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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- lU-s. y /t-r^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, I-iew Y(«k. .'/ CfliS Booft IS RESPECTFULLY DEDICATED TO THOSE WHOSE KINDNESS HAS ENABLED ME TO PRODUCE IT. TO MY PARENTS, WHO GAVE ME THE EDUCATION UPON WHICH IT IS BASED; TO MY WIFE, FOR HER LOVING SYMPATHY, ENCOURAGEMENT, AND ASSIST- ANCE; TO ORLANDO W. NORCROSS OF WORCESTER, MASS., WHOSE SUPERIOR PRACTICAL KNOWLEDGE OF ALL THAT PERTAINS TO BUILDING HAS GIVEN ME A MORE INTELLIGENT AND PRACTICAL VIEW OF THE SCIENCE OF CONSTRUCTION THAN I SHOULD OTHERWISE HAVE OBTAINED. TWELFTH EDITION. The following revisions and additions have been made in this edition : , The chapter on Fireproof Floors has been entirely rewritten and ext.ended to conform to present practice, and several pages of re- visions and additions have been made in Chapter XXV. Several pages of tables relating to iron beams have been omitted, and other tables substituted in their place. New tables have been added in Chapter XI., giving the strength of H -shaped and rectan- gular cast-iron columns, and of the new ** Gray " steel column. A special article on the Strength of Cast-iron Bearing Plates has been added to Chapter X., and new tables are given in Chapter VI. for the Strength of Masonry. There are also several changes in Part III., particularly a revision of the article on Steam-heating, and several new pages giving the cost per square and cubic foot of public and private buildings. Altogether there are about one hundred pages of revised and new matter in this edition. F. E. Kidder. Denver, Mcvreh 1, 1895. PREFACE TO THE NINTH EDITION. Within the past four years the introduction of steel in building construction has been so rapid, and the changes thereby occasioned in the tables relating to the strength of materials so great, that it became necessary to revise all that portion of the book which relates to iron and steel coi^struction. After undertaking this revision, it was found that the changes would be so groat as to necessitate resetting a large portion of the book, and the author then decided to improve the opportunity to rearrange Part 111., and to make certain additions thereto that he has had in contemplation for some time. The present edition, therefore, is largely a new book, all of Chapters XXIIl. and XXY., and nearly all of Chapters XL, Xlll., and XIV., being rewritten, and one hundred pages of new matter added io the second part alone. Part 111. has been rearranged and enlarged by about eighty pages of miscellaneous information of especial value to architects, and a glossary of sixty pages added as an appendix. The new matter contained on pages 746-773, it is believed, will be of especial interest to architects and draughtsmen, as the data there given are not readily accessible elsewhere. It will be noticed that in the list of Noted American Architects there are many dates wanting; if such readers as may be able to supply them will kindly inform the author, he will be greatly obliged. The author is always pleased to receive criticism and suggestions, and is ever willing to give further explanation of any portion of the book that may not be readily understood. F. E. KiDDEB. Denver, Col., November 3, 1891. PEEFAOE. In preparing the following pages, it has ever been the aim of the author to give to the architects and bnilders of this country a r^erenee hook which should be for them what Trautwine's ** Pocket-Book" is to engineers, — a compendium of practical facts, rules, and tables, presented in a form as convenient for application as possible, and as reliable as our present knowledge will permit. Only so much theory has been given as will render the application of the formulas more apparent, and aid the stu- dent in understanding, in some measure, the principles upon which the formulas are based. It is believed that nothing has been given in this book but what has been borne out in practice. As this book was not written for engineers^ the more intricate problems of building construction, which may fairly be said to 'iome within the province of the civil engineer, have been omitted. Desiring to give as much information as possible likely to be of service to architects and builders, the author has borrowed and ouoted from many sources, in most cases with the permission of the authors. Much practical information has been derived from the various handbooks published by the large manufacturers of rolled-iron beams, bars, etc. ; and the author has always found the publishers willing to aid him whenever requested. Although but very little has been taken from Trautwine's " Pocket-Book for Engineers," yet this valuable book has served the author as a model, which he has tried to imitate as well as the difference in the subjects would permit; and if his work shall prove of as much value to architects and builders as Mr. Traut- wine's has to engineers, he will feel amply rewarded for his labor. viii PREFACE. As it is impossible for the author to verify all of the dimensions and miscellaneous information contained in Part III. , he cannot speak for their accuracy, except that they were in all cases taken from what were considered reliable sources of information. The tables in Part II. have been carefully computed, and it is believed are free from any large errors. There are so many points of in- formation often required by architects and builders, that it is difficult for one person to compile them all; and although the present volume is by no means a small one, yet the author desires to make his work as useful as possible to those for whom it has been prepared, and he will therefore be pleased to receive any in- formation of a serviceable nature pertaining to architecture or building, that it may be inserted in future editions should such become necessary, and for the correction of any errors that may be found. The author, while compiling this volume, has consulted a great number of works relating to architecture and building; and as he has frequently been asked by students and draughtsmen to refer them to books from which they might acquire a better knowledge of construction and building, the following list of books is given as valuable works on the various subjects indicated by the titles: — " Notes on Building Construction," compiled for the use of the students in the science and art schools. South Kensington, Eng-* land. 3 vols. Rivingtons, publishers, London. "Building Superintendence," by T. M. Clark, architect and professor of architecture, Massachusetts Institute of Technology. J. R. Osgood A Co., publishers, Boston. " The American House Carpenter" and ** The Theory of Trans- verse Strains," both by Mr. R. G. Hatfield, architect, formerly of New York. ** Graphical Analysis of Ro Trusses," by Professor Charles E. Green of the University of 3higan. "The Fire Protection '*' ' by C. J. H. Wcjpdbury, in- spector for the Factory urance Companies. John Wiley & Sons, publisl PREFACE. ix ** House Drainage and Water Service,*' by James C. Bayles, editor of "The Iron Age" and "The Metal Worker." David Williams, publisher, New York. "The Builders' Guide and Estimators' Price-Book," and "Plas- ter and Plastering, Mortars, and Cements," by Fred. T. Hodgson, editor of " The Builder and Wood Worker." Industrial Publica- tion Company, New York. "Foundations and Concrete Works" and "Art of Building," by E. Dobson. Weale's Series, London. It would be well if all of the above books might be found in every architect's ofl&ce; but if the expense prevents that, the ambitious student and draughtsman should at least make himself acquainted with their contents. These works will also be found of great value to the enterprising builder. PREFACE TO THE FOURTH EDITION. It is now a little more than two years since " The Architect's and Builder's Pocket-Book" was first introduced to the public. Daring that time the author has received so many encouraging words and suggestions from a large niunber of architects and bonders, that he desires to acknowledge their kindness, and to express the hope that the book will always merit their com-> mendation. When preparing the book for publication, especial care and tiiooght were given to the second part of the book; trusting that, if once well done, it would need but little revision for a number of years. The first part, also, it is believed, is quite complete in its way. For Part III., however, the author found time merely to compile such matter as he believed to be of practi- cal value to architects or builders, thinking that, should the book prove a success, this part could be easily revised and enlarged; and, since the second edition was published, the author has de- voted such time as he could command to revising such portions as upon investigation seemed to require it, and preparing addi- tional matter. It is the intention of the author, seconded by the publishers, to make each edition of the book more complete and perfect than the one preceding, in the hope that it may in time become to the architects of the present day what Gwilt's "Encyclopaedia" was to those of former days. The great diversity of informa- tion, however, required by an architect, or those having to do xii PREFACE. of time to devote to the work, to make such a book as complete as could be desired. In the Preface to the first edition it was requested that those who might have information or suggestions which would increase the value of the book would kindly send them to the author, or advise him of any errors that should be discovered. Several persons generously replied to this invitation ; and several small errors have been corrected, and additional information given, as the result. It is believed, however, that there are yet many who have thought, at. times, of various ways in which the book could be improved, or have in their private note-books practical data or suggestions which others in the profession would be glad to possess; and it is hoped all such will feel it for the interest of. the profession to forward such items to the author. Any records or reports of tests of the strength of building materials of any kind will be especially appreciated. To the list of books given in the former Preface the author would add the following, which have been of much assistance in the preparation of the pages on steam4ieating, and in his professional practice : -^ "The Principles of Heating and Ventilation, and their Prac- tical Application," by John S. Billings, M.D., LL.D., Sanitary Engineer, New York. "Steam-Heating for Buildings; or, Hints to Steam-Fitters, by William J. Baldwin, M.E. John Wiley & Sons, New York. "Steam." Babcock & Wilcox Company, New York and Glas- gow. CONTENTS. PART I. PAOV AbithmbticaIi Sign? and Characters 3 Involution . 3 Evolution, Scjuark and Cube Root, Rules, and Tables . 4 Wkiqhts and Measures 25 Thk Metric System 30 Scripture and Ancient Measures and Weights .... 33 Mbnsuration 35 Geometrical Problems 68 Table of Chords o . 85 Hip and Jack Rafters 04 Trioonombtrv, Formulas and Tables ..•»••«,. 95 PART II. Introdiiction i . , ^ . . 123 CnAPTEK I. Definitions of Terms used in Mechanics 125 CHAPTER II. Foundations •••.. IIM) CHAPTER III. Masonry Walls 149 CHAPTER IV. Composition and Resolution of Forces. — Centre o Gravity ..,,.., XIV CONTENTS. CHAPTER V. p^^^ Bbtainikg Walls • • • . 167 CHAPTER VL StRBNGTH OF MaSONBY 171 CHAPTER VII. Stability of Pibbs and Buttbessbs < 187 CHAPTER VIIL Thb Stability of Abches % 191 CHAPTER IX. Rf^istancb to Tension 206 CHAPTER X. Resistance to Shearing and Strength of Pins . • • • • 238 Pbopobtions of Cast-Ibon Beabino Plates 242a CHAPTER XI. Strength of Posts, Struts, and Columns 2ia CHAPTER XII. Bbnding-Moments 290 CHAPTER XIII. Moments of Inertia and Resistance, and Radius of Gy- ration 2&7 CHAPTER XIV. General Principles of the Strength of Beams, and Strength of Iron Beams 829 CHAPTER XV. Strength of Cast-Iron, Wooden, and Stone Beams. — Solid Built Beams 871 CHAPTER XVI. CONTENTS. XV CHAPTER XVII. „,_ Stbekoth and Stiffness of Continuous Girders .... S92 CHAPTER XVIII. Flitch Plate Girders 401 CHAPTER XIX. Tr^tssbd Beams 404 CHAPTER XX. Riveted Plate-Iron and Steel Beam Girders 410 CHAPTER XXI. Strength of Cast-Iron -Arch-Girders 422 CHAPTER XXII. Strength and Stiffness of Wooden Floors 425 CHAPTER XXIII. Fire-Proof Floors • . 488 CHAPTER XXIV. Mill Construction 466 CHAPTER XXV. Materials and Methods of Firf^Proof Construction for Buildings 467 CHAPTER XXVI. Wooden Roof-Trusses, with Details . 486 CHAPTER XXVII. Iron Roofs and Roof-Trusses, with Details of Construc- tion 510 CHAPTER XXVIII. Thbory of Roof-Trusses 521 CHAPTER XXIX. JqIMTS 550 xvi CONTENTS. PART III. PA Chimneys 5 Rules for Proportioning Chimneys £ Examples of Large Chimneys 5 Wrought-iron Chimneys 5 Flow of Gas in Pipes, and Gas Memoranda 5 Piping a House for Gas 5 Stairs and Tables of Treads and Risers 5 Seating Space in Theatres and Schools 5 Symbols for the Apostles and Saints 5 Dimensions of the Largest Ringing Bells 5 Dimensions of the Principal Domes 5 Dimensions of Clock Faces 5 Height op Buildings, Columns, Towers; Domes, Spires, etc. 5 Capacity and Dimensions of Churches, Theatres, Opera Houses, etc 5) Dimensions of English Cathedrals 5 Dimensions of Obelisks 5' Dimensions of Well-known European and American Build- ings 5 Length and Description of Notable Bridges 6 Lead Memoranda 6 Weight of Wrought-iron and Steel (Rules) 6 Weight of Flat, Square, and Round Iron 6 Weight of Flat Bar Iron 6 Weight of Cast-iron Plates 6 Weight of Lead, Copper, and Brass 6 Weight of Bolts, Nuts, and Bolt Heads 6 Weight of Rivets, Nails, and Spikes 6 Weight of Cast-iron Pipes 6 Weight of Cast-iron Columns 6 Weight of Wrought-iron Pipes and Tubes 6 American and Birmingham Wire Gauges 6 Galvanized and Black Iron, Plain and Corrugated . . 