Google
This is a digital copy of a book that was preserved for generations on Hbrary shelves before it was carefully scanned by Google as part of a project
to make the world's books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.
Marks, notations and other maiginalia present in the original volume will appear in this file  a reminder of this book's long journey from the
publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing this resource, we liave taken steps to
prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make noncommercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for
personal, noncommercial purposes.
+ Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.
+ Maintain attributionTht GoogXt "watermark" you see on each file is essential for informing people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liabili^ can be quite severe.
About Google Book Search
Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web
at http : //books . google . com/
^/.. ^ ^^
.^
..^
V . . i. sf? '
^;»,» /<
/. ^ ..^ A^ ^^^^ A
^ , . ^h^ ^ ^ ^J' ^
t
>rS<^ ' t:^
^/^
 ^j
lUs.
y
/tr^Tf'
4. ^ c^^rcA^'i
'/'^
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, Iiew 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 castiron columns, and of the new ** Gray " steel column. A
special article on the Strength of Castiron 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 Steamheating, 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 746773, 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
** PocketBook" 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
rollediron 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
" PocketBook 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' PriceBook," 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 PocketBook" 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 notebooks
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.
"SteamHeating for Buildings; or, Hints to SteamFitters, 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 CastIbon Beabino Plates 242a
CHAPTER XI.
Strength of Posts, Struts, and Columns 2ia
CHAPTER XII.
BbndingMoments 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 CastIron, 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 PlateIron and Steel Beam Girders 410
CHAPTER XXI.
Strength of CastIron ArchGirders 422
CHAPTER XXII.
Strength and Stiffness of Wooden Floors 425
CHAPTER XXIII.
FireProof Floors • . 488
CHAPTER XXIV.
Mill Construction 466
CHAPTER XXV.
Materials and Methods of Firf^Proof Construction for
Buildings 467
CHAPTER XXVI.
Wooden RoofTrusses, with Details . 486
CHAPTER XXVII.
Iron Roofs and RoofTrusses, with Details of Construc
tion 510
CHAPTER XXVIII.
Thbory of RoofTrusses 521
CHAPTER XXIX.
JqIMTS 550
xvi CONTENTS.
PART III.
PA
Chimneys 5
Rules for Proportioning Chimneys £
Examples of Large Chimneys 5
Wroughtiron 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 Wellknown European and American Build
ings 5
Length and Description of Notable Bridges 6
Lead Memoranda 6
Weight of Wroughtiron and Steel (Rules) 6
Weight of Flat, Square, and Round Iron 6
Weight of Flat Bar Iron 6
Weight of Castiron 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 Castiron Pipes 6
Weight of Castiron Columns 6
Weight of Wroughtiron 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 Rainwater 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 . . . I53
Legal Definition of Architectural Terms 5458
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
• • • « •
decimalpoint if there are decimals; thus, 10286.812(5.
2d, Find the largest square In the lefthand period, and place its
root in the quotient; subtract the said square from the lefthand
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 decimalpoint, 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
TJ9
. 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
Yasntaxs
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.^Jt21
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
.(/.ZTTAfil
.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
A»£UX^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
irj 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
'iCi
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
641 ;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
21.8307J:)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
21.7025344
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
21.0.5419.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()01
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
ooj
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
22.715a331
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
140608000
23 80a50ft5
521
271441
141420701
22.rr>4244
272^184
142236018
22.8473193
52.^
27.'J529
113055607
22.K091933
524
271576
143877824
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)1JM8JlJ
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
.0U18n494
8 1932127
.001Sl8ir3
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.i32
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
10iZ
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 = 7020 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 quarieracction 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 NewEngland 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
customhouses, 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 cloverseed = 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.
Halfdollar = 192 troy grains.
5cent piece (nickel) = 77.16 troy grains.
3cent 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 tenmillionth 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 PostOffice.
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)
lOcentinters = 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 decimalpoint 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. Cra = 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 .• ^ ."■;:;■> ix. ^•.i\«:. i:r ibe maw wdghL
Miscellaneous.
lii' 7: r:lir: f»: = l.IH Hobnw 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
stiaight (Fig. 1).
Parallel lines are such as are wholly in the same plane, and have
the same diiection (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 rightangle triangle is one which has a
right angle. The side opposite the right Fig. 4.
angle is called the hypothenuse; the side on Rightangle 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
rightangle 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 rightangle 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 (cfif (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 onehalf 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 onehalf 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
1023.5381
113.4115
.1
1326.7024
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
1081.0299
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
123.1.'>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
124.4071
.6
1562.2896
140.U63
.7
945.6901
109.0133
.7
12:J7.8582
124.7212
.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
01
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
oi
0.
O.J
01
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
132.7326
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
825.329
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
1209.958
1215.099
1220.254
1225.420
12.30.594
1235.782
1*240.981
1246.188
1251 .408
1256.64
1261.879
1267.i:i3
1272.397
1277.669
1282.955
1288.252
1293.557
1298.876
1.304.206
1309.543
1314.895
1320.267
1.325.628
1331.012
1.3.36.407
1341.810
1347.227
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
1418.629
1424.195
1429.776
1435.367
1440.967
1446.580
Cirfnm.
Ft. In. I
119 4i
119 7j :
119 105
120 2 I
120 5 i
120 Hi 1
120 Ui
121 2A
121 5J
121 8^
121 in
122 31
122 61
122 9^
123 i
123 3ji
123 6J
1*23 9
124 IJ
124 4i 1
124 7H i
124 KU \
125 if
12.)
^
125 7
125 11
1*26 2\
126 bi
126 S4
126 1l
1*27 25
127 5,'
1*27 9
1*28 i
128 3g
128 6j
128 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
1924.426
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
.123
1.03987
.184
1.08797
.245
1.15.308
.002
1.00001
.063
1.01054
.124
1.04051
.185
1.08890
.246
1.15428
.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
1.04722
.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
1.17024
.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
1.17276
.018
1.00087
.079
1.01656
.140
1.05147
.201
1.10447
.262
1.17403
.019
1.00097
.080
1.01698
.141
1.05220
.'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
1.18299
.026
1.00182
.087
1.02006
.148
1.05743
.209
1.11269
.270
1.18429
.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
1.00225
.090
1.02145
.151
1.0)973
.212
1.11584
.273
1.18820
.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
1.06209
.215
1.11904
.276
1.19214
.033
1.00289
.094
1.02339
.155
1.06288
.216
1.12011
.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
1.12225
.279
1.19612
.036
1.00345
.097
1.02491
.158
1.06530
.219
1.123:34
.280
1.19746
.037
1.00361
.098
1.02542
.159
1.06611
.220
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
.222
1.12664
.283
1.20149
.040
1. 00426
.101
1.0289S
.162
1.068.58
.223
1.12774
.284
1.20284
.(Wtl
1.00447
.102
1.02752
.163
1.06941
.224
1.12885
.285
1.20419
.042
1.00469
.10 J
1.02S06
.164
1.07025
.225
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
1.1.3219
.288
1. 20827 1
.0*'>
1.00 ).39
.103
1.02970
.167
1.07279
.228
1.13331
.289
1.20^)64 1
.046
1.0056:3
.107
1.03026
.168
1.07365
.229
1.13444
.290
1.21102
.047
1.00587
.108
1.03082
.169
1.07451
.230
1.13557
.291
1.212.'39
.048
1.00612
.103
1.03139
.170
1.07537
.'231
1.13671
.292
1.21377
.049
1.0033S
.110
1.03198
.171
1.07624
.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
1.22073
.0.>4
1.00776
.115
1.03 J90
.176
1.08066
.237
1.14363
.298
1.22213
.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 .22495
.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
1.22778
.059
1.00928
.120
1.03797
.181
1.08519
.242
1.14951
.30:j
1.22920
.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.:iO224 ,
.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
1.38.322
.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.. 32086
.402
1.38671
.441
1.45697
.480
1.53126
.325
1.26138
.364
l.:j2249
.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
1.39724
.447
1.46815
.486
1.54302
:.m
1.270 U
.3'<0
1.332:34
.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
1.34229
.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
1.41324
.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
.420
1.41861
.459
1.49079
.498
1.56681
.343
1.28895
.382
1.35237
.421
1.42041
.460
1.49269
.499
1.56881
.344
1.29052
.38:3
1.35406
.422
1.42221
.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 ■■••if ,T I,, f r7»M '.'§•* rft'tr^i '^f k'llf th€ arf^ and
." '*'r^9t"L it/ii 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^ , ;;ir 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 ur' ^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* 'rsei sill' v i. xai sobtncc the product from 
.:•• li.i^it*. r j.;i jaijuiur^ "i iiiiarn ot the reiuaiuder from
.^' r. ;:i.~ .r .Air. lia.nrL^r. i:ii ::4Ju uhe ft^aare root of that re
> ■•. 11
3:. 4 ' ..:. — Titt riiOiHCr }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^ rxc 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 532
^,.— ^=^— ^^^ RvLii. — Multiply the number of degrees in
J ' _____r:?i.i ...^ _^ ; ^. .j^^ area of the whole circle, anddi
ExOiPLE. — 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 * 
— '^. .. ■•:'• Tr:":. •"»: 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 semiellipse 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 smface.
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
onelialf 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 squaie 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 onehalf 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 onesixth
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 onethird
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 giavity.
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 onefoiuth
the aiea 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 peipeudicular 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. Diaw 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, ot4 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 80foot mark at
a, and a third person take hold of the tape at the 50foot mark,
with his thumb and finger, and pull the tape taut. The 50foot
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
onehalf 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 pencilpoint 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 wheie 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
pioblem; 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.
iHhlg 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
onehalf 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 pencilpoint 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 (liaw 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 ciuve 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 iadlus 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 twothirds of the
liansveise axfs;
Rg.04
C
Second (willi arcs of three radii, Fig. 85). — On the tiansverse
r.xis AB draw the rectangle AGEB, equal in height to 0C\ half
the conjitgatc axis. Diaw 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 stiaiglit 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 semicircumfer
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.
GROMFTRICAl. 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
3i
.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
.•2824
.'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
.3126
.3129
.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
.3203
.3'206
.3'209
.3212
.3216
.3218
.3221
.3*223
.3*226
.3229
.3'232
.3235
.3238
.3241
.3'244
.3246 j
.3249
.3252
.3255
.3258
.3261
.3264
.3267
.3269
.3272
.3275
.3278
.3*281
.3284
.3287
.3289
.3292
.3295
.3298
.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
.3427
: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
.6aa
.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
.6236
.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 smi
.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
.8294
.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'
i29
.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
1.0026
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
.9423
.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
1.0211
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
1.0216
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
1.0221
1.0370
1.0519
1.0007
28
29
.9310
• J7^rO*T
.9617
.9770
.9922
1.0073
1.0223
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
.9622
.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
1.0236
1.0386
1.0534
1.0082
34
35
.9325
.9479
.9633
.9785
.9937
1.0088
1.0238
1.0388
1.0537
1.0086
36
36
.9328
.9482
.96;}5
.9788
.9939
1.0091
1.0241
1.0390
1.0539
1.0087
36
37
.9330
.9484
.96:JS
.9790
.9942
1.0093
1.0243
1.0393
1.0W2
1.0090
37
38
.9333
.9487
.9640
.9793
.9945
1.0096
1.0246
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
1.0256
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
1.0266
1.0415
1.0564
1.0712
40
47
.9351) .9510
.9663
.9816
.99<>7
1.0118
1.0268
1.0418
1.0566
1.0714
47
48
.9359 j .9512
.9660
.9818
.9970
1.0121
1.0271
1.0420
1.0560
1.0717
48
49
.93<)1 i .9515
.9668
.9821
.9972
1.0123
1.0273
1.0423
1.0571
1.0719
49
50
.9364 1 .9518
.9671
.9823
.9975
1.0126
1.0276
1.0425
1.0674
1.0721
50
51
.93(56 .9520
.9673
.0«2»)
.9977
1.0128
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
1.0283
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
1.02S8
1.0438
1.0686
1.0784
65
56
.9379
.9533
.9086
.9S3S
.9990 j 1.U141
1.0291
1.0440
1.0589
1.0730
66
57
.9382
.95.36
.9689
.9841
.9992; 1.0143
1.0293
1.0443
1.0591
1.0730
57
58
.9384
.9538
.9691
.9843
.9i>95
1.0146
1.0296
1.0446
1.0603
1.0741
68
59
.9387
.9541
.9694
.9846
.9998
1.0148
1.0298
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
1.1621
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
1.1225
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
1.2022
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
1.1323
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
1.2329
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
1.2722
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 hiprafter, and (jf of a jackrafter.
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 hiprafter.
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 jackrafter. The
length of each jackrafter 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./AV then the angle at J will be
the top bevel of the jackrafters, and the one at h the down bevel.
Backhu/ of the hiprnftoy. 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 hiprafter.
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)
' Let 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«»U4P),
T . sin C.
area
area
sin A
tanHUJ3)="^^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 = Bn3 = ♦'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(cos4
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(l4cos']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
.170211.985111 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
.loeJi
.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
.15171
.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 .29818
126022 .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
. 93528 ! .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.) ' . 95107
.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
Cosiasine
78^
.30874,. 951 15 .3i.>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
.91851
.948l.'3
.04832
.94823
.94814
.94805
.91795
.94780
.94777
.94708
.94758
.94749
.94740
.94730
.94731
.94713
.04703
.94093
.04084
.94074
.94005
.940ly
.94037
.94037
.94018
.940t')0
.94509'
.9I.7M);'
.9:3.5801
.91.571
.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
.941K5
.91470
57
56
55
54
.5:3
53
.91400! 51
.9145^
.83H.59;. 944.47
.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:iv;,.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
.99e5
.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
.09170
.99473
.90170
.90107
.9n4<J4
.00101
.90458
.99455
168..9S452
iinsiiie
^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
13n 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,
171011
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
97JO
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:
.ftiUO
9mr ffiR
.U«i3
.aun .maA^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
CoSSBb.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
.80127
.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
.ai218'
.85203!
.a5188,
.85173,
.851571
.851421
.a)1271
.85112 1
.85006'
.85081
.aiOOO
.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
.a5:^>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
66412
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
.80Z2
.80204
.80247
.80230
.80212
.80195
.80178
.80160
.8.)143
.&.)125
.80108
.8*K)r3
.8iwi5(;
.800;id
.80021
.TlO0
.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:«
.02150
.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
.02SI2
.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
.0C0i5
.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
.6348
.63451
.0*473
.03400
.03518
.03510
.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
.77144
.77125
.77107
.77070
.77051
.77a0:3
.77014
.7CO0
.70077
.7CO,'50
.70040
.70021;
.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
33 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)TI6
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
971_H41
S.atffiTB
losses
7.:
10363
! 13131
T.
'.mm
03!a
oioLaw
.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:aii38
: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
4110
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
5.20035
.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
.31013
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:«
2.01210
.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
86163
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
2.01152
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
2.89013
.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
' .42213
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
.43486
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
.Rseia 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
.7125
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
. Xfjt '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^
. .OiCO
1.49566 = .09116 \AV/A
.7a:.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 : .6Xi5 . 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 1.4722  .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^La.;: T^EiJ
w
Vf
:
65'
\
34
_J
118
N.iTURAl, TAXOKNTS AND COTANGENTS.
36«
Tang I C()t:irijf
1
2
3
41
«!
I
8
9
10
11
12
13
14
15:
IG
17
18
19
20
21
22
23
24:
25!
26
28;
29'
30'
32
33
31
35
36
37
38
39
40
41
42
43
44
15
46
47
48
49
51
52
53
54
55
56
57
58
59
60
/ I
.72654 .
