the bearing value of a 3^" pin is but 
37,500 lbs., and that we must increase the size of the pin to 8f 
inches. The shearing strength of a 3|" pin is, from Table II., 
67,500 lbs., so that the size of pin we must use in this case is deter- 
mined by the bearing strain. To be sure of the correct size of the 
pin, one must make the calculation for all three of the strains. 






STBSNaTH OV FINB. 



237 



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238 



STRENGTH OF PINS. 



TABLE III. 

Maximum Bending Moments to he Allowed on Pinafor Maacimum 
Fibre Strains of 15,000, 20,000, and 2^,600 Pounds per tquare 
Inch." 



Diam- 
eter of 
pin. 


Moment 

for 
S = 15,000 


Moment 
i for 
,S'=20,000. 


Moment 
for 

^=22,500. 


Diam- 
eter of 
pin. 


Moment 

for 
^=15,000. 


Moment 

for 
^'=20,C00 


Moment 

for 
>S=22.500. 


Inches. 

1 

ii 


Lbs. in. 
1,470 
2,100 

2,aso 

8,830 


Lbs. in. 
1,960 
2,800 
3.830 
5,100 


Lbs. in. 
2,210 
3,140 
4,310 
5,740 


Inches. 
4 

4i 
4| 


Lbs. in. 

94,200 

103,400 

113,000 

123,300 


Lbs. in. 
125,700 
137.800 
150,700 
164,400 


L1)8. in. 
141,400 
155.000 
169,600 
185,000 


ii 


4,970 
6,320 
7,890 
9,710 


6,630 

8,430 

10,500 

12,900 


7,460 

9,480 

11,800 

14,600 


41 
4f 

4J 


134,200 
145,700 
157,800 
170,600 


178,900 
194,300 
210,400 
227,500 


201,800 
218,500 
286,700 
256,900 


2 
2| 


11,800 
14,100 
16,800 
19,700 


15,700 
18,800 
22,400 
26,300 


17,700 
21,200 
25,200 
29,600 


5 

5i 
5| 


184,1U0 
198,200 
213,100 
228,700 


245,400 
264,800 
284,100 
804,900 


276,100 
297,800 
819,600 
848,000 


2^ 


23,000 
26,600 
30,600 
35,000 


30,700 
35,500 
40,800 
46,700 


34,500 
40,0 
45,900 
52,500 


5J 

51 

5J 


246,000 
262,100 
280,000 
298,600 


826,700 
849,500 
873,800 
898,200 


867,600 
898,100 
410,900 
447,900 


8 
8^ 


39.800 
44,900 
50,600 
5fi,600 


53,000 
59,900 
67,400 
75,500 


59,600 
67,400 
75,800 
84,C00 


6 

61 


818,100 
888,400 
359,500 
881,500 


424,100 
451,200 
479,400 
506,700 


477,100 
507,600 
589,300 
S72300 


31 
31^ 
8J 
31 


63,100 
70,100 
77,700 
85,7C0 


84,200 

93,500 

103,500 

114,200 


94,700 
105,200 
116,500 
128,500 


64 
6f 


404,400 
428,200 
452,900 
478,500 


589,200 
570,900 
608,900 
688,000 


606,600 
642,800 
879,400 
717300 



Remarks — The following is the formula for flexure applied to pins : 



M= 



Sir d» 



or = 



S Ad 



32 ""' ~ 8 

M=moment of forces for any section through pin. 
S=strain per sq. in. in extreme fibres of pin at that section. 
A = area of section. 
d= diameter. 
»r=3.14159. 
The forces are assumed to act in a plane passing through the axis of the pin. 
Tiie above table gives the values cf M for different diameters of pin, and 
for three values or S. 

If ?.I max. is known, an inspection of the table will therefore ehow wliat 
diameter of pin must be used in order that S may not exceed 16,000, 20,000, or 
22,500 lbs., as the requirements of the case may be. 

For Railroad Bridges proportioned to a factor of safety of 6, it is castom- 
ary to make 8 max. = 15,000 lbs. in iron aid =: 20,000 lbs. in steel. 



* Carnegie, Phlpps & Co. 'a Hand-book. 



STRENGTH OF PINa 239 

Bending Moment in Pins. 

The only difficult part of the process of calculating the sizes of 
pins will generally be found in determining the bending moment. 
In cases where the strains all act in the same plane, the bending 
moment can generally be determined by multiplying the outside 
force by the distance from its centre to the centre of the next bar, 
as in the foregoing example. When, however, the forces act in 
several planes, as is generally the case, the process of determining 
the bending moment is more difficult, and can be best determined 
by a graphic process, first published by Prof. Chase Green, and in- 
cluded in his lectures to the students in engineering at the Univer- 
sity of Michigan. 

As the pieces acting on any well-designed joint are symmetrically 
arranged, it is unnecessary to consider more than one-half of their 
number. Fig. 4 shows a sketch of one-half the members of a joint 
in the lower chord of a Howe truss. The pieces are parallel to the 
plane of the paper, and the pin is perpendicular to the same, but 
drawn in cabinet perspective, at an angle of 45° with a horizontal. 

The bars are assumed to be each one inch thick, and the channel 
to have one-half -inch web. The centre of the hanger is }" from the 
centre of the channel. 

The method of obtaining the bending moment is as follows : 

Draw the line A B at an angle of 45° with a horizontal, and, com- 
mencing with c, lay off the distances between the centres of the bars 
to a scale (1^" or 3" to the foot will be found most convenient) ; 
then draw the lines 1-3, 2-3, etc. , parallel to the pieces which they 
represent in the trass, to a scale of pounds. Resolve the oblique 
forces into their horizontal and vertical components (in this exam- 
ple there is but one oblique force). 

Next draw the stress diagram (Fig. 6) as follows : On a horizon- 
tal line lay off 1-2 equal to the first or outer force ; 2-3, equal to 
the next, 3-4 ; and 4-1, being the horizontal component of the 
brace, closes the figure. In the same way, lay off the vertical 
forces 15, 5 6, 61. If the forces are correct, the sum of the 
forces acting in one direction will always equal those acting in 
the opposite direction. From 1 draw the line 1 at 45", equal to 
the same scale of, say, 20,000 pounds, or any other convenient 
length. Draw 2, 3, 4, etc. Then, in Fig. 5, starting at the 
first horizontal force, draw c d parallel to 2, 6^ e parallel to 3, 
«/ parallel to 4, and/^ parallel to 1. 

In the same way, starting at the first vertical force, draw r 8 par- 
allel to 5> s ^ parallel to 6, and t 2 parallel to ' ' '^ 



240 



STREN(iTH OF PINS. 



line c d e fk will represent the boundary of the horizontal ordi. 
nates, and /• ,9 1 'O the boundary of the vertical ordinate?. And to 
find the resultant of these ordinates at any point on the pin, it is 



o ^ 





t 




only nooci^sary to draw tlic diagonal from the ends of the ordinates 
ut that ))<)ii)t. Thr.s. thi> resultant at X^ Fig. ft, will be i»-ii, uid 
it is evidtnt that this is the longest hypothenuM whk^ onn be 



BTRENGTH OF PINS. 



241 



dxawn ; and this hypotheause, multiplied by 0-1 (20,000 pounds), 
gives 62,600 pounds as the maximum bending moment on the pin. 

To obtain the maximum bending moment,, it is necessary to take 
the longest hypothenuse that can be drawn, no matter at what 
place it occurs. 

If one desires to try the effect of changing the order of the bars 
on the pin, it can readily be done. Suppose the diagonal tie to 
change places with the next chord bar. The horizontal stress dia- 
gram then becomes 1-2, 2-si, 3-4', 4-1. The equilibrium polygons 




A Fig. 11. 

will now be (Pig. l)cdef' k' and r' s' f w, and the longest hypoth- 
enuse, w a*, or 3J", which makes the bending moment 75,000 
pounds, showing that the arrangement in Pig. 4 is the best. 

As a rule, in arranging the bars on a pin, those forces which 
counteract each other should be close (ogcthor. 

To further illustrate this method of dotcrniininp: the bending 
moment on pins, we will determine the bending moment for the 
pin at the joint A, Pig. 8. This is the some truss as worked out on 
page 686, the strains given in Pig. 8 being ^ of the strains at the 
joint, as all the pieces are doubled. Pig. 9 shows the size and 
•RBOgMDei of the ties and strat. It is assumed that the web of 



'242 STREN(4Tn OF PINS. 

the channel is reenforced to make it §" thick. Drawing the line 
AB, Fig 11, we lay off the outer force at a; then measaring off an 
incli. the distance between centres of the two outer bars, we lay off 
the next force {)arallel to the direction in which it acts ; and in the 
oame way, the other two forces. The three inclined forces must be 
resolved into their horizontal and vertical components. We next 
draw tlio stress diagram (Fig. 10) to the same scale of i)ounds, mak- 
ing 1 e(iual 20,000 pounds. The lines 4 and 6 ha[)pen, in this 
case, to coincide. Then, in Fig. 11, we draw a d parallel to 2, '> f 
parallel to 8, c d — 4, and d e parallel to 1. In the same way, 
we obtain i\w line hjk B. In this case, it will be seen tlial the 
longest horizontal ordinate is h by while at that point there is no 
vertical ordinate^ ; also, that no hypothenuse can be dra^Ti which 
will he as long as h b, so that we must take A 6 as the greati»st re- 
sultant : and this, multiplied by 20,000 pounds, gives 31,800 {xmnds 
as the inaxirnuni bending moment cm the pin. It will be seen that 
this is just the prmluct of the outer force by its arm to the centre of 
the next bar, so that the greatest bending moment is at that point. 
To determino the sizeof the pin, we find, from Table III., that for 
a steel j;iii to sustain this moment, allowing a flbro strain of 20,<MM) 
pounds, wc shall need a 25" pin. This pin has a bearing value 
of JU,5()) i)()un(ls for a bar an inch thick. The outer bar in this 
case is J thick, and has a strain of J31.800 [>ounds, equivalent to 
42,4'K) pounds for a 1 bar. And we see, from Table II., that we 
shall need to u.'^e a lU' pin to meet this strain. The shearing 
streij;^^th of a ii\" pin is 36 tons, or nn)re than double the strain. 
Hence we must use a lU" pin. or. by increasing the thickness of the 
bars, we might reduce the pin to 3 inches. 



BEARIKQ-PLATES FOB GIBBEBS AKD COLUMKS. 



PROPORTIONS OF OAST-IRON BBARINGkPLAT 
FOR aiRDERS AND COLUMNS (1896). 

If a heavily loaded column or girder should rest directly up 
wall or pier of masonry, the weight would be distributed over 
a small area that in most cases there would be danger of cms 
the masonry, particularly if it were of brick or rubble work. 




Section 



• / ^ * \^ 



Pi an 
Fig I 



<£ 



n — n 




FiqZ 



^P 






Fig 3 



prevent this, it is customary to put a bearing-plate between 
end of the beam or column and the masonry, the size of the j 
being such that the load from the column or girder divided bj 
area of the piate shall not exceed the safe crushing- strength o 
masonry per unit of measurement. 

The load per square inch on different kinds of masonry 
not exceed the following limits : 



242^ BEARIKG-PLAT£8 FOB GIBDfiBS AND COL17MK8. 

For granite 1,000 lbs. per sq. in. 

•• best grades of sandstone 700 ** " " ** 

** soft sandstone 400 " *** '* ** 

'* extra hard brickwork in cement mor- 
tar 150 to 170" " " " 

** good hard brickwork in lime mortar. . . 120 " ** ** ** 

** good Portland cement concrete 150 ** " ** " 

'* sand or gravel 60 *« « " « 

Example 1. — The basement columns of a six-story warehouse 
support a possible load of 212,000 pounds each ; under the oolumn 
is a base-plate of cast-iron, resting on a bed of Portland cement 
concrete two feet thick : What should be the dimensions of the 
base-plate ? 

Answer. — ^As the plate rests on concrete, the bottom of the 
plate should have an area equal to 212,000 -h150= 1,413 square 
inches, or 37 inches square. The column should be about 10 
inches in diameter and 1 inch thick. The shape of the base-plate 
should be as shown in Fig. 1. 

The height K should be equal to the projection P, and D should 
be equal to the diameter of the column. The thickness of all p<w- 
tions of the plate should be equal to that of the column above the 
base. This is not so much required for strength as to get a perfect 
casting, as such castings are liable to crack by unequal cooling 
when the parts are of different thicknesses. The projection of the 
flange G should be three inches, to permit of bolting the plate to 
the bottom of the column. It will be seen that in such a plate no 
transverse strain is developed in any portion of it. 

THICKNESS OF FLAT BASEPLATES. 

For small columns and wooden posts with light loads, plain flat 
iron plates are generally used. They may have a raised ring to fit 
inside the base of an iron column, or for a wooden post, a raised 
dowel, 1^ inches or 2 inches in diameter. If the plate is very 
thick, a saving in the weight of the plate may be made by bevel- 
ling the edge, as shown in Fig. 2, without loss of strength. The 
outer edge, however, should not be less than one inch thick. 

When such a plate is used, it is evident that if the plate is to 
distribute the load equally over its entire area, it must have suffi- 
cient transverse strength to resist bending or breaking, and this 
strength will depend upon the thickness of the plate. It is diffi- 
cult to make an exact formula for the thickness of such platu^ 



IKABiyG-PLkTES 50^ ^1212X5 JlTl/ COLrHTs. 34ic 



bat the writer ]u« -is-ricr-i -JiaE: *>_*: 
will be *lway? :c li* <^*-^ iLie 
strength: 

•r 



'-M«— ' 






ThickZKS Lt Z'J^Z>t -Z. Jl< Jlfr* I 



' -^f 



in which r = TZc . *i c v.- : ^-.- ::~:j*:d '." *> u-»sk r. v/;az^ 
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poet or <s:lX=_i- If ▼- ^z*: ' *-_- ^-*r '.■- .i»- s-* —-»■'> -i^u^^e 
we hare r = 1 a "«f:~-'= tz..; .r = l.i* jiit/^a "^/uvr *.ijv*.j=Ai 



i>i 



= -sr T^-J3afi. 






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= ■_ •■ t*. — 4*- z. ;.';». 



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242/ BEABIKG-PLATES FOB GIBDEB8 AliTD GOLUHKS. 

multiplied by 7,000, gives 42,000 pounds as the safe stiength of 
one bracket. 

The resistance to crushing may be found by multiplying the 
distance X by the thickness of the bracket and the product by 
13,000. Thus, if X is four inches and the thickness one inch, the 
resistance to crushing would be 52,000 pounds. Such a bracket 
would support the end of a 20- inch light steel beam of 16 feet 
span under its full load ; for heavier beams, the thickness of the 
bracket and also the length D should be increased. 



■v^ 



STHEMGTU ( POSTS, STRUTS, AND C0LUMN3. 243 



CHAPTER XL 



STRBirGTH OF POSTS, STRUTS, AND COLUMNS. 



As the strength of a post, strut, or column, depends primarily 
upon the resistance of the given material to crushing, we must 
first determine the ultimate crushing-strength of all materials used 
for this purpose. 

The following table gives the strength for all materials used in 
building, excepting brick, stone, and masonry, which will be found 
in Chap. VI. 

TABLE I. 

Average Ultimate Crushing-Loads, in Pounds per Square Inch, 

for Building-Materials. 



' 


. Crashing 




Crushing 


Material. 


weight, in lbs. 


Material. 


weight, in lbs. 




per sq. inch. 




per sq. inch. 




C. 




C. 


For Stone, Brick, 




Woods (continued). 




and Masonry, see 




Beech 


9,300 » 


Chap. VI. 




Birch .... 
Cedar . . . 






11,600 a 
6,500 a 


Metals. 




Hemlock . . . 






5,400 b 


Cast-iron .... 


80,000 


Locust . . . 






11,720 b 


Wrought-iron . . . 


36,000 


Black walnut 






5,690 


Steel (cast) .... 


225,000 a 


White oak . . 
Yellow pine . 






3,150 to 7,000 
4,400 to 6,000 


Woods. 
Ash 


8,600 a 


White pine . . 
Spruce . . , 






2,800 to 4,500 



The values given for wrought and cast iron are those generally 
Tised, although a great deal of iron is stronger than this. The 
values for white oak, yellow pine, and spi*uce, are derived from 
experiments on full-size posts, made with the government testing- 
machine at Watertown, Mass. ; the smaller value representing the 
strength of such timber as is usually found in the market, and 
the larger value, the strength of thoroughly seasoned straight- 
grained timber. For these woods a smaller factor of safety may be 



a Trautwine. 



b Hatfield. 



ii44 STRENGTH OF WOODEN POSTS AND COLUMNS. 

used than for the others, tlie strength of wlilch was derived from 

experiments on small pieces. 

The values for wood are for dry timher. Wet timlx»r is only 
about one-half as strong to resist compression as dry tindx*r, and 
this fact shouhl be taken into account when using gr«'en timlHT. 

TJk- sfrcntfth of <i ro/«////i, jwat, or Mrut depends, in a large 
nu'asun', uiK)n the pr(>j)ortion of the length to the diameter or 
least thickness. Up to a certain length, they bre^k simply by 
comi)ressi()n, and above that they break by first l>ending sideways, 
and then breaking. 

Wo<Mlen Columns. 

For wooden colunms, where the lengtli is not more than twelve 
times the least thickn(*6s, the strength of the column or strut 
may be computed by the nde, 

area of cross-section x C 
Safe load- - -factor of safety ~ . <1' 

where C* denotes the strength of tlie given material as given in 
Table I. 

The factor of safety to l)e used dei>end8 ujwn the plaoc where 
the cohiinn or strut is used, the load which comes ujion it, the 
<iuality of the material, and, in a large measure, ut>on the value 
takt'H for (\ 

Tims foi- white oak, yellow pine, and spruce, the value C is the 
actual cru.sliiiig-strength of full-size i)OSts of ordinar>' quality: 
hence wc need not allow a factor of safely for these greater tlian 
four. For the other wootls, we shouhl us«* a factor of safety of at 
least six. 

//■ //// ItKhJ ujto)! the rolfunu or iM)st is su«'h as conies upon the 
lloor of a iua<'hine-shoi», or where heavy machinei'y Is us«m1, or if 
the strut is for a railway-bridge, a larger factor of safety sliould 
be used in :ill ciises. 

If tin (judlitf/ of t/ir thntur is <>xce]>tionally goo4l, we may ust* the 
Imui'i- v;iln«'< f(»r the constant (\ in tb** cjise of (he last four WikhIs 
i:i\< n in <iic tabl(>. For (»rdinary bard pine or oiik imwIs, uudtiply 
lilt' ;iri;i n\ cross-section in inches by HMM); for >pru(v. by SiM», und 
t«»r wliite pine, by 7"*) pounds. 

V.\ \Mi'M. 1. —What is the siife load for a hanl-pine pust 10 by 
b) in. h. s, IJ ft'ct long? 

Ans. Ana of cross-section = 10 X 10 = 100 square Incliet; 100 X 

KNNI - lOO.IMM) i)ound». 



STRENGTH OF WOODEN POSTS AND OOLITMNS. 245 

ExAMPm II. — What is the safe load for a spruce strut 8 feet 
long. G" X 8" ? 
An8. Area of cross-section = 48 ; 48 x 800 = 38,400 pounds. 



Stren^h of Wooden Posts over Twelve Diameters 

in Length. 

When the length of a post exceeds twelve times its least thick- 
ness or diameter, the post is liable to bend under the load, and 
hence to break under a less load than would a shorter column of 
the same cross- section. 

To deduce a formula which would make the proper allow- 
ance for the length of a column has been the aim of many 
engineers, but their formulse have not been verified by actual 
results. 

Until within two or three years the formulse of Mr. Lewis 
Gordon and Mr. C. Shaler Smith have been generally used by 
engineers, but the extensive series of tests made on the Gov- 
ernment testing machine at Watertown, Mass., on full-size col- 
umns, show that these formulae do not agree with the results there 
obtained. 

Mr. James H. Stanwood, Instructor in Civil Engineering, Mass. 
Institute Technology, in the year 1891 platted the values of all 
the tests made at the Watertown Arsenal up to that time on full- 
size posts From the drawing thus obtained he deduced the fol- 
lowing formula for yellow pine posts : 

Safe load pec square inch = 1,000 — 10 x . ' ^■,^. . — r-^ 
^ ^ breadth m ins. 

The author has carefully compared this formula with the results 
of actual tests, and with other formulae, and believes that it meets 
the actual conditions more nearly than any other formula, and he 
has therefore discarded the tables of wooden posts given in the 
previous editions of this work and prepared the following tables 
for the strength of round and square posts of sizes coming within 
the range of actual practice 

For other sizes the loads can easily be computed by the 
formula. 

The loads for oak and white pine posts were computed b} the 
following formulse : 



346 3TBENGTH OF WOODBN POSTS AND COLD 

For oak and Norway pine : 



For white pine and spmce posts : 

Safe load per sqnare inch = 635-6 X !!°^^° '^ . 
"^ ^ breudth id ius. 

in which the breadth is the le ast sid e of a rectangular stnit, or the 
diameter of a round post. The round posts were compnted for 
the half-inch, to allow tor being turned out of a square post, of 
the size next larger. 

The formuUe were onl^ used for posts exceeding ISdiameters for 
yellow pine, and ten diameters for other woods. 

For posts having bad knots, or other defects, or which are known 
to be eecentrically loaded, a deduction of from 10 to 35 per cent 
should be made from tbe values given iu the tables. 

8APB LOAD IN POUNDS FOR YELLOW PINB POSTS <IU>ITND 
AND SQUARE). 



STRENGTH OF WOODEN POSTS AND COLCHNS. 347 



248 STRKNGTH OF WOODEN POSTS AND OOLUMNa 



eccentric Loardrng. 

When the load on a post is applied in such a way that it is not 
distributed uniformly over the end of the post, the loading is 
called eccentric and the effect on the post is much more injurious 
than if the load were uniformly distributed. When a post supports 
a girder on one side only, or when the weight from one girder is 
much more than from the other, the load becomes eccentric, and 
an allowance must be made in the safe load varying from 10 to 25 
per cent., according to the amount of eccentricity. 

The exact allowance cannot bj calculated, so that one must 
necessarily use his judgment in the matter, remembering that it 
is best to be on the safe side. 




Iron caps for timber pillars are often used in important con- 
structions, and are an excellent invention, as they serve to dis- 
tribute the thrust evenly through the pillar, and also form a 
bracket, which is often desirable, for supporting the ends of 
girders where a second post rests on top of the first. Fig. 1 shows 
the section of one of the simplest forms of caps. 

The Goetz and Duvinage caps, described at the end of Chapter 
XXIV., are the best shape for mill construction. 



STRENGTH OF CAST-IRON COLl \S, 249 

Cast-Irou Columns. 

For cast-iron columns, where the length is not more than six or 
eight times the diameter or breadth of colunm, the safe load may 
be obtained by simply multiplying the metal area of cross-section 
by ()'i tons, which will give tons for the answer. 

Above this proportion, that is, where the length is more than 
eight times the breadth or diameter, the following formulas should 
be used. These formulas are known as Gordon's and Rankine's. 

Formulas — 

For solid cylindrical -cast-iron columns, 

Metal area x 13330 

Safe load in lbs. = fi n — = — : — \ . (4) 

so. of length in inches ^ ' 

1+ ^ 




sq. of diam. in inches X 266 
For hollow cylindrical columns of cast-iron, 



O 





, . „ Metal area x 13330 

Safe load in lbs. = sg. of length in inches ' <^) 

400 X sq. of diam. in inches 

For hollow or solid rectangular pillars 
of cast-iron, 

Metal area X 13330 

Safe load m lbs. = fi n — : — : — i • (6) 

sq. of length m mches ' ' 

500 X sq. of least side in inches 

For cast-iron posts, the cross-section being a cross 
of equal arms, 

^ , , , . , Metal area X 1.3330 

Safe load m lbs. = sq. of length in inches ^^^ 

133 X sq. of total breadth in inches 

Example I. -What is the safe load for a hollow cylindrical 
cast-iron column, 10 feet long, 6 inches external diameter, and 1'' 
thickness of shell ? 

Ans. We must first find the metal area of the cross-section of 
the column, which we obtain by subtracting the area of a circle of 
four inches in diameter from the area of one six inches in diameter. 
The remainder will be the area of the metal. The area of a six- 
inch circle is 28.27 square inches, and of a four-inch, 12.56 square 
inches; and the metal area of the column is 15.71 square inches. 




250 



STRENGTH OF CAST IKON COLUMNS. 



Then, substituting known values in fonnnla. 5, we liave 

15.71 X 18830 
Safe load = .^^^^^^ = 104700 pounds. 

^"^40(rx"36 

There is no use in carrying tlie result farther than the nearest 
hundred pounds, because the accuracy of our formulas will not 
warrant it. 

Example II. — What is the safe load for a cast-iron column 12 
feet long, the cross-section being a cross with equal arms, one inch 
thick, the total breadth of two anns being 8" ? 

Ana, The area of cross-section would* be 8 + 7 = 15 square 
inches. Then, by formula 7, 

15 X 13330 
Safe load in lbs. = 20736 ~ 58300 pounds. 

^■^ 133 X 04 



Projectingr Caps. 

Hollow columns calculated by the foi-egoing formulas should not 
be cast with heavy projecting mouldings round the top or bottom, 






Fig. 2 




as in Fig. 2, at a and 6. It is obvious that these are weak, and 
would break off under a load much less than would be requhredto 



STRENGTH OF CAST-IRON COLUMNS. 251 

cnish the column. When such projecting ornaments are deemed 
necessary, they should be cast seimrately, and be attached to a pro- 
longation of the cohimn by iron pins or screws. Ordinarily it is 
better to adopt a more simple base and cap, which can be cast in 
one piece with tlie pillar, without weakening it, as in Fig. 3. 

In all the rules and formulas given for cast-iron colunms, it is 
supposed that the ends have bearings planed true, and at right 
angles to the axis of the column. 

When the columns are used in tiers, one above the other, the end 
connections of the columns should be made by projecting flanges, 
wide enough to received-inch bolts for bolting the columns together, 
as shown in Fig. 4, page 242^, and the entire ends and flanges 
should be turned true to the axis of the column. The end joints 
are generally placed just above the floor beams, for convenience in 
erecting the work. 

The basement columns should be bolted to cast-iron base plates, 
as shown in Fig. 1, page 242a. The author does not consider it 
advisable to use cast-iron columns with hinged ends, or in build- 
ings whose height exceeds twice their width. 

Tables of Cast-iron Columns. 

By an inspection of the foregohig fonnulas for cast-iron columns, 
it will be seen, that, all other conditions being the same, the strength 
per square inch of cross-section of any column varies only with 
the ratio of the length to the diameter or least thickness. Thus 
a column 15 feet long and 10 inches diameter would carry the same 
load per square inch as a similar column 9 feet long and 6 inches 
diameter, both having the ratio of length to diameter as 18 to 1. 

Owing to this fact, tables can be prepared giving the safe load 
per square inch for colunms having their ratio of length to diame- 
ter less than 40. 

On this principle Table IV. has been computed, giving the loads 
per square inch of cross-section for hollow cylindrical and rectangu- 
lar cast-iron colunms. 

To use this table, it is only necessary to divide the length of the 
column in inches by the least thickness or diameter, and opposite 
the number in column I. coming nearest to the quotient find the 
safe strength per square inch for the column. Multiply this load 
by the metal area in the cross-section of the column, and the result 
will be the safe load for the column. 

Example III. — Wliat is the safe load for a 10-inch cylindrical 
cast-iron column 15 feet long, the shell being 1 inch thick ? 

Ans. The length of the colunni divided by the diameter, botn 
in inches, is 18, and opposite 18 in Table lY. we find the safe load 



252 



STRENGTH .OF CAST-IRON COLUMlSrS. 



per square inch for a cylindrical column to be 7,360 pounds. The 
metal area of the column we find to be 28.27 inches ; and, multi- 
plying these two numbers together, we have for the safe load of the 
column 208,236 pounds, or about 104 tons. 

Besides this table, we have computed Table V. following, which 
gives at a glance the safe load for a cast-iron column coming within 
the limits of the table, and of a thickness thei*e shown. 

Thus, to find the safe load for the column given in the last 
example, we have only to look in the table for columns having a 
diameter of 10 inches and a thickness of shell of 1 inch, and oppo- 
site the length of the column we find the safe load to be 10^ tons, 
the same as found above. 

The safe load in both tables is one-^ixth of the breaking-load. 



TABLE IV. 

Strength of Hollow Cylindrical or Rectangular Cast-Iron Pillars, 

(Calculated bt Formulas 5 and 6.) 



Length 


Breaking-weight in pounds 


Safe load 


in i)ound8 


divided by 


per square inch. 


per square inch. | 


external 
breadth or 
















diameter. 


CyJindrical. 


Rectangular. 


Cylindrical. 


Rectangular. 


5 


75,294 


76,190 


12,549 


12,698 


6 


73,395 


74,627 


. 12,232 


12,438 


7 


71,269 


72,859 


11,878 


12,143 


8 


68,965 


70,922 


11,494 


11,820 


9 


66,528 


68,846 


11,088 


11,474 


10 


64,000 


66,666 


10,666 


11,111 


11 


61,420 


64,412 


10,236 


10,735 


12 


58,823 


62,111 


9,804 


10,352 


13 


56,239 


59,790 


9,373 


9,965 


14 


53,859 


57,471 


8,976 


9,578 


15 


51,200 


55,172 


8,533 


9,195 


16 


48,780 


52,910 


8,130 


8,817 


17 


46,444 


50,697 


7,741 


8,440 


18 


44,198 


48,543 


7,366 


8,090 


19 


42,050 


46,457 


7,008 


7,748 


20 


40,000 


44,444 


6,666 


7,407 


21 


38,050 


42,508 


6,341 


7,085 


22 


36,200 


40,650 


6,033 


6,776 


23 


34,455 


38,872 


5,742 


6,479 


24 


32,787 


37,174 


5,464 


6,195 


25 


31,219 


35,555 


6,203 


5,926 


26 


29,741 


34,014 


4,957 


5,660 


27 


28,343 


32,547 


4,724 


5,423 


28 


27,027 


31,152 


4,504 


5,192 


29 


25,785 


29,828 


4,297 


4,971 


30 


24,615 


25,571 


4,102 


4,761 


31 


23,512 


27,310 


3,918 


4,818 


32 


22,472 


26,246 


3,745 


4,374 


33 


21,491 


25,172 


3,581 


4405 


34 


20,565 


24,154 


3,427 


4,026 


35 


19,692 


23,188 


3,282 


8,814 



STRENGTH OF CAST-IRON COLUMNS. 



253 



TABLE V. 

Showing Scrfe Load in Tons for Cylindrical Cast-Iron Colvmns, 



Thickness of Shell | Inch. 

1 


Length 






Diameter of column (outside). 






of 
column. 












Gins. 


7 ins. 


Sins. 


9 ins. 


10 ins. 


11 ins. 


12 ins. 


13 ins. 


Feet. 


Tone. 


Tons. 


Tons. 


Tons. 


Tons. 


Tons. 


Tons. 


Tons. 


6 


60.6 


78.1 


94.0 


110.8 


128.6 


144.9 


161.7 


180.0 


7 


55.7 


72.2 


88.9 


106.9 


124.2 


140.1 


166.4 


176.0 


8 


60.7 


66.3 


83.8 


101.1 


117.7 


136.2 


151.1 


170.3 


9 


45.8 


61.9 


78.7 


95.2 


113.4 


130.4 


146.8 


164.5 


10 


40.8 


56.0 


73.5 


89.4 


106.8 


123.2 


140.5 


168.7 


11 


37.1 


51.5 


68.4 


83.6 


100.1 


118.3 


135.2 


153.0 


12 


33.4 


47.1 


63.3 


79.7 


95.9 


113.5 


129.9 


147.2 


13 


30.9 


44.2 


58.1 


73.9 


89.4 


106.3 


124.6 


141.4 


14 


27.2 


39.8 


54.7 


70.0 


86.0 


101.4 


119.2 


135.6 


15 


24.7 


36.8 


49.6 


64.1 


78.5 


96.6 


114.0 


129.9 


16 


22.3 


33.9 


46.2 


60.3 


71.9 


91.8 


108.7 


124.1 


18 


- 


29.0 


41.0 


52.5 


67.6 


84.6 


103.4 


118.3 


20 


— 


24.4 


36.0 


44.7 


63.3 


77.2 


98.1 


112.6 




Metal area of croes-eection. 


sq. ins. 


sq. ins. 
14.73 


sq. ins. 


sq. ins. 


sq. ins. 


sq. ins. 


sq. ins. 
26.51 


sq. ins. 




12.37 


17.10 


19.44 


21.80 


24.16 


28.86 


Thickness of Shell 1 Inch. 


Length 






Diameter of column (outside). 






of 
column. 






• 






6inB. 


7 ins. 


8 ins. 


9 ins. 


10 ins. 


11 ins. 


12 ins. 


13 ins. 

! 


Feet. 


Tons. 


Tons. 


Tons. 


Tons. 


Tons. 


Tons. 


Tons. 


Tons. 


6 


77 


100 


121 


143 


167 


188 


211 


234 


7 


71 


92 


118 


138 


161 


182 


204 


230 


8 


64 


85 


108 


131 


153 


176 


197 


222 


9 


58 


79 


101 


123 


147 


170 ' 


190 


215 


10 


52 


72 


95 


116 


138 


161 


183 


207 


11 


47 


66 


88 


108 


130 


154 


175 


200 


12 


42 


60 


81 


102 


124 


147 


169 


192 


13 


39 


57 


75 


95 


116 


138 


162 


184 


14 


35 


52 


69 


90 110 


1.32 


155 


177 


15 


31 


47 


64 


83 104 


126 


148 


170 


16 


28 


43 


69 


78 j 96 


119 


142 


162 


18 


25 


39 


53 


68 ; 88 


105 


128 


151 


20 


22 


35 


46 

1 


6S 1 79 


94 


114 


136 




Metal area of cross-section. 


sq. ins. 


sq. ins. 


sq. ins. 


sq. ins. 


sq. ins. 


sq. ins. 


sq. ins. 


sq. ins. 




15.71 


18.82 


22.00 


25.14 


28.27 


31.41 


34.66 


37.70 



255a STBEl^GTH OF CAST-IRON COLUMNS. 

The principal disadvantage, as found in practice, is the difficulty, 
if not impossibility, of making rigid connections with the beams 
and girders. In buildings of not more than five or six stories, 
however, this is not of great importance. 

(Jast-iron is, of course, subject to flaws, and the columns are 
liable to be cast of uneven thickness of metal, but by careful inspec- 
tion these defects can be discovered, and any columns containing 
them rejected. 

For unprotected columns, c'ast-iron is unquestionably better than 
steel, as has been quite conclusively demonstrated by the experi- 
ments of Prof. Bauschinger, of Munich. Cast-iron, three quarters 
of an inch or more in thickness, is also practically uninjured by 
rust, while it is clnime<l that wrought-irOn or steel may be almost 
destroyed by it. 

Although cast iron columns may be made in a great variety of 
shapes, the hollow cylindrical and rectangular columns have thus 
far been the principal shapes used, and for interior unprotected 
columns the cylindrical column probably meets the usual require- 
ments better than any others. Every year, however, the require- 
ments of building regulations are being made more strict, so that 
at the present time it is required in most of our large cities that 
all vertical supports in buildings over five stories in height shall 
be protected by fireproof material, and for such buildings the 
author would call attention to the H-shaped column, as offering 
the following advantages : 

1. Being entirely open, with both the interior and exterior sur- 
faces exposed, any inequalities in thickness can be readily discov- 
ered, and the thickness itself easily measured, thus obviating any 
necessity for boring, and rendering the inspection of the columns 
much less tedious. 

2. The entire surface of the column can be protected by paint. 

3. When built in brick walls, the masonry fills all voids, so that 

no open space is left, and if the column is 
placed as shown in Fig. 4, only the edge of 
the column comes near the face of the wall. 

4. Lugs and brackets can be cast on such 
columns better than on circular columns, 

■pjQ 4 especially for wide and heavy girders. 

5. The end connections of the columns dp 
not require projecting rings, or flanges, which are often objection- 
able in circular columns. 

The cost of columns of this shape should not exceed that of cir- 
cular columns of the same strength. 




STBBKOTH 07 OABT-IBON COLDKKS. 2SBb 

As to the strength of such columns, the onl; experimental data 
which we have on the subject is that obtAined from the experiments 
of Mr. Eaton HodghinsoD, which give them about theaame strength 
as cyliudrieal columns of the same diameter, when the length does 
not exceed thirty diameters and the thicliness ia not less than three- 
quarters of an inch. When surrounded bj- masonry they would 
probably be stronger than the cylindrical column. 

The column may be flreprocfed in the 
same way as the Z-bar column, which 
it much resembles. The space occupied 
by the column 8lig}itly exceeds that ot 
both the cylindrical and Z-bar column. 



Fio. 6. PiQ. B. 

hut not enough to be of any serious consequence. Figs. 5 and 6 
show details of end connections and brackets, and ot baseplate. 

The beams running at right angles to the web should be tied 
togeihcp by wrought-iron straps passing through holes in the web 
of the column. 

The following table has been calculated with the same stress per 
square inch of metal as allowed for the columns in Table V. 



STBBNGTH OF CA8T-1B0N OOLUHNS. 



TABLE V.a 
Siy% Loads in That of 3.000 Poand« fo 
B-*haped 


















U 


18 


20 




ISi 


a 


87 


48 


30 






















an 




48 


S4 


80 
49 
































sst 


S8 








58 


















SO 












a** 


97 


79 


77 




«4 










1«! 


n 








48 










2a 








ra 




















88 




72 








3J 


m 


88 








«t 








S»! 


las 








85 










3** 




124 


114 


107 




ea 








25 








90 




7S 


or 






8? 












92 


m 












138 


129 


120 




9t 










104 










TO 






811 


ja 




128 






96 








Slk 




160 


144 


136 


125 


111 


101 








lar 


138 








98 


u 






841 


1«8 




4I( 






lie 


101 






40| 








1S8 




IBS 








461 




204 


w 




170 




141 




30 




183 








108 


91 




m 










IM 


m 






JSi 








189 




1B0 


1S2 




491 


»T4 


li! 


207 
232 




^ 


s 


IS 








16G 




1S2 






125 


lis 




vsi 


205 


1% 


IK7 






IM 


188 


4Bi 


















Mi 








251 






209 


18> 








300 


286 




200 


280 


2ia 


4JI 


ess 


air 


%*■ 


198 


189 


180 


183 


147 


; m 








296 


3^ 




198 


1T4 


! 801 


830 


aos 


283 


271) 






tat 


«n 




m 


«M 








278 


«) 


u« 


\ ?5| 


399 




353 


337 


.121 


aot 


277 





STBBKGTH OF CAaX-IBO^ COUJMNB. 265d 



Hollow Rectangular Cast-iron Columns. 

The increasing use of hollow rectangular cast-iron columns in 
buildings, particularly when enclosed in brick walls, has led the 
author to compute Table V.6, which gives the safe loads for a large 
number of sizes and lengths, the application of the table being 
readily apparent. The loads correspond with and are based upon 
those given in the last column of Table IV. 

The author would recommend that the various sizes be not used 
for greater lengths than those given in the table. 



266e 



STRENGTH OF CAST-IBON COLUHKa 



TABLE V.6 

iSafe Loads in Tons of 2,000 Pounds far HoUovo Bertnngular 

Cast-iron Columns. 











LENGTH 


or COLUMN IN 


FBBT. 






W C * a. y. 


















- x >:• = 6< 


















H"^ ut^. h-y: 
U.?5i =-- w^"" 
C t- X 

1 


















10 


12 


13 


14 


15 


16 


18 


90 


6x0 J 151 


.^8 


48 


44 


40 











" 1 


20 


74 


61 


56 


51 










" li 


Si\ 


H7 


7.3 


66 


61 










6x8; J 


18i 


♦5<> 


.58 


52 


48 










" ", 1 


34 


88 


74 


67 


62 










-, ij 


2S} 


106 


88 


80 


74 










6 X 10 J 


21} 


m 


67 


61 


56 


51 








it .. 1 < 


28 


u« 


86 


78 


72 


66 








'• " 1 1} 


33J 


124 


104 


94 


87 


80 








< X « , } 


18S 


78 


67 


62 


58 


58 








t> Ik 1 


24 


100 


86 


80 


74 


68 








7 X 1); J 


21} 


91 


78 


78 


67 


68 








(i i(| 1 


28 


117 


100 


93 


86 


79 








8x8' { 


2U 


100 


87 


81 


76 


71 


65 






", 1 


28 


128 


118 


105 


98 


98 


84 






" li 


33J 


155 


185 


186 


118 


110 


101 






8 X 10 i 


24} 


113 


90 


92 


m 


HO 


74 






Ik il 


1 


32 


147 


128 


180 


118 


105 


96 






it 4k 


li 


mi 


178 


155 


145 


136 


185 


n« 






8 X 12 J 


21} 


127 


111 


104 


97 


90 


88 






41 Ik -t 


36 


IJW 


144 


135 


188 


117 


108 






"It 43} 


201 


175 


164 


158 


144 


125 






10 . 10 J Tt\ 


14.3 


19U 


123 


117 


111 


105 


94 




'•! 1 36 


186 


169 


160 


151 


144 


136 


188 




"1 li i 13} 


220 


ao5 


194 


1H4 


175 


166 


148 




"i 'i 1 •'>i 


•^^ 


239 


227 


215 


804 


198 


m 




10 > 1'^ i ' 3()j 


1.59 


144 


137 


130 


122 


116 


1(M 




"1 10 


2(Ni 


18S 


17H 


168 


160 


158 


186 




•• 1- 1 Hj 


252 


229 


217 


2<t5 


195 


1H5 


I«^ 




•* n , r>r 


2U3 


267 


253 


240 


828 


316 


198 




11)  u } :»| 


174 


I.5S 


1.50 


143 


135 


138 


III 


1 

1 


••11 n 


227 


2i;6 


196 


1K5 


176 


167 


119 




111 ir, 1 1 IS 


218 


225 


214 


802 


192 


183 


168 




M l^ 1 ' .VJ 


2ris 


241 


2:J1 


219 


808 


197 


ITli 




in  '*\ I ' »'»! 


3:jo 


3<NI 


2S-) 


870 


3:^*1 


843 


317 




I'J . 1-J ; :W, 


1^7 


171 


HkS 


161 


154 


IH 


186 


184 


•1 11 


214 


227 


219 


210 


801 


193 


177 


lil2 


•• •• i; •'•••<; 


■.flH 


278 


2»;7 


2515 


346 


236 


317 


l\9i 


•' ]. k\:\ 


349 


32r) 


812 


800 


889 


8T7 


8M 


383 


IJ 11 . :ic. 


2«»;i 


1S9 


1S2 


175 


IfiH 


161 


14K 


I* 


1 iK 


2ti(> 


•J.IK 


23J» 


229 


320 


811 


las 


l~ 


I-.'  !•; I .v: 


2SS 


2GS 


2.-is 


848 


8SH 


88S 


810 


I'.tt 


l-j -.I 1 ♦^s 


371 


."151 


.338 


335 


812 


899 


874 


<:>l 


u  It 1 :.«•• 


.WO 


.31 >H 


2tr7 


2HK 


8iK 


86K 


8BII 


05 


If. it'i 1 r»<) 


:{.-.! 


.^3li 


:») 


324 


S18 


810 


891 


87S 


i«-.  i" 1 i\\ 


37; 


.35M 


358 


345 


389 


880 


814 


tBS 


l'^  J'' 1 ».s 


lit 


401 


391 


3HI) 


874 


887 


8ii 


W 


1" -jj 


1 


HI 


|K.v< 


472 


460 


44M 


440 


«8 


408 


888 



STEEJi^QTH OF CAST-IRON COLUMNS. 255/* 



Wrought-Iron and Steel Columns and Struts 

(1891). 

Within the past three years wrought-iron and steel columns have 
been gradually taking the place of cast-iron columns in fire- proof 
buildings, and the time is probably not far distant when wrought- 
iron oi" steel columns will be used almost exclusively for the inte- 
rior supports of all largo buildings. 

In iron or steel trusses the struts are invariably made of the same 
material, though, of course, the strut bars are of a different section 
from that usod for ties. 

There are many contingencies which may arise in the manufact- 
ure of cast-iron columns which preclude anything approaching 
uniformity in the product. 

Among these are unevenness in the thickness of the metal, which 
has sometimes been found to be very different on one side of a 
round column from that on the opiK)site side. The presence of con- 
fined air, producing '* blow holes ** and *' honey-comb," and the col- 



356 STRENGTH OF WROUGHT-IRON POSTS. 

lection of impurities at the bottom of the mould are aAso frequent 
sources of weakness in cast iron. 

The most critical condition, however, is that due to the unequal 
contraction of the metal during the process of cooling, thereby 
giving rise to initial strains, at times of sufficient force to produce 
rupture in the column or in its lugs on the slightest provocation. 

In many cases the trouble is due to faulty designing or careless- 
ness in the execution of the work ; yet, even under favorable condi- 
tions, it is so difficult to secure equal radiation from the moulds in 
all directions that castings entirely exempt from inherent shrink- 
age strains are probably seldom produced. 

As a protection against these contingencies, resort must be had 
cither to the uncertain expedient of a high factor of safety, or a 
material such as wrought iron or rolled steel must be adopted of a 
more uniform and reliable character than cast iron. 

Columns built up o* rollcl socLioiis alsj offer better facilities for 
fire-proof covering ; and for columns where extreme loads are to be 
supported, as in the lo.ver sLorieii oi' very high buildings, wrought- 
iron and steel columns wiU occupy less room than a cadt-iron 
column, and in many instances will be found to be cheaper. 

The forms of rolled columns now in general use in buildings are 
the ** Phoenix," '* Larimer," " Gray," and *• Z-bar" columns, illus- 
trated on pages 267-389A. 

For the strut bars of trusses two-channels bars, angle or T-bars, 
are generally used. 

In trusses with pin connectiotis the channel bar offers the best 
shape for the struts. I-beams are also often used. 

Streiigrtli of Wroiijjflit-iron Posts. 

The formulas most generally accepted by engineers of the present 
day for the strength of irre^^ular- shaped sections (such as nearly all 
these struts are) are as follows : 

Column — Square Bearing, 

Ultimate strength / _ 40,000 

in lbs. per sq. inch i "~^ sq. of le ngth in inches ^ ' 

' 36,000 X r* 



1 + 



Column — Pin and Square Bearing^ 

Ultimate strength | _ 40,000 

in lbs. per sq. inch ) ~"I sq. of le ngt h"lnlnches ^^ 



STRENGTH OF WROUGHT-IRON POSTa 257 

Column — Pin Bearing, 

Ultimate strength ) _ 40,000 

in lbs. per sq. inch ) ~' sq. of length in inches ' ' 

18,U00 X r» 

in which r denotes th»j radius of gyration. 

A column is square hearing when it has square ends which butt 
against, or are firmly connected with, an immovable surface, such as 
the floor of a building, or riveted connections : it is pin and square 
hearing when one end only is square bearing, and the other end 
presses against a close-fitting pin ; and it is pin bearing when both 
ends are thus piti-jointed with the axis of the pins in parallel direc- 
tions (for example, the posts in pin -connected trusses). 

To shorten the process of computation by this formula, Table 
VI. has been computed, which gives the ultimate strength per 
square inch of cross-section for different proportions of the length 
in feet, divided by the radius of gyration. 

The radius of gyration of the principal patterns of rolled bars now 
on the market may be obtained from the tables given in Chapter 
XIII. 

To use these tables, it is only necessary to divide the length of the 
strut (in feet) by the least radius of gyration, if the strut is free to 
bend either way, and from the table find the load per square inch 
corresponding to this ratio. The area of the cross-section, multi- 
plied by the load, taken from the table, will give the ultimate 
strength of the strut or column. To find the safe load, divide by 4 
for columns used in buildings, and 5 for trusses. 

Example 1. — What is the greatest safe load of a pair of Carnegie 
angles, 6" x 6", 33 pounds per foot, riveted together, 12 feet long, 
with square or fixed ends ; the angles being used as a strut bar in a 
truss ? 

Ans, The least radius of gyration is 1.85. which is contained in 
12, 6.5 times. The strength for a column, with square ends, for 

this ratio of _ is, from Table VI., about 34,200 pounds per square 
r 

inch ; this, divided by 5, gives a safe strength of 6,840 pounds per 

scjuare inch, or a total safe load for the two angles of (6,840 x 

'.9.90) 136,116 pounds, or 68 tons. 

When two or more angles, channels, or I-beams are connected 
together by lattice work, the radius of gyration for the whole sec- 
tion should first be obtained, and then the method of calculation is 
the same as for a single bar. 

Channel bars are generally used in pairs, either connected by lat- 
tice work, or, where additional strength is required, by wrought-iron 



258 STRENGTH OF WROUGHT-IRON POSTS. 

plates riveted to the flanges of the channels. In sach cases, the 
channels should be spaced far enough apart so that the colomu will 
be weakest in the direction of the web ; i.e., with neutral axis at 
right angles to the web, for which case the radius of gyration of the 
column is the same as that of a single channel. 

In Table VII. the quantities d and D show the distance that the 
channels should be separated to have the same radius of gyration 
about either axis. 

If the radius of gymtion is wanted for the neutral axis through 
the centre of section paraliei with web, it can readily be found, as 
the distance between the centre of grjivity of channel and centre 
of section with the aid of Column VI., in tables, pages 301-21. 

If two channels are connected by means of two plates, instead of 
lattice bars, it is necessary to obtain, fii*st, the moment of inertia of 
the section, whence the radius of gyration is found as the square 
root of the quotient of the moment of inertia divided by the area of 
the section. 

This moment of inertia, for a neutral axis, through centre of sec- 
tion perpendicular to the plates, is ecjual to the cube of the width 
of the plate, multiplied by ,'2 of tiie thickness of the two plates 
added, plus the combined area of the two channels multiplied by 
the square of the distance from their centres of gravity to the neu- 
tral axis. For a neutral axis in a direction parallel to the plates, 
it is equal to the moments of inertia of the channels as found in the 
tables, increased by the area of the two plates multiplied by the 
square of the distance between the centre of the plate and the centre 
of the section. 

The strength of such a strut may, however, be calculated with 
suflBcient accuracy for most purposes, by taking the radius of gyra- 
tion of a single channel, and getting the strength per square inch 
of cross-section, and then multiplying by the total area of the sec- 
tion. If the channels are s[)aced according to Table VII., or even 
greater, the true radius of gyration will be a little larger than that 
of the single channel, so that what error there is will be on the saf^ 
side. 

Table VII. has been computed on this basis, giving the strength 
of two channels, used as a strut. The heavy figures give the safe 
load (factor of saf('ty of 5) for the two channels latticed together, 
and the figures in italics give the safe load per square inch of sec- 
tion ; so that, in case the pair of channels alone do not give sufficient 
strength, one can readily tell how much additional area will be 
required. Table VIII. gives the safe load of Carnegie T-bara» used 
singly. 



STRENGTH OF WROUGHT-IRON POSTS. 259 

Example ?. — A certain strut in a roof truss (18 feet Jong) has to 
withstand a stress of 50 tons, and it is desired to use two channels 
for the purpose ; what sized channels will be required, the strut 
baviiiij pin joints ? 

Ans. Looking down the column headed 18 (Table VII.), we find the 

nearest load under 50 tons is 40.8, for two 10" channels, pin bearing, 

and the safe strength per square inch is WA tons. As the load in 

the table lacks 9.2 tons of that required, the section of the channels 

9 2 
must be increased by -^, or 2.7 square inches, which is equivalent 

to 9 pounds per foot additional weight for the two channels ; so 
that we should use two 10" channels, weighing 24^ pounds per foot 
each, and the channels should be spaced 9.1" out to out, the 
flanges being turned in. 

In pin-connected trusses, two channels make the most practical 
form of strut bar. 

A common form of column or strut to be recommended for com- 
paratively light loads is that formed simply of two angles riveted 
together back to back, or four angles united either with a single 
course of lattice bars or a central web plate, as in Fig. 4, page 264. 

The radii of gyration for such struts are tabulated on pages 319-21. 

In cases where four angles are used, the two pairs should be 
spaced far enough apart to make the column weakest about a neu- 
tral axis parallel to the central web or latticing. The radius of 
gyration will then be the same as that given in the tables for a 
single pair of angles, since the moment of inertia of the web plate 
about such an axis is so small that it may be disregarded entirely. 

Example 3. — A strut 16 feet long, to be fixed rigidly at both ends, 
is needed for supporting a load of 80,000 pounds. It is to be com- 
posed of two pairs of angles, united with a single line of i" lattice 
bars along the central plane. What weight of angles will be re- 
quired, with a safety factor of 5 ? • 

Ans. We will assume four W x 4" angles, and determine the thick- 
ness of metal required. The angles must be spread ^" in order to 
admit the latticing. From the table on page 321, we find the radius 
of gyration of a pair of light 3" x 4" angles with the 3" logs par- 

l 16 

alleland^"aparttobe 1 97 '. Hence the value of - = Y~Q7 — ^ 1» 

for#vhich the ultimate strength, as per Table VI. = 31,680 pounds. 
The allowable strain per square inch with a safety factor of 5 
will therefore be 31,680 ^ 5 = 6.34 ) pounds, and the area of the re- 
quired cross-section 80,000 -t- 6,340 = 12.62 square inches, or 3.16 
square inches for each angle. Hence the weight per foot of each 



260 



STKENGTU OF WROUGHT-IRON COLUMNS. 



TABLE VI. 

Ultivuite Strength of Wrought-iron Columns. 

For diflerent proportions of loiigtli in feet ( = O 
To leawt rudiua of L'yratiou in iiiclu'H ( — r). 
I'o obtain Safe lie^JHtance : 

P'or quics<>ent loudt*. an in hnil(lin<;H, divide by 4. 
For moving loads, as in bridges, divide by 5. 



I 
r 



3.0 
3. J 
3.1 
3.0 
3.S 

4.0 
4.-J 
4.4 

4.(i 

4.K 

5.0 

5.4 

5.n 
5.K 

6.0 
(i.L> 
6.1 

6.»i 
6.S 

7.0 
7.2 
7.4 

7.H 

7.H 

H.O 

S.I 
H.»l 

'.».•• 
'.I •-' 
'.'.I 
'.' '• 
'.« *» 

III i) 
lo.j 
lo I 

lO.ii 
It's 



I'ltimatc Ktrength in pounda 
per fr(jnare inch. 



' Square. I P^" ai»<l 
• square. 



:3H,610 

3s.4;m) 
3s,2:^o 
3sj);io 
3r,--^u 

3:.r.i»o 
3r,3r^) 

3T,I:A) 
36.S71) 
3li.«»-.'0 

3«;,:«'.0 
.'IC.OiiO 

.•i:..s*jo 

3:),.-)40 

3i,!»ro 

3t,r.70 
3 1.370 
34.or.0 

:i::,r:)0 

•X\. \ 10 
••{:{, -.-{O 
3-^*.'^10 
3J. UK) 

:«.i;o 

3I.*'.'»0 
3I..VJ|> 

:'.l.!:«o : 

3i.^7o 

:ii:..Mo 

3 •.•.•!(» I 

'I I :.:.' I  
'."» '.'.;( I 

•> !». 



IMn. 



•.'..'.fjll 
^1 . • 1 1' 



I 



37,J»r)0 
37,«H<) 
37,4(H) 
37,110 
:i0,sio 

36,500 
36, KO 
a").H40 

;i"),5oo 

35.140 

34,7SO 
:i4.4i>o 
:M,(K)0 
:^3,670 
3:j,-280 

3--»,SflO I 
3'J,o<H)  
3v»,110 
31.7111 
31.310 

30,910 
3<»v'>10 
80,110 
iil>.710 
:i9,3lO 

2*<,'.I00 
iN,5mi 

;rr.70i» 
•*':.3io 

•J''..'^»o 

•.'•..110 

•.»."•.:•■,(» 
•j:).3rn 

'Jl.lKtO 

'Ji, 60 



87,210 
86,!I70 
36,610 
3'i.a40 
35,860 

a5,«60 
3.V50 
31,(^0 

33,770 

33,3.30 

:w,4n» 

3I,«S0 
31,5:^0 

31,Ofio 
3t).5'.)0 
30.1.30 
:2!),6ro 

88,740 
a8.-*>70 

t>r,8:i0 
'J7..KiO 

2(S,<no 

!M,OM 
iJ.*>,.')7o 

•r..i:t» 

'J4,7«iO 

v>l.-,»70 

■i-i.i;{i» 
•3.o:jii 

^ « t ^^^* ' 

'J 1. 140 

*JI.INiO 



/• 



11.0 
11.2 
11.4 
11.6 
11.8 

12.0 
12.2 
12.4 

12.6 
12.8 

13.0 
1.3.2 
13.5 
13.8 

14.0 
14.2 
14.5 
14.8 

15.0 
1.V2 
15.5 
1.'>.8 

16.0 
16.2 
16 5 
16..S 

17.0 
17.2 
17.5 

17.S 

18.0 
IS.'J 
IK.-, 
Ivs 

r.».2 
v.\ .-I 

l».8 

21 Ml 
2i».2 
20 5 
2l».K 



Ultimate str(>ntfth in jiounds 
I>cr square incli. 



I 



Sqnarc. 



26,950 
26,(>44) 
26,.3iO , 
26,(KM) I 
25,6!)1i . 

25,380 i 

25,070 ' 

24.770 ; 

24.170 ' 

*1,170 j 

23.870 : 
28,570 
23,114) 
22,700 

2.\ 120 
22,l.-rf> 
21.710 
2 1, .320 

21,050 
20,7110 
2l»,21W 
20,<l20 

19,7f*0 
1!).510 

I'M 50 

iK.r.t) 

18,.\'S«» 
1S.320 

I7!i8i» 

17.120 

I r. 21 Ml 

Iti.SNi) 

i6..'»:o 
1 6.3:0 

Hi. 1:0 

i:i.87o 
I.V><i> 

l.\8sii 
I.VJI4I 
IJ/rJl) 

H.ftTiU 



Pin and 
nquart*. 



23.170 

22,S20 I 

22,170 ! 

22, i:*) ' 

21. 8U) i 

31.460 i 

2i,iao : 

20,810 
20.4.10 . 
20, ISO ! 

19,860 ' 
19.560 
19.110 
18,t>70 ! 

18,:W) I 
lS,Hr0 

K.H'.'O ' 

17.290 j 

ir,02() 
16.7tt> 
16,3UI» > 
16,010 I 

15.77»J 
15.M0 
15.190 j 

11.680 > 

11.410 I 

14.1i!0 I 

13,790 i 

13.5iiO i 
13.:t!ii> 
13.1«4) 
IJ.»'20 

12.ii:ii) I 
12. ni 
12, 190 I 
ll.Wi) 

ii.;in 

11.600 
11,3110 
1I.11W 



Pin. 



20.2:» 
19.960 
19.6!0 
19.2n> 
1S,1«30 

18.590 
I8.2ii0 
17.1M0 
17.IU0 
17,310 

17.000 
Itt.riO 
ia,2H0 
15.KS0 

1.V580 
15.310 
14.920 

i4,&ao 

14.290 
14.<»40 
l.-S.lil«t 
I3.3S0 

13.120 
12.910 
12.590 
12.2H0 

I2.«W 
ll.K*« 
11.590 
ll.«IU 

11.140 
lO.'.itiO 
10.700 
10.4:10 

10.290 

lo.i:fl) 

9.A20 

8,in) 



STRENGTH OF CARN£6IE IRON CHAN2 S. 261 



•-3 






9 



fej 






2 

H 

< 

a 
o 



OS 

o 



^ O 
$ Eh 



4S 



1 J 



I 



Da 

3Q 



•1; 






M 






§ 



s a a 

Cfi 0L, GG 



£ 



Pui 02 P^ (/) Pi QQ A4 



'/) p^ 






I a 
at 



a 



s 



»-■» 



^ t>» «4 30 ?0 *^ 



^stooo 



S^S2§^ 



*a«9»ac0»a 



S*=S8 



 • • • 



• • • 



fe*^«^S^S*'i;'^g*'^''SJ 



• • • 



^ 






*'^'^§8*^^''8?''S''^^S5 






•yjos »» 



8 



00 



1^ c. <»^ «* 



'O 



OS 55 aoao '^«o ?o 



g^g^g*ig*5^»jg^g»5g*:gj>5^*»^ •5g*» 



00 



t^ >.-iO «OQ0<t»o 5cS *»»-• -«© S» otSao 



00 »^ 

• • • 






CO »» '-^ ^i ^ i.-^ » "^ » «--^ go oi 



<>» 






OOOi^ <0 00 ^ 00 OS <0 TO t^CC^Cl 



• • • • 



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Jg^g^g<tg<t0g^«j|*SQ§^gg0^g*i3*5g »S^ 



*5i-*5 









04 





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i 


• 

CO 



a 



00 



. II II II . II II II . II II II '^ II II II '" II II II "^ II II II 

1-^ 1-^ n 0> *•* 



262 STRENGTH OF CARNEGIE IRON CHANNELS. 



.i^ 



\f 



1 



TS 
O) 

o 



bo 

a 

pq 



2. C 3 C 






2! 2: 

X 0L, X 



2. = 



gj 






•V c, 55 ^ <o *^ »-j «•■: *^ 'o 55 J. « >: Oi *i S N -r »-. oo ~? » • - i 
V -nI m .>J ec **: TM* *» a ^< *> "^ o "ij I- !*! et ''J ac "^ t- "« «ft -4 | 



! I 



I 



ig ■*» r , -^c l' 5? 2? •"• 



5J 0* f-i f^ 91 »-" »^ 



Si'SS te s'jt ZJ $ii:?» $ -c 



'»x"^i ©•^x'^i ee ■*»o "'•t- ■'!« *» 





a: 


7. 






'«v 








•^ 


"* 






»« 


'ji 


*•> 




•»* 


y. 


1/ 










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'J~ 


y. 


«i=^ 1 


't 




«i; 


1 


"^ 


K 

*- 


^ 




» 


>«^ 


— 1 




• 


it 


ii 


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^ 


7. \ 


e 


'^ 


M« 


t: ! 




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1 




«^ 


^ 


^ 1 




■^ 


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•/. 


tc  


HH 


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r 








r^ 












H 


^ 
&« 


/: 




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i^ 


»* 




i-M 








PQ 


i. 


^ 




<J 


5^ 


^ 




(-H 


<%> 


1-^ 


1 












^H 








jk 










'*. 






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< 





*^ •••••■■■ 

00 « "- »-i 






"■C 'f '-I 

• • • 

•^3: >» 



T NC* '»« >itt ■*» 



>- •/- 



2: 



• • • •  • 

81 '^iS*^"^^*^ 



• ••• •••• ••■••••■ 



91 



tc5e».S;c2». «^:3:-! «o.S-. 5=?S.':g^,SS 



• ••••••• 



^*:g*sifi*n^'n 5R"^S''» '♦^'^*- 2 ">: lO ">i » "*; X »• 



• ••••••• •••• •••• ■••■••■• 



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X 



• ••••••• •••• •••• •■>*■»•• 



5c:9*$$5rJ5i:'^:::5^?^:c.^: ."*^«?:ff.^ 



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|g*3g'»:jO'»:jongi"*:fi': oorw*. S'^g**^'^*'" 



n 




ri 









W 



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Ki 



^ I *" II II II** II 11 II 'I It II ■- '■ '1 !l !I II 



/. 



STRENGTH OF CARNEGIE IRON T-BARS. 



263 



. 

V 

a> 

a 

1 

t 




• • •• •••• 

t'. •» « '-i 0» «« !.« «t 


• • • • • 
00»<«O«) 


• ••• •••••• 

tO>-iiO«) »0 »-• SO »-i 00 »-< 








1^ 
1-^ 


S25 


Sec ?S»»S^ o§?S^ 


88IJ5S2 


S§^j58S 






00 »l 


«o '^ 9^ t' e« OS 9« CO «) 


>0»«I0») 


iO»-iOO»-«00*-« . 






00 


• • 

00 «) 

• • 


• • •■•• •■•• #••• 
t-«) ri»)QO«) o«tt«*t e«4>0») 


• ••••• •• 


• • •••• •••• 

00 «) e»^as«) o«)^«•) 

1-1 1^ 


• • • • 


• ••••• .• 

1-- ♦« ^ »< ^ •? 00 »-» 


to t^ 


«-* ^aoSoo QO^Sos 


«o 2os «>. 


S*3S^^5 « S 






00 «N ©»5MO<M r^e>»00i>» 
1-1 rH »-i 


I, <»1«{M 


00 *»'«"•» ^ •» 93 VI 


I'. 


^:^ 


R?2S^g^ 


8:5SJ§ 


»^5S 








OS »i 00 "50 O »J 

1-1 Tl 


^*iOOCO 


ao ®» i- «» 


00%tlO»)^«) ^%l 


<0 




s^ 




S Sos ^» 


38^Sa»ffi^ S?c 




1-1 


1-^ 


^ *5i-i nn eo*50oc^ 

1-1 T-1 11 


00 00 to 0^ 


os>dto»)in«) "^•i 


« 


OS ».-i 

• • 
TH 

• • 

CO -iT) 


m • •••• •••• 

1-^ 1^ 1"i v^ 


.... 
W *5 00 "50 


10.82 
S.ll 
6.20 
S.18 
5.8 


• • 


-■TJ< 


• • •••• •••• 


• • • • 

o "^s 00 "o 

1-t 


• ••••• •• 


Least 

radins 

of 

gyration. 


• 

O 


0.64 

0.79 
0.78 

0.84 
0.86 


• • 
o o 


S ^ S 13 

• • • • 

o o o o 


08 


• 


QC 
CO 


11.9 3.57 

15.2 ' 4.56 
11.8 3.54 


• • 


• • 

Gi Gi 


'9' OS S X 
« . • • 

« 1-1 rH T-1 


4^ 

'3 


I- 

• 

0» 

1-1 


1" 1-1 

• • 
CO OS 


I, eo 

OS* QC 


«C lO o o 

• • • • 

1-1 CO CO «o 
1-1 


s 

1 


'a 

OD 


X 

to 


of 

X 


X X 


•^ eo 

X X 


"* CO 
X X 

oS" iS* 


1* 00 ^ 
XXX 

•0 eo eo 


00 
X 

of 





WROUGHT-IRON AND STEEL <X)LU1IK8. 



angle vill be 3.16 -i- 0.8 = lO.S Ibe. This weight will be foood b 

agree with a thioknesa of 1 inch for a 4" x 8" aug-le. 








iTZ'SaV CDlumn *'Z:B>r OclWMl 



^ of Ooliinuu bjr IVflootlon or Bnckllimr*— 

ri>nt,'tjt-in)ii l-oIuidds fail either by deflecting bodily out 
111 line, or l>f the bui-kling of the metat botwaen rivata 
iLDli) uf supjiurt. Both actions maj take place at tfa* 



\ lOUGHT-raON AND STEEL COLUMNS. i!Ot> 

same time, bat if the Latter occurs alone, it maj be aa indication 
that the rivet spacing or the thickness ol the metal is losufficient. 

The niJe has been deduced from actual eiperiments upon wrought- 
fron columns, that the distance between centres of rivets should not 
exceed, in the line of strain, sixteen times the thioltness of metal of 
the parts joined, and that t]je distance Ijetween rivets or other 
points of support, at right angles to the lino of strain, should not 
exceed tbirt;-tiTO times the thickness of the metal. 

On page 244 sections are shown of some of the most common 
forms of steel and wrougbt-lron columns. Figs. 5 and 6, as well as 
the Ph<Bnix and Keystone Columns illustrated on piiges 267 and 377, 
belong to the type known as Cloeed Columns. As it is impractica- 
ble to repaint the inner surfaces of such columim, they should pref- 
erably be used only for interior work, where tlie clianggs in tem- 
perature are not considerable, and the air is comparatively dry. In 
places exposed to the extremes of temperature and unprotected 
from the rain, the paint on the inner surface of the column will, 
sooner or later, cease to be a protection to the iron, corrosion will 
set in, and, once begun, will continue as long as there is unoxidjzed 
metal left la the column. 

Figures 4 and 8 on page 264 represent types of 

}]=- columns with open sections, which readily admit 
-^-J of repainting, and are therefore suitable for out- 

Ik. door work. 

J 0( these, the latter, designed bj C. L, Strobel, 
C.B., and known as the Z-bar Column, is believed 
to oSer advantages equal, if not superior to those 
t any other steel or wrought-iron column in the 

f market. 
Bracing of Channels. — When chaqnels are 
i oonnected by hittioe work (as in Fig. 1). that there 
^ may not be a tendency in the channels to bend be- 

£__jl tween the points of bracing, the distance I should 
jr be made to equal the total length of strut, mul- 
P,g i_ tiplied by the least radius of gyration of a single 

colunm, and the product divided by the least radius 

of gyration for the whole section ; or, I = „ where the letters 
have the following significance : 
/ = length between bracing, 
L = total length of stmt. 
r = least radios of gyration for a single channel. 
B =■ leaat ntdiu of gyration for the whole section. 



i 



266 



STRENGTH OF STEEL COLUMNS. 