6 Memoranda for Excavators and Well Diggers .... 6 Memoranda for Bricklayers, Tables, etc ^ 6 Measurement of Stone Work 6 Description and Capacity of Drain Pipe 6 Tables of Board Measure of Lumber 6 'iling Memoranda e BANDA FOR PLASTERERS 6 CJONTENTS. XVU PASS IXDA FOR Roofers 653 :lics of Plumbing 6S9 LXDA FOR PaIXTERS 666 tSQ COXDCCTORS 667 [CAL DEFixmoys and Formula ? . . 660 AND Requirements for Lkcandbscext Lightinu . . 675 f Glass : Price List, etc .... 687 TUM 6QS lsphalt 6M T of Freight Cars 607 • of Substances per Cubic Foot 697 OSS AND Weight of Church Bells TOO ' AN'D Cost of Buildings 701 LSD Tear of Building Materials 7TO T of Cisterns axd Tanks T08 • AND Composition of Air T06 isoN of Thermometers 706 OF Iron caused by Heat 707 J Point and Expansion of Metals 708 toPERTiES of Water TOO PTioN of Water in Cities 711 bscence on Brickwork 712 noN OF Rain-water Conducttors to Roof Surface . 712 TE Strength of Sulphur, Lead, and Cement . . . 713 ient of Friction 714 vE Blue Prints of Tracings 715 L Wool 716 TE ILvrdness of Woods 718 ooD LuMHER Grades 718 x)wer 719 ' OF Castings (Rules) 719 )F Drums and Pulleys (Rules for) 7t30 • of Grindstones 720 .ANEOUs Memoranda 721 IONS of Pianos, Wagons, Carriages, etc 722 ' of Sash Weights, Lumber, etc 723 [VK FoRCK OF Blasting Materials 724 OF the Wind 725 iutes 725 erators 726 AL MoULiiiNOS 728 jissicAL Orders ... 729 XVlll CONTENTS. PAGS List of Noted Foreign Architects 740 List of Noted American Architects , » 746 Architects of Noted Buildinos 753 Cost of Buildings per Cubic Foot 700 Cost of Buildings per Square Foot leOg Charges and Professional Practice op Architects . . 7607* Standard Building Contract 764 Architectural S(-hools and Classes in the United States 769 Travelling Fellowships and Scholarships 772 List of Architectural Books 774 ^fTEAM Heating 776 Residence Heating 807 APPENDIX. Glossary of Technical Terms, Ancient and Modern, used BY Architects, Builders, and Draughtsmen . . . I-53 Legal Definition of Architectural Terms 54-58 PART L PRACTICAL Arithmetic. Geometry, and Trigonometry. Rules, Tables, and Problems PEACTICAL ARITHMETIC AND GEOMETRY. SIGNS AND CHARACTERS. The following signs an() cliaitictjrt; 3tre generally nsed to denote and abbreviate the several mathematical operations : — The sign = means equal to, or equality. — means minus or less, or subtraction. + means phis, or addition. X means midtiplied by, or multiplication, -r means divided by, or division. 2 ( Index or power, meaning that the number to which * c they are added is to be squared (^) or cubed {^), : is to 1 :: so is [ Signs of proportion. : to J J means that the square root of the number before which it is placed is required. A^ means that the cube root of the number before which it is placed is required. ' the bar indicates that all the numbers under it are to be talien together. {) the parenthenis means that all the numbers between are to be taken as one quantity. . means decimal parts; thus, 2.5 means 2^^, 0.46 means ^^. ® means degrees, ' minutes, '' seconds. •*. means hence. INVOLUTION. To square a number, multiply the number by itself, and the product will be the square; thus, the square ofl8 = 18xl8 = 324. The cube of a number is the product obtained by multi-* plying the number by itself, and that product by the number agftin; thus, the cube of 14 = 14 x 14 x 14 = 2744. 4 EVOLUTION. The fotirtli power of a number is the product obtained by multiplyini; tlie number by itself four times; thus, the fourth power of 10 = 10 x 10 x 10 X 10 = 10000. EVOLUTION. Square Boot. — Rule for determining the square root of a ^umber. 1st, Divide the given number into periods of two figures each, conunencing at the right if it is a whole number, and at the • • • « • decimal-point if there are decimals; thus, 10286.812(5. 2d, Find the largest square In the left-hand period, and place its root in the quotient; subtract the said square from the left-hand period, and to the remainder bring dowu the next period for a new dividend. 3d, Double the root already foiuid, and annex one cipher for a trial divisor, see how many times it will go in the dividend, and put the number in the quotient; also, in place of the cipher in the divisor, multiply this final divisor by the number in the quotient just found, and subtract the product from the dividend, and to the remainuer bring down the next period for a new dividend, and proceed as before. If it should be foiuid that the trial divisor cannot be contained in the dividend, bring down the next perio<l for a new dividend, and annex another cipher to the trial divisor, and put a cipher in the quotient, and proceed as before. KxAMPLB. 10236.8126 ( 101.17 square root. 1 20l]0236 201 2021 ) 3581 2021 20227 ) 156026 141589 14437 Cube Root. — To extract the cube root of a number, point off the number from right to left into periods of three figm*es each, and, if there is a decimal, commence at the decimal-point, and point off into periods, going both ways. Ascertain the highest root of the first period, and place to right of number, as in long division; cube the root thus found, and sub- fi-* he first period ; to the remainder annex the next period : lae root already found, and multiply by three, and annex CUBE ROOT. two ciphers for the trial divisor. Find how oftrn this trial divisoi is contained in the dividend, and write the result in the root. Add together the trial divisor, three times the proiuct of the first figure of the root by the second with one cipher annexed, and the square of the second figure in the root; multiply the sum by the last figure in the root, and subtract from the dividend ; to the remain- der annex the next period, and proceed as before. When the trial divisor is greater than the dividend, write a cipher in the root, annex the next period to the dividend, and proceed as before. Desired the ^493039. 493039 ( 79 cube root. 7 X 7 X 7 = 343 7x7X3 = 14700 150039 7X9X3= 1890 9X9= 81 16671 150039 Desired the ^4035a3.419. 403583.419 ( 73.9 cube root. 7 X 7 X 7 = :343 7x7x3 = 14700 7X3X3= 630 3X3= 9 15339 73 X 73 X 3 = 1598700 7a X 9X3= 19710 9X9= 81 1618491 Desired the ^158252.632929. 60583 46017 14566419 14566419 158252.632929 ( 54.09 cube root 5 X 5 X 5 = 125 5X5X3 = 7500 5X4X3= 600 4X4= 16 8116 540 X 540 X 3 = 87480000 540 X 9X3= 145800 9X9= 81 87625881 33225 32464 788632929 788632929 TABLE OF SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND RECIPROCALS, yroiii 1 to lOS"^* The following table, taken from Searle's " Field Engineering,'* will be found of great convenience in finding the square, cube, square root, cube root, and reciprocal of any number from 1 to 1054. The reciprocal of a number is the quotient obtained by dividing 1 by the number. Thus the recipixxjal of 8 is 1 -r 8 = 0.125, SQUARES, CUBES, SQUARE ROOTS, No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10 17 10 10 20 Ll 22 23 24 25 26 27 28 29 30 31 82 33 34 85 36 87 38 39 40 41 42 43 44 45 46 47 48 49 50 61 52 53 54 55 56 57 58 59 Squares. Cubes. Square lioots. 1 1 1.0000000 4 8 1.4142136 9 27 1.7020508 16 64 2.0000000 25 125 2.2300680 36 216 2 4494897 49 343 2.6457513 64 512 2.8284271 81 T-J9 . 3.0000000 100 1000 8.1622777 121 1331 8.3166248 144 1728 8.4641016 169 2197 8.6055513 196 2744 8.7a3E74 225 8375 8.8729833 256 4096 4.0000000 289 4913 4.1231056 324 5as2 4.S42C407 361 6859 4.3588989 400 8000 4.4721360 441 92G1 4.1825757 484 10348 4.0904158 529 12167 4.7958315 576 13824 4.8009795 625 15025 6.G0C0000 676 17576 5.C0C0195 729 19683 6.1CG1E24 784 21952 6.2015026 ^1 24389 5.3851648 900 27000 6.47T2256 961 29791 5.5677614 l(m 82768 5.C568&42 1069 85937 6.7445626 1156 89304 6.8309519 1225 42875 6.9160798 1296 46656 6.0000000 1369 50653 6.0G27625 1444 54872 6.1644140 1521 69319 6.2449980 1600 64000 6.3245553 1081 68921 6.4001242 1764 74088 6.40)7407 1849 79507 6.5574C85 1036 85184 6.f>l;r496 2025 91125 6.703!D039 2116 97236 6.7G23G00 2209 10G823 6.C55G&46 2304 110592 6.9282032 2401 117049 7.0000000 2500 125000 7.0710G78 2001 132651 7.1414284 2704 140608 7.2111026 2809 148877 7.2801099 2916 1574G4 7.3484692 3025 16C375 7.41G1C85 8136 175016 7.4830148 3249 185103 7.5498344 8364 195112 7.6157731 8481 205379 7.6811457 8600 216000 7.7469667 3721 226081 7.810^197 3844 238328 7.8740079 Cube Roots. Reciprocals. 1.0000000 1.000000000 1.2599210 .500000000 1.4422496 .S333333J^ 1.5874011 .250000000 1.7099759 .200000000 1.8171206 .166006667 1.9129312 .142857143 2.CCC0000 .125000000 2.0800637 .111111111 2.1544347 .100000000 2.2239601 .090909091 2.2894286 .083233333 8.8513347 .076923077 8.4101422 .0714!C8571 S.4G62121 Mimm&t 2.5198421 .062500000 2.5712816 .05882Rr29 2.6207414 .05.5555556 2.6684016 .C5i2631579 2.7144177 ,050000000 2. -J 589243 .C4701SC48 2.8020393 .04M54545 2.8438670 .04C478£G1 2.C844991 .041GCe667 2.C340177 .c-:ccooooo 2. £624900 .l£8461538 8.CC0G00O .0&70C7037 8.CCG5669 .035714286 8.07^168 .C3448275© 8.1072325 .033333333 8.1413806 .032.968065 8.1746021 .03J250000 8.207C343 020303030 8.23S0118 .029411765 8.2710603 .0:^571429 8.S019272 .027777^78 8.332£218 .027027027 B.ZQIOTU .0£C315789 8.3912114 .025641026 8.4199519 .025000000 8.44021':"2 .0:4390244 8.47CC266 .023809524 8.C03G981 .023255814 8.5C0C483 .02272727^ 8.55C8C33 .022222222 8.58£Oi79 .021739130 8.G088261 .021276600 8.G342411 .fl£G833aS3 8.6593057 .020406163 8.6840314 .020000000 8.7G&4298 .019607843 8.7325111 .019280769 8.7502858 .018867925 8.7797031 .018518519 8.8029525 .018181818 8.8258624 .017857143 8.8185011 .017543860 8.8708766 .017241379 8.8929965 .016949153 8.9148678 .016666667 8.9364973 .016393443 8.9578915 016129080 CUBE ROOTS, AND RECIPROCALS. 9 No. Squares. Cubes. Square icbots. Cube Roots. Reciprocals. C3 3969 250047 7.9372539 8.9790571 .015873016 64 4096 262144 8.0000000 4.0000000 .015625000 65 4225 2^46^ 8.0a225V7 4.0207256 .015384015 60 4856 267490 8.12403^ 4.011^101 .015151515 67 4489 800763 8.1853528 4.0315480 .014923373 68 4624 314432 8.24G2113 4.0816551 .014;05882 69 4761 328509 8.3066239 4.1015661 .014492754 TO 4900 813000 8.3666003 4.1212S53 .014285714 71 6041 357911 8.42G1493 4.1408178 .014084307 73 51&1 873248 8.4852814 4.1601C76 .0138888GD 78 5329 889017 8.5440037 4.1793390 .013698630 74 5476 405224 8.6023253 4.1988364 .013513514 75 5625 421875 8.6602540 4.2171633 .013333333 76 6776 438976 8.7177979 4.2358236 .013157895 77 5929 45G533 8.7749614 4.2543210 .0121:87013 78 6061 474552 8.8317600 4.2726586 .012820313 79 6241 493039 8.8681944 4.2906404 .012058228 80 6400 B12000 8.9442719 4.3068695 .012500000 81 6561 631441 9.0000030 4.3007487 .0123450; 9 82 6724 5513C8 9.0353851 4.3144815 .012193122 83 6889 671787 9.1101836 4.3320707 .012016193 84 7056 692704 9.1G51514 4.3795191 011901762 8S 7225 614125 9.2195445 4.3968296 .011761706 86 7396 636056 9.2730185 4.4140019 .011027907 87 7569 658503 9.3278791 4.4310176 .011494253 88 T?44 681472 9.3806315 4.4479G02 .011363636 89 7921 7019G9 9.4339811 4.4647451 .011235955 90 8100 ■reoooo 9.4868330 4.4814017 .011111111 91 8281 733571 9.5393920 4.4979114 .010089011 98 8164 778683 9.5916G30 4.5143574 .010369565 93. 8649 801357 9.6430508 4.5306519 .010752688 94 8836 830584 9.6958597 4.5468359 .010638298 96 9025 857375 9.7467943 4.5629026 .010326316 96 9216 884786 9.7979590 4.5788570 .010416667 97 9409 912373 9.8188578 4.5917009 .010309278 98 9604 941192 9.8994019 4.6101363 .010204062 99 9601 970299 9.9498744 4.6260650 .010101010 100 10000 1000000 10.0000000 4.6415888 .010000000 l(Ml 10201 1030301 10.0498756 4.657C096 .OOOJ00990 lOS 10404 1061208 10.0995019 4.6723287 .009803923 108 10609 1092727 10.1488916 4.6875482 .000708738 104 10816 1124864 10.1980390 4.702GG94 .009015385 106 11025 1167625 10.2469508 4.7176940 .009328810 106 11236 1191016 10.2956301 4.732G235 .009133962 107 11449 1225013 10.3440604 4.7474594 .009*45791 108 11664 1259712 10.3923018 4.7622032 .00:259259 109 11881 1295029 10.4403065 4.7768562 .009174312 110 12100 1331000 10.4880885 4.7914199 .009090909 111 12321 1367631 10.5356538 4.8058955 .009009000 112 12544 1404928 10.5830052 4.8202815 .008928571 118 12769 1442897 10.6801458 4.8315881 .003849338 114 12996 1481544 10.C770783 4.8188076 .008771930 116 13225 1520875 10.7238053 4.8629442 .008095652 116 13456 1560696 10.rr03296 4.87C9990 .008C20C90 117 13689 1601613 10.8166538 4.8909732 .008317009 118 139^ 1643032 10.8627805 4.9048681 .008174576 119 14161 1685159 10.9087121 4.9186847 .008403361 190 14400 1728000 10.9544512 4.9324242 .008333333 m 14641 1771561 11 MJXm 4.9160674 .008261463 Itt 14884 1815848 11.0153610 4.9596757 .008196^1 198 16189 1660667 11.0905365 4.9731898 .008130081 IM 15376 1906624 11.1355287 4.9866310 .008064516 10 SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Koots. Cube Roots. Reciprocals. 123 15625 1953125 11.1303399 6.0000000 .006000000 126 15876 20C0376 11.2^9723 6.0132979 .007936508 127 16129 2048383 11.2094277 6.0265257 .007874016 128 1G3S4 2097158 11.3137085 6.0396843 .007818500 129 16641 8146689 U.8578167 6.0537748 .0077S1968 laa 16900 2197000 11.4017548 6.0657970 .007698306 131 17161 2248091 11.4455231 - 6.0787531 .007638588 1C2 174S4 2299968 11.4891253 6.0916434 .007575758 133 17G39 2352637 11.5325G26 6.1044687 .007518797 VA 17056 »106104 11.5758369 6.1172299 .007462687 1C5 18225 8160675 11.6181^00 6.1299278 .007407407 1:3 18196 2515456 11.6619038 6.1425638 .007858041 1:7 18769 2571853 11.7040099 6.1551367 .007S99&nO 108 19044 262807^ 11.7478401 6.1676498 .007^16377 lo9 19321 8685619 11.7896261 6.1801015 . .007194845 140 19600 8744000 11.8321596 6.1924941 .00n48887 111 19881 880S221 11.874&121 6.2048279 .00^02199 112 20164 8863288 11.9168753 6.2171034 .007048854 143 20449 89^1207 11.9582607 6.2293215 .006998007 144 20736 8985984 12.0000000 6.3414828 .006944444 145 2102s 8048625 12.0415946 6.2535879 .006890668 146 21316 8112136 12.0630460 6.2656374 .006848615 147 21609 8170523 12.1243557 6.2776321 .006808781 148 21904 8241793 12.1055251 6.2895725 .0067^6757 14d 82201 8307949 12.2005556 6.3014598 .000711409 150 22500 8375000 12.»174487 6.3132928 •006600067 151 22801 8142951 12.2882057 5.3?.