.72699 '
.72713
87=;
Tang I Cotoiig
880
rsj
.72832
.72877
.72921 :
.?^f66
.7:U)10
.7:i055
.73100
.73144
.'.•:J189
.7:fcJ:«
.7;J278
.73323
.73368
.73413
.78457 I
."3502
.V8W7
.7S602
.730:57
.73681
.78?^
.78771
.78816
.78801
.73906
.73ir)l
.78996
311 .74041
.74086
.74131
.74176
.74221
.742()7
.74312
.T4357
.74409
.74447
.74498
.745:« i
.74.')83 ,
.7«528
.74074 I
.74719
.747M
.74810
.74856
50 .74900
.74946
.71991
.75037 :
.75082
.75128
.75173
.75219
.75264
.75310
.75355
Cotang
.a76:J8
.87554
.37470
.S7m\
.37302
.37218
.37134
.37050
.361V>7
.368Ki
.36800
.36716
.366:«
.36549
.mm*
.36383
.36300
.36217
.36134
.36<151
.35968
.35885
.;i")802
.35719
.35637
.35554
.35472
.35389
.35307
.35224
.35142
.35060
.;MU78
.34896
.34814
.S1732
.34650
.34568
.^4487
.34405
.34323
.a4242
.34160
.a4079
.33998
.33916
.33835
.33754
.3:JG73
.335:12
.33511
.33430
.33349
.33268
.83187
.331f>7
.a'KW6
.32{>I6
.32S('»5
.327^:)
.32704
Tang Cotang
.75:«5
.75401
.75447
.75492
.75538
.75584
.75(K9
.75721
.75767
.75812
.75a58
.75904
.75930
.75996
.76042
.76088
.76134
.76180
.76226
.76272
.76318
.76364
.76410
.76456
.76502
.7^548
.76594
.76640
.76686
.76738
.76779
.76825
.76871
.76918
.76964
.77010
.77057
.77108
.77149
.77196
.77242
.7W89
.77335
.77382
.77428
.7r475
.77
.776(
.77615
.77661
.77708
.777M
.77801
.77}'48
.7:"!J5
.77941
.779H8
.78aJ5
.78082
.78129
'^'^521
,«r608
.32704
..')24(U
.:J2384
..•12:W4
.:)2224
.32144
..*J20l>4
.31(K4
.31904
.31825
.31745
.31666
.31586
.31507
.31427
.31348
.3123)9
.31190
.31110
.81031
.30J).2
.30873
.30795
.80716
.30637
.30558
.30480
.30401
.80323
.80344
.30166
.300.^7
.30009
.29931
.29853
.2U775
.2:»U96
.29<J18
.29541
.29463
.29385
.29307
.29229
.29152
.29074
.28997
.28919
.28842
.28764
.286S7
.2h'C10
.2^T>•«
.2S4.')6
.28:rr9
.2H:W)2
.2:^.»25
.2X148
.2H)71
Tang
Tang
.78129
.78175
.7««2
.78a(i9
.7X316
.78363
.78410
.78457
.78698
.786^46
.78692
.78739
.78786
.78834
.78881
.78928
.78975
.79022
.79070
.79117
.791W
.79212
.79259
.79306
.798M
.79401
.79449
.79196
.79544
.79591
.796:)9
.70() 6
.79734
.79781
.798J9
.79877
.79924
.79972
.80020
.80067
.80115
.80163
.80211
.80S58
.80306
.80354
.80402
.80ir)0
.80498
.8rK>46
.8051)4
.80612
.806<i0
.8(rT;M
.807 H6
.WK^4
.80882
.809:k)
.80978^
Cotaug'
63«
62<
61
J»
89» 1
Cotang
. Tang
.80978
Cotang
1.2;9St4
1.23490
1.27917
.81027
1.23416
1.27841 ;
' .81075
1.23^43
1.2776^4 1
.81123
1.28270
1.27688
.81171
1.23196
1.27611
.81220
1.83123
1.27535
.81868
1.88l%J0
1 .27458
.81316
i.assm
1.273S2 !
.simi
l.:S9()4
1.27306
.81413
1.23H81
1.27260
.81461
1.887^
1.27153
.81510
1.88685
1.27077
.81558
1.82612
1.27001
.81606
. 1.8S539
1.26925
.81656
1.88467
1.26A49
.817T)3
1.88801
1.267r4
.81758
1.88821
1.26698
.81800
1.22819
1.26622
.81&49
1.88176
1.26546
.81898
1.88104
1.26171
.81M6
1.88U81
1.C6395
.81995
l.SlflSO
1.26319
.82044
1.81H86
1.26244
.88098
1.81814
1.26169
.88141
1.81748
1.26093
.88190
1.81070
1.26018
.88288
1.81698
l.l^WS .
.88287
1.81588
1.25867
.82836
1.81464
1.25792
.82385
1.818K2
1.25717
.88131
I.8I81O
1.25(M2
.8848S
1.81288
1.25567
.88531
1.81166
1.25492
.88b80
1.810O1
1.25417
.68629
1.21088
1.25343
.88678
1.80051
1.25268
.88727
1.80879
1.25193 :
.88776
1.80608
1.25118
.82885
1.80738
1.25044
.88874
1.80065
1.24969
.88983
1.80BB8
1.^4895
.88978
1.80688
1.2^4820
.89028
1.80151
1.1U746
.88071
1.80879
1.^46?^
.88120
1.80808
1.84597
.88160
1.80887
1.84528
.88818
1.80168
l.»4440
.88868
1.90006
1.W875
.88317
1.80081
1.84301
.88866
1.10053
1.24227
.88416
1.19688
1.84168
.88406
1.19811
1.84079 :
.83514
1.19M0
1.84005
.88561
1.19089
1.83931
.88613
1.19609
1.23858
.88668
1.19688
1.23784
.88718
1.19467
1.23710
.88761
1.19887
1.23637
.88811
1.19S18
1.23563
.88880
1.19816
1.23490
.8sno
1.19177S
Tang
Ootang
Tuig
i
»•
NATURAL TANGKNTS ANB OOTANOKNTS.
aog
Cobine
INIO
"■liiro":"
MOOD
»0»
;is
mM
uaxt
.tfwn
tmr
. 8014
axa
. »m
S
'. 135
' 9R6
M3W
. BIB
(I8GG
, ws
lTTT
ueu
.1708
ttOOB
.103S 1
15(67
.lOUB
:iS61
j^
:S
;jK!K
.nsi
jMW
.lWB
»15B
.ITOIO
UGOS
)!»t5B
iwMB
ffiom
.1ft 09
OtiUO
eToi
s
: 6S.T5
!59SS
 8329
amis
MISS
: ma
3(UIS
. sger
soaiT
. soia
3a31H
M3S8
■.Isjm'
B0119
SUTO
BGOSl
6^73
'.iss:i
.I530S
.15340
essm
! 15104
maa
.isro?
Mag
Ting
.euMO '
.Bnss I
,HT1W I
.BT3W
.«7a88
■ffrase
.87441
.B74B3
.87RIS
.B;6!»
.STWS
.BTSW
.B7749
.B7801
.B7E66 I
.eaooT
!»uii}
,sai<»
.turn
isMIT
.R«»4
.saaaa
'.ssiua
.SWltf
.tliUI»
I Tanit
r .MWO" "
.WlOU) ;
!w)TS7 I
.own: '
■tKWM
.DIW7
' .flOOW
. .90993
[ .vimii '
' .SUM
.a9RS4
.OWTO
_.09JOI1
l.OUUI:!
l.WU>»
1.0II5I4
.9I01S
iflim
.91180
.91335
.91290
.MOTU I
.M131 , ,
.IMffiO I
.wi«a :
1 !US1J3
.05139 I SI
..C6SItS»
t.OS:£S!^
t!o5iia .^
i.oaoTS a
i.onos I
l!lU5H3 I
1.042:9 I
l.OlilS 1
1.G41SH I
I7T3S I !•
ITGTO I .1
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 wroughtiron columns have lately been
y changed by some engineers; but as the question of the
th of wroughtiron 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 windpressure 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 windpressure 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 coefficient 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 crosssection.
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 firstmentioned 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 rooftruss.
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 prothKe 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 pullingforre 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 wroiightiron, two inches square and
ten feet long, securely fastened at one end, and to the other end
we apply a pullingforce 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 crosssection
is four s(iuare inches, and hence the pullingforce 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
peifect 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 buildingmaterials may be exposed
are: —
I. Tension, as in the case of a weight suspended from one end
of a rod, rope, tiebar, 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
ingconstruction, 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
" foundationbed."
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^
ingstratum 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 crushingforce; 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 hardsetting 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 oneeighth 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
hardsetting 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 surfacewater 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 frostline,
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 bearingstratum in soft soils.
134 FOUNDATIONS.
Whichever material is employed, the bed is first prepared by ezca^
vating a trench sufficiently deep to place the foundationcourses
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.
Sandpiling 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.
Sandpiling 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, sandpiles are found of more service
than timber ones, for the reason that the timberpile transmits
pressure only in a vertical direction, while the sandpile transmits it
over the whole surface of the hole it fills, thus acting on a large
area of bearingsurface. 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 crosssection. 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 yellowpine are the best woods for piles; though
spruce and hemlock are very commonly used.
Where piles are exposed to tidewater, 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
crushingstrength 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 pilehead, and the resistance
offered by the ground through which the pile is passing; and, as
the lastnamed 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 bearingload on piles have been given,
Perhaps tho nile most commonly given is that of Major Sanders,
UnitedStates 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 pocketbook 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 onehalf the extreme load
should be taken for i)iles thoroughly driven in firm soils, and one
fourth when driven in rivermud 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 onefifth 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 onefourth of an inch at the last blow of a
ram weighing 2500 pounds, falling 80 feet.
In ordinary piledriving 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 nMulr«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 1200pound 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 (141 ) ~ ^'^ ^^^^ ^^^ ^^'^ 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 Sandeis'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 thirtyfive 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 crushingstrength 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 batterpudding." *
V. Still another method is to surround the site of the work with
shccLpiling (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 Injuiy; 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 briiringsiirface 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 preseivation 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 crossstrain
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,
fourInch by sixinch timber, laid flatwise, will answer in ordinary
FouiKlations for Cliimiteys. — As examples of tlie foun
dations i'ciuired 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 rootinggr&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 bedrock ; 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 Ibeams, 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 prepartd (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 Ibeams 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
footingcourses to such a height as to encroach unduly upon the
basementroom .
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 sl 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 Pocketbook.
FOUNDATIONS.
Bquare foot od tho foundation, what dse ftnd length of steet Ibemu
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 Ilieams 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 hardpan,
cap[)ed by wooden sleepers, with heavy wooden crossbeams 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 puddinglike 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 bearingsurface, 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 crossstrains 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 crossstrain 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 bearingsurface
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 batteringface, 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 basestones 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 onefourth brick beyond the one
>?e ity exo^ in. the case of an eightinch 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. 155157.
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 airspace 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 terracotta, or by
plastering threequarters 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 thirtysiz
and thirtyseven, for every addition of twentyfive 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 onehalf 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 eightinch wall in Boston is probably as strong
(to resist crushing) as a thirteeninch 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 . . .
*»
,
a
>>
Of
t^
B
o
cc
2
■4.9
«
13
8
13
13
17
13
22
17
22
17
26
22
t
&
r
c
o
o
•M
■*j
■*a
4>
GQ
CO
QQ
,d
xs
ji
'(H
U
^
to
8
13
13
13
13
13
17
13
13
13
22
17
13
13
13
One story. ..
Two atones..
Three stories.
Foor stories. .
Five stories..
Six stories...
BUILDINGS OTHER THAN THE ABOVE.
13
8
17
13
13
22
17
13
13
26
22
17
13
13
26
22
17
17
13
13
SO
26
22
22
17
n
13
00
CO
z
o
OQ
o
n
m
CD
n
z
M
O
t2
HI
p
g
<
Q
H
a
OQ
h9
<
O
'd
5 ®
.d S a
« 2S
"2 a 2
S
g 2 2
^ a 9,
22
Wl ^ ♦*
O O ^
• Q. ft. O
o O «iX
^ ^ :: ^ 3J
c c  r o
^^_J_I.
"^ '> .^ s
^•' <o ti 00
tn be t« 60
"e — C —
I «M
4> « O
£ c s s
•d "O •«
cos
o «s «s •
•g ^ 4i tS
SQ S 8 S
I
c o
« • 9
GO
o
Hl
(4
: S
OQ
:a
Q
o
o
o
CO
Q
<
s
a
o
GO
O
I—
u
PQ
"S
c8
h
o
o
. . «a
eight
eight
pper
ja 43 a
9 SS •
2 1 ^ i
1 i A^
s £2^
 5 ** s
'^ '^ ■** vT
•« "S >, X
floor,
floor,
.floor,
of the
« 0* '^ SJ
%4 ^< %« «^
o o o ^
a e. p. '^
to to
to to
to to
Ithin
H ^ ^ ^
^ ^ %
iissi
^ ^ ^
^sk
I
MASONBY WALL&
U1
Gi
CO
I
g
o
o
CD
00
e
3
O
o
m
o
QO
S
n
a
QO
s
D
O
n
i
o
<
g
o
"S js J4 th
• c fl S
§11 I
ai "aS "a^ s
?•* flO ^O
o<s •£»« "ca »
SS *pts ao M
r^5 Sc Sfl c
X g ^ ^
I I 1
^ ^ s
£•
5*
^t'S
— &
: — '. — : — ; — ; — : — ■" oo
4A ^^ ^A fc^ 4^ ^J Q)
S h h kl li k ki ^S
S O O o O o o ^t»
Kf'^' 'M '^ «■ '^ ^ V hA
=l§§Si§ il
5 a « « S s * g Q
»
P
O
Pi
o
I
s
0) V
^ K. V ♦*
* * ^
sa^gss
^^ ^ >— ^ ^
. 0) <li 1> V
' (li O V o
a) ^
O In ti *4 If
>. k > >
o o o o
SFo o o o
~ c c p e
^ a a c ti
O S CS cS 08
M «i» «•*«■> *•
^
1
8
d
s.
3
a
o
•3
>
o
a
2
I
s
«2
•^a as
^1
— D.
<■ CO
is
ii
is'a
«a
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
arrowhead, 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 arrowhead 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 inchpounds.
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 figme ; 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 semiellipse, 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".
3X49
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.
FigI
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 " PocketBook
igineers," for the thickness at the base of vertical retaining
with a sandbacking 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
crushingforce, 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 TerraCotta, 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 fourtenths 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 onehalf
of their crushingload.
CrushingHeight 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' shottower 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 squaie 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.
WorldngStrengtli of Masonry.— The worlringstreiigth
of masonry is generally taken at from onesixth to onetenth of the
crushingload 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 firstclass pressed brickwork in cement to more tlian thirteen
or sixteen tons' pressure per square foot, or good handmoulded
brick to more than twotliirds 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 Pocketbook, 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 firstclass 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 rubblestone or brick, or
under any iron beam or archgirder, 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 crosssection of the pier.
In Boston, it is required that '*all piera shall be built of good,
hard, wellburned 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 capiron at least two inches thick, or a granite
capstone 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 testingimacliliie
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 onehalf 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 TerraCk>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 TerraCotta Company^ a Brick were manufactured of
a rather fine clay, and were such as are often used for face brick.
The NewEngland 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
bardburnt 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
onefourth 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 188889, 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 couises, 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 UnitedStates
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
twentysix days old. As the piers were built in cold weather, the
bricks were not wet.
The piers were built by a skilled bricklayer, and the mortars
were mixed under his superintendence. ITie tests were made with
the government testingmachine 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
crosssection 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
,IIWJII.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 Hmemortar
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 limemortar, ^
it ti it i(
It it ii it
• • • • • • •
804
196
175
100
110
lOi
9a
9b
9c
1 part Rosendole cement, 2 parts limemortar,^
ti it ti ti
it ti it it
PlasttTofParis.
• • • • • •
194
198
162
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 Limemonar, 1 part lime, 8 parui
STRENGTH OF MASOKBT. 181
out andae waste of materials. For the size of oastiron bearing
plates on masonry, see page 342&. For strength of architeotnral
terracotta, 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 fireclay tiles 80 lbs.
*• ordinary clay tiles 60 **
Porous terracotta tiles 40 **
Mortars.
(In 4inch 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 4inch joints for bedding
ixonp tea 70
182 8TBENGTH OF MASONRY.
Actual Tests of the CrushingrStren^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 reddishbrown 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 stoneyard 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 crosssection 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 crosssection 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 crushingstronjrth.
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 arianged to show the sectional area and
strength of each specimen, and the average for each variety of
^tone. The crackingstrength, 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 Lwoliiuli cubes. The r
suits obtalnetl by bliii laiigtvl fiotii 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 twoinch 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 sixinch 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 twoinch 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 rocksalt 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 liinemortar 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 terracotta, 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.
Terracotta 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, wellburned terracotta cannot safely
be made of more than about 1^ inches in thickness, whereas, when
required to bond with brickwork, 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 terracotta, although alone it is able to bear a very heavy
weight. *• A i'Olid block of terracotta 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 terracotta blocks :
Crushing wt.
per en. ft.
1. Solid block of terracotta 523 tons.
2. Hollow block of terracotta, unfilled 186 *'
8. Hollow block of terracotta, slightly made and unfilled. 80 "
Tests of terracotta 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.
Terracotta block, 2inch square, red 6,840 lbs. or 492 tons.
Terracotta block, 2inch square, buff 6,236 *' '* 449
Terracotta block, 2inch square, gray 5,126 " " 369
( (
((
Prom these results, the writer would i)lace the safe working
strength of terracotta blocks in the wall at 5 tons per square foot
when unfilled, and 10 tons per square foot when filled solid with
brickwork or concrete.
The weight of temootta 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 fireproof buildings, terracotta 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 onehalf 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 iotate: 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 desiied 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
pioceed 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) (haw 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 onefourth 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 " setoffs," 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 crossmarks. 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 aiea = 11 sq. ft. Xi = 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 <Mnal the
whole \\«'ii:iil of the pier.
Ilaviiii: itioeeeded 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 'neaier than onequarter 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 hammerbeam 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 fiuni 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 sprinyhKjUne ; 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 aie the simplest to construct.