When the radius of gyration of channels, about an axis parallel 
with the web, is not ffiven, it will be sufficiently accurate to use for 
r tlie distance given in CoJumn VL in the tables on pages 801- 
321. 

Example 4. — We will determine the distance l^ for the strut calcu- 
lated in Example 2. In this case 2/ = 18 feet, or 216 inches, R = 
3.85 : and in Column VI., page 804, the distance d for a 20-pound 
channel is .70, for a 35-pound channel .75, so that we will assume 
.72 as the proper distance for a 24-pound channel ; or r = .72; 
216 X .72 



then I — 



3.85 



.- - = 40 inches. This same rule will also apply for 



angles, though with them the lattice work is generally doubled, 

in Fig. 2. 



ii 



11 




Steel Columns. 

'' Exi)oriments thus far made upon steel stmts indicate that fof 
Icnfifths up to 90 radii of gyration," (or 7.5 in Table VI„) '* their ulti- 
mate stnngth is alx)ut 20 |)or cent, higher than for iron. Beyond 
this )N)int. th<' excess of strength diminishes until it becomes zero at 
about 200 nulii. After passing this limit, the compressive re^ist- 
anco of <\y'v\ and iron seems to Iwcome practically equal.*** In 
Tables Vil and VIII. the loads to the left of heavy black line are 
for ratios less than 90 diameters. an<l those to the right for ratios 
alH)V(' that limit. 



Sp<M*ial Forms of AVr<MiKlit-iroii aiul Steel Coluiiiiui. 

7'A/ 1* In mix Sifjincnfal ('obtmn\ has now been on the market 
fi>r a iiiiiiitMr of years, and is very extensively used in buildings, 

ami al.-^o lor posts in bridges. 



^ .Mauiiiucturfd by the Phoenix lYoo Comptny, FliUaMphlik 



ROUGHT-IRON AND STEEL COLUMNS. 



267 





CO 



bages are : Economy of metal, simplicity of construc- 
bility to the requirements of building construction, and 
i. 

limns are made up of the rolled segments *'(','* which 

are riveted together, by rivets about six 
inches apart, by moans of flanges along 
their sides, as shown at * ' A " (Fig. 18). 
Between every two segments an iron bar 
is frequently inserted, through which 
the rivets pass. These bars, or '* flats *' 
as they are called, increase the area of 
the cross-section, and contribute much 
to the strength of the pillar. Table IX. 
gives the sizes of the columns rolled by 
the Phoenix Iron Company, as pub- 
lished in their book of sections. 

The interior surfaces of all Phoenix 
columns are thoroughly painted before 
riveting the segments together. After 
twenty years of service in exposed situ- 
" ations, columns have been cut open and 
l5 found uninjured by rust, and the paint 
still in good condition. 

The illustrations on pages 270 and 
271 show methods of joining the several 
tiers of columns in a building, and the 
connections with girders, etc. 

Bearings for girders or beams at ir- 
regular heights are provided by project- 
ing brackets that are properly riveted 
to a segment, or by a plate passing 
transversely through the column be- 
tween the flanges, with seating angles 
alon^r its upper edge. 

For joining columns at the levels of 
different tiers, inside sleeves of wrought 
iron may be usod. They are riveted to 
the segments of the lower column, and 
cting tenon which is fastened by diagonal through bolts 
colamn when it is put in place. 

line the actual value of Phoenix columns under loads, 

have been made at different times and on various 

id especially that of the United States (Government at 






STRENGTH OF WRODQHT-lR< COLDMNB. 

TABLE VI. 

TJliiraate Strength of Wroiight-iron Cotumiu. 
Pordiaen-iit propunioiiaorkiigth in feet ( = I) 



SSS.l.Hl 


iS 


21, AM) 
M,7«l 

acMSB 
ao.uao 


i 


1S.7C0 


RS 


isItbo 


H.KW 


i7.'jao 

17.6W 


14,W> 


li:H8U 


1S.M0 
18,380 
IS.ICO 


li«70 
15,5T0 


ili 


is.sflo 
u'.fso 

14,«N> 


II.TXI 



10,190 

un 



STRBNQTH OF CABNSQIB IRON CHANNELS. 



rn 
t 



. i 
.11 

^ If 

s ij 

1 1 

? I 






268 



PHCBNIX WROUGHT-IKON COLUMNS. 



TABLE IX. 



Sizes of Phcenix Columns. 








One Segment. 


One Column. 












Least 




1 






radius of 


Mark. 


Thicknes? 
in inches. 


Weight 
in ponndB 
per yard. 


Area in 
sq. inches. 


Weight 

in pounds 

per foot. 

12.6 


iryration 
in inches. 


A 


A 


9i 


3.8 


1.45 


4 segment. 


i 
h 


12 
14i 


4.8 

5.8 


16.0 
19.8 


1.50 
1.56 


3|" inter, diam. 


8 


17 
16 


6.8 


22.6 


1.59 




i 


6.4 


21.3 


1.92 


B' 


h 


19^ 


7.8 


26.0 


1.96 




23 


9.2 


30.6 


2.02 


4 segment. 


iV 


26i 


10.6 


85.8 


2.07 


4|f" inter, diam. 


1 
1% 


30 
83i 


12.0 
13.4 


40.0 
44.6 


2.11 
2.16 




i 


87 


14.8 


49.8 


2.20 




m 


7.4 


24.6 


2.84 


B' 


A- 


22i 


9.0 


30.0 


2.89 


I 


26^ 


10.6 


35.8 


2.48 


4 segment. 


A- 


m 


12.2 


40.6 


2.48 


^le.'/* j_ !• 


i 


34i 


13.8 


46.0 


2.62 


5if inter, diam. 


ft 


38i 


15.4 


51.8 


2.57 




s 

8 
i 


42i 


17.0 


66.6 


2.61 




25 


10.0 


88.8 


2.80 




>'b 


30 


12.0 


40.0 


2.a5 




2. 


35 


14.0 


46.6 


2.90 




,'b- 


40 


16.0 


58.8 


2 94 




i 


45 


18.0 


60.0 


2.98 


C 


,\ 


48 


19 2 


64.0 


8.08 




i 


53 


21 2 


70.6 


8.08 


4 segment. 


li 


58 


23 2 


77.8 


8 12 


7ft" inter, diam. 


i 


63 


25.2 


84.0 


8.16 




iJ 


68 


27.2 


90.6 


8.21 




1 
» 


73 


29.2 


97.8 


8.26 




1 


83 


33.2 


110.6 


8.84 




H : 93 


37.2 


124.0 


8.48 




11 103 

1 


41.2 


187.8 


8.6d 



PH(BNIX WROUGHT-IRON COLUMNS. 



TABLE IX,— Concluded. 
Sizes of Pluenix Columns. 



>islied colnmoe. 



the VVatortown (Mass.) Arsenal. Prom these enjierimonts formu- 
las have been deduced Irom which the aeeompanyinfc tables have 
boun prepared, in which are shown the safe loads in net tocjs for 
each size and length of the several patteros made. 



272 PHCENIX WROUGHT-IRON COLUMNS. 

columns are unequally loaded, then it will be adyisable 1 
the tabular figures or use heavier sections for the case, a 
indicated hj the circumstances. 

Steel Columns. — These tables have been prepared 
columns. If it is desired to use steel, it will be proper to 
for loads from 15 to 20 per cent, more than those giv( 
tables, the greater value being for short, and the lesser 
columns. 



PHOENIX IRON COLUMNS. 



273 



SAFE LOADS IN TONS OP 2,000 POUNDS. 
PHOESNIX IRON COLUMNS. 

Square Ends. 



4 Segment, A Column, 8f Inside Diameter. 



Length of 


^" 


\" 


h" 


f" 


column hi 


12.6 11)8. per ft. 
8.8 D in. 


16 lbs. per ft. 


19.8 lbs. per ft. 


22.6 lbs. per ft. 
6.8 a m. 


feet. 


4.8 □ in. 


5.8 n in. 


10 


17.29 


22.17 


27. -W 


32.36 


12 


16.87 


21.65 


26.57 


31.63 


14 


15.99 


20.54 


25.23 


30.05 


16 


15.08 


19.30 


23.84 


28.48 


18 


14.17 


18.24 


22.45 


26.79 


20 


13.29 


17.12 


21.10 


25.21 


22 


12.39 


15.99 


19.73 


23.61 


24 


11.57 


14.95 


18.47 


22.13 



4 Segment, B* Column, m" Inside Diameter. 



Length 


i" 


21. 3 lbs. 


column 


per ft. 


in ft'ei. 


6.4 Din. 


10 


30.30 


12 


29.45 


14 


28.49 


16 


27.46 


18 


28.40 


20 


25.28 


22 


24.14 


24 


23.00 


26 


21.88 



Bibs 



26 lbs. 

per ft. 

7. Sain. 



37.40 
36.36 
35.20 
33.94 
32.64 
31.27 
29.89 
28.50 
27.14 



jf 



It 



30.^ lbs. 

per ft. 

9.2niii. 



44.67 
43.44 
42.07 
40.59 
39.05 
37.44 
35.80 
34.17 
32.56 



_7 
T«.. 

35.3 11)8. 

per ft. 

10. Gain. 



52.10 
50.68 
49.10 
47.40 
45.08 
43.77 
41.90 
40.01 
88.16 



k" 
40 lbs. 

per ft. 

12 Din. 



59.71 
58.10 
56.31 
54.88 
52.38 
50.28 
48.15 
46.02 
43.92 



9 // 

44.6 lbs. 

per f r,. 

13.4nin. 



67.47 
65.68 
63.69 
61.53 
59.29 
56.95 
54.57 
52.19 
49.84 



49.3 lbs. 

per ft. 

14.811 in. 



70.41 
73.43 
71.28 
68.84 
66.37 
63.78 
61.16 
58.53 
55.94 



\ Segment, B^ Column, 5^g" Inside Diameter. 



Lenj;th 


\" 


30 lbs. 


\" 


J " 

IB 


k"" 


■^b" 


\" 


of 


24.6 lbs. 


85.3 lbs. 


40.6 lbs. 


46 lbs. 


51.3 lbs. 


56.0 lbs. 


column 


per ft. 


per ft. 


per ft. 


per ft. 


per ft. 


per ft. 


per ft. 


in feet. 


7.4 a in. 


9 D in. 


10.6 3 in. 


12.2 Din. 


13.8a in. 


15.4 Din. 


17Din. 


10 


a"), or 


44.30 


52.r9 


61.14 


60.85 


78.72 


87.75 


12 


85.25 


43.33 


51.56 


59. 9d 


68.51 


77.23 


8ii.l0 


14 


34.43 


42.32 


50.38 


58.59 


66.97 


75.50 


84.20 


16 


33.^3 


41.23 


49.09 


57.12 


65.30 


73. H4 


82.14 


18 


32.57 


40.06 


47.72 


55.53 


63.50 


71.04 


7^.93 


20 


31.55 


38.83 


46.26 


53.86 


61.61 


69.52 


77.60 


22 


80.48 


87.58 


44.73 


52.09 


59.61 


67.29 


75.14 


34 


29.41 


3^.22 


43.19 


50.32 


57.61 


65.06 


72.67 


26 


28.31 


84.89 


41.62 


48.51 


55.57 


62.78 


70.15 


28 


27.23 


33.57 


40.06 


46.72 


53.54 


60.52 


67.66 



214 



PHCENIX IRON COLUMNS. 



SAFE LOADS IN TONS OF 2,000 POUNDS. 
PHCSNIX IRON COLUMNS. 

Square Ends, 



4 Sboment, C Column, 7^'' Inside Diaxbter. 



Length y 

of 33.3 lbs. 
column per ft. 
in feet. [ lOoin. 


8 // 

per ft. 
12 a in. 


46.6 lbs. 
per ft. 
14 Din. 


7 // 

53.3 lbs. 
per ft. 
16 Gin. 


60 lbs. 
per ft. 

18 a in. 


641b6. 

per ft. 

19.2oin. 


70.6 IbB. 

per ft. 

21 .2 Din. 


10 
12 
14 
16 
18 
20 
22 
24 
26 
38 
30 
32 
34 
36 
38 
40 


50.97 
50.33 
49.62 
48.91 
47.87 
46.93 
45.92 
44.86 
43.77 
42.63 
41.48 


61.16 
60.40 
59.54 
58.59 
57.46 
56.31 
55.11 
53.83 
52.63 
51.16 
49.78 
48.42 


71.35 
70.46 
69.46 
68.48 
6;. 02 
65.70 
64.29 
62.81 
61.28 
59.68 
58.07 
56.4!) 
54.85 


81.55 
80.53 
79.30 
7H.2I) 
76.t)0 
75.08 
73.48 
71.78 
70.04 
68.21 
06.37 
64.56 
02.69 
60.88 


91.74 
90.60 
89.31 
88.04 
86.17 
84.47 
82.66 
80.75 
78.79 
76.74 
74.67 
72.63 
70.53 
68.43 
66.37 


97.86 
96.64 
95.87 
08.91 
91.92 
90.10 
88.17 
86.14 
84.04 
81.85 
79.65 
77.47 
75.23 
7:^.00 
70.80 
68.61 


106.05 

10i».71 

105.19 

103.69 

101.49 

99.49 

97.36 

95.11 

92.80 

90.38 

87.94 

85.54 

83.07 

80.60 

78.17 

75.75 













Lenffth 


TT.Slbs. 

per ft. 

2:12 D 

in. 


84 lbs. 


90.0 lbs. 


97.3 Ibe. 


110.6 lbs. 


124 IbB. 


187.3 lbs. 


column 
in feet. 


per ft. 
25.2cin. 

128.45 


per ft. 
27.2Din. 

138.65 


per ft. 
29. 2 Din. 


per ft. 
33.2Din. 


per ft. 
Sf.Soin. 


per ft. 
41. 2 a in. 


10 


118.2(5 


148.84 


169.23 


189.QS 


210.01 


12 


116. n 


120.84 


13»i.91 


140.97 


167.11 


187.94 


207.38 


14 


115.11 


125.04 


134.90 


144.89 


164.73 


184.68 


904.43 


10 


113.48 


123.20 


133.04 


142.83 


162.39 


181.96 


201.59 


IS 


111.07 


120.04 


130.22 


139.79 


158.94 


178.00 


197.94 


20 


108.87 


118.20 


127.04 


137.03 


155.80 


174.67 


193.85 


22 


100.54 


115.73 


124.91 


134.10 


152.47 


170.84 


189.91 


24 


104.08 


113.U5 


122 M 


131.01 


148.95 


166.89 


184.84 


26 


101.5.-) 


110.31- 


119.00 


127.82 


145.33 


168.84 


180.85 


28 


98.91 


107.44 


115.90 


124.49 


141.54 


158.0O 


175.65 


30 


90.24 


104.54 


112. S3 


121.13 


137.71 


164.» 


iro.9i 


32 


93.01 


101.08 


109.75 


117.82 


133.91 


160.10 


166.94 


34 


90.90 


98.74 


10<).5S 


114.42 


130.00 


146.78 


161.44 


36 


RS.iJO 


95.81 


103.41 


111.01 


126.22 


141.48 


156.64 


38 


85.55 


92.<>2 


100.30 


107.67 


122.42 


187.17 


161.98 


40 


82.90 


90.05 


97.19 


104.34 


118.64 


ias.96 


147.23 



PHGBNIX IKON OOLCJMNS. 



275 



SAFE LOADS IN TONS OF 2,000 POUNDS. 
PHCSNIX IRON COLUMNS. 

Square Ends, 



6 Segment, E Column, 11'' Inside Diameter. 



Lens^th 
of 


56 lbs. 

per ft. 

16.8D 

in. 


641 bs. 


72 lbs. 


801b-. 


88 lbs. 


9611)8. 


1" 
106 lbs. 


column 


per f I. 


I>er ft. 


per ft. 


per ft. 


per ft. 


P'^r ft. 


in feet. 


19.2a in. 


21. 6 Din. 


24 Din. 


26.4Din. 


28.8Din. 


31.8 a in. 


10 


86.94 


99.36 


111.78 


124.20 


186.62 


149.04 


164.56 


12 


86.41 


98.76 


111.11 


123.45 


135.80 


148.14 


163.57 


14 


85.79 


98.06 


110.31 


122.56 


134.82 


147.08 


162.40 


16 


85.09 


97.24 


109.40 


121.56 


13:3.71 


145.87 


161.06 


18 


84.30 


96.34 


108.88 


120.48 


132.47 


144.51 


159.66 


20 


83.44 


95.36 


107.28 


119.20 


131.12 


143.04 


157.95 


22 


82.52 


94.81 


106.09 


117. ^s8 


129.67 


141.46 


156.20 


24 


81.51 


93.15 


104.80 


116.44 


128.00 


139. 7^ 


154.29 


26 


80.47 


91.96 


103.46 


114.^ 


126.45 


137.95 


152.82 


28 


79.88 


90.72 


102.06 


118.40 


124.74 


18tl.08 


150.25 


30 


78.28 


89.41 


100.59 


111.76 


l-.?2.94 


184.12 


148.09 


32 


77.02 


88.08 


99.08 


110.04 


121.04 


132.04 


145.80 


34 


75.76 


86.50 


97.41 


108.24 


119.06 


129.88 


143.41 


86 


74.50 


85.15 


95.79 


106.44 


117.0R 


127.72 


141.03 


38 


73.21 


as. 67 


94.13 


101.59 


115.05 


126.51 


138.58 


40 


71.90 


82.17 


92.44 


102.72 


112.99 


123.26 


180.10 



Leno^h 


116 lbs. 

piT ft. 

34.8 D 
in. 


1" 


\l" 


1" 


V 


H" 


U" 


126 lbs. 


186 lbs. 


146 lbs. 


166 lbs. 


186 lbs. 


206 lbs. 


column 


per ft. 


per ft. 


l)er ft. 


per ft. 


per ft. 


per ft. 


in feet. 


87.8Din. 


40.8Din. 


43.8 Din. 


49. 8 Din. 


55.8Din. 


61 .8 Din. 


10 


180.09 


195.61. 


211.14 


226.66 


257.71 


288.76 


819.81 


12 


179.01 


194.44 


209.87 


225.30 


256.17 


287.03 


:317.89 


14 


177.71 


193.04 


20S.3I) 


2:23.68 


254.32 


284.97 


315.61 


16 


176.26 


191.45 


206.65 


2:31 .84 


252.23 


282.62 


:313.01 


18 


174.63 


189.68 


204.73 


219.78 


249.89 


280.00 


310.10 


20 


l':2.85 


187.75 


202.6,'3 


217.55 


247.35 


277.15 


:^06.96 


22 


170.93 


185.67 


200.40 


215.14 


244.62 


274.08 


:30:3.50 


24 


168.84 


18:3.40 


197.96 


212.51 


241.62 


2,0.74 


i299.85 


25 


166.69 


181.06 


195.43 


209.80 


288.54 


207.28 


21)0.02 


28 


164.4:3 


178.60 


192. 7S 


200.95 


235.30 


203.05 


292.00 


30 


162.06 


17'?. 08 


190.00 


203.97' 


231.91 


259.80 


287.80 


32 


15^.55 


17:3.31. 


187. 0<5 


200.82 


22S.33 


255.84 


283.35 


84 


156.94 


170.47 


184.00 


197.53 


•.'24.59 


251.05 


278.71 


86 


154.88 


107.64 


180.94 


194.25 


220.86 


247.47 


274.08 


88 


151.H5 


104.78 


177.80 


190.88 


217.02 


243.17 


2^9.32 


40 


148.94 


161.78 


174.62 


187.46 


213.14 


238.82 


264.50 



276 



PHCENIX IRON COLUMNa 



SAFE LOADS IN TONS OP 2,000 POUNDS. 
PHCENIX IRON COLUMNS. 

Square Fnda. 



8 Segment, G Column, 14j" Inside Diameter. 



Length 


801be. 


93.3 ll)s. 


iV 

106. ti lbs. 


120 lbs. 


13:^.8 Ihs. 


1" 
146.6 lbs. 


16Ulb8. 


column 


per ft. 


per ft. 


per ft. 


per ft. 
36 Din. 


, per ft. 


per ft. 
44 Dill. 


per ft. 


In feet. 


24 Din. 


28Din. 


32 D in. 


40 Din. 


48 Din. 


10 


124.92 


145.74 


166. .56 


187.38 


208.20 


229.02 


240.84 


1-2 


124.44 


145.18 


165.92 


186.66 


207.40 


228.14 


^48.88 


14 


123.5H 


144.56 


165.21 


18'). 8() 


206.. 52 


227.17 


247.82 


16 


123. 2.S 


143.83 


161.38 


iai.98 


205.48 


226.02 


'Zm.57 


18 


122.59 


143.02 


163.45 


183.88 


204.82 


224.75 


245.18 


20 


121.82 


142.12 


162.43 


182.73 


208.04 


223.84 


243.64 


22 


120.98 


141.14 


161.81 


181.47 


201.64 


221.80 


241.96 


24 


120.04 


140.05 


160.06 


180.07 


200.06 


220.as 


240.09 


26 


119.11 


18S.96 


158.81 


178.66 


198.52 


218.87 


288.22 


28 


118.08 


137.76 


157.44 


177.12 


196.80 


216.48 


23(>.16 


80 


117.00 


13>i.50 


156.(0 


175.50 


195.00 


214.60 


234.00 


82 


115.84 


135.15 


154.40 


178.77 


198.08 


212.86 


231.69 


84 


114.fi4 


133.75 


152.86 


171.97 


191.06 


210.18 


229.29 


86 


113.28 


132.16 


151.04 


169.92 


188.80 


207.68 


226.56 


88 


112.08 


I30.7() 


149.44 


168.12 


186.80 


205.48 


2^.16 


40 


110.80 


129.27 


147.74 


166.21 


184.68 


908.14 


221.61 



Length 

of 
column 
in feet. 



10 
12 
14 
16 
18 
20 
22 
84 
26 
28 
80 
82 
81 
Si) 
38 
40 



J/, 

173.3 
lbs. 

per ft. 

'^2 n in. 



270.66 
269.62 
2i)8.47 
267. 1-' 
26.-). 61 
26:^.95 
262.13 
2(K).10 
:l'58.07 
25.'>.84 
'J53.50 
:>51 .00 
248.40 
245.44 
243. K4 
240.08 



186. «) lbs. 
por ft. 
5(5 a in. 



291.48 
290.36 
289.12 
287.67 
28H.04 
2g4.25 
2S2.29 
280.11 
277.92 
275.52 
273.00 
2; 0.31 
26r.51 
264.32 
2(n.52 
258.55 



200 lbs. 
per ft. 
60nin. 



312.30 
311.10 
309.78 
808.22 
30«).48 
:iOI.56 
302.46 
300.12 
297.78 
295.20 
292.50 
289.(2 
28.i.r)2 
28{.:ii0 
2S0.20 
277.02 



1" 

226.6 lbs. 


258.8 lbs. 


980 lbs. 


806.6 Ibfl. 


per ft. 


per ft. 


per ft. 
84 Din. 


per ft. 


68Din. 


(6 Din. 


9iain. 


358.94 


895.58 


487.29 


478.86 


3.V2.58 


394.06 


48.'^.54 


477.09 


351.08 


892.88 


488.69 


474.99 


349.81 


80;). 41 


481.60 


472.60 


:i47.84 


388.90 


4WJ.07 


469.03 


345.10 


88.1.77 


496.88 


466.99 


342.78 


383.11 


498.44 


468.77 


34). 13 


380.15 


420.16 


460.18 


387.48 


877.18 


416.89 


4S6.m 


334.56 


373.98 


418.28 


452.64 


381.50 


370.50 


409.60 


448.60 


328.28 


866.85 


40r>.46 


444.08 


324. t^a 


883.06 


401.96 


489.48 


:V>().96 


35-).T8 


896.48 


4^.24 


317.56 


854.92 


399.« 


49>).64 


313.95 


850.89 


887.89 


424.78 



STOl OCTA »K COLI F. 



211 



Keystone Octagon Column. 

Another special form of wrought-irou column is that known as 
the Keystone Octagon Column, manufactured by Carnegie, Phipps 
& Co. It is made of four rolled segments of wrought iron, riveted 
together as shown in Fig. 5. 




mmr///M 



Fie. 5. 




The table oo the following page giyes the diameters, areas, and 
weights of these columns as rolled. To compute the strength of 
these columns it is first necessary to find the radius of gyration 
(r), when the strength per square inch can then be determined from 
Table VI. 

The radius of gyration may be found by the following formulo : 



J= 

7= 
A 
r 



12 



r = 



/4.' 



in which 



moment of inertia ; D 
area of column ; d 
radius of gyration. 



= outside diameter ; 
= inside diameter ; 



278 



KEYSTONE WROUGHT-IRON COLUMNS. 





1- 














ll. 




tf 




u 




Q. 




CO 




1- 




z 


m 


o 


^ 
S 


hJ 





o 


J 


z 


o 


< 


o 


CO 




< 


^ 


hJ 


o 


OC 


o 


< 


< 


o 


tH 


z 


o 


Q 


o 


z 







H 


Q. 


^ 


CO 

llJ 


o 


q: 


Eh 


oc 


C/} 





tH 


o 


^ 




M 


Q 
Z 




< 




CO 




hJ 




CO 




CO 




UJ 




Z 




^ 




o 



•8saa3[3iqx 


"i «e-^ 


•e^fstnc 


^■^ 


^^ 


1 i 






1 


•-» 














• 
















js Si a 














• 


t* ^ S) 


^ COM 

5 «T.; 


O Ci 

• • 

too 


oct- 

• • 


1 1 


1 1 


* ' i 

1 


» 




















• 














J 




■^ 












1 


o 




JS 


. eo 


5<ll- 


cooa 






1 


a 




to 

I 


5 ceo 


SS 


^^ 


1 1 


1 1 


, 1 1 


le, 


Q 
















t-* 


p) 
















1 




< 


• • • 


82 

• • 


QCC) 


1 1 


1 1 






• 














« 

as 

as 


js it a 


3 -^o 


• • 


00 1- 

• • 
1-H 


COCO 

*ico 


1 1 




;j 




• 














,j 




*< 














o 




ja 


. r-x 


C50 


^•N 


00 ^ 






O 

a 


• 

a 




J5 xr5 

— ^^ 2^ 


tt 


t-i 


^s 


1 1 




I 


9> 






























(O 


OQ 


1 


• • 


• • 


• • 


• • 


1 1 








-<{ 


j*^^ 


xo 


1-1 CO 


-^CD 


  








»^ 


r-i 1^ 


1-H l-H 








• 


% 1 QD 


x-»o 

• • 

Oi-i 


(NO 


• • 

CDQO 






5K 




1-1 


1-1 1^ 


1^ 1-H 


»-hO* 




\J 
















»j 


*^ 














c 


•s. 


O 


coo 


x« 


C40 


Oif 




i ^ 


2 ;s 


5 1 3^ 

- CO 


^% 


• • 

(NO 


s^ 


t-t 




■n 


a »>' 














o 


o -^ 














7: 


c 














H- 


^ ! 














1 * 


0; . 


«- 1 


Sao 


ss 


ss 


ss 






< 

1 


? ^ 


• • 

^co 

T^ 1-1 


• • 

i-« f-i 


^5 


^d 




j 

1 


2 oi 


• 


c:x 


XX 


r-t- 


o« 


oo 


1 ^ 


.2F 


5 a 


1 1 1 


• • 


• • 

T-i 1-H 


1-H(N 


tt 


t;^ 




• 


















J3 




-^co 


-*o 


Ofth- 


• ■» 


CO-N 


a 


• 

5 


13 


i 1 1 


• • 


• 

or- 


J^ig 


3^ 


ood 


s 


> 










*^ 


•iH »H 


o 

X 


















101 




'NX 


38 


So 


^V. 


S!3i 




^ 


< 


1 1 


• • 


• • 

X ^ 
1=.^ 


tt 


t^ 


Sl^ 


'Mtia 





4.L 


ua .- 


■c^"^ 




*- 


:>• 


5^» 



Z-BAR COLUMNS. 279 



Z-Bar Columns. 

Within the past three years, what is known as the Z-bar column 
has been introduced, and is now manufactured by all the leading 
iron mills. It is built up of four Z-bars, riveted together, as shown 
in Figs. 7 to 12, page 264. 

The dimensions of the different shapes manufactured will be 
found in the tables given in Chapter XIII. 

This column possesses so many advantages for building purposes 
that it is undoubtedly destined to be extensively used. 

Its claims for superiority are based mainly on the following 
qualities : 

1. Cheapness, — The Z-bars are furnished at a lower price per 
pound than channels and I-beams, and only two rows of rivets are 
required, while four or more are used for any other column of an 
equal sectional area. 

2. High Ultimate Resistance to Gompressioii. — Careful tests made 
upon fifteen full -sized (Carnegie) specimens, in which the web plates 
were replaced by lattice bars, showed an average ultimate resistance 
per square inch of 35,650 pounds for lengths ranging from 64 to 88 
radii. These results are as favorable as have been obtained for 
closed cylindrical columns, and are more favorable than have been 
obtained for any other open columns. For detailed report of the 
tests referred to, see paper by C. L. Strobel, in Trans. Am. Soc. 
C. E., April, 1888. 

3. Great Adaptability for Effecting Connections with I-heams. — 
When used in buildings, for supporting single floor beams, or 
double beam girders, this quality is of the greatest importance. 
The illustrations on pages 280 and 281 show different methods of 
making the connections, as employed by Carnegie, Phipps c: Co. 
This column may bo easily covered with terra-cotta blocks, for 
fireproofing, and finishing with plaster or cement, and the air-space 
between the tiling and the metal adds to the protection of the latter 
in the event of fire. The recesses in the columns may bo used to 
good advantage for conducting water and gas pipes, electric wires, 
etc. 

4. Favorable Form for Inspection and Repairing. — This is a 
very desirable feature when used for out door work. 

When unusually heavy loads must be provided for, as in the case 
of columns for tho Iov,cr stories of very high buildings, the stand- 
ard sections of Z-bar columns may be reenforced to the required 
strength by using either a double central web plate, or by the addi- 
tion of outside cover plates, or, if need be, both, forming thus a 



Z-BAK COLUMNS. 



Connaotient ol 



IBomu is"in<I ij"I Bwu ni3fr'°."9">nd B" T'^nd S' 

I Tods. J] Too). ^ Buiw  I Booh 

17.6 Tom,  B.I Tool 
Ccnnsotlonl d >dDubl« Som glnittt« Fluign otZB«r*< 



88T0111. IBMioi IBhum IBaoM 

S3 Tons. 35 Tom. >7>< Ttm, 

n*Htimitre//eii(iidlcaitd,aftutttAibaJtfinimfirttmmttf 
girdtrtjir ahick Iht ennKiiimt arr prr^triiuud, 

BivtUamtBtlttHdta.—AUBtlUtui.vtirt,tMllta4t. 



Z-BAB OOLl rs. 281 

DETAILS OP BTANDABD CONNECTIONS 
OPI^EAMSTO Z-BAR COLUMNS. 



leot Z-SarColun: 



Numitr »f riv4U rtf Hired /sr nntactiannef differ, 
c/l^^mi 10 »t*i o/Zbi<r>.-wiU be the !a.«t ni jEokph . 



282 Z-BAR COLUMNa 

closed or box column. A form of column, offering advantages in 
some cases, especially if the column is to be finished circular in 
form, is shown by Fig. 3 on page 281. Pig. 8 on the same page 
shows the manner of splicing columns, whether of equal or unequal 
size. 

•* The standard connections for double I-beam girders and single 
iloor beams to Z-bar columns, detailed on pages 280 and 281, were 
designed to fairly cover the lunge of ordinaiy practice. When the 
maximum loads in tons indicate<l for each case are exceeded, the 
connections may be correspondingly strengthened by simply using 
longer vertical angles for the brackets and increasing the number 
of rivets. In proportioning these connections, the shearing strain 
on rivets was assumed of a maximum intensity of 10,000 pounds per 
square inch. For steel Z-bar columns, the maximum loads given 
for these ccmnections may be safely increased 15 per cent.'*  

The following tables give the safe load in tons for standard Z-bar 
columns of different lengths, as manufactured by Carnegie, Phipps 
&Co. 

The values for steel Z-bar columns should be used only for cases 
in which the loads are for the most part statical, and equal, or very 
nearly so, on opposite sides of the columns. When there is much 
eccentricity of loading, or the loads are subject to sudden changes, 
the tabulated values must be n^duced according to circumstances. 

The Carnegie Steel Co. has discontinued the manufacture of iron 
bars of all kinds, and their product is now confined entirely to steel, 
which has practically superseded iron in structural work, being 
sold at the same price per pound, while 20 per cent, stronger. 

(The steel here referred to is what is knovrn as "mild" steel, 
having an ultimate strength of about 60,000 pounds per square 
inch, and containing a comparatively low percentage of carbon.) 

Example. — What size of Z-bar column, 30 feet long, with square 
bearing ends, will be required to carry a load of 200 tons, using a 
safetv factor of 4 ? 

A7is. Referring to table of steel Z-bar columns, page 287. we 
find that for a length of 30 feet, a 12-inch column with |-inch 
metal, weighing 118 4 lbs. per foot, will support with safety 202.6 
tons, which is slightly in excess of the load. 



* Carnegie, Phipps & Co.'t) Pocket Companion, 1890. 



E-BAB COLUHN DIUENSI0K8. 



Z-BAR COLUMN DIHBHeiONS. 






fOf J>- fOl 



M 



^m 



- ^-y* 



% of Z-BsT columns in inches for mil 
mum thicknesaes. 



Note. — In columns A. B, C. D, E, and F, the thickness of the 
Z-bars iind web plates does not vary, the variations in the strength 
of the eoliimn being mode in the thickness of the side plates. 

Columns G. H, K, and L, have no side plates, and the variations 
are in the thickness of the bars snd web plate. 

All of Column B and part of A have four side plates, two on each 
side, the others have but one plate on eacli side. 



STBBL Z-BAK C0LDMN8. 
BAFB LOADS IN TONS OP 3,000 LBS. 

BTBBI. Z-BAR COLUMNS. 

Square Endt. 

ine per BQQflrB inch  1 18.000 Ibn., for length of TO rsdll ornode 
ilely fsclor 4 : ' ^ n.lOO-SI-^, for lenglhe over so ndU. 

90" Z-BAR COLUHNS.-A. 

4Z-B«riSi"  1". 1 Web Plate U" y I". Side Plata SO" w 



SO" Z-BAK COLUMNS.— B. 

Secllan: 4Z-Birsei" > 4". 1 Web Plate 14" > 1". 4 SMe Plata *0" wlda. 






BZBBL Z-BAB COLUHNB. 

BAS% LOADS IN TONS OF 1,0I» LBS. . 
STBEI. Z-BAR OOLnMNS. 

Sgttare Endt. 



Allows 


d«™in*per.quflretacl 
Bsfely factor 4 ; 


I,.iia,0001b8., fi 


jrlenellHofBOradiiornnder. 
.torlcni^thioterBOradU. 



BTEEL Z-BAB OOLUHNS. ' 



, . <, ll,«aO IbB., rar hnglbe oT 90 ndll 01 
'"( 17,100-ST J. , tor li^iigUiB owr SO ni 



Section : 4 Z-Bara fll 



STEEL Z-BAR COLUHM& 

8AFB LUADS IN TON» OP 3.000 LBS. 
STBBL Z-BAR OOI.nMN8. 

Square End*. 



w\ 


w! 








i|*,«. 






















«,r 










1SS.6 


SOI 


;iis- 








IBO.g 


m 


4 194. 



STEEL ZBAK COLUHHS. 

SAPB LOAl>S IN TONS OP 1,0110 LBS. 
STBBI. Z-BAR OOIiUMNS. 

Square End*. 



ved slraina per square 
ttee\ -. HSfet; faclor 



Length tf( colnmn In r«l. 


mm 

^ Ml 




|?!IS 


...m,.,.n,k.r 


47:t I wis 

ST.- «> 

st'.s te'.t 


si 

4C.4 

4i.a 


»4.0 IM.B 




Si 

7B.7 

i:l 

U.1 












•JK .. 


81 
S:i 



lABIMEBS PATENT ALL STEEL COLtTMN. 389 

i^matBBm patent aza steel oolomn. 

(MiaufactxKd by Jonea & LmghllnB, Pilteburgb.) 

This cotumn was patented Jutio 3, 1891. It is made by bending 
two I-beams at ngbt angles in tha middle of the web and riveting 




LARIMItn'e PlTEl 



theiu together as in tbe illustration. The porumn is very light and 
com pact, aad has but one row of rivets. The fallowing table gives 
tbe strength of tbe eolumn. 



289a luAKIMKRS PATENT ALL STEKL COLUMN. 



s 

Hi 

o 

D 
Hi 

n 

n i I 

Hi Z 

h) f 

^* 

n I 
<i ^ 

CO •= 

< 5 
h] ^ 

of : 
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55 I 

21 

o >, 
o = 

O }- 

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O - 
H :- 

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Q •> 
< 

o 

H 
< 

CO 



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y 



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coc:= cere ■cc-so 



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71 



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i-til '>iiii.jjs 
P'"''ll\ 

MilMJl I Jl. I'T.I.I.VV 

' = "■■'11 I J'»"iS 



r- ■■« < - 51 "^ r ■- — r. x SCI - — £ — — 15 

SsTir "r;; =-rJ— S:-*?? 



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:=>..= c e = ii7 

•  I  



2* 7 — ii / i'^ 7 T« -- 7> -• j. ^ a: ^ t- n 
 ■-»-« w - M  •" ".' S ,» Z^ !£ '- ^ 

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f r -i / I - / S 1 - K is — *ft 



 lllllil •. I Jo -•/IS 



urimee'b patent all steel column. 289J 



289c THE GKAY ST££L COLUMNS. 



The Gray Column* 

The fibres on the opposite page show a perspective Tiew and 
section of a column which was patented in December, 1892, by Mr. 
J. H. Gray, C. E., and which has since been used in some promi- 
nent buildings. As may be seen from the illustratioDS, this column 
is made of angle- bars riyeted together and braced every few feet 
in height by flat iron ties, as shown in the perspective view 

The angles may be reenforced by cover-plates riveted to their 
faces, when necessary to increase the strength of the column. Any 
bridge shop can make these columns by paying a small royalty to 
the patentee. 

As angles are the cheapest shape of rolled steel that is manufact- 
ured, this should be an economical column. 

The special advantages claimed for this column are : 

1. A strong, economical section. 

2. Provides continuous pipe space from basement to rool 

3. Has four flat sides for connections. 

4. Size of column does not vary when section is iucreased or 
diminished. 

5. Does away with ** cap-plates," and joins sections of colamns 
firmly together, making a continuous column. 

Tests made in the hydraulic machine of the Keystone Bridge 
Works on 14-inch columns, 11 feet long, developed a resistance to 
crushing of from 38,000 to 40,000 pounds per square inch of section, 
and a modulus of elasticity of from 24.030,000 to 27,750.000 
pounds. 

The tables on pages 2SQe-2S9h give the safe loads of several siaea 
of square, wall, and corner columns as computed by Mr. Gray. 

By varying the thickness of angles and adding cover-plates, the 
strength of the column can be greatly increased. 

Tables of wall and corner columns, and further particulars, maj 
be obtained by addressing Mr. J. U. Gray, C. E., Chicago. 



TBB GRAY STEEL COLUMKS. 



269e 



THE GRAY STEEL COLUMNS, 



SAFE LOADS IN TONS OF 2,000 LBS. BY FORMULA 17,100 LBS. - 57 -. 

SQUARE OOIiUMNS WITHOUT OOVBR PI1ATB& 

10" COLUMN. 



No. 
Pieces. 



8 
8 
8 
8 
8 
8 
8 











2k" 


X 2i" Lb. 


i 


9.52 


44 


it 


A 


11.76 


(t 


»» 


I 


13.84 


ti 


tt 


16.00 


H 


ti 


k 


18.00 


2k" 


X 3" La. 


k 


20.00 


ti 


t» 


A 


22.24 



r. 


13 ft. 


S.16 


69.0 


8.15 


85.2 


3.13 


100.0 


3.12 


116.0 


3.11 


130.1 


3.00 


143.4 


2.98 


159.5 



16 ft. 


90 ft. 


64.1 


60.8 


80.1 


75.0 


94.2 


88.1 


108.8 


101.7 


122.2 


114.3 


134.5 


125.4 


149.3 


139.1 



89 ft. 



55.8 
68.8 
78.0 
84.2 
94.5 
102.6 
118.6 



12" COLUMN. 



8 


8" 


X 8" Ls. 


i 


11.52 


3.81 


86.1 


81.9 


77.8 


67.6 


8 


(t 


ii 


A 


14.24 


3.79 


106.3 


101.2 


96.1 


83.2 


8 


ti 


ii 


f 


16.88 


3.77 


125.9 


119.9 


118.7 


96.4 


8 


3" 


X 4" Ls. 


» 


19.84 


3.57 


149.2 


141.6 


138.7 


114.4 


8 


ii 


ii 


^ 


22.96 


3.55 


169.8 


160.9 


158.1 


129.9 


8 


8" 


X 5" Ls. 


/« 


26.48 


3.36 


194.1 


183.2 


178.5 


145.5 


8 


it 


ii 


i 


30.00 


8.34 


219.0 


207.8 


195.1 


164.8 


8 


it 


ii 


A 


33.44 


3.32 


244.6 


230.8 


217.6 


188.8 


8 


ii 


ii 


i 


36.88 


3.30 


♦J69.5 


254.2 


S:«.9 


800.7 


8 


ii 


ii 


H 


40.24 


3.28 


293.7 


276.9 


260.1 


818.8 


8 


ti 


ii 


f 


43.52 


3.26 


317.3 


299.0 


280.7 


285.0 


8 


ii 


ti 


H 


46.72 


3.24 


340.3 


320.6 


800.8 


851.5 



14'' COLUMN. 



8 


4" 


X 8" Ls. 


^ 


16.72 


4.63 


128.2 


123.5 


118.4 


105.8 


8 


ii 


ii 


f 


19.84 


4.61 


152.0 


146.1 


140.2 


185.4 


8 


4" 


X 3i" Ls. 


t 


21.36 


4.50 


163.1 


156.9 


150.2 


188.9 


8 


4" 


x4" Ls. 


X 


22.88 


4.40 


174.3 


167.2 


160.1 


142.8 


8 


ii 


ii 


26.48 


4.39 


201.7 


193.4 


185.2 


164.5 


8 


4" 


X 5" Ls. 


^ 


30.00 


4.12 


226.6 


216.7 


206.7 


181.8 


8 


(i 


ti 


i 


34.00 


4.10 


256.7 


245.3 


834.0 


205.6 


8 


4" 


X 6" Ls. 


k 


38.00 


3.93 


285.2 


272.0 


258.7 


1^.5 


8 . 




ii 


A 


42.48 


3.92 


321.7 


806.7 


891.8 


854.8 


8 


ii 


it 


f 


46.88 


3.91 


851.6 


a35.2 


318.8 


877.6 


8 


(i 


ti 


H 


51.29 


3.89 


384.4 


866.3 


84S.4 


808.3 


8 




it 


i 


55.52 


3.88 


416.0 


396.3 


876.8 


887.9 


8 


it 


tt 


\l 


59.76 


3.87 


447.6 


426.4 


405.8 


808.5 


8 




if 


i 


63.92 


3.86 


478.5 


455.9 


488.8 


876.6 



THE GllAY STEEL COLUMNS, 



289/ 



SAFE LOADS IN TONS OP 2,000 LBS. BY FORMULA 17,100 LBS. - 57 -. 

r 

SQUARE OOX.UMNS WITHOUT OOVZSR PLATBS. 

16" COLUMN. 



No. 
Pieces. 


Dimensions. 


Thick. 


Area 
Sq. In. 


r. 


12 ft. 


16 ft. 


20 ft. 


30 ft. 


8 


5" 


X 3" Ls. 


1 


22.88 


5.45 


178.4 


172.7 


166.9 


152.6 


8 


5" 


X 34" Ls. 


f 


24.40 


5.85 


190.8 


184.6 


178.3 


162.6 


8 


5" 


X 4" Ls. 


f 


25.84 


5.24 


200.7 


194.0 


187.2 


170.4 


8 


it 


(I 


iV 


30.00 


5.21 


232.8 


225.0 


217.1 


197.4 


8 


5" 


X 6" Ls. 


^ 


33.44 


5.01 


258.5 


249.4 


240.2 


217.5 


8 


ii 


(t 


* 


88.00 


5.00 


293.7 


283.4 


272.8 


246.9 


8 


It 


(t 


A 


42.44 


4.98 


338.2 


316.5 


804.9 


275.7 


8 


it 


it 


f 


46.88 


4.96 


362.1 


349.2 


336.2 


303.9 


8 


It 


t( 


H 


51.36 


4.94 


396.4 


382.3 


868.0 


832.5 


8 


ti 


tt 


i 


55.52 


4.93 


428.5 


413.1 


897.7 


359.2 


8 


ti 


tt 


H 


59.68 


4.92 


460.5 


443.9 


427.3 


385.9 



18" COLUMN. 



8 


6" 


X 8i"L8. 


f 


27.86 


6.15 


215.7 


209.6 


208.5 


188.3 


8 


6" 


X 4" Lb. 


k 


28.88 


6.07 


227.4 


220.9 


214.4 


198.1 


8 


tt 


tt 


38.44 


6.05 


268.2 


2.55.7 


248.1 


229.3 


8 


tt 


tt 


i 


38.00 


6.03 


299.0 


290.4 


281.8 


260.2 


8 


6" 


X 6" Ls. 


^n 


40.48 


5.64 


316.6 


806.8 


297.0 


272.5 


8 


tt 


tt 


i 


46.00 


5.63 


359.8 


348.6 


837.4 


309.6 


8 


tt 


tt 


^ 


51.44 


5.62 


402.5 


389.7 


8n.2 


346.9 


8 


t. 


tt 


f 


56.88 


5.60 


444.6 


480.7 


416.8 


332.1 


8 


tt 


(t 


H 


62.24 


5.59 


486.5 


471.3 


456.1 


417.9 


8 


tt 


tt 


i 


67.52 


5.57 


527.3 


511.0 


494.4 


452.9 


8 


tt 


tt 


H 


72.72 


5.55 


56S.0 


550.0 


632.0 


487.3 


8 


tk 


It 


i 


77.92 


5.54 


608.5 


589.2 


569.9 


521.9 



22" COLUMN. 



8 


8" 


X 6" Ls. 


i 


54.00 


7.30 


431.4 


421.3 


411.1 


385.8 


8 


tt 




T?B 


60.48 


7.29 


483.1 


471.8 


460.4 


4a2.0 


8 


it 




f 


66.88 


7.27 


534.2 


521.5 


508.9 


477.5 


8 


<t 




\h 


73.38 


7.26 


585.2 


571.3 


557.5 


5-23.0 


8 


tt 




f 


79.52 


7.24 


634.8 


619.8 


604.8 


507.3 


8 


tt 




H 


• 85.76 


7.23 


684.6 


638.4 


652.1 


611.6 


8 


tt 




i 


91.92 


7.22 


783.6 


716.3 


698.8 


655.3 


8 


tt 




H 


98.06 


7.21 


782.7 


764.2 


745.6 


699.0 


8 


tt 




1 


104.16 


7.20 


881.2 


812.4 


791.6 


742.2 



289iir 



THE GKAY STEEL COLUMNS. 



SAFE LOADS IN TONS OF 2,000 LBS. BY FORMULA 17,100 LBS 



-.1. 



WAIiZi COLUMNS WITHOX7T OOVBR PZJLTB& 



10" COLUMN. 



No. 
Pieces. 


• 

Dimensions. 


Thick. 


Area 
sq. in. 


r. 


12 ft. 


16 ft. 


20 ft. 


80 ft. 


6 


2i" X 2i"L8. 


i 


7.14 


2.25 


48.0 


48.7 


89.3 


26.5 


6 


it (t 


A 


8.82 


2.25 


59.3 


58.9 


48.6 


86.2 


6 


II it 


I 


10.88 


224 


69.7 


68.4 


67.1 


41.2 


6 


it It 


12.00 


2.24 


80.0 


78.8 


65.0 


47.6 


6 


It - it 


i 


13.60 


2.28 


90.6 


81.9 


74.0 


68.8 


6 


2k" X S" Ls. 


i 


15.00 


2.17 


99.9 


90.4 


81.0 


67.0 


6 


it it 


fk 


16.68 


2.16 


110.9 


100.8 


89.8 


68.8 



12" COLUMN. 



6 


3" 


X 3" Ls. 


i 


8.64 


2.71 


60.8 


66.4 


68.0 


41.8 


6 


It 


it 


A 


10.68 


2.70 


75.1 


69.7 


64.8 


60.7 


6 


it 


it 


t 


12.66 


2.69 


88.9 


82.5 


76.1 


60.9 


6 


3" 


X 4" Ls. 


i 


14.88 


2.56 


103.4 


95.4 


87.4 


07.6 


6 


it 


it 


h 


17.22 


2.55 


119.7 


110.4 


101.1 


78.0 


6 


8" 


X 5" Ls. 


S 


19.86 


2.47 


186.8 


125.8 


114.8 


87.8 


6 


tt 


it 


k 


22.50 


2.47 


155.0 


142.5 


180.0 


98.0 


6 


t( 


it 


A 


25.05 


2.46 


1^2 6 


158.6 


144.6 


100.8 


6 


It 


it 


* 


27.66 


2.46 


190.8 


174.9 


169.6 


181.1 


6 


tt 


it 


H 


30.18 


2.45 


207.4 


190.6 


178.8 


181.6 


6 


tt 


it 


i 


32 64 


2 44 


224.1 


205.8 


187.6 


141.8 


6 


tt 


ti 


H 


35.04 


2.43 


240.4 


220.7 


801.0 


161.6 



14" COLUMN. 



6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 



4" 


x8" Ls. 


^ 


12.54 


it 


it 


1 


14.88 


4" 


X sy Ls. 


f 


1(5.02 


4" 


X 4" Ls. 


k 


17.16 


ii 


ii 


19.86 


4" 


X 5" Ls. 


/e 


22.50 


it 


ii 


i 


25.5U 


4". 


X 6" Ls. 


i 


28.50 


it 


it 


A 


31.86 


tt 


it 


« 


35.16 


tt 


it 


ii 


38 47 


it 


it 


i 


41.64 


it 


tt 


H 


44.82 


tt 


ti 


i 


47.94 



3.83 
3.31 
3.25 
3.19 
3.18 
8.06 
8.05 
2.97 
2.96 
2.95 
2.95 
2.94 
2.94 
2.93 



91.8 


86.7 


81.6 


108.8 


102.7 


06.6 


116.7 


110.0 


108.8 


124.6 


iir.3 


100.9 


144.2 


135.6 


127.1 


ll;6.2 


152.1 


142.1 


183.7 


172.8 


160.9 


204.3 


191.2 


178.1 


228.2 


213.5 


198.7 


251.7 


235.4 


819.1 


275.3 


257.6 


289.7 


297.9 


278.6 


259.8 


820.7 


299.7 


878.7 


342.7 


320.4 


298.1 



74.8 

88.6. 

86.4 

01.4 

106.6 

110.0 

188.8 

146.8 

101.9 

178.8 

196.1 

810.7 

886.8 

848.0 



IH£ QBAY STEEL COLUHMS. 



SAFE LOADS IN TONS OF 8,000 LBS. BY FORMULA 17,1«» - BT -■ 
CORNER OOIiUMNS WITHODT COVER PLATBa 

11' COLUMN HEDUCKD FROM H" COLUMN, 



15" COLUMN KBDUCED E 



1 IB" COLUUH. 



1 

it; 

1 


Vi 

i;iS 


i 

f 


li i 

iflois' 1*S 
i:».4 i5« 

III 



290 BENDING-MOMENTS. 



CHAPTER Xir. 

BENDINGMOMENXa 

Tmk bonding-niomont of a beam or tnws represents the destnic- 
live energy of the load on the l)eani or truss at any point for which 
tlie ]>en(hng-nionicnt is computed. 

The moment of a force around any given axis is the product of 
the force into the pen^^ndicular distance between the line of action 
of the force and the axis, or the product of the force into its arm. 

In a I^eam the forces or loads are all vertical and the arms hori- 
zontal. 

The bending-moment at any cross-section of a beam is the alge- 
braic siun of the moments of the forces tending to turn the beam 
ai*ouud the horizontal axis passing through the ceuti'e of gravity 
of the section. 

Example. — Suppose we have a beam with one end securely 
fixed into a wall, and the other end projecting from it, as in Fig. I. 

]jet us now 8upix>se wc liave a 
weight, which, if placed at tlie end 
of the beam, will cause it to break 
at the point of support. 
/^^v, ^ Then, if we were to place the 

^\^ ^^^ weight on the Ix^am at a point 

^^v, ^\^ ^-x near the wall, the beam would 
^>.^ ^^V/ support the weight easily; but, as 
^\^ we move the weight towards the 
outer end of the beam, the beam 
bends more and more; and, wh<»n 
^' the weight is at the end, the beam 

breaks, as shown by the dotted lines. Fig. 1. 

Now, it is evident that the destructive eneigy of the weight la 
greater, the farther tlie weight is removed from tlie wall-end of the 
beam, thouixh the weight itself remains the same all the time. 
Tlie reason for this is, that the moment of the weight tends to 
turn the beam alwut the point A, and thus producer a pull on the 
ui>i)er fibres of the beam, and compresses the lower fibres. As the 
weight is moved out on the beam, its moment becomes greater, and 
hence also the pull and compression on the fibres; and, when tlie 







^rm 



^^ 




BENDING-MOMENTS. 



291 



moment of the weight produces a greater tension or compression 
on the fibres tlian they are capable of resisting, they fail, and the 
beam breaks. Before the fibres break, however, they commence to 
striitch, and this allows the beam to bend: hence the name "Ixmd- 
ing^nionient" h«s been given to the moment which causes a beam 
to bend, and perhaps idtimately to break. 

There may, of course, be several loads on a beam, and each one 
having a d liferent monvent, tending to bend tlie beam; and it may 
ilso occur that some of the weights may tend to turn the beam in 
different directions: the algebraic sum of their moments (calling 
those tending to turn the beam to the right +, and the others — ) 
would be the bending-moment of the beam. 

Knowing the bending-moment of a beam, we have only to find 
the section of the beam that is capable of resisting it, as is shown 
in the general theory of beams. Chap. XIV. 

To determine the bending-moments of beams mathematically, 
requires considerable training in mechanics and niathematics; but, 
as most beams may be placed under son\e one of the following 
cases, we shall give the bending-moment for these cases, and then 
show how the bending-moment for any other methods of loading 
may be easily obtained by a scale diagram. 



Examples of Bencliugr-Momeuts. 

Case I. 

Beam fixed at one end^ and loaded 
with concentrated load W. 

Bending-moment = W X L. {L 
may, or may not, be the whole length 
of the beam, according to where the 
weight is located. ) 



Case II. 

Beam fixod at one end, loaded with ^^^ 
u dt-'itribntt'd load \V. ^-'^ 




Bending moment = W x - • 



Note. — The length L mast always* he taken 
In the same unit of measurement «>« is listed for 
the breadth and depth : thus, if B and D are in 
inches, L must be in inches. 




292 



BBNDING-MOMENTS. 



Case III. 

Jionm fixpd at one end, loaded with both a concentrated and a 

distributed load. 




/., 



Bending-moment = P X Lj + JK x -^ 



Casr IV. 

licam supported at both ends, loaded with concentrated load lU 
centre, 

W 

J Bcnding-raoment 




Case V. 
Beam s^iipported at both ends, loaded with a distributed load W. 



V -, - ■■: 




'm% 



<?; 



'n 



Fig.6 




Bending-moint;nt 



Cask VI. 

livam supported at both tnids, loaded with concentrated load nol 

at ('('litre 



Bending-moment 




= Wx 



m X n 



BKNDING-MOMENTS. 



293 



Cask VTI. 

Beam supported at both encU, loaded rcith two equal concen- 
trated loadSy equally distant from the centre. 



Bending-moment 
= W X nu 




m-^ 



v^m 



<rm 



Flg.8 



iiW 




From these examples it will be seen that all the quantities which 
enter into the bending-moment aro the W?ight, the span, and the 
distance of point of application of concenti-ated load from each 
end. 

The hendin{i-moment for any case other than the above may 
easily be obtained by the graphic method, which will now b« 
explained. 



Graphic Method of Determining Benclin^- 

Moments. 



The bending-moment of a l)eam supported at both ends, and 
loader! with one concentrated load, may be shown graphically, as 
follows : — 

Let W be the weight applied, as shown. Then, by rule under 
Case VI., thebeuding- 
niomeut directly under [<« f^ J^ jjj 



W = IF X 



in X n 





Draw the beam, with ^ 
the given span, accu- 
rately to scale, and 
then measure down 
the line AB equal 
to the bending - mo- 
iiuMit. Connect B 
with each end of the beam. If, then, we wished to find the bending- 
moment at any other point of the beam, as at o, draw the vertical 
line y to BC ; and its length, measured to the same scale as ABf 
will give the bending-moment at o. 

Beam with two concentrated loads. 

To draw tlie bending-moment for a beam with two concentrated 
loads, first draw the dotted Hues ABl) and ACD, giving the outline 



294 BENDINO-MOMENTS. 

of Ihe bending-nioment for each loful separately; KB heing eqiwl 

toWx It^^ „„(, rr pqjial to P X '-^ 




Fi9.IO 



Now, Ihe beiuting-momnnt at Uie pclnt E equals RJi, doe to tha 
loud ir, anil Kb, clue to tlie load P: liunce the l)Uiiilii^(-nioiuent at 
i'slioHld lie drawn ainul to En+ Kh — Kll, ; anil at Fthe beiMl- 
Iiig iiioiiienl shoHkt equal H,'+ Fc= FC,. The otOUne lor the 
bendlng-niomunt due to both loaits, tlieu, would be the Uiie 
AIl^C'iD, anil the greatest bending-iuoment would In this parUe- 
ular tasu be FC'i- 

Jleam with three concentrated load*. 



Fiy.tl 

Pmcpficl as in the laat ease, and drawthp hending-moment for 
eaoli load separately. Then make AD = A\ + A2 + AS, BB = 
m + m + /J:t, and i:F= (I + r2 + f73. The line IIDEFI urill 
then Ih- the c)ut1hie for the Ix'iiding-ii'ouiunl due to all tl)e wt^lghts. 
The iH>iidln(>-nioiuent for a lieani loadeil tvith nnj- number of «w 
<«ntiiited weiglita uiav be drawn In tlie oanie way. 



BENDING -HOHENT& 
Beam with untformly distrVnUed load. 




Draw the beam with the given spaii. accurately to a scale, m 
before, and at the middle of the beam draw the vertical line AH 
I 

equal to If x gi W representing the whole distributed load. 

Then connect the points C, it, D by a parabola, and It will ^ve 
the outline of the bendlng-moraents. if, now, we wanteil the 
bendlng-Qiomeat at the point a, we have only to draw the vertical 
line ab, and measure It to th<! same scale as ^ B, and it will be the 
moment deatred. Hethoda for drawing the parabola may be found 
in " Geometrical Problems," Part I. 

Beam loaded viith both diatritruted and concenlrated loads. 

To determine the bendlng-momcnt in this case, we have only to 
combine the methods for concentrated loads and for the distributed 
load, as shown in „ 

the accompanying 
figure. The bend- 
ing-moment at any 
point on the beam 
will then be lim- 
ited by the line 
ABC on top, and 
CHEFA on the 
bottom ; and the 
gii'atesi bendfiig- 
moraent will be 
the longest verti- 
cal tine that can 

be drawn between Ha.ia 

these two bounding lines. 

For example, the tiending-momeiit at X would be BE. The posi- 
tion of the greatest ben ding-moment will depend upon the position. 
of the concentrated loads, and it may aud may not occur at tlie 



296 



BENDING-MOMENTS. 



Example. —What is the greatest bending-ittmneilt In a hektk of 
20 feet span, loaded with a distributed load of 800 pounds and a 
concentrated load of 500 pounds 6 feet from one end, and a con- 
centrated load of 600 pounds 7 feet from the other end ? 

L 
Ans. 1st, The moment due to the distributed load is W X ^* 

800 X 20 
or y = 

2000 pounds. We 
therefore lay off 
to a scale, say 
4000 pounds to 
the inch, Bl = 
2000 pounds, and 
draw a parabola 
between the 
points Af B, and 
C. 

2d, The bend- 
ing-moment fbr 




the concentrated load of 500 pounds is 



50 X 6 X 14 
20 



, or 2100 pounds. 



Hence we draw E2 = 2100 pounds, to the same scale as Bly and 

then draw the lines AE and CE, 

3d, The bending-monient for the concentrated load of 600 pounds 

600 X 7 X 13 

— , or 2730 pounds; and we draw i)8 = 2780 pounds, 



IS 



20 



and connect D with A and C. 

4th, Make EII = 2 — 4, and DG = 3 — 5, and connect O and H 
with C and A and with each other. 

The greatest bending-nioment will be represented by the longest 
vertical line which can be drawn between the parabola ABC and 
tlu* broken line AHGC. In this example we find the longest veitl- 
cal line which can be drawn is xy ; and by scaling it we find the 
greatest bending-nionient to be 5550 pounds, applied 10 feet 11 
inches from the point A. 

In this case, the position of the line Xy was determined by 
drawing the line TT\ parallel to IIG, and tangent to ABC, The 
line Xy is drawn through the point of tangency. 



Note. — As the measurements ased for determining the bending-momeiit \ 
in feet, we must multiply the moment by 12. to get it into inch poands; otfaar- 
wise, in working out the dimenaione of the beam, they would be in feot Inntfiad 
of inches. 



MOMENTS OF INERTIA AND RESISTANCE. 297 



CHAPTER Xm. 

MOMENTS OF INERTIA AND RESISTANCE, AND 

RADIUS OF GYRATION. 

Moment of Inertia. 

The strength of sections to resist sti-ains, either as girders or as 
posts, depends not only on the area, but also on the form of the 
cross-section. The property of the section which represents the 
effect of the form upon the strength of a beam or post is its mo- 
ment of inertia, usually denoted by I. The moment of inertia for 
any cross-section is the sum of the products obtained by multiply- 
ing the area of each particle in the cross-section by the square of 
its distance from the neutral axis. 

Note. — The ueutral axis of a beam is the line on which there is neither 
tension nor compression; and, for wooden or wronght-iron beams or posts, it 
may, for all practical purposes, be considered as passing through the centre of 
gravity of the cross-section. 

For most forms of cross-section the moment of inertia is best 
found by the aid of the calculus; though it may be obtained by 
dividing the figure into squares or triangles, and multiplying their 
areas by the squares of the distance of their centres of gravity 
from the neutral axis. 

Moment op Resistance. 

The resistance of a beam to bending and cross^breaking at any 
given* cross-section is the moment of the two equal and opposite 
forces, consisting of the thrust along the longitudinally compressed 
layers, and the tension along the longitudinally stretched layers. 

This moment, called "the moment of resistance," is, for any 
given cross-section of a beam, equal to 

• moment of inertia 

extreme distance from axis* 

In the general formula for strength of columns, given on p. 281, 
the effect of the form of the column is expressed by the square 
of the radius of gfyration, which is the moment of inertia of 

the sectiou divided by its area; or -r = r^. The moments of 
inertia of the principal elementary sections, and a few common 



206 



MOMENTS OF INERTIA AND RESISTANCE. 



forms, are given below, which will enable the moment about any 
given neutral axis for any other section to be readily calculated 
by merely adding together the moments about the given axis of 
the elementary sections of which it is composed. 

In the case of hollow or re-entering sections, the moment of the 
hollow portion is to be subtracted from that of the enclosing area. 

Moments of Inertia and Resistance, and Radii of 

Gyration. 

I = Moment of inertia. 

R — Moment of resistance. 

G = Radius of gyration. • 

A = Area of the section. 

Position of neutral axis represented by broken line. 



 

1 
1 


. --1 — rf 


w~— 




• 

i 

ui — 





Y-h--* 




I 


bcP 
■" 12* 


R 


b(P 
= 6' 


& 


-12' 


I 


6(i» 
= 3' 




<P 



3 





1 

« — 


i- 


— » 






z 


1 

1 
1 

1 

? 

 


» 






1 




1 



T 



I 



E 



6 — 



\ r 



/ = 



h(p - bii^ 



12 



i 21 

I— ^ = ;/ ' 
2X (^ = 



bd - b,d. 



I-Ream (another fonnula). 

Let a denote area of one flange, 
a' denotes area of w(»b, 
cT = effective depth between centres of gravity of flanges; 



then 



-v'+6;2 



This is the formula generally used by the engineers for the iioiir 
companies. 



MOMENTS OF INERTIA AND RESISTANCE. 



299 



y-h'i 



Ie--6~-^ 



T- 




1 — 


T 

1 


1 




fy 


Ik 






_^ 




J._- 


,. . 1 



^S 10 



L_li 1 



n. 



ihi 





». — h- 




<b 



•t 



■t 



J 



!■ 



h 



O- 




I 


6# 
"■ 3 " 


-M. 


<!2 

4 


~ 12' 


Gf2 


I 








/ 


6d« 
= 36' 








It 


3/ 


24' 






G^ 


i 


d^ 
18 






I 


6d« 
-12- 








G^ 


= 6' 








I 


6d« 
= 4' 








C2 


~ 2* 








/ 


_ bd^ 4- 6,(Z,« 


j__ 


{b,-b)dj 






3 




/? 


J 








G2 


/ 
" A' 








I 


= 0.7854)-*. 






R 


= 0.7854r3. 







(?2= -r. 



7 = 0.7854 ()•*- 
i? 

G2 



= 0.7854 U'S-^J 



r 



1 r* - r* 



BOO TABLES OF INERTIA AND GYRATION. 



Moments of Inertia and Radii of Gyration of 
Mercliant Sliapes of Iron and Steel. 

For the sections of rolled iron beams and bars to be found in the 
tnarket, the moments of inertia are given in the '* Book of Sections " 
published by the manufacturers. The following tables give the 
moments of inertia and radii of gyration for the principal sections 
manufactured by ( amegie, Phipps & Co., the New Jersey Steel and 
Iron Company, and the Phoenix Iron Company (revised to October 
1, 1891). The Pencoyd Iron Works have recently made changes in 
a number of their sections, and some of the old seotioDS of iron 
beams and channels have been abandoned, and they are not at 
present prepared to furnish the revised data. 

The tables give the least weight for each section of iron beam, 
and the minimum and maximum weights for channels, deck beams, 
and angle irons. These shapes can be rolled for any weight 
between the two given, while the weight of the beams can also be 
greatly increased. With the quantities given in these tables, one 
can find all the data required in usual calculations. 

The tables on pages 322-24 will be found very oonTenie&t in 
computing the strength of struts formed of two or four angle bart. 



TABLES OF INERTIA AND GYRATION. 



301 



MOMENTS OF INERTIA AND RADII OF GYRATION 
OF CARNEGIE BEAMS— IRON. 



V 



u 



IB 



A 



\1 



-Oi 







I. 


n. 

• 


III. 


IV. 


.V. 


Size, in 


Weight 

per lw)t, 

in lbs. 


Area of 

cross- 

bection, 


Moments 


of inertia. 


Radii of j 


gyration. 


inches. 






 






in sq. in 
24.0 


Axis A B. 


Axis C D. 


Axis A B. 


Axis CD. 


15 


80 


813.7 


38.8 


5.82 


1.27 


15 


60 


18.0 


625.5 


23.0 


5.90 


1.13 


15 


50 


15.0 


522.6 


15.5 


5.90 


1.02 


12 


56.5 


17.0 


348.5 


17.4 


4.53 


1 01 


12 


42 


12.6 


274.8 


11.0 


4.67 


0.94 


10^ 


40 


12.0 


201.7 


12.0 


4.10 


1.00 


m 


31.5 


1^.5 


165.0 


8.01 


4.17 


0.92 


10 


42 


12.6 


198.8 


13.74 


3.97 


1.04 


10 


36 


10.8 


170.6 


10.02 


3.97 


0.96 


10 


30 


9.0 


145.8 


7.43 


4.03 


0.91 


9 


38.5 


11.6 


150.1 


12.84 


3.61 


1.05 


9 


28.5 


8.6 


110.3 


6.79 


3.59 


0.89 


9 


2J.5 


7.1 


92.3 


4.64 


3.62 


0.81 


8 


34 


10.2 


102.0 


10.2 


3.16 


0.99 


8 


27 


8.1 


82.5 


6.30 


3.19 


88 


8 


21.5 


6.5 


66.2 


3.95 


3.20 


0.78 


7 


22 


6.6 


51.9 


4.58 


2.80 


0.83 


7 


18 


5.4 


44.2 


3.28 


2.86 


0.78 


6 


16 


4.8 


29.0 


2.87 


2.46 


0.77 


6 


13.5 


4.1 


24 4 


2.00 


2.46 


70 


5 


12 


3.6 


14.4 


1.46 


2.00 


0.64 


5 


10 


3.0 


12.5 


1.15 


2.04 


0.62 


4 


7 


2.1 


5.7 


0.67 


1.65 


0.57 


4 


6 


1.8 


4.6 


0.36 


1.61 


0.45 


3 


9 


2.7 


3 5 


0.85 


1.15 


0.56 


3 


5.5 


1.7 


2.5 


0.44 


1.24 


0.52 



80a 



MOMENTS OF INERTIA 



MOMENTS OF INERTIA AND RADII OF aYRATION 
OF CARNEGIE BEAMS-STEEL. 