>0740 .0066SS517 153 23104 8611808 13.3288280 6.3368088 .006678047 153 23409 8581577 12.3693169 6.3481818 .006536048 154 23716 8652264 12.4096736 6.8601064 .006498606 155 ^1025 8723875 12.4496996 6.3710R54 .006461018 156 24336 8796416 13.4899060 6.3632126 .000410860 157 24649 8869603 12.5299611 6 3916907 .000909187 158 24964 8944813 12.5096051 6.4061208 .000839114 159 25281 4019679 12.6095203 6.4175015 .006880806 160 85600 4096000 12.6491106 6.4888368 .000850000 IGl 25921 4173281 12.0885775 6.4101218 .000311180 1G2 26244 4251528 13.7279221 6.4513618 .000178640 1G8 86569 4330747 13.7671453 6.4625550 .000184800 164 26896 4410944 13.8062486 6.4787087 .000007801 1G5 27225 4492125 13.^52326 6.4848065 •006000600 166 87556 4574296 12.8840987 6.4958647 .006081000 167 27889 4657463 12.9228480 6.5068784 .005866084 168 88224 4741633 12.9014814 6.5178484 .O0G9S8881 169 28561 4826809 13.0000000 6.6887748 .006017100 170 88900 4913000 13.03«<4048 6.5396588 .006008058 171 29241 6000211 13.07GC968 6.5501991 .006847968 172 29584 5068448 13.1148770 6.5618978 .006818868 173 29929 6177717 18.1529164 6.5780546 .006780617 174 80276 6268024 18.1909060 6.6827703 .006747180 175 80625 6359375 13.2287566 6.6984447 .008714866 176 80976 M5irr6 18.2664992 6.6040787 .000661818 177 81329 6545233 18.3041847 6.6146734 .OO6O40n8 178 81684 6639753 13.3116641 6.6352268 .006017998 179 82041 6735339 13.8790683 6.685740B .006686608 180 82400 6832000 13.4164079 6.6468109 tymmmmmmm 181 82761 6929741 13.4536240 6.6566588 .0068tMB08 182 83124 6028568 18.4907376 6.6670511 .O0O4O4B05 1{» 83489 6128187 13.5277493 6.6774114 .006104481 184 83856 6229504 18.5646600 6.6877840 JXMMTSO 185 84225 6331685 18.6014705 6.6060198 J000lfl040^ 186 84596 6481856 13.6381817 6.7088675 jOQBoni' Ct'BE ROOTS, AND RECIPROCALS. a No. Squares. Cubes. Square Roots. Cube Roots. ' ! Reciprocals. 187 84969 6639203 13.6747943 5.7184791 .005347594 188 85814 6644672 13.7113092 5.7286543 .005319149 180 85721 6751260 13.7477271 5.7387936 .005291005 190 86100 6R50000 13.7840488 6.7488971 .005263158 191 86481 6067871 13.8202750 6.758CGn2 .Cai235602 103 86864 7077888 13.8564065 5.7689982 .0052C8383 108 87249 7180057 13.8924440 5.77899G6 .005181347 194 87636 7801384 13.928S883 5.7889604 .005154689 195 88025 7414875 13.9642400 5.7988900 .005128205 106 88416 7529536 14.0000000 5.8087857 .Oa5102041 197 88809 7645373 14.035C088 5.818&179 .005076142 193 89204 7762392 14.0712473 5.8284767 .OCr,050505 199 89601 7880599 14.10673G0 5.6382?^ .005025126 200 40000 6000000 14.1421356 5.8460855 .005000000 2C1 40401 8120601 14.17744C9 5.8577660 .004975124 203 40604 8^42408 14.2126704 5.^674643 .004950495 208 41209 6365427 14.2478068 6.8771307 .004926106 204 41016 ^89664 14.28285G9 5.8867653 .0(M901961 205 42025 8615125 14.3178211 5.8963685 .004878049 2oa 42436 8741816 14.3527001 5.9050400 .004854369 207 42849 8869743 14.8874946 5 9154817 .004880918 203 43264 8098912 14.42J:2C.'31 5 9^9921 .004807692 200 48681 0128820 14.45683i:3 5.9344721 .004784689 210 44100 0281000 14.4918767 5.9439220 .004761905 211 44521 0308031 14.525&90 5.£CcS4l8 .004739386 212 44C44 0528128 14.5GQ2196 5.9G2';&20 .004716981 218 45369 0668597 14.5945195 5.9720926 .004694836 214 45796 06C0344 14.6287288 5. £814240 .004672897 215 46225 9988875 14.6G28783 5.91:07204 .004651163 216 46656 100776C0 14.69CC8e5 6.CC0C0C0 .004629630 217 47089 10218313 14.73C9109 6.CC£24C0 .0046C8295 218 47524 108GC2S3 14.7(Via:£l 6.C184C17 .C04587156 219 47961 10606459 14.798C4£6 6.G27G5G2 .004566210 220 48400 10648000 14.8323070 6.0868107 .004545455 21:1 48841 10708801 14.eCCCGC7 6.C459435 .0045248.87 223 49284 10041048 14.898C&44 6.Cn50489 .004504505 2:^8 49128 11060507 14.9331845 6.CC41270 .004484805 2;:4 fiOlTB 11230424 14.96(K£05 6.0731779 .004464286 225 60025 11890C25 15.C0CCC00 C.C822020 .004444444 226 51076 11548176 15.03S2CG4 6.0911994 .004424779 227 61529 11G07(«3 15.0CC5192 6.1C01'<02 .004405286 298 61964 11C52C52 15.CCCCC89 6.1C01147 .004385C65 229 62441 12006969 15.1E274C0 6.1180S32 .0043CC812 290 62000 121G7000 15.1657509 6.1269257 .004347826 281 533G1 12326391 15.108C&12 6.1857924 .004829004 im 68824 124enG8 15.2315402 6.144C337 .004310345 238 54289 12649337 15.264S375 6.1{:S4495 .004291845 234 547n6 12G129M 15.2970585 6.1C22401 .004273504 2ii5 55225 12977875 15.3297007 6.171C068 .004255319 236 55096 18144256 15.3622915 6.1707466 .C04237288 287 601C0 13812053 15.3948043 6.1884628 .C04219409 238 £6044 18481272 15.4272486 6.19n544 .004201681 2S9 57121 18651919 15.4596248 6.2058218 .004184100 240 57600 18824000 15.4919334 6.2144C50 004166667 241 58061 18897521 15.5241747 6 2i:ccr43 .CC414C378 2i2 58564 14170488 15.. '5503402 6.2310797 .C04182231 243 59040 14848907 15.5HR4573 6.2402515 .004115226 244 50536 14526784 15.G204994 6.^487008 .004008861 243 60025 14706125 15.C524758 6.2573248 .004081083 'VM 60516 14886036 15.C84*)871 6.2058206 cc-^ccrx4i r 61G00 15069223 15.71C0:>:G ! 6.274S054 .C0404r:r3 ^ ^504 152S2092 15.7480157 1 6.2827613 .004a32258 SQUAKES, CUBES, SQUARE HOOTS, „. ftsr Cube Boots. S iJlfl 1S.77W7338 e.WllDM wwimu sso 15.8113883 <I.W9<)a53 00400000Q t.dcmaas oosgstOM e.aesaM 15:b058737 a.8iM703B lS,»3rjr!3 B.xaoM SiU e.su^r UK lO! 031:2] ue siaajsoii 0CO8910S1 IB.0CS3;»1 fl.3G009«8 WB7SM0 m m 10.13U1KI B.ssaaota OOSSUIM 003831418 xa ittiauofos ist. offiw i6.i.T!aa» i.mim mnw» aa; 0(B74O3ie m in.aroTOJi 8,*173037 OOJiMSU aea 1S.40I3185 B.4K3148 oo3;m» ■m IB!4a!»77S 8^4712736 f l«.4»itiW 8,4ri«J3« «H0T04n ie!fi[W).<i IKI3UieS85 10.B831S40 ODSKsesM 2Tfl la.fliaaiTT 8,5106300 taxasm 10.6433i:0 B.siama oaxioioe a;s 0(Ki58uai 280 16.7S3i!0OS o.seisss ocispwo iB.Tastow ani 283 001533668 «81 wisoiiiifao oisvaia^a oosaiiK 2d5 Ht.8319l30 6.5aCWH3 ooBosra SSS m OLtuiijjoe m 17.tO»38M 6.C10I0CO oosMsm iTinrei* 8;8418Ma oojiiaiao r,i40iii»i e.wBSOM xH4oi3ai Wi £9e M33;s3;s S97 Y-asntaxs e:oriBio3 0(B.-)07«B SZB r.aciWTOs e.orowx) Oft-;i-->T05 T..T!CW<1 €.C3*3KI5 toi -,;yy^iu CTOlTMa 303 7.3.-|!urii 8.70B17S9 003311SS8 803 .iom^ G.nSfiJOO ae3(n330 C0398M74 ;4e4»t'ia oliaiaias 0U:378(>8S ,4M8.V.7 0.7380011 (X0387W4 li.74j!ni6T (Mao^aa* 308 !ftiui»« oonsiuras 3je )i3«*i3<B .soisius e.TBWJifi 1 CUBE ROOTS, AND RECIPROCALS. 13 No. Squares. 811 96721 312 97344 313 97969 314 98596 315 99225 316 99656 317 100489 818 101124 319 101701 830 102400 821 103041 a22 103G84 323 104329 324 104076 325 105625 326 106276 327 106029 328 107584 329 10B241 390 108900 331 109561 332 110224 338 110389 334 111556 335 112225 336 112806 337 113569 338 114244 339 114921 Z40 115600 342 343 344 315 346 U7 »8 249 350 351 852 353 354 £55 356 357 G58 SCO CCl 862 363 364 3t3 366 367 8G8 300 116964 117C49 118336 119025 119716 120109 121104 tiimi 12^j00 l;3o^01 123004 124G09 125316 120025 12ci736 127449 1J:J881 120G00 l.^Jt-21 131044 1317G9 102136 133225 133956 l.T^l^^l 13G1C1 130900 137641 138984 Cubes. Square llootB. 80080231 17.6351921 80371328 17.6635217 80664297 17.6918000 80959144 17.7200451 81255875 17.7482398 81554496 17.7703888 81855018 17.8044938 82157432 17.8G25515 82461759 17.8605711 82768000 17.8885438 8307G161 17.9104729 83386248 17.9448584 83698267 17.9722008 84012224 18.0000000 84328125 18.0277504 84645076 18.0554701 84965788 18.0831418 85287552 18. 1107703 85011280 18.1383571 85087000 18.1659021 862&10i)l 18.19&4054 86594G08 18.2206072 86UJXC37 18.2482S76 87250704 18.2750009 87595371> 18.30G0052 87983056 18.33aXK3 88272753 18.35755i« 88614472 18.3847768 88958219 18.4119526 89304000 18.4390889 89051821 18.46CiJw3 40001CC3 18.4932120 40363007 18.520e.:92 40707584 18.54r;J;V0 41063025 18.5741756 414217S6 18.60107:2 41781023 18.C27C:,00 42144102 18.G547L81 42508549 18.0815417 42875000 18.7062860 43243551 18.7340040 43614206 18.7G1CC30 43966977 18.788^042 44361864 18.814^>077 44788875 18.8414137 45118016 18.8679G23 4'>499203 18.894 4- Jo6 4.>'«2712 18.0:»>/i9 46268279 :0.9472953 Cube Boots. Reciprocals. 46666000 4701.':-.M 474370Ji 47832117 482285!! 48627125 40027?v90 5024^109 50653000 510G?fll 6147rj8l8 t 18.9736660 . )O.(XX>0C/j0 I 19.0262076 I 19.0525.'i89 10.0787^0 ■i9.l04a7r:2 19.1311CV, ia.l5?^Ml 10.2093727 19.2353W1 19.2Gl.r.08 19.2fJ7S^15 I r 6.7751690 6.782;;J29 6.7896013 6.7968844 6.8040921 6.8112847 6.8184620 6.8250242 6.8327714 6.8899087 6.8470213 6.8641240 6.8612120 6.8682855 6.6753443 6.8y23e88 6.8694188 6.690i.'345 6.9034;;59 6.C1042S2 6.91739&4 6.9213556 6.0313006 6.9Ui2S21 6.9451496 6.9520583 6.9589434 6.0058198 6.9?26826 6.9795S81 0.0^03081 6.9031006 7.0(KXX)00 7.00GrOC3 7.(mo',vi 7.C20a400 7.0»n053 7.0388197 7.0405806 7.0472967 7.0M0(M1 7.060C0f;7 7.0C737C7 7.0740440 7.080r/JH8 7.0H7.'>ni 7.C039^(J9 7.ioor/>c> 7.1071087 7.11378G6 7.1200074 7.1260360 7.13aiai> 7.1400370 1405695 7.1030901 7.15950f« 7.1660X7 7.1725809 7 1790544 7.1tV>K2 7.1919663 .003215434 .003205128 .003191888 .003184718 .003174608 .003164557 .0CJJ151574 .00:il44(i51 .003134796 .008125000 .00311526S .003105590 .0()::C05976 .0(3X80420 .00PXJ70923 .00.'XX;74fc5 .000058104 .00.'X>187a) .003009514 .00;»30808 ,co::x;i204« .000003003 .crj:W4012 .0(U;b5075 .002976100 .W)20073i:9 C029585J:0 .002949853 .002941176 .002932551 .C02923977 .00291545a .ar290C977 .002898551 .002890173 .002881844 .002873503 .002865330 .002857143 .C02849003 .caAioooo .cojjC2j;gi .C..:<'.iilfJ59 .002810001 .C-'.-^/Jlli.'O .c/)<j7««j:>i .0^f.r785515 .002777778 .f.0J7700«? .a/;w^e2431 .(/.ZTTAf-il .C;r/47253 .r. ,2739728 .C 0/732210 .r//:r72i706 .002710027 .002702708 .0026.75172 ^.--i i\ 11 f 1 1 No. 1 CtA«. ' BqiMra 1 CUbaBootiL Becfprocak i zn 13919 5i^sb::7 i»si»n9 : T.lSBtfGO .000680065 ' a-.i la*^ 5iilir.;ii 19 33Wnt i Tjmsn 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 ir-j i 14A*4 &;..i';ia 19.4t2S21 T.»>ttflB .ttEGisecs &T5P 14£k41 : 5UiabUUA i.>.4d:ae8S T.Msna .000688622 5SN) lUlO) ' WtTSWO 19.4995807 T.MSISK .00001679 Tvl i«:»:i 5r;>i«;>«i 19.5lStS>l3 T.MSOMB .0QEBE9l6i8 5 &"2 i UZJ^ t5T42«8 19.M4San T.S556I15 .000617801 ! &<J ! \V^j:Q Ki?i*r 19.570RKV) 7.aa2i6a .0QB610066 fr>4 1 Uli^A \ &>^1«>4 19.5956119 7.2;4;;sN . .OQB0M167 3Ki \4?-:£& 67'.»>.':25 19.6214169 .00007408 3^ 1 14^SW 6751iM.-.« 19.646fct27 T.2B10TM .000600674 Zhl I4'y7e9 6::'>.»><3 19.C:2?15« T.SSTSfilT .QOKBOOTO &v3 \:*i:M £ :-!i-.-r2 19.6iK7l5« 7.;£9d63a0 .000577380 doSf 15132^ 5cou:ibC9 l^.T^BXSSi T.29980a8 .QOOBRMOft 300 152100 5050«i«) 19.7484177 T.ao6i4as .OOOBOdOO &01 i.:j^:.i w^r<'^ii 19.7:37199 7.3123a» .OaBBffi45 'iC-i 1.>>X>4 G^i302S8 19.7569609 7.318G114 OQOB6U0D 3:>j l&i449 6o«^.^7 19.8!24iS78 7.S2I828K .00064080 8!4 155.^ ClKii:^ 19.8494333 7.3310900 .000606071 a5 15(Xr25 61t>2ah75 19.8746069 7.3372390 .000631646 3ri6 15CS16 62U09136 19.8997487 7.8431906 .O0E82BO68 J>i;7 15TC09 6257U773 19.9^M8588 7.3195966 .000618000 £JH 15>404 63.>44;92 19.9499373 7.855TK54 .OOESlOaB 31/9 15U201 68521199 19.9749t>44 7.36191:8 .000500006 400 IWOOO 61000000 20.0000000 7.3680680 .008800000 Hil l(>A\ijl 64-1 ;ion 2U.024SI>44 7.&;41979 .000498^6 4(r2 161tX>4 &l'J04.-.-08 20.0499:377 7.St03227 .OU018S66O 40:) 10:^409 6545a>27 20.074i:,':09 7.Sl^64373 .000481800 4r>i 1C:£216 C5939204 20.0997512 7SU25418 .060478018 405 1GU^25 60130125 20.1:^113 r.3&86363 .000160186 4iA lOiKiO 6002:3116 20.1494417 7.404?,»6 .GQ0468O64 407 ](35m9 67410143 20.1742410 7.4107950 .000457900 4<i8 lGr>lf>4 G;yi«':il2 20.1«)aU9 7.41Ce.:95 .COMSOKO 409 107281 6&41?d29 20.2237464 7.4)^29142 .000444868 410 168100 68921000 20.^&15C7 7.4289589 .000480001 411 1G<''I*;J1 604^:0531 20.^531849 7.4*49938 .000138000 412 10'J744 60aS4528 20.2977b31 7.4410189 .00242n84 413 170509 70444097 20.S2iM014 7.4470342 .00^21866 414 insoo 70967944 20.a469e99 7.45S0S99 .002415400 415 172225 71473375 20.8/15488 7.4590859 .002400680 410 173056 71991296 20.8960;81 7.4050223 .002408846 4!7 1738(<9 72511713 20.4205779 7 4709991 .CG2S980e2 413 174724 730JM(:32 20.4450483 7.47C9G04 .002392844 419 1755G1 78560059 20.4694896 7.4829242 .002386685 430 170400 74088000 20.4939015 7.4888TO1 .002880962 4^1 177-^1 74M84C1 20.5182845 7.4948118 .002375297 4:ii i:roH4 75151448 20.5426386 7.6007406 .002369663 4lKl 17R{)29 775G80CG7 20.5069038 7.5066607 .002364066 4^i 179776 70225021 20 5912603 7.5125715 .002358491 425 1W)025 7G7a'3025 20.6155281 7.51&4780 .002352941 4:(i 1814:6 77008776 20.6897674 7.5218652 .002»47418 4^7 IP^'X'O 778M483 20.6039788 7.5806M82 .002*11920 42\ IKJIKI 7W02753 20.6881609 7.5861221 .002336440 4^J i»-1041 7)»fia580 20.7128152 7.5419667 .002331002 4.'X) 1&1900 79507000 20.7364414 7.6478488 .002^2558' 431 1857C1 8(X)G2991 20.7C05395 7.5536888 4US IBGCa* 80621568 20.7846097 7.5595268 «» 1874K9 81182787 20.8086520 7.5038548 4»l 188356 81746501 20.8826667 7 5ni743 -"\ ^^^ ^SAIIB • "-. ^J CUBE ROOTS, AND RECIPROCALS. 15 i No. 4C5 Squares. Cabes. Square Boocs. 1 Cube Root& BeciprociUs. 1302S3 ^312^^75 i)a.&-'665JK 7.570»^ .0ai£9t^M 43a 1'jG0l)6 83tiK18j6 20.biM>ldO 7.&iK>*65 .OOfciftWSrS 437 100009 83^3453 20.9O4M5a 7.5fW57ya .v.X]fcSJi^0:5) 433 1j1S44 &ia2;c?2 d0.9ei844;o 7.5i^4;>i;53 .vX122S3UK> 43d iJjt^iSl 81604519 20.9ai^(>3 7.U)01385 .OOB^fTTOOA 440 19^500 85181000 20.97617:0 7.6060019 .WS73757 441 101481 85766121 21.0000UOO 7.6116626 .a)22G7o74 412 ia>15l 86350333 21.(123^)00 7.6174116 .ae»2443 443 193249 8698S907 21.047565^ 7.6231519 .0^^2257336 444 19na6 87528384 21.0713075 7,62888:57 .0<)225aj52 445 19^025 83121125 21.0050231 7.6346007 .a)224n91 446 193916 887ir»536 21.1187121 7.640^13 .a)iM2152 447 199300 89314623 21.1423745 7.6400272 .a>2237i:50 448 203704 80J15392 21.1600105 7.6517217 .o.>:i52iiJ 440 201601 90518349 21.1896201 7.657413J .002227171 450 20K00 91125000 21.?1320f54 21.