Klliptical and threecentred 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 bedjoints 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 wedgeshaped 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 figiue of the arch by making the bedjoints 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 halfbrick rings together.
This may be done either by thickening the joints of the outer of
a pair of halfbrick 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 halfbrick 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 bedJoints of the backing and spandrels.
196
THE STAlilLlTY OF ARCHES.
By the aid of hoopiron bond. Sir Marclsanibard Brunei
halfarcli 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 onelialf 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 castiron
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 tiansverse 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 fiist 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 archring 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 extiados to the intra
dos of the arch; and if the keystone projects above the archring,
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 onehalf 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 archring 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 archrfi^ Is divided into
voussoirs of equal size, this Is easiest done tiy computing th« ana
of the archring, and dividing by the number of voussoira.
Fls.3
Ridi' for 'W'li of •iiifhiiif vf urdirim; is as follows: —
Area in square feet = 0.7854 X (outside radius squared — itaW.c
radius squared).
In this example the wholi' area equals 0.78Vl 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 onehalf of the archring, 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 archring. 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 archring.
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 archring, 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 archring ; the horizontal snrtece of
rhc wall being 3 feet C inches above the archring at the crown.
1st Stei. — Find centre qfgraHty,
Commencing at Ibe crown, divide the load and aFchrlng 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 arehring.
In this case, the areas of the slices are as shown by the Ognnt on
their faces (Fig. 5}.
Now <]lvlde the arehring 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 archrlog
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 semielliptical 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
archring 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 archring 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 40loot 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 TIEBOD8, 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
crosssection 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 crosssection.
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 crosssection; and hence, if we know the tenacity
of the material per square inch of crosssection, we can obtain the
total strength by multiplying it by the area of the section in
inches.
The tenacity of different buildingmaterials per square inch hM
been found by pulling apart a bar of the material of known dimen
sions, and dividing the breakingforce 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 crosssection, 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 NewYork 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 tiebar 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 wroughtiron : 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, bearingplates, 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 biidges 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 crosssection is
reduced, the tangent between shoulders shall be at least twelve
times its shortest dimension, and the area of minimum crosssec
tion in either case shall be not less than onequarter 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 fullsize 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 :
frt rxr^n. 7,000 X arca of original bar , ,, . . , .
52,000 r^ — , i ^.—ra (a^ ^ inches),
circumference of onginal bar
with an elastic limit not less ttian onehalf 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 onehalf 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 rollingmill. 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 crosssection 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 threequarters 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 onehalf 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 rollingmill. 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 crosssection
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 firstclass. 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 rivetholes 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 rollerbeds
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 machinedriven 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 Pinhole, and Pilot Nuts.— 82. All
pinholes 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
thirtyse3oncl of an inch will be allowed in the length between
centres of pinholes ; tlio diameter of the pinholes shall not exceed
that of the pins by more than one thirtysecond inch, nor by more
than onefiftietli inch for pins under three and onehalf 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 onethird 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 onehalf 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 onetblrty.second inch
below and onesixteenth 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 onesixteenth inch the
diameter of the livcts to be used. Rivetholes must be accurately
spaced ; the use of driftpins 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 rivetholes in steel work, if punched, shall be made with a punch
oneeighth inch in diameter less than the diameter of the rivet in
tended to be used, and shall be reamed to a dluneter onesixteenth
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 WBODsOTlHoir 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^htImm.
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 twentyfive 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 aorkingiarerirjtb 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 : —
Wroughtiron, ru^no of its length per ton per square inch.
Castiron, ^,,^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 wroughtiron, and that a crystalline fracture showed
that the iron was bad, hard, and brittle. Mr. Kirkaldy's experi
ments, however, show conclusively, that, whenever wroughtiron
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 wroughtiron 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 wroughtiron,
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 breakingstrain does not indi(uite the quality, as hitlK'ito
assumed.
** A hUjh breakingstrain 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 breakingstrain 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 Wroughtiron 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 breakingstrain 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 breakingstrain 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 breakingstrain 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 breakingstrain 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 wiredraw
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 wircdniwini;: 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 breahingstrain, 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 onehalf. 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 workimjHtrain.
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 breakingstrain 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 breakingstrain per square
inch of fractured area, and not to the breakingstrain 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."
EyeBars and ScrewEnds*
Iron ties are generally of flat or round bars attached by eyes
And pins, or by screwends. In either case, it is essential that the
proportion of the eyes or screwends 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 wellformed eye is made from the
iron of the bar itself.
A similar process is adopted for enlarging the screwends 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 eyebar, 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 onefourth 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 screwends for dif
ferent sizes of rods, as adopted by the keystone Bridge Com'
pany.
Castiron has only about cnothirJ the tensile strength of
wroujj:! Itiron ; and as it is liabk* to airholes, 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, DIEFOEGSD 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 aic 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 " PocketBook 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 ScretBEnd» fm Round and Square Bars.
StINDAHD PKOFORTIOm OP THE KETBTOKK BRIDGE COUPAKr.
RESISTANCE TO TENSION.
TABLE VI. (concluded).
Upset SrrewEnda.
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 twentyfive 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 twentyfive 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 thirtyfive to fortyfive 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 : —
Boltrope . . . .18 per cent
Shrouding . . 15 to 18 per cent
Cables 21 per cent
Spunyarn . . 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 UnitedStates navy test is 4^00 pounds for a white rope, of
three strands of best Ri(/a hemp, of one and threefourths 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 onethird, 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 teslod in comparison with a similar link
which hiid been preserved in the slockhousc 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 aifliEtectural construction in vrbicb
tbe resistance to siiearing tms to lie provided for. The one moat
frequently met witii is at the end of a tiebeam, 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 tiebeam; 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 tiebeam 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
tiebeam, 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 stiap 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 o57565.
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 crosssection ; 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
II
Tin
I
8
^J
12 £
?^£
iS2 ' o
fl * * * *
• • •
oooo
JO » » ak aK
ss
fooo
•b * tfte «%
S
to
iO
tfr»00
C3CD CO CPvD
«D»Ol>il
e*«oaQCQ
Ol ^T QO Od
00 f* o>c^
00'
Sc#SS
(.^ ^oSco
CO 00 00 CO
i
CD CO CO CO
ooOt^OiO
§coapTJoo
H04Ol8OeO
©t^THri Othcqjo
CO CO '^ H*
e*ococD
CO O 00 o
SCO CO o
'■^ "^C 3C 3C
• oooo
JBOOOS
*■ » • •. •»
CO 00 CO ^
gfSS
C^ lO o o
®§88
o _
to
00 Ct *4 09
^5
lO lOiO <o
toa"^ CQCQosa i>cooaoo oooit*
COOIA <OeOOt« lOCOrHO AOOtCO
gt> «>Qdo6 oJOfH— c^ooTjlio iocet^oo
tf CO 00 00 oo
CO OO OO 00
^5' ^^ ^T ^5'
^» ■^l ^1 ^<
lOQO)
^3 9^ 0^ ^ ^
,J IT. t^ 0> tH
lO lO CO o
Of «D O t'
•* ^ r "k
00 »o ooo
tH tH tH Ol
00 CO Od 90
<M<M W 00
QCIO OCO
ceo;^ OO
00 ■^ '^ ■^
fooo
ooo
lO OlO
hP Oi CO lO CO
ooc
o»o ~ »o
00 OS th c<r
— rHG<l Ol
o  o o
OO c? 5
o»oo»o
Tti jC t 00
^ooo
O lOO »o
O — CO'^
cr CO CO CO
o .
<B P
• lO Tf t^ iC
^X) OSCQ 00
& •
CO t> =; CO
o o ^ t'
CQtCO ^
^ '^ r^ 00
OOIOO*
o — ^ o
Oi "^ Oi '^
t^tH tHO^MW COCOOOTf rfttOtet^
CO
ni»
a
Hs'XhXMKH*
0Q^C9C9
09 09 0)01
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 Handbook.
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 welldesigned joint are symmetrically
arranged, it is unnecessary to consider more than onehalf of their
number. Fig. 4 shows a sketch of onehalf 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 onehalf 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 13, 23, 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 12 equal to the first or outer force ; 23, equal to
the next, 34 ; and 41, 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 01 (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 12, 2si, 34', 41. 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.
BEARIKQPLATES FOB GIBBEBS AKD COLUMKS.
PROPORTIONS OF OASTIRON 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 bearingplate 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^ BEARIKGPLAT£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 sixstory warehouse
support a possible load of 212,000 pounds each ; under the oolumn
is a baseplate of castiron, resting on a bed of Portland cement
concrete two feet thick : What should be the dimensions of the
baseplate ?
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 baseplate
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^
IKABiyGPLkTES 50^ ^1212X5 JlTl/ COLrHTs. 34ic
bat the writer ]u« isricri 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^
inches, azati P j:± pr ;.r; •  f m. ^i^ ^ os s a'.* > ioi Vjt
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.
* .»r*.
' " /* A
i — ^ T Lr: :■ if '.. r. n.— e.L*. * ** » Tat '
= ■_ •■ t*. — 4* z. ;.';».
V, ' J
, A;,
■i?^.
Thicdesff = % ^ ^ .■*:_
1 TH'flr*
t  •
^IjJftTrr*
e'^rrHj.'" ill. ^
The ^. *
* "
t^\.
s> r
^Ir
:i.?ri ..
^ . ^
■*•
V
M*
242/ BEABIKGPLATES 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 crushingstrength 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 CrushingLoads, in Pounds per Square Inch,
for BuildingMaterials.
'
. 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
Castiron ....
80,000
Locust . . .
11,720 b
Wroughtiron . . .
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 fullsize 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 onehalf 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 crosssection 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.sliiiigstrength of fullsize 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<'hineshoi», or where heavy machinei'y Is us«m1, or if
the strut is for a railwaybridge, 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\ crosssection 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 hanlpine pust 10 by
b) in. h. s, IJ ft'ct long?
Ans. Ana of crosssection = 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 crosssection = 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 fullsize 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 = 6356 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 halfinch, 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 CASTIRON COLl \S, 249
CastIrou Columns.
For castiron 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 crosssection
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 castiron 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 castiron,
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 castiron,
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 castiron posts, the crosssection 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
castiron column, 10 feet long, 6 inches external diameter, and 1''
thickness of shell ?
Ans. We must first find the metal area of the crosssection 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 fourinch, 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 castiron column 12
feet long, the crosssection being a cross with equal arms, one inch
thick, the total breadth of two anns being 8" ?
Ana, The area of crosssection 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 foiegoing 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 CASTIRON 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 castiron 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 receivedinch 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 castiron base plates,
as shown in Fig. 1, page 242a. The author does not consider it
advisable to use castiron columns with hinged ends, or in build
ings whose height exceeds twice their width.
Tables of Castiron Columns.
By an inspection of the foregohig fonnulas for castiron columns,
it will be seen, that, all other conditions being the same, the strength
per square inch of crosssection 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 crosssection for hollow cylindrical and rectangu
lar castiron 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 crosssection of the column, and the result
will be the safe load for the column.
Example III. — Wliat is the safe load for a 10inch cylindrical
castiron 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 CASTIRON 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 castiron 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 breakingload.
TABLE IV.
Strength of Hollow Cylindrical or Rectangular CastIron Pillars,
(Calculated bt Formulas 5 and 6.)
Length
Breakingweight 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 CASTIRON COLUMNS.
253
TABLE V.
Showing Scrfe Load in Tons for Cylindrical CastIron 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 croeseection.
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 crosssection.
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 CASTIRON 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.
(Jastiron 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'astiron is unquestionably better than
steel, as has been quite conclusively demonstrated by the experi
ments of Prof. Bauschinger, of Munich. Castiron, three quarters
of an inch or more in thickness, is also practically uninjured by
rust, while it is clnime<l that wroughtirOn 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 Hshaped 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 OABTIBON 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 Zbar column, which
it much resembles. The space occupied
by the column 8lig}itly exceeds that ot
both the cylindrical and Zbar 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 wroughtiron 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 CA8T1B0N 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 CAaXIBO^ COUJMNB. 265d
Hollow Rectangular Castiron Columns.
The increasing use of hollow rectangular castiron 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 CASTIBON COLUHKa
TABLE V.6
iSafe Loads in Tons of 2,000 Pounds far HoUovo Bertnngular
Castiron Columns.
LENGTH
or COLUMN IN
FBBT.
W C * a. y.
 x >:• = 6<
H"^ ut^. hy:
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 . 1J ; :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
lj .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 CASTIRON COLUMNS. 255/*
WroughtIron and Steel Columns and Struts
(1891).
Within the past three years wroughtiron and steel columns have
been gradually taking the place of castiron 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 castiron 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 *' honeycomb," and the col
356 STRENGTH OF WROUGHTIRON 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
fireproof 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 cadtiron
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 *• Zbar" columns, illus
trated on pages 267389A.
For the strut bars of trusses twochannels bars, angle or Tbars,
are generally used.
In trusses with pin connectiotis the channel bar offers the best
shape for the struts. Ibeams are also often used.
Streiigrtli of Wroiijjflitiron 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 WROUGHTIRON 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 closefitting pin ; and it is pin bearing when both
ends are thus pitijointed 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 crosssection 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 crosssection, 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 Ibeams 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 wroughtiron
258 STRENGTH OF WROUGHTIRON 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 30121.
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 crosssection, 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 Tbara» used
singly.
STRENGTH OF WROUGHTIRON 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 pinconnected 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 31921.
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 crosssection 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 WROUGHTIRON COLUMNS.
TABLE VI.
Ultivuite Strength of Wroughtiron 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
■ii.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
• • • •
• • •
Jg^g^g<tg<t0g^«j*SQ§^gg0^g*i3*5g »S^
*5i*5
04
4.8
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* fi f^ 91 »" »^
Si'SS te s'jt ZJ $ii:?» $ c
'»x"^i ©•^x'^i ee ■*»o "'•t ■'!« *»
a:
7.
'«v
•^
"*
»«
'ji
*•>
•»*
y.
1/
^ •
'J~
y.
«i=^ 1
't
«i;
1
"^
K
*
^
»
>«^
— 1
•
it
ii
■^
^
7. \
e
'^
M«
t: !
•^
^^
1
«^
^
^ 1
■^
S^
•*^
•
^
•/.
tc
HH
>»;
;«r
»«
^^
"< !
>"
r
r^
H
^
&«
/:
■^
i^
»*
iM
PQ
i.
^
<J
5^
^
(H
<%>
1^
1
^H
jk
'*.
.^
<
*^ •••••■■■
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 »•
• ••••••• •••• •••• ■••■••■•
__i
X
• ••••••• •••• •••• •■>*■»••
5c:9*$$5rJ5i:'^:::5^?^:c.^: ."*^«?:ff.^
• • •
g*3g'»:jO'»:jongi"*:fi': oorw*. S'^g**^'^*'"
n
ri
W
I
Ki
^ I *" II II II** II 11 II 'I It II ■ '■ '1 !l !I II
/.
STRENGTH OF CARNEGIE IRON TBARS.
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«)^«•)
11 1^
• • • •
• ••••• .•
1 ♦« ^ »< ^ •? 00 »»
to t^
«* ^aoSoo QO^Sos
«o 2os «>.
S*3S^^5 « S
00 «N ©»5MO<M r^e>»00i>»
11 rH »i
I, <»1«{M
00 *»'«"•» ^ •» 93 VI
I'.
^:^
R?2S^g^
8:5SJ§
»^5S
OS »i 00 "50 O »J
11 Tl
^*iOOCO
ao ®» i «»
00%tlO»)^«) ^%l
<0
s^
S Sos ^»
38^Sa»ffi^ S?c
11
1^
^ *5ii nn eo*50oc^
11 T1 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
1t
• ••••• ••
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
« . • •
« 11 rH T1
4^
'3
I
•
0»
11
1" 11
• •
CO OS
I, eo
OS* QC
«C lO o o
• • • •
11 CO CO «o
11
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
WROUGHTIRON 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" augle.
iTZ'SaV CDlumn *'Z:B>r OclWMl
^ of Ooliinuu bjr IVflootlon or Bnckllimr*—
ri>nt,'tjtin)ii loIuidds fail either by deflecting bodily out
111 line, or l>f the buikling of the metat botwaen rivata
iLDli) uf supjiurt. Both actions maj take place at tfa*
\ lOUGHTraON 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 wrougbtlron 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 Zbar Column, is believed
to oSer advantages equal, if not superior to those
t any other steel or wroughtiron 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 20pound
channel is .70, for a 35pound channel .75, so that we will assume
.72 as the proper distance for a 24pound 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<MiKlitiroii 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
ROUGHTIRON 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 crosssection, 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 WRODQHTlR< COLDMNB.
TABLE VI.
TJliiraate Strength of Wroiightiron Cotumiu.