U 



71 



~~i — J 

IB 



Size, in 
inches. 



24 

20 

20 

15 

15 

15 

15 

12 

12 

10 

10 

9 

9 

8 

8 

7 

7 

6 

6 

5 

5 

4 

4 



1 



n 



m. 



Weight 

per foot, 

in lbs. 



80 
80 
64 
75 
60 
50 
41 
40 
32 
83 

25.5 
27 
21 
22 
18 
20 
15.5 
16 
13 
13 
10 
10 
7.5 



Area of Momcntfl of inertia 

cross- 
sec lion, 
ill sq. in. 



23.2 
23.5 

18.8 
22.1 
17.6 
14.7 
12.0 
11 

9 

9 



7, 



7 
4 
7 
5 
9 




Rsdii of gyradon. 



2,059.3 
1.449.2 
1,146.0 
75r 7 
644.0 
529 . 7 
424.1 
281.3 
222.3 
161.3 
12:^.7 
110.6 



6.2 


84.3 


6.5 


71 9 


5.3 


57.8 


5.9 


49.7 


4.6 


38.6 


4.7 


28.6 


3.8 


23.5 


8.8 


15.7 


3.0 


12.4 


2.9 


7.7 


2.2 


5.9 



sis CD. 


Axis A B. 


41.6 


9.42 


45.6 


7.86 


27.8 


7.80 


40 1 


5.86 


80.4 


6.04 


21.0 


6.00 


14.0 


6.94 


16.8 


4.90 


10.8 


4.85 


11.8 


4.08 


7.32 


4.06 


9.10 


8.72 


5.56 


8.70 


6.62 


8.38 


4 35 


8.80 


5.52 


2.91 


8.47 


2.91 


3.24 


2.47 


2.27 


2.48 


1.99 


2.08 


1.29 


2.06 


1.22 


1.62 


0.75 


1.68 



Axis CD. 

1.34 
1.89 
1.20 
1.85 
1.82 
1.20 
1.08 
1.20 
1.04 
1.10 
0.99 
1.07 
0.96 
1.01 
0.91 
0.97 
0.87 
0.83 
0.77 
0.72 
0.6(S 
0.66 
0.58 



AND RADII OF GYRATION. 



803 



MOMENTS OF INERTIA AND RADII OP GYRATION OF 
CARNEGIE DECK BEAMS-IRON. 



[I 



_J : /^ D 



d-*\^ 







I. 


II. 


in 


IV. 


V. 


Size, in 


Weight 

per foot, 

in lbs. 


Area of 
cross- 
section, 
in sq. in. 


Moments of inertia. 


Radii of 


gyration. 


inches. 


Axis A B. 


Axis C D. 


Axis A B. 


Axis CD. 


10 


26.9 


8.1 


118.4 


6.12 


3.83 


- 

0.87 


10 


85 2 


10.6 


139.9 


7.41 


8.64 


0.84 


9 


28.2 


7.0 


77.6 


2.45 


3.34 


0.59 


9 


29.8 


8.9 


01.0 


3.15 


3.19 


0.59 


8 


21.4 


6.4 


52.1 


2.23 


2.85 


0.59 


8 


28.0 


8.4 


63.2 


2.96 


2.74 


0.59 


7 


17.0 


54 


34.4 


1.81 


2.60 


0.59 


7 


22.8 


6.9 


41.8 


2 34 


2.47 


0.58 



Deck Beams— Steel. 



9 


26 


7.6 


85.2 


4.61 


3.35 


0.76 


9 


30 


8.8 


93.2 


5.18 


3.25 


0.75 


8 


20 


5.9 


57.3 


4.45 


3.12 


0.82 


8 


23.8 


7.0 


63.5 


5.21 


8.01 


0.82 


7 • 


20 


5.9 


42.2 


4.50 


2.67 


0.82 


7 


. 23.5 


6.9 


46.6 


4.87 


2.60 


0.82 



304 



MOMKNTS OF INEBTIA 



MOMENTS OF INERTIA AND RADII OF GYRATION OF 
CARNEGIE OHANNEL-BARS—IRON. 



n 



IB 



^ 







I. 


n. 

Moments 


IV. 


VI. 








R»dii of 


Distance of 


Siz<', in 
inches. 


Weight per 
foot, in 11)8. 


Area of 

cross-section, 

in sq. in. 


of inertia. 


gyraticm. 


centre «'f 

gravity fhxn 

oatdde of 












Axis A B. 
473.1 


AxIr a B. 


web. 


15 


60 


18 


5.12 


0.88 


15 


40 


li 


360.6 


5.48 


0.82 


12 


50 


15 


247.3 


4.10 


0.88 


12 


30 


9 


17.]. 7 


4.40 


0.76 


12 


20 


6 


120.2 


4 48 


0.70 


10 


35 


10.5 


126.3 


8.47 


0.75 


10 


20 


6.0 


88.8 


8.85 


0.70 


10 


16 


4.8 


62.8 


8 62 


0.55 


9 


30 


9.0 


87.8 


8.12 


0.73 


9 


18 


5.4 


63.5 


8.48 


0.67 


8 


28 


8.4 


63.9 


2.76 


0.78 


8 


20 


6.0 


45.5 


2.75 


0.69 


8 


16 


4.8 


39.1 


2.85 


0.57 


8 


10 


3.0 


28. :J 


8.07 


0.50 


7 


20 


6.0 


37.7 . 


2.51 


0.67 


7 


18i 


4.0 


25.5 


2.51 


0.53 


7 


8^ 


2.5 


19.0 


2.73 


0.49 


6 


16 


4.8 


?2.3 


2.16 


0.08 


6 


10 


3.0 


16.9 


2.R8 


0.62 


6 


7i 


2.2 


12 2 


a 84 


0.48 


5 


14 


4.2 


13.10 


1.77 


0.61 


5 


8^ 


2.5 


8.72 


1.85 


0.49 


4 


9 


2.7 


5.75 


1.46 


0.56 


4 


5 


1.5 


3.69 


1 57 


0.45 


3i 


8.1 


2.4 


3.82 


1.25 


0.52 


3 


6 


1.8 


2.23 


1.15 


0.51 



AND RADII OF GYRATION. 



305 



FOMENTS OF INERTIA AND RADII OF GYRATION OF 
CARNEGIE CHANNEL-BARS— STEEL. 



;b 







I. 


II. 


IV. 


VI. 








Moments 


Radii of 


Distance of 


Size, in 


Weight per 
foot, in lbs. 


Area of 

cross-section, 

in bq. iu. 


of inertia. 


gyration. 


centre of 

Lravily from 

oatside of 


incbes. 












Axis A B. 


Axis A B. 


web. 


15 


82 


9.4 


284.5 


5.53 


0.75 


15 


51 


15.0 


390.0 


5.13 


0.77 


12 


20 


5.9 


117.9 


4.49 


0.62 


12 


80i 


8.9 


153.9 


4.17 


0.62 


iio 


15i 


4.5 


63.8 


3.80 


0.63 


10 


23i 
12| 
20i 


7.0 


84.6 


3.50 


0.61 


9 


8.7 


43.3 


3.42 


0.58 


9 


6 


58.5 


3.14 


0.56 


8 


lOi 


3.0 


28.2 


3.05 


0.53 


8 


17i 


5.0 


38.9 


2.78 


0.52 


7 


Sk 


2.5 


17.4 


2.67 


0^49 


7 


m 


4.3 


24 6 


2.42 


0.48 


6 


7 


2.1 


11.1 


2.31 


0.48 


6 


12 


8.6 


15.6 


2.09 


0.47 


5 


6 


1.7 


6.5 


1.94 


48 


5 


lOi 


3.0 


9.1 


1.75 


0.47 


4 


5 


1.4 


3.5 


1.57 


0.48 


4 


Si 


2.4 


4.8 


1.81 


0.48 



Deck Beams — Steel. 



9 


26 


7.6 


85.2 


3.85 




9 


30 


8.8 


93.2 


8.25 




8 


20 


5.9 


57.3 


3.12 




8 


28.8 


7.0 


63.5 


3 01 




7 


20 


5.9 


43.2 


2.67 




7 


28.5 


6.9 


46.6 


2.60 





306 



MOMENTS OF INERTIA 



MOMENTS OP INERTIA AND RADII OP GYRATION OF 

CARNEGIE ANGLE-BARS. 

For minimum and maximum thickneeses and weight. 

ANGLES WITH EQUAL LEGS — IRON OR STEEL. 




Weights in Table are for Iron; for Steely add 2 per cent. 







I. 


VI. 


n. 


IV. 


V. 






Distance 








Sizi*. in 
inches . 


Weight, 
per foot. 


Area of 

crosp- 

pection. 


of centre 
of gravity 
from out- 
Hide of 


Moments 
of inertia. 


Raclli of gyntioii. 












in sQ. in. 


flange, 
in inc.lietit. 


Axis A B. 

17.68 


Axis A B. 


AxIbOD. 


6 xG 


J16.0 


5.06 


1.66 


1.87 


1.19 


(33.1 


9.95 


1.85 


34.09 


1.85 


1.17 


5 x5 


J12.0 


3.61 


1.39 


8.74 


1.56 


0.99 


127.0 


8.28 


1.61 


20.00 


1.56 


1.00 


4 x4 


j 9.5 
120.1 


2.86 


1.14 


4.36 


1.28 


0.79 


6.03 


1.33 


9.00 


1.22 


0.88 


3ix3^ 


j 8.3 
(17.4 


2.48 


l.Ol 


2.87 


1.07 


0.68 


5.22 


1.20 


5.90 


1.06 


0.72 


3 x3 


4.8 


1.44 


0.84 


1.24 


0.98 


0.68 


^11.7 


3.50 


1.01 


3.00 


0.93 


0.62 


2^x2^ 


j 4.4 
) 9.0 


1.31 


0.78 


0.98 


0.86 


0.64 


2.69 


0.95 


2.22 


0.91 


0.06 


2i X 2A 


\ 4.0 


1.19 


0.72 


0.70 


0.77 


0.50 


\ 7.9 


2.37 


83 


1.44 


0.78 


0.60 


2i X 21 


j 3.5 
) 7.0 


1.06 
2.11 


0.66 
0.78 


0.51 
1.04 


0.69 
0.70 


0.46 
0.49 


2 x2 


\ 2 4 


(».71 


0.57 


28 


0.62 


040 


'( 5.5 


1.65 


0.60 


0.06 


0.68 


0.64 


1^x1! 


j 2.1 


0.6i 


. 51 


0.18 


0.54 


0.22 


4 9 


1.47 


0.64 


0.44 


0.56 


0.40 


l^xli 


1.8 


0.53 


0.44 


0.11 


0.46 


0.29 


\ 3.6 


1 06 


. r,4 


0.24 


48 


0.88 


li X \{ 


j 1.0 


0.80 


35 


0.044 


0.38 


0.22 


1.9 


0.56 


0.40 


0.077 


0.3V 


0.24 


HxU 


S 0.0 
/ 1.9 


0.27 


0.32 


0.032 


0.84 


0.19 


0.55 


0.40 


0.077 


0.37 


0.25- 


1 xl 


j 0.8 
\ 1.5 


0.23 


0.30 


0.022 


0.81 


0.21 


0.44 


34 


0.037 


0.29 


0.18 


i x} 


J 0.6 

( 0.8 


0.17 


0.23 


0.009 


0.28 


0.14 


0.25 


: 0.26 


0.012 


0.22 


0.16 



AND RADII OP GYKATION. 



307 



MOMENTS OP INERTIA AND RADII OF GYRATION OP 

CARNEGIE ANGLE- BARS. 

Forminimam and maximnm thicknesses and weight. 

UNEVEN LEGS — IRON OR STEEL, 




Wei 


ghts in 


TcUde are for 


Iron; . 


for Steel, add 2 per cent 


 






I. 


II. 

Mom€ 
inei 


III. 

mts of 
rtia. 


IV. 


V. 




VI. 


VI. 


Size, in 


Weight, 
per 
foot. 


Area of 
c roes- 
section, 


Radii of gyration. 


Distance from 

hate to 
neutral axis. 


inches. 










1 








int'q.in. 


Axis 


Axis 


Axis 


Axis 


Axis 


d. 


/. 








AB. 


CD. 


AB 


CD. 
1.17 


EP. 


6 x4 


J12.0 


3.61 


13.47 


4.90 


1.93 


.88 


1.94 


0.94 


(27.3 


8.18 


29.58 


10.68 


1.90 


1.14 


.88 


2.15 


1.16 


6 x8i 


jll.4 


3.42 


12.86 


3.34 


1.94 


0.99 


.77 


2.04 


0.79 


^25.8 


7.75 


28.20 


7.25 


1.91 


97 


.78 


2.25 


1.00 


5 x4 


jlO.8 


3.23 


8.14 


4.67 


1.59 


1.20 


.86 


1.53 


1.03 


(22.8 


6.83 


16.75 


9.57 


1.57 


1.19 


.88 


1 72 


1.22 


5 x3i 


jlO.2 


3. Oh 


7.78 


3.18 


1.60 


1.02 


.76 


1.61 


86 


I2I.4 


6.42 


15.99 


6.52 


1.58 


l.Ol 


.77 


1.80 


1.05 


5 x3 


j 9.5 


2.86 


► 7.37 


2.04 


1.61 


0.85 


.66 


1.70 


0.70 


(20.1 


6.02 


15.19 


4.18 


1.59 


0.83 


.66. 


1.89 


0.89 


4ix3 


j 8.9 


2.67 


5.50 


1.98 


1.44 


0.86 


.66 


1.49 


0.74 


(18.7 


5.62 


11.26 


4.06 


1.42 


0.85 


.67 


1.08 


0.98 


4 x3i 


j 8.9 


2 67 


4.18 


2 99 


1.25 


1.06 


.73 


1.21 


0.96 


(18.7 


5.61 


8 53 


6.10 


1.23 


1.04 


.74 


1.39 


1.14 


4 x3 


j 7.0 


2.09 


3.38 


1.G5 


1.27 


0.89 


.65 


1.26 


0.76 


(17.4 


5.21 


8.09 


3.92 


1.25' 


0.87 


.66 


1.47 


0.97 


3ix3 


i 6.5 


1.93 


2.33 


1.58 


1.10 


0.90 


.63 


1.06 0.81 


(16.0 


4.80 


5.54 


3.76 


1.07 


0.89 


.65 


1.27 1.02 


3ix2i 


( 4.8 


1.44 


1.80 


0.78 


1.12 


0.74 


.55 


1.11 


0.61 


1 9.8 


2.92 


4.0s 


1.81 


1.17 


0.78 


.58 1.27^ 0.77 


3ix2 


i 4.2 


1.25 


1.36 


0.40 


1.04 


0.57, 


.44! 1.09 0.48 


) 8,3 


2.48 


2.70 


0.81 


1.04 


0.57 


.45i 1.22 0.59 


3 x2i 


i 4.4 


1.31 


1.17 


0.74 


O.Ooi 


0.75; 


.53 0.91 


66 


} 8.7 


2.60 


2.34 


1.49 


0.951 


0.70 


.54 


1 03 


0.78 


3 x2 


j 4.0 


1.19 


1.09 


0.39 


0.90' 


0.57 


.44 0.99 


0.49 


( 8.0 


2.31 


2 27 


0.84 


0.1)9 


0.60 


.47 1.12 0.63 


2ix2 


j 2.7 


81 


51 


0.29 


0.79, 


('.60 


.43 


76 51 


( 7.2 


2.18 


1.38 


0.80 


80; 


0.61 


.44 


0.87 


0.67 


2 xll 


j 2.6 
( 4.6 


0.78 


0.37 


0.12 


0.63 


0.39 


.30 


0.69 


0.37 


1.^9 


0.56 


0.*^2 


0.63 


0.40 


.31 


0.79 


0.47 


l}xl 


0.9 


0.28 


05 


0.02 


0.44 


0.29 


.22 


0.44 


0.26 



308 



MOMENTS OF INEBTIA 



MOMENTS OF INERTIA AND RADII OF GYRATION 
OF CARNEGIE T-BARS— IRON OR STEEL. 

c 




8 



Weights in Table are for Iron ; for Steely add 2 per cent. 




ni. 



Moments of 
inertia. 



Azi» : Axis 
A B. CD. 



5 x3 

5 x2i 
4ix3i 
4 x5 
x5 

x4i 
x4" 
x3 
x2i 
x2 
3.i X 4 
3iLx4 
3ix3* 

^x^ 

iJixS 
Ux% 
3x4 
x3i 
x3 
x3 
x2i 
x2i 
2Ax3 
2i X 2i 
2ix2i 
2 x2 

2 xH 
If'x 1} 

Uxli 

1 xl 



5.5 
4.9 
3.7 

2 8 

2.1 

2.8 

2.1 

2.5 

2.1 

1.8 

1 8 

1.89 

1.42 

1.89 

1.42 

1.88 

1.18 

1.21 

1.20 

1.20 

0.75 

0.89 

0.':5 

0.44 

0.44 

0.2o 

0.18 

0.18 

VI 

0.08 

0.02 



IV. 



^ I 



V. 



RadUof 
gyratifin. 



AzIh 
AB. 



0.76 
0.64 
1.04 
1.54 
1.56 
1.87 
1.88 
1.20 
0.86 
0.70 
0.51 
1.21 
1.22 
1.04 
1.05 
0.87 
0.89 
1.23 
1.06 
0.88 
0.90 
0.72 
0.7;? 
0.94 
0.74 
0.67 
60 
0.42 
0.51 
0.49 
0.29 



Axis 
CD. 



1.21 
1.26 
90 
0.79 
0.78 
0.81 
0.80 
0.88 
0.88 
0.91 
96 
0.72 
0.70 
0.74 
0.78 
0.77 
0.76 
0.59 
0.62 
0.64 
62 
66 
0.65 
0.51 
0.52 
0.47 
0.42 
0.45 
0.37 
0.84 
0.21 



VL 



Distance 

f/from 

ba.<te to 

neatnl 

azia. 



0.67 
0.87 
t.ll 
1.06 
1.61 
1.87 
1.81 
1.15 
0.78 
0.00 
0.51 
1.25 
1.19 
1.06 
1.01 
0.88 
0.78 
1.88 
1.18 
0.98 
0.86 
0.71 
0.68 
0.92 
0.74 
0.66 
0.60 
0.42 
0.64 
0.42 
0. 



AND BAUII OF GYRATION. 



Weighh in Tabh are for Iron ; for Steel, add 2 per eent. 



310 



MOMENTS OF INERTIA. 



MOMENTS OF INERTIA AND RADII OP GYRATION OP 

TRENTON BEAMS— IRON. 



7 



I* 



.JL. 



B 



\J 





Weight 
per foot, 

in IViM 


I. 

Area of 


n. 


m. 


IV. 


V. 


Size, in 
IncheH. 


Moments ( 


>f inertia. 


Radii of gyration. 




IIL J vO • 


section, 
in sq. in. 






















90.6 


27.20 


Axis A B. 


Axis C I). 


Axis A B. 


AxiaCD. 


20 


1,650.3 


46.50 


7.79 


1.30 


20 


66.6 


19.97 


1,238.0 


26.62 


7 88 


1.15 


15 


66.6 


20.02 


707.1 


27.46 


5.94 


1.17 


15 


50 


15.04 


523.5 


15 29 


5.90 


1.01 


15 


41.6 


12.36 


434.5 


11.64 


5.98 


1.02 


12i 


56.6 


16.77 


391.2 


25.41 


4.88 


1.28 


12i 


41.6 


12.33 


288.0 


11.54 


4.80 


.Vt 


13 


40 


11.73 


281.3 


16.76 


4.90 


1.20 


12 


32 


9.46 


2-29.2 


11.66 


4.92 


1.11 


m 


45 


13.36 


23:J . 7 


15.80 


4.18 


1.10 


loi 


;J5 


10.44 


185.6 


9.43 


4.22 


.96 


lOA 


30 


8.90 


164.0 


8.09 


4.29 


.95 


9' 


41.6 


12.33 


150.8 


11.28 


3.47 


.95 


9 


28.3 


8.50 


111.9 


7.35 


8.63 


.98 


9 


23. :J 


7.00 


93.9 


4.92 


8.66 


.84 


8 


26.0 


H.03 


83.9 


7.55 


3.28 


. vO 


8 


21.6 


6.37 


67.4 


4.55 


3.24 


.85 


7 


18.3 


5.50 


44.3 


3.90 


2.84 


.84 


6 


40 


11.84 


64.9 


18.59 


2.86 


1.25 


6 


30 


8.70 


49.8 ' 


10.78 


2.39 


1.11 


C 


16.6 


4.97 


29 . 2 


2.86 


2.42 


.70 


6 


18.3 


3.98 


23.5 


1.61 


2.48 


.64 


5 


13.3 


3.90 


15.4 


1.68 


1.94 


.66 


5 


10 


2.99 


12.1 


1.04 


1.99 


.59 


4 


12.3 


3 6() 


9.2 


1.74  


1.59 


.69 


4 


10 


2.91 


7.5 


1.11 1 


1.60 


.62 


4 


6 


1.77 


4.5 


.31 


1.60 


.48 



AKP RADII OF GYRATION. 



311 



)MENTS OF INERTIA AND RADII OF GYRATION OF 

TRENTON BEAMS— STEEL. 



►G- 



7 



r 



iB 



4 







I. 


II. 


III. 


rv. 


V. 


Size, in 


Weight 

per foot, 

in lbs. 


Area of 
cross- 
section, 


Moments 


of inertia. 


Eadii of 


gyration. 


inches. 












in sq. in. 


Axis A B. 


Axis CD. 


Axis A B. 


Axis CD. 


15 


50 


14.70 


529.7 


20.96 


6.00 


1.19 


15 


41 


12.02 


424.4 


13.94 


5.94 


1.07 


13 


40 


11.73 


281.3 


16.76 


4.89 


1.19 


12 


82 


9.46 


229.2 


11.64 


4.93 


1.10 


10 


45 


13.14 


216.1 


17.94 


4.05 


1.17 


10 


33 


9.67 


1(J1.3 


11.81 


4.08 


1.10 


10 


25.3 


7.50 


123.6 


7. 82 


4.06 


.98 


9 


27 


7.98 


110 6 


9.13 


3.73 


1.07 


9 


21 


6.15 


84.3 


5.56 


3.70 


.95 


8 


22 


6.47 


71.9 


6.62 


3.34 


1.01 


8 


18 


5.28 


57.7 


4.36 


3 30 


.91 


7 


20 


5.87 


49.7 


5 51 


2.91 


,97 


7 


15.5 


4.55 


38.6 


3.47 


2.91 


.87 


6 


16.6 


4 97 


29.2 


2.86 


2.42 


.76 


6 


13.3 


3.97 


23.4 


1.63 


2.42 


.64 


5 


13 


3.80 


15.7 


1.98 


2.03 


.73 


5 


10 


2.96 


12.4 


1.30 


2 04 


.67 


4 


10 


2.94 


7.7 


1.22 


1.62 


.04 


4 


7.3 


2.21 


5.9 


.75 


1.63 


.59 



.l4i. _:.^_i-.- 



312 



MOMENTS OF IKERTIA 



MOMENTS OF INERTIA AND RADII OP GYRATION OP 
TRENTON CHANNEL AND DECK BEAMS— IRON. 



w 



IC 



d-r^^B 








I. 


II. 


m. 


IV. 


V. 


VI. 


Size, in inches. 


Weight 
per 


Area 

of 
crosg- 


Moments of 
inertia. 


Radii of 
gyration. 


Distanced 

of centre 

of gravity 


foot, 
lbs. 


sectioR, 
8q. iu. 


Axis 
AB. 


1 
Axis ', Axis 
C D. A B. 


AxlH 
CD. 


from oat- 
aide of 
web. 



Channel Bars. 



15 


63.3 


18.85 


15 


40 


12.00 


12i 


40.6 


14.10 


12i 


23.3 


7.00 


lOi 


20 


6.00 


10 


16 


4.77 


9 


23.3 


7.0-2 


9 


16.6 


5.08 


8 


15 


4.48 


8 


11 


3 30 


7 


12 


3.60 


7 


8.5 


2.54 


6 


15 


4 82 


6 


11 


3.20 


6 


7.5 


2 . eo 


5 


6.3 


1.92 


4 


5.5 


1.65 


3 


5 


1.45 



586.0 


32.25 


5.57 


1.31 


1.26 


376.0 


14.47 


5.60 


1.10 


0.25 


291.6 


17.87 


4.65 


1.12 


1.120 


153.2 


5.04 


4.68 


.86 


0.755 


88.4 


3.84 


3.84 


.80 


0.628 


64.0 


2.20 


3.68 


.68 


0.666 


82.1 


5.35 


8 42 


.87 


0.86 


58.8 


2.53 


3.40 


.70 


0.08 


44.5 


2.54 


8.15 


.75 


0.76 


32 9 


1.44 


3.16 


.66 


0.68 


27.1 


1.96 


2.74 


.88 


0.716 


17.3 


.8;^ 


2.61 


.67 


0.611 


21.7 


2.12 


2.24 


.70 


0.725 


17.2 


1.30 


2 32 


.64 


0.68 


12.6 


.70 


2.37 


.66 


0.64 


7.2 


.44 


1.98 


.48 


0.464 


3.9 


.32 


1.54 


.44 


46 


2.0 


.29 


1.17 


.45 


0.61 



Deck Beams. 



8 

7 


21.6 

18.3 


6.25 
5.35 

1 


54.7 
35.1 


8.7 
3.6 


2.96 
2.56 


.76 

.82 


• 



AND RADII OF GYRATION. 



313 



MOMENTS OF INERTIA OF TRENTON ANGLE-BARS. 



Size, in inches. 



Weight per 
foot, in J be. 



I. 



Area of 

erosB- 

section, 

iu 
sq.ins. 



II. 



Moment 
of inertia. 



VI. 



Distance 

d from 

base to 

neutral 

axis, 

in inches. 



EVEN 




LEGS. 



6 in. 
4i " 
4 ** 
3} " 
3 
25 
2i 
2i 
2 

13 

li 

1 

1 



<< 



X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 



6 in 

4i " 

4 

3i 

3 

2J 

2i 

2i 

2 

13 

H 

li 

1 

J 
I 



a 
(( 
(( 



19 to 32i 

12i to 20^ 

9i to 18 

8} to 14i 

4.8 to 12i 
5.4 to 9i 

3.9 to 7i 
3^ to 
si to 
2 to 
IJ to 
1 to 

3 to 



0.6 to 

T%tO 



6 

4i 
3i 
2i 

li 

1 
0.8 



5.75 


19.910 


1.685 


Axis A 


3.75 


7.200 


1.286 




2.86 


4.360 


1.138 




2.48 


2.860 


1.013 




1.44 


1.240 


, 0.842 




1.62 


1.150 


0.802 




1.19 


0.700 


0.717 




1.06 


0.500 


0.654 




0.94 


0.350 


0.592 




0.62 


0.180 


0.507 




0.53 


0.110 


0.444 




0.30 


0.044 


0.358 




0.23 


0.022 


0.296 




0.20 


0.014 


0.264 




0.17 


0.009 


0.233 





B 



UNEVEN -=- 




LEGS. 



6 in. X 4 in. 
i 5 " X 3i " 

; 4i " X 3 " 



i( 



X 3 



(< 



3^ " X H " 
3 " X 2> " 
8 " X 2 " 



14 to 23 


4.18 


j 15.460 
] 5.600 


1.964 
0.964 


10.2 to 19i 


3.05 


j 7.780 
] 3.190 


1.610 
0.8(K) 


9 to U\ 


2.67 


S 5.490 
\ 1.980 


1.490 
0.740 


7 to 14^ 


2.09 


j 3.370 
1 1.640 


1.260 
0.760 


4.0 


1.19 


j 1.500 
] 0.170 


1.320 
0.320 


4i to 9i 


1.31 


j 1.170 
1 0.740 


0.910 
0.660 


4 to 7i 


1.19 


( 1.090 
] 0.390 


0.990 
0.490 



Axis C 
" A 



16 
ii 
il 

a 
a 
n 
n 
il 
(t 



c 

A 
C 
A 
C 
A 
C 
A 
C 
A 



D 
B 
I) 
B 
D 
B 
D 
B 
D 
B 
D 
B 



C D 
A B 



3U 



MOMENTS OF INERTIA 



MOMENTS OF INERTIA OP TKBNTON T-BABSL 

c 




B 



^fe 



B 



Size, 
in inches. 



4*x4 
3^x31 
3 x3 
2ix2i 

2 x2 
6 x2i 

3 x2 
2 xli 
aixli 
2 xl 





I. 


Weight 

per foot, 

in lbs. 


Area of 
croBs- 

Bection, 

in 
sq. in. 


m 


3.75 


9.6 


2.87 


7 


2.11 


5 


1.46 


3i 


0.94 


11.7 


3.50 


4.8 


1.45 


3.00 


0.91 


2.40 


0.74 


2.15 


0.65 


1.86 


0.56 



IL 



Moment 
of inertia. 



IV. 



i 5.560 
^2.620 

j 3.260 
<1&30 

i 1.760 
^0.970 

(0.850 
(0.400 

j 0.350 
10.160 

a.500 
^5.090 

i 0.470 
'i 0.680 

(0.170 
"(0.180 

(0.060 
) 0.180 



tS: 



040 
140 



0.040 
0.070 



Radii of 
gyration. 



VI. 



1.22) 

.84j: 

1.06) 

.73 f 

.91) 

.62 f 

.76) 
.52 y 

.60^ 
.43 f 

.65) 
1.20f 

.571 

.68 f 

.43/ 

.45 f 

.29) 
.49 1) 



.26 
.46 

.26 
.35 



[ 



Distance 
d from 
base to 
nentral 
axis, 

In ineheB. 



1.180 
1.030 
0.890 
0.740 
0.590 
0.610 
0.520 
0.500 
0.290 
0.260 
0.280 



jAxisAB. 
i Axis CD. 

< AxisAB. 
) Axis CD. 

jAxisAB. 
(Axis CD. 

jAxisAA 
(Axis CD. 



\ 



Axis AB. 
Axis CD. 



jAxisAB. 
] Axis CD. 

j Axis A B. 
(Axis CD. 

( Axis A B. 
(Axis CD. 



t 



AxisAB. 
CD. 



jAxIs AK 
(Axis CD. 



I 



AB. 
Axis CD. 



* The flret dimension Ib the width. 



AND BADII OF GTKATIOK. 



TRENTON IRON OE STEEL Z-BARS. 



PHCENIX IRON Z-BARS. 



316 



MOMENTS OF IKEBTIA. 



MOMENTS OP INERTIA AND RADII OP GYRATION OP 
JONES & LAUGHLIN'S, LIMITED, STEEL BEAMS. 



17 






IB 



4 







I. 


II. 


III. 


IV. 


V. 


!4i7P 


Weight 


Area of 
cross- 


Moments ol 


r Inertia. 


Radii of GyratioiL 


in inches. 


per foot, 
in lbs. 


section, 

in 
sq. in. 












Axis A B. 


Axis C D. 


Axis A B. 


AxiHCB. 


15 


70 


20.6 


731.1 


37.8 


5.95 


1.85 


15 


59 


17.3 


640.9 


30.3 


6.08 


IM 


15 


48 


14.1 


495.9 


19.2 


5.98 


1.16 


15 


39 


11.5 


403.3 


13.1 


5 92 


1.06 


12 


50 


14.7 


302.0 


18.1 


4.53 


1.11 


12 


38 


11.2 


265 4 


15.6 


4.86 


1.18 


12 


30 


9.1 


211.7 


10.2 


4.82 


1.05 


10 


32 


9 4 


152.6 


10.8 


4.02 


1.07 


10 


28.8 


7.0 


117.7 


7.09 


8.88 


.05 


9 


24.5 


7.2 


101.1 


7.80 


8.74 


1.04 


9 


19.75 


5.8 


79.8 


5.03 


8.71 


0.03 


8 


25 


7.3 


71.8 


6.66 


8.18 


95 


8 


18 


5.3 


57.3 


4.27 


3.28 


0.89 


7 


18.8 


5.4 


40.4 


5.02 


2.98 


0.96 


7 


15.25 


4.5 


37.9 


3.38 


2.89 


0.86 


6 


16.6 


4.9 


2S.4 


3.39 


2.40 


0.88 


6 


12.75 


3.7 


23.1 


2.22 


2.49 


0.77 


5 


13 


3.8 


15.7 


1.83 


2.02 


0.00 


5 


10 


2.9 


13.5 


1.40 


2.16 


0.60 


4 


10.2 


3.0 


7.7 


1.20 


1.42 


O.W 


4 


6.85 


2.0 


5.8 


0.71 


1.70 


0.60 


8 


7 


2.0 


3 1 


65 


1 24 


0.50 


8 


5.1 


15 


2.3 


0.;;5 


1.28 


0.47 



AND BADII OF GYRATION. 



• 317 



IfOMENTS OP INERTIA AND RADII OP GYRATION OP 

PHGSNIX BEAMS— STEEL. 



fl 



U 



^ 



B 



71 



\1 



--0. 



Siase, 
tn inches , 



15 
15 
15 
15 
12 
12 

m 
m 

9 

9 
8 
8 
7 
7 
6 
6 
5 
5 
4 



Weight 


I. 




per foot, 


Area of 


in lbs. 


cross- 




section, 




in sq. in. 


75 


22.05 


60 


17.64 


50 


14.70 


41 


12 05 


40 


11.76 


82 


9.41 


83 


9.70 


m 


7.47 


27 


7.93 


21 


6.17 


22 


6.47 


18 


5.29 


20 


5.88 


15i 


4.55 


16 


4.70 


13 


3.82 


13 


3.82 


10 


2.94 


10 


2.94 



II. 




Moments of inertia. 



Axis A B. 



757.7 

644.0 

529.7 

424.1 

281.3 

222.3 

179.6 

137.3 

110.6 

84.3 

71.9 

57.8 

49.7 

38.6 

28.6 

23.5 

15.7 

12.4 

7.7 



Radii of gyration. 



Axis A B. 


Axi?CD. 


5.86 


1.35 


6.04 


1.32 


6.00 


1.20 


5.94 


1.08 


4.90 


1.20 


4.85 


1.04 


4.54 


1.10 


4.52 


0.99 


3.7SI 


1.07 


8.70 


0.95 


8. as 


1.01 


8.30 


0.91 


2.91 


0.97 


2.91 


0.87 


2.47 


0.83 


2.48 


0.77 


2.03 


0.72 


2.05 


0.66 


1.62 


0.66 



31- 



M03fENTS OF DTEHTIA 



MOi£E>'TS OF lyERTlA ASD RADU OP GYRATION OP 

PHCELNIX DECK-BEAMS AND T-BARS. 

c 



c ^ 



=o^ 




IIL 



IV. 



V. 



VI. 





"T.'iini 




A4 »a^ ..C"?. 


n L>» 


u 



V-jnufSLtB jf 



SM. Bihlii of gyration. I>isUnee 

d from 

IWMtO 

neatnl 
«{. n. A^z^JlB. ^Ti*CD. AtjwJlR AxJgCD. 



* 


s: s 


I.I 


^« s 


4 


s 


^ 

^ 


j' 


"• 


* ^ 


i 


'♦ 




V i 



.K.S 3£^ 



i05L 





T'S-'K ?raxs — Iho». 


k 






* 5 


:*i "5 


5 IT 


4*1 


0.74 


4.37 


> 5 


.5: ^ 


5 H 


4£ 


O.TO 


8.77 


< * 


-^j -» 


4 >4 


3 27 


0.84 


2.96 


< .1 


TO ;• 


3 -jS 


2.90 


0.81) 


2.» 




■« ■^^ 


3 >H 


2.53 


0.77 


2.96 


'fc i 


:i « 


^35 


2.17 


•.75 


1.88 


1 ^ 




^ :i» 


1.7» 


0.51 


2.41 



-4 


il 


^ *» 




4 41 


9.n 


•.« 


4.06 


> 


^  




^ -7 


4 Z« 


2.« 


0.73 


8.«8 


« 


* - 


" i? 


54 11 


? U 


2.« 


• 85 


2.85 


5 


3S 


? i 


i*? \i 


i *? 


a. IS 


•.76 


2.89 


i 


'"A 


* . . 


:* m 


i 15 


d.ltf 


•.73 


2.78 



^-v :i 


> 


« « 


m m 


J 24 


V-» aV 


1.23 


0.77 


• * 


4 » 


: * 


: 5» 


i M 


f.6» 


1.17 


0.66 


«* ' ' 


« 
X ^ 


^ 
% 


 -^ 


d J8» 


# * 


•.•8 


0.78 


* ^ : 


> 


< 


. V. 


: i^ 


« m 


1-01 ; 


0.57 


* ** 


• 


< 


• 


• » 


4 s4 


• a 


0.84 


'»■. s •'" . 


* 


 N- 


I ft 


: ** 


I « 


•.76 . 


1.03 


^ N '• 


^ 


^ 


"S 


' 5» 


4 •» 


•.•9 


0.86 


* - 


* X 


> 


 -e 


' 4t 


« :s 


•.S 1 


0.75 


• 




* 


• « 


i :t 


*.» 


••"i 


0.68 



*^ O*. w:)m«aK«'a > 



AND RADII OF GYRATION. 



319 



MOMENTS OF INERTIA AND RADII OF GYRATION OF 
PHOENIX CHANNEL-BARS— IRON. 



A Ia n ^ 

IB 





Weight 


I. 


II. 


in. 


IV. 


V. 


VI. 
















Size, in 
inches. 


per 
root, 


Area of 


Moments 


of inertia. 


Badii of gyration. 


Distance d 




in lbs. 


cropD- 
section, 

in 
sq. in. 










from base 












to neutral 






Axis A B. 


AxifiCD. 


Axis A B. 


Axis CD. 


axis. 


15 


66.6 


20 


554.57 


23.61 


5.27 


1.09 


1.08 


15 


60 


15 


421.87 


12.39 


5.30 


0.91 


0.86 


15 


38.3 


11.5 


351.56 


10.01 


5.53 


0.93 


0.83 


12 


50 


15.0 


235.73 


8.44 


3.96 


0.75 


0.80 


13- 


29.8 


8.8 


159.44 


4.19 


4.26 


0.69 


0.82 


12 


20 


6.0 


123.50 


8.01 


4.54 


0.71 


0.86 


10 


37 


11.1 


128.61 


5.26 


3.40 


0.69 


0.76 


10 


25 


7.5 


97.36 


8.51 


3.60 


0.69 


0.66 


10 


16 


^•^^ 


63.67 


2.21 


3.64 


0.68 


0.56 


9 


38.8 


10.0^ 


94.27 


5.24 


3.07 


0.73 


0.76 


9 


23.8 


7.0 


75.29 


8.69 


8.28 


0.73 


0.70 


9 


15 


4.5 


61.01 


2.36 


8.49 


0.69 


0.70 


8 


19 


5.7 


43.99 


2.14 


2.76 


0.61 


0.66 


8 


10 


8.0 


26.20 


0.85 


2.96 


0.53 


0.45. 


7 


19 


5.7 


32 69 


2.00 


2.40 


0.59 


59 


7 


8.8 


2.5 


17.62 


0.75 


2.66 


0.55 


0.47 


6 


15.6 


4.7 


23.12 


2.5 


2.22 


0.73 


0.73 


6 


7 8 


2.2 


10.42 


0.62 


2.18 


0.53 


0.40 


5 


9 


2.7 


9.52 


0.84 


1.88 


0.56 


0.55 


5 


5.6 


1.7 


6.35 


0.43 


1.9:3 


0.51 


0.47 


4 


8 


2.4 


5.53 


0.79 


1.52 


0.57 


0.60 


4 


5 


1.5 


8.74 


0.4 


1.58 


0.52 


0.62 


8 


6 


1.8 


2.26 


0.86 


1.12 


0.45 


0.53 


8 


5 


1.5 


1.98 


0.29 


1.15 


0.44 


0.50 



320 



MOMKNTS OF INERTIA 



MOMENTS OF INERTIA AND RADII OP GYRATION OF 

PHCENIX ANGLE-BARS— IRON. 

ANGLES WITH EQUAL LEGS. 








I. 


n. • 


TTT. 


IV. 


V. 


VI. 




Wefght 

per 

foot, 

in lbs. 


Area 

of 

croBS- 

t^ection, 

sq. in. 


Moments of 
inertia. 


Badii of 
gyratiou. 


Distance 

dtTOOk 


Size, in inches. 


Axis 
AB. 


Axis 
CD. 


Axis 
AB. 


Axis 
CD. 


base to 

ikentral 

axis. 


6 x6 
6 x6 
5 x5 
5 x5 
4 x4 
4 x4 
3ix3i 
3ix3i 
8 x3 
3 x3 
2|x2} 
2Jx2i 
2ix2i 
2i X 2i 
2J X 21 
2i X 2i- 
2 x2 
2 x2 
l}xlj 

lixH 


33.3 

16.8 

20.6 

12.3 

17.2 

9.4 

13.6 

6.8 

9.4 

5 

8.6 
4.5 
7.9 
3.5 
6.1 
2.6 
4.6 
2.5 
2.0 
1.5 


10 

5.03 
6.2 
3.7 
5.16 
2.81 
4.1 
2.05 
2.81 
1.5 
2.58 
1.34 
2.36 
1.05 
1.83 
0.8 
1.4 
0.75 
0.61 
0.44 


35.17 
17.22 
14.70 
9.35 
7.18 
4.39 
4.35 
2.30 
2.23 
1.33 
1.65 
1.01 
1.22 
0.62 
0.82 
0.40 
0.49 
0.29 
0.18 
0.9 


13.98 
6.77 
6.07 
8.77 
3.01 
1.71 
1.84 
0.95 
0.95 
0.54 
0.62 
0.41 
52 
0.25 
0.35 
0.17 
0.20 
0.12 
0.07 
0.04 


1.87 
1.85 
1.54 
1.59 
1.18 
1:25 
1.08 
1.06 
0.89 
0.94 
0.80 
0.87 
0.72 
0.77 
0.67 
0.71 
0.59 
0.62 
0.55 
0.46 


1.18 

1.16 
0.99 
1.01 

0.76 

0.78 

0.67 

0.68 

0.58 

0.6 

0.49 

0.55 

0.47 

0.49 

0.44 

0.46 

0.88 

0.40 

0.85 

0.29 


r.84 

1.08 

1.55 

1.46 

1.22 

1.16 

1.06 

0.96 

0.98 

0.87 

088 

0.8S 

0.77 

0.7 

0.74 

069 

0.08 

0.0 

0.08 

0.44 



AND BADII OK GYBATION. 



,r 






Nora.— E P b parallel to Uds tluongh ends ut aides. 



322 



MOMENTS OP INERTIA 



RADII OF GYRATION FOR A PAIR OP CARNEGIB 
ANGLES PLACED BACK TO BACK. 

ANGLES WITH EQUAL LEGS. 



n 



ymMmm 




\x 



^^^^ ^vummm 



n 




Hadii of Gyration given, correspond to directions indicated by arrow-heads. 



Size, in 
inches. 



6 x6 

6 x6 

5 x5 

5 x5 



4 x4 


5.72 


4 x4 


12.04 


^x^ 


4.96 


8^x3^ 


10.44 


8 x8 


2.88 


3 x3 


7.00 


2^x2J 


2.62 


23x2J 


5.88 


2ix2i 


2.88 


2ix2i 


4.74 


2i X 2i 


2.12 


^4,x2i 


4.22 



*Area of 

croBS- 

section, 

in inches. 



10.12 



16.56 



Weitjbt 
per foot 
of 8ingle 




Hadii of gyration. 




angle, 
in lbs. 


^0. 

1.87 


n. 


»•«• 


r». 


16.9 


2.50 


2.67 


2.76 


33.1 


1.85 


2.62 


2.80 


2.89 


12.0 


1.56 


% 09 


2.20 


2.85 


27.6 


1.55 


2.24 


2.42 


2.62 


9.5 


1.28 


1.68 


1.86 


l.d5 


20.1 


1.22 


1.81 


2.0(> 


2.10 


8.8 


1.07 


1.47 


1.66 


1.76 


17.4 


1.06 


1.60 


1.80 


1.00 


4.8 


0.9] 


1.25 


1.43 


1.58 


11.7 


0.98 


1.37 


1.56 


1.66 


4.4 


0.85 


1.15 


1.84 


1.44 


9.0 


0.91 


1.31 


1.60 


1.61 


4.0 


0.77 


1.05 


1.24 


1.84 


7.9 


0.78 


1.14 


1.38 


1.48 


8.5 


0.69 


0.96 


1.14 


1.84 


7.0 


0.70 


1.05 


1.24 


1.85 



AND RADII OF GYRATION. 



323 



RADII OF GYRATION FOR A PAIR OF CARNEGIE 
ANGLES PLACED BACK TO BACK. 

ANGLES WITH UNEQUAL LEGS. 




'2 

M * V 

r.fV 




Radii of Gyration given, correspond to directions indicated by arrow-heads. 



Size, in 
incties. 



6 
6 
6 
6 

5 
5 
5 
5 



x4 
x4 
x3i 
x3i 

x4 
x4 
x3i 
x3i 



5 x3 
5 x3 
4^x3 
4ix3 

4 x3i 

4 x3i 

4 x3 

4 x3 

3^x3 
3^x3 

3ix2i 
3^x2^ 

3ix2 

3ix2 
3 x2i 
3 x2i 

3 x2 
8 x2 
2^x2 
2ix2 



♦Area of 
cross- 
section, 
in inches. 



7.22 
16.36 

6.84 
14.50 

6.46 
13.66 

6.10 
12.84 

5.72 
12.04 

5.34 
11.24 

5.34 
11.22 

4.18 
10.42 

3.86 
9.60 

2.88 
5.94 

2.50 
4.96 
2.62 
5.20 

2.38 
4.62 
1.62 
4.36 



Weight 

per foot 

of single 

angle, 

in Ids. 



12.0 
27.3 
11.4 
25.8 

10.8 
22.8 
10 2 
21.4 

9.5 
20.1 

8.9 
18.7 

8 9 
18.7 

7.0 
17.4 

6.5 
16.0 

4.8 
9.8 

4.2 
8 3 
4.4 

8.7 

4.0 
8.0 
2.7 
7.2 



Radii of gyration. 



ro. 



1.93 
1.90 
1.94 
1.91 

1.59 
1 57 
1 60 
1.58 

• 

1.61 
1.59 
1.44 
1 42 

1.25 
1.23 
1.27 
1.25 

1 10 
1.07 
1.12 
1.17 

1.04 

1.04 

.95 

.95 

.96 
.99 
.79 

.SO 



n- 


r,' 


1.50 


1.67 


1.62 


1.80 


1.26 


1 43 


1.39 


1.58 


1.58 


1.75 


1.70 


1.89 


1.33 


1.51 


1.45 


1.61 


1.10 


1.27 


1.22 


1.41 


1.13 


1.31 


1.26 


1.45 


1.43 


1.60 


1.54 


1.74 


1.17 


1.35 


1.30 


1.50 


1.22 


1.40 


1.35 


1.55 


0.96 


1.13 


1.10 


1.28 


0.74 


0.92 


0.82 


1.02 


1.00 


1.18 


1.09 


1.28 


0.75 


0.93 


0.87 


1.06 


0.79 


0.97 


0.90 


1.10 



1.76 
1.90 
1.53 
1.68 

1.85 
1.98 
1.60 
1.74 

1.37 
1.51 
1.41 
1.56 

1.70 
1.84 
1.44 
1.60 

1.49 
1.05 
1.23 
1.39 

1.02 
1.12 
1.28 
1.38 

1.03 
1.17 
1.07 
1.21 



* 1 > flfmrpR in this column give the area of both angles. 



324 



MOMENTS OF INERTI4 



RADII OP GYRATION FOR A PAIR OP GARNEGIB 
ANGLES PLACED BACK TO BACK. 



4 



.e-il 



ANGLES WITH 'UNEQUAL LEGS, 



%5;%:^^^;5^^ S$5SSSS55S5JSSS: 






Radii of Oyration given, correspond to directions indicated by arrow-headt. 



Size, in 
inches. 



♦Area of 

cross- 

pectiou, 

' in inches. 



6 
6 
6 



5 
5 
5 
5 



x4 
x4 
x3^ 

x3i 

x4 
x4 
x3^ 
x3^ 



5 x8 
5 x3 

4^x8 
4^x3 

4 x3i 
4 x3i 
4 x3 
4 x3 



Six 2 
3ixa 
8 x2^ 
8 x2^ 



8 

3 



x2 
x2 



2ix2 
2^x2 



7.22 
16.36 

6.84 
14.50 

6.46 
13.66 

6.10 
12.84 

5 72 
12.04 

5.34 
11.24 

5.84 
11.22 

4.18 
10.42 



2.50 
4.96 
2.62 
5.20 

2.88 
4.62 
1.62 
4.86 



Weight 

per foot 

of single 

anffle, 

in lbs. 



8^x3 


3.8J 


34x3 


9.60 


3ix2^ 


2.88 


8Jix2^ 


5.91 



12.0 
27.3 
11.4 

25.8 

10.8 
22.8 
10.2 
21.4 

9.5 
20.1 

8.9 
18.7 

8.9 
18.7 

7.0 
17.4 

6.5 

16.0 

4.8 

9.8 

4.2 
8.3 
4.4 
8.7 

4.0 
8.0 
2.7 

7.2 



''o- 



1.17 
1.14 
0.99 
0.97 

1.20 
1 19 
1.02 
1.01 

0.86 
0.83 
0.86 
0.85 

1.06 
1.04 
0.89 

0.87 



' 0.90 

: 89 

i 0.74 

\ 0.78 

0.57 
0.57 
O.7.-) 
0.76 

0.57 
60 
0.60 
0.61 



Badii of gynttion. 



n- 


ri. 


2.74 
2.87 
2.81 
2.95 


2.92 
8.06 
8.00 
8.14 


2.20 
2.88 
2.27 
2.39 


2.88 
2.52 
2.45 
2.59 


2.35 
2.47 
2.07 
2.20 


2 52 
2.66 
2.25 
2.89 


1.74 
1.86 
1.79 
1.98 


1.92 
2.05 
1.97 
2.12 


1.52 
1.66 
1.58 
1.72 


1.71 
1.H6 
1.76 
1 91 


1.51 
1.00 
1.31 
1.40 


1.70 
1.80 
1.60 
1.59 


1.88 
1.49 
1.10 
1.18 


1.57 
1.69 
1.28 
1.87 



8.01 
8.16 
8.10 
8.24 

2.48 
2.62 
2.65 
2.69 

2.62 
2.77 
2.86 
2.40 

2.08 
2.15 
2.07 
2.22 

1.80 
1.96 
1.86 
2.01 

1.80 
1.91 
1.69 
1.69 

i.er 

1.79 
1.89 
1.48 



* The figures in this colnmn give the area of both anslea. 



AND RADII OF GYRATION. 



325 



For compound sections made up of two or more beams or bars, 
the moments of inertia are found by combining those of the several 
shapes as given in the preceding tables. Thus : ~ 



/ = 



G2 = 



Twice the moment of inertia 
for l)eam a (col. II.) + that for 
beam 6 (col. III.). 

I 

sum of areas of beams a and b 
(col. I.) 



1^^^ 



a. 




rtS^^ 



I = 



B 



Twice area of beam a (col. I.) x 

d^ + twice moment of inertia 

for beam a (col. III. ) + that for 

beam b (col. II.). 

I 

d + i width flange of beam a 

L 

sum of areas of beams a and b 
(col. I.) 



^ a i 



^„ r%^iii 



t^--*- 



A^ 



a 



I = 



G^- 



Twice area of channels (col. I. ) 
y. d^ + moment of inertia (col. 
III. ), in which d = distance of 
centre of gravity of the channel 
from centre line of the combi- 
nation. 

7 

area of the two channels (col. I. ) 



t 

^s-^ 



J 



fOJ..n.imm.m.<.i^ 



 ■I 11 111 L mm 






XatHce 



I = Twice the moment in col. II. 



G'^ = Same as for single channel. 




When a section is employed alone, either as girder or post, the 
neutral axis passes through its centre of gravity. When rigidly 
connected with other sections forming part of a compound section, 
the neutral axis passes through the centre of gravity of the com- 



pound section; and therefore the moment of inertia of the elemen- 
tary section will not be that around its own centre of gravity, but 
around an axis at a distance from that point. The moment tjif 
inertia of a section aitout an axis other than that thronyh its mi- 
tre of (jravitjj is (Mjuai to the moment aixmt a i>arallel axis iHitt^ing 
tliruu^h its coiilrt' of ^Tavity plus the area of the section mulli 
plied by tlif square of tlu* distance Ix^tween the axes. 

The first step, then, in findinji; the moment of inertia, is to find 
the position of the »'entre of i^ravity of the se<^tion. For all sym- 
nirtrieal sections, this, of course, lies at the middle of the depth. 
For triani^lrs. it is found on a line j)arallel with the hjiso, and tlis- 
tant ont'-third the heii;ht of the triangle above the bast». For other 
sections, it is found by supposing the area divided up into elemen- 
tary sections, and nndti])lying the area of each such section by the 
distance of its centn* of gnivity from any convenient line. The 
sum of these products ilivided by the total area of the si»ction will 
give tile distance of the centre of gravity from the line from which 
the distances w<'re measured. 

KxAMPLK. — Find the neutral axis of a X siH'tion having the 
fo)l(>wini; dimensions : wi<lth, 8 inches ; depth. 10 inches : thick- 
ncs'< of metal, 2 inches. The area of the vertical flange, considering 
It as running through to the l)ottom of the section, would lx> 10 X :>, 
or 2n scjuare inches; and the distam'C of its centre of gravity alcove 
the l)ottoin line, 5 inches. The product of thest' quantities, lhen»- 
forc, is 1(H). The area of the bottom flange, not included in tlie 
Vertical Mange as above taken, is (> times 2, or 12 squan* inches: the 
distance ut' its centre of gravity above the l)ottom line. 1 inch; and 
the product of the two, therefore, 12. The sum of thesk* pHnUicls 

112 

divided l>v the total area is .,.r« or .'}.r> Inches, which is the distance 

•»- 

ot" till- centre of i^ravitv ab(»ve the lK>ttom line of the MVtion. 

11a villi: tound the neutral axis of this .section, its moment of 

iiwiiia I-- readilv fomid bv the fornnda before given. Thus, in the 

la- i'l^-' >Upl»n«sed, f/ WOldd be 10 ~ :»..">= «i.."i. (/^ = .*{.5; »/.^ = l.o; 

ami tiif iiiouieiit WOldd be (see ]i. 2!hM, 

(2 X r...v«) -}- (S X ;l..v«) — (ti X 1.5-*) 
/ - .J = 2t»«»l. 

The iiioMieiit «)f resistance of this sei'tion as a ginler would b(> 

. . . ■•! tl : and it* a Mrain on the tibres of the iron of 12,UM) 

poiimU i« i- sijiiure inch be allowed, then, sinet> the moment of 
n-^i^iani-f lit the ginler multiplied by hirain |ier s«|uaro iuoh musi 



AND BADII OF GYRATION. 327 

«qiMl the bcndlng^moment of the load, it will be able to support a 
kmd whose bending-moment is 44^ times 12,000 pounds, or 536,000; 
Le., if used as a girder secured rigidly at one end, and loaded at the 
other, it would support a load, in pounds, of 

536000 



length in inches 



Or if supported at both ends, and the load uniformly distributed 
over the span, It would support a load eight times as great; the 
bending-moment in such case being one-eighth that in the former 
case (see pp. 291, 292). 

NoTs.— The formulas and fignree on pp. 296, 299, and 325. are taken, by per* 
iniP<)ion of The New-Jersey Steel and l9xm Company ^ from a hand- book which 
they pnblieh, entitled ** Usefal Information Tor Engineers and Architects,'^ and 
containiog fall information pertaining to the forms of iron which they mannfac- 
tnie. 

Radius of Gyration of Compound Shapes. 

{Ninth Edition.) 

In the case of a pair of any shape without a web the value of 
B can always be readily found without considering the tiioment 
of inertia. 

The radius of gyration for any section around an axis parallel 
to another axis passing through its centre of gravity, is found 
as follows : 

Let r = radius of gyration around axis through centre of grav- 
ity ; B = radius of gyration around another axis parallel to 
above ; d = distance between axes. 



When r is small, E may be taken as equal to d without mate- 
rial error. Thus, in the case of a pair of channels latticed to- 
gether, or a similar construction. 

Example 1 —Two 9-inch, 15-pound PhoB- c 

nix channel bars are placed 4.6 inches apart, ^^ ! i"^"1 



K-44J-- 



as in the figure ; required the radius of gyra 

tion around axis C D for combined section. j 

Ans. Find r, in Column V., p. 819= ^ |-H— -i — H— B 

0.69; and r^ = . 4761. 

Distance from base of channel to neutral 
axis. Column VI.. is .7. One-half of 4.6 = 
2.3-1- .7 = 8, the distance l)etween neutral ^ 

axis of single channel and of combined section ; henoe, 

B = y9 + .4761 = 3.077 ; or, for all practical purposes, R = d, 



328 RADIUS OF GYRATION OP COMPOUND SHAPES 

Example 2.-*Four 8x8 inches, 5-pound Phooniz angles 
as shown form a column 10 inches square ; find the rad 
gyration. 

Ans. From Column IV., p. 830, we find r = 0.94 aiu 




A 



.8886. The distance from base of angle to neutral axis, O 
VI.. is .87 ; hence, c? = 5 - .87 = 4.18 ; or, (f = 17.0609 

.8= /i7.0509 + .8836 = 4.28. 



PRINCIPLES OF THE STRENGTH OF BEAMS. 829 



CHAPTER XIV. 

GZSNERAL PRINCIPIiES OF THE STRENGTH OF 
BEAMS, AND STRENGTH OF IRON BEAMS. 

By the term "beam" is meant any piece of material which 
supports a load whose tendency is to break the piece across, or at 
right angles to, the fibres, and which also causes the piece to bend 
before breaking. When a load of any kind is applied to any beam, 
it will cause it to bend by a certain amount; and as it is impossible 
to bend a piece of any material without stretching the- fibres on 
the outer side, and compressing the fibres on the inner side, the 
bending of the beam will produce tension in its lower fibres, and 
compression in its upper ones. This tension and compression are 
also greatest in those fibres which are the farthest from the neutral 
axis of the beam. The neutral axis is the line along which the 
fibres of the beam are neither lengthened nor shortened by the bend- 
ing of the beam. For beams of wrought-iron and wood the neutral 
axis practically passes through the centre of gravity of the cross- 
section of the beam. 

To determine the strength of any beam to resist the effects of 
any load, or series of loads, we must determine two things: first, 
the destructive force tending to bend and break the beam, which is 
called the " bending-moment ; " and, second, the combined resist- 
anceof all the fibres of the beam to being broken, which is called 
the **ni oment of resistance ." 

The methods for finding the bending-moments for any load, or 
series of loads, have been given in Chap. XII. ; and rules for finding 
the moment of resistance, which is equal to the moment of inerlia 
divided by the distance of the most extended or compressed fibres 
from the neutral axis, and the quotient multiplied by the strength 
of the material, have been given in Chap. XIII., together with 
tables of the moment of mertia for rolled iron sections of the usual 
patterns. 

Now, that a beam shall just be able to resist the load, and not 
break, we must have a condition where the bending-moinent in 
the beam is equal to the moment of resistance multiplied by the 
strength of the material. That the beam may be abundantly safe 
Uy resist Ibe given load, the moment of resistance multiplied by 



y / 



3150 rillNCIPLKS OF THE STRKNOTIl OF BKAMS. 



Atrcimtli of material must be several times as cjeat as the bendiiif;- 
nioinent; and the ratio in which this pnKlnet exe<'o<ls I lie ImmhI- 
ini:-iiH)inrnt, or in whirh the breaking- load exceeds the safe load, 
is known as the "factor" of sjifcty. 

r.y "ih*' strength of the material" is meant a certain constant 
(]iiantity, whiiji is dctermim^l by exiM'rinicni. and wliicli is known 
a> thf *• Mo:lu]us of Rupture." Of course this value isdifTerent for 
each ditlVn'nt mat<Tial. The following table contains the values 
of (I I is constant divided by the factor of safety, for most of the 
nianriais used in buildini;-const ruction. The moment of n^sistanee 
nmuiplied by these values will give the sttfe reiiiti(imj-\)OweT of the 
beasii. 



MoiU'Lrs OF lU'PTURE FOR SaFE STRENGTH. 





Vahu' <>r 


M.itt-rijil. 


f{. 




in \hj*. 


(':i-t Iron 


:..%« 


\Vrnii"|ii Iron 


l;!.i)i)0 


si.r; 


Ki.iXK) 


Ain«Ti«;Mi :i*h 


'.».(KK) 


.\ni'-rir:iii r-cl IhtcIi 


l..H<)J) 


Am i<- III \i!lM\v Itirrh 


i.ii-a) 


■\iiifiii-ni u lii'i" rciljir . 


l.(MN) 


Aii\i' ii in ili'i 


I.KI) 


Niu Kii. "iinil lir 


\.m) 


II'IIli'"' i» 


l.',1K» 


\mii : 1 .III \\ hill- o.ik 


!.:«► 



Material. 



!v,:. 



I 



lie of 
ill Urn. 



l.Osil 



AiiuTiraii wliitr pine.. 

.Anierimn yollnw |iim> l.sm 

Anii-rican fpnirr i l.'J»iil 

Om';;<)Ii pine I l.ViO 

niu(*Hti<iii> ll:ii;i;ii)K (liiiil-, 

son Rivni ' 

(•i:iniii>. avi'ntire 

Liine>ti>nf 

Marbli- 

Saiii|>toii4> 

Slatr 



I 



.TTR 

art) 

»IN) 



'ri;.- ;ili..\.' valiH's I if R for wi*ou«:hi inui and st«H'l are one-fmirrh 
th.ii f"" t'l.- bri'akiFiir-l'»ads ; lor ca.st- iron. «me-sixth : for wt mm I, une- 
tliirl : jii'l i'"'!' >iniii-, <ine-sixih. Th*' constant'^ lor wimmI an- bJl^4•«l 
wj.-m ill- f.iiiii I'-t-i iiiaili" al the Massacliusetis Institute of Teeli- 
i,<i|<i-\ -Ml' :i l'!iil-^i/i- liiidH-rs of the usual i|uality found in build- 
■n.r-. ! 'i-' tu'iir*-* ;;i\'ii "ii lh«' above labb- are bellev(»il to )>e amply 
sil".- ■■:■ i":rii> in ll<»nr-. ni" ilwrlliuirs. public balls. n»of.«*, etc.: but, 
fiir tl'X'i- III iiiiiN arnl warehouse- lb hm's, the niitbor ret -on in lends 
that iHit !ii>:i- than two-tbinls nf ilu- aUivr values Ih- iiM-d. The 
-:\U' ioj'l I T iIm' 'rn-ntiMi. iMio'nix. and »*arne«;ie s4><-tions. ustii as 
ih-ar::-. :ii'-.ill cDiiipiiIrd wjlii l'2.l>(M) pnuiids bir the >:ife value of 
/,'. ..■■ w"'- r.*.« OD |HHii)d>> libri' '^train, as it is •rcnerally calleti. 
:.-: 1-.::. iii| H'i.CMKI piiiinti> for >tcii. 

'!'•.'■ .■■ i-i riain ( a^« -« of be.ini> which most fri'ipieiitly occur 
:ti i<ii;l-i.:ij •••ii^t nn-i inn. f«ir which ftirmulas can In- given by wtiich 
tin -at I'ad^ fur llie bi-aiMS ma\ In- determiuetl ilirectlv ; hut U 

 

fieri liapp- u> ihal we may have either a iX'gularly nliapud bttun 



« • 



FBINCIPLES OF THE STRENGTH OF BEAMS. ^JSl 

Inregiilarly loaded, or a beam of irregular sectioh, but with a com- 
mou method of loading, or both ; and in such cases it is necessary 
to determine the bending-moment, or moment of resistance, and 
find the beam whose moment of resistance multiplied by R is 
equal to this bending-moment, or wliat load will give a bendinp:- 
moment equal to the moment of resistance of a beam nuiltipliod 
by R. 

For ezainplej suppose we have a rectangular beam of yellow 
pine loaded at irregular pomts with irregular loads: what dhnon- 
sions shall the beam be to carry these loads ? We will suppose that 
we have found the bending-moment caused by these loads to be 
480,000 inch pounds. 

Then, as bending-moment equals moment of resistance multiplied 
by li, 

480,000 pounds = —\^ x 1800 = J? x 2>« x 800 ; 

_ • , 4H00()0 
or B X D^= " SOO ~ 

If we assume i> = 12 inches, then B =  ^  =11 inches ; or, 

144 

the beam should be 11 inches by 12 inches. 

If, instep of a hard-pine beam, we should wish to use an iron 
beam to carry our loads in the above example, we must find a 
beam whose moment of resistance nuiltiplied by 12,000 equals 
480, OOC) inch pounds. We can only do this by trial, and for the 
first trial we will take the Trenton I2:t-inch 125-pound beam. Tlie 
moment of inertia of this beam is given as 2S8; and its moment of 
resistance is one-sixth of this, or 48. Multiplying this by 12,000, 
we have 576,(X)0 pounds as the resisting-force of this beam, or 
96,000 pounds over the bending-moment. Hence we should prob- 
ably use this beam, as the next lightest beam would probably not 
be strong enough. Fn this way we can find the strength of a beam 
of any cross-section to carry any load, however irregularly disposed 
it may be. 

Strength ol' Wrouglit-Iron Beams, Clianiiels, Aiijyle 

and T Bars. 

It is very seldom that one needs to compute the strength of 
wrought-iron beams, channels, etc. ; because, if he uses one of the 
regular sections to be found in the market, the computations have 
already been made by the manufacturers, and are given in their 
handbook. There might, however, be cases where it would be 
necessary to make the calculations for any particular beam; and to 
tneel such^cascs we give the following formulas. 



332 PRINCIPLES OF THE STRENGTH OF BEAMS. 



Beams fixed at one end, and loaded at the other (Fig, 1). 



Safe load in pounds = 

1000 X moment of inertia 



length in feet x y 



. (1) 




Beams fixed at one end, loaded with vniformly distributed load 
(Fig. 2). 



Safe load in pounds = 

2000 X moment of inert ia 
length in feet X y 



. (2) 




Fig. 2. 

Beams supported at both ends, loaded at middle (Fig. 3). 

W 




Safe load in pounds = 



Fig. 3. 
4000 X moment of inertia 



(81 



span in feet x y 

Beams supported at both ends, load uniformly distributed 
(Fig. 4). 




Safe load in pounds = 



Fig. 4. 
8000 X moment of inertia 



span in feet X y 



w 



PBINGIPUSS OF THE STRENGTH OF BEAMS. 333 



Beaifis supported at both ends, loaded with concentrated load 
not at centre (Fig. 5). 




Safe load in pounds 



Fig. 6. 

1000 X moment of inertia X span in feet 



^(5) 



m X nX y 

Beams supported at both ends, loaded with W pounds, at a dis- 
tance m from each end (Fig. 6). 




Fig. 6. 

Safe load W, in pounds at each point = 

1000 X moment of inertia 



(6) 



m in feet x y 

The letter y in the above formulas is used to denote the distance 
of the farthest fibre from the neutral axis; and, in beams of sym- 
metrical section, y would be one-half the height of the beam in 
inches. These formulas apply to iron beams of any form of cross- 
section, from an I-beam to an angle or T bar. For steel beams, 
increase the value of W one-third. 

Weight of Beam to be subtracted from its Safe 

Load. 

As the weight of iron beams often amounts to a considemble 
proportion of the load which they can carry, the weight should 
always be subtracted from the maximum safe load : for beams with 
concentrated loads, and for beams with distributed loads, one-half 
the weight of the beam should be subtracted. 

Example 1. — What is the safe load for a Trenton 12i-inch light 

I-beam, 125 pounds per yard, having a clear span of 20 feet, the 

load being concentrated at a point 5 feet from one end ? 

1000 XIX span 1000 X 288 ;^ ^0 
Ans. Safe load (For. 5) = - 

12,500 pounds. 



mX nX y 



5 X 15 X 6i 



334 



STRENGTH OF IRON AND STEKL BKAXB.' 