£30;0;\i 7.6630943 .0^3322222 451 203401 91733151 7.66870;.5 .002217205 45fi2 204304 8eJ3454« 21.2002016 7.67443LI) .0112212389 453 2(B209 92^50377 21.28370C7 7.680aCi7 .rtK2075(K5 1 454 206116 93570304 21.3072753 7.6S57a>3 .002202(H3 455 207085 94i9G;:ro 21.8307-J:)0 7.6913717 .002197802 456 207036 94818310 21.3541505 7.6970023 .0021029ti2 457 203349 95443003 21.3775583 7.7036246 .003188184 453 2o:r;64 9oonni2 21. 400*5 JO 7.70t^2583 .00218:5 UKi 453 210681 9o70257J 21.4242853 7.7138443 .002178649 460 211600 97336000 21.4476106 7.n94426 .002173013 4'Jl 212531 9rJ7;3181 21.4709106 7.?250:)25 .O;)2100107 4G3 213444 080111^:3 21.4041853 7.7:500141 .002101502 463 214369 902a2:U7 21.5174:543 7. 7301877 .ar215082r 464 215296 90307344 21.5406502 7.7417582 .002155172 465 216225 100544025 21.5fl3a')87 7.7478109 .0021505315 466 2m56 101 1940 JO 21.5870331 7.7528006 .00211592:) 467 218089 101847553 21.6101823 7.7584023 .002141328 468 210024 102503232 21.03.3:3077 7.7G;59361 .0021:50752 469 219961 103101703 21.6564078 7.7604620 .002132106 470 220900 103828000 21.6794834 7.7749801 .002127860 471 221841 104487111 21.702534-4 7.7804004 .a)2123142 473 222784 10515404.5 21.7255010 7.7850028 .002118644 473 223729 105823817 21.748JC32 7.7014875 .002114165 474 S24676 103406424 21.7715411 7.7060745 .00210l»705 475 225625 107171875 21.7044W7 7.80215:58 .(X)21 06263 476 223576 107850176 21.8174212 7.H070254 .(X)2100810 477 227529 108531&33 21.8403207 7.81:5:5892 .(X)2()96I30 473 223 134 100215.>52 21 8632111 7.8ir4iI56' .()>■.■»( )92«.»:»() 479 229441 109002230 21.8800680 7.8212042 .(X»'J()K7(yia 480 230400 110592000 21.0089021 7.8297353 .002083:5.33 481 251301 111234(;U 21.9517122 7.8'5510W5 .u)i^)7m)^ 482 23i3:^ lli0801O;5 21.0.5-4-19.U 7 Hia5910 .(X)J0710.S'.) 48;} 2:iVH9 112078587 21.9772010 7.8100134 .iA)nniY.m 4H4 251258 113379904 23.0000000 7.8514214 A%innmui 4a'3 2]5ii5 11408412.5 22.02271.55 7.850H2;a .<);)2(MJ1K50 480 2:50196 114791250 22 0151077 7.8022212 .002057013 487 2:571 GO 115501303 22.0(V<();05 7.80701:50 ,(j020.5:j.'i>« 4S8 25S144 1102i:r2 2,v()^r;r2n 7. 8720'.) 14 JHWiiUHf) 4S'3 2:50121 11003010) LMUJin 7.87K5(;sl AHMHMM 490 210100 117040000 22.1350430 7.8837:552 .002040816 1 401 2110S1 im3707il 22.1585103 7.8800016 .(XWOJWMKiO i 492 24J0r>t 110095483 22.1810730 7.8044463 .002^X52.720 493 24-'5040 110823157 22.203608:) 7.8097917 .002028:508 '94 241036 12nfMr784 22.2261108 7.9061204 .002021201 % 245025 121287375 22.2485055 7.9104.599 ,002020202 i^ 240010 122028086 22.2710575 7.9157832 .002010129 16 SQUARES, CU15ES, SQUARE ROOTS, :>\7 r.H 51'.) n.v) :..-,! .v»i rci5 r>5rt No. Squares. Cubes. Square Roots. 497 217009 12276a473 22.2934908 4,U 21<S()0-1 123505993 22 315913a 4J0 :W9001 124251499 2;S.338S079 500 2.50000 125000000 22.3606793 r>,;i 251001 125751501 22.3830293 252)01 120506008 22.4033365 50.) 25.*J009 127203527 22.4276015 501 251010 128024064 22.449»443 51)5 2.55025 128787623 22.4722051 503 2.5<)0.*JG 129554216 22.41M4438 mr 2,57049 130323843 22.5166003 5o;j 25S(X)-4 131096512 22.5388553 oo-j 259081 1318?2229 2J.5GI0283 510 200100 132651000 22.5831796 511 201121 133432S31 22.0033001 512 2G2144 134217728 22.6274170 513 2G:U()9 1.35005097 22.6495033 511 2<H19a 135790744 22.0715081 515 2(5.5223 136590f;75 22 6036114 516 2002.50 13r38S096 22.715a-331 617 207289 l.*)8188413 22. 7370.3 JO 518 2()H;W4 1:389918.32 22.7590134 519 209.301 139798359 £2.7815713 620 270100 1-40608000 23 80a50ft5 521 271441 141420701 22.rr>4244 272^184 142236018 22.8473193 52.^ 27.'J529 113055607 22.K091933 524 271576 1-43877824 22.8910403 525 275025 144703125 22.9128783 52« 2^(>076 14.55.31,576 22.9:340899 527 277?29 14(>36:3ia3 22.9504800 528 2mS784 147197972 22.r.7r.2.5(XJ 52U 279811 148035889 23.0000000 5.30 2S0(N>0 148877000 23.«17289 5;u 2^1901 1497212:)1 £3.0134:372 5:i2 2S;5()24 1.5(V)08708 23.0051252 rm 2S1089 151419437 23a^67928 5:U 2S.51.50 1.5227'3;304 2:5.1084400 5.15 2S«;2*i5 1.5.3130375 25.1300670 WW 2sr2^»<i 153990650 2:5.15167:38 5;J7 2^s;}«;«) 1.548341.5.3 25 1732603 5.18 2s:)114 1,55?20S72 23 194WrO iJiiQ 2«.*i)521 156590819 23.21G3r33 5!0 201C,(» 1.57464000 23.23T9001 5U 2'.fJ.J.si 1.5S;M0421 23.25941X57 512 2.i:{7i;4 15'W20088 23.2S089:]5 M.i 2*.) I ' 19 16.1030(17 2:3..3t>23<504 5U 2'.i.-:i:ii> 1609891K4 23.3238076 5 15 29 ."'>•.'.> 161878025 2:3.34.52:351 5HJ 2'.tM0 1627n:33<J 23.3606429 ,31 ''4 3ul . '1 :i'>:;;',iti :3<m;i)4 ::ii.-,s<)'j .'iiMi'.llO ;ii»N)25 :«)9i:)6 3HV219 311:3<>1 lt>;366732.3 164.566,592 16U6»14J 106375000 107281151 l(}.^19(}«i(H 169112:377 17(K»146t 1T0953875 inK7»)16 1?2H0K6IM 173741112 23.3880:311 2.3.-l(y:):/'9i t:a. 4307490 23.4520788 23.473J3892 23.4tH(5.sift> 23.51.59.521) 2:5..53?20I6 23.55K13H0 83.6706S23 S3.600M74 ss.esssosao Cubo Roots. Rcclprooala 7.9210994 .00201207;3 7.9264085 .002008032 7.9317104 .0Q20O4OUJ 7.937005S .002000000 7.9422931 .001996008 79475739 .001992002 7.0528477 .001988073 7.9381144 .001984isr/ 7.9633743 .001980196 7.9686Sn .001976286 7.9738r31 .001973387 7.9791122 .001968.')(>1 7.9843444 .001964637 7.9895697 .0019607&1 7.9947883 .00105694/ 8.0000000 .001953125 8.C032049 .0011M9318 8.0104032 .001945525 8.0155946 .001M1740 8.0207794 .001937984 8.0259574 .001934236 8.C3112M7 .C0193a503 8.0362935 .001926783 8.0414615 .001933077 8.0466030 .001919383 8.0.517479 .001915709 8.0508862 .001912046 8.0020180 .001908397 8.0671432 .001904762 8.0722020 .0)1901141 8.0773743 .001897533 8.a"24K00 .0)189.3930 8.0875794 .001890359 8.0920?23 .001886793 8.a»77.589 .0)1883233 8.1028.390 .001879699 8.1079123 .001876173 8.1129803 .a)l«?2659 8 118a414 001869153 8.12309(52 0)1863073 8.1281447 .0)1862197 8.1331«70 .0)l858r33 8.1382230 .001855283 8.14.32.529 .001851853 8.14t^V03 .0)1JM8Jl-J 8 1.532939 .On.»U5013 8.1.5S.3(X51 .0;iH416£l 8.16.3:3102 .OM838235 8.i(5.s:>o::2 .0)18*48(:3 8.i7:i::< ^) .OUW31503 8.1782^ 3 .0M8281.51 8.18.32;:. 5 .c :-^:jsn 8.1882441 .0U18-n494 8 1932127 .001Sl8ir-3 8.19t317,5.3 .0»1814«ii 8.2031319 .OI181159& H. 2080823 .0)l.s<K3H 8.2130371 .INUHO5031 8.21790.57 .0)1801803 8.233R9K5 .0017118561 8.237H351 .001793333 8.2327463 .(X)1?J3115 CUBE KOOTS, AND IIECIPROCALS. 17 No. ^Squares. Cubes. 559 313481 174676879 5G0 313600 175616000 5G1 314721 176558481 562 315844 177504328 6G3 316969 178453547 504 318096 179406144 6C5 319225 180362125 5C6 320356 181321496 EG7 321489 182284263 5G3 322624 183250432 509 323761 184220000 570 324900 185103000 571 32G041 186109411 572 327184 187149248 573 828329 188132517 574 329476 189119224 575 330625 190109375 576 331776 191102976 577 332929 192100033 578 334084 193100552 579 335241 194104539 580 836400 195112000 581 337561 196122941 582 338724 19713r368 583 330389 198155287 684 341056 199176704 585 342225 200201625 586 343:96 201230056 587 344569 202262003 588 345744 2032974?2 589 346921 204336469 590 348100 205379000 591 349.^1 206425071 592 850464 207474688 593 351649 208527857 594 352836 209584584 595 354025 210644875 590 355316 211708736 P97 356409 212776173 t08 a->7604 213847192 C39 358801 214921799 603 360000 216000000 CO I SG1201 217081601 C02 SG2404 218167208 C33 CG3G09 219256227 604 SG4S16 220348864 605 3GG025 221445125 60G CGr236 222545016 6'J7 SG3449 223648543 COB 8C9G64 224755712 609 370881 225866529 610 372100 226981000 cn 373.J21 228099131 612 <:71544 229220928 613 3757G9 230346397 614 37G996 231475544 615 37S225 232608375 G16 379456 23^3744896 617 380G89 2348a5113 618 3«1924 236029032 619 383161 237176659 &30 384400 238328000 Square Icoots. Cube Roots. 23 6431806 8.2876614 23 6643191 82425706 23.6854386 8.2474740 23.7065392 8.2523ri5 23.7276210 8.2572633 23.7486842 8.2621492 23.7697286 8.2670294 23.7907545 8.2719039 23.8117018 8.2';bY/26 28.8327506 8.2816355 2J.853?S09 8.2864928 23.8746728 8.2913444 23.8956063 8.2961903 23.9165215 8.S010304 23.9374184 8.3058651 23.9582971 8.3106941 23.9791676 8.3155175 24.0000000 83203853 24.0208243 8.8251475 24.0416300 8.3299542 24.0624188 8.3347558 24.0831891 8.3895509 »4. 1039416 8.3443410 24 1246762 8.3491256 24.1453929 8.3539047 24.1660919 8.8586784 24.1867732 8.2634466 24.2074369 8.3682095 34.2280829 8.8729668 24.248ni3 8.8777188 24.2693222 8.8824653 24.2899156 8.8872C65 24.3104916 8.3919423 24.3310501 8.8966729 24.3515913 8.4018981 24.3721152 8.<061ie0 ^.3926218 8.4108S26 ^.4131112 8.4165419 24.4335834 8.4S02460 24.4540385 8.4240448 24.4744765 8.4296883 24.4948074 8.4348267 24.5153013 8.4390098 24.5356883 8.4486877 24.5560583 8.4483605 24.5704115 8.4530281 24.5907478 8.4576006 24.6170073 8.4623479 34.63r3,00 8.4670001 24.a576560 8.4716471 24.0779254 8.4762892 24.6981';«1 8.4809261 24.7184142 8.4855579 24.7386338 8.4901848 24.7588368 8.4948065 24.7790234 8.4994233 24.7991935 8.5040350 24.8193473 8.5086417 24 8394847 8.5132435 24.8596058 8.5178403 24.8797106 8.5224321 24.8997992 8.6270189 Reciprocals. 001788909 .001785714 .001782531 .001779359 .001776199 .001773050 .001769912 .001766784 .001768668 .001760563 .001757469 .001754386 .001751313 .001748252 .001745201 .001742160 .001789130 .001736111 .001783102 .001730104 .001727116 .001724138 .001721170 .001718213 .001715266 .001712329 .001709402 .001706485 .001703578 .001700680 .001697793 .001694915 .001692047 .001689189 .001686341 .001683502 .001680672 .001677852 .001675042 .001672241 .001669449 .001666667 .001668894 .001661130 .001658375 .001655629 .001652893 .001650165 .001647446 .001644737 .001642036 .001639344 .001636661 .001633987 .001631321 .001628664 .001626016 .001623377 .00162C746 .001618123 .001615509 .001612903 ^.^ 20 SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Sqiiaro Boots. Cube Roots. Reciprocals. 745 555025 413493625 27.2946881 9.0653677 .001342282 V4G 650510 415100936 27.3130006 9.0o94^20 .001340483 T47 •558009 416832723 27.3313007 9.0734726 .COl&'WfiRR 743 559504 418508992 27.3495887 9.0775197 .001836898 749 561001 420189749 27.3678644 9.0815031 .001335113 750 562500 421875000 27.8861279 9.0856030 .C01fi38883 751 504001 423564751 27.4043792 9.0696392 .0. 1331658 752 565504 425259008 27.4226184 0.0936719 .G01o20787 753 507009 42695YYVV 27.4408455 9.0977010 .C01828D21 754 ' 5G35I6 428661064 27.4590604 9.1017265 .C01S26260 .755 570025 430368875 27.47'r263S 9.1057485 .1015:24503 756 571536 432081216 27.4954542 0.1097669 .t<;lS22751 757 573049 4337930i>3 27.5186330 9.1137818 .C01821004 v:>8 . 574501 435519512 27.6317998 9.1177931 .001319261 759 576081 43?i45479 27.5499546 0.1218010 .101317623 730 B77600 438976000 27.5680975 0.1258053 .C01315789 7G1 579121 440711081 27.5862284 9.1298061 .C01314060 732 580044 442450728 27.604*475 9.1338a34 .101312886 703 532109 444194047 27.62.4540 9.1377971 .001310616 7(>4 583096 445943744 27.6405499 9.1417874 .001306001 7J5 585225 447697125 27.6586:334 9.1457742 .001807190 706 580756 449455096 27.6767050 0.1497576 .C01805483 7G7 583289 451217063 27.6947648 9.1C37375 .001803781 708 63D824 452984832 27.7128129 0.1577189 .001802068 709 591361 454750009 27.7808492 0.1616809 ..C01300890 770 592900 45653;3000 27.7488?39 C. 1656505 .001298701 .101297^17 771 594441 458314011 27.7668668 0.1fc962x;5 772 595984 400099648 27.7848880 0.1735852 .C01296837 773 597529 401889917 27.8028775 0.1775445 .001293661 774 599076 403684824 27.8208555 0.1815003 .001291990 7.5 000625 405484375 27'. 8388218 0.1854527 .C012C0823 7r6 002176 467288576 27.8507706 0.1894018 .C01288660 777 603729 469097433 27.8747197 0.1983474 .001287001 778 605284 470910952 27.8926514 0.1972897 .C012fc5347 779 606841 472729139 27.9105715 0.2012286 .C01288697 780 608400 474552000 27.9284801 0.2051641 .001282061 781 609901 470379541 27.9463772 9.2090962 .001280410 732 611524 478211708 27.9642029 9 2180250 . 001278772 733 613039 480048687 27.98213?2 0.2109505 .001277189 7.S4 614056 481890304 28.0000000 0.2206726 .001275510 735 610225 483730625 28.0178515 0.2247914 .C01273886 'm 617796 485587656 28.a356915 9.2287068 .001272265 737 619369 4874434C3 28.a535203 2826189 .C01270648 788 620944 489303872 S8.(ynii:i77 9 2365277 .C0126C036 789 622521 491109069 28.0891438 2404833 .001267427 790 6^4100 493039000 28.1069386 9.2443855 .0012a')823 791 825081 494913071 28.1247222 9 2482344 .C01C64223 •592 02?264 490793088 28.1424946 0.2521300 .0015:62626 75)3 628a49 49867?257 28.1602557 9.25602^ .001261034 794 630436 600566184 28.1780056 0.2599114 .C0K.';0446 795 632025 602459875 28.1957444 9.2C87973 .101257862 796 633616 604358336 28 2134720 9.207(;7'.;8 .C012E6281 797 . 635209 600261573 28.2311884 9.2715592 .C0i:c£4705 798 636804 608169592 28.2488938 9.2754*52 .001253133 799 638401 610082399 28.2665881 9.2798081 .001251564 830 640000 512000000 28.2842712 0.2831777 .001250000 801 641601 51392i»01 28.30194*4 9 2870440 .001248439 orv) 643204 615849608 28.3196045 9.2909072 .001246883 644809 517^1627 28.3372546 0.2W7071 .001245830 646416 CI 0718464 28.a548938 9.2980239 .001243781 648025 6210C0125 28.3725219 9.8024775 .001242236 649636 623606616 28.3901391 9.3063278 .001240695 CUBE ROOTS, AND RECIPROCALS. 21 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. S07 651249 625557943 28.4orr4r>4 9.3101750 .001239157 {.03 652864 527514112 28.4253408 9.3140190 .0012:37624 809 654481 529475129 28.4429253 9.3178599 .001236094 010 656100 531441000 . 28.4604989 9.3216975 .001234568 811 657721 533411751 28.4780617 9.3255320 .001233046 812 659344 635387328 28.4956137 9.3293634 .001231527 813 660969 537367797 28.5131549 9.3331916 .001230012 8lt 662596 539353144 28.5306852 9.3370167 001228501 815 664225 641343375 28.5482048 9.340aS86 .001226994 810 665856 &43a38496 28.5657137 9.3446575 .001225490 817 667489 545338513 28.5832119 9.3484731 .001223990 813 660124 54734^432 28.6006993 9.3522857 .001222494 819 670761 549353259 28.6181760 9.3560952 .001221001 830 672400 651368000 28.C)856421 9.3599016 .001219512 621 674041 553387661 28.6530976 9.3637049 .001218027 G22 675684 555412248 28.6705424 9.3675051 .001216545 823 677329 557441767 28.6879766 9.3713022 .001215067 GU 678976 559476224 28.7054002 9.3750963 .001213592 8,35 680625 661515625 28.7228132 9.3788873 .001212121 826 682276 563559076 28.7402157 9.8826752 .001210654 62? 