Pordiaeniit 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 WROUGHTIKON 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
4f" 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 WROUGHTIRON 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 WROUGHTIRON 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
12
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 wroughtirou 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 WROUGHTIRON 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
• •
1H
COCO
*ico
1 1
;j
•
,j
*<
o
ja
. rx
C50
^•N
00 ^
O
a
•
a
J5 xr5
— ^^ 2^
tt
ti
^s
1 1
I
9>
(O
OQ
1
• •
• •
• •
• •
1 1
<{
j*^^
xo
11 CO
^CD
»^
ri 1^
1H lH
•
% 1 QD
x»o
• •
Oii
(NO
• •
CDQO
5K
11
11 1^
1^ 1H
»hO*
\J
»j
*^
c
•s.
O
coo
x«
C40
Oif
i ^
2 ;s
5 1 3^
 CO
^%
• •
(NO
s^
tt
■n
a »>'
o
o ^
7:
c
H
^ !
1 *
0; .
« 1
Sao
ss
ss
ss
<
1
? ^
• •
^co
T^ 11
• •
i« fi
^5
^d
j
1
2 oi
•
c:x
XX
rt
o«
oo
1 ^
.2F
5 a
1 1 1
• •
• •
Ti 1H
1H(N
tt
t;^
•
J3
^co
*o
Ofth
• ■»
CON
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^»
ZBAR COLUMNS. 279
ZBar Columns.
Within the past three years, what is known as the Zbar column
has been introduced, and is now manufactured by all the leading
iron mills. It is built up of four Zbars, 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 Zbars are furnished at a lower price per
pound than channels and Ibeams, 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 Iheams. —
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 terracotta blocks, for
fireproofing, and finishing with plaster or cement, and the airspace
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 Zbar 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
ZBAK 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.
ZBAB OOLl rs. 281
DETAILS OP BTANDABD CONNECTIONS
OPI^EAMSTO ZBAR COLUMNS.
leot ZSarColun:
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 ZBAR 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 Ibeam girders and single
iloor beams to Zbar 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 Zbar 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 Zbar
columns of different lengths, as manufactured by Carnegie, Phipps
&Co.
The values for steel Zbar 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 Zbar 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 Zbar columns, page 287. we
find that for a length of 30 feet, a 12inch 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.
EBAB COLUHN DIUENSI0K8.
ZBAR COLUMN DIHBHeiONS.
fOf J> fOl
M
^m
 ^y*
% of ZBsT columns in inches for mil
mum thicknesaes.
Note. — In columns A. B, C. D, E, and F, the thickness of the
Zbars 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 ZBAK C0LDMN8.
BAFB LOADS IN TONS OP 3,000 LBS.
BTBBI. ZBAR COLUMNS.
Square Endt.
ine per BQQflrB inch 1 18.000 Ibn., for length of TO rsdll ornode
ilely fsclor 4 : ' ^ n.lOOSI^, for lenglhe over so ndU.
90" ZBAR COLUHNS.A.
4ZB«riSi" 1". 1 Web Plate U" y I". Side Plata SO" w
SO" ZBAK COLUMNS.— B.
Secllan: 4ZBirsei" > 4". 1 Web Plate 14" > 1". 4 SMe Plata *0" wlda.
BZBBL ZBAB COLUHNB.
BAS% LOADS IN TONS OF 1,0I» LBS. .
STBEI. ZBAR OOLnMNS.
Sgttare Endt.
Allows
d«™in*per.quflretacl
Bsfely factor 4 ;
I,.iia,0001b8., fi
jrlenellHofBOradiiornnder.
.torlcni^thioterBOradU.
BTEEL ZBAB OOLUHNS. '
, . <, ll,«aO IbB., rar hnglbe oT 90 ndll 01
'"( 17,100ST J. , tor li^iigUiB owr SO ni
Section : 4 ZBara fll
STEEL ZBAR COLUHM&
8AFB LUADS IN TON» OP 3.000 LBS.
STBBL ZBAR 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. ZBAR 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 Ibeams 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 :
Q ^'
55 I
21
o >,
o =
O }
CO .T
!5 ^
O 
H :
H4
Q •>
<
o
H
<
CO
7.
y
I HUOJ,
p.)A\(/l(V I
•MIOJ, I
I III piJo'l .»iWS i
I
r
p,).\vt/[iv 1
I J<Ki HIIIUJIS j
ui pno'i oj«tj
I xj'i III ,, J '
I J.kI SMlj.'JlS
?/ X — «rt X
St " 0» — ii
coc:= cere ■ccso
^^ ■ ^^ ■*
2: t<>»i I 11 ^ii —
i ^ — i 3
c» »» c* ci ■?»'
at«»aK wSrS w*^3*
cj I? ?/ ?»' ?* *?* •:» c* c» r* 7» *:»
1 "" '" ! '"""
?J /. — . at
c *c •* ~.
i^Xj* —
1^ r^ ft rm
1 ^^ ^
*c i 3^ W a^
at — ■^ ar
si ^ ?r j»
^  ■« "^
ci:*^* ?*":*
T» if ti ci
• » » >
^ ^ •■ ^«
T» ci C» ti
f^r
7» X •^ I . x I'; X 7' — it k". ». ^» c; ■?» X e
— — jrT"— I— ^«^ xjiv jBvirK I
fJK'Ejrx — I — ■« — »— — VX'— w I
'It 'I ^i '' T* ?»■:»■:»'?» cfc^T*?* 1/ •:>■:'?» 
«•••• ••■• >■«■
*i 7 — » : r. 1^ i. ■:» * ?» o r. ?» c ?» x c
**
•s«n III ,, :j I
I p»A\onV I
I "r""'! i
I HI piio'i .)J«s.
I
sq'I Uj . :J
jiflllliMiS
p.i\\«ill\
?»' ?i *:* Tf ■:
1
1
■. "li ?»?•
* * •
Z! * ' 3i if
    i
w ^ ^ ^
c = rc
w e C i.*?
5» 7 — i.'i ^
■J* — W » "
1— ^ ^ »—
^ 2 'y C*
iC ^ » K «
?s = i
SK a a «
s = co
2 7 i J
■r* 7' T» "?» ?♦
c» :» ?* CI
• « •
c* ?' rt —
c» — ■:# —
www
w _ V? t..
T!a I
71
xi'j III . ;i
I j.Ml>uii:.is
p.iAvo'nv
Ml piiD'l .)jis
"' ! H'l 111
itil '>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:*??
_  _ I
:=>..= c e = ii7
• I
2* 7 — ii / i'^ 7 T«  7> • j. ^ a: ^ t n
■»« w  M •" ".' S ,» Z^ !£ ' ^
": r 1 " ; " T i •■:?. ^' V T y
71 7» 7' 7» 7* 7» 7* ,■ 7»
J , r. • 
_.i. __#i7 • wl*»~
^^ •« ■** ^« i* A ' "l  • * ^ XI *1
i~ i". i" •" •" 7» 7» 7» 7'
■•..•,1 ..........
x f f x ■/#—■*_ X y 9? *
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 coverplates 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 ** capplates," and joins sections of colamns
firmly together, making a continuous column.
Tests made in the hydraulic machine of the Keystone Bridge
Works on 14inch 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 2SQe2S9h 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 coverplates, 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
523.0
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 BENDINGMOMENTS.
CHAPTER Xir.
BENDINGMOMENXa
Tmk bondingniomont 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(hngnionicnt 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 bendingmoment at any crosssection 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 wallend 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
^^
BENDINGMOMENTS.
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 bendingmoment of the beam.
Knowing the bendingmoment 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 bendingmoments 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 bendingmoment for these cases, and then
show how the bendingmoment for any other methods of loading
may be easily obtained by a scale diagram.
Examples of BencliugrMomeuts.
Case I.
Beam fixed at one end^ and loaded
with concentrated load W.
Bendingmoment = 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
BBNDINGMOMENTS.
Case III.
Jionm fixpd at one end, loaded with both a concentrated and a
distributed load.
/.,
Bendingmoment = P X Lj + JK x ^
Casr IV.
licam supported at both ends, loaded with concentrated load lU
centre,
W
J Bcndingraoment
Case V.
Beam s^iipported at both ends, loaded with a distributed load W.
V ,  ■■:
'm%
<?;
'n
Fig.6
Bendingmoint;nt
Cask VI.
livam supported at both tnids, loaded with concentrated load nol
at ('('litre
Bendingmoment
= Wx
m X n
BKNDINGMOMENTS.
293
Cask VTI.
Beam supported at both encU, loaded rcith two equal concen
trated loadSy equally distant from the centre.
Bendingmoment
= 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 bendingmoment aro the W?ight, the span, and the
distance of point of application of concentiated load from each
end.
The hendin{imoment 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 bendingmoment 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 bendingmoment at o.
Beam with two concentrated loads.
To draw tlie bendingmoment for a beam with two concentrated
loads, first draw the dotted Hues ABl) and ACD, giving the outline
294 BENDINOMOMENTS.
of Ihe bendingnioment for each loful separately; KB heing eqiwl
toWx It^^ „„(, rr pqjial to P X '^
Fi9.IO
Now, Ihe beiutingmomnnt 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
bendlngniomunt due to both loaits, tlieu, would be the Uiie
AIl^C'iD, anil the greatest bendingiuoment would In this parUe
ular tasu be FC'i
Jleam with three concentrated load*.
Fiy.tl
Pmcpficl as in the laat ease, and drawthp hendingmoment 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'iidingii'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 bendlngmoraents. if, now, we wanteil the
bendlngQiomeat 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 bendlngmomcnt 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
ingmoment 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 tiendingmomeiit at X would be BE. The posi
tion of the greatest ben dingmoment will depend upon the position.
of the concentrated loads, and it may aud may not occur at tlie
296
BENDINGMOMENTS.
Example. —What is the greatest bendingittmneilt 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
ingmoment 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 bendingmonient 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 bendingnioment 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 bendingnionient 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 bendingmomeiit \
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 stiains, either as girders or as
posts, depends not only on the area, but also on the form of the
crosssection. 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 crosssection is the sum of the products obtained by multiply
ing the area of each particle in the crosssection 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 wronghtiron beams or posts, it
may, for all practical purposes, be considered as passing through the centre of
gravity of the crosssection.
For most forms of crosssection 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* crosssection 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 crosssection 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 reentering 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 —
Yh*
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.
IReam (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
yh'i
Ie6~^
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 32224 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 BEAMSSTEEL.
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 BEAMSIRON.
[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 OHANNELBARS—IRON.
n
IB
^
I.
n.
Moments
IV.
VI.
R»dii of
Distance of
Siz<', in
inches.
Weight per
foot, in 11)8.
Area of
crosssection,
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 CHANNELBARS— STEEL.
;b
I.
II.
IV.
VI.
Moments
Radii of
Distance of
Size, in
Weight per
foot, in lbs.
Area of
crosssection,
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 ANGLEBARS.
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 TBARS— 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
229.2
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
dr^^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
7.02
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 ANGLEBARS.
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 TBABSL
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 ZBARS.
PHCENIX IRON ZBARS.
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 DECKBEAMS AND TBARS.
c
c ^
=o^
IIL
IV.
V.
VI.
"T.'iini
A4 »a^ ..C"?.
n L>»
u
VjnufSLtB 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
101 ;
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 CHANNELBARS— 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 ANGLEBARS— 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
2x2}
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 arrowheads.
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 arrowheads.
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
.eil
ANGLES WITH 'UNEQUAL LEGS,
%5;%:^^^;5^^ S$5SSSS55S5JSSS:
Radii of Oyration given, correspond to directions indicated by arrowheadt.
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^ianif 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 bendingmoment 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
bendingmoment in such case being oneeighth 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 NewJersey 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 9inch, 15pound PhoB c
nix channel bars are placed 4.6 inches apart, ^^ ! i"^"1
K44J
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. Onehalf of 4.6 =
2.31 .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, 5pound 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 wroughtiron 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 " bendingmoment ; " 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 bendingmoments 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 bendingmoinent 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.ittrijil.
f{.
in \hj*.
(':it Iron
:..%«
\Vrnii"ii Iron
l;!.i)i)0
si.r;
Ki.iXK)
Ain«Ti«;Mi :i*h
'.».(KK)
.\ni'rir:iii rcl IhtcIi
l..H<)J)
Am i< III \i!lM\v Itirrh
i.iia)
■\iiifiiini 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
Aniirican 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 onefmirrh
th.ii f"" t'l. bri'akiFiirl'»ads ; lor ca.st iron. «mesixth : for wt mm I, une
tliirl : jii'l i'"'!' >iniii, <inesixih. Th*' constant'^ lor wimmI an bJl^4•«l
wj.m ill f.iiiii I'ti iiiaili" al the Massacliusetis Institute of Teeli
i,<i<i\ Ml' :i l'!iil^i/i liiidHrs of the usual iuality 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 twotbinls nf ilu aUivr values Ih iiMd. The
:\U' ioj'l I T iIm' 'rnntiMi. iMio'nix. and »*arne«;ie s4><tions. ustii as
ihar::. :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.
'!'•.'■ .■■ ii riain ( a^« « of be.ini> which most fri'ipieiitly occur
:ti i<ii;li.:ij •••ii^t nni inn. f«ir which ftirmulas can In given by wtiich
tin at I'ad^ fur llie biaiMS 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 bendingmoment, or moment of resistance, and
find the beam whose moment of resistance multiplied by R is
equal to this bendingmoment, 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 bendingmoment caused by these loads to be
480,000 inch pounds.
Then, as bendingmoment 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 hardpine 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:tinch 125pound beam. Tlie
moment of inertia of this beam is given as 2S8; and its moment of
resistance is onesixth of this, or 48. Multiplying this by 12,000,
we have 576,(X)0 pounds as the resistingforce of this beam, or
96,000 pounds over the bendingmoment. 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 crosssection to carry any load, however irregularly disposed
it may be.
Strength ol' WrouglitIron Beams, Clianiiels, Aiijyle
and T Bars.
It is very seldom that one needs to compute the strength of
wroughtiron 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 onehalf the height of the beam in
inches. These formulas apply to iron beams of any form of cross
section, from an Ibeam to an angle or T bar. For steel beams,
increase the value of W onethird.
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, onehalf
the weight of the beam should be subtracted.
Example 1. — What is the safe load for a Trenton 12iinch light
Ibeam, 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 12inch Carnegie iron channelbar, 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^ebars, 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 12iinch beam at 125 pounds per yard, or by two
9inch beams at 85 pounds per yard. But the 12Hnch beam, 21 feet
long, would weigh only 875 pounds, while the two 9incfa 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 IBeams.
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 TBars
— 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 Ibeams.
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 24inch
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 IBEAMS— 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 IBEAMS— 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
CHANNELBARS AND DECKBEAMS— 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.
ChannelBars.
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
DeckBeams.
ch
65
55
91,800
63,500
54.7
35.1
4i
4*
6.29
I
5.35
* For coefficient of steel bars add onethird.
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
TB.
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 onethird. For any other tfma dMdo tilli
foeiBcient by span.
SISENGTH OF IBON AND STEEL BBAMS.
341
TRENGTH, WEIGHT, AND DIMENSIONS OF CARNEGIE
IBEAMS— 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
CHANNELBABS— IRON.
STRENGTH OF IRON AND STEEL BEAMS.
34a
STRENGTH, WEIGHT, AND DIMENSIONS OF CARNEGIE
CHANNELB 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
IBEAMS— 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 15inch 145lb. 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
IBEAMS— 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
IBEAMS— 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 .nKFNirrEi <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.nI 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*"
uiiiiout
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 BBAMSmON.
.1 »«■ d.'llprll<>1» In Inc
onwiMjndins i
~.uia markwr* taT 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 '. ■5g
1 £ M ^
Mil
t !!l
ss ; r
§ Z^^M
— c i §
1 " sIC^
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 onebair 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 onefourth 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 bendingstress at the centre of the beam, mulling
from a brick wall of the above shape, is the same as that caused by
a load onesixth less, concentrated at the centre of the beam, or
twothirds 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 bendingstress 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 bendingstress 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 4inch heavy beam would just about
support this load; but as a 5inch light beam would not weigh any
more, and would be nmch stiffer, it would be better for us to use
two 5inch 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 onethird
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
floorbeams, they are often placed side by side, and should in such
Fig. 8. Rg. 9. Fig. 10. Fig. 11.
cases be furnished with castiron 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 boltboles being iimmI
the 15ineh and ]2iinch 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 angleiron 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 Ibeams 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
Ibeams 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 IBEAMS,
WHERE STANDARD COxVNECTION ANGLES MAY BE
SAFELY USED, WITH BEAMS LOADED TO THEIR
FULL CAPACITY.
Stbbl IBeams.
Iron IBbams.
^«
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 IBEAJfS.
%
(H Ha ten* _rm
■III d++l
+
+ +
4 ^4&tl.».»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.