Example 2. — A 12-inch Carnegie iron channel-bar, wdgliing 90 
pounds per yard, and having a clear span of 24 feet, supportBA 
concentrated load at two points, 6 feet from each end. Wliat is 
the maxiinuin load that can be supported at each point consistent 

with safety ? 



Avs. Safe load at each point = ;;-— 

^ 6x6 



4825 pounds. 



The moment of inertia for channels and an^e-bars, and other 
sections, will be found in Chap. XIII. 

Deepest Beam always most EconomicaL 

Whenever we have a large load to carry with a given span, It will 

be found that it can be carried with the least amount of iron by 

using the deepest beams, provided the beams are not too strong for 

the load. Thus, suppose we wish to support a load of tons with 

a span of 20 feet, by means of Trenton beams. We oould do this 

either by one 12i-inch beam at 125 pounds per yard, or by two 

9-inch beams at 85 pounds per yard. But the 12Hnch beam, 21 feet 

long, would weigh only 875 pounds, while the two 9-incfa beams 

would weigh 1190 pounds; so that, by using the deeper beam, we 

save 315 pounds of iron, worth from three to five cents per pound. 

C 
The following table, under the heading |F?, gives the relative 

strength of Trenton beams in proportion to their weighty thns 
exhibiting the greater economy of the deeper patterns. 

Trenton Rolled I-Beams. 



Strength of each Beam in Proportion to its Weight. 





c 






c 


Bbam. 


w 




Bbam. 


W 


15 inch, heavy .... 


37.41 


8 inch, light 


».75 


15 '♦ light . . . 




36.76 


7 " 


55 pounds . 






19017 


124 " heavy . . . 
12 " light . . . 




28.41 


6 •' 


120 «• 






14^ 




30.61 


6 " 


90 " 






44.07 


loX ♦* heavy . . . 




26.64 


6 " 


heavy . . 






Mjas 


10 1 •• light . . . 




27.':0 


6 • 


light . . 






16.05 


10| '* extra light . 




27.78 


5 ** 


heavy . . 






18.S7 


9 " extra heavy . 




21.44 


6 ' 


' light . . 






1S.90 


P •• heavy . . . 




23.41 


4 • 


' heavy . . 




. - . 


Mi 


" - • light 




2:5.86 


4 • 


' light . . 






lOM 


" heavy .... 


20.99 


4 •' 


extra light . . 




IOjQO 



STRENGTH OF IRON AND STEEL BEAMS. 335 

Another important advanta^ in the use of deeper beams is their 
greater stiffiiess. By referring to the tables, it will be seen 
tiiat a beam twenty feet long, under its safe load, if 6 inches deep 
will deflect 0.95 inch ; 9 inches deep, will deflect 0.63 inch ; 12 i^ 
inches deep, will deflect 0.46 inch ; and 15 inches deep, will de- 
flect only 0.38 inch. 

A floor or structure formed of deep beams will therefore be much 
more rigid than one of the same strength formed of smaller sections. 

There are, of course, cases where the use of deep beams would be 
inconvenient, either from increasing the depth of the floor, or from 
the fact that, with a light load and short span, they would have to 
be placed too far apart for convenience. In general, however, it 
will be best to employ the deep beams. 

Inclined Beams, — The strength of beams inclined to the horizon 
may be computed, with suflBcient accuracy for most purposes, by 
using the formulas given for horizontal beams, taking the horizon- 
tal projection of the beam as its span. 

Steel and Iron Seams. — The relative efficiencies of steel 
and iron beams depend upon the conditions under which they are 
used. The transverse strength of beams of the same length and 
section is proportional to the tensile strength of the material, or 
beams made of steel, of 65,000 pounds tenacity, will possess an 
ultimate stren^h about 80 per cent, greater than similar beams 
made of iron of 50,000 pounds tenacity. But the steel beam will 
not be stiffer than the iron beam — that is, it will deflect under 
working loads as much as the iron beam of the same length and 
section ; the steel beam merely bending farther than the iron beam 
without injuiT to its elasticity. Therefore, if strength without 
regard to stiffness is sought, the steel beam is the best ; but if 
stiffness without regard to ultimate strength is desired, beams of 
either material would probably prove of equal utility. 

Steel beams should not be used for their full load when the span 
in feet exceeds tivice the depth of the beam in inches. 

Note.— Since 1893 the Carnegie Steel Company has discontinued 
the manufacture of iron beams and bars for structural work, and 
now manufacture all their shapes in steel only. As steel beams, 
angles, etc., are sold at the same price per pound, and are about 
20 per cent, stronger than iron, steel has naturally almost entirely 
superseded iron in rolled sections. 

Strengrth of Trenton, Pencoyd, Phoenix, and Car- 
negrie Rolled Beams, Channels, Angle and T-Bars 
— Iron and Steel. 

The foUowing tables ^ve the strength and weight of the various 
sections to be found m the market, together with the general 
dimensions of the I-beams. 

The tables are in all cases made up from data published by the 



386 STBElfGTH OF IB017 AND 8TEBL BBAMB. 

respectiye manufacturers. The deflection of the beams under their 

maximum safe distributed load is also given in some of the tables. 

The tables on pages 849 to 363 will to found very convenient, for 
they can be used for the spans indicated, without any computations 
whatever. In these tables, the loads to the nght of and below the 
heavy line will crack plastered ceilings. When 12- to 24-inch 
beams are used to their full capacity for spans less than 10 feet^ 
the web should be stiffened at the ends. 



STRENGTH OF IBO^ AND STEEL BEAMa 



887 



tENGTH, WEIGHT, AND DIMENSIONS OP TRENTON 

ROLLED I-BEAMS— IRON. 



Blgnation of beam. 



;h, heavy 

light 

heavy 

light 

light 

heavy 

light 

heavy 

light 

heavy 

light 

extra light . 
extra heavy 

heavy 

light 

heavy 

light 

55 lbs 

120 " 

90 " 

heavy 

light 

heavy 

light 

heavy 

light 

extra light.. 



Weight 

per yard, 

in lbs. 



872 

200 

200 

150 

135 

170 

125 

120 

96 

195 

105 

90 

125 

85 

70 

80 

65 

55 

120 

90 

50 

40 

40 

30 

37 

30 

18 



n. 



Safe 
distributed 
load for one 
footof span, 
in lbs.* 



1,320,000 

990,000 

748,000 

551,000 

460,000 

511,000 

877,000 

875,000 

806,000 

360,000 

286,000 

250,000 

268,000 

199,000 

167,000 

168,000 

185,000 

101,000 

172,000 

132,000 

76,800 

62,600 

49,100 

38,700 

36,800 

30,100 

18,000 



m. 



Moment 
of inertia. 



IV. 



Neutral 

uxi8 
perpen- 
dicular to 
web. 



Width of 
flange, 
ill ins. 



V. 



707.1 

523.5 

434.5 

891.2 

288.0 

281.3 

229.2 

283.7 

185.6 

164.0 

150.8 

111.9 

93.9 

83.9 

67.4 

44.3 

64,9 

49.8 

29.0 

23.5 

15.4 

12.1 

9.2 

7.5 

4.6 



6.75 

6.00 

5.75 

5.00 

5.00 

5.50 

4.79 

5.50 

5.25 

5.00 

4.50 

4.60 

4.50 

4.50 

4.00 

4.50 

4.00 

8.75 

5.2r 

5.00 

3.50 

3.00 

3.00 

2.75 

3.00 

2.75 

2.00 



Area of 
cross- 
section, 
ininii. 



27.20 

20.00 

20.02 

16.04 

12.86 

16.77 

12.33 

11.78 

9.46 

13.36 

10.44 

8.90 

12.88 

8.60 

7.00 

8.08 

6.87 

5.50 

11.84 

8.70 

4.91 

4.01 

3.90 

2.S9 

3.66 

2.91 

i.rr 



* For any other span divide this coefficient by span in feet. 



838 



STRENGTH OF lAON AND STEEL BICAMGL 



STRENGTH, WEIGHT, AND DIMENSIONS OF TRENTON 

ROLLED I-BEAMS— STEEL. 





I. 


n. 


ra. 

Moment of 
inertia. 


IV. 


V. 


Deeijifnation 


Weight 

per yard, 

in lbs. 


Safe distribated 
load for one 

foot of span in, 
lbs. Fibre 

strain of 16,000 
lbs.* 


Width of 

Hange, 
in incnes. 


Aieaof 


of beam, 
in inches. 


Neatral axis 
perpendicu- 
lar to web. 


cnjoB- 

iectioii, 

iniuchet. 


15 


150 


753,000 


529.7 


5.75 


14.70 


15 


123 


603,000 


424.4 


5.5 


12.02 


12 


120 


500,000 


281.3 


5.5 


11.78 


12 


96 


407,000 


229.2 


5.26 


9.48 


10 


135 


461.000 


216.1 


5.25 


18.14 


10 


99 


344,000 


161.8 


5.0 


967 


10 


76 


264,000 


123.6 


4.75 


7.50 


9 


81 


262,000 


110.6 


4.75 


7.98 


9 


68 


200.000 


84.8 


4.5 


6.16 


8 


66 


192,000 


71.0 


4.5 


6.47 


8 


54 


154,000 


57.7 


4.d5 


5.28 


7 


60 


151,000 


49.7 


4.25 


5.87 


7 


46.5 


118,000 


38.6 


4.0 


4.55 


6 


50 


104,000 


29.2 


8.5 


4.07 


6 


40 


83,300 


23.4 


8.0 


8.27 


5 


39 


67,000 


15.7 


8.18 


8.80 


5 


30 


52,900 


12.4 


8.0 


2.90 


4 


30 


41,200 


7.7 


2.75 


2.24 


4 


22.5 


31,400 


5.9 


2.62 


2;2i 



* For any other span divide this coefficient by 



STRENOTH OP IRON AND STEEL REAMS. 



339 



lENUTH, WEIGHT, AND DIMENSIONS OF TRENTON 
CHANNEL-BARS AND DECK-BEAMS— IRON. 



esi^ation of bar. 



I. 



Weight 

per yard, 

in IbB. 



II. 



Safe 

distributed 

load, in lbs., 

for one foot 

of span.* 



III. 



Moment 

of inertia 

I. 



IV. 



Width of 
flange, 
in ins. 



V. 



Area of 
cross- 
section, 
in ins. 



Channel-Bars. 



ch, heavy 

light 

heavy 

light 

light 

heavy 

heavy 

light 

light 

extra light. 

light 

extra light 

heavy 

li^t 

extra light 
extra light 
extnt light 
extra light 



190 
120 
140 
70 
60 
48 
70 
50 
45 
33 
36 
25i 
45 
33 

19 

16i 

15 



625,000 

401,000 

381,000 

200,100 

134,750 

102,500 

146,000 

104,000 

88,950 

65,800 

62,000 

89,500 

68,300 

45,700 

aS,680 

22,800 

15,700 

10,500 



586.0 

376.0 

291.6 

153.2 

88.4 

64.0 

82.1 

58.8 

44.5 

32.9 

27.1 

17.3 

21.7 

17.2 

12.6 

7.2 

3.9 

2.0 



4f 

4 
4 
8 

2f 

2i 

2.2 

2 

2i 

2i 

n 

H 

u 



18.85 
12.00 
14.10 
7.00 
6.00 
4.77 
7.02 
5.06 
4.48 
3.30 
3.60 
2.54 
4.32 
3.20 
2.25 
1.92 
1.65 
1.45 



Deck-Beams. 



ch 


65 
55 


91,800 
63,500 


54.7 
35.1 


4i 
4* 


6.29 


I 


5.35 



* For coefficient of steel bars add one-third. 



340 



STRENGTH OF IRON" BEAMS. 



STRENGTH, WEIGHT, AND DIMENSIONS OF TRENTON 

ANGLE AND T BARS. 





I. 


n. 




I. 


n. 


Designation of 
bar. 


Weight 

per foot, 

in ibs. 


Safe 

diBtributed 

load for one 

foot of span, 

in lbs. 


Designation of 
bar. 


Weight 

per foot, 

inlba. 


Safe 

distriboted 

load for onej 

footofspao, 

Inlbf. 


ANGLEf 


) Even Li 


EGS. 


Anolbs 1 


[Jme^ual LBG8. 


6 in. X 6 in. 
44 " X 44 " 


19.00 
124 


36,900 
18,000 


6 in. X 4 in. 


14.00 


( 80»680 
14,7S0 


4 " X 4 " 
34 •• X 34 " 


94 
81 


12,184 
9,200 


6 " X 84 " 


10.20 




3 " X 3 " 
2| " X 2| " 


4.80 
5.40 


4,611 
4,710 


44 «» X8 " 


9.00 


f 14.680 

( T.oao 


24 " X 24 " 
2J " X 2i " 


3.90 
3.50 


8,156 
2,530 


4 " X 8 ** 


7.00 


( •,860 

( ».8n 


2 " X 2 " 
11 " X 1| « 


3.13 
2.00 


1,970 
1,150 


34 " X 14 «« 


4.00 


r 6.616 
( 1,148 


14 " X 14 « 
IJ " X l| " 


1.75 
1.00 


832 
393 


3 " X 24 " 


ua 


( M90 
\ S,S88 


1 "XI " 
1 " X J « 


0.75 
0.60 


246 
186 


3 " X 2 " 


4.00 


I 4.884 

1 8,080 


J " X g « 


0.56 


133 












T-B. 


kR8. 






4 in. X 4 in. 


12.50 


15,800 


3 in. X 2 in. 


4.80 


2.640 


34 " X 34 " 


9.60 


10,550 


2 " X 14 " 


8.00 


1.866 


3 " X 3 ♦• 


7.00 


6,680 


2\ " X IJ " 


2.40 


604 


24 " X 24 " 


5.00 


3,850 


2  X 1 " 


2.15 


467 


2 •♦ X 2 ♦• 


3.13 


1,970 


14 •• XI •♦ 


1.86 


421 


5 •• X 24 " 


11.70 


6,044 









** 7or coeflicient of steel barn add one-third. For any other tfma dMdo tilli 
foeiBcient by span. 



SISENGTH OF IBON AND STEEL BBAMS. 



341 



TRENGTH, WEIGHT, AND DIMENSIONS OF CARNEGIE 

I-BEAMS— STEEL. 



Depth 

of 

beam, 

in inches. 


Weight 

per 

foot, 

in lbs. 


Thickness 

of 

web, 

in inches. 


Width 

of 

flange. 

in inches. 


Safe dis- 
tributed load 
for one foot 
of span, in lbs. 
16,000 lbs. 
fibre strain 

for 
buildings.* 


Safe dis- 
tributed load 
for one foot 
of span, in lbs. 
12,500 lbs. 
fibre strain 

for 
bridges.* 


24 


100 


.75 


7.20 


2,086,600 


1,670,000 


24 


80 


.50 


6.95 


1,830,500 


1,486,000 


ao 


80 


.60 


7.00 


1,545,600 


1,207,500 


ao 


64 


.50 


6.25 


1,222,400 


955,000 


16 


75 


.67 


6.31 


1,077,800 


841,700 


16 


60 


.54 


6.04 


916,800 


715,800 


15 


50 


.45 


5.75 


7.v3,aoo 


588,500 


15 


41 


.40 


5.50 


603,200 


471,800 


12 


40 


.39 


5.50 


500,100 


390.700 


12 


S2 


.85 


5.25 


395,200 


3083)0 


10 


33 


.37 


5.00 


344,000 


268,800 


10 


25.6 


.32 


4.75 


263,800 


206,100 


9 


27 


.31 


4.'?5 


262,200 


204,900 


9 


21 


.27 


4.50 


199,900 


156,100 


8 


22 


.27 


4.50 


191,600 


149,700 


S 


18 


.25 


4.25 


154,000 


120,300 


7 


SO 


.27 


4.25 


151,400 


118,300 


7 


15.5 


.23 


4.00 


117,600 


91,900 


6 


16 


.26 


3.fi3 


101,800 


79,500 


6 


13 


.23 


3.50 


83,500 


65.300 


5 


13 


.26 


3.13 


67,000 


52,400 


5 


10 


.22 


3.00 


52,900 


41,800 


4 


10 


.24 


2.75 


41,200 


32,200 


4 


7.6 


.20 


0.63 


31,400 


24,600 



* For any other span divide tliis coefficient by span. 



343 STRBNGTU OF IBOM AXSi 8TBEL 

STRENGTH, WEIGHT, AND DIMENSIONS OP CAENEGIl 
CHANNEL-BABS— IRON. 



STRENGTH OF IRON AND STEEL BEAMS. 



34a 



STRENGTH, WEIGHT, AND DIMENSIONS OF CARNEGIE 

CHANNEL-B ARS- STEEL. 











Safe dis- 
tribured load 


Safe dis- 
tributed load 


Depth of 
cbaDnel, 
in inches. 


Weight 

per foot, 

in lbs. 


Thickness 

of web, 
in inches. 


Width 
of flange, 
in iuches. 


for one loot 

of span, in lbs. 

16,000 lis. 

fibre strain 

for 
buildings.* , 


for one foot 

of span, in lbs. 

12,500 lbs. 

fibre strain 

for 
bridges.* 


15 


32 


.40 


3.40 


464,700 


316,200 


15 


51 


.775 


3.775 


554,700 


433,400 


12 


20 


.30 


2.90 


209,600 


163,800 


12 


30i 


.55 


3.15 


273,600 


213,800 


10 


15i 


.26 


2.66 


136,100 


106,300 


10 


23} 


.51 


2.91 


180,500 


141,000 


9 


m 


.24 


2.44 


102,700 


80,200 


9 


2(H 


.49 


2.69 


138,700 


108,400 


8 


lOi 


.22 


2.22 


75,n00 


58,800 


8 


17i 


.47 


2.47 


103,700 


81.000 


7 


8i 


.20 


2.00 


53,100 


41,500 


7 


m 


.45 


2.25 


75,000 


58,600 


6 


7 


.19 


1.<S5 


39,400 


80,800 


6 


12 


.44 


2.14 


55,400 


43,300 


5 


6 


.18 


1.78 


27,900 


21,800 


5 


lOi 


.43 


2.03 


39,000 


30,500 


4 


5 


.17 


1.G7 


18,700 


14,600 


4 


8i 


.42 


1.92 


25,700 


20,100 



* For any other span divide this coefficient by span. 



344 



STBENGTH OF IBON AlSfD STBEL BEAHS. 



STRENGTH, WEIGHT. AND DIMENSIONS OP JONES ft 
LAUGHLIN'S, LIMITED, STEEL BEAMS. 











Safe dis- 


Safedis- 










tributed load 


tribnted load 


Depth of 

beam, 
In inches. 


Weight 

per foot, 

in lbs. 


Thickness 

of web, 
in inches. 


Width 
of flange, 
in inches. 


for one foot 

of span, in Iba. 

ltf,000 Ibe. 

fibre strain 

for 
buildings.* 


for one foot 

of span, in lbs. 

12,000 lbs. 

fibre strain 

for 
bridges.* 


15 


70 


0.64 


6.366 


1,089,700 


810,700 


15 


59 


0.468 


5 968 


910.000 


710.900 


15 


48 


0.406 


5.726 


705,200 


650,900 


15 


39 


0.375 


5.475 


673,600 


448,000 


12 


50 


0.598 


5.723 


536,800 


419,400 


12 


38 


0.343 


5.468 


471,800 


868.600 


12 


30 


0.312 


5.218 


876,400 


294,100 


10 


32 


0.3125 


4.937 


826.500 


254,800 


10 


23.8 


0.281 


4.72 


251,100 


196.200 


9 


24.5 


0.296 


4.671 


239.700 


187,800 


9 


19.75 


0.266 


4.39 


189.100 


147,700 


8 


25 


0.287 


4.537 


101,600 


149,600 


8 


18 


0.25 


4.25 


15i,800 


119.400 


7 


18.3 


0.2G6 


4.266 


141,400 


110,600 


7 


15.25 


0.25 


4.0 


115,500 


90,200 


6 


16.6 


0.265 


3.765 


100.900 


78.800 


6 


12.75 


0.25 


3 5 


8-^.100 


64,100 


5 


13 


0.31 


3.06 


07,000 


62,800 


5 


10 


0.22 


2 845 


67,600 


46.000 


4 


10.2 


0.28 


2.78 


41,100 


82,100 


4 


7.9 


0.25 


2.(59 


32.000 


26,000 


4 


6.85 


0.19 


2.56 


31.000 


24,200 


8 


7 


0.19 


2.152 


22,000 


njioo 


3 


5.1 


0.156 


2.03 


16,800 


12,700 



* For any other span divide this coefficient by spaa. 



STRENGTH OF IBON AND STEEL BEAMa 



345 



STRENGTH, WEIGHT, AND DIMENSIONS OF PHCENIX 

I-BEAMS— STEEL. 



Depth of 

beam. 
In inches. 


Weight 

per yard, 

in lbs. 


Thickness 

of web, . 

in inches. 


Width 
of flange, 
in inches. 


Safe dis- 
tributed load 
for one foot 
of span, in lbs. 
16,000 lbs. 
fibre strain 

for 
buildings.* 


Safe dis- 

tribnted load 

for one foot 

of span, in lbs. 

12,500 lbs. 

fibre strain 

for 
bridges.* 


15 


225 


.62 


6.375 


1,076,000 


840,600 


15 


180 


.50 


6.125 


920,000 


718,750 


15 


150 


.45 


5.75 


752.000 


587,500 


15 


123 


.40 


5.50 


602,000 


470,300 


12 


120 


.39 


5.50 


500,000 


390,600 


12 


98 


.35 


5.25 


394,000 


307,800 


10^ 


99 


.35 


5.00 


368.000 


287,500 


lOi 


764 


.30 


4.75 


284,000 


221,800 


9 


81 


.31 


4.75 


262,000 


204,600 


9 


63 


.27 


4.50 


200.000 


156,200 


8 


66 


.27 


4.50 


190,000 


148,400 


8 


54 


.25 


4.25. 


154,000 


120,300 


7 


60 


.27 


4.25 


142,000 


110,900 


7 


m 


.28 


4.00 


114,000 


89.060 


6 


48 


.26 


3.625 


100,000 


78,120 


6 


39 


.23 


3.50 


82,000 


64,060 


5 


39 


.26 


3.125 


66,000 


51,560 


5 


30 


.22 


3.00 


52,000 


40,620 


4 


30 


.24 


2.75 


40,000 


31,'250 



* For any other span divide this coefficient by span. 



346 STB^GTH OF IRON AND STBBL BBAM& 



Peucoyd Beams and Cliaiinels* 

The coefficient for strength of the Pencoyd sections has been 
calculated for a fibre strain of 14,000 lbs. for iron, and 16.500 lbs. 
for steel. 

These tables also contain the maxim am load that should be 
placed on the beam, whatever the length, unless the web is stiffened 
at the points of support. 

Example. — What should be the maximum distributed load for 
a 15-inch 145-lb. iron beam of 10 feet span ? Ans. The coefficient 
of this beam is 648,600 lbs. Dividing by 10, we have 04.860 lbs., or 
32.4 tons as the safe load ; but we see, by the last column, that it 
will not be safe to put more than 22.1 tons on the beam without 
stiffening the web. Hence, the safe load for that span is 22.1 tons. 
It is only for very short beams that this condition will apply. 



STRENGTH, WEIGHT, AND DIMENSIONS OF PENCOYD 

I-BEAMS— STEEL. 



Depth of 

beam, 
in inches. 


Weight 

per yard, 

in lbs. 


Thickness 

of web, 
in inches. 


Width 
of flange, 
in inches. 


Safe dia- 

tribated load 

for one foot 

of span, in lbs. 

14,000 lbs. 

fibre strain 

for 
baildlnfiB.* 


Maxlmnm 

]oad in tons, 

witlioat 

atlffeniiif 

welK 


10 


70.1 


* .30 


4.50 


248,260 


18.06 


9 


GO.l 


.28 


4.80 


198,010 


10.44 


8 


51.7 


.26 


4.00 


146,360 


&g8 


7 


48.4 


.24 


3.75 


106,840 


7.60 


6 


34.9 


.22 


3.40 


76,160 


6.18 


5 


27.3 


.20 


3.00 


49,000 


4.04 


4 


•25.0 


.22 


2.6 


&5,860 


6.05 


4 


18.6 


.16 


2.8 


27.180 


8.16 * 


3 


20.5 


.22 


2.4 


21,480 


8.77 


8 


15.9 


.16 


2.2 


17,880 


%.7% 



' For any other span divide this coefllcient by span. The load, 
be greater than that in next column, unless the web is stiflenad aft aoppoita 



STRENGTH OP IRON AND STEEL BEAMa 



347 



STRENGTH, WEIGHT, AND DIMENSIONS OP PENCOYD 

I-BEAMS— IRON. 



Depth 

of 

beam. 

In inches. 


• 

Weight 

per 

yard, 

in lbs. 


Thickness 

of 

web, 

in inches. 


Width 

of 

flange, 

in inches. 


Safe dis- 
tributed load 
for one foot 
ofspan,inlb8. 
14,000 lbs. 
fibre strain 

for 
buildings. 


Maximam 

load in tons, 

without 

stiffening 

web. 


15 


190.0 


.562 


6.687 


844,560 


89.57 


15 


145.0 


.437 


5.125 


648,600 


22.10 


16 


124.1 


.406 


5.609 


541,980 


18.59 


12. 


1680 


.656 


5.5 


578,640 


88.63 


12 


120.0 


.453 


4.80 


424,440 


22.22 


12 


89.5 


.343 


5.0 


817,440 


13.60 


10* 


134.4 


.468 


5.25 


429,560 


22.13 


10* 


108.3 


.406 


4.87 


347,420 


17.71 


10* 


89.3 


.343 


4.5 


288,460 


13.35 


10 


111.7 


.5 


4.625 


324,0^ 


23.68 


10 


90.4 


.343 


4.375 


276,860 


13 18 


9 


90.0 


.406 


4.75 


246,420 


16.53 


9 


70,6 


.312 


4.25 


195,880 


9.94 


8 


80.0 


.406 


4.375 


188.840 


13.88 


8 


61.0 


.297 


4.0 


161,400 


10.46 


7 


65.8 


.437 


3.20 


132,760 


15.69 


7 


51.4 


.234 


3.61 


114,880 


6.17 


6 


115.5 


.625 


5.25 


196,740 


21.19 


6 


90.1 


.5 


4.87 


160,000 


16.42 


6 


55.5 


.281 


3.84 


103,480 


7.75 


6 


40.0 


.218 


3.47 


76,500 


5.25 


5 


29.7 


.26 


3.0 


46,560 


4.91 


4 


24.6 


.22 


2.6 


30,000 


4.33 


4 


18.2 


.16 


2.3 


23,000 


2.71 


8 


20.1 


.22 


2.4 


19,340 


3.23 


8 


16.6 


.16 


2.2 


14,740 


2.33 



n.-.n .-n-KF-N-irrEi <if runs anti stkei, beamp. 



II) II II 111 i- ai a : 

..;a;.iin4i..ioi!>.«i6.«ris."iw.:<iia.i7 ii 
.%!i!>«(i)<!inir<:mi3!H;Njniii!!ii ■j'.-m v 

. ■ii.ai !■ riio.'ri m.-n ?.!«i r.m li.i- :■ 



> j!is am i!bs| i!: 

: .;« »:« ■:'■ «' 



i.!jH i.nir 

■■•.-■1 



snimfiTB OF irok and stkkl beaiis. :iol 



li.wriftniod. E\ts 



10 U ■< 



» su ss.rf M.-o «is n.c: ^.m m.m, si.«> -^.m 4>.nj m. 
lb i3> *■.;* ii,i3 «.M is,«s i«,w is.n-I i».:i w.-v u.iJ ui. 



348 



STRENGTH OF IRON AND STEEL BEAH8. 



STRENGTH, WEIGHT, AND DIMENSIONS OP PENCOY] 

CHANNELS. 

For Steel. 



Depth 

of 

channel, 

in inches. 


1 
! 

Weight 

per 
vard. 
in lbs. 


Thickness 

of 

web, 

in inches. 


Width 

of 

flange, 

in inches. 


Safedis- 

tribnted load 

for one foot 

of span, in lbs*. 

14,000 IbH. 

fibre strtiin 

for bnildini^. 


1 

1 
1 

Maximnm 

load ill ton*" 

u-iiiiout 

St ffeiilug 

web. 


8 


81.8 


.22 


1 

i 2.27 


79,0S0 


6 55 


7 


26.6 


.21 


2.11 


79,080 


6.91 


6 


22.2 


.20 


1.95 


42,600 


6.25 


6 


18.1 


.19 


1.79 


29,360 


4.65 


4 


14.7 


.18 


1.P8 


19,800 


8.79 


Foil IltON. 


15 


139.0 


.562 


8.94 


539,940 


84.84 


15 


106.0 


.375 


8.87 


437,600 


16.88 


12 


88.5 


.406 


2.94 


284,280 


18.49 


12 


60.0 

• 


.281 


2.61 


192,440 


9.14 


12 


61.5 


.2.S1 


8.09 


206,460 


9.06 


10 


59.7 


.328 


2.75 


164,740 


18.67 


10 


47.5 


.25 


2.5 


133,660 


8.46 


9 


52.7 


.812 


2.69 


125,740 


18.90 


9 


37.2 


.234 


2.36 


92,640 


7.17 


8 


43.0 


.281 


2.28 


96,83» 


8.77 


8 


39.5 


.25 


2.50 


80,800 


7.66 


8 


30.7 


.218 


2.28 


68,940 


4.66 


7 


41.0 


.297 


2.30 


78,700 


9.07 


7 


25.0 


.171 


1.95 


49,aaao 


8.42 


6 


81.9 


.25 


2.25 


67,160 


6.60 


6 


22.7 


.20 


1.7S 


86,820 


5M 


5 


28.9 


.23 


2.06 


34,120 


5.14 


4 


21.5 


.25 


1.69 


24,060 


6.19 


4 


16.5 


.19 


1.26 


19,800 


4.99 


8 


15.2 


.22 


1.68 


12,640 


8.49 


8V 


11.8 


.25 


1.87 


0,660 


8.90 


1 


8.8 


.22 


1.09 


4,600 


9.49 



SAFE DBTRIBUTED LOADS AND DEFLECTIONS OP 
PENCOYD BBAMS-mON. 



.1 »«■ d.'llprll<>1» In Inc 



onwiMjndins i 



~.uia markwr* ta-T be idled in ^Ic^il, ivlieii i\«- wplghis will be Incpeat 
otr cent. -faff loail aboul ao pvr cent. Detleclluii (.rarilcslly Ihe BUoe w 
Etn vKb aQiwl loadt. 



STRENGTH -OF TRENTON STEEL I 



<TI.— The flenreH tn Italic arc thndefli'cClnnH. In InchM. 
-ds above. For the dcllMHono or graileM nafB loiula In i 
DofttleUbiilarflgDrealii iUlliM. 



STREIfOTH OF OAHNEGIE IKON BEAHB. 



E DISTRIBUTED LOADS OP CARNEGIE IRON BEAMS. 



e loadi' In net tons 



In mlddlii, m 



In 


Weldrt 1 
1^. 


Length of »pB 


n, in feet. 


IB 


3W 




B 


U 


ISO 




Ml 


u 


tn 




Q 


u 


Mat 






V 


m 






»( 


uo 







M rlgbt and below bcavy lii 



STRBVOTH OF CARNEGIE BTEEL 



O 5 1 8 


1 




s 


g'l i| 


1 1^. 


I Hi 


Hi 


ej- S 1 i 


6. S ^^ 


1 1*1 


° t i-^ 


J =?l,- 


CO '. ■5-g 


1 £ M ^ 


Mil 


t !!l 


ss ; r| 


§ Z^-^M 


— c i § 


1 " s-IC^ 


2 ill 


lli^ 


C = 3 „ 


siiin 


2 S ^-^ 


' !r = 1 ^ . "f 


s '1 


^ ■; ■§ -1 i ^■ 



||S J?|j|| 






BTBBNaTH OF CABNEOIE STEEL BKAU8. 8S7 



1 i ' " "I 1 

s s s ^Is S u \ s 



^ s 



STBENOTH or OABKEQIE 8TEBL ] 



8TBENGTH OF OAKNBGIli: STEKL BKAMS. 



STBENQTH OF IRON BEAJK. 



STRENGTH OF IRON BEAMS. 



test safe load in IbB. iiDlforoil; dtsttibated. Including weight a 

ir 13,000 Ibe. fibre stnitn. 

lonceDtTBled load in middle of beam allow one-bair o[ (hat givu 



In Inches. 






«t'B*-l 



STRENGTH OF IBON BBAHB. 



Angles with UnegwU Legs — Long Leg VerHeal. 

vatceceafe In&d Id Ibn. untfomily distributed, inclndliis welifht of aiula- 
. For K.noo \bf. fliire atr^n. For coucuntnled IobU Iu middlB of b««m »&m 



STI :QTH or IRON ] JJ8. 



Attglet ailh Uaequal Leg* — Short Leg TeHietd. 

GraateBtrsfe ^•"•/> in ih- ..nif-^iiy diatribnled, Inclodine welabt of aosle- 
iron, f.ir 18,000 1 oonceolouad iod in mtadle of bom allow 



364 



BEAMS SUPPORTING BRICK WALLS. 



Beams Supportingr Brick Walls. 

In the case of iron beams supporting brick walls having no 
openings, and in wliich the bricks are laid with the UBual bond, the 
prism of wall that the beam sustains will be of a triangular shape, 
tlie height being one-fourth of the span. Owing to freqaenft iirogn- 
larities in the bonding, it is best to consider the height as one4hinl 
of the span. 




Fig. 7. 



The greatest bending-stress at the centre of the beam, mulling 
from a brick wall of the above shape, is the same as that caused by 
a load one-sixth less, concentrated at the centre of the beam, or 
two-thirds more, evenly distributed. 

The weight of brickwork is very nearly ten pounds per square 
foot for one inch in thickness ; and from tlds data we find that 
the bending-stress on the beams would be the same as that caused 
by a uniformly distributed load equal to 

25 X square of span in feet X thickness In inches 

- 1» 

J) 

Having ascertained this load, we have merely to determine from 
the proper tables the size of beams required to carry a distrfbuted 
loail of this amount. 

£xAMi>LK. — It is proposed to support a solid brick wall IS 
inches thick, over an opening 12 feet wide, on rolled Iron beams: 
*. should be the size and weight of 1)eams ? 
x. Hy the rule given alH>ve, the unifonnly distributed load 



FRAMING AND CONNECTING IRON BEAMS. 



365 



which would produce the same bending-stress on the beam as the 
wail, equals 



25 X 144 X 12 
9 



= 4800 pounds. 



As the wall is twelve inches thick, it would be best to use two beams 
placed side by side to support it, as they would give a greater area 
to build the brick on ; then the load on each beam would be 2400 
pounds, or 1.2 tons. From the preceding tables for safe distributed 
loads on beams, we find that a 4-inch heavy beam would just about 
support this load; but as a 5-inch light beam would not weigh any 
more, and would be nmch stiffer, it would be better for us to use 
two 5-inch light beams to support om- wall. 

If a wall has openings, such as windows, etc., the imposed weight 
On the beam may be greater than if the wall is solid. 

For such a case consider the outline of the brick which the beam 
sustains to pass from the points of support diagonally to the out- 
side comers of the nearest openings, then vertically up the outer 
line of the jambs, and so on, if other openings occur above. If 
there should he no other openings, consider the line of imposed 
brickwork to extend diagonally up from each upper comer of the 
jambs, the intersection forming a triangle whose height is one-third 
of its base, as described above. 

When beams are vsed to support a wall entirely (that is, the 
beams run under the whole length of the wall), and the wall is more 
than sixteen or eighteen feet long, the whole weight of the wall 
should be taken as coming upon the beams ; for, if the beams should 
bend, the wall would settle, and might push out the supports, and 
thus cause the whole structure to fall. 



Framingr and Connecting Iron Beams. 

When beams are used to support walls, or as girders to carry 
floor-beams, they are often placed side by side, and should in such 







Fig. 8. Rg. 9. Fig. 10. Fig. 11. 

cases be furnished with cast-iron separators fitting between the 

flanges, so as to firmly combine the two beams. These separators 

"may be placed from four to six feet apart. Such an arrangement 

iB shown by Figs. 8 and 10, Figs. 9 and 11 showing fonus of sepa- 



ooo 



rnAJ»i:>ij ainu uuin w liu i ixn u ittuiM t5iSAM». 



rators usually employed; that with two bolt-boles being iimmI 
the 15-ineh and ]2i-inch beams, and that with a single hole 
smaller sizes. 





Fig. 12. Fig. 13. 

When beams are required to be framed together, it is usu 
done as shown by the accompanying cuts, in which Fig. 12 sli 
two beams of the same size fitted together. Fig. 13 shows a b 
fitted flush with the bottom flange of a beam of larger size. 
14 shows a smaller beam fitted to the stem of a larger beam, al 
the lower flange. 





Fig. 14. Fig. 16. 

Wooden heanis may be secured to an iron girder in the si 

manner as an iron beam, by framing the end, and securing it b] 

^-bracket; or an angle-iron may be riveted to the web of 

3n eirder to afiford a flat bearine on which the wooden faeun i 



FRAMING AND CONNECTING IRON BEAMS. 



367 



The different rolling mills have standard connection? for con- 
necting iron beams with each other. 

The standard connection angles for all sizes and weights of steel 
and iron I-beams manufactured by Carnegie, Phipps & Co., 
Limited, are illustrated on page 3(58. These connections were 
designed on the basis of an allowable shearing strain of 10,00;) lbs. 
per square inch, and a bearing strain of v*(),000 lbs. per square inch 
on rivets or bolts, corresponding with extreme fibre strains in the 
I-beams of 16,000 and 12,00') lbs. per square inch, for steel and 
iron respectively. The number of rivets or bolts required was 
found to be dependent, in most instances, on their bearing values. 

The connections have been proportioned with a view to covering 
most cases occurring in ordinary practice, with the usual relations 
of depth of beam to length of span. In extreme instances, how- 
ever, where beams of short relative span lengths are loaded to their 
full capacity, it may be found necessary to make provision for 
additional strength in the connections. The limiting span lengths, 
at and above which the standard connection angles may be used 
with perfect safety, are given in the foUowing table : 



TABLE OP MINIMUM SPANS, FOR CARNEGIE I-BEAMS, 
WHERE STANDARD COxVNECTION ANGLES MAY BE 
SAFELY USED, WITH BEAMS LOADED TO THEIR 
FULL CAPACITY. 





Stbbl I-Beams. 






Iron I-Bbams. 








^« 










I 








S$ 






S3 a> 




at V 




rS. (U 










cc a 




li gj 




K a> 


Designation 


Su 


Designation 




Designation 


^y 


Depignation 


VT. 


of 




§.H 


of 






of 


§.£ 


of 


S.h: 


beam. 




Is 


beam. 




C OB 
•- P. 

9.5 


beam. 


10. 


beam. 


'= 5 


20" -80. 


lbs. 


17.0 


9"— 27. 


lbs. 


15"-^. Ibp. 


9"— 2S.5 lbs. 


8.0 


*' 64. 




16.0 


♦' 21. 

8"-22. 




8.C 


•' 60. " 


13. ( 


*' 23.5 " 


8.0 


15"-75. 




12. 0' 




8.0 


" 50. " 


13. ( 


8"— 34. '' 


7.0 


" 60. 




11.5 


" IS. 




7.0 


12"-56.5 " 


9.( 


" 27. " 


7.0 


" 50. 




11. C 


7"— 20. 




6.0 


" 42. " 


8.0 


" 21.5 " 


6.5 


♦• 41. 




10.5 


" 15.5 




5.5 


lOi'MO. '' 


9.( 


7"— 22. '' 


5.0 


12"- 40. 




8.5 


6"— 16. 




6.5^ 


" 31.5 " 


10.01 '' 18. " 


6.5 


" 38. 




7.5 


" 13. 




6.d 


10''-42. " 


10.5 


6"— 16. •' 


5.0 


10" 88. 




lO.S 


5" 18. 




4.0^ 


'• 36. " 


10.5 


" 13.5 " 


4.5 


"- 25.6 




9.0 


" 10. 




4.0 


" 30. " 
9"— 38;5 " 


10.5 
6.5 


5"-12. " 
'♦ 10. ♦' 


3.0 
3.0 













i STANDABU CONNECTION AH6LB8 KOE I-BEAJfS. 



% 







(H Ha ten* _rm 

■III d++l 






+  
+ +  



4 ^4&t-l.-»-.»tf n. 

«x nt<x''-~°- 'rf''t- 






H-'iiil- 









fi 






SSPAaATOBS FOR CAKNEGIE STEEL BEAMS. 3<{9 

SIZES AND WEIGHTS OF SEPARATORS FOR CARXECilE 
STREL BEAMS. 

Separators for 20" lieains arc maile nf I" nii'IHl. 



WITH TWO BOLTS, 



IS 

s 



SEPARATORS WITH 



870 SEPARATORS FOR CASNBaiE IRON 



SEPARATORS WITH TWO BOLTfl. 



HEI-AR^TORS WITH OKB BOVt. 



la 36 1 66i 


lOJ 


S 




12 8a 43 


9? 


n 




lOi 4A 40 


10,'„- 


5 




101 4/1 3n 


Bt 


6 




10 


7 


42 


10 


6i 




10 


56 


;i8 


9i 


5 




10 


5f» 


:iO 


9,'r 


4 




9 


6c 


381 


10 


5 




9 


t» 


28 


85 


4 




9 


»a 


a3i 


Si 


4 


3! 


8 


Se 


31 


91 


e 




8 


86 


37 


84 


^ 




8 


8.1 


311 


8 


5 




7 


96 


2i 


8rV 


4J 




7 


9a 


18 


7i 


4 




e 


105 


IS 


!^'- 


4 s 


- 1 


6 


10a 


13J 




lit 1 





116 


13 


«.) 


Si 1 




6 


llu 


10 


61 


8i K 




i 


13 


7 


6t 


8 1 





STRENGTH OF CAST IKON BEAMS. 



371 



CHAPTER XV. 

STRENGTH OF CAST-IRON. T7700DEN, AND STONE 
BEAMS — SOLID BUILT BEAMS 



Cast-iron Beams. — Most of our knowledge of the strength 
of oast-iron beams is denved from the experiments of Mr. Eaton 
Hodgkinson. From these experiments he found that the form of 
cross-section of a beam which will resist 
the greatest transverse strain is that shown 
in Fig. 1, in which the bottom flange con- 
tains six times as much metal as the top 
flange. 

When cast-iron be^ms are subjected to 
very light strains, the are^s of the two 
flanges ought to be nearly equal. As in 
practice;* it is usual to submit beams to 
strains less than the ultimate load, and yet 
beyond a slight strain, it is found, that 
when the flilnges are as 1 to 4, we have a proportion which 
approximates very nearly the requirements of practice. The thick- 
ness of the three parts — web, top flange, and bottom flange — 
may with advantage be made in proportion as 5, 6, and 8. 

If made in this proportion, the width of the top flange will be 
equal to one-third of that of the bottom flange. As the lesull of 
his experiments, Mr. Hodgkinson gives the following rul(» for the 
breaking-weight at the centre for a cast-iron beam of the above 
form :—- 




Fig 1 



Breaking-load in tons = 



Area of hot. flange ^ depth ^ o 426 
in square inches in ins. 

clear span in feet 



(1) 



Cast-iron beams should always be tested by a load equal to that 
which they are designed to carry. 

Wooden Beams, — Wooden beams are almost invariably 
square or rectangular shaped timbers, and we shall therefore con- 
sider only that shape in the following niles and fonnulas. 



372 



STRENGTH OF WOODEN IJEAMS, 



For beams willi a rectangular cross-secticHi, wo can simplify our 

formulas for strength by substituting for the moment of inertia 

}, X ip 
its value, viz., ~r:>~~ , where h = breadth of beam, and d its depth. 

Then, substituting this value in the genenil formulas for beams, 
W(> have for rectangular beams of any material the following 

foniiulas : — 

B V an LS fixed at one end, and loaded at the other (Fig. 2). 








Fig. 2, 



W 



or 



Safe load in pounds = 



Iheadth in inches = 



breadth x square of deptli X A 
4 X length in feet 

4 X load X length in feet 
s<|uarH of depth X A ' 



(2| 



(3) 



ficatns fired at one end, and loaded with uniformly dUdrihuled 

load (Fig. ;5). 




■'^^y^y 



Fig 3 



breadth x snuan^of depth X A 

Safe luad in j>ounds = ., ^ , . -.--; — \--i » 

* 2 X lengrh in feet 

or 

2 X Icmjrih in fivt X loocl 

Iheailth in inches = — ^ .. ,. e~\r:zr\r^ — i — . 

8(|uare of deplli X ^1 



14) 



(&( 



STRENGTH OF WOODEN BEAMS. 



§73 



Beams supported at both ends, loaded at middle (Fig. 4). 

W 




Safe load in pounds = 



Fig 4. 

__ broadlli X square of dopth x A^ 



span in feet 



or 



Breadth in inches 



_ span in feel x load 



(6) 



(7) 



square of <leptli X A' 

Beams supported at hoik andsj had. uniformly distributed 
(Fig. 5). 




Fig. 5. 

2 X breadth x square of depth x A 
Safe load in pounds = span in feet ' ^^^ 



or 



Breadth in inches = : 



span in feet x load 



2 X square of depth X A' 



(0) 



Beams supported at hidh ends, loaded with concentrated load 
yOT AT CENTRE (Fi^. (>). 




K-n—> 



m 




»w 



. / 

. / 



Fig 6 



breadth x sf|. of depth X span X A 
Safe load in pounds — 



4 X //< A // 



or 



BreaiUh in inches — 



4 X load X /;/ X )i 
square of dcptli x span x A' 



(101 



(11 



374 



STRKNGTH OF WOODEN BKAMS. 



Beams supported at both ends, and loaded wiUi W pounds at 

a distance m /row. each end (Fig. 7). 






^■■^ :.:■■■■■ ' 



■m-*- 



WM 



<rW 




^W ^ 



Fig. 7. 



Safe load M' in pounds _ breadth X sciuare of depth X A ^ 



or 



at each point 



Breadth in inches = 



4 X m 
4 X load at one point X m 



(12) 



(13) 



scj. of depth X A 

Ndte. — Iti the lUKt two c-aflCH the ieiigthB denoted by tn and n should b« takeu 
in feet, the Huinc us the ripiiUH. 

Valuks of the Constant A, 

The letter A denotes the safe load for a unit beam one inch 
scpiare and oik! foot si>an, loaded at the eentre. This is also one- 
eii^hteenth of the modulus of rupture for safe loads. The follow- 
ing are tlie values of .1, which are obtained by dividing the moduli 
of rupture in Chap. XIV. by 18. 

TABLE I. 
Values of .4.— Co-kfficient for Beams. 



MuteriMl. 


.1 lbs. 

;W8 

888 
KM) 

«) 

TO 


Matcrhil. 


.4 \\m. 


( "nst iron 


Pino, white. Wentem 

'• Texue yellow 

S^)ruce 

\N hltewocKi (poplar) 

; Rluostoiie tlagiiiii!; iHudvoii 
1 Kiver) 


05 


Wrou'hi-iroM 


90 


Steel 


TO 


Aineriean wood.-* : 
(M:,.-fmit 


05 


Ilt'iiilix'k 


ti 


<);ik. \^ hiu- 


< Jr.inite, averaire 

Limestone 


17 


I'iiii-. • itor-'iM vellow 


15 


( )1CMI11 


Marble 


17 


iid or NDrway 

\\ hitc. Ka^te^n 


Sail' stone 


M 
60 







'lIu'M- v:ilnes for the ccwnieionl .1 are one-thinl of tlio hn»aking- 
u<iL:iii ot tiiiilMTs of the same si/.«> and i|U:tlity as that iisi'd in flrst- 
rla*-- Inii Minus. Tlii'< i»< a siiMirirnt allo'vanc** for timlM»rs in roof 
trii^^*'^, and lM'ani«« wliirh do not have to carry a nion* w^ven* Umd 
than that on a dwrllini: hou<(> floor, and small halls, etc. Wliori' 
tJMTi' i^ likely to \w M>ry much vihraiion, as in the lloor <if a mill, 
or a L;\niiiaNium tloor. or tlitoi-s of lari;*' public hail^i. llii* uiillitir 
r<rnmm«iid- I hat oidv foiu'-tifth.s of the :i1n»vc values of .1 In* usmmI. 



RELATIVE STRENGTH OF BEAMS. 376 

« 

ExAMPLV 1. — What load will a hard-pine beam, 8 inches by 18 
inches, securely fastened into a brick wall at one end, sustain with 
safety, 6 feet out from the wall ? 

Ans. Safe load in pounds (Formula 2) equals 

8 X 144 X 100 



4x6 



= 4,800 lbs. 



EXA.MPLE 2. — It is desired to suspend two loads of 10,000 pounds 
each, 4 feet from each end of an oak beam 20 feet long. What 
should be the size of the beam ? 

Ans. Assume depth of beam to be 14 inches ; then (Formula 13\ 

breadth .— ^ * — ==^ — = 11 inches, nearly ; therefore the beam 
should be 11 x 14 inches. 



Helative Streng:th of Rectang:iilar Beams. 

From an inspection of the foregoing forniulas, it will be found 
tliat the relative strength of rectangular beams in different cases 
is as follows : — 

Beam supported at both ends, and loaded with a uniformly 
distributed load 1 

Beam supported at both ends, and loaded at the centre ... i 

Beam fixed at one end, and loaded with a uniformly distributed 
load . . . . ; \ 

Beam fixed at one end, and loaded at the other | 

Also the following can be shown to be true : — 

Beam firmly fixed at both ends, and loaded at the centre . . 1 
Beam fixed at both ends, and loaded with distributed load . . li 

These facts are also true of a uniform beam of any form of cross- 
section. 

When (I Hqiiare beam is supported on Us ethje^ instead of on its 
side, — that is, has its diagonal vertical, — it will bear about seven- 
tenths as great a breaking-load. 

The stronfjest beam which can be cut out of a e^ "^^ & 

round log is one in which the breadth is to the / ^ 



depth as 5 to 7, very nearly, and can be found / 



\ / 




yd 
/ 



r^ 



\ 



\ 

I 

/ 
/ 



graphically, as shown in margin. Draw any [ 

diagonal, as ah, and divide it into three equal \ 

parts by the points c and d ; from these points 

draw perpendicular lines, and connect the points " 

#? and/ with a and h, as shown. ^' 

1 
CYLiNl>UI<''Af. Bkam.s. — A cylindrical beam is oidy .^ as 

1 • I 



SIQ STRENGTH OF WOODEN BEAMS. 

• 
strong as a square beam whose side is equal to the diameter of the 
cirolo. [lonco, to find the load for a cylindrical beam, Hrst finil 
tlio propter load for the corresponding square beam, and then divide 
it by 1.7. 

77/ r hcnrUiri of the ends of a 1>eam on a wall beyond a certain 
amount does not strengthen the beam any. In general, a beam 
slioulil have a bearing of four inches, though, if the beam be very 
short, the bearing may be less. 

Wv'ujUt of the Benin itHelf to be taken into Account. — The for- 
nuilas we have given for tlie strength of beams do not take into 
account the weight of the beam itself, and hence the safe load of 
tli(>. formulas includes both the external load and the weight of the 
material in the beam. In small wooden beams, the weight of 
th(i beam is generally so small, compared with the external load, 
that it need not be taken into account. But in larger wooden beams, 
and in metal and stone beams, the weight of the beam should be 
subtracted from the safe load if the load is distributed ; and if 
the load is applied at the centre, one-half the weight of the beam 
should be subtracted. 

The weight per cubic foot for different kinds of timber may be 
found in the table giving the Weight of Substances, Part III. 

Tables for the stren^li of yellow aud wliite pine» 
spruce, aud oak beauis, are given below, for beams one inoh 
wide. 

To find the strength of a given beam of any .other breadth, it is 
only necessary to multiply the strength given in the table by the 
breadth of the given beam 

Example. — What is the safe distributed load for a yellow-pine 

beam, supported at both ends, 8 inches by 12 inches, 20 feet clear 

span ? 
Alls. From Table II., safe load for one inch thickness is 1,440 

pounds. 1,440 x 8 = 11,520 pounds, safe load for beam. Far a 

concentrated load at centre, divide these figures by 2. 

To find the size of a beam that will support a given load with a 
given span, find the safe load for a beam of an assumed depth .one 
inch wide, and divide the givcm load by this strength. 

KxAMPLK.— Wh.it size spruce beam will be required to carry a 
distributed load of S,64() pounds for a clear span of 18 feet ? 

Ann. From the table, we find that a beam 14 inches deep and 1 

inch thick, 18 feet span, will support 1.524 pounds ; and diridiiig 

the load, 8.640 pounds, by 1.524, we have 5) for the breadth of the 

*t in inches : hence the V>eam should be 6 by 14 inohea, to oany 

ibuted load of 8,640 pounds with a span of 18 feei. 

*■■  



STRENGTH OF HABD-PINE BEAMS. 



311 



GQ 

n 

111 

I 

n 



I 8 

S) s 



a 

03 



I 

i 

•SI 

o C 

'a 5 
Si "Q 

o 
Jg 
2 



9i 



OQ 



$ 



OD 



«1 









CO 



OC 



J5 



1^ 0< 



IS 



t-t Ol 0» 



^ rH rH of e* 



^ o ^ 



rH O f- O 

1-1 W TH K5 

*> *k •» *> ^ 

ri T-i e< e< ot 



g 



s i § 



» «> 



f-i 11 e» e< 




Ol 



^ s S^ s 

n iQ Q a 

o et OD -« 

rH tH T-l 01 




kO 



OD 

Xi 



no 


o 


09 


Bi < 


^5 


r^ 


Ito 


^ 


yf 








Id 


CO 


CO 



^ lO t> 



I S I P § s 

i-H VI In 09 00 



eo 



<M 



OD 



:0 
00 




s s s 

o» t- <?*. 

00 eo -* 



OD 



3 



S QO g p « 

i-i" o» o» 00 -^ *o 



OB 

a 

s 

u 








> 

V 






c 

> 

o 

eS 

09 

O 

h3 



X 



Xi 



© o rfS^ «o 
.-T i-T of of cc 



i 



>A CO 



? S5 8 

«o i-< c* 



00 < 
'^^^©lOIOO^Ol-OO 



jog 



<Eto{^aoa»oo<'^*0(0 

fl »M »-< »H n »-< 



372 



STRENGTH OF WOODEN 11EAM8, 



For beams wilh a rectangular cross-section, wo can simplify onr 
fonmilas for strength by substituting for the moment of inertia 

its vahie, viz., — t^", where h = breadth of beam, and il its deptli. 

Then, substituting tliis value in the general formulas for 1)eaMis, 
wo liave for recUingular beams of any material the following 
forniulas : — 



Beams fixed at one end, and loaded at the other (Pig. 2). 




or 



Safe load in pounds = 



Breadth in inches = 



Fig. 2, 

breadth x square of depth X A 
4 X length in feet 

4 X lo ad X length in feet 
scpiare of depth X A ' 



(2| 



(8J 



lieams fixed at one end, and loaded with nn{foTuHy dUArihiuUd 
load (Fig. 3). 




or 



Safe load in pounds = 



Breadth in inches = 



Fig 3. 

breadth X square of depth X A 
~2'^lengMi in feet 

2 X h'ugth in ft?et X load 
8(juare of depth X A ' 



U) 



m 



STRENGTH OF WOODEN BEAMS. 



§73 



Beams supported at hoik evds^ loaded at middle (Fig. 4). 

W 




Safe load in pounds = 



FI9 4. 

breadth x square of depth X A^ 
span in feet 



span in feet x load 
Breadth in inches = s,,„are of depTh x~7r- 



(6) 



(7) 



Beams supported at both ends, had uniformly distributed 
Fig. 5). 




Safe load in pounds = 



Fig. 6. 
_ 2 X breadth x STfuare of depth x A 



span in feet 



_ span in feet x load 

Breadth in inches = .> ^ . ..^..^^r .1 ^*u v < » 

2 X •Kjiiare of depth x A 



(8) 



(») 



Beams sujtported at both ends, loaded with concentrated load 
^OT AT CENTEE (Fiir. «)- 




y////,///y''* 



Safe load hi poancb = 



>x 



Brpailtli m \Tif\\c^. ~ 



Fig 6 

breadth x vf. of depth x span X A 

4 X ;<» X H 

4 y Uy\t] y />/ y u 

ft'inar** of *\<'\A\\ / .s|>vif» y A' 



(101 



(11 



!•■ 



374 



STRENGTH OF WOODEN BEAMS. 



Beams supported at both ends, and foaded with W pounds (U 

a distance m from each end (Fig. 7). 




or 



FIfl. 7. 



Safe load H' in pounds __ breadth X square of depth X A ^ 
at each point 4 X m 

^ 4 X load at one point X m 
Breadth in inches = sq. of depth X A * 



(12) 



(18) 



Note. — In the last two cascft the leugthf) denoted by m and n sliould tn taken 
in feet, the same as the spans. 

Values of the Constant A. 

The letter A denotes the safe load for a unit beam one inch 
square and one foot span, loaded at the centre. This is also one- 
eighteenth of the modulus of rupture for safe loads. The follow- 
ing are the values of A, which are obtained by dividing the moduli 
of rupture in Chap. XIV. ..by 18. 

TABLE I. 
Values of J.— Co-efficient fob Beams. 



Material. 


.4 lbs. 


Cast-iron 


308 


Wrou«;ht-iron 


()0(5 


Steel 


888 


American woods : 
Cies^tnnt 


60 


Hemlock 


55 


Oak, while 


75 


Pinu. (iCorL'ia yellow 

" Oreiron 


100 
90 


" red or Norwav 


70 


" white, Eastern 


00 







Material. 



Pine, white. Western . . . 
•• Texas yellow .... 

Spruce 

I W hi te wood (poplar) . . . . , 



{ Bhicstoiie flagging (Hndson 

j River) 

I Granite, average 

i Limestone 

Marble 

' Sannstouc 

I Slaie 



AWm, 



00 
90 



n 

17 
16 
17 
8 
BO 



Tlu\s<; values for the co-oflicient A are one-third of the breaking- 
weiixht of timbers of th(> same si/.e and quality as that used in firat- 
class buildings. This is a sutticient allo'.vance for timbers in roof 
trusses, an<l beams whi<:h do not have to carry a more severe load 
than that on a dwelling-liousc floor, and small halLs, etc. Where 
there is likely to be very much vibration, as in the floor of a mill, 
"* gymnasium-floor, or floors of larg(> public halU, the author 
uenils that only four-tifths of the above values of ^ be used. 



BELATIVE STRENGTH OF BEAMS. 375 

Example 1. — What load will a hard-pine beam, 8 inches by 12 
inches, securely fastened into a brick wall at one end, sustain with 
safety, 6 feet out from the wall ? 

Ans. Safe load in pounds (Formula 2) equals 

8 X 144 X 100 



4x6 



= 4,800 lbs. 



Example 2. — It is desired to suspend two loads of 10,000 pounds 
each, 4 feet from each end of an oak beam 20 feet long. What 
should be the size of the beam ? 

Ans. Assume depth of beam to be 14 inches ; then (Formula 13^ 

breadth — ' — -^ — = 11 inches, nearly ; therefore the beam 

should be 11 X 14 inches. 



Relative Strengrth of Rectangular Beams. 

From an inspection of the foregoing formulas, it will be found 
that the relative strength of rectangular beams in different cases 
is as follows: — 

Beam supported at both ends, and loaded with a uniformly 
distributed load 1 

Beam supported at both ends, and loaded at the centre ... ^ 

Beam fixed at one end, and loaded with a uniformly distributed 
load . . . . ; 4 

Beam fixed at one end, and loaded at the other i 

Also the following can be shown to be true : — 

Beam firmly fixed at both ends, and loaded at the centre . . 1 
Beam fixed at both ends, and loaded with distributed load . . li 

These facts are also true of a uniform beam of any form of cross- 
section. 

When a square beam is supported on its edffe, instead of on its 
side, — that is, has its diagonal vertical, — it will bear about seven- 
tenths as great a breaking-load. 



The sironf/est beam which can be cut out of a e^ "^^.^ 



round log is one in which the breadth is to the / ^x / 



depth as 5 to 7, very nearly, and can be found / 
graphically, as shown in margin. Draw any | 
diagonal, as ab. and divide it into three equal \ 



yd 
/ 



/^N 



\ 

\ 

\ 

/ 
/ 




parts by the points c and d ; from these points \ J/ \ 

draw perpendicular lines, and connect the points " "^^ — -^/ 

f; and/ with (t and ?>, as shown. '^' 

1 
Cylindrical Bkams. — A cylindrical beam is only .^ as 

1*1 



382 »ULll> BUILT WOOUBN BBAH8. 

When a beam is built of several pieces la lengtb afl well i 
(Icptb, tbej sliould break joints with each other, Tlie layen b 
the neutral a^U should be lengthened by tlie scarf or Rati y 
iiseJ for resisting tension; and the npperoues should have the 
abut against each other, using plain buU joints. 



'I* 

Si 

i'l 



UM 



Many builders prefer using a hiiiU heam of selected Umber 
single solid oni', on acisjunt of the great dlfticutty of getting 
latter, whi'ii very lai^i', frep fi'orn defeotsr moreover, the atrei 
of the former is to l>e relieil u[>on, althougli it caimol be stro: 
than the corresponding solid one, if perfectly sound. 



STIFFNESS AND DEFLECTION OF BEAMS. 383 



CHAPTER XVI. 
STIFFNESS AND DEFLECTION OF B£AMS. 

In Chaps. XIV. and XV. we have considered the strength of 
beams to resist breaking only ; but in all first-class buildings it is 
desii-ed that those beams which show, or which support a ceiling, 
should not only have sufficient strength to carry the load with 
safety, but should do so without bending enough to present a bad 
appearance to the eye, or to crack the ceiling : hence, in calcu- 
lating the dimensions of such beams, we should not only calculate 
them with regard to their resistance to breaking, but also to bend- 
ing. Unfortunately, we have at present no method of combining 
the two calculations in one operation. A beam apportioned by the 
rules for strength will not bend so as to strain the fibres beyond 
their elastic limit, but will, in many cases, bend more than a due 
regard for appearance will justify. 

The amount which a beam bends under a given load is called its 
deflection, and its resistance to bending Is caUed its stiffness: 
hence the stiffness is inversely as the deflection. 

The rales for the stiffness of beams are derived from those for 
the deflection of beams; and the latter are derived partly from 
mathematical reasoning, and partly from experiments. 

We can find the deflection at the centre, of any beam not strained 
beyond the elastic limit, by the following formula: — 

_ load in lbs. X cube of span in inches X c 
Def. in inches - ^duius of elasticity X moment of inertia* ^^^ 

The values of c are as follows : — 

Beam supported at both ends, loaded at centre . .0.021 
" " *' uniformly loaded . . 0.01:3 

** fixed at one end, loaded at the other .... O.-^Vi 
" ♦* *' unifonnly loaded .... 0.125 

By wM^lgi"e the proper substitutions . in Formula 1, we derive the 



384 STIFFNESS AND DEFLECTION OF BEAMS. 

following formnla for a rectangular beam ^supported at bath ends, 

and loaded at the centre : — 

. _ load X cabe of span X 1728 

Def. in inches - 4 x breadth X cube of depth X E^ *^' 

the span being taken in feet. From this fommla the value of the 
modulus of elasticity, E, for different materials, has been circu- 
lated. Thus beams of known dimensions are supported at each 
end, and a known weight applied at the centre of the beam. The 
deflection of the beam is then carefully measured; and, substituting 
these known quantities in Formula 2, the value of £ is easily 

obtained. 

1728 
Formula 2 may be simplified somewhat by representing a^e ^ 

■^, which gives us the formula 

WX L^ 
Def. in inches = j^ x I^x F^ ^^^ 

For a distributed load the deflection will be five-eighths of this. 
Note. — The constant i'^ correBponds to Hatfield's F, in Us Tnuisreiae Stimiiu. 

If we wish to find the load which shall cause a given deflection, 
we can transpose Formula 2 so that the load shall fdrm the left- 
hand member. Thus : — 

Load at centre _ 4 X breadth X cube of depth X def. in ins. X E 
in pounds ~ cube of span X 1728 ' * ' 

Now, that this formula may be of use in determining the load tb 
put upon a beam, the value of the deflection must in some way be 
fixed. This is generally done by making it a certain proportion 
of the span. 

Thus Tredgold and many other authorities say, that, if a flooi> 
beam deflects more than one-fortieth of an inch for every foot of 
span, it is liable to crack the ceiling on the under side; and henoe 
this is the limit which is generally given to the deflection of beams 
in first-class buildings. 

Then, if we substitute for ** deflection" the value, length in feet 
-r 40, in the above fornmla, we have, 

breadth X cube of depth X e 
Load at centre = ^^— ^ ,-^jj , (5) 

E 
letting e = p=^- 

y engineers and architects think that one-thiriieUk qfan inch 
)t of span is not too much to allow for the defleetton of floor 



STIFFNESS AND DEFLECTION OF BEAMS 



385 



beams, as a floor is seldom subjected to its full estimated load, and 

then only for a short time. 

If we adopt this ratio, we shall have as our constant for deflec- 

_ E 
tion, €i - J2900- 

In either of the above cases, it is evident that the values used for 
Ef F, e, or Ci, should be derived from tests on timbers of the same 
size and quahty as those to be used. It has only been within the 
last three or four years that we have had any accurate tests on 
the strength and elasticity of large timbers, although there had been 
several made on small pieces of various woods. 

The values of the vaiious constants for the fii*st three woods in 
the following table have been derived from tests made by Professor 
Lanza and his students at the Massachusetts Institute of Tech- 
nology, and the values for the other woods are about six-sevenths 
of the values derived from Mi*. Hatfield's experiments. The author 
believes tliat the values given in this table may be relied upon for 
timber such as is used in first-class construction. 



TABLE I. 

Values of Constantn for Stiffness or Deflection of Beams, 

E = Modulus of elasticity, pounds per square inch. 

F = Constant for deflection of beam, supported at both ends, and 

loaded at the centre. 
€ = Constant, allowing a deflection of one-fortieth of an inch per 

foot of span, 
e, = Constant, allowing a deflection of one-thirtieth of an inch per 

foot of span. 



Material. 



Cast iron . . 
Wrought-iron 
Steel . . . 
Yellow pine . 
Spruce . . . 
While oak . 
White pine . 
Hemlock . . 
Whilewood . 
CheHtaut . . 
A«h. . . . 
Muple . . . 



E. 



15,700,000 

26,000,000 

31 ,000,0.00 

1,780,000 

1,294,000 

1,240,000 

1,073,000 

1,045,000 

1,278,000 

944,000 

1 ,48-.\000 

1,902,000 



F^ 



E 

432" 



36,300 
60,000 
71,760 
4,120 
3,000 
2,S70 
2,480 
2,420 
2,960 
2,180 
3,430 
4,400 



E 



17280 



907 

1500 

1794 

103 

75 

72 

62 

60 

74 

54 

86 

no 



E 

^1 " 12960' 



1210 

20:k» 

23o8 

137 

100 

95 

82 

80 

98 

72 

114 

146 



394 CONTINUOUS GIRDBR8. 

Contimtons Girder of Three Equal Spans, Concentrated Load <^ 
W Poitnda at Centre of Each Span. 

Re-action of either abulment, 

R,=R, = i\W; (7) 

Re-action of either centi-al support, 

B, = A'j = U yV; (81 

r 

or the re-action of the end supports is lessened three-tenths, and 
that of the central supports increaseil three-twentieths, of that 
which they would have been, had three separate girders of the samp 
cross-section been used, instead of one continuous girder. 

D 




Fig.2 

Continuous Girder of Three Equal Sjmns uniformly loaded with 
w Pounda per Unit of Lenyth. 

Re-action of either end support, 

R,=R, = Uol; m 

Re-action of either central support, 

R^ = R, = \htol; (10) 

hence the re-actions of the end supports are one-fifth less, and of 
tlie central supports one-tenth more, than if the girder were not 
continuous. 

Strength of ContiuHous Girders, — Uviymg determined the re- 
action of the supports, we will now consider the strength of the 
girder. 

Tlu; strength of a beam depends upon the material and shape 
of the l)eain, jind upon the external conditions impose<l upon the 
beam. The latter j;ive rise to the bemling-moment of the beani, or 
tlu> amount by which the external forces (such as the load and 
supporting forces) tend to bend and break the beam. 

It is Ibis bonding-moment which causes the difference In the 
Ijoaring-strength of continuous and non-continuous girders of 
the same cross-section. 

Continuoua Girdtrs of Tico .s>«».s. — When a rectangular beam 
is at the point of breaking, we have the following conditions :^ 

Bendim;- _ Mod, of rupture x breadth X sq. of depth . 
moment "~ 6 ' '"' 

:hat the lieam may carry its load with perfect safety^ 
the load by a proper fac^tor of safety. 