683929 565609283 28.7576077 9.3864600 .001209190 823 685584 567663552 28.7749891 9.3902419 .001207729 829 687241 6097^2789 23.7923601 9.3940206 .001206273 830 688900 671787000 28.8097206 9.3977964 .001204819 8U 690561 573a56191 28.8270706 9.4015691 .001203369 832 692224 575930308 28.8444102 9.4053387 .00120192:3 833 693889 678009537 28.8617:394 9.4091054 .001200480 a3i 695556 680093704 28.8790582 9.4128690 .001199041 a35 697225 582182875 ■28.8963066 9.4166297 .001197605 836 698896 6&4277056 28.9136646 9.4203873 .001196172 837 700569 686376253 28.9309523 9.4iU1420 .001194743 833 702244 588480472 28.9482207 9.4278936 .001193317 839 703921 590589719 28.9654967 9.4316423 .001191895 840 705600 592704000 28.9827535 9.4353880 .001190476 841 707281 694823321 29.0000000 9.4391807 .001189061 &i2 708964 696947683 29.0172363 9.4428704 .001187648 &i3 710649 699077107 29.0344023 9.4466072 .001186240 844 712336 601211584 29.0516781 9.4503410 .001184834 845 714025 603351125 29.0688837 9.4540719 .001183432 846 715718 605495736 29.0860791 9.4577999 .001182033 &47 717405 607645423 29.1032644 9.4615249 .001180638 848 719104 609800192 29.1204396 9.465^70 .001179245 849 720301 611960049 29.1376046 9.4689661 .001177856 850 722500 614125000 29.1547595 9.4726824 .001176471 851 724201 616295051 29.1719043 9.4763957 .001175088 '852 725904 618470203 29.1890:390 9.4801061 .001173709 853 727009 620650477 29.2061637 9.4838136 .001172333 854 729316 6228358G4 29.2232784 9.4875182 .001170960 855 731025 625026:375 29.2403830 9.4912200 .001169591 856 732736 627222016 29.2574777 9.4949188 .001168224 857 734449 629422793 29.274502:3 9.4986147 .001166861 858 7:36164 631628ri2 20.2916:370 9.5023078 .001165501 fm • 737881 6338397; 9 • 29.3087018 9.5059980 .001164144 860 739600 636056000 29.3257566 9.5096854 .001162791 861 741321 638277:381 29.3428015 9.5ia3699 .001101440 862 743044 64050:3028 29.359a3a5 9.5170515 .001160093 863 744709 042735G47 29.3768610 9.5207:303 .001158749 864 746496 644972544 29.3938769 9.52440<;3 .001157407 865 748225 647214625 29.410882:3 9.5280794 .001156069 860 749956 649461896 29.4278779 9.5317497 .001154734 867 751689 651714:363 29.4448637 9.5354172 .001153403 868 7534^4 0539?2032 29.4618397 9.5390818 .001152074 22 SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Hoots. Cube Rootq. Reciprocalfl. 8G9 7551C1 056234909 29.4788069 9.i>127437 .C01150748 870 756900 C58503000 29.4957624 9.5464027 .001149425 871 758641 G607/()311 29.5127091 9.5500089 .001148106 873 760384 G63054&18 29.52^^1 9.5537123 .001146789 873 762129 6«5;iS8G17 29.5465734 9.5578630 .001145475 874 763876 667627021 29.5634910 9.5610106 .001144165 875 V65625 669921875 29.580;««» 9.5646569 .001142867 876 767376 672221376 29.5972972 0.5682982 .001141553 877 769129 674526i:« 29.6141858 0.6719377 .001140251 878 7708&4 676836158 29.6:310648 9.5755745 .0011380ri2 879 7?^641 679151439 29.6479342 0.5792065 .001137650 880 774400 681472000 29.6647939 0.5828397 .001186864 881 776161 683797841 29.6816442 ,9.5864682 .001135074 882 7779^^ 6HJ128968 29.6984848 9.5900939 .001138787 883 779689 68W65387 29.7163159 9.5937169 .001182803 884 781456 690807104 29.7:321375 9.5973373 .001131222 885 783225 693154125 29.7489196 9.6009548 .001129944 886 781996 695506456 29.7657521 9.6045C96 .001128668 887 786769 697864103 29.7825452 9.6081817 .001127896 888 788.'>44 700227072 29.79932B9 0.6117911 .001126126 889 790321 702595369 29.8161090 0.01539r/ .0011^4650 890 792100 704969000 29.8328678 9.6190017 .001128606 891 793881 707347971 2:9.8496231 9.(2i26C30 .001122834 S»i 795661 709732288 29.8663690 9.6262016 .001121076 893 797449 712121957 29.8831056 9.0297975 .001110621 894- 799236 714516984 29.8998328 9.6333007 .001118668 895 801025 716917:375 29.9165506 9.G369812 .001117818 896 802816 719323136 29.9332591 9.C40oC90 .001116071 897 804609 ?217a4273 29.9499583 9.044ir>42 .001114827 898 806404 ?241 50792 29.9666481 9. 0477^67 .001113686 899 808201 ?-«}5?2699 29.983328/ 9.051S1G6 .001112847 900 810000 729000000 80.0000000 9.6.'>48938 .001111111 901 811801 731432701 30.0166620 9.05846^4 .001109678 902 813604 733870808 30.0333148 9.6G2O403 .001108647 903 815409 736314327 30.04995&1 9.0656096 .001107420 901 81?216 738763264 30.0665928 9.6691762 .0011C6195 905 819025 741217625 30.0832179 9.0727403 .001104072 906 820836 743677416 30.0998339 9.0768017 .001103758 907 822649 746142643 30.1164407 9.6798601 .001102580 908 624464 748613312 30.1:330383 9.6834166 .001101822 909 826281 751089429 30.1496269 9.0869701 .001100110 910 828100 753571000 30.1662063 0.G905211 .001006001 911 829921 756058031 30.1827765 0.(5940094 .001097695 912 831744 758550528 30.1093377 9.6976151 .001096491 913 883569 76104R497 30.2158899 9.7011.583 .001096200 914 835396 763551944 30.2:3^1329 9.7046989 .001094002 915 837^25 766060875 80.^489669 9.70S»369 .001092896 916 839056 768575296 80.2654919 9.7117r23 .001001708 917 840889 771095213 80.2820079 9.71.5:30.51 .001090618 918 842724 773()20632 30.2985148 9.71H?C>51 .001089335 919 844561 776151559 30.3150128 9.7223631 .001088130 920 84(M00 778688000 30.3315018 ( 9.72588a3 .001086967 921 848241 781229961 30.3479818 9.7294109 .001085rr6 922 850084 783'/7V448 30.3614529 9.7820:309 .0010^4599 923 a51929 786330467 30.380915i 9.7:3(M484 .G010K3423 924 853776 788889024 30.3973(W3 9.7:399634 .001082251 925 855625 79145:^125 30.41.38127 9.7434758 .001081081 926 857476 794022776 80.4302481 9.7469857 .001079914 927 a59329 796.597983 80.4466747 9.7.504930 .001078749 928 861181 799178752 30.4630921 9.7539J)79 .001077586 929 863041 80176.')0«9 30.4795013 t). 7575002 .001076426 'WO 864900 804357000 30.4956014 9.7610001 001075269 CUBE ROOTS, AND RECIPROCALS. 23 f No. Squares. Cubes. Square Roots. Cube Boots. Reciprocals. 031 866761 806954491 30.5122926 • 0.7644974 .001074114 932 868624 809557568 30.5286750 0.7679922 .001072961 933 870489 8121l>6237 30.5450487 0.7714845 .001071811 {m 872356 814780504 30.5614186 0.7749743 .C01070664 935 874225 817400375 30.57r;697 0.7784616 .C010C9519 93G 87(3096 820025856 80 5941171 0.7819466 .001068376 937 877969 822656953 30.6104557 0.7854288 .001067236 938 879844 8252936ra 80.6267857 9.7689087 .001066098 939 881721 827936019 30.6431069 9.7923861 .001064963 940 883600 830584000 80.6594194 9.7958611 .001063830 941 eii5481 as;3237621 30.6757233 9.7993336 .001062699 942 887364 835896888 80.6920185 9.8028036 .001061571 943 889249 888561807 30.7083051 9.8062711 .001060445 944 891136 841232384 }.0 7245830 9.6097362 .001059322 945 893025 843906625 80.7408523 9.8131989 .001058201 946 894916 846590536 80.7571130 0.8166591 .001057082 947 896809 849278123 30.7733651 0.8201169 .001055966 948 898704 851971392 80.7896086 9.6235723 .001054852 949 900601 854670349 30.80J:8436 9. 62'. 0252 .001053741 950 002c00 857375000 £0.8220rOO 9.6CC4757 .001052632 951 904401 860085351 £0.8382879 9.62o9238 .001051525 952 906304 862801408 £0.8544972 9.&37Se95 .001050420 953 908209 865523177 30.870€e81 9.84C8127 .CC1049318 954 910116 868250664 30.8866904 9.6442526 .001048218 955 912025 870983875 80.9020743 9.8476C20 .001047120 956 913936 873722816 £0.9192497 9.6511260 .C0104€025 957 915849 876467493 •£0.9354166 9.6545017 .C01044932 958 917764 879217912 80.9515751 9. 657 9929 .001042641 959 919681 881974079 30. 9677251 9.6614218 .CC1C42753 960 921600 884736000 50.9828668 9.6648483 .C01041667 961 923521 C87503681 31.0C00C0O 9.6662724 .CG1C4C563 962 925444 £90277128 31 .0161248 9.6716941 .GClOSSSOl 963 927869 898056347 31.0322413 9.8751135 .C01038422 064 929296 £95841344 31.0483494 9.8785305 .C010S7344 965 931225 898632125 31.0644491 9.6819451 .C01C2G269 966 933156 901428696 31.0805405 9.6863574 .C01C25197 967 935080 904231063 31.0966236 9.6687673 .C01C34126 968 937024 907039232 31.1126984 9.6921749 .001022058 969 938961 909853209 31.1287648 9.6955601 .€01031192 OTt) 940900 912673000 31.1448230 9.686C6E0 .C0103C928 C71 942841 915498611 31.1608729 9.{;Ci:£6£5 .C01C298C6 972 944784 918330048 31.1769145 9.CC57817 .001C26607 973 946729 S21 167317 31.19294';9 9. £0917 76 .C01C27749 974 948676 924010424 31.2089731 9.9125712 .C0102GG94 975 950625 926859375 31.2249900 9.915CC24 .C010i:5G41 976 952576 929714176 SI. 2409987 9.9102513 .C01G24CC0 977 954529 932574883 31.2569992 9.9227379 .C01023541 978 956484 035441352 SI. 2729915 0.C2C1222 .001022495 979 958441 938313739 £1. £889757 9 12C5042 .C01021450 960 960400 941192000 31.S04D517 9.S328839 .001020408 961 962361 . 044076141 31.£2C91£'5 9.C2G2013 .001019268 082 964324 046966168 S1.33CS7fi2 9.9SC6263 .C01016S20 983 966289 949862087 31.3528SC8 9.9430C92 .001017294 964 968256 952763904 31.3687743 9.1MG3797 .001010260 965 970225 955671025 31.2&47097 9.9497479 .00101C228 966 972196 958585256 31.4CGC3G9 9.9531128 .001014199 967 974109 961504803 31.416.'35G1 9 9564775 .001012171 mo 976144 964430272 31.4324673 9 9598389 .001012146 060 978121 967361669 31 .4483704 9.9631981 .001011122 090 980100 970299000 31.4642654 9.9665549 .001010101 001 982081 973242271 31.4801525 9.9699095 .001009062 092 984064 976191488 81.4960315 9.9732619 .001006005 24 SQUARES, CUBES, SQUARE ROOTS, ETC. No. Squares. Cubes. Square Roots. Cube Root!. Reciprocals. 903 930)40 979140057 31.5119025 9.9766120 .001007049 901 9.5303J 9:?2107784 31.5277055 9.9799599 .00100603G 905 900025 983074875 31.5436:^ 9.9833055 .0010(»Q25 906 932010 9330479:30 31.5594077 9.9866488 .001004016 937 904000 991020973 31.5753008 9.9899900 .001008009 903 930004 994011902 31 591i;.i80 9.9933289 .001002004 909 903001 9070029:)0 81.C009813 9.996665G .ooiooion 1000 1030000 lOOOOOvWJO 31.G22r7GG 10.0000000 .001000000 1001 1002001 1003003001 81.6885840 10.0033322 .0009990010 1003 1004004 1006012008 31.6543836 10.0006022 .0009960040 1003 1006009 1009027027 81.6701752 10.0099899 .0009970090 1004 1008016 1012.)48064 31.6859590 10.0133155 .0009960159 1005 1010025 1015075125 81.7017349 10.0166889 .0009950249 1006 1012036 1018108216 81.7175030 10.0199601 .0009^)358 iao7 1014049 1021147343 81.7332033 10.0232791 .0009880487 1003 1010004 1024192512 81.7490157 10.0265958 .0009990635 1009 1018031 1027243729 81.7647603 10.0299104 .0009910603 1010 1020100 1033301000 31.78049r2 10.0332228 .0009900990 1011 1028121 1033364331 81.7962232 10.0365330 .0.309891197 1012 1024144 1036433r2.3 31.8119474 10.0396410 .0009681423 1013 1026169 1039503197 81.8276609 10.0431469 .00098n663 1014 1028196 1042593744 31.8433666 10.0404506 .0009861933 1015 1030225 1045673 J75 81.8590646 10.0497521 .00096»2217 1016 1032256 1048772336 31.8747549 10.0580514 .00Og642S») 1017 1034289 1051871913 81.8904374 10.0568485 .0009688843 1013 1033324 1054977.i-32 81.9061123 10.0596435 .0009823188 1019 1038361 1058039850 31.9217794 10.0629364 .0009818643 1020 1040400 1061203000 81.9374388 10.0662271 .0009608982 1021 1042441 1064332201 31.9530906 10.0695156 .0009794319 10-iZ 1044484 1067402643 31.9837347 10.0728020 .0009784796 1023 104a529 1070539167 31.9843712 10.076086.3 .0009775171 1024 1048576 1073741824 32.(000030 10.0793884 .0009766625 1025 1050325 1076390025 82 0156212 10.0826484 .0009766098 1028 1052576 1080045576 82 0312:313 10.0859282 .0009746689 1027 1051729 1083200S33 32.0483407 10.0892019 .0009787098 1028 1056734 10383733/2 iJ2.0824'331 10.0924755 .0009727886 1029 1053S11 10335473 D 82.0783333 10.0a57469 .0009718173 1030 1060900 1092727033 32.0936131 10.0990163 .0009708738 1031 1062961 1095312731 32.1091837 10.1022835 .0009699381 laiJ 1085024 1099104703 32.1247503 10.1055487 .00 9689988 1033 1067039 1102302337 32.1403173 10.1088117 .0009680548 1034 1089156 1105507334 82.1558704 10.1120726 .0009671180 1035 1071225 1103717375 82.17141.59 10.1153314 .0009661886 1036 1073296 1111934053 82.1869539 10.11&5882 .0009658510 1037 1075:369 1115157653 32.2024314 10.1218428 .0009643808 1038 1077444 llia333372 82.2180374 10.1250953 .0009633911 ia39 1079521 1121622319 82.233.5229 10.1283457 .0009624639 1040 1031600 1124364000 32.2490310 10.1315941 .0009616885 1041 1083681 1128111921 82.2645316 10.1348403 .0009606148 1042 1035764 1131:306038 82.2800248 10.1380845 .0009596929 1043 1087349 11*4626507 82.2955105 10.1413286 .0009587738 1044 1089936 11378a31J?4 82.8109888 10.1445667 .0009578544 1045 1092025 1141166125 82.3264598 10.1478047 .0009569378 1046 1094116 1144445336 82.8419233 10.1.510106 .0009.560229 1047 1096209 11477130823 32 a573794 10.1.542744 .0009551096 1048 1098304 1151022592 32 37289m 10.1575002 .0009541985 1049 1100401 1154320649 32.3882605 10.1607:359 .0009532888 1050 1102500 1157625000 82.40:37035 10. 1839836 .0009523810 1051 1104601 11609:35651 82.4191:301 10.1671893 .0009514748 ::o2 1106704 1164252(k)8 a2.4345l?r> 10.1704129 .0009505708 105.3 1108809 1167575877 32.44^)615 10.1738:344 .0009490676 1054 1110916 1170905464 32.4653662 10.1768639 .OOO9487n06 WEIGHTS AND MEASURES. 25 WBIGHTS AND MEABURBEl Measures of Len^b. : Inches = 1 foot- feet = 1 yard — 38 inches, i yards = 1 rod = 188 inches = 18i ft. ' rods = 1 turlong = 70-20 inches = fiflO ft. = 220 yds, furlongs = imile = 63360 inches - 13280 f t. = 1760 yds-, yard = 0,0006682 of a mile. [= 320 rods, ounteb's chain. 7.92 Inches = 1 link. 100 links = 1 cliain = 4 rods = 00 feet. 80 chains = 1 mile. 6 feet = 1 fathom. 120 fathoms = 1 cable's length. I Deoimals of a 26 MEASURES OF SURFACE ANT) VOLUME. GEOGRAPHICAL AND NAUTICAL. 1 degree of a great circle of the earth = GO. 77 statute miles. 1 mile = 2046.58 yards. .siio?:makers' measure. No. 1 is 4.125 inches in length, and every succeeding number la ' -^Mii of an inch. 'J'here are 28 numbers or divisions, in two series of numbers, vis., iroui 1 to 18, and 1 to 15. MISCELLANEOUS. 1 palm = 3 inches. 1 span = 9 inches. 1 hand = 4 inches. 1 meter = 3.2800 feet. Measures of Surface. 144 square inches = 1 squanj foot. 9 square feet - 1 square yard = 1296 square inches. 100 square feet = 1 square (architects' measure). LAND. 30i square yards ~ 1 stjuare roJ. 40 square roils = 1 square rood =1210 square yards. 