HEIAR^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 CASTIRON. T7700DEN, AND STONE
BEAMS — SOLID BUILT BEAMS
Castiron Beams. — Most of our knowledge of the strength
of oastiron beams is denved from the experiments of Mr. Eaton
Hodgkinson. From these experiments he found that the form of
crosssection 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 castiron 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 onethird of that of the bottom flange. As the lesull of
his experiments, Mr. Hodgkinson gives the following rul(» for the
breakingweight at the centre for a castiron beam of the above
form :—
Fig 1
Breakingload in tons =
Area of hot. flange ^ depth ^ o 426
in square inches in ins.
clear span in feet
(1)
Castiron 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 crosssecticHi, 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^. (>).
Kn—>
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 caflCH 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.— Cokfficient 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'hiiroM
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 onethinl of tlio hn»aking
u<iL:iii ot tiiiilMTs of the same si/.«> and iU: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 tlitois 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 hardpine 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 breakingload.
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, onehalf 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 yellowpine
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 HABDPINE 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
tt Ol 0»
^ rH rH of e*
^ o ^
rH O f O
11 W TH K5
*> *k •» *> ^
ri Ti e< e< ot
g
s i §
» «>
fi 11 e» e<
Ol
^ s S^ s
n iQ Q a
o et OD «
rH tH Tl 01
kO
OD
Xi
no
o
09
Bi <
^5
r^
Ito
^
yf
Id
CO
CO
^ lO t>
I S I P § s
iH VI In 09 00
eo
<M
OD
:0
00
s s s
o» t <?*.
00 eo *
OD
3
S QO g p «
ii" o» o» 00 ^ *o
OB
a
s
u
>
V
c
>
o
eS
09
O
h3
X
Xi
© o rfS^ «o
.T iT of of cc
i
>A CO
? S5 8
«o i< c*
00 <
'^^^©lOIOO^OlOO
jog
<Eto{^aoa»oo<'^*0(0
fl »M »< »H n »<
372
STRENGTH OF WOODEN 11EAM8,
For beams wilh a rectangular crosssection, 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.— Coefficient fob Beams.
Material.
.4 lbs.
Castiron
308
Wrou«;htiron
()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 cooflicient A are onethird 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 dwellingliousc floor, and small halLs, etc. Where
there is likely to be very much vibration, as in the floor of a mill,
"* gymnasiumfloor, or floors of larg(> public halU, the author
uenils that only fourtifths of the above values of ^ be used.
BELATIVE STRENGTH OF BEAMS. 375
Example 1. — What load will a hardpine 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 breakingload.
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 firstclass buildings it is
desiied 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 fiveeighths 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 onefortieth 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 firstclass 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 onethiriieUk 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 sixsevenths
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 firstclass 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 onefortieth of an inch per
foot of span,
e, = Constant, allowing a deflection of onethirtieth of an inch per
foot of span.
Material.
Cast iron . .
Wroughtiron
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.
Reaction of either abulment,
R,=R, = i\W; (7)
Reaction of either cential support,
B, = A'j = U yV; (81
r
or the reaction of the end supports is lessened threetenths, and
that of the central supports increaseil threetwentieths, of that
which they would have been, had three separate girders of the samp
crosssection 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.
Reaction of either end support,
R,=R, = Uol; m
Reaction of either central support,
R^ = R, = \htol; (10)
hence the reactions of the end supports are onefifth less, and of
tlie central supports onetenth 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 bemlingmoment 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 bondingmoment which causes the difference In the
Ijoaringstrength of continuous and noncontinuous girders of
the same crosssection.
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 bendingmoment of a beam under
any conditions, we can easily determine the required dimensions of
the beam from Formula 11.
The greatest bendingmoment 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 bendingmoment in a continu
ous girder of two spans, / and /i , loaded with a unifonuly distributed
load of w pounds per unit of length, is
Bendingmoment = o /# , > » (12)
V/hen i = f , , or both spans are equal,
Bendmgmoment = g, (12a)
which is the same as the bendingmoment 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 noncontinuous.
Concentrated Load, — The greatest bendingmoment 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
Bendingmonaent •=^ h^(W+Wx). (13)
When W = W\^ov the two loads are equal, this becomes
Bendingmoment = ^WU (13a)
or onefourth less than what it would be were the beam cut at the
middle support.
Continuous Girder of Three Spans^ Distributed Load. — The
greatest bendingmoment 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,
Bendmgmoment = .,.>. , ^. v . (14)
When the three spans are equal, this becomes
Bendingmoment = 7a» (14a)
or onefifth 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
• s s g
£2 Ai( S
s % s
i ^ ^ § § R
i s
00
i =
C8
'J
Ei
CQ
m
I
Q
5 t;
5 t:
~ S j
= tc I
c a
N —
es a
'• :s
c "s
if /
5 i*
•^ " _r ^ «r
. I
«^
Jl I I I a ^
— a — c^ 'f "rf * "'
X
X
B
<0 CD O O W
© 25 ^ JO •
»^ ^4 *4 *M P«
STin SS AKD DBFT^ECnON OF BBAJI&
389
m
o
p
OQ
I
s
« 2
OEl
§ o
K
p. a
O o
fl ^
^1
OS «
^§
§ ^
I?
'« 2
o o
t ^
o :S
^*
OD CD*
fp "a
S c
O V
•° s
•E «*
J2 ^
g a
B
9
8
«8
2 a>
&*■
cox
Q0«
a 9
QOO
CO
1^
a
«D QO
O M ^ lO «
^H T^ »^ »^ 1^
00
^ o
I
i
OD
s
«8
i § §
I §
s
e«
s S i § i
t^^ r4^ 9f
CO t
(O o
00 "^
g s
9i
O '
»l 11
00 •
Ti Oi G) 01
00
1H
00
IS 8 S ^ S 3
•• ^ O t *< fi5
»• ^ r • _ ^ ^
11 11 ©I <ji 00 eo
O
P t Q O
©» 1i Wt «
r1^ ^ l^ O
ri Tl »i 01
CC 00 •^
i
f2
f2
Tt ir^ (?» CO ■Tf lO O
00
•»r CO 5 '
1^ c^ •» iB < ^
»<" e* c* CO »o « t 00
8 '■J'
00 X
1
9
a
a
o
ao
gcOXOiOMTfOCO
STIFFNESS AND DEFLECTION OF BE A J.
Eh
o
a
CD
c
83
'O
e3
O
tii
o o
a ^
o ^
C V
•= y
^ s
C «3
rr *■
E :
? 
^ *•
& i
^ < ^a 
W ? fe
m
c c
C tf
M
<<j
^•^
o
^ 5
2 r
y\
OS
Depth
of
beam.
1
«D GO e> o e« ^
rN »iN tH
si
ig ^ s s s ^
'^ Tt 0« ^ t»
891
1,081
94 D
X
g s
eo ^
J a s g
CZi
E
O
 a
2 g w
CO ^ So
II.
I § s
r ^ ^
rl il ©»
§
• CO S
» 11 00
.a
§11
of of of
s «f
fi S M
5 >2 ***
OD ^ 00
•■ » »
•^ « «
g s
§ 1 1 g I
^' M of Cd «f
»i CO
2 g 8 e 22
lo 00 r « 01
» ^ r m ^
»• 1^ Ot •© f
1 S § k s
to o_ in o o)
iI c*" 91 ef ^ rf frf"
'81
X
§ 5 i s
g 1$ §
»■'?» CO W lO I 00
STc g
» O X
«>4 S v« S «4
o
ja
i 9
o
ja
IS
ft*
o
' I
STIFFNESS AND DEFLECTION OF BEAMS 391
ExABCPLE 2. — What should be the dimensions of a yellowpine
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 rollediron 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 fie
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 iioncontinnous 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 gnder 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
reaction of the support Ri will be represented by the fonnula
R =: *
32
(i)
the reaction of the support R.^ ^Y
«2 = j^(ir + ^r,),
and the reaction 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.
Reaction of left support,
10 r ^« + / « 
2L' 4/(fh/.)J
Reaction of central support,
R, = w{lhl,)R,  /?3.
Reaction of right support,
^^  2U' 4/, (/ + /.)]•
(4)
(5)
(6)
When both spans are equal to /, the reaction of each end support
is i Kj/i, and of the central support t '«' ' hence the girder, by being
contuuious, reduces the reaction of the end supports, and increases
thai of the central support by onefourth, or twenty five per cent.
394
CONTINUOUS GIRDERS.
Continuous Girder of Three Equal Spans, Concentrated Load of
W Pounds at Centre of Each Span.
Reaction of either abutment,
R,=R, = ;\,}V; (7)
Roaction of either central support,
liz = H, = U ^V; («)
or the reaction of the end supports is lessened threetenths, and
lliat of the central supports increased threetwentieths, of that
which they would have been, had three separate girders of the sam^
crosssection been used, instead of one continuous girder.
D
Continuous Girder of Three Equal Spans uniformly loaded with
w Pounds per Unit of Lent/ th.
Reaction of either end support,
/r = /?4 = i tot; (9)
Reaction of either central supi>ort,
/?, = /^^ = ^,( ,o/; (10)
hcnco the reactions of the end supports arc onefifth less, and of
the central supports onetenth 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<lingmoment 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 noncontinuous ginlers of
tie* >ame crosssection.
('(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 bencUngmoment of a beam under
any conditions, we can easily determine tlie required dimensions of
tlie beam from Formula 11.
The greatest bendingmoment 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)endingmoment in a continu
ous girder of two spans, / and /, , loaded with a uniformly distributed
load of w pounds per unit of length, is
Bendingmoment = o /^ ^ > » (12)
V/hen Z = i I , or both spans are equal,
top
Bendingmoment = g, (12a)
which is the same as the bendingmoment 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 noncontinuous.
Concentrated Load, — The greatest bendingmoment 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
Bendingmonaent ^ ^iHW+Wy), (13)
When W = ITi , or the two loads are equal, this becomes
Bendingmoment = A IT/, (13a)
or onefourth less than what it would be were the beam cut at the
middle support.
Continuous Girder of Three Spans^ Distributed Load. — The
greatest bendingmoment 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^
Bendingmoment = .,... , ^i y (14)
When the three spans are equal, this becomes
xol^
Bendingmoment = Tqi (14a)
or onefifth less than what it would be were the beam not con^
tinuous.
396 CONTINUOUS GIRDERS.
Conconfrated Loads. —The greatest bendingmoment in a con
tinuous girder of three equal spans, each of a length 2, and each
loaded at the centre with [V pounds, is
Bendingmoment = ^,^ Wl, (15/
or two fifths less than that of a noncontinuous 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 noncontinuous 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 seventlsteenUit
of the nuneontinuous 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 onehalf that
of a ncncontinuous 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 eleventwentieths of the noncontinuous 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 noncontinuous girder.
First as to the Su2)ports* — We see from the formulas given for
the i*eaction 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 canied 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 bendingmoments, we can easily calculate the
.^9S CONTINUOUS GIRDERS.
strongth of a continuous girder, knowing the formulas for its bend*
ingnioni(Mit. From the values given for the bendingnioments 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 buildingconstruction, 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
crosss(»etion. The fonnulas for strength are deduced from the
fornuda,
Bendingmoment = ;. * (22)
where 1i is a (constant known as the modulus of rupture, and la
ei^litecu times what is generally known as the coefticicnl of
stn'nijth.
SiKKNJ.Tn. — (.'ontinnoits tjirder of two equal Hpana^ loadtd
nnij'nnnhj oi'cr ((ir/i span^
2x nx U^x A
lirealvingweight = 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
Breakingweight = 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
Breakingweight = 3 x j • (26)
Stiffness. — The following formulas give the loads which the
beams will support without deflecting more than onethirtieth 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)endinginoinent by the for
mulas given, and, from tahles saving the stiength and sections of
rolled beams, find the beam whose moment of inertia =
bendingm oment X depth of beam
2000
•.vhen tli«* beuilinsj moment Is in foot pounds.
For (^xjunphs we have a continuous llwam 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
Bemlingmoment = 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 teninch lK*am, we should have 10 X 10 = 100; and we
sec from Tal)lcs, i)p. 2(50272, that no teninch beam lias a moment of
Inertia as small as KM): so we will take a nineinch beam. W X 10
~ INK and the lightest nineinch beam has a moment of inertia of
\Y,\: so we will use that beam. In tluj case of continuous Ii)eams
of three e(nal spans, (upially load(>d with a distributed ItKid. wi*
may take fourfifths 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 Ibeam 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 eoefll
ciint is nearest to this is the nineinch 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 reactioiis of the sup]M)rts ami for the
(l<tliiri«>ii oi (v)iitiiiMoiis Lcirders with eoneentnili'd NhmIsi, were
vnitiil bv Mm aulboi b> means of careful experiments on small
sr«'! bai>> IIm other forinulas have Inn'ii veriH«Ml hy <>oni]iAri9un
witli iitbir iiiilboi'it ies, wliei'i* it was His.sible to do so; though one
or iwo ot tbf la^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
fouithsinch 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 wroughtiron 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 hardpine beam
to bend the same as the iron plate, we must put only onethirteenth
as much load on it. If the wooden beam is eleven times as thick
as the iron one, we should put eleventhirteenths 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 onetwelfth of the
braidth oi the beam, approximately : —
402 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 = \/ 2/7if 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 railwaystation in which the second story is devoted to
offices, and where we must use girders to support the second floor,
of twontyliyc 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<iuirea8eventeenlncli by four
teeninch beam. If we were to use an iron Ix'ani, we find tliat a
fifteeninch ln^iivy iron beam would not have the requisiti^ strength
for this span, and that we should be obliged to use twotwelve4nch
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 oneeighth. 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 twelveinc4i by fourteenincb girder. NoVt
FLITCHPLATE GIRDERS. 40;^
for a comparison of the cost of the three girders we have considered
in this example. The seventeeninch by fourteeninch hardpine
girder would contain 515 feet, board measure, which, at five cents a
foot, would amount to $25.75.
Two twelveinch iron beams 25 feet 8 inches long will weigh
2083 pounds; and, at four cents a pound, they would cost $83.82.
The FlitchPlate 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 hardpine beams. FlitchPlate 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 ironphite 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 bellyrod 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 Merod, 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 crosssection required to resist these stiesses.
Foi: sixciu: srui t iiellykod 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 bellyrod 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 (MMal 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 erosss(>eti(>ns hy (he folluwliig
rules ; —
eonip. in strut
Area of crosssection 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 tierods = 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 crosssection 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 crosssection 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 castiron.
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 wroughtiron.
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 strutbeams and two rods. We van
allow the bellyrod 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 striitheams we will make of spruce. Tlie compression in
the two strutbeams = 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 crosssection, 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 strutbeams three 65inch by 12incli 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 castiron;
but, if we are to build but one, we might make the strut of oak. If
22500
of castiron, the strut should have ^.w^q , or 1.8 square inches of
crosssection 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 stmtl)eanis
7 inclu's by 12 inches (of spruce), about 31 feet long, two bellyrods
U inches diameter, and a castiron 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 lectureroom 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 thiee equal spans of 13 feet each. The tietaun will
•TUUSSED BEAMS 409
consist of two hardpine 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 floorarea 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 tiel)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 crosssection m the strut,
which is equivalent to a 9inch by 12inch timl)er . or, as that is
not a merchantable size, we will use a 10inch by 12inch strut.
The tiebeams will each have to carry onehalf of 106000, or 58000
5800()__
pounds ; and the area of crosssection to resist this equals ^j^ —
27 inches, or 2^ inches by 12 inches. The distributed load on
one section of each tiebeam coming from the floorjoist 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 tiebeam must be 84^ inches + 21 inches = 6 inclies : hence
the tiebeams 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 tiebeams 6 inches by 12 inches; two rods at each joint, J
inch diameter i and strutpieces 10 inches by 12 inches.
A\0
BIVETEU PLATKIHON GIKDKHS.
CHAPTER XX.
RIVETED PLATEIRON 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 impiacticable, we must employ a riveted ironplate 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
'' webplates/" are made of a single plate of wronghtiron, rarely
less than ont^fourth, or more than fiveeighths, of an inch thick,
and geiKMaliy tliivei>ightlis of an inch thick. Under a distributed
load, the web of threeeighths of an inch thick is generally snfll
ciently stiong to resist tlu^ shearingstress Ln the girder without
ng, provided that two vertical pieces of anglelroD ; r ivebed
>^eb, near each end of the girder. Tliese ve ii i !«■ of
>n or Tiron, whichever is used, are c ";
ten the girder is loaded at the centre, ana :
• If
K4": .
RIVETED PLATEIRON 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 shearingstress 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 angleirons. It has been found,
that in nearly all cases the best proportions for the angleirons 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 angleirons are
joined together should ho, threefourths of an inch in diameter,
unless the girder is light, when fiveeighths 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 shearingstrain. To calculate the
strength of a riveted girder very accurately, we should allow for
tilt* rivetholes in the flanges and angleirons ; 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 rivetholes. 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 webplate, or the distance between the flangeplates. The
w(^b we may make either onelialf or threeeighths 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 PLATEIRON GIRDERS.