CONTINUOUS GIRDERS. 395 

Hence, if we can determine the bending-moment of a beam under 
any conditions, we can easily determine the required dimensions of 
the beam from Formula 11. 

The greatest bending-moment for a continuous girder of two 
spans is almost always over the middle support, and is of the oppo- 
site kind to that which tends to break an ordinary beam. 

Distributed Load. — The greatest bending-moment in a continu- 
ous girder of two spans, / and /i , loaded with a unifonuly distributed 
load of w pounds per unit of length, is 

Bending-moment = o /#  , > » (12) 

V/hen i = f , , or both spans are equal, 

Bendmg-moment = -g-, (12a) 

which is the same as the bending-moment of a beam supported at 
both ends, and uniformly loaded over its whole length: hence a 
continuous yirder of two tfpans uniformly loaded is no stronyer 
than if non-continuous. 

Concentrated Load, — The greatest bending-moment in a con- 
tinuous girder of two equal spans, each of length /, loaded with W 
pounds at centre of one span, and with W^ pounds at the centre of 
the other span, is 

Bending-monaent •=^ h^(W+Wx). (13) 

When W = W\^ov the two loads are equal, this becomes 

Bending-moment = ^WU (13a) 

or one-fourth less than what it would be were the beam cut at the 
middle support. 

Continuous Girder of Three Spans^ Distributed Load. — The 
greatest bending-moment in a continuous girder of three spans 
loaded with a uniformly distributed load of w pounds per unit of 
length, the length of each end span being /, and of the middle 
span Ij is at either of the central supports, and is represented by 
the formula, 

Bendmg-moment = .,.>. , ^. v . (14) 

When the three spans are equal, this becomes 

Bending-moment = 7a» (14a) 

or one-fifth less than what it would be were the beam not con^^ 
tinuous. 



388 



STIFFNESS AND DEFLECTION OF BEAMa 



a 

as 



O 

u 

o 

c 



o 

a ^ 

o r 

if -2 

c *- 



Is 

C ^ 


















g«OQoaoo««io«o 



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s % s 



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STin SS AKD DBFT^ECnON OF BBAJI& 



389 



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f2 



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gcOXOiOM-TfOCO 



STIFFNESS AND DEFLECTION OF BE A J. 



Eh 



o 

a 

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a ^ 

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m 


c c 




C tf 


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of 
beam. 


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«D GO e> o e« ^ 

rN »iN tH 




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ig ^ s s s ^ 

'^ T-t 0« ^ t» 


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1,081 



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STIFFNESS AND DEFLECTION OF BEAMS 391 

ExABCPLE 2. — What should be the dimensions of a yellow-pine 
beam of 10 foot span, to support a concentrated load of 4250 pounds, 
without deflecting more than ^ of an inch at the centre V 

Ans. A deflection of i of an inch in a span of 10 feet is in the 
proportion of y?, of an inch per foot of span; and as the load is 
concentrated, and applied at the centre, we should use Fomiula 7, 
employing for e the value given in the fourth column, opposite 
yellow pine. 

Formula 7 gives the dimensions of the breadth, and to obtain it we 
must assume a value for the depth. For this we will first try b inches. 

Substituting in Formula 7, we have, 

4250 X 100 
Breadth = 512 x l'j7 ~ ^ inches, nearly. 

This would give us a beam 6 by 8 inches. 

Example 8. —What is the largest load that an inclined spruce 
beam 8 by 12 inches, 12 feet long between supports, will cari7 at 
the centre, consistent with stiffness, the horizontal distance between 
the supports being 10 feet ? 

An9. Formula 12 is the one to be employed, and we will use the 
value of e given in the third column, opposite spruce. Making 
the proper substitutions, we have, 

^ , . 8 X 1728 X 75 

Safe load = — r2 x"To — ~ ^^^^ pounds. 

Cylindrical Beams. 

For cylindrical beams the same fonnulas may be employed as 
for rectangular beams, only, instead of #■, use 1.7 X e, : that is, a 
cylindrical beam bends 1.7 times as much as the circumscribing 
rectangle. 

Deflection of Iron Beams. 

For rolled-iron beams the deflection is most ac(;urately obtained 
by Fonimla 1. The following ap])roximate formula gives the de- 
flections quite accurately for the maximum safe loads, 

s^iuar(» of si)an n\ feet 
Deflection in inches = 



70 x iht: depth of bv.am 

The deflections for tlie PJuvnix, Pencoyd, Trenton, and Car- 
negie beams, arc given in the tables for strength of beams, in 
Chap. XIV. 

In using iron beams, it should be n*membered that the deepest 
btatu is aJways the most economical; and the stiffness of a floor is 
almtys gi!Qftt!er wheu a suitable number of deep beams are used. 



302 CONTINUOUS GIRDERS, 



CHAPTER XVII. 

STRENGTH AND STIFFNESS OF CONTINUOUS 

GIRDERS 

Girders resting upon throe or more snpiwrts arc of quite fi-e- 
qiieiiL octurreiioi* in buiUling construction; anti a great variety of 
oi)inions is held as to the relative strength and stiffness of continu- 
ous and iion-continnous girders: very few i)ersons, probably, having 
any coiic'Ct knowledge of tin* subject. 

In almost every building of importance, it is necessary to employ 
girdtrs iisiing ui)on jiiers or columns placed from eight to fifteen 
feet ai)ari ; and in many cases gndcrs can conveniently bo ubtaiueil 
wliitli will span two and even three of the spaces l)etween the piera 
or columns. When this is tlie case, the question arises, whether it 
will be heller construction to use a long continuous girtler, or to 
have each ii:irdcr of only one span. 

Most aiehitects an? probably aware tliat a girder of two or more 
sj)aii> is sirougi'i and stifTer than a gn-der of the same section, of 
only one s])an. but just htnn much stronger and stiffer is a question 
they are unable to answer. 

As it i> -eldoin ihai a iiirderof more than three spans ih employed 
in (Utlmaiy bni Idlings, we shall c<»nsid(»r only these two caM*.s. hi 
all struelures, the first point which slumld Ih» considennl is the 
n'sistaiiee require<| of (Im* su])poris, and we will first cimsider 
the resistance offered by the siq)iH)r(> of a continuous ginler. 

In this elia)>ter we shall iioi go into the mathematical flisciission 
of the "subject, but leier any readi'is inter«»sle<l in the derivation of 
the toiniulas for (-ontinnous girders to an article on thai sulijt*<-l, 
b\ the author, in the .Inly (J8^;ij number of Van NostrandV 
" Knmneeiiiig Magazine.'' 

Supporting- Forces. 

(iinhrs <tj Two spuuM, lotuh'tl al lUf. Centre qf Baeh Span. —If 
a ginler ot two si>ans, / and /,, is loaded at tho centre of the tpui I 



CONTINUOUS GIRDERS. 



393 



with W pounds, and at the centre of ^ with Wi pounds, the 
re-action of the support Ri will be represented by the fonnula 



R =: -* 



32 



(i) 



the re-action of the support R.^ ^Y 



«2 = j^(ir + ^r,), 

and the re-action of the support R^hy the formula 

13 IK, -:]W 



(2) 



A»» = 



32 



(3) 



If H^ = IF,, then each of the end supports would have to sustain 
1^ of one of the loads, and the centre support V- of W, Were the 
girder cut so as to make two girders of one span each, then the end 
supports would carry ^ or tb W', and the centre support |g ]V: hence 
we see, that, by having the girder continuous, we do not require so 
much resistance from the end supports, but more from the central 
support. 

ABC 




m 



m 



m 

R2 



Fig 1 






Girder of Two Spans, uniformly Distributed Load over Each 
Span, — Load over each span equals lo pounds per unit of length. 
Re-action of left support, 



10 r ^« + / « -| 

-2L' 4/(f-h/.)J- 



Re-action of central support, 

R, = w{l-hl,)-R, - /?3. 

Re-action of right support, 

^^ - 2U' 4/, (/ + /.)]• 



(4) 



(5) 



(6) 



When both spans are equal to /, the re-action of each end support 
is i Kj/i, and of the central support t '«' ' hence the girder, by being 
contuuious, reduces the re-action of the end supports, and increases 
thai of the central support by one-fourth, or twenty -five per cent. 



394 



CONTINUOUS GIRDERS. 



Continuous Girder of Three Equal Spans, Concentrated Load of 
W Pounds at Centre of Each Span. 

Re-action of either abutment, 

R,=R, = ;\,}V; (7) 

Ro-action of either central support, 

liz = H, = U ^V; («) 

or the re-action of the end supports is lessened three-tenths, and 
lliat of the central supports increased three-twentieths, of that 
which they would have been, had three separate girders of the sam^ 
cross-section been used, instead of one continuous girder. 

D 




Continuous Girder of Three Equal Spans uniformly loaded with 

w Pounds per Unit of Lent/ th. 

Re-action of either end support, 

/r = /?4 = i tot; (9) 

Re-action of either central supi>ort, 

/?, = /^^ = ^,( ,o/; (10) 

hcnco the re-actions of the end supports arc one-fifth less, and of 
the central supports one-tenth more, than if the ginier were not 
continuous. 

str'iKjtli of' ('(nitiuuous Girders. — Having detemiineil the re- 
action of the supports, we will now consider the strength of the 
irinler. 

The streiij^th of a beam depends upon the material and shai)e 
of the beam, and ii]K)n the external conditions imiH)s<Hl ii{H>n the 
beam. Tlie lattei ijive rise to the ben<ling-moment of the l)eani, or 
the amount by wbieh the external forces (such as the load and 
support iiiu forces) tend to ben<l ami break the l)eam. 

It is tliis bendini^-monient which causes the ^liflTerenee in the 
l>eaiiiiLj-^tiemitb of continuous and non-continuous ginlers of 
tie* >ame cross-section. 

('(mfiiiii'iiis (iirdti's o/* Tii'n spiois. — When a rectangular heam 
is at the point of breakiuir, we have the following (^mditions : — 

Hendini:- _ ^lod. of rupture X bre mllh X s<|. of depth , 
moment ~ " «" ' *"* 

and. that the beam may carry its load with perfect safety, wemiut 

divide the load b> a proper factor of safety. 



CONTINUOUS GIRDERS. 395 

Hence, if we can determine the bencUng-moment of a beam under 
any conditions, we can easily determine tlie required dimensions of 
tlie beam from Formula 11. 

The greatest bending-moment for a continuous girder of two 
spans is almost always over the middle support, and is of the oppo- 
site kind to tliat which tends to break an ordinary beam. 

DiMtrlbuted Load. — The greatest l)ending-moment in a continu- 
ous girder of two spans, / and /, , loaded with a uniformly distributed 
load of w pounds per unit of length, is 

Bending-moment = o /^  ^ > » (12) 

V/hen Z = i I , or both spans are equal, 

top 
Bending-moment = -g-, (12a) 

which is the same as the bending-moment of a beam supported at 
both ends, and uniformly loaded over its whole length: hence a 
continuous girder of two ifpams uniformly loaded is no stromjer 
than if non-continuous. 

Concentrated Load, — The greatest bending-moment in a ron- 
tinuous girder of two equal spans, each of length if, loaded with \V 
pounds at centre of one span, and with Wi pounds at the centre of 
the other span, is 

Bending-monaent ^ ^iHW+Wy), (13) 

When W = ITi , or the two loads are equal, this becomes 

Bending-moment = A IT/, (13a) 

or one-fourth less than what it would be were the beam cut at the 
middle support. 

Continuous Girder of Three Spans^ Distributed Load. — The 
greatest bending-moment in a continuous girder of three spans 
loaded with a uniformly distributed load of w pounds per unit of 
length, the length of each end span being /, and of the middle 
span I, is at either of the central supports, and is represented by 
the formula, 

wl^ 4- wli^ 
Bending-moment = .,... , ^i y (14) 

When the three spans are equal, this becomes 

xol^ 
Bending-moment = Tqi (14a) 

or one-fifth less than what it would be were the beam not con-^ 
tinuous. 



396 CONTINUOUS GIRDERS. 

Conconfrated Loads. —The greatest bending-moment in a con- 
tinuous girder of three equal spans, each of a length 2, and each 

loaded at the centre with [V pounds, is 

Bending-moment = ^,^ Wl, (15/ 

or two- fifths less than that of a non-continuous girder. 

Deflection of Continuous Girders. 

CoutiniiOHs Girder of Two Eqvdl Spniis. — The greatest deflec- 
tion of a continuous girder of two equal spans, loaded with a 
uniformly distributed load of w pounds per unit of length, is 

id* 
Deflection = 0.005416 ^. (16) 

{E donotos modulus of elasticity; /, moment of inertia.) 

The deflection of a similar beam supported at both ends, and 
uniformly loaded, is 

Deflection = 0.01:3020 ^. 

ITencc the deflection of the continuous girder is only about two- 
fifths that of a non-continuous girder. The greatest deflection 
in a continuous girder is also not at the centre of either span, but 
betweer. the centre and the abutments. 

The greatest deflection of a continuous girder of two equal spans, 
loaded iit the centre of one span with a load of IV pounds, and at 
the centre of the other span with IV i pounds, is, for the span with 
load \V, 

(28n'-0lF,)/« 
Deflection = - 153^.^^ ' <"> 

for the si)an with load ir,, 

(2:ur, ~oir)/« 

Deflecti(m = — I'l'A' EI * (Ha) 

When ImMIi si)ans have the same load, 

7 ir/« 

Deflection = >.^■^ ^^ • (17'>) 

T]i<> drilci tion of a beam su])ported at l>oth ends, and loaded at 

tin* (•»'iiin- with \y pounds, is 

Deflection = .^ j^-,-j\ 

or tlir ditlrction of the continuous girder is only seven-tlsteenUit 

of the nun-eontinuous one. 



CONTINUOUS GIRDERS. 397 

Continuous Girder of T/tree Eqiial Spans, — Uniformly distrib- 
uted load of 10 pounds per unit of length, 

Deflection at centre of middle span = 0.00052 ^ ( 18) 

Greatest deflection in end spans = 0.006884 j^ (10) 

or the greatest deflection in the girder is only about one-half that 
of a ncn-continuous girder. 
Concentrated load of W pounds at centre of each span, 

I \Vl^ 
Deflection at centre of middle span = t^ -^t- (20) 

II Wl» 
Deflection at centre of end spans = kqk -^j- (21) 

or only eleven-twentieths of the non-continuous girder. 

Several Observations and Formulas for Designing: 

Continuous Girders. 

From the foregoing we can draw many observations and conclu- 
sions, which will be of great use in deciding whether it will be best 
in any gi\^n case to use a continuous or non-continuous girder. 

First as to the Su2)ports* — We see from the formulas given for 
the i*e-action of the supporting forces in the different cases, that in 
all cases the end supports do not have as much load brought upon 
them when the girder is continuous as when it is not; but of course 
the difference must be made up by the other supports. This might 
often be desirable In buildings where the girders run across the 
building, the ends resting on the side walls, and the girders being 
supported at intermediate points by columns or piers. In such a 
case, by using a continuous girder, part of the load could be taken 
from the walls, and transferred to the columns or piers. 

But there is another question to be considered in such a case, 
and that is vibration. Should the building be a mill or factory in 
which the girders had to support machines, then any vibration 
givea to the middle span of the beam would be carried to the side 
walls if the beam were continuous, while if separate girders were 
used, with their ends an inch or so apart, but little if any vibration 
would be can-ied to the side walls from the middle span. 

In all cases of important construction, the supporting forces 
should be carefully looked after. 

Strength, — As the relative strength of continuous and non- 
oontinuouB girders of the same size and span, and loaded in the 
•aoie wny, is as their bending-moments, we can easily calculate the 



.^9S CONTINUOUS GIRDERS. 

strongth of a continuous girder, knowing the formulas for its bend* 
ing-nioni(Mit. From the values given for the bending-nioments of 
the various cases considered, we see that the portion of the girder 
most strained is tliat which conies over the middle supports; also 
that, except in llie single case of a girder of two spans uniforndy 
loaded, tlie strength of a girder is greater if it is continuous than if 
it is not. But tlie gain in strength in some instances is not very 
great, altliough it is generally enough to pay for making the girder 
continuous. 

Stijrnc's^i. — The stiffness of a girder is indirectly proportional to 
its deflection; that is, the less the deflection under a given load, the 
stiffen the girder. 

Xow, from the values given for the deflection of continuous 
girders, we see that a girder is rendered very much stiffer by being 
made (continuous ; and this may be considered as the principal 
advantage in the use of such girders. 

It is often the case in building-construction, that it is necessary 
to usi^ beams of nmch greater strength than is required to carry 
the superimposed load, because the deflections would be too great 
if i\ui beam were made smaller. But, if we can use continuous 
girders, we may make the beams of just the size required for 
strength; as the deflections will be lessened by the fact of the gird- 
ers being (continuous. It should therefore be remembered, that, 
wh(>re great stiffness is required, continuous beams or girders 
should be used if possible. 

Foriuulas for Strciigtli and Stiffhess. 

For eonvenienee we will give the proper formulas for calculating 
the streni;tli and stiffness of continuous ginlers of rectangidar 
cross-s(»etion. The fonnulas for strength are deduced from the 

fornuda, 

Bending-moment = ;. * (22) 

where 1i is a (constant known as the modulus of rupture, and la 
ei^litecu times what is generally known as the co-efticicnl of 
stn'nijth. 
SiKKNJ.Tn. — (.'ontinnoits tjirder of two equal Hpana^ loadtd 

nnij'nnnhj oi'cr ((ir/i span^ 

2x nx U^x A 

lirealving-weight = i ' (23) 

where li d«>ri()tes the breadth of the ginler, D the depth of the 
girder (botli in inches), and L the length of one span, in/eef. The 



CONTINUOUS G1KDEB8. 399 

values of the oonstant A are three times the values given in Table 
L, p. 874. For yellow pine, 800 pounds ; for spruce, 210 pounds ; 
and for white pine, 180 pounds, — may be taken as reliable values 
for A. 

Continuous girder of two equal spans, loaded equally at the 
centre of each span, 

4 B X D^x A 
Breaking-weight = 3 X r • (24) 

Continuous girder of three equal spans, loaded uniformly over 
each span, 

« , . . , f) Bx D2x A 

Breaking- weight = 9 ^ L * ^^^ 

Continuous girder of three equal spans, loaded equally at the 
centre of each span, 

5 B X D^x A 
Breaking-weight = 3 x j • (26) 

Stiffness. — The following formulas give the loads which the 
beams will support without deflecting more than one-thirtieth of 
an inch per foot of span. 

Continuous girder qf two equal spans, loaded uniformly over 
each span, 

Bx l>^x e 
Load on one span = q 26 x L-^ ' '^'^^ 

Continuous girder of two equal spans, loaded equally at centre 
of each span, 

16 B X D^x e 
Load on one span = "7" x j-^ • (28) 

Continuous girder* of three equal spans, loaded uniformly over 
each span, 

B X Z)'^ X e 
Load on one span = q 33 x L^ ' ^^^ 

Continuous girder of three equal spans, loaded equally at the 
centre of each span, 

20 B X D^x e 
Load on one span = TT ^ jo • (oO) 

The value of the constant e is obtained by dividing the modulus 
of elasticity by 12,1)(50 ; and, for the three woods most commonly 
used as beams, the following values may be taken : — 

Tellow pine, 187 ; white pine, 82 ; spruce, 100. 



400 CONTINUOUS GIRDERS. 

For iron beams we may find the l)ending-inoinent by the for- 
mulas given, and, from tahles saving the sti-ength and sections of 
rolled beams, find the beam whose moment of inertia = 

bending-m oment X depth of beam 
2000 

•.vhen tli«* beuilinsj moment Is in foot pounds. 

For (^xjunphs we have a continuous l-lwam of three equal spans, 
loaded ovtM- each span, with 2(KM) pounds per foot, distributeil. 
Each span being 10 feet, then, from fonnula 14(r, we have 

2(KX) X 100 
Bemling-moment = rr^ = 20000. 

2(XNK) 
Moment of inertia = ~:^^^ x depth of beam; 

20,(XM) -^ 2(MM) = 10, and we must find a beam whose depth multl- 
plic 1 hy ten will c(jUJil its moment of inertia. 

If \\v try a ten-inch lK*am, we should have 10 X 10 = 100; and we 
sec from Tal)lcs, i)p. 2(50-272, that no ten-inch beam lias a moment of 
Inertia as small as KM): so we will take a nine-inch beam. W X 10 
~ INK and the lightest nine-inch beam has a moment of inertia of 
\Y,\: so we will use that beam. In tluj case of continuous I-i)eams 
of three e(|nal spans, (upially load(>d with a distributed ItKid. wi* 
may take four-fifths of the load on one siKin, and find the iron 
beam which would support that load if with only one span. 

KN.VMri.i:. — if we have an I-beam of three equal siNins of 10 
feet each loadcil with 20,000 pounds over each span, wliat Hize 
beam should we use? 

Ans. -! of 20.<MM)= 10,000. Tlie ecpiivalent load for a span of 

oui- foot would be 10,000 X 10= UMMMM). 

rrnin Tables, Chap. XIV., we find that the beam whose eo-efll- 
ciint is nearest to this is the nine-inch light lM*ani, — the s;inie 
beam wbicb we found to carry the same load in the prt*(*e«Iing 
c\aiii|>lc. Tor iK'anis of two equal spans loiidtMl uniformly, the 
>nciii:ib <»t the beam is the same as though the beam were not 
colli iinioiis. 

rin- t'oi iinila^ ui\en for tbe re-actioiis of the sup]M)rts ami for the 
(l<-tli-iri«>ii oi (-v)iitiiiMoiis Lcirders with eoneentnili'd NhmIsi, were 
vnitii-l bv Mm- aulboi- b> means of careful experiments on small 
sr«'-! bai->> IIm- other forinulas have Inn'ii veriH«Ml hy <>oni]iAri9un 
witli iitbi-r iiiilboi'it ies, wliei'i* it was |His.sible to do so; though one 
or iwo ot tbf l-a^e^ uJMMi, tli(* auliiur has never seen dlicuaa»d in 
ail) woiU on tbe .subject. 



FLITCH PLATk GIRDEttS. 40 J 




CHAPTER XVITT. 
FLITCH PLATE GIRDERS. 

In framing large buildings, it often occurs that the floors must be 
supported upon girders, which themselves rest upon columns ; and 
it is required that the columns shall be spaced farther apart than 
would be allowable if wooden girders were used. In such cases 
the Flitch Plate girder may be iron flate 
used, oftentimes with advan- 
tage. A section and elevation of 
a Flitch Plate girder is shown in 
Fig. 1. Fig. 1. 

The different pieces are bolted together every two feet by three- 
fouiths-inch bolts, as shown in elevation. It has been found by 
practice that the thickness of the iron plate should be about one- 
twelfth of the whole thickness of the beam, or the thickness of the 
wood should be eleven times the thickness of the iron. As the elas- 
ticity of iron is so much greater than that of wood, we must propor- 
tion the load on the wood so that it shall bend the same amount as 
the iron plate: otherwise the whole strain might be thrown on the 
iron plate. The modulus of elasticity of wrought-iron is about thir- 
teen times that of hard pine; or a beam of hard pine one inch wide 
would bend thirteen times as much as a plate of iron of the same 
size under the same load. Hence, if we want the hard-pine beam 
to bend the same as the iron plate, we must put only one-thirteenth 
as much load on it. If the wooden beam is eleven times as thick 
as the iron one, we should put eleven-thirteenths of its safe load on 
it, or, what amounts to the same tiling, use a constant only eleven- 
thirteenths of the strength of the wood. On this basis the follow- 
ing formulas have been made up for the strength of Flit(;h Plate 
girders/ in which the thickness of the iron is one-twelfth of the 
braidth oi the beam, approximately : — 



40-2 FLITCH PLATE GlRDEES. 

Let 1) = Depth of beam. 

B = Total thickness of wood. 
L = Clear span in feet. 
i = Thickness of iron plate. 
f __ i 1^^> pounds for hard pine. 

f 7o pountls for spruce. 
W = Total load on girder. 

Then y for beams supported at both ends, 

Saf<» load at centre, in pounds = j- (/B-\-*JnOt), (11 

22)2 
Safe distributed load, in pounds = —f— (/B + 7500. (21 



For distributed load, D = \/ 2/7i-f Kiitbt ' 

I irZ 

For load at centre D = \/ >^"j_7^' 



(3) 



(4) 



As an example of the use of this kind of girder, we will take tl«(* 
case of a railway-station in which the second story is devoted to 
offices, and where we must use girders to support the second floor, 
of twonty-liyc feet span, and not less than twelve feet on centres. If 
we can avoid it. This would give us a floor area to be supported by 
the girder of 12 X 25 = :300 square feet; and, allowing 105 i>ounds p«T 
s()iiare foot as the weight of the suiKjrimposed load and of the floor 
itself, we have ol,r>00 pounds as the load to be supported by the 
ginlcr. Now we find, by computation, that if we were to us«» a 
M)li(l girder of hard pine, it would re<iuirea8eventeen-lncli by four- 
teen-inch beam. If we were to use an iron Ix'ani, we find tliat a 
fifteen-inch ln^iivy iron beam would not have the requisiti^ strength 
for this span, and that we should be obliged to use twotwelve-4nch 
beams. 

We will now see what size of Flitch Plate ginler we would 
recpiire, sliould we decide to use such a girder. We will assume 
tlie total breadth of both beams to be twelve inches, so that we can 
use two six -inch tind)ers, whi<'h we will have hanl pine. The thick- 
ness of the iron will he one inch and one-eighth. Then, substi- 
tuting in Formula JJ, wt* have. 



/ :{!.')( M) X 25 . — 

^' = V- X KM) X 12 + I;V)T7^rHt = VIW, or 14 inches. 

Hence we sliall require a twelve-inc4i by fourteen-incb girder. NoVt 



FLITCH-PLATE GIRDERS. 40;^ 

for a comparison of the cost of the three girders we have considered 
in this example. The seventeen-inch by fourteen-inch hard-pine 
girder would contain 515 feet, board measure, which, at five cents a 
foot, would amount to $25.75. 

Two twelve-inch iron beams 25 feet 8 inches long will weigh 
2083 pounds; and, at four cents a pound, they would cost $83.82. 
The Flitch-Plate girder would contain 364 feet, board measure, 
which would cost $18.20. The iron plate would weigh 1312i 
poimds, which would cost $52.50; making the total cost of the 
girder $70.70, or $13 less than the iron beams, and $45 more than 
the solid hard-pine beams. Flitch-Plate beams also possess the 
advantage that the wood almost entirely protects the iron; so 
that, in case of a fire, the heat would not probably affect the iron 
until the wooden beams were badly burned. 



404 



TRUSSED BEAMS. 



CHAPTER XIX. 



TRUSSED BEAMS. 



AVhexkveti wo. wish to support a floor upon ginlers having a 
span of more than thirty feet, we must use eitlier a trussed ginler, 
a riveted iron-phite fjinU^r, or two or more iron beams. The clieap- 
esi and most convenient way is, probably, lo use a large woo<leu 
girder, and truss it, either as in P'igs. 1 and 2, or Figs. JJ ami 4. 

In all these forms, it is desirable to give the girders as much <!epth 
as the conditions of the case will permit; as, the deei)er the ginler, 
the less strain there is in the pieces. 

In the belly-rod truss we either have two beams, and one rod 
which runs up between them at the ends, or three beams, and two 
rods runnini^ up between the beams in the same way. The beams 
should be in one continuous length for the whole span of the ginler, 
if they can be obtained that length. The requisite dimensions of 
the Me-rod, struts, and beam, in any given case, must be deter- 
mined by lirsi tindiui^ the stresses which come ui>on these picH»t»s, 
and then the area of cross-section required to resist these sti-esses. 
Foi: sixciu: srui t iielly-kod tkisses, sucli as is represented 
by FJLi. 1, the strain ni)on the pieces may be obtained by the foUow- 
ini: formulas : — 

For DisTiniu'iEi) LOAD ir over whole (jiriJeTf 




'1 


'ension 


in r 






^ 


o 

10 


w 


X 


( 


oinpression 


in 


r 


^^ 


s' 


w. 




( 


'ompH'ssion 


in 


li 


zz 


10 


ir 


X 



length of T 
length of C 



le ngth of B 
length of Cf 



(1) 
(2) 

m 



TRUSSED BEAMS. 



405 



For CONCENTRATED LOAD W 09€r C, 

,«.,«, ^ length of T 

Tension in T = y x ,^„g,i, ^^ ^T 

Compression in C = W. 

. . „ H^ length of B 
Compression in B = g- x ^^^^^^ ^^ ^ 



W 



(5) 



For girder trussed as represented in Fig. t under a distributed 
LOAD W over whole girder, 





3 length of S 

Compression in S = j^ »' x lengthof C" 



(6) 



Tension in R 


- ^w. 


Tension in B 


_ 3 length of B 
10 "^ length of C 



(7) 



For CONCENTRATED LOAD, W at centre, 

, ^ W length of S 
Compression in S = ^ X i^ng^j^^f^- 

Tension in 1? ^ W, 

W length of B 
Tension in B = y x j^^^pT^f^. 



(8) 



(9) 



For double strut belly-rod truss (Fig. 3), with distributed 
i,OAD W over whole girder. 





B 




Tension in T 



Fig.3 

length of T 
= 0.307 W x 7- '^ 



length of C 

Compression in C = 0.367 W. 

^ length of B 

Comp. in iJ or D = 0.367 H^ X i,„^„ ^f p - 



(101 



(11) 



406 



TRUSSED BEAMS. 



Fo7' coNCKNTRATEi) LOAD W over cQch of the HtruU C, 

leneLli of T , , 

Conipression in C = W, 

leiigtli of Ji 
Coiiip. in B or tension in /) = \V x ip,iiwj7Qf7"'' (**^) 

For (jinlcr trusffvd, as in Fly. 4, under a distkiuuted load H' 

over whole (jlrder, 



r^ 



^Jp 





-v 



^^-TU 



Fig.4 



lon^tli of .S , , , , 
= 0.307 irx,^.^g,^-^jr-,^. (14) 

= 0.307 1 r. 



ConipR'ssion in S 

Trnsion in R 

lonslh of W .,,. 
Tension in li or conip. in D = ().:>07ir X \7r{^(u~^i~fy '*^' 



igtli 

Under ('ON<'KNTnATKi) loads W applied (H 9 and 3. 

len^h of iS 



('oinpivssion in S 
Tension in H 



= W X 
= W. 



len<;tli of H 



(16) 



lon^li of Ji 
Tension in /; or conip. in I) = M' X i^^^^jT^fT;- (17) 

Trusses sneh as shown in Figs. 3 and 4 should Iw divided so that 
the rnds li, (»r I lie struts (', shall divide the lont^th of Iho ginler into 
three (M|Mal oi* n*'arly e<|ual parts. The len<;ths of the pi«»ci»s T", 
(\ li, li, >, rt<'.. should he measured on the <'entrt»s of the pleees. 
Tiius iIk* lrui;th of li should he taken from the eeiitre of llie lie- 
heaui r» lo the <-eutre of the strut I) : and the leii«;tli of Cshoiilil Im 
inraviiiTil from the eentre of the rod to the ivntre of the strut- 

IXMMI li. 

After dt'terminiui: the strains in the pieees hy these formulas, 
we may compute the areii of the eross-s(>eti(>ns hy (he folluwliig 



rules ; — 



eonip. in strut 
Area of cross-section of strut = — -r, • 



(18) 



<. . . . , , /tension In rod 

Dianu'ter of smjjle th^nMl » = \/ i^^i . {\9) 



^ Al:<>^^ inL' 1'J.(MNi iioiiiidrt Hufo ifiiHiuii iN*r Miii«rc tiieh In Ibo rod. 



TRUSSED BEAMS. 407 



^* . . , , . , /tension in rod 
Diameter of each of two tie-rods = a/ T^gso * (20) 

For the beam B we must compute its necessary area of cross-, 
section as a tie or strut (according to which truss we use), and 
also the area of cross-section required to support its load acting as 
a beam, and give a section to the beam equal to the sum of the two 
sections thus obtained. 

Area of cross-section of B to / tension comp. 

resist tension or compression j T C ' ^ ' 

In trusses 1 and 2, 

Wx L 
Breadth of iJ (as a beam) = o x Z>=^ ~x~A' ^^^ 

In trusses 3 and 4, -^ ^'■' ^/'/^■-* - 

2 X If X L 
Breadth of B (as a beam) = 7 ^ n^ x A ' ^^^ 

Id these formulas, 

C — 1000 pounds per square inch for hard pine and oak, 
800 pounds per square inch for spruce, 
700 pounds per square inch for white pine, 
13,000 pounds per square inch for cast-iron. 

T = 2000 pounds per square inch for hard pine, 
1800 pounds per square inch for spruce, 
1500 pounds per square inch for white pine, 
10,000 pounds iDei' square inch for wrought-iron. 

A = 100 pounds per square inch for hard pine, 

76 pounds per square inch for oak and Oregon pine, 
70 pounds per square inch for spruce, 
60 pounds per square inch for white pine. 

Examples. — To illustrate the method of computing the dimen- 
sions of the different parts of girders of this kind, we will take two 
examples. 

1. — Computation for a (jlrder snch as is shown in Fig. 7, for a 
span of 30 feet, the truss to be 12 feet on centres, and carrying 
a floor for which we should allow 100 pounds pi^r sc^uare foot. The 
girder will consist of three strut-beams and two rods. We van 
allow the belly-rod T to come two feet below the beams B, and we 
will assume that the depth of the beams B will be 12 inches; then 
the length of C (which is measured from the centre of the beam) 
would be 80 Inches. The length of B would, of course, be 15 feet, 
and by computation, or by scaling, we find the length of T to be 
15 feet 2i inches. 



408 TRUSSED BEAMS. 

The total load on the girder equals the span multiplied by the 
distance of girdei*s on centres, times 100 pounds = 90 X 12 X 100 = 

3(KX)0 pounds. 
Then we find, from Fonnula 1, 

Tension in nxl = f», of 30000 X g^V"^^ = 65664 pounds; 
and, from Fornuda 20, 



/6y064 
Diameter of each rod = x/jM^g = Ij inches, nearly. 

The striit-heams we will make of spruce. Tlie compression in 

the two strut-beams = i% of 36000 X '/,P = 64800 pounds, or 21600 

pounds for each strut. To resist this compression would require 

21600 

-^g^ , or 27 square inches of cross-section, which corresponds to a 

beam 2^ inches by 12 inches. The load on B = i of 36000. or 18000 
pounds; and, as there are three beams, this gives but 6000 pounds' 
load on each beam. Then, from Formula 22, 

6000 X 1.5 _ . ^ . 1^ 
^ ~ 2 X 144 X 70 " • incbea^ 

and, adding to this the 2} inches already obtained for compression, 
we have for the strut-beams three 65-inch by 12-incli spruce beams. 
The load on C= ^ Fl', or 22500 pounds. If we are to bave a num- 
ber of trusses all alike, it would be well to have a strut of cast-iron; 
but, if we are to build but one, we might make the strut of oak. If 

22500 
of cast-iron, the strut should have ^.w^q , or 1.8 square inches of 

cross-section at its smallest section, or al)out 1 inch by 2 inches. If 

22500 

of oak, IL would require a section equal to "Tqqq • or 22i square 

inohos, = 4^ inches by 5 inches, at its smallest section. Thus we 
hav(> found, thai for our truss we shall require three stmt-l)eanis 
7 inclu's by 12 inches (of spruce), about 31 feet long, two belly-rods 
U inches diameter, and a cast-iron strut 1 inch by 2 inches at the 
smallest end, or else an oak strut 4i inches by 5 inches. 

2. — It is desired to support a floor over a lecture-room forty feet 

wide, by means of a trussed girder; and, as the room above is to be 

used foi- electrical i)uri>oscs, it is desiretl to have a truss with very 

little iron in it, and hence we use a truss such as is shown in Fig. 4. 

re the girders rest on the wall, there will be brick pilasters 

g a projection of six inches, which will make the span of the 

10 feet ; ^nd we will space the rods /if /^ so as to diTldeUieUe- 

into thi-ee equal spans of 13 feet each. The tie-taun will 



•TUUSSED BEAMS 409 

consist of two hard-pine beams, with the struts cominjGf between 
them. We will have two rods, instead of one, at i?, coming down 
each side of the strut, and passing through an iron casting below 
the hoanis, forming supports for them. The height of truss from 
centre to centre of timbers we must limit to 18 inches, and we will 
s})ace the trusses S feet on centres. Then the total floor-area sup- 
ported by one girder equals 8 feet by 89 feet, equal to .*U2 square 
feet. Tin; heaviest load to which the floor will be subjected wiii 
be the weight of students, for which V) pounds per square foot 
will be ample allowance; and the weight of the flooi* itself will be 
about 25 pounds; so that the total weight of the floor and load will 
be UK) pounds per square foot. This makes the total weight liable 
to come on one girder 81,200 pounds. 
Then we find, Formula 14, 

157 ins. 
Compression in struts = 0.;^>7 W x .o. ,., = 106800 pounds. 

156 ins. 
Tension in both tie-l)eams = 0.867 ir X ^^ .^^.^ = 106000 pounds. 

Tension in both rods i? = 0.807 W = 1 1450 pounds. 

The timber in the tniss wdl l>e hard pine, and hence we must have 

10(>8(X) 

—TTwTTT-, or 107 square inches, area of cross-section m the strut, 

which is equivalent to a 9-inch by 12-inch timl)er . or, as that is 

not a merchantable size, we will use a 10-inch by 12-inch strut. 

The tie-beams will each have to carry one-half of 106000, or 58000 

5800()__ 
pounds ; and the area of cross-section to resist this equals ^j^ — 

27 inches, or 2^ inches by 12 inches. The distributed load on 
one section of each tie-beam coming from the floor-joist equals 
i:J X 8 X 100 = 10400 pounds; and from Formula 28 wo have 

^ = ^ 7T. 7 = ^ — ^Mj ^/w> = 3? inches. Then the breadth 

5 X JJ- X A 5x 144x 100 

of each tie-beam must be 84^ inches + 21 inches = 6 inclies : hence 

the tie-beams will be 6 inches by 12 inches. Kach rod will have to 



/57/i5 
^..^ = } inch, 

nearly. 

Thus we have found, for the dimensions of the various pieces of 
the girder: — 

Two tie-beams 6 inches by 12 inches; two rods at each joint, J 
inch diameter i and strut-pieces 10 inches by 12 inches. 



A\0 



BIVETEU PLATKIHON GIKDKHS. 



CHAPTER XX. 

RIVETED PLATE-IRON GIRDZSR8, 

Whenever the load upon a girder or the span is too great to 
admit of using an iron beam, aiul the use of a tmssed wooden 
girder is impi-acticable, we must employ a riveted iron-plate girder. 
Ginlers of this kind are quito commonly used at the present day ; 
as they can easily be made of any strength, and adapted to any 
span. They are not generally used in buildings for a greater span 
than sixty feet. These girders are usually made either like Fig. i 




tlWW 



n'A'AMyVitf.wj 



Fig. 2. 



or Fig. 2, in section, with vertical stiifeners riyeted to the web- 
plates (»very few feet along their length. The vertical plates, called 
'' web-plates/" are made of a single plate of wronght-iron, rarely 
less than ont^-fourth, or more than five-eighths, of an inch thick, 
and geiKM-aliy tliive-i>ightlis of an inch thick. Under a distributed 
load, the web of three-eighths of an inch thick is generally snfll- 
ciently sti-ong to resist tlu^ shearing-stress Ln the girder without 

ng, provided that two vertical pieces of angle-lroD ; r ivebed 
>^eb, near each end of the girder. Tliese ve ii i !«■ of 
>n or T-iron, whichever is used, are c "; 

ten the girder is loaded at the centre, ana : 



• If 



K4":- . 



RIVETED PLATE-IRON GIRDERS. 411 

under a distributed load, it is necessary to use the stiffeners for 
tlie whole length of the girder, placing them a distance apart equal 
to the height of the girder. The web is only assumed to resist 
the shearing-stress in the girder. The top and bottom plates of tlie 
girder, wliich have to be proportioned to the loads, span, and lieiglit, 
are fastened to the web by means of angle-irons. It has been found, 
that in nearly all cases the best proportions for the angle-irons is 
:i indies by 3 inches by .J inch, which gives the sectional area of two 
angles five and a half square inches. The two angles and the plate 
taken together form the flange; the upper ones being called the 
'* upper flange," and the lower ones the ** lower flange." 

RiVKTs. — The rivets with which the plates and angle-irons are 
joined together should ho, three-fourths of an inch in diameter, 
unless the girder is light, when five-eighths of an inch may l)e sutti- 
cient. The spacing ought not to exceed six inciies, and should be 
closer for heavy flanges : and in all cases It should not be more than 
three inches for a distance of eighteen inches or two feet from the 
end. Rivets should also not be spaced closer than two and a half 
times their diameter. 

Rules for the Strength of Riveted Girders. 

In calculating the strength of a riveted girder, it is customary to 
consider that the flanges resist the transverse strain In the girder, 
and that the web resists the shearing-strain. To calculate the 
strength of a riveted girder very accurately, we should allow for 
tilt* rivet-holes in the flanges and angle-irons ; but we can com- 
pute the strength of the girder with sufficient accuracy by taking 
the strength of the iron at ten thousand pounds per square inch, 
instead of twelve thousand pounds, which is used for rolled beams, 
and disregardnig the rivet-holes. Proceeding on this consideration, 
we have the following rule for the strength of the girder : — 

10 X area of one flange x height 
Safe load in tons = :] x span in feet ' ^ ^ ) 

Area of one flange I _ 3 x load X span in f eet 
in square inches ) 10 X height of web in inches' ' 

The height of the girder is measured in inches, and is the height 
of the web-plate, or the distance between the flange-plates. The 
w(^b we may make either one-lialf or three-eighths of an inch 
thick ; anil, if the girder is loaded with a concentrated load at llie 
centre or any other point, we should use vertical stiffeners the whole 
length of the girder, spaced the height of the girder apart. 



412 



RIVETED PLATE-IRON GIRDERS. 



If the load is distribvted^ divide one-fourth of the whole load on 
the girder, in tons, by the vertical sectional area of the web-plate: 
and if the quotient thus obtained exceeds the figure given in 
the following table, under the number nearest that wlifcli wouhl 

1.4 X height of ginler 
bo obtained by the following expression, " thickness of wci7 ' 

then stiffening pieces will l)e required up to within one-eighth of 
tho span from the middle of the girder. 



c/ X 1 .4 
t 


31) 
3.08 


35 
2.84 


40 
2.61 


45 
2.39 


50 
2.18 


55 
1.99 


60 
1.82 


65 
1.60 


70 
1.52 


75 
1.40 


80 
1.28 


85 
1.17 


90 
1.08 


9;) 
1.00 


100 
0.92 



Example. —A brick wall 20 feet in length, and weighing 40 

tons, is to be supported by a riveted plate-girtler with one web. 

Tho girder will be <24 inches high. What should be the area of 

each flange, and the thickness of the web ? 

3 X 40 X 20 
4ns. Area of one flange = — m x 2^ ~ ^^ square inches. 

Subticicting 5 squai*e inches for the area of two 8-inch by 8-inch 
angle-irons, we have 5 s(iuare inches as the area of the plate. If 
we make tho plate 8 inches wide, then it slK>uId be5-r8,orfofan 
Inch thick. The web we will make J of an inch thick, and put two 
stiffonors at each end of the girder. To find if it will be necessary 
to use more stiffeners, wo divide J of 40 tons, equal to 10 tons, by the 
area of the vortical section of the web, which eqimls f of an inch X 
24 inches = sciuaro inches, and we obtain 1.11. The exin^esslou 
1.4 X lioii^ht of girder 

 thioknoss of \vA) — ' *" ^^^'^^ **^**^' ^^^^^^"^ ^'^' ^* number near- 
est this in the table is 00, and the flgure under it is 1.06, which is a 
little less than 1.11 ; showing that we nnist use vertical stiffeners 
uj) to within i\ feet of tho centre of the girder. These vertical stiff- 
eners we will make of 2i-ineh by 2j-inch angle-irons. From tlie 
fonnuhl for th(> area of flanges, the following table has been coni- 
piilei). wliiel) greatly faeilitato.s the process of finding the necessary 
area of flanges for any given girder. 



RIVETED PLATE-IRON GIRDERS. 



Co-efficient for deLenninin;; Ihe area required in flanges, allowing 
10,00IJ pouiiils ]wr siiuare incb of cross-section fibre strain ; — 

1U:lk. — Mnlliply Use load, in tons ot -JOOIl i)Oiinii» unffomily 
ilistribiitetl, by tlie co-?fbcient, and dividu by 1000 pounds. Tlie 
quotient will be the gross area, in square inches, required for each 
llan^. 



I im ms. 



ExAMl'l.E. — l.ol iLS take the same giriler that we have jiisl 
c'0]iipiite<l. Here llie a]>an was 20 feet, and the depth of girder 24 
iuehes. From the table we find the eo-efli<!ient to \»: 2-~)0 ; and 
multiplying this by the loail, 40 tons, and ilividlng by 1000, we 
have lU square inches as the area of oue Sange, being the same 
result as thai obtained before. 



4U RIVETED PLATE-IRON GIRDERS. 

Girders intended to carry plastering should be limited in depth 
(out to out of web) to one-twenty-fourth of the span-length, or 
half an inrh per foot of span: otherwise the deflection is liable to 
eau<e the plastering to crack. In heavy girders, a saving of iron 
may often i)e made by nMlucing the thickness of the flanges towanls 
the ends of th(^ i^irder, where t\w strain is h'ss. The bendinir- 
moment at a number of points in the length of the girder may Ix' 
detJMiiiined, and the area of the flange at the different i)oints nia<h' 
propoilional to the bending-moments at those points. The thick- 
ness of the llanges is easily varied, as required by forming them of 
a sutticiiMit numlu'r of plates to give the greatest thickness, and 
allowing them to extend on each side of the centre, only to such 
distanc'es as may be nt'cessary to give the required thi<:kness at each 
point. The deflection of girders so formed will be greater than 
those of uniform cross-sectiou throughout. 



TABLES OF SAVE LOADS FOR RIVETED PLATE- 

IROX GIRDERS. 

The tables given on pp. 414 and 415 have I)een computed ac- 
cording to the fonnula on p. 411, to give an idea of the siz«* of 
girder that will be reiiuired for a given load, of the heights and 
siKin^ inlieiited. 

If i; i- r(nuinl)ered that the strength of a girder depends tUrectly 
as tlh- ;i!( a of its llanges and its height, the width and thickness of 
the tl,in.r<- pi ite may be changed, inttrided the area rcniahis the 
.sn,in . witlnni* altering its strength. Thus a girder ii(5" liigh, with 
tlaiu.- tni.i,.- i of 4.r' X 4f' X ^" angles, and f X 24" plate, would 
be as vT,.ni:, as one with th«' same aniilos and 1" X 12" plate, pro- 
vi«l.' I iIm' u»'!> plates are ])r()perly stitTened, as described on p. ;i47. 
In eompuiiiv,' li:e weight of the ninlcrs in the tables, no allow- 
ancf b.l«^ h«M'ii made for siitT<»ners. In computing the stn^ngth of 
rivet*'. 1 uiidrr'*, it will be convenient to know that — 
The ana of two :V' x ;}" X |" angle-irons = ').iy stpiare niches. 

:U X ;)f' X f *' =({.4 *• 

4' X 4" X f *' =7.4 " 

4f' X 4f' X f ** =v{.4 «-• 



RirKTBD PLATR-IRON QIBDKR8, 41 



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STEEL BEAM GIBDEB8. 417 



Steel Beam Girders. 

An economical style of box girder, well adapted for short span 
lengths, is one composed of a pair of I-beams with top and bottom 
flange plates. Such girders are commonly used for supporting 
interior walls in buildings. 

The following tables give the safe loads for ** Carnegie " beams, 
with different thicknesses of plates. They were prepared for steel 
girders on account of the advantages possessed by steel beams over 
beams of iron. The former are more economical of section and 
permit the use of a higher unit strain than the latter. 

The values given in the tables are founded upon the moments of 
inertia of the various sections. Deductions were made fgr the rivet 
holes in both flanges. The maximum strain in extreme fibres was 
limited to 13,000 lbs. per square inch, while in the tables on- rolled 
steel beams a fibre strain of 16,000 lbs. was used. This reduction 
was made in order to amply compensate for the deterioration of the 
metal around the rivet holes from punching. 

Box girders should not be used in damp or exposed places, since 
the interior surfaces do not readily admit of repainting. 

ExajIPle. — A 13' brick wall, lo feet high, is to be built over an 
opening of 24 feet. What will be the section of the girder re- 
quired ? 

Ans. — Assuming 25 feet as the distance, centre to centre of 
bearings, the weight of the wall will be 25 x 15 x 121 = 45,375 lbs., 
or 22.68 tons. 

On page 420 we find that a girder composed of two 12" steel 
beams, each weighing 32.0 lbs. per foot, and two 14" x i" flange 
plates will carry safely, for a span of 25 feet, a uniformly dis- 
tributed load of 23.23 tons, including its own weight. Deducting 
the latter, 1.42 tons, given in tho next column, we find 21.81 tons 
for the value of the safe net load, which is 1 . 07 tons less than re- 
quired. From the following column we find that by increasing the 
thickness of the flange plates ,^j" we may add 1.52 tons to the 
allowable load. This will more than cover the difference. Hence 
the required section will be two 12" steel beams 32.0 lbs. per foot, 
and two 14 ' x ■^%" steel cover plates. 
27 



418 bterl beam qibderb. 

stki':l beam girdbbb. 
safe loads in tons, uniformlt disnubhtbel 

S-X" eti.«l (Caiiiogle) I-beama and 3 aleel platw 18" x J" 



it 



liii" lit in.'iu lb*, prraq. !■. 



STEBL BEAM OIRDEBS. 419 

STKBL BEAM QIRDBRg. 

SATB LOADS IN TONS, UNIFORMLT DI8TRIBUTXIO. 

X-IS" Bieel (Carnegie) I-beams and 9 utee] plates 14" > |" 



420 



STEEL BEAM GIRDERS. 



STEEL BEAM GIRDERS. 
SAFE LOADS IN TONS, UNIFORMLY DISTRIBUTBIX 

si-lS" steel (Carnegie) I-beams and 2 steel plates 14" x i" 



« 






tt 










c 




/> 


jf — 6 — >r«, 




^ 1 




^^ 


a) 


^ 


b=^'„„ . , 








Si 




Vi" steel 


. . ^ 


r, uJ 




X> 




steel 




I-beamn. 




^^ *^-» 


" 12" steel 


^Ui 


O 


plates, 




40.0 lbs. 


2 


steel 




I-beams, 




£ 


14" X J" 




per foot. 


.JA*^?',- 1 




83.0 lbs. 


•§« 


centi 
eet. 




4 


^r-^ 




14 


Xf- I 


^1I^ 


per foot. 


•a'5 


0<4- 














ti S 


**^ .i^ 






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I 


JS 9 


«.s 


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 «k-«M 


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s o o 


' a; 0) C .iM S X 


s 


s o o 


' 0) V o 


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4— t 

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*^4J XI 


.U o 


^s 


1) 
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75 


•^ < «-■ 


«5: tt - . "" J, 3 - 
- = * id - - 


S3 
5 


5-3 r 

— "S Ti Ti 




•S.5 

si 

.5.S 


10 , 




(U.IM 


' <).r.5 3.':.') 




58.08 


0.57 i 8.81 


0.06 


11 




5.>.(W 


0.71 3.40 




32. SO 


0.63 i 3.45 


0.08 


12 




r>4.1.> 


O.T". 3.12 




4S.40 


0.68 3.17 


0.08 


13 1 




41). IC) 


0.S4 2.8S 




44. (W 


0.74 2.W . 


0.04 


14 




4()..i'.) 


n.!)l 2.«S 




41.48 


0.80 2.7« 


0.04 


ir. 




13. vM) 


U.')7 2..-0 




38.72 


0.85 2.58 


0.04 


K 




4 ).:.!) 


1.04 2.34 




3<)..30 


0.91 2.88 


0.U5 


K ; 




;«.-.>() 


1.10 2.21 




i^.m 


0.97 2.34 


0.05 


is 




3(».i>S 


1.17 2.08 




82.27 


1.03 3.11 


0.06 


IS) 




31. 1« 


1.2.3 1.97 




30.57 


1.08 > 3.00 

i 


0.05 


iM 




3i.47 


1.3«) 1.87 




29.04 


1.14 : 1.90 


U.06 


iil 




3). '.13 


1.3«; 1.7S 




27.r.« 


1.20 ' l.Hl 


0.06 


.>.) 

-v^ 




•>I!)..V2 


1.43 1.70 




2(5.40 


1.25 1.78 


0.06 


i>.3 




^s.\>:j 


1.1'.) i.r»3 




25.2.'> 


1.31 1.65 


0.07 


i»l 




'.■:■.<)»» 


!..')<; i.r><» 




21.20 


1.37 1.5H 


0.07 


U.-) 




.::>.i»s 


i.r.i \.rii\ 




23.2.3 


1.42 1.52 


U.07 


•,M) 




:i.'.ts 


1. »•.'.» 1.14 




2-2. :« 


1.4S 1.46 


O.08 


•^>r 




!!.'•.') 


1.7.") i.:is 




21.r.l 


1.54 1.41 


0.06 


'.»S 




,'3. 1) 


1.S2 1.34 




2.1.74 


1.1,0 1.86 


0.08 


•^.t 




•,»•,». 31) 


l.SH 1.21) 




20.03 


1.K5 1.31 


0.08 


;j' 




'JI.JJ-, 


l.li.-. 1.2.". 




11). :W 


l.;i 1.27 


0.(« 


;n 




•J ».!).'i 


2.111 1.21 




18.7:1 


1.77 1.23 


0.09 


3j 




•JO.-JI 


■J. OS 1.17 




IS. 15 


1.82 1.19 


O.OO 


;{:$ 




I'l.'N 


2.M 1.14 




17.J»o 


l.SM 1.15 . 


0.10 


Ml 




l!».l.t 


2.21 1.10 




i;.«>8 


1.1»4 1.13 


0.10 


.-i.-) 




IS.-.:) 


2.-,: l.i>7 




10..-..) 


1.19 1.09 


0.10 


.*i'i 




IS. .11 


2.::4 1.04 




in. 13 


2.05 i.r6 


0.10 


3: 




17..-.-. 


2.4" 1.01 




15. :o 


2.11 1.08 


0.11 


i^ 




K.iM 


2.4-: o.ii'J 




1.').2S 


2.17 l.ilO 


0.11 


i** 




i"i.»ir. 


.> r 


Vl o.w I 




11. M) 


3.32 


0.98 


0.11 



.\)mi\i- \.tiu('>< .ire WaM* i nn nia.\iniuin lUirc i^lniins of 18,01)0 Ibi*. pvr M|. to. 
Uiv'-i iiii!i-< III l.i.ili ilHiit{«'?< deducteil. Weif^hti* uf KirUvn corKiipuiid tolcngtba 

I'fri re ft. rriiJn- i.r licurin^'!*. 



STEEL BEAU OIRDEBS. 431 

STEEL BEAU GIRDERS. 

SAFE LOADS IN TONS, UNIFORMLT DISTBIBUTBDl 

S-IO" steel (Camegic) I-beama and 2 eteel plates IS" n )" 



Atvore TmlneH ore ba^cd on m'jximu 
Blvet holes In botb naDgee deducted. 
«mA» to emtae of beuinge. 



422 



CAST-IRON ARCH-GlRDKUa. 



CHAPTER XXT. 

STRENGTH OP CAST-IRON ARCH-GIRDERS, "WITH 
WROUGHT-IRON TENSION-RODS. 

Oast-iko.v jircli-girders are now (juite extensively enii)loyeil to 
support tlic front or rear walls of brick buildinfj^s. Fig. 1 shows the 
usual form of such a girder, the section of the casting and roil 

hv'uvj; shown in Fig. 2. 



— ii .' 



Fig. 1. 





Fig. 2. 



Th<^ casting is niad(» in one ])iece with l)ox ends, the latter having 
LCroov*'^ and scats to nMH'ivc the wrought-iron tie-rod. 

rii«' ti('-!().l is madi* from one-eighth to three-eighth8 of an Ineh 
sliorirr tlian the casting, and has scjnare ends fonning shoulilers 
so as t(» li! into the castings. The rod has usually one weld on its 
liimtli. and ixrcat <*arc shouh' he taken that this weld lie jXTftH't. 

Tlir r<> I is ('X])and«'d hy heat, and then pla<'etl in i)Osition in the 
(•as; in.:, and allowed to eontra<'t in cooling; thus tying the two enils 
()!" iln' ca^'ing together to form abutments for receiving the hori- 
zontal iliiiist of tie* areji. If the rod is too long, it will not n*eeive 
till- full ]i'-oportion of the strain un'il the east-iron \iha so far dr- 
ll««t«i|. tliai its lower edge is >ubji'eted to a severe tensile strength, 
whirh cast-iron <'an feebly resist. If the ti(»-rod Is made too short, 
the ea^tinu is eambered up, and a sev(>re initial strain put Upon 
both the east and wrought iron, which enf(*4>hles lK>th for carryii|g 



CAST lEON ARCH-GIRDERS. 



423 



a load. The girders should have a rise of about two feet six inches 
on a length of twenty-five feet.i 

Rules for Calculating^ Dimensions of Girder and 

Rod. 

A cast-iron arch-girder is considered as a long column, subject 
to a certain amount of bending-strain ; and the resistance will be 
governed by the laws affecting the strength of beams, as well as 
by those relating to the strength of columns. 




Fig. 3. 



If we regard the arch as flexible, or as possessing no inherent 
stiffness, and the rod as a chord without weight, we can deduce the 
following formula for the horizontal thrust or strain : — 

Hor. thnist _ ^^^^ P^^ ^^^^ ^^ span x span in feet, squared^ 
or strain "" 8 x rise of girder in feet ^ ' 

From this rule we can calculate the required diameter of the 
tension-rod, which may be expressed thus : — 



Diameter in inches 



Vloail 



on girder X span in feet 



8 X rise in feet X 7854 



(2) 



The rule generally used, however, in proportioning the wrought- 
iron tie to the cast-iron arch is to alloiv one square inch of crosa- 
section of tie-rod for every ten net ton.^i of load impoaed upon the 
span of the arch. 

The following table, taken from Mr. Fryer's book on " Architec- 

I _ I — 

1 Andiiteflliml Iron- Work for Buildings. — William J. Frter, Jun. Pp. 38. 



4J4 



OAST-IHON ARCII-GIRDERS. 



tiiral Iron- Work," shows ihe section of the cast-iron arch requirptl 
to supinrrt solid hrick icallsy and haciny a span of from 13 tot6 
feet. 



I<it{lil of 
wall. 


Tliifkncss 
wall. 


of 


Di 
Top flange 


4u ftM't. 
-III " 


I 12 iuchi'H 

12 " 
1 ir» 

1 10 " 

1 


. 


12" X 1" 
12" X 1 y 
12" X 1, " 
10" X ij'' 






_ 






DiMEXHioxH OF Section. 



Centre web. 



1-2" X 3'' 

i— '^ 8 

12" X »•' 

12" X 1" 



Bulb. 



;i" X 2" 

1// y ,>// 

X 2" 



3i" 
4^' 



I 



Substitute for Cast-iron Areli-Ciirder, 

In tlu* cast-inm arch-j;inior with wrou.uhl-iron tcnsioii-roil. the 
ca^tin;^ only serves to resist coinnn'ssion. Its place can as wi*!! be 
till«Ml l»y a l)rick arch foottMl on si \mv of cast-iron skewbacks, 
wliich an* thcnis(»lvcs held in ixwilion by a pair of tie-rods, as in 

In I Ills case, Fornniht 1 will still jjivc the horizontal pull to be 
resistci by the tie-rods ; i)Ut, as vvc nnist have two rotls instoad of 
ont , tlic diameter of eac.'h will bo obtained ))y the Ibnniila, 

Diameter o. each _ /Tj*!*.") )*>»''^"" arch X sjian ^.j 

rod in inclics \ Hi X rise of arch in fiM'l X 'tSTA 

N.r.. — TJu- rlH- Ik nie!i!>.i:r«Hl from thi* cent a' of the nnJ to the eentre of the 
:t:i-li. It will alM) ln' rem*>ini)<3n'd that the hpan iK tti \h.'. (i/irt/^jr taken In feet* 

mile--. DiliiTwi-if spti'iti-'d. 

Kx.vMPi.i-: I. — It is desired to siipiM)rt a 12-incb brick wall Ai) 
til-* liiuli «'\ci- an n]H'nin.Lr -'► l'e«»i wide, with a easi-iri>n an-li-iiinliT. 
''.'Ii;t! -imidd lie ilic dim 'n-^inji-! of lln' u:ir<ler'.* 

I'nr !!•« riistin;;. we lind from the tabic that the eross-sei'tion of 
;li.' llanv:"- hnnid be li: iin-b:" l>y 1 ineb : of tli«' web, TJ inebi's b\ 
: in-li : and of I lie bnlb. :! incbes by li inches. W'v will make llw 
ri'^e nt I'lic udrder *J feet and <> incbes. and fnon Forninla 2 Wf HniP 

\\ei<.;bt of wall X s]ian 

i)iim. **i I .kJ in im-bes - \ j*^ • ^ ...  :.. «• , v -w?i"i 

\ s X riNc ot areli in feel X iK>4 

Miio X -JO X Ml') X :io_ , — 
\ s X lM X 7s:h " ^ •■'•" = '-^* ^^ 



> I '••I:-! I>!i-;L; Ibai ihi* uinli-r wonlii o:ll^ Kup|N>rt atNiiit twcnly feet of Ihf 
■\)k\\ in lii-iL'ht, thi> will! abiiM- l>i,tt ^uppnrtillK tlMfif. 



WOODEN FLOORS. 425 



CHAPTER XXII 

STRENGTH AND STIFFNESS OF WOODEN 

FLOORS 

Strengrtli of Floors. — In calculating the strength of floor- 
beams, the first thing to be decided is the span of the beams, which 
is generally determined by the size of the opening to be covered ; 
and the second is the load which is to come ui)on the floor. 
Wooden floor-beams should not have a span of more than twenty- 
five feet (if it can be so arranged ) : for, if they are of a greater length 
than this, it is difficult to stiffen them sufficiently to prevent vibra- 
tion under a heavy or moving load When the distance between 
the l)earing- walls of a building is greater than the above limit, par- 
tition-walls should be built, or else the beams should be supported 
by iron or wooden girders resting upon iron or wooden columns. 

The Building Laws of the cities of New York and Boston require 
that m all buildings more than thirty feet in width, except churches, 
theatres, schoolhouses, car-stables, and other public buildings, the 
space between any two of the bearing- walls shall not be over twenty- 
five feet, unless ginlers are substituted in place of the partition- 
wall. Floor-beams, when supported at three or more points, 
should always be made continuous if possible, as the strength of 
each portion of the beam is thereby greatly increased. 

Superimposed Loads. — There is some difference of opinion 
among authorities as to what should be allowed for thc^ suprrim 
posed load upon the floor of a dwelling or upon the floors of public 
buildings. The New- York Building Law requires that in all build- 
ings every floor shall have sufficient strength to bear safely upon 
every superficial fool of its surface seventy-five pounds, and, if used 
as a place of public assembly, one hundred and tvv«^nty pounds. 

In dwelling-houses, where the maximum load consists of nothing 
but ordinary furniture and the weight of some ten or twelve people, 
it is not necessary to allow more than forty pounds per square foot 
for the superficial load ; and, in most cases, eighty pounds per s(|uare 
foot 18 ample allowance for the weight of an assemblage of peopl(^ 
Only in cases where people are liable to be jammed together during 



426 



WOODEN FLOORS. 



a jMinio or some unusual circumstance, is it possible to p;pt a weight 
on the tl<«>r of one humlriMl ami twenty pounds per Rr{uarp fool. 
Tlu' follt)\vin^ tablt* iiivcs tlie weight per squaiv foot which shouM 
l»e assume* I, in addition Lo the wciglit of tlir floor, for thcso various 
cases : — 

For stHM't l^ridges for general public traffic, S*t lbs. per s<{uarp foot. 

For tln»)r'5 of dwellings 4H lbs. {-ter s<|uaiiMont. 

Ft»r iliunh»'>. theatres, and ball-rooms, SO to V2i) lbs. i>er si|iian* foot. 

For s'hools ^<0 lbs. per sqiiar** foot. 

Fur hay-l«»tts S() lbs. per square loot. 

For si(>rai;e of gram HH) lbs. \n*r st{uare fool. 

For wan'houses anil general merchandise, '1')^) lbs. jHir siiuare foot. 

For fa<tories 1(M> to 4(X) lbs. per square foot. 

F'or oHirt- buildings liH> His. per square foot. 

Wan 'ho list '-floors are sometimes very heavily loaded, and for 
lhe>f a >iMMial compulation should bt? made in each case. 

Til- i.»ll.nvhig table, compiled by Mr. 0. J. H. Woodbury.' gi'<*« 
the tli>«>r areas, cubic space, and weights of merchandise, as usualty 
siori'd in warehouses. If the goods are piled two or luore cas^ 
hiu'h. I lie weight per square foo: of lloor will of course he increased 
in proportion. " The measuremenis were always taken to the 
outside ol case or package, and gross weights of such packages are 
given." 



Matkkial. 



W»>ol. 

Ha!.- K.i^- I- .li.i . . . . 
*• A;:*'.:. I i.i . . . . 
" S 11*. !; \iiiiTii"a . . 

I •* o ,_:n 1 

I •• < '.I ;:■■;• li.i .... 

 Kiii W ■'. 

>;.u^ . :" S. ■iind Wiml . 

Wiiolll'll (ifMXls. 

' ';i"'f }• .% . x-t" 

•• K r ;:• -. hi':»\ y . . 

I >ri'-- « i■nlll^ . 

'• < ".i-- iii>-*i ■«.... 
•• 1"'. :•  u r.ir .... 

•• li I A :» .... 
•• II - W .iiikii.-. . . 



Ci»!ttii». I'tr. 

i;.i- 

 • -^-1 ii . . . 

! •■ . i> « ■■•in:ni-«-i\l 
•• .1 -.- ..'... 



•• -1 :"■ I i-hiiiir-* 
■' VI I i: I . . 
X -I I 

 •^«il • • • I 



 a 



MkaSI- HEME NTS. 



Floor 



...0 

.'•.s 
7 

7.5 



7.1 
.'i . .'» 

M..-» 

» ■! 

 .•! 

111.:; 
4.0 



N.l 

4.1 

•J. 4 

•J.n 
■• .t 



•*».S 



I • II Mo 
M. 

 ■o. 

3i). 



1J.7 

'J-J.rt 

■JS.il 

•Jl.O 
14.0 

44.-2 

■« ,.• 
.1. 1  

iii..'i 

:;4.7 
17.11 



Wbiuuts. 



(fPiM*. 


M. ft. 

113 


IVr 

cubic f :. 


;mo 


28 


s-^ 


m 


lA 


lUiJO 


14S 


29 


4S-2 


70 


lA 


.'mO 


73 


17 


•M 


40 


7 


— 


- 


5 


■*jrt 


40 


17 


.^•.0 


40 


22 , 


4i*iO 


M 


21 ! 


.vm 


ft2 


•» 


;;.-> 


4)« 


10 


4.'H> 


44 


13 


■j..i» 


63 


18 


M.*, 


64 


12 


:»-.o 


134 


25 


l-V. 


100 


40 


:«k) 


125 


4:.i» 


174 


43 


•>rt 


88 


« 


TOO 


Rl 


» 


41 «) 


7» 


M 



I Dif Kirt> PriM.u-tiitii ttf Mills, ii. lift 



WOODEN FLOORS. 



427 



Iatbrial. 



I €k>ods. 

leached Jeans . 

3k 

vn Sheetings 
«hed Sheetings . 

t8 

t Cloth. . . . 

ts 

ings 

>tton Yam . . 

?*ng 

in Bales. 

nen 

tton 

)tton . . . . 
ivings . . . . 



td Book . . 
endered Book 



er . . 
ard . 
toard 



Bags 
Bulk 



« 



mean 
lour on side 

•• on end 
tags . . . 

in Barrels 
ags . . . 
lay . . . 
lerick Compressed 



« 

« 






Measurements. 



tiiflf'*, etc. 

I Bleaching Powder, 
Soda Ash . . . 

?" 

rh 

ac 

oda in iron drum . 

arch 

>arl Alum .... 
act IvOgwood . . 

ime 

.'raent, American . 

" English . . 

aster 



Floor 
space. 

4.0 
1.1 
3.6 
4.8 
7.2 
4.0 
4.5 
3.3 



1.4 



8.5 
9.2 
7.6 
7.5 

16.0 
7.5 

2.8 



4.2 



4.1 

3.1 

3.6 

3.7 

3.3 

5.0 

1.75 

1.75 

1.75 

1.75 



11.8 
10.8 
3.0 
4.0 
1.H 
4.3 
3.0 
3.0 
1.06 
3.G 
3.8 
3.8 
3.7 



Cubic 
feel. 