4 square roods | — 1 acre = 4840 s<^iuare yards. 10 square chains S = 100 sfiuare rods. 040 acres ~ 1 scjuare mile = 3007000 square yards = 208.71 feet square = 1 acre. 1 102400 sq. rods = 25C0 sq. roods. A Heciion of land is a square mile, and a quarier-acction is ICO acres. Measures of Volume. 1 gallon liquid measure = 231 cubic inches, and contains 8.330 avoir.liii)o:s pounds of distilled water at 39.8° F. 1 gallon dr>' measure = 208.S cubic inches. 1 bushel ( WlncheHicr) contains 2150.42 cubic inches, or TJ.CSft ],ounils distill«Ml water at 39. ^° F. A heape.l bushel contains 2747.715 cubic inches. DRY. 2 pints = 1 quart = 07.2 cubic inches. 4 quarts = 1 gallon = 8 pints = 20H.8 cubic inches. 2 gallons = 1 pe<^k = 10 pin Is = 8 quarts = 537.0 cubic inches. 4 pecks = 1 bushel = 04 pints = 32 quarts = 8 gals. = 2150.42 1 chaldron = 30 heaped bushels = 57.244 cubic feet. |cu. ia 1 cord of wood =128 cubic feet. MEASURES OF VOLUME AND WEIGHT. 27 IJQUID. 4 gills == 1 pint. 2 pints = 1 quart = 8 gills. 4 quarts = 1 gallon = .32 gills = 8 pints. In the United States and Great Britain I barrel of wine or brand]^ = 31i gallons, and contains 4.211 cubic feet. A hogshead is 03 gallons, but this term is often applied to casks ftf various capacities. Cubic Measure. /^r^^r 1728 cubic inches = 1 foot. 27 cubic feet = 1 yard. In measuring loood, a pile of wood cut 4 feet long, piled 4 feet high, and 8 feet pn the ground, malting 128 cubic feet, is called a cord. /^--^ "/ /i'>- -/^. -- S'i^>^ 16 cubic feet make one cord foot. A perch of stone is lOJ feet long, 1 foot high, and li feet thick, and contains 242 cubic feet. A perch of stone is, however, often computed differently in dif- ferent localities; thus, in Philadelphia, 22 cubic feet are called a perch, and in some of the New-England States a perch is computed at 16i cubic feet. A ton^ in computing the tonnage of sliips and other vessels, is 100 cubic feet of their internal space. Fluid Measure, 60 minims = 1 fluid drachm. 8 fluid drachms = 1 ounce. 16 ounces ~ 1 pint. 8 pints = 1 gallon. Miscellaneous. Butt of Sherry = 108 gals. Puncheon of Brandy, 110 to 120 gals. Pipe of Port = 115 gals. Puncheon of Bum, 100 to 110 gals. Butt of Malaga = 105 gals. TTo'?=?hoad of Brandy, 55 to 00 gals. Puncheon of Scotch Whis- Hogshead of claret, 4(5 gals. key, 110 to 130 gals. Measures of Weiglit. The standard avoirdupois pound is the weight of 27.7015 cubic inches of distilled water weighed in air at 39.83^, the barometer at 30 inches. 28 MEASURES OF WEIGHT. AvoirdupoiSy or Ordinary Coiumercial Weight. 16 drachms = 1 ounce, (oz.). 16 ounces = 1 pound, (lb.). 100 pounds = 1 himdred weight (cwt. ). 20 hundred weight = 1 ton. Tn collecting duties upon foreign goods at the TJnite<l Sta custom-houses, and also in freighting coal, and selling it by who «jale, — 28 poimds = 1 quarter. 4 quarters, or 112 lbs. = 1 himdred weight. 20 hundred weight = 1 long ton = 2240 poimds. A stone = 14 pounds. A quintal = 100 pomids. The following measiu*es are sanctioned by custom or law : 32 poimds of oats = 1 bushel. 45 poimds of Timothy- seed = 1 bushel. 48 poimds of barley = 1 bushel. 50 pounds of rye = 1 bushel. 56 poimds of Indian corn = 1 bushel. 50 poimds of Indian meal = 1 bushel. 60 pounds of wheat = 1 bushel. 60 pounds of clover-seed = 1 bushel. 60 pounds of potatoes = 1 bushel. 56 pounds of butter = 1 firkin. ^ 100 pounds of meal or flour = I sack. 100 pounds of grain or flour = 1 cental. 100 pounds of dr>' fish = 1 quintal. 100 pounds of nails = 1 cask. 196 pounds of flour = 1 barrel. 200 pounds of beef or pork = 1 barrel. Troy Weij^ht. USED IN WEIGHIXG GOLD OR SILVER. 24 grains = 1 pennyweight (pwt.). 20 pennyweights = 1 ounce (oz.). 12 ounces = 1 pound (lb.). A carat of the jewellers, for precious stones, is, in the Uni States, 3.2 grains: in London, 3.17 grains, in Paris, 3.18 grains i divided into 4 jewellers' grains. In troy, apothecaries', and av( dupois weights, the grain is the same. MEASURES OF VALUE AND TIMK. 29 Apothecaries' Weiglit. USED IN COMPOUNDING MEDICINES, AND IN PUTTING UP MEDICAL PRESCRIPTIONS. 20 grains (gr.) = 1 scruple ( 3 ). ;^ scruples = 1 drachm ( 3 ). 8 drachms = 1 ounce (oz.). 12 ounces = 1 pound (lb.). Measures of Value. UNITED STATES STANDARD. 10 mills = 1 cent. 10 cents = 1 dime. 10 dimes = 1 dollar. 10 dollars = 1 eagle. The standard of gold and silver is 900 parts of pure metal and 100 of alloy in 1000 parts of coin. The fineness expresses the quantity of pure metal in 1000 parts. The remedy of the mint is the allowance for deviation from the exact standard fineness and weight of coins. *e»' Weigrlit of Coin. Double eagle = 516 troy grains. Eagle = 258 troy grains. Dollar (gold) = 25.8 troy grains. Dollar (silver) = 412.5 troy grains. Half-dollar = 192 troy grains. 5-cent piece (nickel) = 77.16 troy grains. 3-cent piece (nickel) = 30 troy grains. Cent (bronze) = 48 troy grains. Measure of Time. 365 days = 1 common year. 366 days = 1 leap year. 60 seconds = 1 minute. 60 minutes = 1 hoiu*. 24 hours = 1 day. A solar day is measured by the rotation of the earth upon its ji :1s with respect to the sun. in astronomical computation and in nautical time the day com- mences at noon, and in the former it is counted throughout the 24 hours. In cixil coinputation the day conunences at midnight, and is divided into two portions of 12 hours each. A solar year is the time in which the earth makes one revolution around the sun; and its average time, called the mean solar year, is 305 days, 5 hours, 48 minutes, 49.7 seconds, or nearly 365i days. A mean lunar month, or lunation of the moon, is 29 days, 12 hours, 44 minutes, 2 seconds, and 5.24 thirds. 30 THE CALENDAR. — ANGULAR MEASURE. The Calendar, Old and New Style. The Julian Calendar was established by Julius Csesar, 44 B.C., and by it one day was inserted in every fourth year. This was the same thing as assuming that the length of the solar year was 305 (lays, 6 hours, instead of the value given above, thus introducin;: an accumulative error of 11 minutes, 12 seconds, every year. This calendar was adopted by the church in 325 A.I>., at the Council of Nice. In tlie year 1582 the annual error of 11 minutes, 12 seconds, had amounted to a period of 10 days, which, by order of Pope Greg- ory XIII., was suppressed in the calendar, and the 0th of October reckomnl as the 15th. To prevent the repetition of this error, it was decided to l(^a.ve out three of the inserted days every 400 years, and to make this omission in the years which are not exactly divisi- ble by 400. Thus, of the years 1700, 1800, 1900, 2000, all of which arc leap years according to the Julian Calendar, only the last is a leap year according to the licfoinned or Greyorian (/alendar. This Ileformed Calendar was not adopted by England until 1752, when 1 1 days were omitted from the calendar. The two calendars are now often called the Old Sft/lc. and the New Style. The latter style is now adopted in every Cliristian country except liussia. Circular and Ang^iilar Measures. tSEl) FOK MEASUUINO ANGI^ES AND ARCS, AND FOR DBTSH- MININO LATITUDE AND LONGITUDE. CO seconds (") = 1 minute ('). 00 minutes = 1 degree (°). 360 degrees = 1 circumference (C). Herouds are usually subdivided into tenths and hundredths. A iiilnute of the circumference of the earth is a geographical mile. D('(j}'pes of the earth's circumference on a meridian average 69.7.6 common miles. THE METRIC SYSTEM. Thf nn'frir. fii/Moni is a system of weiu^lits and measiu'es based r.pon a unit called a meter. The meter is one ten-millionth part of the distance from the equator to either pole, measured on the earth's surface at the level jl the sea. THE METRIC SYSTEM. 31 The names of derived metric denominations are formed by pre- fixing to the name of the primary unit of a measure — Milli (miU'e), a thousandth, Centl (sent'e), a hundredth, Dec! (des'e), a tenth, Deka (dek'a), ten, Hecto (hek'to), one hundred, Kilo (kil'o), a thousand, Myria (mir'ea), ten thousand. This system, first adopted by France, has been extensively adopteq by other countries, and is much used in the sciences and the arts. It was legalized in 1866 by Congress to be used in the United States, and is already employed by the Coast Survey, and, to some extent, by the Mint and the General Post-Office. Linear Measures. The meter is the primary unit of lengths. Table. 10 millimeters (mm.) = 1 centimeter (cm.) = 0.393*7 in. 10 centimeters = 1 decimeter = 3.937 in. 10 decimeters = 1 meter = 30.37 in. 10 meters = 1 dekameter = 393.37 in. 10 dekameters = 1 hectometer = 328 ft. 1 in. 10 hectometers = 1 kilometer (km.) = 0.62137 mi. 10 kilometers — 1 myriameter = 6.2137 mi. The meter is used in ordinary measurements; the centimeter or jnillimeterf in reckoning very small distances; and the kilometer y for roads or great distances. A centimeter is about ^ of an inch ; a meter is about 3 feet 3 inches and | ; a kilometer is about 200 rods, or $ of a mile. Surface Measures. The square meter is the primary unit of ordinary surfaces. The are (air), a square, each of whose sides is ten wicie/vs, is the unit of land measures. Table. 100 square millimeters (sq. mm.) = 1 square ) _. ^ -^j-- j^^l^ centimeter (sq. cm. ) S 100 square centimeters = 1 square decimeter = 15.5 sq. inches. 100 square decimeters ^ 1 square I ^ ^55^ .^^ ^^ j jgg y^^ IMTEB (sq. ni. I ) Axao 100 centUrea, <x sq. meters, = 1 AR A square meter, or one emttari, tl Bquare yards, and a hectare Is ftboot St X' CnMol The cubic meter, or itert (stair), t> the Tablk. 1000 cubic inillimM«ra (en. mm. ) = 1 cut The atere is the tuune given to the i wood and timber. A t«ittli of & itae Is are a JefciMtere. A cubic meter, or etere, Is about 11 cub feet. Liquid and Dry M< The liter (leeter} is the primary unit and is a cube, each of whose edgee is a t( The kectnliter Is the unit In meaanring fruits, roota, and liquids. Table. 10 milliliters (ml.) = 1 centiliter (d-) lO-centinters = 1 decUlter 10 deciliters = 1 lttbb (1.) 10 liters = 1 dekaliter 10 dekaliters = 1 BECTOLITEB (hi 10 hectoliters = 1 kiloUter A centiliter is abotit i of a flidd oonee; a (Iter Is about liV H quarts, or I'.f of a dry quart; mJieetoUter Is about 2) bmheb; a The gram it the primary unit of wel^its, and Is tbs vacuum of a cubic ceutlmeter of dlaUUed water at Uw «f SU.2 degrees FkhrenbdL ANCIENT MEASURES AND WEIGHTS. SZ Table. 10 milligrams (mg.) = 1 centigram 10 centigrams 10 decigrams 10 grams 10 dekagrams 10 hectograms 10 kilograms 10 myriagrams 10 quintals 0.1543 troy grain. 1.543 troy grains. 15.432 troy grains. 0.3527 avoir, ounce. 3.5274 avoir, ounces. 2.2046 avoir, pounds. 22.046 avoir, pounds. 220.46 avoir, pounds. = 1 decigram = = 1 GRAM (g. ) = = 1 dekagram = = 1 hectogram = = 1 KILOGRAM (k.) = = 1 myriagram = = 1 quintal = = 1 TONNE AU (t. ) = 2204.6 avoir, pounds. The gram is used in weighing gold, jewels, letters, and small quantities of things. The kilotjram, or, for brevity, kiloy is used by grocers; and the tonneau (tonno), or metric toji, is used in find- ing the weight of very heavy articles. A gram is about 15i grains troy; the kilo about 2i pounds avoir- dupois; and the metric ton, about 2205 pounds. A kilo is the weight of a liter of water at its greatest density; and the metric ton, of a cubic meter of water. Metric numbers are written with the decimal -point (.) at the right of the figures denoting the unit; thus, 15 meters and 3 centimeters are written, 15.03 m. When metric numbers are expressed by figures, the part of tha expression at the left of the decimal-point is read as the number of the unit, and the part at the right, if any, as a number of the lowest denomination indicated, or as a decimal part of the unit; thus, 46.525 m. is read 46 meters and 525 millimeters, or 46 and 525 thousandths meters. In writing and reading metric numbers, according as the scale is 10, 100, or 1000, each denomination should be allowed one, two, op three orders of figures. SCRIPTURE AND AKCIfilTT MEASURES AKD "WEIGHTS. Scripture Long: Measures. Inches. Feet. Inches. Digit = 0.912 Cubit = 1 9.888 Palm = 3.648 Fathom = 7 3.552 Span = 10.944 Egryptian Longr Measures. Kahad cubit ^ 1 foot 5.71 Indies. Royal cubit s= 1 foot 8.66 inches. -J^.ZOrZ IdJL^'iZ^ AyZv "TTE^aHT^ >^. = -'trr. C-ra = 1 a.406 ^-± A!kxiziirUi5 Tiiia* = 11.11912 '■»•- f 415.1 l-:r.-^:i^- z:±^ = -.>^ I 431 J! ^ = ;,-i^.' r>r.w>-. = ije.5 : .• :i.*r. -jk .tr . ■» ^ ": tk .-• -^ .-"■;:;■> i-x. ^•.i\«:. i:-r ibe maw wdghL Miscellaneous. lii' 7: -r:lir: f-»: = l.I-H Hobn-w o::b!X =1.817 MENSURATION. - DEFINITIONS. 