If the load is distribvted^ divide onefourth of the whole load on
the girder, in tons, by the vertical sectional area of the webplate:
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 oneeighth 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 plategirtler 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 8inch by 8inch
angleirons, we have 5 s(iuare inches as the area of the plate. If
we make tho plate 8 inches wide, then it slK>uId be5r8,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 2iineh by 2jinch angleirons. 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 PLATEIRON GIRDERS.
Coefficient for deLenninin;; Ihe area required in flanges, allowing
10,00IJ pouiiils ]wr siiuare incb of crosssection 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 eoefli<!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 PLATEIRON GIRDERS.
Girders intended to carry plastering should be limited in depth
(out to out of web) to onetwentyfourth of the spanlength, 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 bendingmoments 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 crosssectiou 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 " angleirons = ').iy stpiare niches.
:U X ;)f' X f *' =({.4 *•
4' X 4" X f *' =7.4 "
4f' X 4f' X f ** =v{.4 «•
RirKTBD PLATRIRON QIBDKR8, 41
PSPSSBsasss I
SBSSESSSS38K
SSSSSSSSSSgg
SSSKSSSSKSaS '
gS3SS3SS5
!SS
Si!5
S
SSSS*5:;«S3(3;
as
55 '
g'SSSSSSSS
^gv
s:i^sssgs;3ss$$$:is
RlVbrrKD PI^TEIBON UlKDEHa
1^
t I' '
n I
a ■'
sisssas^psisus >
SSiSSiSSESESS I
gSgSSSSfiSSSSSSS
SSS£«SS$$9:;
ESESSSSSSSSSiS '
gSSiS^SSSSSS!
Si«S^S8Sgi88 I
s3:.^«iSS3iS«3«««S
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 Ibeams 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
SX" eti.«l (Caiiiogle) Ibeama 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.
XIS" Bieel (Carnegie) Ibeams and 9 utee] plates 14" > "
420
STEEL BEAM GIRDERS.
STEEL BEAM GIRDERS.
SAFE LOADS IN TONS, UNIFORMLY DISTRIBUTBIX
silS" steel (Carnegie) Ibeams and 2 steel plates 14" x i"
«
tt
c
/>
jf — 6 — >r«,
^ 1
^^
a)
^
b=^'„„ . ,
Si
Vi" steel
. . ^
r, uJ
X>
steel
Ibeamn.
^^ *^»
" 12" steel
^Ui
O
plates,
40.0 lbs.
2
steel
Ibeams,
£
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^
1 1
I
JS 9
«.s
1
«k«M
« ♦» 'a 1 o • »
(
j:«ii*i
»e'2
SCJ4
kl
r^
s o o
' a; 0) C .iM S X
s
s o o
' 0) V o
•^*o
4— t
a
c
— ^ 00
*^4J XI
.U o
^s
1)
o
S
75
•^ < «■
«5: tt  . "" J, 3 
 = * id  
S3
5
53 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
22. :«
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
SIO" steel (Camegic) Ibeama 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
CASTIRON ARCHGlRDKUa.
CHAPTER XXT.
STRENGTH OP CASTIRON ARCHGIRDERS, "WITH
WROUGHTIRON TENSIONRODS.
Oastiko.v jircligirders 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 wroughtiron tierod.
rii«' ti('!().l is madi* from oneeighth to threeeighth8 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 eastiron \iha so far dr
ll««t«i. tliai its lower edge is >ubji'eted to a severe tensile strength,
whirh castiron <'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 carryiig
CAST lEON ARCHGIRDERS.
423
a load. The girders should have a rise of about two feet six inches
on a length of twentyfive feet.i
Rules for Calculating^ Dimensions of Girder and
Rod.
A castiron archgirder is considered as a long column, subject
to a certain amount of bendingstrain ; 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
tensionrod, 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 castiron arch is to alloiv one square inch of crosa
section of tierod 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
OASTIHON ARCIIGIRDERS.
tiiral Iron Work," shows ihe section of the castiron 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.
12" X 3''
i— '^ 8
12" X »•'
12" X 1"
Bulb.
;i" X 2"
1// y ,>//
X 2"
3i"
4^'
I
Substitute for Castiron AreliCiirder,
In tlu* castinm archj;inior with wrou.uhliron tcnsioiiroil. 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 castiron skewbacks,
wliich an* thcnis(»lvcs held in ixwilion by a pair of tierods, as in
In I Ills case, Fornniht 1 will still jjivc the horizontal pull to be
resistci by the tierods ; 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:ili. It will alM) ln' rem*>ini)<3n'd that the hpan iK tti \h.'. (i/irt/^jr taken In feet*
mile. DiliiTwiif spti'iti'd.
Kx.vMPi.i: I. — It is desired to siipiM)rt a 12incb brick wall Ai)
til* liiuli «'\ci an n]H'nin.Lr '► l'e«»i wide, with a easiiri>n anliiiinliT.
''.'Ii;t! imidd lie ilic dim 'n^inji! of lln' u:ir<ler'.*
I'nr !!•« riistin;;. we lind from the tabic that the erosssei'tion of
;li.' llanv:" hnnid be li: iinb:" l>y 1 ineb : of tli«' web, TJ inebi's b\
: inli : 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 imbes  \ 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* uinlir wonlii o:ll^ KupN>rt atNiiit twcnly feet of Ihf
■\)k\\ in liiiL'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 floorbeams 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
titionwalls 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, carstables, 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. Floorbeams, 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 seventyfive pounds, and, if used
as a place of public assembly, one hundred and tvv«^nty pounds.
In dwellinghouses, 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 ballrooms, SO to V2i) lbs. i>er siiian* foot.
For s'hools ^<0 lbs. per sqiiar** foot.
Fur hayl«»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 }• .% . xt"
•• 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 ihiiiir*
■' 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
'JJ.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
4S2
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
lV.
100
40
:«k)
125
4:.i»
174
43
•>rt
88
«
TOO
Rl
»
41 «)
7»
M
I Dif Kirt> PriM.utiitii 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
(':ik Crockery
I>aie Liailier
" (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 !>«> lememhtMed that the efTt>et of a load bud
dinlx applied uiK)n a Ix'am is twiei> as i:nat 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 imwd of priipir i^ usually ron>ider«Ml to pn>ihii'«' an
<th<i vxliiiii i» a iiiiaii iHiWft'ii thai nl llir sinH* ItKid wheli ;;ratlu
a)l\ and w Inii sitild<nl\ atplled ; 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 faihuH of safi'ty for lltNiriindM'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 Floorbeams. — 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 floorbeams 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 onehalf 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
iiil)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'
nonrliiMins in !i liwlliii;:. Ilii' luaiuM lo limv ii 8)Ht]i of 13 reel, and
tub,, j.lmoil llliiKOus. i.L1': 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 lUsimi 1.. us, 3 by 10 imli VLlIow.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' sm<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..».
Utiici' III,' MiKir vrillbeMitli(ii'ullyslrouj;if thclH'iiiuaiiru pluoottlt
IlrUl^iiiK of l''t(>'>rlM'anis. — lly "luidcina" i<* ini>nnt
syd'iii of liraiiiii; fl«Hirl¥iiui«,
i>iibi'r by iiuiiiis of siuitll Htnil*.
iis 111 Kif;. I, or lij iiuniH of siuitli
lii.i^'s of iHianlH at ri^lit aiiuh^
III llii joists, mill titliii); jil Ih^
l«<vn IlK'iii.
Tl mit of tliU l>ni<in; \» il.
iliMdlmti'J loiid. Tlii
ii: :ils« siiiTiiis tU>' joints.
I'M'iiis Miiin fnoii tiiniiiii;
.>'. It \* cMliIoiiiiirv III
niuM of •hisHliridi:iii..: M
iiiylivi t<iilf<hl fiiliiiilic
' 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 "crossbridging," 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 liinch by 3inch stock.
Carriagebeams, Headers, and Tailbeams.— 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
** tailbeams : " the beams jEF and O//, which support the tail
beams, are called the ** headers : " and the beams AB and ('D, the
"carriagebeams," or "trimmers."
The tailbeams 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 tailbeams 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 onelialf of the load
rarrit'd l»y KF plus onehalf the load carried by ^»7/, and also one'
half ot tin* load su])p<)ried by the ordinary joist. The l)esl way in
wliKli to (aliiilalc 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^ onehalf 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 irnnnieis, 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
StirriipIroiis.— 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 Inaxoided. It is now tlie etis
tniti. Ill til !<j;is»< r»)ii>iiMetinii, !<► support the (>nds <if Inniders l>y
nit,i:i •'! »• ;rriipiroMs, mn nIiowii in Fiu. '•'*. Tin* ISoston ami New
\ Ml k I'.ii ! '.Jul: Laws !»'inire 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 iipiroii'^ of suitable tbiekness for the sixe of tlie
t JndM I.'"
It 1^ ividi'iit that t'aeh vertical part of the stirnip will liave to
WOODEN FLOORS. 438
carry onefourth of the load on the header ; and we can easily
deduce the rule,
, . load borne by header
Area of crosssection of stirrup =  — Sfijoo * W
The stirrupirons are generally made of iron bars about two inches
wide and threeeightlis or onehalf 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 floorjoist, 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 floorjoist, 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 floorplank 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 suM»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 :Winch 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
MillFloors,*' shows the dimensions of Ix'ams, and thickness of plank
for waichousefloors 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 firstclass 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 onefortieth or onethiilieth 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 tloorlx'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<iiiim1 al)o\e, a <ietlection of onethirt 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 plankfloors 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
2M:io : ,: r: jieie 2 1218 ; 3, Jo 20 * ' ^* ■**
>. « ifl 2 10 21 »2>1012 „ ,, . t 2 '1211 8<1«2J ,. ,
1 n irt "'. "\ "9 '' 9ii>it; 9.19— sn 9'Hiii .'"'i*~'J ••U— 14
riii.'. ' ^ « 10  "* (2 10 24 * '^^ ' 3'1S 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
12
14
U\
IS
20
22 ! 24
Whltf » „ J ,r o 11 iJ I 2 ' 10 Hi 9 1213 ,„ , ,^,, ,. ,
1'1,„. > 2H 10 2 K' 1" ,2 12 2.. ."5 12 19* '* '* SXllH. 
Spru'i. •.' — \'J .„.,,. „o ...10 .>A 2 12 Irt 8 12 17 . , . ,i_,i» tAl414
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
2ioirt ;.;; ''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
Siirmi' •.'
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 314 !»' j
.., • 2  12 .,
I lo 'J: '
:\
12 12 3 It IS
iM li 2 12 17 :;.\i_\; ,.i; ;; i a>i4 u it^uii
WOODEN FLOORS.
437*
D.FOR STORES AND FACTORIES.*
(Total Weight, 180 lbs. prr Square Foot.)
Clear Span in Fekt.
10
2X1016
\tx 8ia
"(SXlO17
SX 817
12
r2xioii
(2XH1
2x1012
2x1218
2X1019
14
txitii
3X1217
•^X1218
3X1220
*2X1018
( ^X 1219
16
18
8X12
8X14
18
18
8X121«
1X121)
8XMI4i
(8x1212
« 8x1416
( 2<1212
■( 8x1218
20
8X1411
22
8x14181 8X1411
8X1416
8x1214
8X1419
24
SX1418I
* 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, Schoolrooms,
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
810
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
1014
12 .14
_ 1
( 12. It;
( 14 •• 14
1
12x11)
14x16
14x16
16^16
YKLLOW PINE.
Sl'\N IS
Ki;i T.
10
Distance aiaiit on Tkntekh in Feet.
12
14
16
'  l
itt
> () 10 \ )') 10
/ '^  s , S *<
»; 10 810 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.' 1J
V* \i
10 11
8 12 10 12 10 . 12 ; }i; ' \\
10 12 ' II!" \i 10. 14 1011
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:j14 )}J;{5 W"!"
24
s.
12
10
1".»
10.
12
10
:t
12
12
1211
12*11
12. 1«
1414
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'JxlJ ^^^1* 1X14 1^X14 jj^^jj
» 10 « 14
12 » 1J
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>ir\\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
1214 13x14 1114
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.
FiHKPllOOF FLOUKS.
CHAPTER XXUI.
FIREFROOF FLOORS.
TnE tPrin " fireproot floor '' is hert unrlenstcXK] to mean a flool
rfin^triii'liil of (irtproof mnterial, RupiMirtcil on or betwe n iron iit
9ti'il ixaiiiH or gmlcrs, or fireproof 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 iitliir
i'i'~. I'i'iikrii iil>'. ~i<iiii'. Ill' tirii'k; iiiui iiiac) eoin[H)iiilioiiii niiiile
lilaiiT ol' I'^iri^ .IS II (iiiiiiitiiif MiiiliTint. The flr^l tlim'
ri;il~ III'" p'tirnilly ii..<l ill ill.' r<iv I iiRlies 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', ix[iiinii'<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.
FIREPROOF 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 trimmerbeams 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. AnchorHtrapa 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.
KlllErUOOF KLOOKK.
Itrick Arc'li«s,
I vviiy of iiiiikiiiu i> lln'iU'oKf floor of hrirk \s to fill
.■■11 111.' ji.isls«illi Lrii^k unln's. n■^Ii1l;,• <>ii llli low.T
;.'iT;i.'.>n:i <ir biitk sk'wliitrks. M'hiii tliis mctlio<l
'^hoiiM lu' tiiki'ii Ilial llii l<rliks of niiiHi tli.' .irvli.s
aiv of ^1111.1 sliai.'. :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> ktyiil n'itli
lill.M ttUll 111.' IHSt
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..
FIRBPROOF 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 fireproof unless the lower
flanges of the beam are protected by porous terracotta, fireclay
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 fireproof floor may be more cheaply constructed of other
material.
Hollow Porous Terracotta and Hollow Dense
Terracotta 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 sidemethod skewback (or
abutment) is used, with endtoend interiors and keys, or endtOr
end interiors and sidemethod 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.
FIBEFROOF FLOORS. 443
beams, and like centrepieces above, crosdng the beams. The
ptanka on whieh tiles arc laid shfiuld be twoinch, 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 keyshaped 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 Tboits, until the
sofflt tile are hard against the beams and the planking has a crowa
' not esc«&diag onefourth 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 tliirtysix 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 lightweight densetile design, nol so gvinerally useil as fonncrjy.
Figs. It and lli show the simplest endmethnd design for porous
tiling, which has become known as iho ■'Leo endmethod 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 fireproof 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 soundproof, 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 onehalf 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 rollingmill cinders, coke screenings, broken flreproo€
14(5
FIREF»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 ('in
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 firstclass
fire 1 1 root' \. ork.
'1 li« VMi'iition in width of spans between beams is pn>vidi*ii for
by ^ui'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 puiichMl in the tiles, and T
h. a; (i i ol". ;iif inserted and secured a> >h«»wn in Fig. 17.
'■ Inn ;..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 irikuirk. rn'am^ plaeed out of parallel make il
\.n I \.. !.;\«' III Mi 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,
FIREPROOF 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 Densetile 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 Fireproofing 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
poroustile arches of the Lee end method design, which may be
FIREPROOF FLOORS. 446c
olted together with finch 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 hollowtile 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 fireproof 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
3rtyeight 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 seventytwo 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</ FIKEPUOOF FLOOUS.
arclies withstand this impact for three continiious 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(Huired
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 r202o, 1890, oi" which a
tuil reporl was ])ublislied in the A /."trioii' Architect and littiUUug
\rirs, M.'injh "Js. IbiOl. Kight anhes built of hollow bum«*d lin*
elay til«', and four of ])orous terraiottu, were subjei^ted to four kinds
of te1s. under as nearly the same comlit ions as p(»ssible. Thraifrhes
wri» earrie*! on 10inch steel Mn^'ims, set 5 feel apail on centres, and
were built of 10 inch tile. The tcrraeotta 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 anhos were
till I>\ •onlinnons tire of high heal, until the arches were
Ill !<■ iin fii!s t:st.'iiii' ••; the llreelay 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 ine
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»iiM> le:,;: i "fi ali'll. wl'ih 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 diimi
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. uNiii the inahile of the anli. Ikjlh of thi* hoU
FIRKPROOF FLOORS. 447
low flreolay 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 terracott: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 fireclay arches was destroyed by three ap[>lications
of the water ; the other withstood fourteen applications of the
water, alternating with extreme heat.