12.5 

2.3 

10.1 

11.4 

19.0 

9.3 

13.4 

8.8 



5.3 



39.5 
40.0 
30.0 
34.0 
65.0 
30.0 
11.1 



4.2 



5.4 
7.1 
3.6 
5.9 
3.6 
20.0 
5.25 
5.25 
5.25 
5.25 



39.2 

29.2 

9.0 

3.3 

4.1 

0.8 

10.5 

10.5 

.8 

4.5 

5.5 

5.5 

6.1 



Weights. 



Gross. 


Per 


Per 




sq. ft. 


cubic ft. 


300 


72 


24 


75 


68 


33 


235 


65 


23 


330 


60 


30 


296 


41 


16 


175 


44 


19 


420 


93 


31 


325 


99 


37 


— 


— 


11 


130 


— 


30 


100 


70 


24 


910 


107 


23 


715 


78 


18 


442 


50 


15 


507 


68 


15 


450 


28 


7 


600 


80 


20 


400 


143 


36 






50 


— 


— 


69 


~ 


_ 


38 




_ 


33 


_ 


_ 


59 


_ 


_ 


64 


. 


_ 


10 


— 


- 


37 


165 


39 


39 


_ 




44 


_ 


_ 


39 


— 


— 


41 


218 


53 


40 


218 , 


70 


31 


112 


31 


31 


218 


59 


37 


96 


29 


27 


284 


57 


14 


125 


72 


24 


100 


67 


19 


150 


86 


29 


100 


57 


19 


1200 


102 


31 


1800 


167 


62 


385 


128 


43 


1.50 


38 


45 


160 


100 


39 


600 


140 


88 


250 


83 


23 


350 


117 


33 


55 


52 


70 


225 


63 


50 


325 


86 


59 


400 


105 


73 


325 


88 


53 



4L>h5 



WOODEN FLDOKS. 



Matbbial. 



I 



I 



Dye RtnflDB, etc—OonVd. 

Barrel KuHiii 

•• LardOU 

Uope . 

Miftcellaneoos. 
Box Till 

•' GhL*^ 

C rate ( 'rockery 

(':i-k Crockery 

I>aie Li-ailier 

" (rnatr<kin8 

" iiaw Hides 

" " '• compref8ed, 

'• Sole Leather . . . 

Pile S.ilf Leather . . . 

I>arrel Granulated K^ugar. 

Brown Sugar . . 

Cheese 



Measubexents. 



Floor 
space. 

3.0 
4.3 



2.7 

9.9 
1U.4 

7.3 
11.2 

rt.O 

0.0 
lli.tt 

3.U 
3.0 



1 Cubic 
feet. 

9.0 
12.3 



0.5 

39.6 
42.5 
12.2 
16.7 
3<).0 
30.0 

s.y 

7.5 



Wbiohts. 



OroBB. 

430 
422 



139 

1600 
600 
190 
300 
44X) 
700 
200 

317 
340 



Per I Per 
BQ. ft. cubic ft. 



143 
98 



48 
4A 



09 


278 


. 


60 


102 


40 


52 


14 


26 


16 


27 


18 


67 


13 


117 


23 


22 


16 


— 


17 


106 


42 


113 


45 


- 


ao 



AV<Mj4:Iit of tlie Floor itself. — Having <lecided upon the 
span of the Moor boanl^ an. I upon the siiiH.Tinii)Oseil load, we must 
nt'xl consithi- the weiijjlit of th»? tlix^r itsrlf. 

WoodtMi floors in (hvellinjxs wiMirh. on thcavprago, from 8eveni.»H»n 
to twrnty two i>oiimls ]kt vS(|uai(' foot of floor, incluiling tht* weight 
of tin* plastt'rini: on the nmh'r sn\v. For onlinarj' spans tho Wiight 
may l)»' takrn at twontv pounds iH»r squan* fool. Jind, for lorn; spans, 
twnity two pounds por squan* f<K)t. For floors in public bulldins^, 
tin* \\«'ii:lit piM- sq nan' foot seldom oxcoeds twenty -five pounds, and 
it nia\ NMti'Iy ))e assumed at that amount. 

In wanliouse floors, whieh havi- to sustain ver\* hoavv loads, the 
w»'iu'lii iM'i- sqiian- fool may souH'times 1h» as gnMt as forty or fifty 
lHinnd>: and m Mieh ease*^ the a]»pro>Limate weight of the floor ^kt 
Miuan* \\n*l >hoiild l>e tirst caltulateil. 

FjU'tor of Safety to be used.— In eonsiderlni; tho load 
on a tliiiir. it siiould !>«> lememhtM-ed that the efTt>et of a load bud 
di-nlx applied uiK)n a Ix'am is twiei> as i:n-at as that of the Hanie 
ln:ii| i:i'.idnaliy applieii: and hrnrc the fa<'toi of s;ifety utH*4l for llu* 
fiiiimr »Li)Mld In* I win' a<« ijiral ax> that for the latter. The loail 
i-.iiio.l li\ a i-mwd of priipir i^ usually ron>ider«Ml to pn>ihii'«' an 
<th-<-i vxliii-ii i» a iiii-aii iH-iWft'ii thai nl llir sinH* ItKid wheli ;;ratlu- 
a)l\ and w Inii sitild<-nl\ a|tplled ; ami hmer a faelor of safKy IS 
iiii|>lii\fd wliirji i.s a mean lH*tw«'en that for a live and for a dead 
load. 

Tin- faihu-H of safi'ty for lltNir-iindM'rs adoptetl by the best engfn- 
citn \ar\ troni -i to 't. For short s|»ans hi onllnary dwelllngSi 
pnhlic Ituijilinu^. and Moivs, :{ is probably amply HUlHcieiil for 



I'JI ' 

- WOODEN FLOORS. 429 

strength ; but. for long spans, and flooi*s in factories and machine- 
shops, a factor of safety of 5 should often be used.' 

Rules for the Strength of Floor-beams. — In consid- 
ering the strength of a floor, we assume it to be equally loaded over 
 its whole surface, as this would be the severest strain to which the 
timbers could be subjected. Hence, in calculating the dimensions 
of the floor beams, we use the formula for a distributed load. That 
formula i^ for rectangular beams, 

2 X bread th x depth squared X A 
Safe load - span in feet x S ^^^ 

*S being the factor of safety. 

For floor-beams the safe load is represented by the superimposed 
load and weight of floor supported by each beam. 

The areA of floor supported by each beam equals the length of 
beam multiplied by the distance between centres. If we. let f de- 
note the weight of the superimposed load per square foot of floor 
surface, and/' the weight of one square foot of the floor itself, then 
the total weight per square foot will be (/+/') pounds, and the 
total load on each beam will ecjual 

Length of beam X distance between centres x (/4-/'). 

Now, if we substitute this expression in place of the safe load in 
the above formula, and solve for the depth, we shall have, 

Square of __ S x dist. bet, cen tres x length squared x (/ + /') 
depth. - 2 x~ bread thlT^ ' ^^^ 

or, if we solve for the distance between centres, we shall have, 

Distance between _ 2 x breadth x depth squared x A 
centres in feet - ^sVlength"^ared x (/ + /') ° ^^ 

N. B.— The length and distance between centret* must be taken in feet' and 
the length meanB only the distance between sapports, or the clear span. 

The values of the constant A for the four woods in general use 
are as follows : 



/ 



Spruce 210 

Eard pme 300 



Oak 225 

White pine 180 



Formulas 3 and 8 apply to all floors supported by rectangular 
beams, whatever be the factor of safety employed, the weight of 

> Until very recently It has been our custom to use factors of safety twice as 
great as these : bat, as we have had occasion to reduce the constants for strength 
to abont one-half of that formerly used, we have reduced the lactors of safety 
■ecofdins^y. It will be found that the result is the same as that obtained by the 
n3M«f odborirMtefs.  



llic sii[«TiiiiiM)sf.l limil, or of the Htm- ilstll. To illustrete the 
ii|il)liculi(iii (if tlicsc tnrniulas. ve will i;ive two examples such as 

K\Ai;i'i.K 1.— IVIiiii sliiiiilil In- ilu- <1iiui'nsiniiB of thi- HpruLi' 
nonr-liiMins in !i liwlliii;:. Ilii' lu-aiuM lo limv ii 8)Ht]i of 13 reel, and 
tub,, j.lmoil llliiKOu-s. i.L-1': fis'l. iinci'ii1n.sV 

-l».. Iti lliisciisi' w,-ninililusi' a TiLcKir ..f siifet)- of -I : / sl.oiil.l 
In- iHki'ii III Id ix>iiiuis. / ,11 i I iHiiiiiils, ami .'I is 210 jKimids, A*. 
suizii' 3 iiiilics fur till' lirmilli. Tlieii, by Fiirmulu •. 



. eo_ 

2 . 2 X aio 



•■''H.ti.-'V' r,:™'!0.5 



Till' ilopfli A "■^.■'  " 'il"c '"■'■■■ fl ifitlies. Tkiifo. to haw the 
miuisil.' s1n>ii},'lb. 1bi> IxMins sli.mlc! lit '> x 111 iTich.'s. 

KxAMi'i.i: a. -11 is lU-simi 1.. us,- 3 by 10 im-li VL-lIow.i.inr 
(•.■iuns in til,. n.«.r of ii ■■l.iircb. ill,- lu'U.iis to Imve a spin of IB 
ii','t. What ili-^tiini'i' sbiiiilil lli,-y Ix' s|m<ril ,>ii i:uln» i 

nils. / ■•', iioiuiilii, and A -■ 300 



I"' 



„ls, 'n,.'ll, by F„ 



\ SIM) 



l.isri,n.v b,.,w.rTi ...jntivs -. ' 1 ' .;.;;- J,,": ^ ^ 0.73 ft.. ..rft i..». 
Ut-iic-i' III,' MiKir vrillbeMitli(-ii'ullyslrouj;if thclH'iiiuaiiru pluoottlt 

IlrUl^iiiK of l''t(>'>r-lM'anis. — lly "luidcina" i<* ini>nnt  
sy-d'iii of lirai-iiii; fl«Hir-l¥iiui«, 
i>iibi'r by iiu-iiiis of siuitll Htnil*. 
iis 111 Kif;. I, or lij- iiu-niH of siuitli- 
lii.i^'s of iHianlH at ri^lit aiiuh^ 
III llii- joists, mill titliii); jil Ih^ 
l«-<vn IlK'iii. 
Tl m-i-t of tliU l>ni<-in|; \» il.- 



iliMdlmti'J loiid. Tlii- 
ii: :ils« siiiTi-iis tU>' joints. 
I'M'iiis Mii-in fnoii tiiniiiii; 
.>'. It \* cMliIoiiiiirv III 
niuM of •-hisH-liridi:iii..: M 
iiiylivi- t<ii-lf<hl fii-liiiilic 
' Iliiy -IkiiiM U- ill »tnii).'l>t 
I imiy iibul tllrwtly uiwd 



W UUUlfi JN I'L.UUKB. 



4b 1 



those adjacent to it. The method of bridging shown in Fig. 1, 
and known as "cross-bridging," is considered to be by far the 
l>est, as it allows the thrust to act parallel to the axis of the strut, 
and not across the grain, as must be the case where single pieces 
of board are used. 
The bridging should be of li-inch by 3-inch stock. 

Carriage-beams, Headers, and Tail-beams.— Fig. 2 

represents the plan of the timbera of a floor, liaving a stairway 
opening on each side. The short beams, as KL, are called the 
** tail-beams : " the beams jEF and O//, which support the tail- 
beams, are called the ** headers : " and the beams AB and ('D, the 
"carriage-beams," or "trimmers." 

The tail-beams are calculated in the same way as ordinary floor- 
joist; but it is evident that the headers and trimmers will require 
separate computations. 




It would be very difficult to give formulas that would serve for 
'•.vei'y case of trimmers and headers ; and the best way in any case 
is to find the load which the trimmer has to carry, and then, from 
the formulas already given, determine the required dimensions. In 
a floor such as is represented in Fig. 2, it is evident that the floor- 
area supported by EF or Gil = y X ^n. Multiplying this area by 
(f'\-f), we should have the load which each header would be 
required to support ; and then, by Formula 9, Chap. XV., we could 
determine its necessary dimensions. 

As the headers are wcakene.l by the tail-beams being mortised 
into them, a certain allowance should be made for mortising in 
calculating the dimensions, in ordinary cases it would probably 
be enough to make the breadth from one to two inches more than 
the calculated dimensions. 



4:VJ 



WOODEN FLOORS 



The tritnmerft, A B and C'/A have to support one-lialf of the load 
rarrit'd l»y KF plus one-half the load carried by ^»7/, and also one' 
half ot tin* load su])p<)ried by the ordinary joist. The l)esl way in 
wliK-li to (-ali-iilalc siuh a triiiiiiirr i> to <'on>ider it to Im' made up 
iH two l»ain^ plact'd >ide by side, oiH' to earry the end of th«» he:ul 
ns KF \\\u\ (wll, and the second bein^ one-half the thi<-kness of th»» 
(H«linarv joinI The breadth of tlu» part carryinj: the ends of 
tlh' tiiiniiKi-; ruul.l then be calculated by Foruuda V-\, ("hap. XV., 
and the ti)ial breadth of the trimmers found by addiniz tot^*tht>r 
the bnadihs of the two })arts into which it is supixi^iiHi to Ik? 
divided. We have not the sjiace here to consider further the 
slun^ih of headtTS and irnnniei-s, but would lefer any readers 
dcsiriuL: further informatu>n on the subje<'t to IlatHehrs *• Trans- 
vei"se Strains,*' where they will tind the subji*<*l fully discussed. 






Fig. 3 



Stirriip-Iroiis.— At the iM)int of eonmn'tion of the end of 
\\ir li«-.i«i« I with the trimmer, tlu' load on th«' trinun<T (^onun? 
fioiii tli«- ixadrr is a conrcntrated one : and all mortising at this 
|iniiii. In nrtlv*' ihr header. sh(»uld In-axoided. It is now tlie etis- 
tniti. Ill til -!-<j;is»< r»)ii>iiMetinii, !<► support the (>nds <if Inniders l>y 
nit,i:i- •'! »• ;rriip-iroMs, mn nIiowii in Fiu. '•'*. Tin* ISoston ami New- 
\ Ml k I'.ii ! '.Jul: Laws !»'i|nire tlia' '"I'verx trinnui'r or lieiuler nion» 
tli.iii titiii t'«->'t lolc^^ u^rd in any builiiiiii; e\e(>pk a dwi^lHnf;, shall 
bf liiiiiL.' Ml ^tin iip-iroii'^ of suitable tbiekness for the sixe of tlie 
t JndM I-.'" 

It 1^ i-vidi'iit that t'aeh vertical part of the stirnip will liave to 



WOODEN FLOORS. 438 

carry one-fourth of the load on the header ; and we can easily 

deduce the rule, 

, . load borne by header 
Area of cross-section of stirrup = --- — Sfijoo * W 

The stirrup-irons are generally made of iron bars about two inches 
wide and three-eightlis or one-half inch thick. 

The headers are also generally bolted to the trimmer, as shown 
in the same figure; so that the trimmers shall not spread, and let 
the headers fall. 

Joist Hangers. — On page 437/ are shown two styles of 
patented joist hangers, which are intended to take the place of 
the stirrup iron, at less cost. . 

Oirders. — Formulas 2 and 3 will also apply to wooden girders 
supporting the floor-joist, neglecting the weight of the girder itself. 
In this case the distance between centres would, of course, mean 
the distance between the centres of the girders. The application 
of thijse formulas to girders being the same as for the floor-joist, it 
seeujB hardly necessary to illustrate by examples. 

• 

Solid or Mill Floors. 

By Solid or Mill Floors we mean a floor constructed of large 
beaniS spaced about eight feet on centres, and covered with plank 
of suitable thickness, and this, again, covered with maple or hard- 
plue flooring as desired. Such floors will be found fully described 
in Chap. XXIV. 

For calculating the large timbers, the best method is to compute 
the greatest load that the beam is ever liable to carry, and then 
determine the necessary size of timber by means of the proper 
formula, which may be found in Chap. XV. ; or if the beams are 
spaced a regular distance apart, and have only a uniformly dis- 
tributed load to carry, they may be computed by Formulas 2 and 3, 
given above. 

The floor-plank may be computed for their strength by the fol- 
lowing tonnula, supposing the load to be unifoi*mly distributed:^ 



V weight per square foot x X'^ x. 8 
— ' yT x~l ' ^ ^ ^ 

They would, however, bend too much, when proportioned by this 
fommla, for use in mills, and in buildings where the under side of 
the plank must be plastered. 

For such buildings the thickness of the plank should be propor- 
tioned by the formula for stiffn(>ss, which is, 



434 WOODEN FLOORS. 



Thickness of plank = ?/weight per squ are foot x U (gj 

y 19.2 X c 

e being the constant for deflectiou given in Chap. XVI. 

For s])riice, o — KM) pounds, and for hard pine 187 iwunds, for a 
defli'olion of on»'-tIiirti('th of an incli per foot of span. 

The \v»-ii;nt i)«'r sriuare foot should include the su|M»rfioial load on 
tlu* ll(>(u- and tin* wcii^ht of the ])lank and upper flooring. 

KxAMiM.K. — AVliai sliouhl be the thickness of the spnice plank 
in a mill where the ])eanis are spaced 8 feet on centres, and the 
superficial load may attain 12t) pounds ix»r square foot ? 

J//N. 'i'he weight of the plank and flooring, with deafening 
iM'tweeu. will weigh a])<)ut I.") i)()unds jM'r S(iuare foot, making tlie 
total load per scjuare foot 185 pounds. Then, from Formula 0, 



Thickness of plank = \/- ^{:,- u>^~ = ^ .>j • i i i 
^ \ 1U.2 X 100 or :W-inch plank. 

Tlie ])iaiik would j)rol)ably come in two or three lengths, which 
would iiiakc the lloor considerably St iffer; but, as there nught oiMnir 
eases when the Ih^or wouhl have to sustain heavy conoentrate<l 
loads foi- a short lime, it would Ik^ hardly wise to use a less tlii(*k- 
ness of plank. 

The following table, taken from Mr. C. J. H. Woodbury*8 excel- 
lent work on "Tlie Fire Protection of Mills, and Construction of 
Mill-Floors,*' shows the dimensions of Ix'ams, and thickness of plank 
for waichouse-floors loaded with from fifty to three hundred pounds 
]H*r s(|uarc foot, the ])eanis ])eing spaced eight feet on ci?ntres. The 
])lank is supposed to b«* of spruce, and the beams of hard or 8outli- 
eni ]>iii('. 

Scv«'! a! si/.cs ui h<'ams are given ; so that a selection of those which 
will appl> m(»st convenieuily to any specific case may be made. 



WOODEN FLOORS. 



435 



STRENGTH OP SOLID TIMBER AND PLANK FLOORS. 

(By C. J. H. Woodbury.) 



Weight per Square Foot op Floor. 



Super- 
ficial 
load. 



50 



75 



100 



125 



i50 



! 175 



200 



225 



250 



275 



300 



Weight 

of b^m, 

iu lbs. 



3.00 

4.08 
5.33 

3.00 
4.08 
5.33 

3.00 
4.08 
5.33 

3.00 
4.08 
5.33 

3.00 
4.08 
5.33 

3.00 
4.08 
5.33 

3.00 
4.08 
5.33 

3.00 
4.08 
5.33 

3.00 
4.08 
5.33 

3.00 
4.08 
5.33 

3.00 
4.08 
5.33 



Weight 
of floor- 
plank. 



6.07 I 



7.40 



8.55 



9.55 



10.45 



11.26 



12.05 



12.75 



13.45 



13.55 



14.72 



Total. 



59.07 
60.15 
61.40 

85.40 
86.48 
87.73 

111.55 
112.6:3 
113.88 

137.55 
138.63 
139.88 

163.45 
164.53 
165.78 

189.26 
190.34 
191.59 

215.05 
216.13 
217.38 

240.75 
241.83 
243.08 

266.45 
267.53 
268.78 

291.55 
292.63 
293.88 

317.72 
318.80 
320.05 



Dimensions oi 


Depth, 


1 
Breadth 


in 


in 


inches. 


inches. 


12 


6 


14 


7 


16 


8 


12 


6 


14 


7 


16 


8 


12 


6 


14 


7 


16 


8 


12 


6 


14 


7 


16 


8 


12 


6 


14 


7 


16 


8 


12 


6 


14 


7 


16 


8 


12 


6 


14 


7 


16 


8 


12 


6 


14 


7 


16 


8 


12 


6 


14 


7 


16 


8 


12 


6 


14 


7 


16 


8 


12 


6 


14 


7 


16 


8 



Span, 

in 
feet. 



20.95 
26.16 
31.63 

17.42 

21.82 
26.46 

15.25 
19.12 
23.23 

13.73 
17.23 
20.96 

12.59 
15.82 
19.25 

11.71 
14.70 
17.91 

10.98 
13.80 
16.81 

10.38 
13.06 
15.90 

9.86 
12.40 
15.08 

9.43 
11.86 
14.46 

9.03 
11.36 
13.85 



Thickness 
of floor- 
plan k, 
in inches. 



2.43 



2.96 



3.42 



3.82 



4.18 



4.51 



4.82 



5.11 



5.38 



5.62 



5.89 



Stiffness of Wooden Floors. 

Floors in first-class buildings should possess something more than 
mere strength to resist fracture : they should have sufficient stiff- 
ness to prevent the floor from bending, under any load, enough to 
cause the ceiling to crack, or to present a bad appearance to the 
eye. To obtain this desired quality in floors, it is necessary to cal- 
colate the requisite dimensions of the beams by the formulas for 
nttffniyft ; and, if. the dimensions obtained are larger than those 



£j.. 



i'M) WOODKN KIXK)US. 

obtained by the. formulas for strength, they should be adopted, 
instead of those obtained by the hitter fonnulas. The only way 
in which we can b(^ sun^ tliat a beam is botli stronj^ <'non«^h and 
stiff enou^li to bear a ^iveii load is to calculate the re<]ulred dimen- 
sions liy both tlie formula for streni^th and th;* fornuda for stiff n»'ss, 
and take the lari^iM* dimensions obtained. Asagenenil rub;, thos** 
lieanis in vhicii the proportion of <lrpfh to Innjth is very mmiH 
siioiild !)(' calculated l)y the form uhis for .s/>y'//f/^//, and ricf rfi'sn. 
Foiinula 10, (hap. XVI., giv(?s the load which a given Ix'iun will 
carry without dellecting more than one-fortieth or one-thiilieth of 
an inch per foot of sj)an, according to tlu^ vahu^ of e which we 
use. Kornuda II, Chap. XVI., gives thii dimensions of the beam 
to carry a given load under the same conditions. 

In the cas«' of tloor-lx'ams, the load is given, and is represented, 
as wc saw under tin* Sfrcntfth of Flttors, by the expression. 

Distance between centres in feet X length in feet X ( /'+/'). 

Tlicn, if we substituti^ this expression in place of the load in 
Kormula 11, (hap. XVI., we shall have the fornmla, 

T) X (list, between centres X cube of length X (f+f) 

Jreadth — u \y , \ ^-.i .4i s^ (<) 

8 X cuIm; oi depth X c ^** 

or 

S X breadth X cube of dei)th X e 
Dist. between centres = . ^ ^.^^j^^. ^^^. ,^^^^^^^,^ ^ {f^rT <*^^ 

The piojKT valu<'s for./' and/' have been given under the Stmnjth 
of ll<>oi> in the i)receding part of this cha])ter, and the value of f 
foi- any iriven casci may be found in Chai). XVI. ^ 

In oiiiinarv floors, when the values of /" used are thos4' n^-oin- 
im<iii|im1 al)o\e, a <ietlection of one-thirt i<>th of an inch {kt f(H>t of 
l>aM Mia> saftly be allowed, as the lloors would probably Ik» \ery 
raicJN loailed to their utmost ca])acity, and then but for a short 
tiiiH-: so ili.it it would have no injurious effects. 

As ail .".ample .showing the ai)plicjiti(»n of Fonnula 7, we will 
taivi- i!\aiiiplc I under the strength of wooden floors. 

Ill iiii> e\aMipIe, the l)eams were to have a sj)an of 15 fe«»t, and 
be , la.-.il I . t<et on centres: ./' was taken at -lo pounds, and./' at 'JUi 
I.oiiii.i-. What should be the dimensions of the In'anis. that thi'V 
iiia\ ^at- l\ cairy the Itiad upon them without deflecting more than 
;. I'l a:i 'ii<|i |)cr foot of span ? 

> i >.■ -.1' .<- t')!' '-. fur r<|irii('(.*, hard pltic, iiiui ifuk, uns 

l>.f. jlnA. !>*/.- 4*0^ 

hM 76 

11  :•> • l:i7 108 

< '•*' M TS 



s 



WUODEJN FLOOtttt. 437 

Ans, We have simply to substitute our known quantities in 
Formula 7, assuming the depth at 10 inches, and taking the value 
of e at 100 pounds, the beams being of spruce. 

Performing the operation, we have, 

^ . ^ 5 X li X 15» X (40 + 20) 

Breadth = s X W x lOO =1.0^ inches. 

This gives us about the same dimensions that we obtained when 
considering the beam in regard to its strength only : hence a Ix^am 
two by ten inches would fulfil both the conditions of strength and 
stiffness. 

In the case of headers, stringers, etc., where the joist has to carry 
not only a distributed load, but also one or more concentrateci loads 
applied at different points of the beam, the required dimensions 
can best be obtained by considering the beam to be made up of a 
number of pieces of the same dei)th, placed side by side, and {;om- 
t)uting the required bre^adth of beams of that depth to carry each of 
the loads- singly, and then taking the sum of the breadths for the 
breadth required. 

The formula for stiffness of plank-floors has already been given 
on p. 484. 

Dimensions of Joists and Girders for Different 

Loads and Spans. 

To enable an architect to tell at a glance the size of joists and 
girders required for the ordinary classes of buildings, the author 
has computed the following tables, which give the dimensions 
required for spans from 10 to 24 feet for joists, and also the maxi- 
mum distance that the joists should bo sjmced on centers. Dimen- 
sions for girders are given for different spans and spacings. 

The beams and girdera in the first three classes were comjmted 
from the tables on pages 888, 389, and 390, and in class I) from 
the tables on pages 377 and 379. 

The application of the tables will doubtless be (wident to all. 

When the girders are not s[)Mce(l uniformly, or then^ is only one 
row of girders, take the width of flcK)r area supported by the 
girder, for the distance apart. In several cases two sizes an^ given, 
both of which have sufflcient strength, although one contains less 
lumber than the other. In most oases the deeper b(»am has a little 
excess of strength, but for convenience the shallower beam might 
be preferred. 



437^ 



WOODKN FLOORS. 



TABLE L 
Dimensions of Floor Joists for Different Loads and Spans. 

[Note.— Tlu» iiiinibcr followin;^ the dash diMiotcs the distance aiMurt of joiBts in 

inches on centerf<.| 

A. FOU DWELLINGS. 

(Total Weight, 70 lbs. per Square Foot.) 



Timber. 



Yellow I 



10 



12 



( LE.\u Span in Fket. 
14 10 IS ' t»0 



•»•> 



24 



H Ui 



Whtto I 

I'liic. 1 ■.-• —  ^ ., . io_o4 
« 2  r> n 



1 2  12 IS 



2-M-:io : ,: r: j-ie-ie 2 12-18 ; 3, Jo 20 * ' ^* ■** 



>. « ifl 2 10 21 »2>10-12 „ ,, -. t 2 '12-11 8<1«-2J ,. , 



1  n irt "'. "\ "9 '' 9-ii>-it; 9.19— sn 9'H--iii .'"'i*~'J ••U— 14 

riii.'. ' ^ « 10 - "* (2 10 24 -* '^^ ' 3'1-S XO ^ IS 10 ,3,<,j_|g .!^U_-.o 



B FOR IIOTKLS, SCHOOL- HOOMS, LKJIIT OFFICES, ETC. 

(Total Wei^'lit, 1(H) lbs. per S(iuare Foot.) 



Ttmukk. 



CLKARSPAN is rKKT. 



10 



1-2 



14 



U\ 



IS 



20 



22 ! 24 



Whltf » „ J ,r o 11 iJ I 2 ' 10 Hi 9 12-13- ,„ ,- ,^,, ,. , 

1'1,„. > 2-H 10 2 -K' 1" ,2 12 2.. ."5 -12 19* '* '* SXll-H. | 

Spru'i-. •.' — \'J .„.,,. „o ...10 .>A 2  12 -Irt 8 -12- 17 -. , . ,i_,i» tAl4-14 



i'lii'' 



■* Jl 



'i  \i> 22 "J  12 24 
2 A 1 .-, 
2 • ID 20 



I 8 • 14—19 



t 2 • 10 12 



IS* It Ifl 



2-io-irt ;|.|;; ''2.12 154 3.11- !• .;-;;j_;j «.i4i» 



C. Foi: oKFKM-: I?riLl)IN(iS, A^SKMBLV ROOMS, AND LIGHT 

STORKS. 

(Total Wciu'ht, m) Wx. ptT Sciuare Foot.) 



'llMUKI. 



Wt.lti- , ( ■.> 

I'ln." I -J 

Siirm-i' •.' 

Yi'liow - 

I'liii-. > 



10 



12 



<'LKAK Sj'AN IN l""KICT 

II U\ IS 



20 



tt ! <« 



2 10 H -• 12 1"' S  12 !.-> 3  14 1« j 

2 • I- 17 2 12 Hi ]l'.]l ]l •''■12 IS 3-14- !»' j 



.., • 2 - 12 ., 
I lo 'J: ' 



:\ 



12 12 3  It IS 



iM li 2 12 17 :;.\i_\; ,.i; ;; i a>i4- u it^u-ii 



WOODEN FLOORS. 



437* 



D.-FOR STORES AND FACTORIES.* 
(Total Weight, 180 lbs. prr Square- Foot.) 



Clear Span in Fekt. 



10 



2X10-16 

\tx 8-ia 

"(SXlO-17 
SX 8-17 



12 



r2xio-ii 

(2XH-1 
2x10-12 
2x12-18 

2X10-19 



14 



txit-ii 

3X12-17 

•^X12-18 

3X12-20 

*2X10-18 

( -^X 12-19 



16 



18 



8X12- 
8X14- 



18 
-18 



8X12-1« 



1X12-1) 



8XM-I4i 

(8x12-12 
« 8x14-16 
( 2<12-12 
■( 8x12-18 



20 



8X14-11 



22 



8x14-181 8X14-11 
8X14-16 



8x12-14 
8X14-19 



24 



SX14-18I- 



* Calculated for strength only. 

TABLE II. A. 
Dimensions of Wooden Girders for Dwellings. 

(Total Weight, 70 lbs. per Square Foot.) 
SPRUCE. 



IN 



Distance apart on Centers in Feet. 



10 



i 6x10 

■« 8x 8 

6x10 
6x10 

8x10 

J 6x12 
I 10 X 10 

8x12 



12 



6x10 

8x 8 

6x10 

8x10 

\ 6x12 
I 10x10 

8x12 
9x12 



14 


16 


6x10 


8x10 


8x10 


8x10 


) 6x12 
/ 9x10 


8x12 
10x10 


8x12 


8x12 


10x12 


10x12 


10x12 


1 10x14 
(12x12 



18 



8x10 

\ 8x12 
( 10x10 

8x12 

10 X 12 

\ IOxhI 
"111x121 



20 



9x10 

8x12 
10x10 

10x12 

10x12 

10x14 
12x12 



10x14 10x14 



22 



I 8x12 
1 10x10 

8x12 

10x12 

\ 10x14 

) 12 X 12 

10x14 
12x14 



24 



8x12 
10x10 

9x12 

10x12 

10x12 
12x14 

11x14 
12x14 



YELLOW PINE. 



IN 

r. 




Distance apart on Centers in Feet. 




10 

6x 8 


12 

6x 8 


14 


16 


18 

6x10 
8x 8 


t»0 


22 


24 




\ 6x10 
/ 8x 8 


6x10 
8x 8 


6x10 


8x10 


8x10 




6x 8 


\ 6^10 

< 8x 8 


6x10 


6x10 


8x10 


8x10 


8x10 


9x10 




6x10 


6x10 


6x10 


8x10 


8x10 


S 6x12 
1 10 X 10 


8x12 
10x10 


8x12 




6x10 


8x10 


8x10 


1 6x 12 

'i 10 X 10 


8x12 
10x10 


8x12 


8x12 


10x12 




8x10 


S 6x12 
■/ 10 X 10 


6x12 
10x10 


8x12 


8x12 


10x12 


10x12 


10x12 




i 6x12 
1 10 X 10 


6 X 12 


8x12 


8x12 


10x12 


11x12 


i 10x14 
) 11 x 12 


10x14 




8x12 


8x12 


10x12 


10x12 


< 10x14 
) 12 X 12 


10x14 


10x14 


10x14 



437c 



WOODEN FLOORS. 



B. 

DiMExsioxs OP Wooden Girders for Hotels, School-rooms, 

Light Offices, etc. 

(Total Weight. 100 lbs. per Square Foot.) 











SPKICE. 










bl'AN IN 






Dis 
12 


TANCE APAKT ON' 

14 , 1« 1 

, 1 


Centei 
IS 


:s IN Fk 
20 


ET. 




Fkkt. 


10 


 


2« 


24 


10 

11 


(3. 

^ 8. 
( i\  
> 8v 
/ 10 . 


10. 

1 


12 

 t 

id 


8-10 

8 . 10 
♦i  !•.> 

8- 12 


\ 8 ^ 10 

8 ^ 12 
10- 10 

10 X 12 


8>12 
10. 10, 

8xli> 
h)>. 12. 


8x12 

10 X 12 

\ 10 . 14 
"( 12 X 12 


10x12 

10x12 

10x14 
12x12 


10 X 12 

 10x14 
'1 12 X 12 

10x14 


10x12 

10^14 
12 -IS 

12' 14 


v: 


8  


•-» 


10  12 


10  12 


\ 10. 14 
/ 1-  12 


10x11 


12x14 


12x14 


14 '14 


v.\ 


10 X 


■i 


10 . 12 


\ 10- 14' 
, 12  12 


10 > 14 


12x14 


12x14 


» 12x16 
»14x 14 


12 • IC 


14 


10-- 


2 


< 1" < 11 
. 12- 12 


10 - 14I 


12 • 14 


14 A 14 


12x16 


12x16 


14*^16 


i:> 


> 10 . 

. 1 '? • 


1 
2 


10-14 


12 .-14 

_ 1 


( 12. It; 
( 14 •• 14 

1 


12x11) 


14x16 


14x16 


16^16 



YKLLOW PINE. 



Sl'\N IS 

Ki;i T. 



10 



Distance ai-aiit on Tkntekh in Feet. 



12 



14 



16 



' - l 



itt 



> () 10 \ )')  10 

/ '^ - s , S  *< 



»;  10 8-10 H. 10 



\ 8 ' 12 



ti 



H- 12 



('»  10 

iS . Id 



10 



10 



(1 • 1(1 

r>  111 

»« .«• 

. \-i 

10 - M 

^  12 






8 . 10 

» r, . 12 
. 10. 10 



8  10 

8  12 
10  10 



N 12 

in ^ I-.' 



I 10 • 12 
10 . 10 10 . 10" H - 12 

8- IJ 10- 12 li». 12 

I 



I'l 1 J 
1-.' 1-J 



 V*  \i 
10 11 



8 12 10  12 10 . 12 ; }i; ' \\ 

10- 12 ' II!" \i 10. 14 10-11 

11 -12 *ll]" I:* 10- 11 10x14 12*14 

( 12 - IJ I 

''•■I* 111 11 i.>^i« to 11 »12*16 

12x12 ^^''^* ^-''^* '-*^^ -,14.14 

10- 11 :2- 11 l:j-14 )}J;{5 W"!" 



24 



s. 


12 


10- 


1".» 


10. 


12 


10  


:t 


12- 


12 



12-11 

12*11 

12. 1« 
14-14 

i2»ie 



WOODEN FLOORS. 



43Vc? 



C. 

Dimensions OF Wooden Gibdebs fob Office Buildings, Assem- 
bly Rooms, and Light Stobes. 

(Total Weight, 130 lbs. per Square Foot.) 
SPRUCE. 



Span in 


• 

Distance apart on Centers in Feet. 


Feet. 


10 


12 


14 


16 


18 


20 


22 


24 


9 
10 
11 
12 
18 
14 


8x10 

j 8x12 
110x10 

8x12 

10x12 

(10x14 
1 12 X 12 

10x14 


(8x12 
1 10x10 

8x12 

10x12 

(10x14 
n2xl2 

10x14 
12x14 


8x12 

10x12 

j 10 X 14 
1 12x12 

10x14 

12x14 

( 12 X 16 
"1 14 X 14 


10x12 

(10x14 
) 12x12 

10x14 

12x14 

(12x16 
1 14x14 

12x16 


10x12 
10x14 
10x14 
12x14 
12x16 
14x16 


(10x14 
112x12 

10x14 

12x14 

( 12 X 10 

1 14 X 14 

18x16 
15x16 


10x14 
12x12 

12x14 
18x14 
12x16 
14x16 


10x14 
12x14 
14x14 
14x16 



YELLOW PINE. 



Span in 




Distance apart on Centers in Feet. 


Pket. 


10 


12 


14 


10 


18 

8x12 
10x10 

8x12 

10x12 

(10x14 
) 12 X 12 

10x14 

12x14 

(12x16 
1 14 X 14 


20 


22 

8x12 

10x12 

10x14 
12x12 

12x14 
14x14 
12x16 
14x16 


24 


9 
10 
11 
12 
13 
14 
15 


6x10 
8x10 
8x10 
8x12 
8x12 

10x12 

(10x14 
1 12 X 12 


8x10 

8x10 

(8x12 
/ 10 X 10 

0x12 

10x12 

(10x14 
1 12 X 12 

10x14 


8x10 

( 8x12 
/ 10 X 10 

8x12 

10x12 

) 10x14 
/ 12 X 12 

10x14 
12x14 


(6x12 
) 10 X 10 

8x12 
10x12 
11x12 
10x14 
12x14 
12x14 


8x12 
10x12 

( 10x11 

'i 12 V 12 
10x14 

12x14 

14x14 

12x16 


10x12 

( 10x14 
■/ 12x12 

10x14 
12x14 
14x14 
12x16 

14x16 



4376 



WOODEN FLOORS. 



D. 



Dimensions of Wooden Girders for Stores and Factories. 



^Total Weijrht, 180 lbs. per Square Foot.) 



SPRUCE. 



Stan iv 
Fkkt. 



9 
10 
11 

13 



10 



DljiTANC K APART ON CENTERS IN FbET. 



12 



14 



16 



IS 



SO 



8vl> UK le 10x12 *]S^]:1 10x14 12x14 



10 



,,y > 10x14 jn^i. 10^,4 10^ ti » 10x16 
'"' -I'Jxl-J ^^^1-* 1-X14 1^X14 jj^^jj 



» 10 « 14 
1-2 » 1-J 



10 ^ 14 12x14 



» 10 X 16 
14x14 



12 X 16 13 X 16 



10 ^ 14 V2 > 14 14 X 14 12 x 16 14 x 16: 
l"2v'. I 14x14 12x16 14x16 ' 



22 



24 



18x14 



» 10 X 16 



M4xl4 
12 X 16 14 X 16 

14x16 



YKM.OW riNE. 






10 



l>i-r\\iK AivKT i»\ Crx-^KUs IN Fket. 

I "" 



14 



16 



IS 



20 



22 24 



^ • 1  

*- • i '.' 



10 



10 > 1 



...... • '-'^^ 1» 



1- 1 J 



 * 



'.I 



: 1 



S X IC 



114 



< • 12 10 X 12 



1-' 



1 1 



10  14 



Iv! V II 12 ' : 1 

:s - ] I 






li 14 

12 ".i". :i- it» 



1(1x12 10x14 12x14 

10x14 11x14 12>14 

12-14 13x14 11-14 

I4xll' 12.16 ISxlA 

I 

12. li. 13x16 15x16 

1 1 • 16 13 « 16 



WOODEN FLOORS. 437/ 



JOIST HANOBRa 



Fio. 4.— Ddflei Joibt Hanqeii. Fia, B.— Ooetz Joibt Haxbeb. 

Pig:s. 4 and S show two styles ot joist hangers that have been 
put on the mwkot within a few years. Both these aneliors are 
warranted to be stronger than the timber they support. 'I'hey 
are made in numeroiis sizes, and are inserted in holes bored la 
the sides of the girder, or trimmer. 
While these hangers themselves, 
however, have ample strength, they 
mu-^t weaken t« some extent the tim- 
ber into which the holes are bored, 
which is not the caso with the stirrup 

Fig. G shows a similar hanger made 
to support the wall end of floor joist. 
The writer believes this to be much 

superior to the method of building the p,^ 6.-Dtt«.e» BaitrK Wall 
joist into the wall, as it absolutely Hanoer, 

prevents dry rot. and permits the joist 

to fall in case of fire, without throwing the wall. It also gives the 
weight a good bearing on the wall. 



FiHK-PllOOF FLOUKS. 



CHAPTER XXUI. 

FIRE-FROOF FLOORS. 

TnE tPrin " fire-proot floor '' is hert unrlenstcXK] to mean a flool 
rfin^triii'liil of (irt-proof mnterial, RupiMirtcil on or betwe n iron iit 
9ti'i-l ix-aiiiH or gmlcrs, or fire-proof wiiUs. anil entirely ]irot«cliij); 
tin- ironwork from tlie action of fire. The various materiais si 
|iri-." ril iiMil ill the <'on struct ion of absolately firu^proof floors lire 
bri'k. iinlliiu' [Mjruiis liJu, liullou' dense tlie, ibin pl&tea uf dense tile 



I ;ir..ilLii'i~ iif I'l.'iy: iiiiil (-iiiiiTi'te of Piirtlnnrl in^ment nnd i-itlii-r 
i'i'~. I'i'iikrii iil>'. ~i<iiii'. Ill' tirii'k; iiiui iiiac) eoin[H)iiilioiiii niiiile 
lila-iiT ol' I'^iri^ .-IS II (i-iiii-iitiiif MiiiliTint. The flr^l tlim' 

-ri;il~ III'" p'tirnilly ii-..-<l ill ill.' r<iv I iiR-lies net iHrtwiTn tiic 

II'.. 'I li.' iliiii |.liii'." -f i1>'iiM- till' iiri' iisi'd f.ir (oniiin^' vuiiltK 
..i; uirl.T- »'..ii.h.|,'isuv.-.l.'iili,i'inllii'f"riiiof iin«rcli...t 
I.I -:.\-. n'riniiiL' <1<"<r ami .'I'iliii^'. »'illi liolliiw iiileriur : ill Die 
-. ir'ii liiir', i-x[iiini|i'<l nii'iiil. or wiiv lirs iin- iiiilii'Udiil. Inin 

\.: I. 1 111^ iin- <:>'n< rilly liiiil in tliK>rs nr uliiiwh in Fifr. I, Iho 

- ittl.,'1' jvsliiis nil 1..]. of llie ninlern. Kb in Kig. 3, or lioltei] to 
~iili'^ <if ilii'i^rders. 



FIRE-PROOF FLOORS. 



,430 



Fig. 3 shows the detail of connection when the under sides arc 
made flush ; Fig. 4, the joint to bring tlie upper sides flush; and 
Fig. 5 shows the form usually adopted when the beams are of the 
same size, or the centre lines are brought together. Arrangements 
of this kind are also used to connect the trimmer-beams of hatch' 
ways, jambs, and stairways.^ 




P 




][ 



Fi... \///m 





Fig. 6. 



The wall ends of the joists and girders should be provided with 
nhoes or beariny plates of iron or stone, as the brickwork is ant to 
crush under the ends of beams, unless the load is di^tribjuted by this 
means over a sufficient surface. Anchor-Htrapa should be bolted 
to the end of each r/irder and to the wall end of every alternate 
joist, binding the walls firmly from falling outwards in the event of 
fire or other accident. 

Several simple modes of anchorage are shown in Figs. 3, 4, 
and 5. 

When one beam docs not give sufficient strength for a girder, it 
is customary to bolt tosjcther two or more with cast separators 
between them, as shown in Fig. 6. 



*■ The details of the coiiuectioDs aud framing of iron beams kre more clearly 
shown on pp. 366, 366. 



KlllE-rUOOF KLOOKK. 



Itric-k Arc'li«s, 

I vviiy of iiiiikiiiu i> lln'-iU'oKf floor of hrir-k \s to fill 
.■-■11 111.' ji.isls«illi Lrii^k un-ln's. n■^Ii1l;,• <>ii llli- low.T 

;.'iT;i-.'.>n:i <ir biit-k sk'-wliitrks. M'hi-ii tliis mctlio<l 
'^hoiiM lu' tiiki'ii Ilial llii- l<rli-ks of niiiHi tli.' .irvli.-s 
aiv of ^1111.1 slia|i.'. :iii.l v.'iy liiir.1. Tli.'V sIiomM I>' 

wiilL .■iLili oilirr, iiiiliuiLt liiir: i.ii.l iii: IW ji.iiit* 

trviU. mill Ih> kt-yiil n'itli 



lill.M ttUll 111.' IH-St 



i.T f..iir iii.h.^ llii.'k fi.r f]-^n> l«'tw.--H 

. llii.-k l'..i' -iMiis l..-i».'.-ii »:\ :iii.l .'k'lit 
III tl,.' -k.« l.:i.k- .,iri.- -..U.l.ima > .imn- 
I:, li-.-.'l tin :llvl,-h..|IMI.|-:l^..ll..■l.- 







..; ^.i...tl. ai..| 



FIRB-PROOF FLOORS. 441 

angle- bar or channel serving as a wall plate for distributing the 
strain produced by the thrust of the first arch (Fig. 7). 

The weight of n brick arch with cement filling is about seventy 
pounds per superficial foot of floor. Experience has shown that 
such a floor cannot be considered as fire-proof unless the lower 
flanges of the beam are protected by porous terra-cotta, fire-clay 
tile, or wire lathing, kept an inch away from the beam. 

Brick floor arches are largely going out of use, owing to the fact 
that a fire-proof floor may be more cheaply constructed of other 
material. 

Hollow Porous Terra-cotta and Hollow Dense 
Terra-cotta Floors.— For convenience, these materials will 
be referred to as Porous Tiling and Dense Tiling. A description 
of the materials, their nature and manufacture, will be found in 
Chapter XXV. They consist principally of clay, which is manu- 
factured into hollow blocks, generally with angles on side or ends, 
according to whether the arches of the floors are to be of end- 
method design or side method design. In some instances, to a 
limited extent, rectangular blocks have been successfully used, 
but this shape is not approved. The general practice in flat con- 
struction is to make bevel joints — radius joints are seldom used ; 
the best workmanship) and best results are found to be obtained 
with a bevel joint of about one inch to the foot. There are two 
general schemes of flat construction : one in which the tile blocks 
abut end to end continuously between the beams, and one in which 
they lie side by side, with broken joints, between the beams. In the 
end systems, it is not the practice to have the blocks in one row 
break joints with those in another, as it entails extra expense in 
setting. When this is done, however, the substantialness of the 
floors is increased. 

In some forms of flat construction a side-method skewback (or 
abutment) is used, with end-to-end interiors and keys, or end-tOr 
end interiors and side-method keys. Experience has shown that in 
the side method of flat construction the skewback, or abutment, 
was the weakest — in case of failure, sometimes collapsing, but gen- 
erally shearing off at the beam flange ; consequently, the side- 
method skewback is not approved in the end- method construction 
unless provided with partitions runninc^ at ris^ht anoflos to the 
beam. Keys should be end to end, or solid. The latter, when 
made very small, are preferable. 

A practice which has become somewhat general, especially In 
the East, is for the owner or general cqntractQr tp buy tjles, and the 
tnasofi ^{itraoter on the job to build them in plaee in the building. 



FIBE-FROOF FLOORS. 443 

beams, and like centrepieces above, crosdng the beams. The 
ptanka on whieh tiles arc laid shfiuld be two-inch, dressed on one 
ode to uniform thickness, and should lie on lower centres, at right 
Angles to beams anil placed close together. J'he soffit tlto should 
be a separate key-shaped pieue. oC ei[iial width of beam, and laid 
directly under tbe beam on the planking, aftor whicb the eontring 
is tightened by screwing down tlii) nuts on the T-boits, until the 
sofflt tile are hard against the beams and the planking has a crowa 
' not esc«&diag one-fourth oC an inch in spans of sis feet. This sys- 
tem gives what is very essential— a lirin and steady centre on which 
to construct the flat tile worlt. The tiles should be '■ shoved" in 
jilace with close joinb'. and keys should fit close. The centres 
should remain £n)m twelve to tliirty-six iiours, according to condi- 
tions of ireather, depth of tiling, and moj'tar used. When centres 



are "struck," the ceiling should be straight, even, free from open 
joints, creTices. and cracks, ready to receive the plastering. 

Figs. til 12 show types of flat constructions in use. Different 
tfianufacturers have various modifications of these. Pig. 9 is the 
most general design for dense tiling, although porous tiling, very 
similar in design, may be had from some manufacturers. The end- 
method design is preforahlo, however, for porous tiling. Fig. 10 
is a light-weight dense-tile design, nol so gvinerally useil as fonncrjy. 
Figs. It and lli show the simplest end-methnd design for porous 
tiling, which has become known as iho ■'Leo end-method areh." It 
was first brought into general use by Mr. Thotnas A. I.ee, now of 
New York City. It was used by him in the tests conducted at 
Denver in Dceember, t"S(}, by Messrs. Andrews, Jaques & Ran- 
toul, architects In those tests the design ^ihowod superiority over 
the Dtberdesigns. It has the advantage of simplicity and economy, 
both Id mannfaoturo and construction. Tbe manufacttirer can 



FIRE PROOF FLOORS. 445 

reduced and the stability of construction mcreased.- The reduo- 
tkm in price of all tiling makes the cost rather in favor of increas- 
ing the thickness of tiling and reducing the thickness of concrete. 

Among the advantages possessed by hollow tiles in their ap[)lica- 
bion to fire-proof floors, between steel or iron beams, are these : 

They are absolutely incombustible, because made of clay and 
laving withstood a white heat in the course of manufacture. 

They are sound-proof, from fact of being hollow. 

They are superior to any concrete material used for the same pur- 
jose, owing to their being free from shrinkage, thereby avoiding 
ihe unsightly cracks often seen in ceilings laid with concrete blocks. 

They are proof against rats and vermin. 

Floors made of them are forty per cent, lighter than by the old 
system of segmental solid brick arches levelled with concrete. 

They offer a flat surface on the bottom and top after being laid, 




Fig. 16.—** Austria " Arch, Patented by Pr. von Emperoer. 

ihereby giving a flat ceiling ready for plastering, and a flat founda- 
:ion for the floor strips. 

The flat arches should in all cases be capable of sustaining, with- 
)ut injurious deflection, after being set in place, an equally distrib- 
ited load of 500 pounds upon each superficial foot of surface. 

In laying the tile, a mortar composed of lime mixed up with 
joarse screened sand, in proportions of four to one, and richly tom- 
3ered with hydraulic cement, should be used. This makes a strong 
nortar, and works well with the tile. In no case should a joint 
jxceeding one-half inch in thickness be permitted 

The laying of flat construction in winter weather without roof 
protection should not bo practised in climates where frequent 
tevere rain and snow storms are followed by hard freezing- jind 
;h!iwii)g, as tho mortar joints arc liable to be weakened or ruptured, 
'esulting in more or less deflection of the arches. 

The upper su rface of these arches is generally covered withcon- 
jrete of a sufficient depth to allow for bedding in it the wooden 
(trips to which the floor board-; are nailed. The concrete can be 
nade of light and cheap materials, such as lime or native cement 
knd clean rolling-mill cinders, coke screenings, broken flre-proo€ 



14(5 



FIRE-F»RO()F FLOORS. 



tiling, etc. The floor strips should be of sound and seasoned wood, 
2 inches thi(»k by 2 inches wide on top. bevelled on each hide, 
to 4 inches wide on Iwttoni, paced about 1(> inches on ('i-n- 
tros 'rh(^ coiicrct(» should ix; firmly bodded beneath and ugiiin^t 
oMch <[(\l'. Instead ol' coiicr.'tc filling. tKt?., a filling is soinetiines 
made l)y layiii*; lidllow p.iitition bloc^lis on top of the arches. 
Tlicsc loiin excellent toundations tor marble or other linished tile 
liuoiin^^. 

Tlic j)i}icticc ol* puttini; in comparatively thin flat arcfh eonstruc- 
tiuij U) form ceiling's, then heavy wood strips from lx»am to btjani to 
carry the v. ciiriit ol' the floor, leaving a hollow s^iace between top of 
arclns and under side of wood flooring, ij« not approved. The 
amount of wood contained in such a floor is sufficient to produce u 
Very (lamairing lieal. The hollow space enables the wood to burn 
readily, and niakes a Are very difficult to fight. Such coDstruction, 




Fu.. 17. 



thereiore. i< danufcrous. and sIkjuM not be considere<l as first-class 
fire 1 1 root' \-. ork. 

'1 li«- VMi'iition in width of spans between beams is pn>vidi*ii for 
by ^u|i|'l\ iil: tiles of dilTerent sizes, both for interiors and keys, 
wihii'hy ): \arie»y of eond)i!iations can be sj'cured. 

When i!i-^ii'el to aitaeh iron oi" wood work to the soillts of the 
hoiioA 111. iloo- archer. sli>t holes are puiich-Ml in the tiles, and T- 
h. a-; (i i ol". ;ii-f inserted and secured a> >h«»wn in Fig. 17. 

'■ In-n ;..fiiie. terracotta tile are used, cleats nuiv Im' naile<l or 

S !• U. .| i'.l.ctiv to th.' tile. 

I:: ;■: ! :'ij' ii« n work, too «;r«'at can- cann<»t l)i' exercis^'tl that all 
1 ai'i- ii.- |>I.i. -d paiallil. e»<pe«-ially '.her- (».ie or both emb «if 
1' .:i:.- !■.-• •■•1 i-ri-kui-rk. rn'am-^ plaeed out of parallel make il 
\.n I \|.. !.-;\«' III M-i tile lire |«r-'i'li!i-..r. «»f'en nnpiirin:; cutting nf 
II!'-. aIi:< h -^ tiaiiiauin.;: ami injurious, and shoidd not U* tloue. 

\\ '■ I. -|..iN. - LTineni.!! hollow liie arches isee Figs. IS and IViarv 
>o!ii- t :...- - n-»-'i in wai'ehouscs, factories, ami fur mofs, in thick 
ne.^M.-^ of i.\ and eight inches. I'sually the tiles an* 0x6 inches, 



FIRE-PROOF FLOORSw 



446a 



or 6 X 8 inches, and 12 to 16 or 18 inches long. Spans may be any 
width up to 20 feet, rise about one inch to foot of whole span, 
in some instances the joints are pointed after the centres have been 
removed, and the whole under side painted.- This form of hollow- 
tile work in wide spans from girder to girder is cheaper and lighter 
than flat construction with floor beams. 




4" to U c)t^'<n,t«CU)b\ ^vck X!Uu^ 0^^>^v ^X» 5l>\lH. . 5vmi'«iUo'Iq^o' ac&oram% Xo »a.c o^ 

Fig. 18. 

Weights and Safe Spans for Dense-tile Arches.— 

The following table gives the weight and span of flat hollow dense- 
tile arches made bv the Raritan Hollow and Porous Brick Com- 
pany. This is about an average for spans given by different manu- 
facturers. The Pioneer Fire proof Construction Company, and 
some others, make a lighter grado of tile than this, but their heavy 
tiles correspond very closely with the table below. Dense tiles may 
also be had from Lorillard Brick Works Company and Henry 
Maurer's Son. New York ; the Empire Fire-proofing Company, 
Pittsburg ; Parker & Russell Company, St. Louis ; and others. 



WEIGHTS AND SPANS OF FLAT HOLLOW DENSE- 
TILE ARCHES. 



Depth of Arch. 


Span, between Beams. 
3 ft. 6 in. to 4 ft. 


Weight per sq. ft. 


6 in. 


29 lbs. 


Tin. 


4 ft. to 4 ft. Gin. 


3? lbs. 


•8 in. 


4 ft. 6 ill. to 5 fr. 6 in. 


35 lbs. 


9 in. 


5 ft. to 5 ft. in. 


87 lbs. 


10 in. 


5 ft. 1) in. to ft. () in. 


41 lbs. 


12 in. 


6 ft. 6 in. to 7 fi. in. 


48 lbs. 



The following table gives the weight and span of flat hollow 
porous-tile arches of the Lee end -method design, which may be 



FIRE-PROOF FLOORS. 446c 

olted together with f-inch tie rods, secured to the web of the 
cams near the bottom flanges, and drawn tightly to place by nut 
Ad thread. These tie rods should be set from five to seven feet 
bpart. 

The cost of hollow-tile arches of either kind, set in place ready 
br plastering, in lots of 20,(MM) square feet , ranges from 14 cents to 
•6 cents per square foot, according to size and weights of the tile, 
n Chicago the average price may be taken at 20 cents. 

Specifications for Transverse System of Elnd- 

Pressure Floor Arch. 

The following form of specification may be of assistance to 
rchitects in preparing their specifications for tile floors : 

Contractors submitting proposals for fire-proof floor arches shall, 
hen required, prepare detail drawings showing the sjrstem and 
^plication of floor arch proposed to be used. The general require- 
lents of such design shall be as follows : 

1st. Arches to be level top and bottom, filling space between the 
Bams from a point not less than seven eighths of an inch below 
le soffit of beam up to within one inch of the top of the beam. 

2d. The abutment tile adjoining or resting upon the floor beams 
lall have its hollows run parallel with the beams, but the vous- 
)irs shall be laid transversely, with hollows running at right 
Dgles to the floor beams, so that the tile blocks forming the arch 
lay receive the pressure resulting from imposed load on their end 
.Kstion and distribute it lengthwise of their respective web members. 

3d. Soffits of all beams shall be covered with tile slabs keyed 
5curely in place, flushing with under surface of arch. 

Tests. 

Each arch shall be subjected to a test of a moving load consisting 
f a roller weighing 1 ,000 pounds to each lineal foot, and applied 
3rty-eight hours after the centres have been struck and before the 
oncrete has been filled in. This roller to be rolled over the top of 
be tile wherever the supervising architect or his superintendent 
hall direct. 

In addition to such rolling test, the arches, after being set in 
lace seventy-two hours, shall be subjected to a dropping test made 
1 the following manner : Before the concrete is applied on the 
rches, a bed of sand two inches thick shall be spread loosely over 
le top of the arches, Rud a wooden block or timber, weighing 200 
mnds, shall be dropped thereon from a height of ten feet. If the 



4iG</ FIKE-PUOOF FLOOUS. 

arclies withstand this impact for three c-ontiniious blows without 
breiikin<>: through, the test shall bo considered satisfactory, and the 
floor arches bo accei)ted. Should the floor arches break throu^rh 
under the blows, it >\v.i\\ be deemed (conclusive that the metliod of 
floor arch employed is faulty, and the contractor will Imj r(H|uired 
to remove same from the building and provide arches suitable to 
withsi;nnl the tests recjulred. 

Strt'ii^tii of Flat Hollow l>oiiso and Porous 
T<MTa-('ottJl AiM'lios. — Either of these materials, when prop- 
erly made an<l erected, should have a strcnjy^h of at least 5(:() lliS. 
pcrsijUMie foot. One of the most complete an<l practical tests oi" 
llo(»r arches I'ecorded was made in Denver, < ol., iindtT the direction 
of Messrs. Andrews, Ja(iues & Kantoul. architects, for the Dfiiver 
K.iuit.'blr P>iiilding (N)mpauy, Decendn'r r20-2o, 1890, oi" which a 
tuil reporl was ])ublislied in the A /."trioii' Architect and littiUUug 
\rirs, M.'injh "Js. IbiOl. Kight an-hes built of hollow bum«*d lin*- 
elay til«', and four of ])orous terra-iottu, were subjei^ted to four kinds 
of te-1s. under as nearly the same comlit ions as p(»ssible. Thraifrhes 
wri» earrie*! on 10-inch steel Mn^'ims, set- 5 feel apail on centres, and 
were built of 10- inch tile. The tcrra-eotta tile were manufactured 
by Mr. Thomas A. Lee, and were of the en< I -const met ion type 
shown in Kiirs. 11 and 11//. and it is dtaibtless owing tu thb fact 
that tliesj- arches (h'veloped the strength shown by the testti. 

The U'>ls were as follows : 

l>i. I>y still loatl. increased until the arches broke (h)wn. 

V.M. \'>\ -li<Mk>, repeatiMl until the arches nc re destroyed. 

:M T«-i- by lire and water, aliernaiiug until the an-hos were 

till I>\ •onlinnons tire of high heal, until the arches were 

Ill !<■ iin fii-!s| t:st.'iiii' ••; the llre-elay tile nivhe- bri»k«- at 
.">. U ; Il»s 1)1- "in 11'^. pn- s luari" foot, and the other at H.riTI lb'... 
nj- ]"J^ |i.>. |)iT s.|uar< f'ii>: ; brnh i»! theM* ar«'!ies liad but i-ne 
h<>ri/i»nial wi b, w hieh wa.- at t in- cent n- ul' tin- tilr. I'.oth of »ln>o 
ri'i'-c. ■■■iVi- • !iv sndd'iily. tin- wlmh- .-ir'-li iatlini;down. tbi' failun' 
i:i b !l i-.i-.'- lakinu: plaee m 1 1 r '-LiW baeks. t he remaill'^iT nf I hp 
;i'. : !■ '■'.) . i.iiii.im.il '"li- p»i|i-M> le:-,;: i "fi ali'll. wl'i-h  ad 1»0 

ill 1 1. ■!■■ d wfli^ .-n.-t; iiMil a Iliad ol I."), ll.'i \\'< . i.il b>. per «•■ unre 
:...'. ;. : ! . . Imnr-* willn'Ut breikimr. ^*lll■n the l«ijid war di-imi- 

I i!i:i- •! 

11. ■>("ImI .sriii-^ of tc^i.s wa^ madi- iiy dinppiiig a piece of tim- 
b.i \': MM iii> -^iiuare auii -i iett Inn" weighing 134 ]l)s . tnmi a 
hei>:lit •<] -.ix liei. u|Niii the inahile of the an-li. Ikjlh of thi* hoU 



FIRK-PROOF FLOORS. 447 

low flre-olay tile arches broke at the first blow of the ram, the 
arches dropping from between the beams, the tile breaking *^ like 
a sheet of glass, indicatiu.u: extreme brittleness in the material/' 

The porous terra-cott:i arch withstood four blows from a height 
of six feet, and seven blows from a height of eight feet, the areii 
dropping at the last blow. Pieces of one or more of the tile, how- 
ever, dropped out at nearly every blow. Under the fire and water 
test, one of the fire-clay arches was destroyed by three ap[>lications 
of the water ; the other withstood fourteen applications of the 
water, alternating with extreme heat. 

The porous terra-cotta. arch withstood eleven applications of 
water, alternating with extreme heat, uninjured. The temperature 
of the tile at the time the water was applied varied from 1,300^ to 
1,600° F. Under the continuous fire test, both fire-clay arches 
were destroyed after being subjected to a most intense heat for 
twenty- four hours. The porous tcrra-cotta arch, after having a 
continuous fire under it for twenty-four hours, was practically un- 
injured, as it afterward supported a weight of briyks of 12,5o0 lbs. 
on a space 8 feet wide, in the middle of the arch. 

These tests were conducted with perfect fairness, and unquestion- 
ably show the superiority of the [)orous terra-colta arches. The 
porous terra-cotta tile, new and dry. weighed 34 lbs. to tlio sfiuare 
foot ; the fire-clay tile which stood the tests the best weighed 40^ 
lbs. per square foot, and the other 32 lbs. per square foot. 

Other Tests. — During the construction of the Board of Trade 
building, in Chicago, in 1884, a 6-inch tile arch of 3 feet 8 inches 
span, made by the Wight Fire-proofing Company, of Chicago, 
was loaded up to 7o6 lbs. per square foot without injuring the 
arch. The arch was also severely tested by dropping heavy dry- 
gooils cases upon it from a height of 4 feet, without injury. 

When the large (l6-feet) sfwin arches were laid in the Commerce 
building, on Pacific Avenucj, in Chi(;ago, each arch was tt^sUnl by 
rolling an iron pulley, 6 feet in diameter and 14 inches wide. 
weighing 2,180 lbs., over each square foot, before the concrete had 
been filled in the haunches. This is a convenient method of test- 
ing the strength of a floor after it is laid, and its use is to be highly 
recommended. 

Streivsrth of Briek Arches.— Brick arches, properly built 
betwt»en iron beams, as described on j)age 440, are practically inde- 
stnictible, from any usage or load that could occur in a building. 

When the Western Union Telegrai)h building, in New York 
City, was being erected, Mr. P. C. Merry, the architect, made a 
series of tests on several forms of floor arches, supported by irou 



448 FIRE- PROOF FLOORS. 

beams placed about five feet apart, by dropping a piece of granite, 
li5 inches s(juare and 4 feet lon^j:, with rounded edges, from a 
height of three f(»et. on lop of tlic arches : and. while ail of the 
other jin^lu'S wci*e destroyed, the brick urcli withstooil the nhock 
S(!veral times uninjured, and only after repeated |)oundings in 
the saiiK^ phicc one brick at a time was knocked out until the 
arch was finally hroken down. 

That l>ri(;k floor arch(\s will endure prreat distortion was sliown 
by tin' loiding of an arched fUK)r at the Watertown Arsenal, Mass. 
A flooi- \JI) feel square, was miule of five ir)-inch I-l)eam8, 20U lbs. 
per yard, carrying brick arches. The beams were 7 feet 4.8 inches 
apart on eenlres, and rested on l)ri(!k walls 28 feet inehe.<« apart. 
The rise of the brick arches was y.5 inches. ''Common, rather 
soft-burn(Ml ]>rick were us(mI, laid (m edge with lime mortar. The 
arches were i)acked with concrete, and planked over. The miixi- 
nuiin load carried by t his fl(K)r (when tlie Ijeams, and not the arches, 
failed) was 50:} lbs. per scpiare f(K)t. This load caused a gnulual 
and continuous yielding of the beams, winch was aHowed to con- 
tinue till till' ll(M)r was deflected a distance of 13.07 inulies, meas- 
ured at the centre of I he mi(hlle berims." '*The brickwork en- 
dured this great deflection. an<l apparently wouhl have stood much 
more without failur(>," had it been |K)ssi)>le to carry the test 
further.* 

FiiM'- Proof Floors with Tension Mem born (1805). 

— WitMJii a lew y»'ar-< several styles of nn?-proof floor construction 
havr li. en iiitroduce(l, of whicii there are two general olusi«os ; the 
first ela>s <'onsists of tension memiMM' floors, which in liicmselTei 
furnish tin ne<essary strength for sustaiidng the lhM)r from wall to 
wall, or wall to ginler, without the usi* of (hK)r l)eams; and the 
other ela^s consists of 1 U^ams iivt^ or six feel apart for sustaining 
the fli»)r. with rods or bai*s -usiK^niled or nesting upon the U>ani8, 
su]ip()rting win; cloth, netting, or expanded metal. whi<-li carries 
th<- concrete or plaster filling. I'rondiient among the first ilevici'S 
ineni iniii'.j :ire the II vat t riblnHl metal ties and Portland cement 
conerite ii«i,)r> built by 1*. 11. .ia<-ks<in. Sun {''rnncix-o ; tlh* con- 
crete an<l t w ist -d liar floors built bv the Ransome & Smitli Cciin- 
pany. ot" Cliieagt): and the Lee hollow tile and cabh- nwl fliior«, 
built l»y till' liCc Fii*e pniof Construction ('om|iiiny. of New York. 

ppiiiiiin lit among the l-)N>am and concn'te tiliiiig devices an* 
the sNv;i,Mi«^ nf t Iie .Metropolitan Kin»-I*r«H)Hiig Compiiny. of Tn»n- 
t'li. N. .1.: tliee\{ianiie«l metal con si r u ct I on com |ni nies of St. liiuiis 

* -I I-; n<>\\.iii|. ill .\itniii-a» An'hifn-f ttntl linihliittj .Vf/fA, Mttreb lU, I 



FIRE-PROOF FLOORS. 449 

and New York ; and the New Jersey Wire Cloth Company, of 
Trenton, N. J. 

Hyatt and Jackson Concrete Floors.— Concrete com- 
posed of broken stone, fragments of brick, pottery, and gravel, 
held together by being mixed with lime, cement, asphaltum, or 
other binding substances, has been used in construction to resist 
compressive stress for many ages. 