85 Fig.l A Curved Line. BflllNSTJRATION. Definitions. A point is that which has only position. A plane is a surface in which, any two points heing taken, thfi straight line joining them will be wholly in the surface. A curved line is a line of which no portion is sti-aight (Fig. 1). Parallel lines are such as are wholly in the same plane, and have the same dii-ection (Fig. 2). A broken line is a line composed of a series of dashes ; thus, . fig- 2 An angle is the opening between two Parallel Lines, lines meeting at a point, and is tenued a riyJit angle when the two lines are perpendicular to each other, an acute angle when it is less or sharper than a right angle, and ob- iune when it is greater than a right angle. Thus, in Fig. 3, A A A A are acute angles, O O O O are obtuse angles, K K R R are right angles. Polygons. A polygon Is a portion of a plane bounded by straight lines. A triangle is a polygon of three sides. A scalene triangle has none of its sides equal; an isosceles tri* angle has two of its sides equal; an equi- lateral triangle has all three of its sides equal. A right-angle triangle is one which has a right angle. The side opposite the right Fig. 4. angle is called the hypothenuse; the side on Right-angle Triangle. which the trian^e is supposed to stand is called its bane, and the other side, its altitude. FI9.6. Triangle. Fig. 6. Fig. 7. lso8C«les Triangle. Bquilateral Triangl« GEOMETRICAL TERMS. .1. quadrilateral is a polygon of four sides. Quadrilaterals are divided into classes, as follows, — the irape' zium (Fig. 8), which has no two of its sides parallel; the trapezoid (Fig. 9), which has two of its sides parallel; and the paralleloyram (Fig. 10), which is bounded by two pairs of parallel sides. \ / Fig. 8. Fig. 9. Fig. 10. A parallelogram whose sides are not equal, and its angles not right angles, is called a rhomboid (Fig. 11); when the sides are all equal, but the angles are not right angles, it is called a rhombvtt )Fig. 12) ; and, when the angles are right angles, it is called a rectan- gle ( Fig. 13). A rectangle whose sides are all equal is called a square (Fig. 14). Polygoils whose sides are all equal are called regular. L I Fig. 11. Fig. 12. Fig. 13. Fig. 14. Besides the square and equilateral triangles, there are The i)entaf(ow (Fig. 15), which has five sides; The hexagon (Fig. 16), which has six sides; The heptagon (Fig. 17), which has seven sides; The octagon (Fig. 18), which has eight sides. Fig. 15. Fig. 16. Rg. 17. \ / Rg. 18. The enneagon has nine sides. The decagon has ten sides. The dodecagon has twelve sides. For all polygons, the side upon which it is supposed Co stand h called its base ; the pei*pendlcular distance from the highest side oi GEOMETRICAL TERMS. 37 angle to the base (prolonged, if necessary) is called the altitude ; and a linf. joining any two angles not adjacent is called a diagonal, A perimeter is the boundary line of a plane figure. A circle is a portion of a plane bounded by a curve, all the pointi of which are equally distant from a point witliin called the centre (Fig. 19). The clrcurnference is the curve which bounds the circle. A radius is any straight line drawn from the centre to the cir- cumference. Any straight line drawn through the centre to the circumference on each side is called a diameter. An arc of a circle is any part of its circumference. A chord is any straight line joining two points of the circumfer- ence, as bd. A segment is a portion of the circle included between the arc and its chord, as A in Fig. 19. A sector is the space included be- tween an arc and two radii drawn to its extremities, as B, Fig. 19. In the figure, (U) is a radius, cd a diameter, and db is a chord subtending the arc bed, A tangent is a right hne which / in passing a curve touches without cutting it, as fg, Fig. 19. Fig. 19. Volumes. A prism is a volume whose ends are equal and parallel polygons, and whose sides are parallelograms. A prism is triangular f rectangular, etc., according as its ends are triangles, rectangles, etc. A cube is a rectangular prism all of whose sides are squares. A cylinder is a volume of uniform diameter, bounded by a cm \o 1 surface and two equal and parallel circles. A pyramid is a volume whose base is a polygon, and whose sides are triangles meeting in a point called the vertex, A p3rramid is triangular, quadrangular, etc., ac- cording as its base is a triangle, quadrilateral, etc. A cone is a volume whose base is a circle, from which the remaining surface tapera uniformly to a point or vertex (Fig. 20). P»fl- 20. Conic ucUona are the figures made by a plane cutting a cone. 38 MENSURATION. An ellipse is the section of a cone wlien cut by a plane passing obliquely through both sides, as at «6, Fig. 21. A paxcthola is a section of a cone cut by a plane parallel to its side, as at cd. A hyjnrhola is a section of a cone cut by a plane at a greater angle through the base than is made by the side of the cone, as at i'lu In the ellipse, the tranarerse axis, or loju/ diameter f is the longest line that can be drawn through it. The conjugate axis, or short di- ametery is a line drawn through the centre, at right angles to the long diameter. A frustum of a jyyramid or cone is tliat which remains after cutting off the upper part of it by a plane parallel to the base. A sphere is a volume boimded by a curved surface, all points of which are equally dis- tant from a point within, called the centre. Mensuration treats of the meas:urement of llnesy surfaces, and volumes. ^Flg.21. ^ To compute the area of a square, a rectangle, a rhombus^ or a rhomboid. Rule. — Multiply the length by the breadth or height; thus, in either of Figs. 22, 23, 24, the area = ab X be. Fig.23 To coinpiite the area of a triangle. c Rule. —Multiply the base by the alti- tude, and divide by 2; thus, in Fig. 25, ab X cd area of abc = 2 '^ To find the length of the hypothenuse qfa right-angle triangle when both <idef are knoion. . .. MENSURATION. - POLYGONS. 39 Fig.26 KuLE. — Square the length of each of the sides making the right angle, add their squares together, and take the j^ square root of their suiu. Thus (Fig. 2(3), the length of at* = 3, and of 6c = 4; then a6 = 3 X 3 = 9 + (4 X 4) = + 10 = 25. ^25 = 5, or a6 = 5. a To find the length of the base or altitude of a right-angle triangle, when the length of the hypothenuse and one side is known. Rule. — From the square of the length of the hypothenuse subtract the square of the length of the other side, and take the square root of the remainder. To find the area of a trapezium. Rule. — Multiply the diagonal by the sum of the two perpendiculars falling upon it from the opposite angles, and divide the product by 2. Or, ah X (cfi-f (70 2 = area (Fig. 27). To find the area of a trapezoid (Fig. 28). Rule. — Multiply the sum of the two par- allel sides by the perpendicular distance between them, and divide the product by 2. To compute the area of an irregidar polygon. Rule. — Divide the polygon into triangles by means of diagonal lines, and then add to- jrether the areas of all the triangles, as A, B, and C (Fig. 29). To find the area of a regular polygon. Rule. —Multiply the length of a side by Jie i>eri)endicular distance to the centre (as .((>, Fig. 30), and that product by the nunibcH- of sides, and divide the result by 2. To compute the area of a regular polygon tohen the length of a side only is given. Rule. — Multiply the square of the side by the luoltipUer opposite to the name of the polygon in column A of the following table: — a Fig.30 40 MENSURATION. -POLYGONS AND CIRCLES. A. B. C. D. Name of Polygon. No. of BldeB. Area. liadius of circum- scribing circle. Length of the side. Radius of inscrilxKl circle. Triangle . . . 3 0.43d013 0.5773 1.732 0.2887 Tetragon . . 4 1 0.7071 1.4142 0.5 Pentagon . . . 5 1.720477 0.8506 1.1756 0.0S82 Hexagon . . . 6 2.598076 1 1 O.SOti Heptagon . . . 7 8.633912 1.1524 0.8677 1.0:J8:j Octagon . . . 8 4.828427 1.3066 0.7653 1.2071 Nouagon . . . 9 6.181824 1.4619 0.684 1.3737 Decagon . . . 10 7.094209 1.618 0.618 1.5383 UudecagOD . . 11 9.36564 1.7747 0.5634 1.7028 I>odecagon . . 12 11.196152 1.9319 0.5176 1.86(5 To compute the radius of a circumscribing circle when the length of a side only is given. Rule. — Multiply the length of a side of the polygon by the number in column B, Example. — Wliat is the radius of a circle that will contain a hexagon, the length of one side being 5 inches ? Ans, 5X1=5 inches. To compute the length of a side of a polygon that is contained in a given circle, when the radius of the circle is given* Rule. — Multiply the radius of the circle by the number opposite the name of the polygon in column C Example. — What is the length of the side of a pentagon con- tained in a ch'cle 8 feet in diameter ? Ans. 8 ft. diameter -^ 2 = 4 ft. radius, 4 X 1.1756 = 4.7024 ft. To compute the radius of a circle that can be inscribed in a given polygon, when the length of a side is given. Rule. — Multiply the length of a side of the polygon by tl>« number opposite the name of the polygon in column D, Example. — What is the radius of the circle that can be inscribed in an octagon, the length of one side being 6 inches. Ans. G X 1.2071 = 7.2420 inches. Circles. To compute the circiunference of a circle. Rule. — Multiply the diameter by 3.1416; or, for most purposes, by 3 j is sufficiently accurate. Example. — What is the circumference of a circle 7 inches in diameter ? A\is. 7 X 3.1410 = 21.9912 inches, or 7 X 3} = 22 inches^ tht error in this last being 0.0088 of an inch. MENSURATION. — CIRCLES. 41 To find the diameter of a circle when the circumference is given. Rule. — Divide the circumference by 3.1416, or for a very neai approximate result multiply by 7 and divide by 22. To find the radius, of an arc, lohen the chord and rise or versed sine are given. Rule. — Square one-half the chord, also square the rise; divide their sum by twice the rise; the result will be the radius. Example. — The length of the chord ac. Fig. 30J, is 48 inches, and the rise, ho, is 6 inches. What is the radius of the arc ? Ans, Rad = '-^i±J^ = ?^±^ = 51 ins. "«• 304- 2bo 12 To find the rise or versed sine of a circular arc, when the chord and radius are given. Rule. — Square the radius; also square one-half the chord; sulx tract the latter from the former, and take the square root of the remainder. Subtract the result from the radius, and the remainder will be the rise. Example. — A given chord has a radius of 51 inches, and a chord of 48 inches. What is the rise ? Ans, Rise = rad — ^md^ — ichord2 = 51 — v^2601 - 576 = 51 — 45 = 6 inches = rise. To compute the area of a circle. Rule. — Multiply the square of the diameter by 0.7854, or mul tiply the square of the radius by 3. 1416. Example. — What is the area of a circle 10 inches in diameter V Ans. 10 X 10 X 0.7854 = 78.54 square inches, or 5 X 5 X 8.1410 = 78.54 square inches. The following tables will be found very convenient for finding the circumference and area of circles. 44 MENSURATION. — CIRCLES. ABEAS AND CIRCUMFERENCES OF CIRCLES {Advancing by TentJis, ) Diam. .JO.O Area. Cirenm. Diam. 35.0 Area. Cireom. Dian. 40.0 Area. CireiiB. 706.8583 94.2478 962.1128 109.9557 1256.6371 125.6637 .1 711.5786 94.5619 .1 967.6184 110.2699 .1 1262.9281 125.9779 .2 716.3145 94.8761 .2 973.1397 110.5841 .2 1269.2848 126.2920 .3 721.0662 95.1903 .3 978.6768 110.8982 .3 1275.5573 126.6062 .4 725.8336 95.5044 .4 984.2296 111.2124 .4 1281.8955 126.9203 .5 730.6167 95.8186 .5 989.7980 111.5265 .5 1288.2493 127.2345 .6 735.4154 96.1327 .6 995.3822 111.8407 .6 1294.6189 127.6487 .7 740.2299 96.4469 .7 1000.9821 112.1549 .7 1301.0042 127.8628 .8 745.0601 96.7611 .8 1006.5977 112.4690 .8 1307.4052 128.1770 .9 749.9060 97.0752 .9 1012.2290 112.7832 .9 1313.8219 128.4911 31.0 754.7676 97.3894 .36.0 1017.8760 113.0973 41.0 1320.2543 128.8053 .1 759.6450 97.7035 .1 10-23.5381 113.4115 .1 1326.70-24 129.1195 .2 764.5380 98.0177 .2 1029.2172 113.7267 .2 1333.1663 129.4336 .3 769.4467 98.3319 .3 1034.9113 114.0398 .8 1339.6458 129.7478 .4 774.3712 98.6460 .4 1040.6212 114.3540 .4 1346.1410 180.0610 .5 779.3113 98.9602 .5 1046.3467 114.6681 .5 1352.6520 180.3761 .6 784.2672 99.2743 .6 1052.0880 114.9823 .6 1359.1786 130.6903 .7 789.2388 99.5885 .7 1057.8449 115.2965 .7 1365.7210 ISl.OOU .8 794.2260 99.9026 .8 1063.6176 115.6106 .8 1372.2791 131.3186 .9 799.2290 100.2168 .9 1069.4060 115.9248 .9 1378.8529 131.6827 32.0 804.2477 100.5310 37.0 1075.2101 116.2389 42.0 1385.4424 131.9469 .1 809.2821 100.8451 .1 1081.0-299 116.5531 .1 1392.0476 132.2611 .2 814.3322 101.1593 .2 1086.8654 116.8672 .2 1398.6685 132.6752 .3 819.3980 101.4734 .3 1092.7166 117.1814 .3 1405.30.)1 132.8894 .4 824.4796 101.7876 .4 1098.5835 117.4956 .4 1411.9574 133.2036 .5 829.5768 102.1018 .5 1104.4662 117.8097 .5 1418.0254 133.5177 .6 834.6898 102.4159 .6 1110.3645 118.1239 .6 1425.3092 188.8318 .7 839.8185 102.7301 .7 1116 2786 118.4380 .7 1432.0086 184.1460 .8 844.9628 103.0442 .8 1122.2033 118.7522 .8 1438.7238 1^.4602 .9 850.1229 103.3584 .9 1128.1538 119.0664 .9 1445.4546 1»4.7743 33.0 855.2986 103.6726 38.0 1134.1149 119.3805 43.0 1452.2012 186.0886 .1 860.4902 103.9867 .1 1140.0018 119.6947 .1 1458.9635 136.4026 .2 86).6973 104.3009 .2 1146.0844 120.C088 .2 1465.7415 186.7168 .3 870.9202 104.6150 .3 1152.0927 120..3230 .3 1472.5352 186.0310 .4 8^0.1588 104.9292 .4 1158.1167 120.6372 .4 1479.3446 186.3461 .5 88/. 4131 105.2434 .5 1164.1564 120.9513 .5 1486.1697 136.6598 .() 886.6831 105.5575 .6 1170.2118 121.2655 .6 1493.0105 186.0734 .7 891.9688 10>.8717 .7 1176.2830 121.5796 .7 1499.8670 187.2876 .« 85)7.2703 106.18.58 .8 1182.3698 121.8938 .8 1506.7393 187.6018 .9, 902.5874 100.5000 .9 1188.4724 122.2080 .9 1513.6272 187.0150 34.0 907.9203 106.8142 39.0 1194..5900 122.5221 44.0 1520.5308 188.2301 .1 9i:{.2688 107.1283 .1 1200.7246 122.8363 .1 1527.4502 188.5443 .2 918.63:31 107.4425 .2 1206.S742 1-23.1.'>04 .2 1534.3853 188.8584 .3 924.0131 107.7566 aJ 1213.0396 123.4646 .3 1541.3360 130.1726 .4 929.4088 108.0708 .4 1219.2207 123.7788 .4 1548.3025 199.4867 .5 934.8202 108.3849 .5 1225.4175 124.0929 .5 1555.2847 iao.8000 .6 940.2473 108.6991 .6 1231.6300 1-24.4071 .6 1562.2896 140.U63 .7 945.6901 109.0133 .7 12:J7.8582 1-24.7212 .7 1569.2962 1404202 .8 951.1486 109..3274 .8 1244.1021 125.0354 .8 1576.33ft6 140.7484 .9 956.6228 109.6416 .9 1250.:i617 125.3495 .9 1583U)700 141.0575 MENSl "BATION. - CIRCLBS. AREAS AND CIBCnMFBBENOBS OF CIBCLBS. {Aii»aHcing by Tenthi.) MENSURATION. - CIRCLES. AREAS AND CIBCUMFERBNCES OP CIRCLBa {Adtnncia'j On Tenlli».) s'si«7.4S27 7 M8T,74T4 ■21«,0W,'. ■■' 4UT.4S18 43Se.lM> »a.TSM 43T0.seu warn ii8!ti6eo aujan > MH.sau mmi "'■'■'*" MENSURATION. - C'lRCLBS. &BEAB AND CIRCUMFEBENCES OF CIRCLB8. (Adoancing by Tenths.) 48 MENSURATION. — CIRCLES. AREAS AND CIRGUMFEBENGES OF CIRCLES. {Advancing by Tenths, ) Diam 90.0 Area. Circum. Diam. Area. Circum. Diam. 97.0 Area. CirCBD. 