The porous terracotta. 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 fireclay arches
were destroyed after being subjected to a most intense heat for
twenty four hours. The porous tcrracotta arch, after having a
continuous fire under it for twentyfour 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 terracolta arches. The
porous terracotta tile, new and dry. weighed 34 lbs. to tlio sfiuare
foot ; the fireclay 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 6inch tile arch of 3 feet 8 inches
span, made by the Wight Fireproofing 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 (l6feet) 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 Il)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
softburn(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 {''rnncixo ; 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 ('omiiiny. 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 .\itniiia» An'hifnf ttntl linihliittj .Vf/fA, Mttreb lU, I
FIREPROOF 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 selfsupporting 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 fireproof 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 FIRKPKOOF 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 selfsupf)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 iinch wires threaded
through overy 3 inches. Near the top were bedded two castiron
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
Uinch Inirings at both cn<ls. thus leaving it fed in the clear be
tween snp»«»ris. Tlie beam was loaded with pigirrm piletl luroes
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»sswires 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 hanlpine 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 lianiiuf. a vrry >ui'essrul workiT nf enn('ret«* in Sail Fran
iiii). 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
EPBOO? 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 twentyfive 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
23feet 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
cementfinis I led floors on lop, amounted to tii< cents per square foot
of floor. As a flrei>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 twoinch
FIHEI'UOOF Fl.()i
rinp, ami the fmo iliist, roitjoved by wiishing or screening (washing
prcfLrri'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 typical of thp Lee Hollow Tile anil Cable
Goil FliHir. with a finislicil ceineiil 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 terracotta 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 inil 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'lKUe of ca.li row .if tiles soft eenient in phu'eil, >iiui iin<
or more nils, accoriiiin: lu stnngth rt<i)uireniinli>. 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 Aniliur
lying the lloor lo tin mipimrtii.
tti'l f.>r
ISy II
LT 1 1ll
to^nlhel
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 ief. <•(■
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 onchundredths 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 tenfoot 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, soundproof, and verminproof.
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
itij/i FIKKPliOOF 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 Hortuit' n'»ly lor ilusb.'riu((. Thu
iM iif thu bi'UtDH, tiojcutiu^' ua thuy do bclciw tbo floor
rr/
1.1.': KhUl rry ilu n«>r[>li>t<H.
iirrnnjfi'iiMiil fiii>l(>ytfil whem a flat rciUiif;
PIHEPBOOF FLOORS. 452c
id desired. In this case Ihe floorplate i? the same aa in Pig.
26. Tha ceilingplate 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 nonconductor 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
carefullymade tests
" The loads that so break up the oonpositioa of floors made
452^7 FIllEl'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 icrracotta 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«;veloH'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)is 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 rine eaution and oidy after they havi) buuii t4S8ted
an<l apprcVMl i>y experienced eagineers.
FIREPROOF FLOORS.
4526
Concrete and Wire Netting Floors.
'■ Pigs. 28, 29, 30, and 31, show two styles of fireproof 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.
FIllEi'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 fattor o( safety of si
i»ii in spalls of isix feet bftwei'U oi
 iif tilis HiNir ininKlriK'tion is n mrJeH of rmIs honked over
III tli<> lii'aiiiH. or iiUaitx'rl ti> iIu'mi liy rliiw iltvitnHil for
'I Till rtxU Hre plwcl alxml Iwelvi iiuhiH Ui>«rt, Hnd
in' spnixl sIhtIs of wiri' iHlhiiii; riLntiini; piimllcl with
mr ihc liii of till iHatiiN. Tlii' coiiiH'li is tliin spnad
icivK. ini>iilli of Iwo tr.llirri'iniliis Nn iTiitcrinK w
 Ilii' iTirw iiiislii'.' of Ihi liilliiiii; iin> mi oI<w loKutlior
iK^iii;) iK'nli' Hill •,!•< tiiroii^'li lo llrmly >iiiihiir tb«
■V tl> ('..tun>u W Ml. iW unilir sid. >li<.iilij W iiIkmIiw)
II ^r. i,s to .'lltill.ly >!IIh'.1 till' win >LII<l hHU.
lis slioiilil In iriitiiliil 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.
FIREPROOF FLOORS. 462^
The Fawcett Ventilated FireProof 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 fireproof centering, reducing the dead
weight of the floor twentyfive 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 tireresisting agent in this floor is not
so much the terracotta 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
FIKEPltOOP FLOOitS.
Ml
m
11 1
i
yi
1
FIBEPKOOF 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 ironwork 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 IBeams, 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 hardwood flooring, 4 pounds ; for
454
FIUE PROOF FIDO US.
marble floor tiles, 1 inch thick. 11 pounds. The weight of the
betims may bo taken at 5 pouncls per square foot for 9inch bojuns.
and () pounds lor 10 and 12inch l)eains. Very few fireproof 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 fireproof 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 terracot 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.
Mijn riiiip()~cil l(>;i(l, from Si) t«> s.'> pouiulH jkt wiuun fcwit.)
■'p:!!!.
I ' 1
^ I
l»
r. t 11
1 1 r.
^ • 1
■. "« 1 1
kr."
«»r
iir.
i*M^
ill
•
f..!.
I •
.1.
If.
■ft
ti iiu
hV.
5f
I'Vt.
■)fl
•et
'. inc
h>.
ti fifi.
In
' ill
!.; ill.
/
in.
i:i
11)V
I'l
ill.
1.}
Ihs.
•;
in.
13 H»H.
li
in.
^:^ n»ii
11
i.i ••
1
i:»'.
•
<
i:.'.
1
i;»i
f
i.M ••
ivi
1.". '•
1
i..i
• •
i..l
1
ir,}
1
i:> •■
i:i
. .»
1
!.■.'.
* k
^
IS
s
IH
s
IH ••
n
1.'. ••
8
IK
• k
H
IS
s
IS
i»
.J, ..
!.•)
1 .
* ,
\S
k •
')
•Jl
•.»
;>1
<l
•Ji ••
\<\
.,i ..
i»
'U
»»
•»
•..M
!l
'.M
lo
•.!5i "
r.
•ji ••
v;i
*'
111
i*,'.
111
•ri:
in
sir>' '•
is
1" ••
•.•:. ■•
]i 1
v»".
• •
111
•J.'.'
HI
•ifi.
V*
4} •*
r.'
In ••
•J". •
Id
..1
• •
10
i)i
Vi
aj
\'i
Si ..
•jn
1m •
•.'.". ••
1*1
.'.•J
•
1;'
:w
IJ
.ftj
M
*! ••
•.'1
I'.' ••
.. , . .
:«
•
V2
•.\'i
• • 1
Ii
.iV
la
.« "
• >•>
1 .' ■•
• i"*
i>i
:«
»
Vi
'. '*
1
Vi
:k
].".
■II "
• »•►
^» 1
1 . • •
I'l
• 1 w
•
v:
:i3
1
\:.
11
I.*!
It "
•Jl
]■:
>.» . .
i>i
."K
•"
i:.
11
;; 1
i:.
•11
1.1
41 "
>i ,
\! '■
• ► 1 * •
ITi
11
• •
!:•
11
1..
II
iri
41 '•
FIREPROOF 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 12inch beams, spaced 4 feet apart ; or two girders,
and lOinch beams spiU3ed 6 feet apart. In the former case we
should require 11 beams the full width of the building, weighing
455a FIREPROOF 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 IReaiiis. — The deflection of rolled
iron Ibeams 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 onethirtieth of
an iucli for every foot of span, under the load which they have
been calcnlatcd to support.
T!(»r<)(ls. — Tierods 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 tierod for any floor
is,
. W X span of arch, in feet
Diameter 8«iuared .^., .  i • .» i ;
' 6J8;j2 X rise of anh, 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 ticnxls 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 iontaln
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 nilu< fil, fii rnnf«iriii lo tlio iixc 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 twentyfour inches between
centres, set edgewise.
" Any roofplank 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
twentyfour 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 selfclosing hatches or shutters, or by adequate wooden
firedoors 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 dryrot should ensue. Ends of timbers
ventilated by an inch airspace each side in the masonry.
" Roof nearly flat. Timbers laid across the tops of the walls to
>•
458 MILL CONSTRUCrriON
project eighteen to thirtysix 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 halfinch holes trans
ver..ely through tlu* post n(^ar top and bottom for ventilation.
" Floori)lanks not less than three inches thick for eightfoot
bays, three and a half to four for wider bays. In some cases,
beams have b(?en i)laced twelve feet apart, witli fourinch 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 threequarter inch mortar, or two
thicknesses of rosinsized sheath ingpaiR»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 wirelath, 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 halfinch 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 Inii ^■
iorniid of auN number of su<'h bays pla.'eil one aft4*r the otluT.
Such a buil ling cannot be <'onsidered as firepnH)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 mmI .
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 nndirpiniiinLr. 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 iMahiies 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 twpntyflve or twcncytour 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 floorheams are iwually twelve inches by fourteen inches
hanlpinn tliiihers,' which n>st on twentyinc'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 airspncit around tlie end of the timber, and are
aTH'hon'd to the wall by a castiran plate on which the beam resls.
Tills plate, shown in I'ig. 2, has a transverse projection on the
lllHHT suifaic, 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 dwellingIiouMt
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
lliics fiiini oni'. cnil, &nd nail Uirw.'tly
Fig. 3.
Till' ujipcT tliMiiinir is piii'ially o( sonic liant wooil, an iiirli wml
'liuiriir rliU k. iin'rily .joliiI.'<i.
I'll.' Ill »ir,.^l.. mill ii. ivn.irnil ua(iT(iylit l.y flimfolinlis 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. jiivMiils 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. sii'iinii ilirniili mi.1i a lliM.ra v.> liavi iIiwtMhiI.
h .■:;/ j. :;iii,r:ill> iDiuuii of Uiiiiicli liy 1Hilvi'iiiili lianlI'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 redlead, and well dried before the
sheets are laid.
If a gravel roof is used, it should be equal to the best quality of
tarandgravel roofing over four thicknesses of the best roofingfelt.
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 twelveounce duck, weighing sixteen
ounces to the square yard, and should be thoroughly stretched, and
tacked with seventeenounce tinned carpettacks, the edges being
lapped about an inch. If the roofplanks are rough, or not of an
even thickness, a layer of heavy roofingpaper should be laid before
the duck is put down. After the duck is laid, it should be thoroughly
wet, and then painted with whitelead and boiled linseedoil before
it becomes dry ; which makes it waterproof. To protect from fire,
give it two more coats of whitelead, and over this a coat of iron
clad paint. Instead of the four coats of whitelead and oil, the
duck may be saturated with a hot application of pinetar thinned
with boiled linseedoil. 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 threequarters 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 thrfcfoiiiths of an inch. They nhniild
iiir'lpiiir 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 dijrot. Tlif tytlimins aro
iM.<tirim cuits. as shiiim in V\!i. 4. wliid) support
•• Ilooibeaiiis; ami, "hi'ii" there Is a vcrtlLal line ul
u:i1i iron itinllMi, wlileli ounnert
of tiMoIliir. ;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 threefourths 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 floorplank 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 twoinch 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 floorplanks, 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 castiroa 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 eastiron
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 CONSTHLCTIOS.
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
. piuviiks 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. fiKtiiry 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:.1iiil... 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^ .':il [.rill<i[1lll> ill llli
Miiiii'ii i f I'iii for 111, 1l^.j,.Iillril. 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 siiii.rii>i' til lliiiM' in iniiiiiinii um', Tlu'y iiiiisl iiul lie u*sl, hu«r
evcr, witLuut u IIuvHw fruiu tlic jmluaUHM.
FIBEPBOOF OONSTEUCTION FOB BUIIJ)INQS. 467
CHAPTER XXV.
MATERIALS AND METHODS OF FIREPROOF
CONSTRUCTION FOR BUIIiDINGS.
The terai fireproof 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 noncombustible materials carefully protected
from the action of fire by fireresisting materials. (See also quota
tion from Chicago building ordinance, page 485.)
2d. Those built on the socalled •* mill principle," and protected
by fireproof material.
3d. Those built in the usual manner with wooden construction,
and protected by fireproof 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 fireproof 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 fireproofing
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 fireproof 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 terracotta, terra cotta lumber, cellular pottery, porous til
40^ FIKEPROOF COXSTRrrTTOX FOR BriLDIX"08.
in^, otr. ; the other by fireclay 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 pumicestone. 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 nonc(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
jHTiintaLTf of lireclay 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 nHuin>tl in
m.iiiur.Ki ire that the work of idxing, drying, and burning be
i)i<ii'<Mi:.il\ •joMi*. The bui'uing should be done in downiiniuglit
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'>:. bith <if the Hrepr'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
shi 'I •■;.!' I'r* j»r«M» till if p'aiii ilav'*, w iieu jMirou.s an* iiiore
enduriii.' ih.tn den>'e tilcN, i>\cii it I In* deiiNC iii> Ih tif (larl «ir ail
fire>!'.iy. iNinais tiles are tough and ila.<tie. Men.<«e tiles are hanl
and uriitle The most esMntial reipti^itcs of a fire pnKif filliugand
FIREPROOF CONSTRUCTION FOR BUILDINGS. 469
protecting material are these : It should be tough, not easily shat
tered by impact ; nonexpansive, 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 builtin porous tiling. Shells of
porous tiles should be from seven eighths to one inch thick, and
webs from threequarters to seven eighths, according to size of
hollows.
Dense Tiling;, — Next to porous tiling as a fireresisting 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 fireclay, 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 semiglazed 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 fireclay, broken bricks or tiles, burnt
ballast or slag, and clear sand, is said to resist an intense heat suc
cessfully. It is recommended for fireproof construction by English
writers, and concrete construction has been largely used in Cali
fornia on account of its fireproof 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 redhot 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 imjuis 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 umhr >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;'.iir.
PlaNi.'r Itlocks are not siiilal)le as a tireproof material. In usiiiir
linn [»la>tir ri>r fireju'oof protietion. 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 terrac<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 ailion «: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 iinproof material I'ttr ilie ixieri<»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 liiildJMu^ liavr been Lrr»'ally improved during: tiic past liw
. ■.!!>. alii: it •■•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 inattrials.
Til n.oi apjirovj'd metiuxl of c«inslruiiin^ hijrh buililini;s is lo
■■liiM tlif loundation (U the i>olal«d piir >y^tim. and oii top f)f
liicNi pifis place >ieel nr w rouirliliron cobimi; cMendiui; t)in>UL;h
(^■■i!itir< i '.dii lit till bnihfiinr. bdilinn i he n .tide walls and in
*ihi' iiiii ijiii" of ihe b^ildiiiir Al ca<!i llnir hvel ifoii Lrivdir> an*
bolted I"'!,. ('(.liimn. anil llh* whnle sv«.tem braceil bv diair'Hial
tie> in tin ihnkiiiN'* 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 bikijt irDiii lIn a>iioij of heal and iiMiitiire, will endure
fi»nvir. '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
FIREPROOF 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 terracutta or fireclay 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
ridois, 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 ironwork. 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 firedoors to the openings in the fireproof 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 flreproof 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
wroughtiron and steel posts for the interior supports, and protect
FIBEPBOOP 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, 'Twofoot Coluhh CovuuHae ur tdb Fab«t BinLDDre.
©
Pio. ».— Section or CisTraON Comm
Fib. tc.—VaxSBoor SountB Cots
A'i'lh FIREPKOOF 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 fireproof
partitions apiieais to i)e by tlio use of hollow blocks or tiles, of
either dense or )or()U£> terracotta. Partitions arc sometiinct; built
by using 4inch isteel
beams for studding, and
fastening metal lathing
on each side ; but this is
not as practical a iMir
tition as one made of
torracotta blocks. Par
titions constructed of terracotta blocks, either donso or porous,
have many vMluable features other than their tireproof 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 terracotta. 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 TERRACOTTA
PARTITION BLOCKS.
Densf T«Tracotra.
\Vt. per Ml
flH>t, IbK.
:j inches thick 13
4 '• •' j 17
't ..... «w
r» •• ! JG
7 *■ •• i w^
8 •• •• ' :«
i:
Porous Ti'iracotta.
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<iiartcr iiiche<« in total thicknesb. There are a number of dif
ferent dcviies and methods, all accomplishing substantially Um
FIREPROOF 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 oneinch tile, and burying Lee tension rods ^similar to those
used in the floois) in thv; plastering on each side ; the Doring Fire
proofing Company, using rods, bars or channels, and burlaps ; and
the twoinch porous terracotta [)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 hardsetting 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 Ibeams, set 5 to 7 feet apart, and filled in
between with 3inch 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 terracotta bricks or blocks may be
usi»d for filling bc^tween the Ibeams. For roofs where the pitch is
not over 45 defjrci^s, 8x3 inch Tirons, set 10 inches between cen
tres, and filled in with slabs of porous terracotta, 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
terracotta tile, about 0x8 inches, and  inch thick. This makes
a durable and substantial roof, perfectly watertight and absolutely
FIREPKOOF 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 fireclay linings, as shown in Fig. 6. on which
the plastering may be applied. This not only affords a protection
FIBBPBOOF 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 tlirougli 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 Leilings 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 3inch 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
PIBEPROOF CONSTRUCTION FOR BUILDINGS. 477
Comigratedwire Lathingr consists of flat sheets of
doubletwist warplath, 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 32inch 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 Vshaped 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 fireproof 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.