With the introduction of Portland cement, concrete construction 
has taken a more important position among the various methods 
of building, so that now entire buildings are constructed of con- 
crete, such as the Hotel Ponce de Leon. fi.t St. Augustine, Florida; 
and in (Jalifornia. especially, concrete is largely used in the con- 
struction of floors, sidewalk arches, etc. 

The concrete is not used between iron beams, as are the brick 
and tile arches, but the concrete itself is made self-supporting from 
wall to wall by means of embedding iron in the bottom of the con- 
crete. Portland cement concrete has a great resistance to com- 
pression, but possesses little tensile strength. 

In 187G Mr. Thaddeus Hyatt, the inventor, while considering 
the matter of fire-proof floor construction, conceived the idea of 
forming concrete beams by embedding iron«in the bottom of the 
concrete to afford the necessary tensile strength which the concrete 
lacked. Mr. Hyatt made many experimental beams, with the iron 
introduced in a great variety of ways, as straight ties, with and 
without anchors and washers ; truss rods in various forms ; flat 
pieces of iron set vertically and laid flat, anchored at intervals 
along the entire length. These experimental beams were tested 
and broken by David Kirkaldy, of London, and the results pub- 
lished by Mr. Hyatt for private distribution, in the year 1877. 

By these tests Mr. Hyatt proved conclusively that iron could be 
perfectly united with concrete, and that it could be depended upon 
under all conditions for its full tensile strength. 

The method Mr. Hyatt adopted as the best for securing perfect 
unison of t'.ie iron and concrete was to use the iron as thin vertical 
blades placed near the bottom of the concrete beam or slab, extend- 
ing its entire length, and bearing on the supports at both ends ; 




Fig. 14. 



450 FIRK-PKOOF FLOORS. 

tbcso vortical blades to be anchored at internals of a few inches by 
round win>s threaded through holes punched opposite each other in 
the vertical blades, thus forming a skeleton or gridiron, as shown 
in Fit;. 14 F^>r a perfect combination of these substances, it is 
essential that the one should 1)6 united with the other in such a 
maimer that the iron cannot stretch or draw without the concrete 
extending with it. 

The only person in this country to make practical application of 
the method devised by Mr. Hyatt, so far as the author is aware, is 
Mr. P. II Jackson, of San Francisco, Cal., who has used it quite 
extonisivoly in that city foj: covering sidewalk vaults, and for tl>e 
support of store lintels ; also, for self-supf)orting floors. Mr. Jack- 
son publislicd a pani[)hlot in 189.), entitled Impromment in BuUd^ 
ing ('onnfnfrfit^ny which gives a great amount of information on 
this sul).ject, and on concrete in general construction. 

To sliow the strength of this method of construction, Bfr. Jack- 
son, in Aug\L«*t. 1885, prepared a beam, 7 x 14 inches in section 
and 10 fiM't 6 inches long ; near the bottom were sitven vertical 
blades of iron extending the entire length ; three of these were 
i y \ inch, and four wore i x 1 inch, with i-inch wires threaded 
through overy 3 inches. Near the top were bedded two cast-iron 
rope moulding bars to assist the compressive strength of the con- 
crete, which, however, was siiown to IxMin necessary. The concrete 
at the top and bottom was one ])art cement to one of sand ; centre 
portion, oni' of cement to iwoof siind. Thi' Iwam was supfiorted by 
U-inch In-irings at both cn<ls. thus leaving it fed in the clear be- 
tween snp|»«»ris. Tlie beam was loaded with pig-irrm piletl lu-roes 
it, anil l)n)lv<' un:l(>r a lo.id of 5.>,(ii'>4 llw.. by Kcparating till the 
lon;;itu«Iin.-il bladi's on tlic line of on(> of the cn»ss-wires near the 
centre. .Inst lN>fore breaking, the deflection was measunnl, and 
foumi to Ih' \^_ in<'h. The breaking load of this lM>ani was aU>ut 
oMe-}):iir I hat which would have broken a hanl-pine beam of same 
dimensions and average ipiality. 

Tlu' Kaiisoiiu' and Kinllli Floor. 

\N hile Mr .Ia<'k^i>n was ex)M'iimenling with tlie Hyatt tics, Mr. 
Iv L lian-iiuf. a vrry >ui-'essrul workiT nf enn('ret«* in Sail Fran- 
ii-ii). iin.c.iMd the ide:> (it using siuan* b>irs i»f iron and .'^ti*«-l, 
twi^tni t|.,ii- entire leiurth. in place of the flat Uin* and win*s used 
)>y Mr .l.ick^on. as >)io\vn in Fig. 15. It was found that thest* bars 
Win- !■• Ill ill ihe fimerete i (lUiillv as well, if not UMIer than IIm 
ol liiM*. .ili'i lli'il (hey were niileh le*«s exiH'Msive. Nolle uf thtf in»D 



E-PBOO? FLOORa. 



in the ties is wasted, and it hae been demoastnted by careful ez- 
perirnents that the procesB of twisting the bars to the extent 
desired strengthens the rods instead of weakening them. 



Fig. IS. 

Mr. Bansome patented his improvement in 1884, and since that 
time it ha3i>een used quite extensively in San Franijisco. 

The bars, preferably made from the best quality nf rectangular 
iron, are twisted at an expense not exceeding from twenty-five to 
fifty cents per ton, which constitutes an inaigniflcant item of cost. 
The sizes so far used range from \ inch to 2 inches square. 

Concrete floors, as made by Mr. Eansome, are made in two forms 
— flat, and receesed or panelled. 

It can be and has been used for spans up to 34 feet, A section of 
a flat floor, in the California Academy of Science. 15 x S3 feet, teas 
tested in 1890 with a uniform load of 41,^ lbs, per square foot, and 
the load left on for one month. The deflection at the centre of the 
23-feet space was only ; inch. It was estimated by the architects 
that the saving in this construction over the ordinary use of steel 
beams and hollow- tilo arches of the same strength, and with similar 
cement-finis I led floors on lop, amounted to tii< cents per square foot 
of floor. As a flre-i>roof construction, the concrete and iron con- 
struction above described is undoubtedly equal to any other con- 
Btruction in use. 

Oampotitum of the Concrete. — Regarding the concrete used tor 
these floors, the proportions are given for a cement of good average 
quality, that will develop a tensile strength of 350 lbs. per square 
inch in fourteen days. II a weaker cement is used, the quantity 
should be proportionately increased. 

The aggregates should be of any of the following solmtances, 
which are named about in the order of merit, the first being the 
best: Hard limestone rock, hard clinker brick, hard broken pottery, 
granite or basalt, hard clinker'^, broken flint or other hard rock. 

Care riiould be taken tj> use neither dirty nor soft clayey rock. 
The aggregates should be broken so as to pass through a two-inch 



FIHE-I'UOOF Fl.()i 



rinp, ami the fmo iliist, roitjoved by wiishing or screening (washing 
prcfL-rri'ih In mixing mid sufficient wnter to bring tbe mass into 
a fotl, |msly iiiitilitioii, itnil tam]i it. thoi'(>ii);hly iuto place. 

On (he lH)l.tc)in iif llie iiiiiulil ]>la(:c iilKiut one inch and a half of 
ctiniireic niiulw of ono jjiirl. cement lo two parts of agjrrcKitles vary- 
ing tnjiii ,',, to i ineli in diiinietur. I jiy Hio lower iron liars on this 
niixlure hikI tamp Ihuni Uunn into it ; tiien 1111 uji with a. conrre'" 
ci)ni|>o!<e(l of oni! [Nirl cement and six parts aggregates, making the 
final layer of double atrengtb. 

TIk> L<><- H<>IIow Tile and Cable Uoil Floor. 

Fig. 22 i» » Hk(>leh typic-al of thp Lee Hollow Tile anil Cable 
Goil FliHir. with a finislicil c-eineiil top. The flours are usiuily 
(li'signi'd cm a luisis of ^ inch in cIcpCh foreuuli foot of %paii. The 
spaiiH extend fioni wall to wall or from ginlcr to ginlur, no i- 



i terra-cotta tiles having siiuaru ends and a rod 




griKiri' aliiii;; iini^ hIiIi- near tliP Iwse, hfp iibi^. TIipm! lilea are sim- 
ilar t" IIh' l.i'i- i-nil arrb tiles, 'iVininimry focniH carried im honwa 
uri' iinividiil, and the lilei: are laid with i'ortland crinent niurtar 
in rows, curl in end, fioni wall In ginler. or fnitn ginli'r lo giixler. 
Into ibe ^'I'lKU-e of ca.-li row .if tiles soft eenient in phu'eil, >iiui iin<- 
or more nils, acc-oriiiin: lu stn-ngth rt<i)uirenii-nli>. iin- Imriwl in 

lllesoft e.-1ilel.t. The pilH'.-Bii is n-|>IMlvrl until 111.' wliol.- Hour iS 
fr.rMi.-.l Thr r.Hls iitop at emls of Ibe tili^s at wall lines Ani-liur 
lying the lloor lo tin- mipimrtii. 



tti'l f.>r 
ISy II 



LT 1 1ll 



to^n-lhel 



wliiih mar Ih' appli.'l lo lllling 
t: ,!<„< all Ihrusl i- lakrn ii|. bv the ealile HhI. and .-aeh 
iiid in ils i.l'K'e. CrMr'ks. deneetloDs. nii.l »lli.-r i|ef. <•(■ 
iidiii^' IImi jir<'lie> Hh- iivoi^le.1. Th.- Doors an- firm, rigitl. 
Tbu tloors are Iwwil n[ion the Inkn^vcnv strength uf 



FIRE- PROOF FLOORS. 



452a 



beams. (Computations, verified by actual tests, are made, and the 
use of needless material and weight is thereby avoided. 

The cable rods used in the l«ee system are made of round drawn 
steel rods of about thirty onc-hundredths of an inch in diameter, 
]aid spirally together, usually in two strands, as that form affords 
large gripping surface for the cement. Mr. Lee's patents cover a 
variety of forms, some containing several strands, with different 
shaped buttons, washers, etc., for affording great cement engaging 
surface. The rods being of drawn steel, they have high tensile 
strength, and are specially free from flaws or defects ; hence are 
found to make excellent tension members. The rods are spaced 8, 
10, or 12 inches apart, according to width of tile used. The widths 
and shapes of tiles are varied to suit different spans and loads. 

Fig. 23 shows one design of roof for ten-foot spans. It is a 



f^25 



fty^* 





special adaptation of the system, to cases requiring large protection 
to the metal from heat, as in dust chambers of smelters. 

Fig. 24 shows light design with finished wood top, suitable for 
dwellings, the wood top being more expensive than cement top. 
With a cement top the completed structure is but little more ex- 
pensive than a wood joist structure for the same purpose. The 
floors are absolutely incombustible, sound-proof, and vermin-proof. 

Strength and weight tables are furnished by the builders, giving 
various depths of floor structures for different spans and loads. 



The Metropolitan Company's Floors. 

Under this system, which has heretofore been known as the 
•'Manhattan*' system, and is protected by letters patent, fire- 
proof floors are made as follows : 

Cables, each composed of two galvanized wires, twisted, arc 
placed at given distances apart over the tops of the beams and 
transversely with them, as shown in Fig. 25. These cables i>ass 
under bars in the eenfre of the si)ans, and are thus ^iven a uni- 
form deflection between each pair of beams. The distance between 
the cables is varied with the loads to be provided for. Forms or 
centres are then placed under them, and a composition, made prin- 
cipally of plaster of Paris and wood chips, is poured on. This 
composition solidifies in a few minutes, after which the forms or 



iti-j/i FIKK-PliOOF FLOOKti. 

ceiitros are removed. 1'he rcsultini; lloor is anfitcientlr stroDf; to 
be iiM'il at once uiiilor tiie IoiuIh for which it has been calcukted, 
mill UM ids Kiirfni-'C is imiforin itiid bvul with llie tops of thu bnimx, 
a working Uimt U llius riiniishwl. 'I'LJs iii of (.'riini advarituKi- in 
fiicililaiiujr ihc tinnentt i-(iiislructiuu of builiiinKs. 



Fici. 3S. 



owM tbn urmnf^nient einplo;rod ill caiWR whura a flat 
ntijiiin^I. Id this iirniiij,'eraciit tlie nniler siili! of Ihe 
■■■iiixhfs n coiliiiK Hortui-t' n'»ly lor |ilusb.'riu((. Thu 
iM iif thu bi'UtDH, |ti-ojcutiu^' ua thuy do bclciw tbo floor- 



-rr/ 



1.1.'-: KhUl rry ilu- n«>r-[>li>t<-H. 

iirrnnjfi'iiM-iil fiii|>l(>ytfil whem a flat rciUiif; 



PIHE-PBOOF FLOORS. 452c 

id desired. In this case Ihe floor-plate i? the same aa in Pig. 
26. Tha ceiling-plate is lormed as follows : Dars are placed 
upon the lower Annges of the beams, ami on these wire netting 
is laid. Centres are placed one incli below the beams, and the 
composition is poured thereon. The centres are then removed, and 
the ceiling thus made is readf for plastering. Whether a ceiling 
like that shown in Pig. 36, or a flat ceiling as shown in Fig. ■4'!. is 
osad, the webs of all beams are covered with about three inches in 
thiukness of the Metropolitan composition, which thoroughly pro- 
tects the beams from the etfeeta of heat It is claimed that this ma- 
terial is so remarkable a non-conductor of hat that a moderate 
thickness of it prevents the passage of nearly all warmth. 

" In.sETere Are tests the l)eams have rfmained cold, and conse- 
quentlj were unaffected. When exposed to flame for a long time, 
the Metropolitan composition \b attacked to a depth of from ,\- to 



A of an inch, the remainder being unaffected ; nnd when nater is 
thrown upon it, the mass (iocs not !ly or crack. When made thor- 
oughly wet, as would happen from water thrown into a building 
during a Are. the composition is nofdestroyod." 

In Paris a composition of plaster of Paris and broken brick, 
chips, etc , has liocn used for giineraiions f;.r fniiunig ceilings 
ijutwpon beams, so tliat the question of its durability is there fully 
settled. 

The strength of floor? made under the Metropolitan system has 
been accurately determinB<l for vitrious spans by 11 great number of 
carefully-made tests 

" The loads that so break up the oonpositioa of floors made 



452^7 FIllE-l'ilOOF FLOOUS. 

under this system as to RMjuiro it to be replaced, vary from 1,100 
to 2.00) j)<)un(ls ppT square foot on spans of from 4 to 6 feet. 

'I'he W(M<^lit of ?i floor finislie:!, as shown in Fig. 26, when ready 
for the plaster underneath and the floor above, is about IS pounds 
I>er sfiujire toot ; and for a floor and ceiling such as is shown in Kig. 
27, 24 pounds per sciuaro foot; the thickness of the floor plate is 
alxmt Ji'l iiu'lies. 

T1h» proprietors of this system reconmiend that the floor beams 
be spa('e<l about i> feet apart, as this distance appears to give the 
best results witli the greatest economy. 

P'or further information concerning tliis system, the reader is 
referred to the Metrojx)litan J^'ire Proofing Co., Trenton, X. J. 

There are several styles of floors constructed on the principle of 
the Metropolitan floor, although nearly all of the others use Port- 
land cement concrete instead of the plaster c<)mi)osition. Wire 
lathing, (■xj)anded metal, and various shaped bars are used for the 
t(?nsiun menil)ers. The jn'incipal advantage sought in these floors 
over the icrra-cotta file arches, is a reduction in the weight of the 
fl(K)r, thereby causing a saving in the steel construction. The floors 
themselves are also, as a rule, a littie cheaju'r than the tile floors. 

Another important characteri.stic of all floors constructed on this 
j)rincip]e is, any st^ttling of the anhes. or filling, will tend to draw 
tiie beams (»r girdei's together, instead of pushing them apart, as is 
the case wiih tile arches ; and tie rods are, therefore, unnecessary. 

The strains infl(M>rs of this kind are the same as in those of a 
beam, ilie e!V. c t of tlie load Ixnng to pull the tension members ai>art 
at ilu' 1k):im:ii. and to ciu>li the concrete on top. Wlien the eon- 
(I'ete i> of tlie proper thickness, and of g(XMl ([uality, the stn*ngthof 
th«' llonrwill bedetermined l»ythe strength of the tension n)(Mnl)cr>ii. 

Several ti'^is ot" beams made oi" .ortlaiid cenuMtt. eoncn'te. and 
wile neitiiiLr made by the NciW .lersey Win; Clnih ('om]»any. apjirur 
to show that only about one half the strength of the ten>ion ineni- 
lK'rs'\\h 11 of wire cloili) can be; d«;velo|H'd. In all floors van- 
strueted of coneiTie. plaster, or tile with steel tension nuMnU'rs, it 
is ^ii tiie iir^t imiMirtance that tiie two materials shall 1m* so elosi'lv 
united that the tension memU'r-? will not be dnitrn thror^h, or slip 
ill the eoiiepie : Inr the minute this (K'cui*s, the strength of the 

llniT. lis (I III (I III, is (lest roved. 

\\ lid« >'iiii" of the^r tension memU'r fl<M)i-s liave been ns«»d .sufTi- 
<-ieiiiI\ t(» :iill\ (h'lnonstiate their strength and praetieabiJiiy. yet 
th<- wr.'.ei i't'iieves that new arningements n  devjees should lie 
u^>i| \M!li • At r-ine eaution and oidy after they havi) buuii t4S8ted 
an<l apprcVMl i>y experienced eagineers. 



FIRE-PROOF FLOORS. 



4526 



Concrete and Wire Netting Floors. 

'■ Pigs. 28, 29, 30, and 31, show two styles of fire-proof floors, 
devised by the New Jersey Wire Cloth C'ompany, and described, 
together with several other applications of concrete and wire net- 
ting, in a pamphlet published by them. The segmental arch shown 



^^^r 






^ 


■;*: ' 




^^ 


>— J — 

''Vl 




m 


^^v^^^ 






fiii^'^ 


m^ 










^j 



Fig. 28. 

in Fig. 28 is constructed by forming a centre, made of small rods, 
cut the proper length to form the desired curve, and to just reach 
into the angles of the web and lower flange of the floor beams. 
These rods are inserted between the meshes of wire lathing, and 
the sheets, which would be three feet or more in width, are then 




Fig. 29. 



bent to the curve and sprung into place. A succession of these 
sheets placed side by side fill the entire space from wall to wall, 
and make a continuous network of iron wire and rods, upon which 
concrete can be spread from above without the use of any other 
support. 

The lower flanges of the beams are covered by wire lathing 
attached to a succession of rods hooked over the arch rods and held 
in place by the wedges which are inserted between the beams and 
the rods. 

The under side of the arches and the lathing around the beams 
is then plastered and finished in the usual way. 

It is claimed that with this construction the strength of the arch 
is only limited by the ability of the beams to carry the load. 

The weight of the concrete will vary from 30 to 40 pounds per 
square foot. 



FIllE-i'ROOF PIX)ORS. 



Fi^. BO and 31 ahov u flour coustnuitioi: designed on the com- 
pcjsiic Ikjuth principle. 
It is i'iiii[in.il by till! iiiaimfiiclurerB. that a load ol from 70 to HO 



n lie carried 

■i of I>VIL1UB. 

i^-ljt of the corii^rute. uiru, anil rmls, For both Qoor aad 
ceilinj,', "ill vary troiu 8;Ho 4o jiounds |)iT wjnaru fool. 



a foot, with a fat-tor o( safety of si 
i»ii in spalls of isix feet bftwei'U oi 




- iif tilis HiNir ininKlriK'tion is n m-rJeH of rmIs honked over 
III tli<> lii'aiiiH. or iiUai-tx'rl ti> iIu'mi liy rliiw iltvitnHil for 
'I- Till- rtxU Hre plwcl alxml Iwelvi- iiu-hiH Ui>«rt, Hnd 
in' spnixl sIhtIs of wiri' iHlhiiii; riLntiini; piimllcl with 
m-r ihc lii|i of till- iH-atiiN. Tlii' coiii-H'li- is tlii-n spn-ad 
icivK. ini>-iilli of Iwo tr.llirri'ini-lii-s Nn iTiitcrinK w 

- Ilii' iTirw iiii-slii'.-' of Ihi- liilliiiii; iin> mi oI<w loKutlior 

iK^iii;) iK'n-li' Hill •,!•< tiiroii^'li lo llrmly >iiii-hiir tb« 

■V tl>- ('..tu-n>u- W M-l. iW unili-r sid.- >li<.iilij W iiIkmIiw) 

II ^r. i,s to .'lltill.ly >-!I|Ih'.1 till' win- >LII<l hHU. 

lis slioiilil In- |iriitii-liil liy win- IilIIimlk and pliiitti'rinR, 
KTilal ii'ilhif;, HUjiiHirliHl by linxioH riHla. inay U- hung 

b<>rs <a(i In- con^ttnietetl with eiimndiil iiieUl Iftthlog. 



FIRE-PROOF FLOORS. 462^ 



The Fawcett Ventilated Fire-Proof Floor. 

This is a style of floor construction differing almost entirely 
from any of the floors herein described. It has been used exten- 
sively in England, and to some extent in this country. 

In the construction of this fire proof floor, the special feature is 
a Tubnla/r Lintel^ or hollow tube, made of flre or red chimney pot 
clay, and burned mellow. 

Iron Beams (of sections to suit the spans and loads) are placed at 
two feet centres, and the lintels are fixed between, with their duig- 
onals at right angles to the beams ; the end of each bay is squared 
by cutting (during manufacture) an ordinary lintel, parallel to the 
diagonal ; the piece cut off when reversed goes on the other end. 
Thus the ends and sides of all lintels are open next the walls. 
These are called ** splits." 

The lintels being in position, specially prepared, cement concrete 
is filled in between and over them, which takes a direct bearing 
upon the bottom flange of the beams, thus relieving the lintels of 
the floor load, which is taken by the iron and concrete, the lintels 
forming a permanent fire-proof centering, reducing the dead 
weight of the floor twenty-five per cent, and saving about half the 
concrete. 

Cold Air is admitted (through air bricks in the external walls) 
into any of the open ends or sides of the lintels, and passes through 
them from bay to bay under the beams. Note, only two air bricks 
are absolutely necessary in each room, to insure a thorough current 
of air. 

The flat bottom of the lintel completely incases the bottom 
flange of the beam without being in contact with it, a clear half- 
inch space being left for the passage of cold air. 

It is claimed that the chief tire-resisting agent in this floor is not 
so much the terra-cotta or the concrete as the cold air, and that 
the circulation of air through the floor and around the beams will 
actually prevent the iron from ever getting hot at all. 

The Fawcett Company claims that their floors have never been 
injured by fire and water, Ixiyond what could be repaired bv replas- 
tering the ceiling and redeconiliug the walls. This floor needs no 
centering or any other support from below while in course of con- 
struction, and can be used as soon as finished. It is guaranteed to 
carry fmm 150 to 750 pounds to the square foot, according to the 
requirements of the building, with perfect safety. 
Although the author has never seen this floor put up, it appears 



FIKE-PltOOP FLOOitS. 



Ml 


m 


11 1 


i 


yi 


1 




FIBE-PKOOF FLOORS. 463 

to him to be a very superior floor, although probably more expen- 
sive than the other styles herein described It requires more con- 
structional iron work than the systems generally in vogue in this 
country. 

The Guastavino Tile Arch System. 

Within a few years a method of constructing floors, partitions, 
staircases, etc., by means of thin tile cemented together so as to 
make one solid mass, has been introduced by R. Guastavino, of 
New York. The floors in this system are constructed by cover- 
ing the space between the girders by a single vault, constructed of 
tile about 6" x 8", and ^ inch thick, cemented together in three or 
more thicknesses, depending upon the size of the vault. The thick- 
ness is generally increased at the haunches. The strength of these 
floor vaults, considering their thickness, appears to the author very 
remarkable. This method of forming floors is especially desirable 
where a vaulted ceiling for decorative purposes is wanted, as the 
vault can be made the full size of the room. The iron- work used 
for posts and girders must bo piote:jted as in other methods of fire- 
proofing. The iron-work of the floors must be especially arranged 
for this system when it is desired to use it. As far as the author 
can judge from an inspection of the system, it possesses some ad- 
vantages over all other present methods of construction (and, pos- 
sibly, some disadvantages), and is likely to be largely used in the 
future. It has been employed in a number of buildings in New 
York and Boston, and a few other cities. The new Public Library 
Building in Boston has the Guastavino floor system, which is ar- 
ranged so as to give a fine effect of vaulting in the ceiling. 

Rules for Determining the Size of I-Beams, etc. 

The method of computing the size of the iron beams used in fire- 
proof floors is merely to determine the exact load they will have 
to support, and tlicn to find the required size of beam to carry tliat 
load. 

The weight of the floor itself should be determined for each par- 
ticular case, as it will vary with the kind and size of tile, the 
amount of concrete filling, kind of flooring, etc. 

The weight of the arch itself may be taken from the manufact- 
urer's catalogue, or from the table on page 445, and to this weight 
should be added about 5 pounds per square foot for mortar used in 
setting. For each inch in depth of concrete add 8 pounds; for 
plastered ceiling, 8 pounds ; for hard-wood flooring, 4 pounds ; for 



454 



FIUE- PROOF FIDO US. 



marble floor tiles, 1 inch thick. 1-1 pounds. The weight of the 
betims may bo taken at 5 pouncls per square foot for 9-inch bojuns. 
and () pounds lor 10 and 12-inch l)eains. Very few fire-proof floors 
will be found to woii^li less than 75 pounds per square foot, and 
where marbh^ tiles are used for the flooring? the weight of the (con- 
struction often reaches 1)5 pounds. The superimposed loads will, of 
course, be the sam(^ as those jLriven on page 426. The weight to be 
suj)p»)rted by the beams will be, w = distance between centers x 
span of beams x (/ -f /'); / representing the superimposed load, 
and /' the weight of the floor construction, including an allowance 
for the weight of the beams. 

Having obtained the value of this expression, the size of beam 
required to carry this load may be easily ol)tained from the tables 
in Chapter XIV.' 

To save the labor of making these calculations in the principal 
classes of buildings in which fire-proof floors are used, the follow- 
ing tal)les have been computed, which may be safely relied upon. 

Tables of Floor lioains. 

Tables showing the size and weight of Carnegie steel beams re- 
quired for dilT(»renl spans and sjiacings in different classes of build- 
ings, using hollow tile or terra-cot t a between the arches — the 
l)eams not to deflect so as to crack the phistering: 

TAIiLK I. -F()I{ b'LOOKS IN OKFICKS, IIOTKIjS, AND 

.\P.\kTMKXT IIOI SKS. 

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41 '• 



FIRE-PROOF FLOORS. 



455 



TABLE II.— FOR FLOORS IN RETAIL STORES, 
THEATRES, AND PUBLIC BUILDINGS. 

(Superimposed load, from 125 to 180 ponnds per square foot.) 



Span, 
in 


Distances between Centres of Beams. 












feet. 


4 feet. 


4 feet 6 inch's. 


5 feet. 


5 feet 6 inch's. 


6 feet. 


10 


6 in. -13 lbs. 


6 In.— 13 lbs. 


7 in.-15ilbs. 


7 in.— 16*lb8. 


7 in.— 15*lb8. 


11 


7 " - 15i " 


7 " _15| " 


7 " —15* " 


8 "-18 " 


8 "—18 " 


12 


7 " —15^ " 


8 "—18 " 


8 " —18 " 


8 "—18 " 


8 "—18 '* 


13 


7 " —15^ " 


8 "—18 " 


8 "—18 " 


9 " -21 " 


9 "-21 " 


14 


8 ''—18 " 


9 "-21 " 


9 "-21 " 


9 " -21 " 


10 " —25* »* 


15 


9 "-21 '' 


9 "-21 " 


9 " -21 " 


10 " -25* " 


10 "-25* " 


16 


9 "-21 *' 


10 " —25* " 


10 " -25* " 


10 " -25* " 


18 "-82 " 


17 


10 " — 25J " 


10 " -25* " 


12 "-32 " 


12 " -32 " 


12 " -82 " 


18 


10 " -25* " 


12 " -32 " 


12 "-32 " 


12 " -32 " - 


12 "-82 " 


19 


12 "—32 " 


12 "-32 " 


12 "-82 " 


12 "-32 " 


12 " -82 " 


20 


12 *• —32 " 


12 "—32 " 


12 "-32 " 


12 "-82 " 


12 " -40 " 



TABLE III.—FOR FLOORS IN WAREHOUSES. 

(Superimposed load, from 200 to 210 ponnds per square foot.) 



Span, 


Distances between Centres of Beams. 




in 
feet. 


1 
4 feet. 4 feet 6 inch's. 5 feet. 


5 feet 6 inch's. 


6 feet. 


10 


6 ill.- 13 lbs. 6 ill. —13 lbs. 6 in.— 13 lbs. 


7 in.— 15Alb8. 


7 in.— 15* lbs. 


11 


7 " -15* *' 1 7 " —15* " 7 •• —15* •• 


7 " -15* •' 


H "—18 " 


12 


7 " -15* ' 


8 " —18 " 8 " —18 " 


8 " 18 " 


9 " -21 " 


18 


8 " -18 " 


9 " —21 '' 9 " -21 " 


9 " -21 " 


10 '*-25* " 


14 


9 " -21 " 


9 " -21 " 10 " —25* " 


10 " -25* " 


10 " -251 •* 


15 


10 '• - 2.5* " 


10 " — 25i " 12 " -32 " 


12 " -m " 


12 •• 32 " 


16 


10 " -25i " 


12 " -32 " 12 " -32 " 


12 " 32 '' 


12 " 40 " 


17 


12 " - 32 " 


12 " -32 " ,12 "-32 " 


12 " -40 " 


15 " 41 " 


18 


12 " -32 " 


15 " -41 " 15 " -41 " 


15 ' -41 " 


15 "—41 " 


19 


15 " 41 " 


15 " -41 " 


15 "—41 " 


15 " .'50 •' 


15 " -50 " 


20 


15 •' 41 " 


15 " ^11 " 


15 "—50 '' 


15 " -50 " 


15 " —50 " 



It will bo seen from these tables that it is more coonomioAl to 
space the l^eains farther apart, and use as short spans as the condi- 
tions of the building will {)errait. 

For example, if we have an office floor 48 feet square, to support 
with iron beams and tile arches, wc; may eith(T use one girder down 
the centre, with 12-inch beams, spaced 4 feet apart ; or two girders, 
and lO-inch beams spiU3ed 6 feet apart. In the former case we 
should require 11 beams the full width of the building, weighing 



455a FIRE-PROOF FLOORS. 

16,896 pounds, and in the latter 7 beams weighing 8,568 pounds, a 
saving of nearly 50 per cent, in the steel. From this, however, will 
have to be deducted something for extra girders and columns, but 
tho total saving would probably equal '^5 \)er cent. In rcganl to the 
columns, it will not make much difference in the amount ol' iron 
used, whether there are one or two rows, as the total weight to 1)6 
supported is the same in either case, and if one row of girders is 
used the columns will be closer and heavier than if two rows are 
used. 

l)<»rtecti(>ii of Rolled I-Reaiiis. — The deflection of rolled 
iron I-beams can be computed by Formula 1, under the Stiffne^ts of 
BcatHs, Chap. XVI. 

Accordiiiir to the calculations of Mr C. L. Strobel, C.E., tho 
beams in the foregoing tables will not deflect over one-thirtieth of 
an iucli for every foot of span, under the load which they have 
been calcnlatcd to support. 

T!(»-r<)(ls. — Tie-rods from fivt^-eighths to (me inch in diameter 

are ordinarily em])loyed to take the thrust of the bri(rk arches, and 

to add to th(^ security of the floor. These may l>e spaced from 

eight to ten times the depth of the beams ajiart, and the holes for 

them should always be punched at the centre of tlio depth of the 

l>eam. The formula for the diamet^^r of the tie-rod for any floor 

is, 

. W X span of arch, in feet 

Diameter 8«iuared  .^., . - i • .» i ; 

' 6J8;j2 X rise of an-h, m fi»et * 

irdenotin;: weight of IKh)!*. and superimjH)srd load nesting on the 
arch ii.iU'-way between the tic^-nxls on each side. 

i^x.\.MiM.K. What shoul'l 1h* the diameter of the tii^-HKl to take 
the thni>i of a 1 brick arch, bctwt'en 10 ' iM'ams, spac«*<l 5 fwt 
npart : i he an li having a rise of iJ , and the tic-nxls to Im» sitiicod 7 
U'c\ apart ? I'lie su]M'riinpo.«<ed Imul to U' taken at 100 llw. 

Anf<. in thisca.>^c the span  5 ftn't, nearly ; W  170 x 5 x 7   

5050; and r . fn(.|. Then />- '*".*' "' ,  0, <»r /> 1 in4*h. 

t>ys."j ' k 

nearly. 

( )f «M>ui->^r. where arches abut aL^•tinst each siih- of a iH'um. there 
is no n< I'll -!' iixis tn take thi' ihnist of the arches : but it is alwavs 
safer !•• u-<- •'HIM. as the nntsidt- bay of the thmr might \h' puslied 
 •fV --ifiewise if the who!*' were not tie<l thniugh ; also, if one of the 
arehes >hi)(iid fail, or bn'ak through, the hmIs would keep the other 
arches in place. 



456 ^ILL CONSTliUCTIOXi. 



CHAPTER XXIV. 
MILL CONSTRUCTION.! 

In this ('hai)t('r it is proposed to describe the principal oonstruc* 
tivc f«'atun*s of what, in tli«* Eastern States, is known as the " Mill 
Const ruction," or **Slow-]>nrnina; Construetion." It is a method 
of const rui't ion lirought al)(>nt largely throngh the influence of the 
factory imitiial insurance companies, and especially through the 
efforts of Mr. William i». Wliitinj^, whose mechanical judgment, 
experience, and skill as a manufacturer, have been «levote<lfor many 
years to tli* interests of the factory nnitual comiMinies and to the 
improv(>ni(>nt of factories of all kinds. Mr. K<lward Atkinson, 
presi<lent of tlie IJoston Manufacturers' Mutual Jnsumnce Coni- 
])any. has also done a •xreat deal towards influencing the public In 
favor nt" tlilN mod*' of construction. 

Tlie /// xl'h I'lifmu in this mode of constniction is to have a build- 
inu: wli<»c nuoide walls shall he built of niiisonry (g<*nerallyof brick) 
con<'«ntraici in piers or buttresses, with only a thin wall i-ontaln- 
in^tlie windows l)ctween, and the floors and niof of which shall 
1)e conMructf I of liiri^e tinduTS, covered with plank of a suitable 
tliickn«»: tin' ::ii'ders heiuL; supporte«l In'twi^'n the walls by W(M)deii 
]M><ts. No t'lirrini^ or conc(>aled spa(*es an' allowed, and nothing 
is perniitteil which will allow of the accunndation of dirt, the con- 
cealni* nt of tire, or, in short, any thini^ that is not needed. 

Mr. i\ .1. II. Woodbury, ins]M*ct<»r Utr tin' factory mutual fln»- 
insur.iiH'i' conii>anies of Massachusetts, who has written a \iT>" 
able 1u)ok on the " Kire rroteetion of Mills" (publisheil by .lobn 
Wilev A- SoiiN o;' New York), has ;:iven such conci.se and «*N'ar 
stat<ini*nt^ oi what does ami what does not constitute safe iH>n- 
stniciion lor niills and warehouses, that with his iKTUiission we 
quote thrni nrfnitlin from his wurk. 



> Cuts I In t) in thi» rluiptrr an- inkcii fnmi WiMNlhiiryV Kirc lYolecttoil of 
MIIIh, iiiiil n-ilu< fil, fii rnnf«iriii lo tlio i-ixc of ilif imh^v 



MILL CONSTRUCTION. 457 

r 

'^ Prevailiiig' Features of Bad Constrnetion of 
Mills and Storehouses. — The experience of the Factory 
Mutuals has shown that in mill and storehouse construction, 
where considerations of safety, convenience, and stability are es* 
sential, tlie following prevalent features of bad construction should 
be omitted : — 

" Bad roofs. 

" Rafters of plank, eighteen to twenty-four inches between 
centres, set edgewise. 

" Any roof-plank less than two inches thick (three inches pre- 
ferred) ; any covering which is not grooved and splined. 

" Any hollow space of an inch or more in a roof. 

'^ Any and every mode of sheathing on the inside of the roof so 
as to leave a hollow space. 

" Any and every kind of metal roof, except a tin or copper cover- 
ing on plank. 

" Boxed cornices of every kind. 

" Bad floors containing hollow spaces or unnecessary openings. 

" Thin or thick floors resting on plank set edgewise, eighteen to 
twenty-four inches between centres. 

"All sheathing nailed to the under side of plank or timber, 
making a hollow floor. 

** Bad finish, leaving hollow spaces, or flues. 

"All inside finish which is furred off so as to leave a space 
between the finish and the wall. 

" Wooden dados, if furred off. 

" Open elevators. 

" Iron doors, iron shutters. 

" Any and all concealed spaces, wooden flues, or wooden ven- 
tilators of every kind, in which fire can lurk or spread, and be pro- 
tected from water. 

" Any and all openings from one floor to another, or from one 
department to another, except such as. are absolutely required for 
the conduct of the business (all necessary openings should be pro- 
tected by self-closing hatches or shutters, or by adequate wooden 
fire-doors covered with tin; automatic doors preferred in many 
places). 

" Ji^ssential Features for the Safe Construction 
of Mills an<l Storehouses. — Solid beams, or double beams 
bolted near together, eight to ten feet between centres. Not to be 
painted, varnished, or * filled' for at least three years, after the 
building is finished, lest dry-rot should ensue. Ends of timbers 
ventilated by an inch air-space each side in the masonry. 

" Roof nearly flat. Timbers laid across the tops of the walls to 



>• 



458 MILL CONSTRUCrriON 

project eighteen to thirty-six inches, as may be desired, serving as 
brackets. Plank laid to the ends of the timbers. Neither gutters 
nor boxod cornices of any kind. Wooden ])osts of suitable size, 
not taperctl, unless wlu^n single posts turned from the trunks ot 
trees with tlu^ heart as a centre, following the natural ta]H?r. <.'ore.s 
})()re(l one and a half inches diameter ; two half-inch holes trans- 
ver.-.ely through tlu* post n(^ar top and bottom for ventilation. 

" Floor-i)lanks not less than three inches thick for eight-foot 
bays, three and a half to four for wider bays. In some cases, 
beams have b(?en i)laced twelve feet apart, witli four-inch plank for 
th<* floor ; but in such cases a careful computation of the strength 
should be madt*, based upon the load to be placed thereon, l)efore 
so wide a s})ace between beams is adopted, lest there sliould \ye. ex- 
Cisssive dellection. 'I'he better method, wliere tlie arrangement of 
the machin(»ry reipiires such wide l)ays, is to alter the plan of floor- 
timbers. Toj) lloor one and a quarter inch boards of Southern 
pin<', mjiplc, or some hard wood. The best construction requints 
this top I'ioor to Ih' laid over three-quarter inch mortar, or two 
thicknesses of rosin-sized sheath ing-paiR»r, certain grades of which 
are now made especially for this piU'i>ose. 

••All rooms in which sp<'cial dangers exist, such as hot drj'ing, 
to be ])n)tcct«Ml overhead with jdastering on wire-lath, following the 
•inc of ceilin!^ and timlnT, thus avoiding any cavity in the ceilirg. 
In su'h rooms, the wooilen i)Osts should also be i)rott»cted with tin; 
car«' hciiii: taken to leave the half-inch holes through the ])us'.8 
mar \h*' top and base uncovered, so that dr>-rot may not take 
r:acc. 

Kig. 1 re])»-(»s'nts th<' iM*op<T const met ion of one bay of a thie-"- 
siory ndll, e:it]i bay being like the others, and the building In-ii ^■ 
iorniid of auN number of su<'h bays pla.'eil one aft4*r the otluT. 

Such a buil ling cannot be <'onsidered as fire-pnH)f: but the im'- 
terial In in ^iuli a sha])«' that it would not reailily take tin*, aM-.- 
wonll l»ui n >Io\vly even ihen. Moreover, the construction is mm-I . 
that any jtari of the building can be easily reached by a stream •*: 
wale,- : so ijiat a lin* <"an be n'adilv extinguisheil In'fon* it ha. 
gaine I inueli headway. 

Ill a luiik Imildinii im 'jraiiHr shnnhl fir ?/«cf/. except f«)r sti'pr 
and nndi-rpiniiinLr. a^ it sjdits badly when ex)K)s<Mi to heat, an 1 i.* 
theretme nn -uitablf for <<ills or lintels, or any work liable to Ih» 
exiM)s('ii to an\ intense h«'at in case the building >hoiild In' on tlrt*. 
'I'he hi^t i|Mahiies oi br«>\\n sandstone maybe u.sin! fur sills, aiiik 
for nihei- |)|aecs it would be blotter to use brick or ttTfaHHtttu. 
Mnnl ]•■ I liri<-ks ari> now manufactured in a gn*al variety of fornit. 
and are nell suii,.d for deconilive work. 



MILL CONSTRUCTION. 



45\ 



The best factories and woollen mills Is Husachusetta are now 
generally bulll with the beams eiRht teel. apart from centres, end 
with a span of twpnty-flve or twcncy-tour feet, there being one or 
more rows of posts ai^oriling to the size of the mill. Fig. I repre- 
sents the section of a mill having two rows of posts. 



Fig. I. 

The floor-heams are iwually twelve inches by fourteen inches 
hanl-pinn tliiihers,' which n>st on twenty-inc'h brick piers in the 
basement, antl on wnwleii posts and the outside walls in the other 
Stories. The ends u'liicli rest on the outside wail are arranged so 
BB to iiave an air-spncit around tlie end of the timber, and are 
aTH'hon'd to the wall by a cast-iran plate on which the beam resls. 
Tills plate, shown in I'ig. 2, has a transverse projection on the 
lllHHT sui-fai-c, wlilc'h fit^i into a groove in the bottom of the beam, 
and is turned down alHiut six tuehes into the brickwork at the 
encl. The hrlekwiirk for about five courseji above the beam sliotild 
be laid dry, and the upper edge of the end of the beam slijtlitly 
rounded. In ease of tlie possible hiimlng of the beam, this would 
allow the beam to fall without throwing ont the wall. 

Tlif finnr an top of these iH'ams is ronstnicted, first, of three- 
Inch planks, not over nine Inches wide, planed both sides, and 
grooved on both cdgis, which are filled with splines of hard wood 
(generally haiil pini>) alM>ut lliri>i'-f<iiu'tlis of an inch by an inch 



im MILL CONSTRUCTION. 

[Ill A lialf. In ciailin^ the planks, it la l)etter to "blind nfUl" 
lipiii. !<t'U.'i tlie iiiann<^r of iiailitig miLtrhed floors in dwelling-IiouMt 
ml storfis ; tliat is, ilriviiij; tin- nails obliiiiiely LliroiiRli th« Rroove 
I't'oru t!ie a[itinu la put ill : lliis hIIouh Lite plank to sUrink or 
u't'll without tracking, and wilbout afliliin^ ttic s)ilin(>g. 



Fl(. 2. 

nalli'il In tlih way. rarli pinnk miul 
|iiit <liinii. Tills lakfH (-onstilfrahle 
ay a niiiiilxT of planks, Wiilj;!' Ilii'iu 
lliic-s fiiini oni'. cnil, &nd nail Uirw.'tly 




Fig. 3. 
Till' ujipcT tliMiiinir is p-iii'ially o( sonic liant wooil, an iiirli wml 
'liuiriir rliU k. iin'ri-ly .joliiI.'<i. 

 -I'll.' Ill »ir,-.^l.. mill ii.- ivn.irn-il ua(iT-(iylit l.y flim-folinlis i>( an 
.■liof iii..rkiilu.riv.-.-ri (hr III.].!-!' .111.1 lowri' (l.,..is. Til.' Iiiyi'r ot 
oiuir |.iv.,TV- III.' Iiiiiilirr from il.riiy. jiivM-iils iW ll.M.r fniin 

':ii'ly III" pi'i'iif llian iiny ntli'T I'llii'llral iih'IIliiiI of iMnHlrm- 

liu. :; -lii.iis J. si-i'iinii ilirnii-li mi.1i a lliM.ra- v.>- liavi- iIi-wtMhiI. 
h- .■:;/ j. :;iii,-r:ill> iDiuuii of U-ii-iiic-li liy 1Hi-lvi'-iiii-li lianl-I'llw 



MILL CONSTKUCTIOM. 461 

timbers placed the same as those below; and the outside end is 
allowed to project over the wall from eighteen inches to two feet, 
forming brackets to support the eaves. These timbers are covered 
with two and a half or three inch spruce plank, grooved and 
splined the same as for the floors. The plank extend to the end of 
the overhanging timbers, and form the eaves to the building, no 
boxed cornice being allowed. If the roof is flat, as is generally 
the case in mills and factories, the plank should be covered with 
tin, gravel, or duck. 

If tin is used, it should be the best " M. F." tin, painted on 
the under side with two coats of red-lead, and well dried before the 
sheets are laid. 

If a gravel roof is used, it should be equal to the best quality of 
tar-and-gravel roofing over four thicknesses of the best roofing-felt. 
Cotton duck is gradually coming into use as a roofing material, and 
has for a long time been used for covering parts of vessels. It is 
light, durable, does not leak, and is not readily inflammable. 

The material should be twelve-ounce duck, weighing sixteen 
ounces to the square yard, and should be thoroughly stretched, and 
tacked with seventeen-ounce tinned carpet-tacks, the edges being 
lapped about an inch. If the roof-planks are rough, or not of an 
even thickness, a layer of heavy roofing-paper should be laid before 
the duck is put down. After the duck is laid, it should be thoroughly 
wet, and then painted with white-lead and boiled linseed-oil before 
it becomes dry ; which makes it water-proof. To protect from fire, 
give it two more coats of white-lead, and over this a coat of iron- 
clad paint. Instead of the four coats of white-lead and oil, the 
duck may be saturated with a hot application of pine-tar thinned 
with boiled linseed-oil. This lias been found to work perfectly. 
The ironclad paint should be applied, whichever method is used. 

If the roof is pitched, it should be covered with shingles or slate 
laid over three-quarters of an inch of mortar; which protects the 
slate from the heat, should the building take fire, and rentiers 
the roof cooler in summer, and warmer in .wintei*, whether slate or 
shingles are used. Where there are no buildings near, shingles are 
recommended, as they are warmer than slate (thus saving in the 
cost of heating), and are also cooler in summer. If the shingles 
are painted, which is advisable, they should be dipped in paint 
before being laid, so as to be entirely covered on all sides with 
paint: otherwise, moisture Avill get into the shingle through tlie 
place not painted, and, being prevented from evaporating by the 
paint on the outside, will rot the shingle. 

The columns for such a mill are usually round columns, nine 
incbee diameter in the first story, eight in the second, and seven i& 



4(12 Mll.r, CONSTttUCTION. 

tlie third; thpse l>eine Uie least diamet<'rs of the columns. Ifth< 
(■"luniris iirc tapered, t1»!)- may he half (in inch loss in diameter al 
the top, and oiu: itiHi [iiore at the bottom, making the taper on 



FI9, 4. 
hf I'dliiinn thrfc-foiiiths of an inch. They nhniild 
iiir'l-piiir or 'i'(k tiiiiliiT. tlmLims(lily seasoned, and 
ire.s Imr.'d one ami a half hirhw In illanietcr. with 

lii>li's Iransvi'rsiOy tlinMigh the [mst, npar top ami 
lit ilat ion mill to pri'vi'iit di-j-rot. Tlif tytlimins aro 

iM.<t-irim cuits. as shiiim in V\!i. 4. wliid) support 
•• Ilooi-beaiiis; ami, "hi'ii" there Is a vcrtlL-al line ul 



u:i1i iron itinllMi, wlileli ounnert 
of tiM-oIlii-r. ;in'V<'ntii)grhpKiii^ 
I by the ueigliC oil tin euluiimii 



MILL CONSTRUCTION. 



463 




above. The ends of the pintles and the iron plates against which 
they rest should be turned true, so that the contact will be uni- 
form. Fig. 5 represents a vertical section through the floor and 
the centre of the columns, and Fig. 6 shows a perspective view of 
a pintle with the base of the upper column coming down over the 
top. The brick piers in the basement supporting the columns 
should be capped with an iron plate twenty inches by twenty 
inches, an inch and three-fourths thick. 

The above is the most approved method of construction now in 
vogue for mills, factories, and storehouses; and the dimensions 
given for the various parts will answer for any cotton 
or woollen factory where the bays are not more than 
eight feet long from centres. Where the bays are 
more than this, or the loads on the floors are greater, 
as may be the case in storehouses, the floor-plank and 
timbers should be proportioned according to the rules 
for strength and stiffness given in Chap. XXII., and 
the columns proportioned according to the rule given 
in Chap. XI. 

ff partitions are desired in such a mill or store- 
house, they should be built of two-inch tongued and 
grooved plank placed together on end (forming a solid 
partition), and plastered both sides, either on wire, or 
on dovetailed iron lath. Such partitions have been 
found to work well after a trial of twelve years, and 
offer effectual resistance to fire. 

Mill doors and shutters should be built of two 
thicknesses of inch boards, covered on all sides with 
tin, as described in Chaj). XXVI. 

For a thorough description of the apparatus and appliances used 
for the fire protection of mills, and for a thorough discussion of 
the vibration of mills, the deflection of the floor-planks, and, in 
fact, every thing that refers to the construction and protection 
of mills and factories, the reader is referred to Mr. Woodbury's 
work on Ihe "Fire Protection of Mills," mentioned al)ove. 

The cost, per square foot of total floor area of mills and factories 
at the present time (1884), according to Mr. Edward Atkinson, is 
as follows : — 

Mill with three stories for machinery, and a base- 
ment for miscellaneous purposes 75 to 80 cts. 

Mill with two stories for machinery, and no l)a«ement 65 " 

Mill with one story, of about one acre of floor, with 

basement for heating and drainage only . . . about 85 ** 

The above is for the total area of floors in the building, above 



Fig. 6. 



MILL OONSTBUCTION. 466 

ncrt eTen weakened by the sftace left in the wall, because the anchor 
remalnB, and the crashing strength of this cast-iroa box is much 
greater than that of the wall. No break or breach is made in the 
vail, and. the anchor that remains, securely held, forms a space for 



the easy repiaeoment of joist. The anchor provides a perfect and 
seoiire foundation for each joist. Fire from a defiictive flue cannot 
ignite a joist end, because it is protected by a rentllated east-iron 
box. 

The boxes, or anchors, also have air spaces in the sides, J inch 
wide, which permit a eircujation of air around the ends of the joist, 
effectually preventing dry rot in Che ends of tile timbers. 

If timber is wet or unseasoned it will have a ohanca to dry out 




after it is put in the buildiiin- Tliea; aur^hors are obviously greatly 
superior to the ordinary method of anchoring beams and girders to 
walls, and their use would, in case of fire, undoubtedly save much 
loss by the falling of the walls, which are almost invariably 



MILL CONSTHL-CTIOS. 



pulltnl ciown by tlio ordinary iron anchors. The avenifie wdght ol 
alinx liki; Fig. 7. Tor 2 x li joist, in l.j lu 17 lbs.; of Fig. 8, from 



woihI [Hists. Thw OB]) linlds 
. piuviik-s vuntiliiliim uIh>uI 




T..U..ri} 



ii'l I. 



I .laii 



«<iTilnl liinlior' tn fall. 



■n,.-. ; li..r.. ,in<l .-111^ iir.- r.''-..]iirii.iiiii'il l.v Ih.- fiK-tiiry iiiiiriml 

iiiojr: ,.|„, i,.s <>r N<'»- l':ii:.'liiiL.). :itia <-Mli >'•' iiiii.li- ill aliv 

f..iii. :ry. t.y |.:.yiiii: ii mviilly ..C : r.[ a .vtil jkt |..iiii<l on »lt tliiit 
:ir.'i..;. i.-. '.- ihr ti.^-fx ll..\ Aii'Liir ('..in|.iiiiy. of N'lw AllKiny. 1ml. 

r It.iv ;. r,..;.' \- . ■,.. „f l-.n^iklvTi \. Y,. Ii;nv p:.1i-iil...| l|„>„i„ ti..rs 

iiiirl .'.'.|. -i,:x', i„ ri:r I ]. I (!.,.y |,„v.. lavn lis. > ;i sM.T 

!l)>'. <'\'< II'. Til.' '':it> itiiriT- rilllll 'hr <iiii'l^ .':i|l [.rill<-i[1lll> ill llli- 

Miii-ii'ii i f I'iii- for 111,- |1l^.j,.Iill-ril. I'liii'li li..|iU llu' liliil- r>i. 

ll i- I'i^ii'.i,.] i)i:it III.. |.iii...i.>iiMt ,7iii<i' Ilii- liiiiN'r. (■K'li.'ck Hiiil 

111,, liiiilnr- t.i il ,',|, K.iili..|- „nli,..-. r,.riiis ..f I'lifsulKl 1111,'linrs 

is sii|ii.rii>i' til lliiiM' in i-niiiiiinii um', Tlu'y iiiiisl iiul lie u*sl, hu«r- 
evcr, witLuut u IIuvH-w fruiu tlic jmluaUHM. 



FIBE-PBOOF OONSTEUCTION FOB BUIIJ)INQS. 467 



CHAPTER XXV. 

MATERIALS AND METHODS OF FIRE-PROOF 
CONSTRUCTION FOR BUIIiDINGS. 

The terai fire-proof is applied to various kinds of buildings, 
sometimes correctly, but more often incorrectly. 

The buildings most generally referred to by this term may be 
classed as follows : 

1st. Those in which all the structural parts, both on the interior 
and exterior, are of non-combustible materials carefully protected 
from the action of fire by fire-resisting materials. (See also quota- 
tion from Chicago building ordinance, page 485.) 

2d. Those built on the so-called •* mill principle," and protected 
by fire-proof material. 

3d. Those built in the usual manner with wooden construction, 
and protected by fire-proof material. Of these classes the first is 
the only one that is considered by experts to be absolutely impreg- 
nable to the effects of fire. 

MATERIALS. 

Various materials have been introduced for the purpose of mak- 
ing incombustible buildings, and for the purpose of fire-proof pro- 
tection of other materials in structural parts of buildings, all more 
or less effective. Experience, however, has shown that the only 
materials upon which it is safe to rely are the products of clay, 
some concretes, and lime mortar under certain conditions. Plaster 
blocks have been found to be useless to withstand the effects of fire, 
moisture, and frost. The lime of Teil was for several years used in 
the manufacture of fire proof material, but to the best knowledge 
of the writer this has been discarded. All methods of fire-proofing 
by the use of exposed iron in any form are also acknowledged to be 
ineflicient. Of all materials, burnt clay has the most numerous 
applications in incomI)ustible building. It stands preeminently 
first as the most efiicieni fire-proof material in all departments of 
building, and especially so for interior filling of floors and parti- 
tions. For this it is used in hollow tiles of two general kinds. 
Tliey are known by several different names : the one by such as 
porous terra-cotta, terra cotta lumber, cellular pottery, porous til- 



40^ FIKE-PROOF COXSTRrrTTOX FOR BriLDIX"08. 

in^, otr-. ; the other by fire-clay tile, Iiollow pottery, hard tile, terra- 
coUa, <lonso tiliiip:, etc For convenience, the first is herein referred 
to as porous tiling, and the second as di.*nsL' tiling Tht» terms 
" hollow tiling "' ahd "fireproof tiling" will Ikj usid when Ixiih 
are r< IVrred to in ii general way. They will 'oe descrilxnl in Ihoir 
order. 

l*or<Mis Tiliii$4:. — A substance formed by mixing sawdust with 
pun' clay and submitting it. to nn intense heat, by tho action oi 
which the siwdust is destroyed. leaving the material Jijjhl an>l 
poroiw. like pumice-stone. When prop«*rly mailc it will not cnkk 
or br. ak Irom unerjual heating, or from l»eing suildenly cooUmI l)y 
water wIitMi in a heated condition. It can also bo cut with a 9av 
or edire tools, and nails or screws may be ea>ily driven into it for 
si-.urin;_' interior finish, slates, tiles, etc. For the successful r»'si>t- 
ancc (M li< at, and as a non-c(mdu(-tor. thei*o is no building nmterial 
«<iua! to it. A"^ a casing, covering, or lining for the protection of 
(•tiicr material, it is to 1h.» preierrcfl alx)ve every oihor material. 

li shnul.l bi* manuiactun'd from touixh. plastic clays. A small 
jHTi-i-ntaLTf of lire-clay mixed in is«h'sirable but not essential. 

Till- {)i'oporlioii of sjiwdusl .should be from forty to sixty per 
cent., jic- onling to toughness of clay use<l. ('are is nH|uin>tl in 
m.-iiiur.-K-i ire that the work of idxing, drying, and burning be 
i)i<ii'<Mi:.-|il\ •joMi*. The bui'uing should be done in down-iiniuglit 
kii:i> ii\ I .iek process. Tin* prcxhiet should Ihj compact, tmigh, 
aii'i ii.!;- :. riniring when struck wiih metal. Pixjrly mixed, pn*s«!iHl, 
nr Ii .riie j lil--^, nr tiles from >luiri or sjindy clays, present a nigged, 
^r)^:. .iifl eruiiibly apjiearaiicc, and mv nut desirable. 

.\ ;i:- hPHii fllliui; and protecting material should lie substantial 
a- \\\: .- iti(-<i]riliusiible. In a building made of alisolutely inctini- 
li!i-:iM-- i:i;iieria!< it isnf the first i!n|)ortance t!iat the firi»-pnH»flng 
If- .iM- i<- \vith<ia!id niiii^h usage, for, in th«' event of Hn». daniat:!' 
to til -tnietural parts will lie serious if thefire-))rix)fingisdisliMi>:iHl, 
Hi'ls . I jiMvt. .ir yields to the aetii;:i of fire, or of waliT when a fire 
i> in prii;^' :•■>■■;. or if it- cullajises under sudileii liwiiN, jars, or imfiiu->. 
.-ilih<<u.:li !'-i(> nriier<;<| ii<. If may not burn a! all. In siicti huiM- 
ii;_- iji.liiiiMu' '!ualiii'>:. b-ith <if the Hre-pr'Mif material and its (•••n- 
-;■ . rii'ii. :■.'•>• as vit.il aii'l import mt as the incombustibility «if the 
Hi. I* rial, in the eveu' t'f !i"e. the fir>l ilaiiLT'-r i< fii»m the (-olla{x«* 
of !-. m.i'-ri.ii and imi frum its cundiusiioii. l'!x[H'rii'ner has 
sh-i 'I •■;.!' I'r*- j»r«M»| till- if p'a-iii- i-lav'*, w iieu jMirou.s an* iiiore 
enduriii.' ih.-tn den>'e tilcN, i>\cii it I In* deiiNC ii|i-> Ih- tif (larl «ir ail 
fire>-!'.iy. iNinais tiles are tough and ila.<tie. Men.<«e tiles are hanl 
and uriitle The most esM-ntial reipti^itcs of a fire pnKif filliugand 



FIRE-PROOF CONSTRUCTION FOR BUILDINGS. 469 

protecting material are these : It should be tough, not easily shat- 
tered by impact ; non-expansive, not easily cracked by heating or 
cooling ; slightly elastic, yielding gradually to excessive loads, but 
not breaking or collapsing ; compact and hard burned, but not 
dense ; strong enough, but not of excessive crushing strength. 
Blocks should bo light weight by being porous, but not by having 
thin shell and webs ; should be built in between beams by such 
metiiods as bring all parts of the tiles into position to do the great- 
est service, whereby n. structural eflBciency equal to the efficiency of 
the material is obtained. These requirements are very fully met 
by properly made and properly built-in porous tiling. Shells of 
porous tiles should be from seven -eighths to one inch thick, and 
webs from three-quarters to seven -eighths, according to size of 
hollows. 

Dense Tiling;, — Next to porous tiling as a fire-resisting mate- 
rial must be placed dense tiling, also a product of clay. It is made 
into hollow tiles of much the same shape and size as porous tiling. 
A variety of clays are used. Most manufacturers, though not all, 
use more or less fire-clay, and combine with it potter's clay, plastic 
clays, or tough brick clays. It is very dense, and possesses high 
crushing strength. In outer walls exposed to weather, required to 
be light, it is very desirable. Some manufacturers furnish it with 
a semi-glazed surface for outer walls of buildings. For such use it 
has great durability, and effectually stops moisture. In using dense 
tiling for fire- proof filling, care should be taken that the tiles are 
free from cracks, and sound and hard burnt. 

In the earlier days of fire proof construction dense tiling seemed 
to supply the wants very well, but in later years the improvements 
in the manufacture of porous tiling have resulted in the displace- 
ment of dense tiling to a considerable extent. 

Concrete. — Concrete made of Portland cement mixed with 
broken pieces of burnt fire-clay, broken bricks or tiles, burnt 
ballast or slag, and clear sand, is said to resist an intense heat suc- 
cessfully. It is recommended for fire-proof construction by English 
writers, and concrete construction has been largely used in Cali- 
fornia on account of its fire-proof qualities. 

Thaddeus Hyatt, who invented the process of combining iron 
and concrete so as to resist transverse strains, describes a remarka- 
bly severe test by both fire and water, of concrete construction, in 
a work published by him. entitled, Portland Cement Concrete Com- 
bined mth Iron as a Building Material. The concrete was heavily 
loaded and heated red-hot on the under side, when a stream of 
water was thrown against it for a period of fifteen minutes, and 



47l» FiI{i:-JM:()OI' CON'STKrCTlOX FOR uriLDixos. 

the stren«^lli (jr (lumbilitv of tin* (Mincrote nunaiiKMl unuircottHniv 

tin- tl'>t. 

l*ias(or, or IjIIIU* Mortar, wIhmi dirccily appliei I in brick 
or lilr. will witlistaml llio acliiuior hotli lire iiiid "Mhr; ;.lso \»!nu 
«j»I>Iie«l to tlir suitncc of j)l!inks ami tii!il»i'r> l>y im-juis oi win.- lalli- 
illu^ ]ir(A'icl('(l ii lill.s all thr spaci* l«!t\\'('rii lin- wiir ainl ilu' tiiiiiicr. 
JMa>t«':- oil win- Jatli, appliiMl to a ci'ilin.r "ii tin umh-r >i«lf uf 
\v«>(».lcn joisi s])a('('(l 1:2 or 10 inches on renin's, will sue(vs.-riiily 
ri "-i.-xi an, <»r(!inarv fire, 1 ml is lial)h' to Ik- dania'Ti'il l>v \v;'.ii-r. 
PlaNi.'r Itlocks are not siiilal)le as a tire-proof material. In usiiiir 
linn- [»la>tir ri>r fire-ju'oof proti-etion. il slionM not tM)niain any 
j)l.i>ii'i' of Paris. 

l>ri<'k and St oiio.— Common brick will wiihtJtunW a u'nat 
anii'iirit of Ileal wiiliont malerial (lania«;e, tlion^'li mil in so j^real a 
(ie;;n'i- a- liir brick, jiorous terra-c<itta, ami lire clay tile. S«»nif 
>:iml>i<'nr> «io imi appear to be mncli aifccied by heal, csin'ciiJiy 
ijii.^f c' niaininLT «'«>ii>i«leral)le iron. Marl)le, limcsiono, an«l .i;raiiitr 
biiiiiiH- .•.»iiii)j.t»M\ desir«»ve<l under till- ai-lion «:f inien*^* heat and 
water, and .-liniiid luu be useil in pla<'es when I In- stability <il" the 
biiildiiiLr \\oiild Ix' endan*;* re<l by ilielr demoliiion. Terraroiia is 
undt'wbi.'dly the b«si iin-proof material I'ttr ilie i-xieri<»r <leeoratinn 
<»l biilldiii:,'^. 

MKTIIODS (H-' lOXSTUrCTloN. 

]. niilMiii'^'s (*oiistrti(*t<Mi ol* liiroiiilMistihh* Mat€'- 
riaK projirrly l*rot ortod.— 'i'he mellnnis of construe: iin; 

iii< ir-'H I l-iiildJMu^ liavr been Lrr»'ally improved during: tiic past li-w 
. ■.!!>. alii: i-t •■•iinplilcly !e\ olul ionizJn;; t lie old inetlaNis o] build 
iiij. Tl.- id«al liri priM)f buildin;: should Im« con^tructid cuiinly 
*'\ iidii iir >i-'.l. drf>-M'd <in the oulsidc with lirick. sanil>t<»ni-. or 
lci:a <■■ ta. and j.roU;cle(l on the inside by Iiri'-pnK)f inatt-rials. 

Til- n.o-i apjirovj'd metiuxl of c«inslrui-iin^ hijrh buililini;s is lo 
 ■■liiM tlif loundation (U the i>olal«-d pii-r >y^ti-m. and oii top f)f 
liicNi- pifis place >ieel nr w rouirlil-iron cobimi; cMendiui; t)in>UL;h 
(^■■i!itir< i- '-.dii lit till- bnihfiinr. bdilinn i he n .t-ide walls and in 
*ihi' iiiii ijiii" of ihe b^ildiiiir Al ca<!i lln-ir hvel ifoii Lrivdi-r> an* 
bolted I"'!,.- ('(.liimn-. anil llh* whnle sv«.tem braceil bv diair'Hial 
tie> in tin- ihn-kiii-N'* nf the Ihinr. 'rhu< i< f.irmed an iron «•!• ^teil 
ca;:c rt>liiiL' ' uiirely mi t! .- foundaliitn piei"s aiiu 'hich. sn l«»ii;: as 
ii can bi-kij-t irDiii lIn- a>-iioij of heal and iiMii-tiire, will endure 
fi»n-vi-r. 'I'll'- ••lii'.iile «;dl- are then biiili nf lirick. shine, nr lerni- 
cutla. cn<lo>iii.; il:e biiiliiiuLT and proiectiii^ lis contents fruin thv 



FIRE-PROOF CONSTRUCTION FOR BUILDINGS. 471 

weather. Only sufficient, strength is required in this wall to with- 
stand its own weight, and if any of it should be destroyed it would 
not cause the destruction of the building. The interior columns 
should be encased by porous terra-cutta or fire-clay tiles, finished 
in plaster or Keene's cement, or Portland cement if preferred, and 
the floors should be constructed of iron beams filled in between 
with tile arches, the bottom and top of the beams being carefully 
pi'otected by the same material. 

All partitions for dividing the various floors into rooms, cor- 
ridoi-s, etc., should be built of fire- proof partition tile, or hollow 
bricks, and the roof and upper ceiling should also be constructed 
of the same material, supported by iron-work. In such a building 
it is impossible for the construction of the building to be en- 
dangered by either a local fire or by a conflagration, though the 
inside finish may be entirely consumed. It is possible, however, 
to finish the building in such a way that there will be but little 
wood to consume, which could be easily replaced ; also, by provid- 
ing fire-doors to the openings in the fire-proof partitions, any fire 
originating in the building can be confined to the part of the build- 
ing in which it started. 

DETAILS OF OONSTRUOTION. 

Floors. — The various approved methods of constructing fire- 
proof floors have been described in Chapter XXIII. 

Iron Columns. — The destruction of iron columns by in- 
cipient fires has been the common cause of the loss of vast amounts 
of property ever since iron columns have been useil. Their destruc- 
tion during fires, in buildings supposed to be flre-proof and in 
which incombustible materials of construction have been used, 
has shown the necessity for protecting them from the effects of 
intense heat under all circumstances. These disastrous effects 
have been intensified by the sudden throwing of cold water upon 
the heated columns, causing them to bend suddenly by contraction 
on the side upon which water is thrown, and consequently to break 
with ordinary loads. The expansion which occurs in iron columns 
before they have bec^n materially weakened by heat is another cle- 
ment of weakness. The first result in such cases is to raise the 
floors or walls ; and inasmuch as the strain required to raise them 
is much greater than that needed to hold them, the work to be done 
by the columns is much greater under such circumstances. 

The almost universal practice at the present day is to use 
wrought-iron and steel posts for the interior supports, and protect 



FIBE-PBOOP CONBTKUOTION FOE BUILDISOB. 4T3a 

the floors, the aams material vill generally be beat lor protecting 
tbo girders. Fig. 6e shows several wsjs in which this maj be 



Fib. Oa, -'Two-foot Coluhh CovuuHae ur tdb Fab«t BinLDDre. 



© 



Pio. ».— Section or CisT-raON Comm 



Fib. tc.—Vax-SBoor SountB Cots 



A'i'lh FIRE-PKOOF t'OXSTUUCTION FOU BUILDINGS. 




e:5.¥5".W VS* Toa\v\vo^'Yv\v-»«.\oaa****^ 



Partitious. 

The method at present most in favor for constructing fire-proof 
partitions apiieai-s to i)e by tlio use of hollow blocks or tiles, of 
either dense or |)or()U£> terracotta. Partitions arc sometiinct; built 

by using 4-inch isteel 
beams for studding, and 
fastening metal lathing 
on each side ; but this is 
not as practical a iMir- 
tition as one made of 
torra-cotta blocks. Par- 
titions constructed of terra-cotta blocks, either donso or porous, 
have many vMluable features other than their tire-proof qualities. 