Slieetiron Latliing. — A number of styles of sheetiron
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 fireproof quali
ties, sheetiron 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,
FIBEPBOOF 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 fireclay 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 fireproofing 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 tenacotta 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
fireproof 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 fireproof blocks inanufactur(Ml may be obtained by ad
dressing The Raritan Hollow and Porous Brick Co, , of New York
City ; The Wight Fireproofing Co., of Chicago or New York ;
2'he Pioneer Fireproof Construction Co., of Chicago ; Henry
}Iaurer d; Son, New Tork City ; 2 7ie Lee Fireproof 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() FTKKPnOOF OOXSTKUCTION FOR BUILDINGS.
or cMstiron, and Wm'. treads of slate, marble, or wood. Siicli stnir:
?!'(• iiTiil«.ul)tcdlv UiY better than the ordinarv wooden stairs, but
liicy ni't' iiKMcly iiUM)mluisiibl('. In biiildin^i: such stairs \vro!iErht
iron string sii( uld hi' ust'd with slate tnads 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 liritk
DV Portland «i'nu'nt t'oncntt', witii at least one end sujijiortrd hy a
Itii.i; Wall. If coinMctp >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 thiatn'. 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,trhiin>r
=iriMj i.iiili int.. tiii (M.neretf.
ricT. 1 f ^liow> tw<» Mciions 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 Pinsii'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 niommcnd stairs ntu
I ■' v. . n ..'• ,•;..: ;!•■ !i "iriiiL's. I'n'ifeti'd •*!» I !;t' uiiii' I
.■ :• :'.  ;■.■:;.':■•.:.;■ niio !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
FIEtEPBOOF 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 fireproof.
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( lol:iir l'!iir«». 1''.— ".'i'"!!! \\\ \\\\
'■•••■':••••; . !' ! ".■•'•• *• :!•' ••■ '.ikMn that
I
•:.: <!.." A .. :•• i;ia;f«i iiif.ir bv sltaiii or huC
1 il.. U>f »'Mtl.«t.U fi.r heiitiii;: •>nice> iit dvsicribed
PIRKPRX)P OONSTRUCnON FOR BUILDINGS. 483
In the article on SteamHeating, under DirectIndirect Radiation.
If this method is employed, no hotair 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 threesixteenths 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, castiron, 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
Standpipes. — A very important adjunct to every fireproof
building is a standpipe of 2inch 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.
PIRBPROOP 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 twentyfive hundred square feet,
should be provided with incombustible staircases, enclosed in brick
walls, at the rate of one such staircase for every twentyfive 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 firestops 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 partywalls, and outside walls adjoining neighboring
property, above the roof, and for anchoring* wooden floorbeams 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 Fireproof Construction.
"The term 'Fireproof 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 fireproof 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
terracotta 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 ROOFTRUSgKS.
CHAPTER XXVI.
WOODEN ROOFTRUSSES, WITH DETAIIiS.!
WnK.vKVKR it is rt^uired to roof a hall. room, or ImiMing. where
the flt'ur ST»an is inon* than tweiitvtive 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 ftatures 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 24 T
II
i»tl In L'lnr.i.iiii: tlum in on»» rlas'S. r:illi*»l " n^oftnisses." 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»«>ts usi 1 in :ill kin.l'* of hiiildin;;^ an» itm
>tr;i<t<l '.•riiiiip.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\*' Uin nion' ittini^lctely d«*
Miii*. i. .i:\ \ .1 LinjiTiT \:»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»'Mi> (huii kiok hmvy tai
..«. : 'ilii :iii.'%:. tul the rtUtii'ii «.>f lliv \.irio>i»
WOODEN ROOFTRUSSES.
487
rooftmsses, 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 twentysix degrees,
or six inches to the foot, to sixty degrees, or twentyone 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
fiveeighths 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 wallplate, and holding themselves up at
the top by their own stiffness and strength. A piece of board,
called the "ridgeplate," is generally placed between the upper
ends of the rafters, and the rafters aie nailed to it. In some locali
ties this ridgepiece 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 atticfloor and the ceiling of the room below.
Such a roof can only be used, however, when the distance between
the wallplates is not more than twentyfour 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 walliilat«'*< an* iiiort' tlian twniiyfour ftMt aitait. wv
iiiU'n; alojii 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
liii:::); nt till lniiMiiiLT. aiitl on iliiM' iilar** larm* lifauis, ialli'il ••imr
: '. R
n
2 « 8
CELIN3
J C o T
Fiu.3
I.M'." \\i::'i. :r.;i»i': iln iooi". ur jarkraii«rs. A^ tin* iIistani<' from
!■:. ■; : ' •; :.i :':••• :m\! i«. ni»l L'tmrallN niun* than six or riirht fffi.
i!.' :.ii..:.:;: r* ni;i\ li nia«li a** >niall as iwn in'hr*; hy six iihhr".
\\ .i»i! '!.■■ ^'.•:\\i 111 ill' tin>s is innn than tliirlyfoiir fiHt. t\vi» jijir
l;ii> ii.:«:.: in ]lair.l um «a«Ii >iilf i»f ihi iru^N. or at I and .1. It
i'. aiv.;.> i" :. hi'\\«'\ir. !•» I'hui ihr pnrliiis only ov«'r tht* finl iif a
LiM'. "! :o a jniii}, w lnn it fan In so arranu'fil. Tin" «"fihni: of tin
ri...!M ii«\< !■. I !•> IJM' r«i'*!" i> franpil witji li:;in juisis suiiiortiil li\
W
•* ■'!■ iM.^. 1 ii' • • < . ".1.^ 'nil" '»!ii'"!M ni»! hi* iiiiir
• '■ . I::' ^"i." . ! i ■:••:. .1 ! ■. ■»;u<li h\ funrinili
!.■■■ ; .'  ■• 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 BOOFTBU88B8. 480
beam is lo make it of twoInch 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 tiebeams
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
thirtyfive and fortyfive
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 strainingbeam,
separating tlie upper
ends of the principal
rafters, and a rod at
each end of the strain
ingbeam, 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 Btralniagbeam. This gives the greatest economy in construe
*. *
.« •".. I.
I •'■_
•■ \ I
..I i.
:' I ' I
: i..«. vi .1" ■.j?':ii.r.u i jVAitr .omr it 3. It ^huiiiii V
.. ..'i '.■.i\ ■,.«• 111*;:*:. •'" 1 'i"i.Ni 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 BOOFTRnsSBS. 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 tierods
a.
rhe main rafter, on the contrary, and the tiebeam, 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
seventyfive 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 ROOFTaU88B8. 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 oneeighth of the span, and, it possible, should
he made oiieseveiitli, 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)rTEV>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* ji^'^. This :s a ::iH"1 form of truss for
*!.»atr*. ai'i larj* hall? wli^n ihr» i> a hi.>rizontal ceiling.
< 'oiiiif «Tllrjirrs, — 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»ptn<liii^ a :;all«'ry hy UKans uf rv*\a from the roof
K)D£N ROOFTRUSSES.
495
trasses, — the trass should have additional braces, called ''counter
braces," slanting in the opposite direction to the braces shown.
These counterbraces need only be used when the truss is unsym
metrically loaded.
Wooden 'I russes with Iron Ties. — In all trusses whera
the tiebeam 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
tiebeam.
Fig. 17 shows a form of truss very well suited for the roofs of
carriagehouses, 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
turnbuckles 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 TieRods. — For buildings
where it is desired to liave the trusses and rooftimbers 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 tierod 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 ROOFTRUSSES. 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 ridingroom
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 ROOFTUUSSES.
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 trussis an* invvnitt'i] fruiu h*'\uii imsln*«l <nii l»y two tluveiiu'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:* tlinf Inrltt's liy
• i I". '. ii; i .1 '■ i! '.':>■': :<•!. wliiii 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 noiiiiti tliai tin iiprijflits art Imtli
WOODEN BOOFTKUSSB8. 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.
cnTiiiiber 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 rooftniss 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 BOOFTHL'SSBR.
WOOMK UOOTTBDSSIS.
WOODEN ROOFTRUSSES. 608
Figs. 2820 repreaent tnuses token from old English churches;
but the hammerbeam 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 lenilrmv
wliirli it liuH (u overthrow l.lii! wiill Is not very great, ami may la
easily ri'.sistid l>y a wall iwi'nty itiches or two feet thick, ruCLifoRnl
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'lwork.
Ill a liiiiiimKrbnun truw, in
I'linii of a vault. TnuiM of
Iviliirb bf thittwaliwk hard l>*
WOODEN ROOFTRUSSES.
507
this kind, where there is no bracket under the hammerbeam, are
not as stable as that shown in Fig. 30.
Fig. 33 shows a form of truss used in Emmanuel Chiucli at Shel
bume Falls, Mass., Messrs. Van Brunt & Howe, architects, Boston.
This truss was probably derived from the hammerbeam 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 crosstie 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 ROOFTllUSSES.
Fig. 34 shows a form of hammerbeam 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 lierod: 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 rooftnissos 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<'krafters 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 tiiirtyihr(H> feet, and the trusses an* spaced about
ei;;bl teet apart from <'ent res.
■wnODEN ROOFTRUSSES. 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 ROOFTKUSSES.
CHAPTER XXVri.
IRON ROOFS AND ROOFTRUSSES, WITH DISTAILS
OF CONSTRUCTION.
OwiNc; to the incroasiiifT cost of lumber, and tlie necessity of
oreriinij buildings as nearly firei)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 rooftrusses: 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
rooftrusses, and be familiar with the best forms of trusses for
dilferent spans, conditions of loading, etc.
^
I Beam.
o
^iT
DeckBfam.
ChannelBar.
TBar».
A/ifcM'. Iron*.
Flg.l.
r>eside^ b.'inu: n()ncond)Ustible, iron rooftriLsses 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 ROOFTRUSSES. 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, Trussroofs with straight rafters, which are simply braced
frames or girders.
2d, Bowstringroofs with curved rafters of small rigidity, and
with a tierod 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 strutpieces, and light for ties and rods. )
This truss was built by the Phoenix Iron Company for the roof
of a furnacebuilding. It consists of two straight rafters of chan
nel or T bars, two struts supporting the rafters at the centre, a
main tierod, and two inclined ties assisting the tierod 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 ROOFTRUSSES.
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 wroughtiron 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 Tiron, or four an^^leirons riveted to.G:ether in thc^ fomi of a
iross; for wroui^htiron stmts always tail by bending or l)uoklin^.
and not by direct erusbiui^. In Figs. 210 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 fiftysix 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'KI»lJi.M> AllMENAI..
Fig. 4.
"I'll. i;iii»i> in this truss, for motliTate s])ans, may 1m» Timu^;
Mil. I ii.j l.ircr .,t;ni>. rli:innelhars and Hie ties and Ntriits may !"■
lH.h<d to I lie \«nie:iI rili. For very )ar^M> spans. cliannell>ars uiu\
hf nt 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. iijon of not bciniz in a sliajte to ctmnect n^adily with tlu*
i>; h« t torm of inm. The llanges of an ilH'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 ROOFTRUSSBS.
513
channelbars. The ties are rods of round iron or flat bars; and th«
struts, commonly Tirons or angleirons 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.
BowstriiigRoofs. — 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
Btrillg4XX>fB.
 The prindpal use of the bowstringroof proper is for roofing
514
IRON ROOFS AND ROOFTRtSSES.
very largo areas in one span, such as is often desired in railway
«tations, skatingrinks, riilingschools, drillhalls^ etc
B
MARKETHOUBE, TWKLFTII AND MARKET STREETS, PHILADELPHIA.
Fig. 8.
Fig. n^presents the diagram of a bowstringtruss of a hundred
and fiftyt 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 wroughtiron deckbeam 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 seveneighths of an inch, riveted to the deckbeain on each
side.
Fig. 9.
The St lut s ar<' wroughtiron Ibeams 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 BOOFTBU88BB. Slff
four feet apart. The rlK is onefifth the span, the tierod rising
seventeen feet In tbe middle aimve the springing, and the curved
rafter rising forty feet and a halt. The rafter is a flfteenlncli
wroughtriroii II>esin. The tie is a round rod In ibort lengths,
four inches diameter, thickened at the joints. The tensionbars
of tlie bracing Are of plateiron, five inches to three inches in
width, and flToeighths of an inch thicii. The struts are formed
of bars liaving tbe form of a cross.
The following table, taken from Unwin's "WroughtIron Bridges
and Hoofs," gives the principal proportions of some notable bow
stringtrusses, mostly In England: —
PROPORTIONS OF BOWSTRINGROOFS.
For spans much exceeding a hundred and twenty or a hundred
anil thirty feet the bowstringtrtiss 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 ticrods. As these trusses
embrace the most dlfHcult problems of engineering, and are raiely
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 iweniyflie Hjuuu lodMet
516
IRON ROOFS AND ROOFTRUSSES.
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, channelbars, and Hat plates, and rivet them together at the
joints, so that the truss will consist of a framework of iron bars all
riveted toilet her. The other nuahod is to ust» channelljars, Tirons,
Ibeams, 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 tierod 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 nieans of an iyeeiid 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' ii<: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 ROOFTRU8SE8.
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.
STRUTHEADS.
Fig. 12.
Fig. 13 shows the joints at the foot of the struts, as made in the
STRUTFEET.
XB
Fig. 13.
second method. The pealcs in either method are seciued by means
of fishplates riveted to both rafters (Fig. 14).
PEAKS.
Fig. 14.
Fig. 15 shows the proportions for eyes and screw ends for tension
r—>
TIEBAR.
ROD.
Fig. 16.
btfB as naed in this method of construction.
IRON ROOFS AND ROOFTRUSSES,
""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 nlicrft a raftir.
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 fortiitii 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;.'(lusk<wlwkuf
I'olUii' inlviiHrnHl lu
nin:: llii> unit, a* lu
1 rtH.f ijf xtxty tret
n Iruii ru;l uni
linlli i>f a fiHil fur a diiii^i
tlfty .ti»r<<'!. v.: ami, aa rhla la
iH'ani' anil tihIn in a baiMllli(
< clitiiaK'. niniiH'nsatlan to tllU
fur :ill >iin>u!>M l^or usiy IM( ipaa.
:.'.) ill
IRON ROOFS AND ROOFTRUSSES.
519
the vibration of each wall would then be only* fifteenthousandths
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 sidewalls by firmly fastening to them each shoe of the
rafter. Expansion is also provided against by fastening down one
shoe with wallbolts, and allowing the other to slide to and fro on
ihe wallplate 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 angleirons 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.
anglebar; 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 jackrafters, carrying the sheatliingboards 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 tmnsveis^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 tierods on top, and fjistened to tlie
ralti'is l»y brackets: or hookhead spikes may Ih» driven up into
the i)iiilin. the head of the spike hooking under the flange of the
ln'ani, s»a(inLri)iee('s of woo<l being laid on the top of the iM^ani
fvoui i»inlin to jtuilin. Tin' sheathimiboards and covering are then
nailed down on lop of all in the usual manner.
THEOKY OF ROOFTRUSSES, 621
CHAPTER XXVIII.
THEORT OF ROOFTRUSSES.
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 seventyJive 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 tiebeam .20 "
Roof covered with shingles on J inch board .... 13 '^
n
THEORY OF R00FTEU8SE8.
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 foursided 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 rooftrusses 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>fan*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 onehalf 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 ROOFTEUSSES. 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>, onefourth 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 onehalf 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 tiebeam, 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 tiebeam, the line ad must represent the thrust in the lower
part of the rafter, and the line dc^ the pull in the tiebeam. 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 ROOFTRUSSES.
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 forcelines by the same scale we uscni in laying off the
weiglit, we tuid the strains iis shown by the figures on the lines.
Fig. 1^/.
Having found the stniini)ressure in the different parts of the
truss, it is very easy to determine what should be their dimensions.
Thus tlu» compression in the foot of the rafter is 20,750 pounds.
Now, if we wish to make it of hard pine, we know tliat lianl pine
will safely bear KKKJ i>()unds to thesciuaiv inch; and lience we shall
iK'ed ■,';,',;';," = '2\ scpiare inehes area in the rafter. This would
recpilre only a ;> by 7 timber ; but, as the rafter will need to be cut
into more or less, we will give it more area, and call it a 6 by (I.
The short struts have a pressure of 0.000 pounds, and hence
need not be larger than a 2 by 4, except, that, being so thin, it ia
liabl«' to bend; and so we will make it 4 inches by inches. The
tiebeam resists a inill of 14,700 pounds: and. as lurd pine will
safely withstand a tensile strain of 2<)0<) pounds, we should only
nertl abou! iMLiht MjUitre inelu's of area: but, while tills would resist
tlie pull, we urist add em»m:h more to allow for cutting into the
tie :it the Jninis. and for sagging under its own weiglit; so tliat we
will make the beam (MU of a li iueh by ineh tindxT.
'{'he iriitre i'u\ whieb has to iisist a pull of IHUS pounds, we will
iiiak<' ti\' \\n»iii;htiron inst(>ad of w<mm1. as shown in Fig. 4. ('1ia]i.
\ \VI, : ami. a"! wrougliiiron may be safely tnisted with a pull of
lit.iMH) ]>«iiiiiiN to the square ineh <»f erossseetiim, we shall nt^ed a
roil haviiiix a s