They have the greatest degree of strength combined with light- 
ness. They are entirely vermin pro«jf, and do not reatlily transmit 
cold. heat, or sound. Wiien dense tile are used, courses of porous 
tile should be placed op|)osite the l^ase or any wood mouldings, as 
they will receive and hold the nails while the dense tile are apt to 
be ))rokon by the nails. Several styles of partition blocks are mana- 
factiired. of both dense and porous terra-cotta. some with grooved 
or (love tailed surfaces, and others with plain surfaces. 

Tiie weiu'ht of ))artition tile per square foot will average about 
as follows : 



WF.IGIIT PER SyrARK FOOT OP TERRA-COTTA 

PARTITION BLOCKS. 



Den-sf T«Tra-cotra. 



\Vt. per Ml 

flH>t, IbK. 



:j inches thick 13 

4 '• •' j 17 

't ..... «w 

r»  •• ! -JG 

7 *■ •• i w^ 

8 •• •• ' :« 



i: 



Porous Ti'ira-cotta. 



Wt. per nq. 
foot, IImi. 



3 inches thick 

4 m h  • 

•   • I 

5 " •• ' 

« •• •• ] 

7 ** " ! 

« •• •• 



12 
17 
21 
26 
82 
38 



Til ill rin»-pr<H>l* Partitions.- Tn a considoniblt* extent in 
finicf l)uililiiii;s. sonic hotels and apartment hi>uscs, iiartitioQS are 
n>i'v uscil which flni^ih fninioneatid thn*e-«|uarter inches to two and 
thri'c-<|iiar-tcr iiiche<« in total thicknesb. There are a number of dif- 
ferent dcvii-es and methods, all accomplishing substantially Um 



FIRE-PROOF CONSTRUCTION FOR BUILDINGS. 472c 

same results. Prominent among them are the expanded metal 
companies, using cliannel bars or flat bars and expanded metal 
lathing' ; the Lee Fire Proof Construction Company, using a core 
of one-inch tile, and burying Lee tension rods ^similar to those 
used in the flooi-s) in thv; plastering on each side ; the Doring Fire- 
proofing Company, using rods, bars or channels, and burlaps ; and 
the two-inch porous terra-cotta [)artition made by Henry Maurer & 
Son. The expanded metal system requires a scratch coat of plaster- 
ing on one side, the usual brown coat work on each side, and the 
usual finish coat on each side — altogether, five coats for the com- 
pleted partition. The Lee and Maurer systems require no scratch 
coat, but the usual brown coating on each side, as done with hard- 
setting mortar, and the finishing coats. The Doring requires a 
scratch coat on each side, and then the usual brown and finishing 
coats. 

An essential thing for all thin partiti<ms is that the plastering 
be of hard-setting mortar, such as Acme Cement, King's Windsor, 
Adamant, Rock Wall, and many others. The walls largely acquire 
their stififness from the solidity of the plastering ; hence the firmer 
and harder the plastering, the more substantial the walls. 

Roofs. — For mansard roofs the most economical method of 
constniction is by using I-beams, set 5 to 7 feet apart, and filled in 
between with 3-inch hollow partition tile, provision for nailing 
slate being made by attaching 1^ x 2 inch wood strips to the outer 
face of the tile, the strips being set at the proper distances a[)art to 
receive the slate, the spaces between the strips being then plastered 
flush and smooth with cement mortar. In case of a severe confla- 
gration the slate would probably be destroyed, and the wooden 
strips might be consumed, but the damage could go no farther. In 
place of partition tile porous terra-cotta bricks or blocks may be 
usi»d for filling bc^tween the I-beams. For roofs where the pitch is 
not over 45 defjrci^s, 8x3 inch T-irons, set 10 inches between cen- 
tres, and filled in with slabs of porous terra-cotta, makes a very 
desirable roof. If slales are used they may be nailed directly into 
the tiles, or if it is (h^sired to use hollow tile, strips of wood may l)e 
nailed to the tile for receiving the slate, and the spaces b'.^tween the 
strips filled in with cement. This method may also be used for 
flat roofs. The b(?st construction for flat roofs, however, is to 
build the roof like the floors, with tile arches between ircm beams. 
The arches should then be covered with Portland cement, or rock 
asphalt, flashed around the edges with copper, and then tiled with 
terra-cotta tile, about 0x8 inches, and | inch thick. This makes 
a durable and substantial roof, perfectly water-tight and absolutely 



FIRE-PKOOF CONSTRUCTION FOR BUILDINGS. 473 

proof against fire. Composition, cement, and asphalt have a 
natural affinity for the tile, and adhere readily to it without the 
use of nails or fastenings. If the roof is exposed on the under side, 
it can be plastered and finished the same as the under side of a 
floor. 

TriisscM. — Where steel trusses are used to support the roof or 
several stories of a building, it is very important that they be pro- 
tected not only from heat sufficient to warp them, but so that they 
will not expand sufficient to affect the vertical position of the col- 
umns by which they are supported. 

The following description of the covering of the trusses in the 
new Tremont Temple, Boston, furnishes a good illustration, of the 
way in which this should be accomplished : 

*' The steel girders were first placed in terracotta blocks, on all 
sides and below, these blocks being then strapped with iron all 
around the girders, and upon this was stretched expanded metal 
lathing, covered with a heavy coat of Windsor cement ; over this 
comes iron furring, which receives a second layer of expanded 
metal lath, the latter, in turn, receiving the finished plaster. There 
is. consequently, in this arrangement for fire protection, first a dead 
air space, then a layer of terracotta, a Windsor cement covering, 
another dead air space, and finally the external Windsor cement." 

Ceilings. — In office buildings having a flat roof, there is gener- 
ally an air space, or attic, between the roof and ceiling of upper 
story, ranging from three to five feet in height. This space is 
often utilized for running pipes, wires, etc. Generally the ceiling 
is constructed in the same way as the floors, with the difference 
that lighter beams and filling are used. 

It sometimes occurs that a suspended ceiling is desirable under 
pitch roofs, to form a finish for the upper story, and protect the 
roof construction. If only the weight of the ceilmg itself is to be 
provided for, such a ceiling can be constructed at least expense by 
u.-^ing wire or expanded -metal lathing stretched over light T's or 
angles, suspended from the roof construction. The angles or T's 
may be plac^ed four or five feet apart, and tension rods fastened to 
and under them, to support the lathing ; such a ceiling would 
weigh only about twelve pounds per square foot. Plaster boards or 
thin porous terra cotta blocks, placed between T bars, also make a 
light ceiling, and a goo<l ground for the plaster. 

Walls.— If it is desired to further outside walls they should in 
DO case be strapped with wood, but should be furred or lined with 
porous terracotta or fire-clay linings, as shown in Fig. 6. on which 
the plastering may be applied. This not only affords a protection 



FIBB-PBOOF C NSTBUCTION FOB BtJILDINim 476 

blocka, the same as described under Class 1. Id this method of 
buitding it is also neciessary to protect tiie upper side of ttiu floor 
planii. olticrwiKc tiiu fire would burn tliroug-li Cri>ui tlie top. This 
is best done either by laying; an inch uC mortar between it and the 
upper floor, or by using liollow tiln blocks laid on top of the plaitlt- 
ing, with strips between lor nailing the upper flooring to. 

The flrst method is much the cheapest, and as fire is very slow in 
attacking a floor, suuh a construction would probably resist the ac- 
tion of the fire as long as would the other portions of the bnildlng. 
The first point attacked by any Are is the ceiling of the room or 
story iu which it onginates, and every precaution must be taken to 



Pio. 7— Mill Cohhtbijction, Protkcteu by Plabtbr on Wire LiTHrao. 

make the ceiling imprepiable. Espoeial pains must be taken to see 
that all angles and junction of L-eilings with the walls and parti- 
tions are carefully protected, so that there may be no places in 
which the flre may work its way through the protection back of 
the plastering. 

Partitions. — The partitions in this class of buildings shonid 
be constfueted either of hollow tile partition blocks or bricks, as in 
Class 1, or they may bo built of 3-inch plank, tongued and grooved, 
and covered both skies with wire lathing from floor to ceibng, and 
back of the door jambs. 

The Walls should either be plastered directly on the brick- 
work, or furred with hollow tile blocks, as previously described. 
VTbea carefully built, a building of this kind will be practically 



PIBE-PROOF CONSTRUCTION FOR BUILDINGS. 477 

Comigrated-wire Lathingr consists of flat sheets of 
double-twist warp-lath, with corrugations ^ of an inch deep 
running lengthwise at intervals of 6 inches. These sheets are 
made 8x8 feet in size, and applied directly to the under side of 
the floor timbers, to partitions, or brick walls, and fastened with 
staples. The object of the corrugation is to afford space for the 
mortar to clinch behind the lath, and at the same time do away 
with furring strips. The corrugations alto strengthen the lathing. 
This form of lathing, however, is not as desirable as those fol- 
lowing. 

Stiffened Wire Lathing. — The Clinton stiffened wire lath 
has corrugated steel furring strips attached every 8 inches cross- 
wise of the fabric, by means of metnl clips. These strips constitute 
the furring, and the lath is applied directly to the under side of the 
floors or to brick walls, etc. This lath is made in 32-inch and 36- 
inch widths, and comes in 100 yard rolls. 

The New Jersey Wire Cloth Co. also make a stiffened wire lathing 
by weaving into the ordinary 'wire cloth V-shaped strips of No. 24 
sheet iron every 7.V inches. This is an excellent lath About the 
only difference between it and the Clinton cloth is that the bars in 
the latter are attached to the cloth instead of being woven in. 

Hammond's Metal Furring*. — A combination of shoet- 
metal bearings with steel furring rods, on which ordinary wire cloth 
is applied, makes one of tlio best fire-proof ceilings. By means of 
this furring the plaster may be kept an inch from the bottom of the 
timbers, thus allowing a free circulation of the air over the ceiling. 
It is claimed that t!iis is of importance in connection with fire- 
proofing, and is required by the building ordinance of the city of 
Chicago. The steel wires used for furring are fo small that the 
mortar entirely covers them, thus securely binding the cloth and 
rods together, greatly stiffening the ceiling. This method may be 
applied to any form of construction. 

Slieet-iron Latliing. — A number of styles of sheet-iron 
lathing have been invented and placed on the market, but they are 
objectionable from the fact that, in case of fire, the heat expands 
the iron and contnicls the mortar, so that the latter becomes sepa- 
rated and f:ills off. Even without considering its fire-proof quali- 
ties, sheet-iron latliing is not desirable, as it is difficult to get a 
good clinch on the mortar, so as to securely hold it in place. In 
the wire cloth, the amount of metal in the strands of wire is so 
sniall, and it also becomes so well l)edded in the mortar, that the 
action of intense heat does not affect it, and it has been practically 
demonstrated, both by actual fires in buildings and by fire tests, 



FIBE-PBOOF OOS&TRUUnoS FOR BUILDINGS. 4 79 

elftborate decoration is to be applied, as it affords a much better sur- 
face than any other material. 

The upper surface of the floor must also be protected, either by 
putting an inch of mortar between the under and upper floor board- 
iug, or by filling in between the joist with fire-clay bridging tilo, or 
by brick nogging and covering with cement mortar, on top of which 
the upper floor is laid. As in the previous class, especial pains must 
be taken to see that all corners and angles are well protected. 

Roof. — If the building has a flat roof it should be protected the 
same as the floors, substituting for the upper floor boards, composi- 
tion roofing covered with flat tiles laid in cement. For steep roofs, 
efficient fire-proofing becomes a difficult problem. In the opinion 
of the author no building, five stories high or over, should be cov- 
ered with a pitch roof constructed of wood ; but if such a roof is 
used, it can be protected for a time by covering the roof boarding 
with porous tena-cotta blocks, aoout 15 inches square and 1^ 
inches thick, and nailing the slate directly to them, bedding the 
slate in cement as it is laid ; or the tile may be nailed to tiie 
rafters without boarding. For protection on the under side, if the 
attic space is finished, the under side of the rafters may be pro- 
tected as described for ceilings ; or, if the roof space is unfinished 
and more or less filled with trusses or other supports, a thoroughly 
fire-proof ceiling beneath, without any openings, would probably be 
as good a protection as could be obtained. The walls and partitions 
should be treated as in Class 2. 

Complete information regarding the particidar forms and sizes of 
the various fire-proof blocks inanufactur(Ml may be obtained by ad- 
dressing The Raritan Hollow and Porous Brick Co, , of New York 
City ; The Wight Fire-proofing Co., of Chicago or New York ; 
2'he Pioneer Fire-proof Construction Co., of Chicago ; Henry 
}Iaurer d; Son, New Tork City ; 2 7ie Lee Fire-proof Construction 
Co . N(r.v York ; and llie t^taten Island Terra Cotta Lumber Co., 
New York ( ity. 

Details, Finish, etc. 

After tlie constructive portions of the building are completed and 
the building is plastered, there are yet many details to be arranged, 
so as to afford the least possible material for a fire, and also com- 
bine strength, durability, and often elegance. 

Stairs.— The most important of these are the stairs, which, owing 
to the necessity of their being located in a sort of well or shaft, are 
always fiercely attacke<l by a fire. To construct a thoroughly fire- 
proof stair is nither a difficult undertaking. Many architects con- 
tent themselves by merely making the strings and risers of wrought 



4S() FTKK-PnOOF OOXSTKUCTION FOR BUILDINGS. 

or cMst-iron, and Wm'. treads of slate, marble, or wood. Siic-li stnir: 
?!'(• iiTiil«.ul)tcdlv UiY better than the ordinarv wooden stairs, but 
liicy ni't' iiKMcly iiUM)mluisiibl('. In biiildin^i: such stairs \vro!i|Erht 
iron string- sii( uld hi' ust'd with slate tn-ads ami iron ris«TS 
I'wi iw in .h I lianiirl l>ars inak<' excellent strings, turning llie 
ilanL:«'> <'U.. anl i;«)ilinj,^ tlic n>eis to ih-. stem as shown in Fi^. U. 

Tile 1« <t >iair> lor a fir('-])ro()l" biiildiiiL; are tliose built of lirit-k 
DV Portland «i'nu'nt t'oncn-tt', witii at least one end sujijiortrd hy a 
Itii.i; Wall. If coinM-ctp >tairs an' constructed llic;* should b*- built 
:;'iuari- .-.nd xijid — thai i. , liaviiiLT the same sliapeon the bottom i\-nm 
ihi' loj,. II tin- stairs an- liuill Ix'twciMi two brick walls, as ^iioiilil 
alwa\s :•-■ ihc (•;i>.- in a thi-atn'. thrv will have sullieient strenu'th bv 
ixiiiidinj- ill. in 1 inc!:e< into the hrick wall. If only one enil i*: ^uj^ 
iM'iird l»y M wail, iin- other end can be su}>iM>rte(l by wrou,trhi-in>r 
=iriMj-- i.iiili int.. tiii- (M.neretf. 

ricT. 1 f ^liow> tw<» M-ciions of -i brick stairway. Stairways ^iI:l; 




I.-! :. : 



Fig. 9. 

SECTION OF 

WKO-^v.*H7 il.ON 

STAIRS. 



:■ i!i li- ill the ?i.w Pi-nsii'ii liniiilinj; at NVashincton 
A.i\ ■::.i ■. I ■.:--iiii'-.'il .1^ ;i.i>»«.lut» ly fin'-pniof Ni \f l« 
 .1- Nt >"iii-^. ih-- ainiii'!' U"ulil n-i-ommcnd stairs ntu- 
I  ■' v. . n ..'• ,•;.-.: ;!•■ !i "-iriiiL's. I'n'ifeti'd •*!» I !;t' uiiii' I 
.■   :• :'. - ;■.■:;.':■•.:.-;■ ni-io- !a lii . ami with >l.iU- 



...■'.  »; - _; 



-!■. \ : :;■ w :!i tili-  r j! i-»Ti-.- ■.:! Ill^•.!d^ 
li' II I '1 ;;«!-' ha\<' Jw.M t'"UM'l u!nli-»ir- 



!"■ 



I".,. 



":'■■ ^ \. •■". «.; w :,i :|"::i \ ;; -."iimtl: 

1. :l III' !- ;!  an- "-j'.i'-i- bridri'i r. «.o 'ii.-it 
i- I'll- .I'- \ .!'. «•■ .Id !•<■ far InMit ihiiH 
   '.■ .■■'.- :■ . 

! -•■ i: ■■. !h-.  :_-!i xn h a ^-tairf. Thf •ilrinir* 
'.'.•■ ! i . : ;■-' . Mi'ha^ •■•■i It >>rnaiueutatU)n 



<ii';tMilr >lair>. li: ii..iiiv •: !i.> (i -vi rniiii-.ii buililin;r'< tbr 



FIEtE-PBOOF CONSTRUCTION FOR BQILDINOS. 481 

stairs ftre constracted all of granite, a seotion throni^ the steps be- 
ing like that shown in Fig. 12. One end at the steps is boilt into  
wall, and the other depeods upon the edu^ol tlte steps for support 



Granite and most other natural euin 
stroyed b; the action of fire and water, s 
my be coiisidered as fire-proof. 

As to tho stair railing, if bnckstMrs are used, st 



- - . ' 



.tf 






- \^ 



:•.♦.»:•> 



♦\ r.- :. 



•• iJ. :. . -' 



•* • B B k • M < 



^ 

y 



Fie,, n. 

F.'-l '-'-ZZt 



:• : 



• : :.;• : « r.»- w,. l»y 
• •-. ' K'\ I'.i it • -.ii 



r. :i~ A 



> . 



'■• I y :>»•-: r - f. 



\ (lit il:i( ion :iii(| l|ol-:iir l'!iir«». 1''.— -".'i'"!!-! \\\ \\\\ 
'■•••■':••••; . !' ! ".■•'••- *• :!•' ••■ '.ikMn that 



I 



•:.:- <!..-" A .. :•• i;ia;f«i iiif.i-r bv slt-aiii or huC 
1 il..- U>f »'M-tl.«t.U fi.r heiitiii;: •>nice> iit dvsicribed 



PIRK-PRX)P OONSTRUCnON FOR BUILDINGS. 483 

In the article on Steam-Heating, under Direct-Indirect Radiation. 
If this method is employed, no hot-air flues will be needed, and 
it will only be necessary to provide for ventilation flues. 

In running iron and lead pipes, etc., in the walls and partitions, 
they should run in channels in the brickwork, and be covered with 




d Fig. 12. 
SECTION OF 
GRANITE STAIRS. 

(SELF SUPPORTING.) 



sheets of boiler iron about three-sixteenths of an inch thick, put up 
with screws, in an iron frame fastened to the brickwork. 

This can be painted as desired, and afford ready access to the 
pipes. 

No pipes should be carried in a wall or partition where they are 
not accessible. 

In finishing around elevator doorways, etc., where considerable 
ornamentation is required, cast-iron, painted in color, can be used 
with good results. Where there is no combustible material, there 
can of course be no fire. 

Cement 







Fig. 13. 
SECTION THRO' DOOR JAMB 

Stand-pipes. — A very important adjunct to every fire-proof 
building is a stand-pipe of 2-inch wrought iron, connected with the 
street main and running up above the roof (if flat), and provided on 
e^ch floor with suitable valves, hose, etc., ready for instant use. 



PIRB-PROOP CONSTRUCTION FOR BUILDINGS. 486 

thousand square feet, without special permission, based upon un- 
usual and satisfactory precautions. 

6. That every building to be erected, which shall be three stories 
high or more, except dwelling houses for one family, and which 
shall cover an area of more than twenty-five hundred square feet, 
should be provided with incombustible staircases, enclosed in brick 
walls, at the rate of one such staircase for every twenty-five hun- 
dred square feet in area of ground covered. 

7. That wooden buildings, erected within eighteen inches of the 
line between the lot on which they stand and the adjoining prop- 
erty, should have the walls next the adjoining property of brick ; 
or when built within three feet of each other, should have the walls 
next to each other built of brick. 

8. That the owner of an estate in which a fire originates should 
be responsible for damage caused by the spread of the fire beyond 
his own estate, if it should be proved that in his building the fore- 
going provisions were not complied with. A certificate from the 
Inspector of Buildings 4hall be considered sufficient evidence of 
such compliance, if the building shall not have been altered since 
the certificate was issued. 

In addition to these general propositions, another series of sug- 
gestions was adopted, providing for proper fire-stops between the 
stringers in wooden stairs, and between all studdings and furrings, 
in the thickness of the floors, and for six inches above ; for car- 
rying brick party-walls, and outside walls adjoining neighboring 
property, above the roof, and for anchoring* wooden floor-beams to 
brick walls in such a way as to prevent the overthrowing of the 
walls in case the beams should be burned oft and fall. 



Chicago Definition of Fire-proof Construction. 

"The term 'Fire-proof Constracfion ' shall apply to all bnildings in which 
all parts that carry weights or resist strains, and also all stairs and all elevator 
enclosnres and their contents, are made entirely of incombustible material, and 
in which all metallic structural members arc protected against the effects of lire 
by coverings of a material which must be entirely incombustible and a slow heat 
conductor. The materials which shall be considered as fulfilling the conditions 
of fire-proof covering are : First, brick ; second, hollow tiles of burnt clay 
applied to the metal in a bed of mortar and constructed in such manner that there 
ehall be two air sp ices of at lea-<t three- fourths of an inch each by the width of 
the metal surface to be covered, within the said clay covering ; third, porous 
terra-cotta which i^hall be at least two inches thick, and shall also be applied 
direcT to the metal in a bed of mortar ; foarih, two layers of plastering: on metal 
lath.*' 



JQA 



WOODEN ROOF-TRUSgKS. 



CHAPTER XXVI. 



WOODEN ROOF-TRUSSES, WITH DETAIIiS.! 



WnK.vKVKR it is rt^uired to roof a hall. room, or ImiMing. where 
the flt'ur ST»an is inon* than tweiitv-tive f*t*t. the roof should be 
siil»l»ortf 1 hy a truss of some fonn. Tlie various forms of trusses 
uslmI tnr tliis ]iuri)ose have e«*rtain ft-atures anil ]>rincip]es In fom- 
luon. (litfcring from those in bridge and floor trusses, which have 




^ 



PlC;R seam CR CEiw So. J S' 



V^ 



— >r 



rLATE 



SPANS UP TC 2-4 -T 



II 



i»tl In L'lnr.i.iiii: tlu-m in on»» rlas'S. r:illi*»l " n^of-tnisses." Xeariy 
all II ■i!-Tn>^»'^ in r!nir«lif>. ;i3! ! li.ill-. n! like rhHRi"!er. an«l Ihe 
l.ii«'. r i:«i"."riiiiM ••! ini-»«>t-s usi- 1 in :ill kin.l'* of hiiildin;;^ an» itm- 
>tr;i<-t<l '.•riiii-ip.tlly iit wi^tid. ^\ith unly iron tii'-rods anil Uilts ; 
aii I. ;i«« \ii>. I li 11 iru>'*i"* an* nt nn»n- inifn'sl to \\\o an'^.it«^-t and 
liiiii !• r r-.ti; irnu tru*"'-'*. ilii-> hn\*' Ui-n nion' ittini^lctely d«*- 
Miii*.- i. .i:\ \ .1 Lin-jiTiT \:»rifT\ ni lorm'S an* irivon than for inm 



1:. 



j.r .; 



t-> ^t. V. 



: ihi -.:i:-. -.:« lr<i'»'i-> »!.■ u: .m- .i ! iir:iu!i i>litfhlly iNil of 
 u \: iM il.i \ .I't- ji'ii.ni i> .:>-ihiT. Till- tiii»'M-i> (huii kiok hmvy tai 

..«.-  : 'ilii :iii.'-%:. tul the rtUtii'ii «.>f lliv \.irio>i» 



WOODEN ROOF-TRUSSES. 



487 



roof-tmsses, which are. discussed in another chapter. In the 
Northern States and Canada, where there are often heavy snow- 
storms, experience has taught that the best form of roof for a 
building, except, perhaps, in large cities, is the A, or pitch roof. 

The inclinations of the roof may vary from twenty-six degrees, 
or six inches to the foot, to sixty degrees, or twenty-one inches to 
the foot, but should not be less than six inches to the foot for 
roofs covered with slate or shingles. For roofs covered with com- 
position roofing, tin, or copper, the inclination may be as little as 
five-eighths of an inch to the foot. 



PaiNOIPAL RAFTER 



ACK RAFTER 




CEIUNQ JOIST SPANS FhOM 2C TO 40 FT, 



The simplest form of pitch roof is that shown in Fig. 1. It con- 
sists simply of two by ten or two by twelve inch rafters, supported 
at their lower ends by the wall-plate, and holding themselves up at 
the top by their own stiffness and strength. A piece of board, 
called the "ridge-plate," is generally placed between the upper 
ends of the rafters, and the rafters ai-e nailed to it. In some locali- 
ties this ridge-piece is not used, but the upper ends of each pair of 
rafters are held together by a piece of board nailed to the side of 
the rafters before they are raised. 

The walls of the building are prevented from being puslu'd out- 
ward by the floor or ceiling beams, which are nailed to the i)late. 
The rafters are placed about two feet, or twenty inches, on centres, 
and the boarding is nailed directly on the rafters. The horizontal 
joists support the attic-floor and the ceiling of the room below. 
Such a roof can only be used, however, when the distance between 
the wall-plates is not more than twenty-four feet ; for with a 
greater span the rafters, unless made extremely heavy, will sag 
very coni^derably. 



i^- 



WOODKN KOOl'-TltrsShS 



Kin^ Post Truss. — WIumiovit wv wish to roof a hiiilrtin^i 
ill wliirh till- wall-iilat«'*< an* iiiort' tlian twniiy-four ftM-t aitait. wv 
iiiU'n; a-lojii sniiH" uM'tlunl for sui»]i«>niiiix tlu" rat'ti'i*s at tlic «"«'ntn'. 
TIm- iiii"li»».l ::«'ii«M\!lly fiiP'loNfl i^liDwii ill Kiir. -) is to iisi- Iras'"!- 
ilk.- ili.i! ^linwn ill ill'" liiruii". ^'i.n-!-! n'miir i\vrl»r t"«-ft apart in ih- 
li-ii:::); nt till- lniiMiiiLT. aiitl on ilii-M' iilar** larm* lifauis, i-alli'il ••imr 



: '. R 



n 



2 « 8 

CE-LIN3 
J C  o T 



Fiu.3 



I.M'." \\i::'i. -:r.;i»i': iln- i-ooi". ur jark-raii«-rs. A^ tin* iIistani-<' from 
!■:. ■; : ' •; :.i :':••• :m\! i«. ni»l L't-m-rallN niun* than six or riirht fffi. 
i!.'- :.ii-..-:.:;:- r*- ni;i\ l-i- nia«li- a** >niall as iwn in'-hr*; hy six iih-hr". 
\\ .i»i! '!.■■ ^'.•:\\i 111 ill'- ti-n>s is innn- than tliirly-foiir fiH-t. t\vi» jijir- 
l;ii> ii.:«:.: in- ]-lai-r.l um «-a«Ii >iilf i»f ihi- iru^N. or at -I and .1. It 
i'. aiv.;.>- i" -:. hi'\\«'\ir. !•» I'hu-i- ihr pnrliiis only ov«'r tht* finl iif a 
LiM-'-. "! :o a jniii}, w ln-n it fan In- so arranu'fil. Tin" «"fihni: of tin 
ri...!M ii«\< !■. I !•> IJM' r«i'*!" i> franp-il witji li:;in juisis su|iiiortiil li\ 




W 



•* ■'!■ iM.-^. 1 ii' -• • < . ".1.^ 'ni-l" '»!ii'"!M ni»! hi* iiiiir 

 •  '■ . I-::' ^"i." . ! i- ■:••:. .1 ! ■-. ■»-;u<-li h\ funr-ini-li 

 !.■■■ ; .' - ■• III! .1* "■.••v.!; \',. l.^. •■. 

• ■:.' ;::- - \- ■■:-::.::■. ;;v. I.. 1. ii i- ilitlimll 

■:■ ■■:-!■■■.» •:.«■;_!; I--' : I.I r •.-!•■ -.u., uiihtiut spliriniZ. 

.1- t.".- I.- ;|m- I>i -• ii!i'!i«>-U ■■? liuilihiiL; up Iht' lii»- 



WOODEN BOOF-TBU88B8. 480 

beam is lo make it of two-Inch plank bolted together, the pieces 
breaking joint, so that no two joints aha!! be oppoait* each other. 
This form ot truss is very rarely used where the timbers may be 
se«n from the room below, ami they are therefore generally left 
 rough. If they were to be planed, and maile a part of the finish ot 
the room below, it would be necessary to use solid tie-beams 
spliced together, or else build the truss of hard pine, of which wood, 
timbers may be obtained fifty or sixty feet long. The form of truss 
sliown in Fig. 2 la the modem form of the old king post truss, 
shown in Fig. 4, which was made wholly of wood, excepting the 
iron straps used to connect the piece* at tlie joints. 

Queen Post Truss. — When the span to be roofed is between 
thirty-five and forty-five 
feet, a truss such as is 
shown in Fig. .i is pref- 
erable, for several rea- 
sons, to the king post 

It consists of a hori- 
zontal straining-beam, 
separating tlie upper 
ends of the principal 
rafters, and a rod at 
each end of the strain- 
ing-beam, leaving a 
large space in the cenr 
tre of the beam clear. 
This is a great advan- 
tage in many eases 
where it la desired to 
uliline the attic for 

This form of trass 
should not be used for 
a si>an of over forty 
feet. For spans from 
forty feet to fifty feet, 
anotlier form of the 
same truss, shown in 
Fig. 6, should be used. 

This ia a very strong form of truss, and leaves considerable clear 
apace in the centre. In tliis truss the principal rafter should be 
made of two pieces, — one running to the top, the other only to 
Qie Btralniag-beam. This gives the greatest economy in construe- 



*. -* 






.« •".. I. 
I •'■_  

•■ \ I 

..I i. 



 :' I ' I 



 : i..«. vi .1" ■.j?':ii.r.u i jVAit-r .omr it 3. It ^huiiiii V 

.. ..'i '.■.i\ ■,.«• -111*-;:-*:. •'" 1 'i"i.N-i iiv.iHini*; !iuir«'!v i|.,,n 

^ . • ■-  '• '« ^ I.'- .i/'.'n*'! "••ti"! .ii'!'. iiiti iiar I ■;"'i>- 

• V . , -.> -....-i: .: >■.•- • *,.«• Ill .■ii..i:i:ii:> n .'..mi :• •: 




'»■' • • -« «••' ^ 



••■'.4i I 



\ - 



■■••!! 



I ' • • • •■ I :: ' 
   . 11" 



V I 



• 1 • >  



• I .. 



-I*... :• unci jiuiui til. 



WOODEN BOOF-TRnsSBS. 491 

1 an enUrged detail of It is shown in Fig. 10. This tniM la 
m the Museum of Fine Arts, St. Louis, Ho., Hessn, Peabod; & 
ams, architects, Boston, Moss, 



)'0F Bpana of from forty to eighty feel, a truss such as is shown In 
;. 11 ts one of thit best tonus to adopt, where a pitch I'oof is 

rhe strutB should be largest towards the centre, and ttie tie-rods 

a. 

rhe main rafter, on the contrary, and the tie-beam, have the 

stMt strain at the joint A. Figs. 12 and 13 show details of 



192 



WOODEN ROOF -TRUSSES, 



The tmsses which have thus far been given are the simplest 
forms of nioilern trusses for spanning ox>enings up to sixty or 

seventy-five feet in width, or 
even gn?ater, wliore it is <li»- 
sired to liave a pitcli roof. 

At the present <lay. how- 
ever, flat roofs are very ex- 
tensively used; and, when i; 
is desired to carry a flat ro«if. 
a different fomi of truss iiill 
be found more economical. 





WOODEN ROOF-TaU88B8. 498 

The form of tnua generally employed for SM roofe is that shown 
in Pigs. 14 and 15. This truss may be adapted to any span from 




twenty to one Imndred feet, b; simply 
changing the height of the truss and the 
number of braces, and proportioning 



the various parta to the strains which 
they carry. Tlie hciglit of the truss be- 
tween the centres of the chords ought 



^^ 



not to be less than one-eighth of the span, and, it possible, should 
he made oiie-seveiitli, as the higlier the truss, tlie less will be the 
strain on the chords.' 

It shoulil be noticeil, that in this truss the braces are Inclined in 
the opposite direction to that in which t);oy are placed in the 



•s r -: 



wrXjI/ZN R«X)r-TEV>SE5. 







F.>'. ]►'■ -:.■ \^- ::-►- l-<t iiiKhr"! of fomiinc th»* joints. A, A, A, 
Ji. /;. li. ••*'■. F;j. 1.' . A!*.ln»i:::ii in.'t '.rn" frt^iumily used in roof- 

•..- F".- -Tia::> ov»-r i"rty f»-»-:. th-^ ii».*-N*am should be made up 

or ]•. i.vk l"^;*.-'l :«.iL:»-tli»T. as sh'.»wii in Ki:;. ;\ unless it is possilile 
•'. }..t". - Ml- ?:t-]i*-aiii in nn*- j-i^-'-^-. This :s a ::iH"1 form of truss for 
*!.»-atr*-. ai-'i larj*- hall-? wli^-n- ih-r»- i> a hi.>rizontal ceiling. 




< 'oiiiif «T-llrjirrs, — li it i^iliNJrri! tn Iiku) thf tniMH at any 
iHiiiit u;lii-: liiaii tli*- rcMtn' uith a rtiui'i'iitralnl Ioa«l, — as, for 
in^tan<-(-. ^ii»pt-n<liii^ a :;all«'ry hy UK-ans uf rv*\a from the roof- 



K)D£N ROOF-TRUSSES. 



495 



trasses, — the trass should have additional braces, called ''counter- 
braces," slanting in the opposite direction to the braces shown. 

These counter-braces need only be used when the truss is unsym- 
metrically loaded. 

Wooden 'I russes with Iron Ties. — In all trusses whera 
the tie-beam of the truss is not horizontal, but higher in the centre 
than at the ends, it is better to substitute an iron tie for the wooden 
tie-beam. 

Fig. 17 shows a form of truss very well suited for the roofs of 
carriage-houses, stables, or any place where it is desired to have 
considerable height in the centre of 
the room, and a ceiling is not desired. 

The horizontal iron rod is fastened 
to the two struts at their ends, and 
the other two rods are fastened only 
at their ends, and merely nm over the 
end of a strut in a groove. The iron 
rods are tightened by means of the 
turn-buckles shown on the drawing. 
Fig. 18 shows a detail of the upi^er joint A, A better way of 
making the joint would be to have an iron box cast to receive the 
end of the rafters, and fasten the ends of the tie. 

Arched Trusses with Iron Tie-Rods. — For buildings 
where it is desired to liave the trusses and roof-timbers show, with 




DETAIL OF JOINT "A"FIQ.17 




no ceiling but that formed by the roof, a very pretty and jjraeeful 
form of truss is obtained by tlie use of arched ribs, either for the 
principal chords of the truss, or for braces. In such trusses an 
iron tie-rod adds to the grace and apparent lightness of the truss, 
and may be very conveniently usfd. Fig. \,) shows a form of truss 
used to support the roof of the Metropolitan Concert Hall, New- 
Tork City, George B. Post, architect. The span of the truss in 



WOODEN ROOF-TRUSSES. 4fl7 

tlie purlins aod rafters, and only carries the load directly 
-cU. It does not assist tlie truss in any way In carrying 

lethod of imp- 



relied form of 
i shall give a 
crlption of the 
:ioii of the ["oof 
supports. A 
he riding-room 
ented by Fig. 

and six Feet 
es long, and ' 
hree feet wide. S 
ce U kept en- U 
ar of posts or 
and the en- -r 
is supported | 
large tnisaes, j 
hieh is shown p 
J2. The root ^ 

the trusses 
either side is 
1 by aiiiailei' 
»ting on tliese 
sses ; but each 



I 



ouilt of exint 
rk. ItH'as<l<>- 
rovide for tlie 

these large an^lies witliout having rods showing In the 
d the method aEiopted is very ingenious. Opposite the 
ta of tlie iron posts wliieh receive the arched ribs ai« oak 



498 



WOODEN ROOF-TUUSSES. 



struts, wliirh are lu'M in \t\iiro by i'-on tii»-l)ars and heavy iron 
Iwaiiis, ^\lli(■h toi^t'tluT form a Imriztmtal truss at eai.-h oinl. Tht*si* 
tw.i trussi-s an* invvnitt'i] fruiu h*'\uii imsln*«l <nii l»y two tluve-iiu'h 
liy <»n«'-inch ti»»-]»ars in viivh siile wall srhuwn in tin* plan (Fig. S\). 




I". ".■•■':.•> iif ;lii- :a<' iii»:i j-'^"- :ir-- Ttfil tt>-ji thiT fiy irmi nuls 

;;■.'.■_ .-. . : :!.«■ :!■•■■:■■.>■ -a 1 ■.• •:.,' ■•! :!.f riMiin. AltiiLTflliiT 

» _ «•:".. ;■•■:  !^ ■■: t I ■. :i i^- "u.i l-.j:* tlin-f Inrltt's liy 

•  i I".  '. ii; i .1 '■ i! '.':>■': :<•!. wlii-ii wniiM Iw 

•■■:.'.•'•■ ■• 11 ^ ;1.:. ■• :: " • -. .m 1 tliP- -fniiriliH hv uiu* 

;._■ i *<■[;••:.* ««t 'li'- i:i"». u; : ■.^1::'.. ,iiul Iiruh'S, np" 

-^i.-'vv:. .:. 1 .:. l'J. li slmiilil Ik- noiii-iti tliai tin- iiprijflits art Imtli 



WOODEN BOOF-TKUSSB8. 49fl 

ota and ties, by having ircui rods throogh tlieir centre holding 

TO riba tofielher. 

, 24 showB a detail, or enlarged view, of tbe Iron skewback 

08t at eacti end of tbe tniss sliown in Fig. 22. 

. 25 shows the nietliod adopted for supporting the roof and 

J of tbe Citj Amiory at Cleveland, O. 




cn-Tiiiiber Trusses. — One of the principal charaeter- 
of the (iolhic style of architeeture is that of mftkhig the 
ural piirtiuiis of the liii) Ming ornamental, ami exposing the 
construction of an ediHce to view; and, as the pointed 
I and steep roofs were developed, the roof-tniss liecame an 
tant feature in tlie ornamentation of the intei'ior of the 
c ehiii'ches. 

!Se trusses were built almost entirely of wood, and generally 
ry heavy timbers, to give the ap|>earancc of great strength. 
i the siinpleat forms of these trusses is shown In Fig. 2ft. As 
w se«D in the figure, the truss is really not much more tikan a 



WOODEN BOOF-THL'SSBR. 



WOOMK UOOT-TBDSSIS. 



WOODEN ROOF-TRUSSES. 608 

Figs. 28-20 repreaent tnuses token from old English churches; 
but the hammer-beam tnus Is also frequeatly used in this country 
U> support the roof of Gothic churches. 



Fig. 30 refweaenta half of one of the trusses in the First Cliurch, 
Uoslon, Uasa., Messrs. Ware & V:in Brunt, architects. The truss 
It hnlshed In black wahiut, and has the effi^tt of being very strong 
and hwrj. Pi^ 81 shows tliu fratiiiiig of Uie saiue truss without 
anj caainc cr falsework. It should be noticed that inside the 



606 WOODKN ROOF- 

tnmed ralumn, at tin; upper pnrt of the tniss (Fig. %l), there h in 
Iron roil (Fitr. ^U) wliich holds up tho joint A,^ 

111 iliis form of ti'iiss tlic outward tlirust of the arch enten tlw 
null jiisL iibov.' till' corliM. A'; mid, ns Uii^ diret^lion of the thnui i» 
ini'liiu'fi only ulNiiit lliirty di'i,Ti'cs from a, vprlical, the lenilrm-v 
wliirli it liuH (u overthrow l.lii! wiill Is not very great, ami may la- 
easily ri'.sisti-d l>y a wall iwi'nty itiches or two feet thick, ru-CLifoR-nl 
by u hultress uii Clic utitsitle. 



I .shimlil In* ROFiin'ly ftuliiHsl 

-li lai'li othiT, ami tlu> wlink 

li'|H'ndi'llit> for extra streiiKlll 

iK'l-work. 

Ill a liiiiiimKr-bnun truw, in 

I'linii of a vault. TnuiM of 

Ivi-liirb bf thittwa-liwk hard |l>* 



WOODEN ROOF-TRUSSES. 



507 



this kind, where there is no bracket under the hammer-beam, are 
not as stable as that shown in Fig. 30. 




Fig. 33 shows a form of truss used in Emmanuel Chiu-cli at Shel- 
bume Falls, Mass., Messrs. Van Brunt & Howe, architects, Boston. 
This truss was probably derived from the hammer-beam truss, and 
possesses an advantage over that truss in that it has in eifect a 
trussed rafter, so that there is no danger of the rafter being broken ; 
and, if the truss is securely bolted together at all its joints, it exerts 
but very little thrust on the walls. The rafters and cross-tie are 
formed of two pieces of timber bolted together, and the small 
upright pieces rim in between them. 

The trusses in the church at Shelbume Falls have the hammer- 
beams carved to represent angels. 



508 



WOODEN ROOF-TllUSSES. 



Fig. 34 shows a form of hammer-beam truss sometimes used in 
wooden chiirclies. The braces Zi/i are carried down nearly to the 
floor, so that no outward thrust is exerted on the walls. 




It is LTrnenilly bettor, however, in wocnlen buildin<;:s, to us»» a 
trii>;s witli a lie-rod: and. if an iron rod i.s used, it will not mar tlu* 
«'tV«'<f of ilie heii,dit of tlie room seriously, if the roof-tnissos an* 
l»la(ed only about eii^bt reet apart, the roof may Ik* ooven*<l with 
two and a iiu!f ineji .spinee jdank laid <lin»etly fnmi one triiH.s to 
tin- »tln'r without th«» intervention of ja<'k-rafters or purlins. The 
planlviiii: <'an tbeti be covered with slate or shin<;les on the ont- 
si«i«', ami "^beatiied within. Ki^'. o4 shows the nM»f eovere*! in this 
\sa\. l'iir!iu«^ an' jMit in, however, thish with the rafters of the 
trij-"^ !•• di\idt' tbe eeilinij into ]>anels. 

y'\L'.. ;'.■' allows a ''•■etiini tbron;^h the nM»f of St. .Ianie.H*s C'hureh^ 
: iii:ii VaiiiiiMifh. Knir 

Tlir Np.iri is tiiirty-ihr(H> feet, and the trusses an* spaced about 
ei;;bl teet apart from <'ent res. 



■wnODEN ROOF-TRUSSES.  6 

The siie of the scantllngB are as follows : — 

Primcipals: Rafters 12 inches x inches. 

Collars 9 " X 9 " 

Ridge 12 " X 5 " 

Purlins 8 " X 5 " 

Cradling 7 " x 21 " 

The roof is coustmcted of Memel limber. 




610 



IllON HOOFS AND ROOF-TKUSSES. 



CHAPTER XXVri. 

IRON ROOFS AND ROOF-TRUSSES, WITH DISTAILS 

OF CONSTRUCTION. 

OwiNc; to the incroasiiifT cost of lumber, and tlie necessity of 
oreriinij buildings as nearly fire-i)roof, and with as little inflaniina- 
ble material in the roof, as possible, it is becoming quite a common 
])racti('(" to roof large and expensive buildings with iron roofs, 
wliich, of course, involves the use of iron roof-trusses: lionce it 
is im])ortant that the architect and progn^ssive builder should liave 
a general idea of th(^ constniction and principles involved in iron 
roof-trusses, and be familiar with the best forms of trusses for 
dilferent spans, conditions of loading, etc. 



^ 



I -Beam. 



o 



^iT 



Deck-Bfam. 



Channel-Bar. 



T-Bar». 





A/ifcM'. -Iron*. 

Flg.l. 



r>eside^ b.'inu: n()n-cond)Ustible, iron roof-triLsses are superior to 
Wooden trusses y.i that they may ho built nuich stronger and 
li'jliter. and are much mon* durable. 

Various forms of trusses have I>e4>n c(>nstruct«Ml to suit differenft 



IBON ROOFS AT7D ROOF-TRUSSES. 511 

conditions of span, load, height, etc., and of these fhe following 
examples have been found to be the best and most economical. 

Before proceeding to describe these various forms of trusses, we 
would call the reader's attention to the sections of beams, angle- 
irons, T and channel bars, shown in Fig. 1. It will frequently be 
necessary to refer to these sections; as they are the principal shapes 
of rolled iron entering into the construction of iron roofs, and it 
is of great importance that an architect or builder be familiar with 
their forms and names. 

For convenience in describing the different forms of iron roofs, 
we shall divide them into the following classes: — 

1st, Truss-roofs with straight rafters, which are simply braced 
frames or girders. 

2d, Bowstring-roofs with curved rafters of small rigidity, and 
with a tie-rod and bracing. 

3d, Arched roofs, in which the rigidity of the curved rafter is 
sufficient to resist the distorting influence of the load without 
additional bracing. 

Trussed Koof s. — For small spans, the most economical and 
simplest form of truss is that represented in Fig. 2. (Owing to the 




LEBANON FURNACE. 

Fig. 2. 

small scale to which it is necessary to draw these figures, we have 
represcmtecl the pieces by a single line, which has been drawn heavy 
for strut-pieces, and light for ties and rods. ) 

This truss was built by the Phoenix Iron Company for the roof 
of a furnace-building. It consists of two straight rafters of chan- 
nel or T bars, two struts supporting the rafters at the centre, a 
main tie-rod, and two inclined ties assisting the tie-rod to support 
the end of the struts. The lines on the top of the truss represent 
the section of a monitor on the roof, which is not a part of the 
truss, but only supported by it. 

One of the great merits of this truss is that it has but ftew pieces 
in compression, viz., the rafters and two struts ; which is a condi- 



512 



IRON ROOFS AND ROOF-TRUSSES. 



tion very desirable in iron trusses, owing to the fact that wronght- 
iron n'sists a tensile strain much better than a compressive one, 
and hence it is more economical to use wrought-iron in the fonn 
of ties than in th<»- form of struts. 

It sliould be borne in mind that for ties, rods or flat bars of iron 
are the most suita])le; whili*. for struts, it is necessary to use soiiip 
form of section that olfers considerable resistance to bending, suol. 
a< a T-iron, or four an^^le-irons riveted to.G:ether in thc^ fomi of a 
i-ross; for wroui^ht-iron stmts always tail by bending or l)uoklin^. 
and not by direct erusbiui^. In Figs. 2-10 the piec<.»s which an? 
struts, or resist a comjm'ssive strain, are drawn with heavy lint's, 
and those pieces which act as ties are drawn with a light line. 




Fig. 3. 

FiiT. *> repn^sents a truss similar to that in Fig. 2, but having two 
struts instead of one, which is more economical where the s[)an is 
o\er fifty-six feet, for the. reason that it allows the rafters to !)»' 
made of liizliter iron. 

F<»r s])ans of from seventy to a hundnnl fi^et, the fonn of tnisa 
sliown in Fiii. 4 has ])een found to be about the most economical 
and >atisfactory in <'very resjuict. 




R  

M W YlII.l.. I'iKJ'NIX IKON U'('i<K>, U<)C'K-I»lJi.M> AllMENAI.. 

Fig. 4. 

"I'll.- i;iii»i> in this truss, for motliTate s])ans, may 1m» T-imu^; 
Mil. I ii.j- l.ir-cr .,|t;ni>. rli:innel-hars and Hie ties and Ntriits may !"■ 
lH.h<-d to I lie \«-nie:iI rili. For very )ar^M> spans. cliannel-l>ars uiu\ 
hf n-t -d. |>l:i<i>d iiiiek ti> liMek, with the ends of the bnicing bars lie- 
tu.-Mi tli.iii. I -beams an- also Used for t lie rafters, but they liave 
th<- niii. i-ijon of not bciniz in a sliajte to ctmnect n^adily with tlu* 
i>; h« t torm- of inm. The llanges of an i-lH'am do not offer so good 
.III I >;)}.. .riiijiitN fur rivet iiii! as do tlio.se of angle and T Irons iLiul 



rr" • 



IRON ROOFS AND ROOF-TRUSSBS. 



513 



channel-bars. The ties are rods of round iron or flat bars; and th« 
struts, commonly T-irons or angle-irons bolted together. 




MASONIC TEMPLB, PHILADSLHIA. 

Fig. 5. 

Another form of truss, shown in Fig. 5, derived from the wooden 
queen post truss, is very commonly used for spans of from sixty to 
a hundred and forty feet. A modification of this truss is shown in 
Fig. 6, in which both struts and ties are inclined, instead of only the 




Fig. 6. 



struts, as in Fig. 5. The truss in Fig. 6 has the advantage that 
the struts are shorter, more nearly perpendicular to the rafters, and 
less strained. 

Bowstriiig-Roofs. — In designing iron roofs, it is sometimes 
desired to vary the ordinary straight pitch roof by using a curved 
laf ter. Two examples of such roofs are shown in Figs. 7 and 8, 




ALTOONA STATION, PENNSYLVANIA RAILROAD. 

Fig. 7. 

which were constructed by the Phoenix Iron Company of Phila- 
delphia. These may be considered as the simplest forms of bow- 

Btrillg-4XX>fB. 

- The prindpal use of the bowstring-roof proper is for roofing 



514 



IRON ROOFS AND ROOF-TRtSSES. 



very largo areas in one span, such as is often desired in railway 
«tations, skating-rinks, riiling-schools, drill-halls^ etc 




B  

MARKET-HOUBE, TWKLFTII AND MARKET STREETS, PHILADELPHIA. 

Fig. 8. 

Fig. n^presents the diagram of a bowstring-truss of a hundred 
and fifty-t liree f(H?t span. The trusses in this particular case are. 
spaced tw(;iity-<)ne feet six inches ai)art. The arched rafter con- 
sists of a wrought-iron deck-beam nine inches deep, with a plate, 
 en inches by an inch and a fourth, riveted to its upper flange. 
Towards tlie springing, this rib was strengthenetl by plates, seven 
in<hes by seven-eighths of an inch, riveted to the deck-beain on each 
side. 




Fig. 9. 

The St lut s ar<' wrought-iron I-beams seven inches deep. The tie- 
roils havoix and a half scpiare inches area, and the diiigonal tension- 
l)ia«<>; ar.' an incli and a fonrth diaun^er. These tnisses art» llxinl 
at one cn«l. and rest on rollers at the other, jHTnutting fn»e exiian- 
sion and contract iun of the iron nnder the varying heat of the sun. 




I 



t>i>) 



\2 — 



Fig. 10. 




Ki:;. 10 shows a similar truss having a si»sin of two hundred and 
twelve feei^ it consist.s of lM>WMtnng principals spMwd iwenty- 



IKON ROOFB AND BOOF-TBU88BB. Slff 

four feet apart. The rlK is one-fifth the span, the tie-rod rising 
seventeen feet In tbe middle aimve the springing, and the curved 
rafter rising forty feet and a halt. The rafter is a flfteen-lncli 
wroughtr-iroii I-I>esin. The tie is a round rod In ibort lengths, 
four inches diameter, thickened at the joints. The tension-bars 
of tlie bracing Are of plate-iron, five inches to three inches in 
width, and flTo-eighths of an inch thicii. The struts are formed 
of bars liaving tbe form of a cross. 

The following table, taken from Unwin's "Wrought-Iron Bridges 
and Hoofs," gives the principal proportions of some notable bow- 
string-trusses, mostly In England: — 

PROPORTIONS OF BOWSTRING-ROOFS. 



For spans much exceeding a hundred and twenty or a hundred 
anil thirty feet the bowstring-trtiss is much the niost economical, 
and advantageous to use. 

Arcbed Hoofs. — These roofs consist of trusses in tlie form 
of an arch, having braced ribs, wliicli possess sufficient I'lgidity in 
themselves to reaial the load upon tliom. The thrust of these large 
ribs, however, has to be provided for, as In the case of masonry 
arciics, either by heavy abutments or by tic-rods. As these trusses 
embrace the most dlfHcult problems of engineering, and are rai-ely 
used, we have thought best not to give any examples of such trusses. 
If any reader should have occasion to visit the Boston and Provi- 
dence Railroad Depot at Boston, lie can there see an admirable 
example of this form of truss. 

> At (prlugtug iweniy-flie Hjuuu lodMet 



516 



IRON ROOFS AND ROOF-TRUSSES. 



Details of Iron Trusses. 

After deciding upon the form of tniss which it will be best to use, 
the fihfi))(' of the iron to form the different mc^mbtTS is a matter to 
he eonsith'reil. There are many practical reasons which make it 
desirable to use certain shapes of iron in constructing iron trusses, 
even tlioui^h those shajM'S may not be the most desirable in rt»j^nl 
to streni^tli; so that a knowledge of the details of iron tnisses is 
requisite for any one who wishes to become a master of building 
construction. 

By far the best way to study the details of construction is to ob- 
serve work aheady l)uilt and that which is in process of construc- 
tion: but tills recjuires considerable tim(», and often the thing one 
wants cannot ho found at hand. The following details of the 
various ways of joining the different members of iron tiusses will 
be found us<>ful. 

Tlu'H' ar«' two general methods of constructing iron triiss<*s. 
Olio is to make all the parts of the truss of combinations of angle- 
irons, channel-bars, and Hat plates, and rivet them together at the 
joints, so that the truss will consist of a frame-work of iron bars all 
riveted toilet her. The other nuahod is to ust» channel-ljars, T-irons, 
I-beams, etc., for the rafters and struts, and ro<ls for the ties, which 
are conneeted at the joints by eyes and pins. 




HEELS. 




f^- '-■-!" 



r . - /// /■/, >/,//. .' ,,///////', 



Fig. 11. 



In tin- lir^i nietlKMl the ties are either made of flat bars or anglo- 
ir<»ii^. 

l-'i-. II -JMtw^ two way^ in whieb the tie-rod is seciiriHl to the 
t...i! Ill I;:, i.iih'in tin- «^irond met hot! of eon>! met lou. .V easting. 
!'•: ii:i;i4 .' -it: •>! "• >-1iim-."' is maili', in whieh the rafter fhs. anil the 
; . iv -. :-..i i.i iIh- "Nhor" by niean-s of an i-ye-eiid and pin: or a 
|.|i;i- ;i :\ 111- III >lt«tl to 4-:irh .side, and the whole re>t on an iron plate. 
< M' i-i<:i!->i- rlii- tic nni^t in either ease consist 4)f two t)ara, one on 
c.ich '^i'b- «»f the shoe. 



IRON ROOFS AND ROOF-TRU8SE8. 



617 



Fig. 12 Illustrates two ways of fastening the upper ends of the 
struts to the rafters. In the first method the casting is made to fit 
inside the strut, and is bolted to the bottom of the rafter. 




STRUT-HEADS. 




Fig. 12. 



Fig. 13 shows the joints at the foot of the struts, as made in the 




STRUT-FEET. 




XB 



Fig. 13. 

second method. The pealcs in either method are seciu-ed by means 
of fish-plates riveted to both rafters (Fig. 14). 




PEAKS. 




Fig. 14. 



Fig. 15 shows the proportions for eyes and screw ends for tension- 



r—> 



TIE-BAR. 




ROD. 







Fig. 16. 



btfB as naed in this method of construction. 



IRON ROOFS AND ROOF-TRUSSES, 



""igs. 16 and 17 show the luannnr of forming the Joints in the 
t methoil of construction. Fig. 16 represenla the joint at the 




Fig. 16. 

 iiiiUii rafler: anil Fig. 17. Ilii- joint n-licrft a rafti-r. 
II, lii'. ami stnil ponic 10H.'(li.'r. Ail tlie pipcps an- 
cii to a pii'i'i' of |)Iiit<'-ir<in. wliieli thus hol<ls thpin 
LI- ofiiiT joints ail- fortiit-ii in a. similar way. ^Vliicli 
.letliiHi of consiriic'tiiiii ili'iii'inl.'* voi'v much on circuiu- 




ri-. i-: l.iii ilii- 1. 



il,-l,r 



v,i 11 



nil nf IhP trtm, Ai»e 

in;.'(lu-sk<-wlwkuf 
I'olU-ii' inlviiHrnHl lu 
nin:: llii> unit, a* lu 
1 rtH.f- ijf xtxty tret 
n Iruii ru;l uni- 



li-nlli i>f a fiHil fur a diiii^i- 

tlfty .ti-»r<-<'!. v.: ami, aa rhla la 

iH'ani-' anil tihIn in a baiMllli( 

< c-litiiaK'. niniiH'nsatlan to tllU 

fur :ill |>iin>u!>M- l^or usiy IM( ipaa. 



:.'.) ill 



IRON ROOFS AND ROOF-TRUSSES. 



519 



the vibration of each wall would then be only* fifteen-thousandths 
of a foot either way from the perpendicular, — a variation so small, 
and so gradually attained, that there is no danger in imposing it 
upon the side-walls by firmly fastening to them each shoe of the 
rafter. Expansion is also provided against by fastening down one 
shoe with wall-bolts, and allowing the other to slide to and fro on 
ihe wall-plate without rollers. 




leiaiiiisji 







Fig. 13. 

After the trusses are up, there are various ways of constructing 
the roof itself. If the roof is to be of slate, it is best to space the 
trusses about seven feet apart, and use light angle-irons for purlins, 
which are spaced from seven to fourteen inches apart, according to 
the size of the slate. On the iron purlins the slate may be laid 
directly, and held down by copper or lead nails clinched around the 




Fig. 19. 



angle-bar; or a netting of wire may be fastened to the purlins, and 
a layer of mortar spread on tliis, in which the slates are bedded. 
When greater intervals are used in spacing rafters, the purlins may 
be light beams fastened on top or against the sides of the principals 



520 



IRON ROOFS AND ROOF- TRUSSES. 



with brackets, allowance always being made for longilurlinal ex 
pansion of the iron by changes of temperature. On these purlins 
an^ fastened wooden jack-rafters, carrying the sheatliing-boards or 
laths, on which the nietalUc or slate covering is laid in the usual 
manner; or sheets of corrugated iron may be fastened from purlin 
to purlni, and the whole roof be entirely composed of iron. 

When tlu^ rafters are si)aced at su(di intei'vals as to cause too 
'much deflexion in the purlins, they may be supported by a light 
beam placed midway betwt;en the raft(TS, and trussed tmnsvei-s^dy 
with i)()sts and rods. These rods pass through the rafters, and have 
bevelled washers, scn^ws, and nuts a< each end for adjustment. IJy 
alternating the trussrs on each sid( of the rafter, and slightly in- 
creasing the length of the purlins above them, leaving all others 
with a little play in the notches, si fficient provision will be made 
for any alteration of length in ihi roof, due to changes of tem- 
perature. 




Fig. 20. 



AVhen \v()()d«»n ])urlins are employed, they may be put between 
tli<' rafters, and held in place by tie-rods on top, and fjistened to tlie 
ralti'is l»y brackets: or hook-head spikes may Ih» driven up into 
the i)iiilin. the head of the spike hooking under the flange of the 
ln'ani, s|»a(inLr-i)iee('s of woo<l being laid on the top of the iM^ani 
fvoui i»inlin to jtuilin. Tin' sheathimi-boards and covering are then 
nailed down on lop of all in the usual manner. 



THEOKY OF ROOF-TRUSSES, 621 



CHAPTER XXVIII. 
THEORT OF ROOF-TRUSSES. 

In this chapter it is proposed to give practical methods for com- 
puting the weight of the roof with its load, and the proportion of 
the tiniss and its various paits. 

The first step in all calculations for roofs is to find the exact load 
''vhich will come upon each truss, and the load at the different joints. 
The load carried by one truss will be equal to the weight of a 
section of the roof of a width equal to the distance between the 
trusses, together with the weight of the greatest load of snow that 
is ever likely to come upon the roof. In warm climates, of course, 
the weight of snow need not be provided for. 

It is a very common practice to assume the maximum weight of 
the roof and its load at from forty to sixty pounds per square foot 
of surface ; but, while this may be suificiently accurate for wooden 
roofs, it would hardly answer for iron roofs, where the cost of the 
iron makes it desirable to use as little material in the truss as will 
enable it to carry the roof with safety, and no more. The weight 
of the roof itself can be easily computed, and a sufficiently accu- 
rate allowance can be made for the weight of the truss ; and, if 
the roof is to be in a climate where snow falls, a proper allow- 
ance must be made for that ; and, lastly, the effect of the wind on 
the roof must also be taken into account. 

Mr. Trautwine says, that within ordinary limits, /or spans not 
exceeding about seventy-Jive feet, and with trusses seven feet apart, 
the total load per square foot, including the truss itself, purlins, 
etc., complete, may be safely taken as follows : — 

Roof covered with corrugated iron, unbearded ... 8 pounds. 

If plastered below the rafters 18 *' 

Roof covered with corrugated iron or boards . . . .11 

If plastered below the rafters 18 

Roof covered with slate, unboarded, as on laths t . . 13 
Roof covered with slate on boards \-\ inches thick . . 10 
Roof covered with slate, if plastered below the rafters .26 " 
Roof covered with shingles on laths 10 " 

If plastered below the rafters, or below tie-beam .20 " 
Roof covered with shingles on J -inch board .... 13 '^ 



n 



THEORY OF R00F-TEU8SE8. 



523 



^nd: hence the resultant of the wind pressure must act in a 
lirection normal (at right angles) to the face of the roof. In this 
iountry the wind seldom blows with a pressure of more than forty 
)Ounds per square foot on a surface at right angles to the direction 
>f the wind ; and it is considered safe to use that number as the 
p*eatest wind pressure. ^ But the pressure on the roof is generally 
nucb less than this, owing to the inclination of the roof. The 
ollowing table gives the normal wind pressure per square foot on 
surfaces inclined at different angles to the horizon, for a horizontal 
wind pressure of forty pounds per square foot. 

NORMAL WIND PRESSURE. 



AN6LB OF BOOF. 


Normal 
pressure. 


Angle of Roof. 


Normal 
pressure. 


Degrees. 


Rise in one 
foot. 


Degrees. 


Rise in one 
foot. 


5 
10 
15 
20 
25 
30 

• 


1 inch. 
2i inches. 
3^ " 

4? " 
5i " 
6i% " 


5.2 lbs. 

9.6 " 
14.0 " 
18.3 " 
22.5 " 
26.5 " 


35 
40 
45 
50 
55 
60 


8f inches. 
10 " 
12 " 
14A " 

m " 

20i " 


30.1 lbs. 
33.4 " 
36.1 " 
38.1 '' 
39.6 '' 
40.0 " 



Until of late years it has been the general custom to add the 
fdnd pressure in with the weight of snow and roof ; and, although 
;hls is evidently not the proper way to do, yet for wooden trusses 
t gives results which are perhaps sufficiently accurate for all prac- 
;ical purposes ; and, if caution is taken to put in extra bracing 
vherever any four-sided figure occurs, this method will answw 
rery well for wooden trusses. For iron trusses, however, the 
(trains in the truss due to the vertical load on the truss, and those 
lue to the wind pressure, should be computed separately, and then 
lombined, to give the maximum strains in the various pieces of the 
russ. It should be borne in mind that a horizontal wind j^ressure 
>f forty pounds per square foot is quite an unconnnon occurrence, 
ind, when it does occur, generally is of short duration ; so that a 
russ which would not withstand this pressure, if applied for a long 



> At the obser^'atory, Bidstoii, Liverpool, the following wind pressures per 
quare foot have been regi8tered. 1868, Feb. 1, 70 pounds; Feb. 22, 65 pounds; 
)ec. 27, 80 pounds. 1870, Sept. 10, 65 pounds; Oct. 13, 65 pounds. 1871, 
imrch 9, 00 pounds. 1S75, Sept. 27, 70 pounds. 1877, Jan. 30, 63 pounds; 
Cof. S8» 68^ poundB.— Ambrican Architect, vol. xv. p. 237. 



5i>4 



THEORY OF ROOB'-TRUSSES. 



time, may possess sufficient elasticity to withstand the strain for 
sliort time without injury. 

In veri/ crjtosrd poMtioun, such as on high hills or mountain: 
wluTO the force of the wind is unobstructed, the roofs of all hii; 
biiildini^s should be especially designed to withstand its powerfi 
eltVcts. 

Cirrapliioal Analysis of Koof -Trusses. — The simplest 
anil ill most cases the readiest, way of computing the strains i 
trusses, is by the graphic method, which consists in representin 
the loads ami strains by lines drawn to a given scale of pound 
to the fraction of an inch. 

\V(; think the gra])hic analysis of roof-trusses may be best shuwi 
by examples, and hence shall give a sutticient variety to show th 
method of procedure for most of the trusses already describeil ii 
thes(» articles. 

Example 1. — As the simplest case, we will take the trus 
shown in Fig. 4, Chai). XX VI. 



0,(»8 





Fig. la. 

It \\r should «lra\v a line through tln» centre of each piert» of thi 
tnis^^. we shouid have a diagr.un such as is shown in Kii;. 1. \V 
will .su])iM>se that this truss has a span of .'U fe(>t, and tlie rafU*r 
hav<' an iix'lination of Vt° with a horizontal line. Then tho lenjrt 
of thf rafter would be 24 feet : and. if the trusses wen» I'J feet aiiarl 
<»n>' truss would su]»iK)rt a roi>f-an*a of 12 X 24 X 2 = r»7(t sqiiar 
b I'l. Now, if we hM>k at Fig. 1, we can see that the ]>urlhi or plat 
a I . I <n /•; would carry one-half of the nwi from A to U. The ptii 
Ihi at Ii woidd carry the roof from a iM)int mitlway betwivn .-t an 
/> to a point midway Uawivii Ii and f\ whiclt would he oii«-foiiit 
the area of ii>uf supported by each truss. 



THEORY OF ROOF-TEUSSES. 626 

The pttrlins C and D would also support the same amount of ' 
roof. 

If we consider the roof to be slated on boards an inch and a 
fourth thick, we shall have for the weight of one square foot 16 
pounds ; allowing for snow, 15 pounds ; nonnal pressure of wind, 
36 ; total weight or load on one square foot, 67 pounds ; total weight 
supported by one truss, 67 x 576 = 38,592 pounds ; total load com- 
ijig at each of the points B, C, and Z>, one-fourth of 38,592 = 9648 
pounds. 

The load coming at A and E is supported directly by the walls of 
the building, and need not be considered as coming on the truss at 
all. If, now, we draw a vertical line on our paper, and, commencing 
at the upper end, lay off 9648 ix)unds at some convenient scale, say 
5000 pounds to the inch (in the following figures different scales 
have been used to keep the diagrams within the limits of the page, 
but were first drawn to a large scale to get thes tresses more accu- 
rately), and then one-half of 9(J48 pounds, or 4824 pounds, to the 
same scale, we shall have the line ac (Fig. la) representing just 
half the load on the truss, or the load coming on each of the 
supports. 

Now, that the forces acting in the rafter and tie-beam, and the 
supporting forces, all coming together at the point A, shall balance 
each other, they nmst be in such a pro])ortion, that if we draw a 
line from a parallel to the rafter, and a line through c parallel to 
the tie-beam, the line ad must represent the thrust in the lower 
part of the rafter, and the line dc^ the pull in the tie-beam. If we 
next consider the forces acting on the joint 2?, commencing with 
the rafter, and going around to the right, we find that the first 
force which we know, is the force in the rafter, represented in 
Fig. 1« by the line da. Next we have the weight, 9648 poimds, 
acting down, represented by the line a?>, and there remain two 
unknown forces, — that in the upper part of the rafter and the force 
in the strut. 

To obtain these forces, draw a line through b (Fig. la), parallel 
to the rafter, and a line through (Z, parallel to the strut. These 
two lines will intersect in c; and the line be will represent the force 
in the rafter, and the line ed the force in the strut. Furthermore, 
if we follow the direction in which the forces act, we shall see that 
the force da acts up : hence the rafter is in compression. The 
remaining forces must act around in order : hence ab acts down, 
be acts towards the joint, and cd acts up towards the joint, so that 
both pieces are in compression. 

Next take the forces acting at the point C. The first force we 
know is ebf which acts up ; next we have the weight, 9648 pounds, 



520 THEORY OF ROOF-TRUSSES. 

which would extend beyond « to/; then there remain the forces 
in tli«^ rafter to the right, and the vertical tie, which are determined 
by drawing a line through / parallel to the rafter, and a line 
through (' parallel to the tie. These two lines intersect in /; anil 
the line //will represent the force in the rafter, and ei will repn»- 
sent the pull in the tie. We have now only to measure tlie lines 
>n our diagram of fon'es, and we have the forces acting in ever>" 
part of the truss; as, of course, the (^oiTesxxmding pieces on the 
dilTnvnt sides of the truss would be similarly strained. Measuring 
tin' ditferenl force-lines by the same scale we uscni in laying off the 
weiglit, we tuid the stra