6361.7251 282.7433 93.5 6866.1471 293.7389 7389.8113 304.7345 .1 6375.8701 283.0575 .6 6880.8419 294.0531 .1 7405.0559 305.0486 .2 6390.0309 283.3717 .7 6895.5524 294.3672 .2 7420.3162 305.3628 J3 6404.2073 283.6858 .8 6910.2786 294.6814 .3 7435.5922 305.6770 .4 6418.3995 284.0000 .9 6925.0205 294.9956 .4 7450.8839 305.9911 .5 6432.6073 284.3141 94.0 6039.7782 295.3097 .5 7466.1913 306.3053 .6 6146.8309 284.6283 .1 6954.5515 295.6239 .6 7481.5144 306.6194 .7 6461.0701 284.9425 .2 6969.3106 295.9380 .7 7496.8532 306.9336 .8 6475.3251 285.2566 .3 6984.1453 296.2522 .8 7512.2078 307.2478 .9 6489.5958 285.5708 .4 6998.9658 296.5663 .9 7527.5780 307.5619 91.0 6503.8822 285.8849 .5 7013.8019 296.8805 98.0 7542.9640 307.8761 .1 6518.1843 286.1991 .6 7028.6538 297.1947 .1 7558.3656 308.1902 .2 6532.5021 286.5133 .7 7043.5214 297.5088 .2 7673.7830 308.5044 .3 6.546.8356 286.8274 .8 7058.4047 297.8230 .3 7589.2161 308.8186 .4 6561.1848 287.1416 .9 7073.3033 298.1371 .4 7604.6648 300.1327 .5 6575.5498 287.4657 95.0 7088.2184 298.4513 .51 7620.1293 309.4400 .6 6589.9304 287.7699 .1 7103.1488 298.7655 .er 7635.6095 309.7610 .7 6604.3268 288.0840 .2 7118.1950 299.0796 .7 7651. lOM 310.0752 .8 6618.7388 288.3982 .3 7133.0568 299.3938 .8 7666.6170 310.3894 .9 6633.1666 288.7124 .4 7148.0343 299.7079 .9 7682.1444 310.7085 92.0 6647.6101 289.0265 .5 7163.0276 300.0221 99.0 7697.6893 311.0177 .1 6662.0692 289.3407 .6 7178.0366 300.3363 .1 7713.-2461 311.3318 .2 6676..5441 289.6548 .7 7193.0612 300.6504 .2 7728.8206 311.6460 .3 6691.0347 289.9690 .8 7208.1016 300.9646 .3 7744.4107 311.9602 .4 6705.5410 290.2832 .9 7223.1577 301.2787 .4 7760.0166 312.2743 .5 6720.0630 290.5973 96.0 7238.2295 301.5929 .5 7775.6382 812.5885 .6 6734.6008 290.9115 .1 7253.3170 301.9071 .6 7791.2754 812.9026 .7 6749.1542 291.2256 .2 7268.4202 302.2212 .7 7806.9284 818.2168 .8 6763.7233 291.5398 .3 7283.5391 302.5354 .8 7822.6971 818.6309 .9 6778.3082 291.8540 .4 7298.6737 302.8405 .9 7838.2815 818.8451 93.0 6792.9087 292.1681 .5 7313.8240 303.1637 100.0 7853.9816 314.1503 .1 6807.5250 292.4823 .6 7328.9901 303.4779 .2 6822.1.'>69 292.7964 .7 7344.1718 303.7920 .3 6836.8046 293.1106 .8 7359.3693 304.1062 .4 6851.4680 293.4248 .9 7374.5824 304.4203 MENSURATION. — CIRCLES. 49 AREAS OF CIRCLES. ^ADVANCING BY EIGHTHS.) AREAS. Diam. 0.0 0.1 0-1 o.| H o-# O.f O.J 0.0 0.0122 0.0490 0.1104 0.1963 0.3068 0.4417 0.6013 1 0.7854 0.9940 1.227 1.484 1.767 2.073 2.405 *i.761 2 3.1416 3.546 3.976 4.430 4.908 5.411 5.9.39 6.491 3 7.068 7.669 8.295 8.946 9.621 10.32 11.04 11.79 4 12.56 13.36 14.18 15.03 15.90 16.80 17.72 18.66 5 19.63 20.62 21.64 22.69 23.75 24.85 25.96 27.10 6 28.27 29.46 30.67 31.91 33.18 34.47 35.78 37.12 7 38.48 39.87 41.28 42.71 44.17 45.66 47.17 48.70 8 50.26 51.^ 53.45 55.08 56.74 58.42 60.13 61.86 9 63.61 65.39 67.20 69.02 70.88 72.75 74.66 76.58 10 78.54 80.51 82.51 84.54 86.50 88.66 90.76 92.88 11 95.03 97.20 99.40 101.6 103.8 106.1 108.4 110.7 12 113.0 115.4 117.8 120^ 122.7 125.1 127.6 130.1 13 132.7 135.2 137.8 140.5 143.1 145.8 148.4 151.2 14 153.9 156.6 159.4 162.2 165.1 167.9 170.8 173.7 15 176.7 179.6 182.6 185.6 188.6 191.7 194.8 197.9 16 -201.0 204.2 207.3 210.5 213.8 217.0 220.3 223.6 17 226.9 230.3 233.7 237.1 240.5 243.9 247.4 250.9 18 254.4 258.0 261.5 265.1 268.8 272.4 276.1 279.8 19 283.5 287.2 291.0 294.8 298.6 .302.4 306.3 310.2 20 814.1 318.1 322.0 326.0 330.0 334.1 338.1 342.2 21 346.3 850.4 854.6 358.8 363.0 367.2 371.5 375.8 22 380.1 384.4 388.8 393.2 397.6 402.0 406.4 410.9 23 415.4 420.0 424.5 429.1 433.7 438.3 443.0 447.6 24 452.3 457.1 461.8 466.6 471.4 476.2 481.1 485.9 25 490.8 495.7 600.7 605.7 510.7 515.7 520.7 525.8 26 630.9 636.0 641.1 646.3 551.5 656.7 562.0 567.2 27 572.5 677.8 683.2 688.5 693.9 599.3 604.8 610.2 28 615.7 621.2 . 626.7 632.3 637.9 643.5 649.1 6.54.8 29 660.5 666.2 671.9 677.7 683.4 689.2 695.1 700.9 30 1 706.8 712.7 718.6 724.6 730.6 736.6 742.6 748.6 1 31 754.8 760.9 767.0 773.1 779.3 785.5 791.7 798.0 32 804.3 810.6 816.9 823.2 829.6 836.0 842.4 848.8 33 855.3 861.8 868.3 874.9 881.4 888.0 894.6 901.3 34 907.9 914.7 921.3 928.1 934.8 941.6 948.4 955.3 35 962.1 969.0 975.9 982.8 989.8 996.8 1003.8 1010.8 36 1017.9 1025.0 1032.1 1039.2 10i6.3 1053.5 1060.7 1068.0 37 1075.2 1082.5 1089.8 1097.1 1104.5 1111.8 1119.2 1126.7 38 1134.1 1141.6 1149.1 1156.6 1164.2 1171.7 1179.3 1186.9 39 1194.6 1202.3 1210.0 1217.7 1225.4 1233.2 1241.0 1248.8 40 1256.6 1261.5 1272.4 1280.3 1288.2 1296.2 1304.2 1312.2 41 1320.3 1328.3 1336.4 1344.5 1352.7 1360.8 1369.0 1377.2 42 1385.4 1393.7 1402.0 1410.3 1418.6 1427.0 1435.4 1443.8 43 1452.2 1460.7 1469.1 1477.6 1486.2 1494.7 1503.3 1511.9 44 1520.5 1629.2 1537.9 1546.6 1655.3 1564.0 1572.8 1581.6 45 1590.4 1699 J) 1608.2 1617.0 1626.0 1634.9 1643.9 1652.9 50 MENSURATION. - CIRCUMFERENCES. CIRCUMPBRBNCBS OP CIRCLES. (advancing by eighths.) CIRCUMFERENCES. Diam. 0.0 04 o-i 0.| O.J 0-1 Of O.J 0.0 0.3927 0.7854 1.178 1.570 1.963 2.356 2.748 1 3.141 3.534 3.927 4.319 4.712 5.105 5.497 5.890 2 6.283 6.675 7.068 7.461 7.854 8.246 8.639 9.032 3 9.424 9.817 10.21 10.60 10.99 11.38 11.78 12 17 4 12.56 12.95 13.35 13.74 14.13 14.52 14.92 15.31 5 15.70 16.10 16.49 16.88 17.27 17.67 18.06 18.45 6 18.84 19.24 19.63 20.02 20.42 20.81 21.20 21.60 7 21.99 22.38 22.77 23.16 23.56 23.95 24.34 24.74 8 25.13 25.52 25.91 26.31 26.70 27.09* 27.48 27.88 9 28.27 28.66 29.05 29.45 29.84 30.23 30.63 31.02 10 31.41 31.80 32.20 32.59 32.98 33.37 33.77 34.10 11 34.55 34.95 35.34 35.73 36.12 36.52 36.91 37.30 12 37.69 38.09 38.48 38.87 39.27 39.66 40.05 40.44 13 40.84 41.23 41.62 42.01 42.41 42.80 43.10 48.68 14 43.98 44.37 44.76 45.16 45.55 45.94 46.33 46.73 15 47.12 47.51 47.90 48.30 48.69 49.08 49.48 49.87 16 50.26 50.65 51.05 51.44 51.83 52.22 52.62 63.01 17 53.40 53.79 54.19 54.58 54.97 55.37 66.76 66.15 18 56.54 56.94 57.33 57.72 58.11 58.51 58.90 60.29 19 59.69 60.08 60.47 60.86 61.26 61.65 62.04 62.48 20 62.83 63.22 63.61 64.01 64.40 64.79 66.18 66.58 21 65.97 66.36 66.75 67.15 67.54 67.93 68.32 68.72 22 69.11 69.50 69.90 70.29 70.68 71.07 n.47 71.88 23 72.25 72.64 73.01 73.43 73.82 74.22 74.61 76.00 24 75.39 75.79 76.18 76.57 76.96 77.36 77.75 78.14 25 78.54 78.93 79.32 79.71 80.10 80.50 80.89 81.28 26 81.68 82.07 82.46 82.85 83.25 83.64 84.03 84.48 27 84.82 85.21 85.60 86.00 86.39 86.78 87.17 87.57 28 87.96 88.35 88.75 89.14 89.53 89.92 90.32 00.71 29 91.10 91.49 91.89 92.28 92.67 93.06 93.46 03.85 30 94.24 94.64 95.03 95.42 95.81 06.21 06.60 96.90 31 97.39 97.78 98.17 98.57 98.96 99.35 99.76 100.14 32 100.53 100.92 101.32 101.71 102.10 102.49 102.89 103.20 33 103.07 104.07 104.46 104.85 105.24 105.64 106.03 106.42 34 106.81 107.21 107.60 107.99 108.39 108.78 109.17 109.56 36 109.96 110.35 110.74 111.13 111.53 111.92 112.81 112.71 36 113.10 113.49 113.88 114.28 114.67 115.06 116.46 115.85 37 116.24 116.63 117.02 117.42 117.81 118.20 118.60 118.90 38 119.38 119.77 120.17 120.56 120.95 121.34 121.74 122.13 39 122.52 122.92 123.31 12:J.70 124.09 124.49 124.88 125.27 40 125.66 126.06 126.45 126.84 127.24 127.63 128.02 128.41 41 128.81 129.20 127.59 129.98 130.38 130.77 131.16 181 J5 42 131.95 132.34 132.73 133.13 133.52 133.91 134.30 184.70 43 135.09 135.48 135.87 136.27 136.66 137.05 187.4ft 187.84 44 138.23 138.62 139.02 139.41 139.80 140.19 140.60 l¥iM 45 141.37 141.76 142.16 142.55 142.94 143.34 148.78 tuja MENSURATION. - CIECLES. 51 AREAS AND CIRCUMPBRBNCES OP CIRCLES. From I to 50 Feet. (advancing by one inch.) IHam. Area. Cimim. Diaffl. Area. Circnm. Diam. Area. Circum. Ft. Feei. Ft, In. Ft. Feet. Ft. In. Ft. Feet. Ft. In. 1 0.7854 ill 5 19.635 15 8t 15 llg 9 63.6174 28 3^ 1 0.9217 1 20.2947 1 64.8006 28 6} 2 1.069 3 8 2 20.9656 16 21 2 65.9951 28 9 3 1.2271 3 11 3 21.6475 16 5j 3 67.2007 29 f 4 1.3062 4 2| 4 22.34 16 9 4 68.4166 29 3 5 1.5761 4 5 5 23.0437 17 5 69.644 29 7 6 1.7671 4 8 6 23.7583 17 3 6 70.8823 29 10 7 1.9689 4 11 7 24.4835 17 6 7 72.1309 30 1 30 4 8 2.1816 5 2 8 25.2199 17 9| 8 73.391 9 2.4052 5 ^ 9 25.9672 18 3 9 74.662 30 7i 10 2.6398 5 9 10 26.7251 18 3 10 75.94^ 30 lift 11 2.8852 6 2| 11 27.4943 18 7 11 77.2362 31 Ijj 20 3.1416 6 Si 60 28.2744 18 104 10 78.54 31 5 1 3.4087 6 6 1 29.0649 19 1 1 79.854 31 83 2 3.6869 6 9 2 29.8668 19 43 2 81.1795 31 in 3 3.976 7 3 30.6796 19 7* 19 10 3 82.516 32 2i 4 4.276 7 31 4 31.5029 4 83.8627 32 5| 5 4.5869 7 7 5 32.3376 20 1| 6 85.2211. 32 8j 6 4.9087 7 10^ 6 33.1831 20 ^ 20 8| 6 86.5903 32 111 7 5.2413 8 1 7 34.0391 7 87.9697- 33 2| 8 5.585 8 4l 8 34.9065 20 111 8 89.3608 33 65 9 5.9395 8 7jr 9 35.7847 21 2} 9 90.7627 33 91 10 6.3049 8 10| 10 36.6735 21 5, 10 92.1749 34 f 11 6.6813 9 l| 11 37.5736 21 8i{ 11 93.5986 34 3| 30 7.0686 9 5 7 38.4846 21 llj 11 95.0334 34 6| 1 7.4666 9 8^ 1 39.406 22 3 1 96.4783 34 9.^ 2 7.8757 9 11 2 40.3388 22 61 2 97.9347 35 1 35 43 • 3 8.2957 10 2 3 41.2825 22 91 3 99.4021 4 8.7265 10 5 4 42.2367 23 1 4 100.8797 35 7. 5 9.1683 10 8} 5 43.2022 23 2i 5 102.3689 35 lOJ 6 9.6211 10 llj C 44.1787 23 ^ 6 103.8691 36 l| 1 10.0346 11 3 7 45.1656 23 9| 24 1} 7 105.3794 36 45 s 10.5591 11 6 i; 46.1638 8 106.9013 36 71} 9 n.0W6 11 9 ft 47.173 24 4J 9 108.4342 36 10| 10 r.5403 12 k 12 3 10 48.1962 24 7} 24 lOf 10 109.9772 37 25 11 ij.om 11 49.22,36 11 111.5319 37 51 4 12.5664 12 6J 8 50.2656 25 li 25 4| 12 113.0976 37 83 1 13.0952 12 9| 1 51.3178 1 114.6732 37 111 •; 13.63>3 13 1 2 52.3816 25 7j 2 116.2607 38 ^ 3 14.1862 13 4 n i> 63.4562 25 11 3 117.859 38 5i 4 14.7479 13 7 4 54.5412 26 2 26 5 4 119.4674 38 8| 5 15.3206 13 10 5 55.6377 5 121.0876 39 6 15.9043 14 1 6 56.7451 26 8 6 122.7187 39 3a 7 16.4986 14 4 7 57.8628 26 llJ 7 124.3598 39 ^ 8 17.1041 17.7205 14 7 8 58.992 27 2i 8 126.0127 39 9, 9 14 11 9 60.1321 27 51 9 127.6765 40 1 10 1S.3476 15 2| 16 6j 10 61.2826 27 9 10 129.3504 40 33 11 18.8858 .11 62.4445 28 i 11 131.036 40 6| 52 MENSURATION. - CIRCLES. Areas and Circumferences of Circles (Feet and Inches) . 1 Diam. Aw. Cireiim. Dbni. Area. Cirenra. Diam. Area. Cirenn. Ft. Feet. Ft. III. /Y. Feet. Ft. In. Ft, Feet. Ft. III. 13 132.73-26 40 10 18 254.4696 56 tii 23 415.4766 1> 3 1 134.4:391 41 U 1 256 8303 56 93 1 418.4915 7; 6jj 2 136.1574 41 4^ 2 259.2033 57 2 421 518J IL 9 3 137.8867 41 Ih 3 261.5872 57 4 3 424..')577 4 139.626 41 10.' 42 n 4 263.9807 57 ■^1 4 427.6055 7:3 31 5 141.3771 5 266.3864 57 10 5 430.6658 7:3 6^ 6 143.1391 42 4i 6 268.8031 58 1 6 433.7371 7;J 9^ i 144.9111 42 8 7 271.2293 58 4 7 436.8175 74 I 8 146.6049 42 IJi 8 273.6678 68 7 8 439.9106 74 4i 9 148.4896 43 2\ 9 276.1171 58 lOi 9 443.0146 74 7> 10 150.2943 43 5ft 43 8| 10 278.5761 58 2 10 446.1278 74 10 75 1 11 152.1109 11 281.0472 69 H 11 449.2536 UO 153.9384 43 Hi 19 283.5294 69 81 24 452.3904 75 4| 1 155.7758 44 2j 1 286.021 59 lU 1 455.5362 75 71 2 157.625 44 6 2 288.5249 60 2 2 458.6948 75 11 3 159.4852 44 9} 45 J 3 291.0307 60 5 3 461.8642 76 2| 4 161.3553 4 293.5641 60 sl 4 465.0428 76 5l 5 163.2373 45 Sl 5 296.1107 60 Hi 5 468.2341 76 8 6 165.1303 45 6{ 6 298.6483 60 H 6 471.4363 76 11 7 167.0331 45 9i 7 301.2054 61 el 7 474.6476 77 24 8 168.9479 46 i 8 303.7747 61 9ft 8 477.8716 77 6i 170.8735 46 4 9 306.365 61 ; 9 481.1065 77 9 10 172.8091 46 71 46 111 10 308.9448 61 31 10 484.3506 78 1 78 3} 11 174.7565 11 311.5469 62 6j 11 487.6073 15 176.715 47 1ft 20 314.16 62 9| 25 490.875 78 6ft 78 9i 1 178.6832 47 4 1 316.7824 62 n 1 494.1516 2 180.6634 47 73 2 319.4173 63 4 ••• 2 497.4411 79 1 70 8| 3 182.6545 47 10- 3 322.063 63 3 600.7415 4 184.6555 48 2ft 4 324.7182 63 lU 4 504.051 70 7 6 1S6;6684 48 5 5 327.3858 63 If 6 507.3732 79 n 6 18^6923 48 8, 330.0643 64 n 6 510.7063 80 1 7 19;X726 48 11 7 332.7522 64 7j 7 514.0484 80 4 8 192^7716 49 2 8 :j35.4525 64 11 8 517.4034 80 7 9 194:8282 49 5 9 338.1637 65 2i 9 520.7692 80 10 10 190.8946 49 8| 10 340.8844 66 H 10 524.1441 81 1 11 198.973 50 11 343.6174 66 8} 11 527.5318 81 5 16 201.0024 50 3| 50 (U 21 346.3614 «5 lll 26 530.9304 81 81 81 11} 1 203.161'. I 349.1147 66 'A 1 534.3.'379 2 205.2726 Th) 9 2 351.8804 66 H 2 537.7583 82 2| 3 207.S94li 51 3 3.)4.657l 66 9 3 641.18JKJ 82 5 i 4 209.5264 51 31 4 357.4432 66 4 4 544.6209 82 k} 5 211.6703 51 4 5 360.2417 67 6 648.083 82 112 6 213.82.51 51 10 (i 363.0511 67 6A 6 551. .5471 83 3 7 215.9896 52 n 7 365.8698 67 of 7 655.0201 88 Oft 8 218.1662 52 4i 8 368.7011 68 1 . 8 558.5069 83 OJ 9 220.3537 52 n 9 371.5432 6S 3| 9 662.0027 84 i 10 222.551 52 10ft 10 374.3947 68 1 10 665.5084 84 3 11 224.76J3 53 if 11 377.2587 68 10", 11 569.027 84 0| 17 226.9806 53 41 22 380.13:36 69 1| 27 572.5560 84 OZ 1 229.2105 5:1 8 1 :i8'3.0177 69 41 1 576.0940 86 1 2 231.4525 53 in 2 385.9144 69 7I 2 679.6463 85 4 3 233.7055 54 2 8 383.822 69 lOj 1 8 683.2066 85 8 4 235.9682 .54 5 4 391.7389 70 n 4 686.7796 85 11 5 238.2m 54 85 5 394.6683 70 5 5 600.3637 80 1 6 240.5287 .54 llg 6 .397.6087 70 8] 6 603.0587 80 4 7 242.8241 55 21 7 400.558!J 70 111 1 7 607.5026 80 7 8 245.1316 55 6 8 403.5204 71 A 8 601.1793 80 11 247.45 55 9 66 9 406.49:35 71 ^i 9 004.807 87 4 10 249.7781 10 409.4759 71 85 10 608.4436 87 U 11 252.1184 56 3^ 11 412.4707 71 ni 11 612.0(R)1 87 ^ MENSUIIATION. - CIRCLES. 58 Areas and Circumferences of Circles (Feet and Inches). Dim. Ft. 28 1 2 3 4 5 6 I 8 9 10 11 29 1 2 3 4 5 6 7 8 9 10 11 30 1 2 3 4 5 6 7 8 9 10 11 31 1 2 3 4 5 6 < 8 9 10 11 32 1 2 3 4 5 6 7 8 9 10 11 Area. Feft. 610.7536 619.4228 623.105 626.7982 630.5002 634.2152 6:J7.9411 641.6758 645.4235 649.1821 652.9495 656.73 660.5214 664.3214 668.1:346 671.9587 675.7915 679.6375 6H3.4943 687.3.598 691.2385 695.1028 699.0263 702.9377 706.86 710.791 714.735 718.69 722.654 726.631 730.618 734.615 738.624 742.645 746.674 750.716 754.769 758.831 762.906 766.992 771.086 775.191 779.313 783.440 787.581 791.732 795.892 800.065 804.25 808.442 812.648 816.865 821.090 825.3-29 829.579 833.837 838.103 »42.:«)1 846.681 8o0.»85 €irr«in. Ihim. Ft. Ft. III. 87 \\\ '&\ 88 21 1 88 5^ 2 88 9 3 89 1 89 3j 4 5 89 6j 6 89 9} 7 90 8 90> 3| 9 90 6^ 10 90 11 11 91 n 34 91 H 1 91 n 91 lOf 2 3 92 r 4 92 4 5 92 »! 6 92 \\\ < 93 2| 8 93 ol 9 9:J S| 10 93 11^ 11 94 •>» :l) 94 (5 1 94 9i 95 i 2 3 95 3A 4 95 6| 5 95 9j 6 96 2 i 96 4 8 96 7| 96 lOj 9 10 97 U 11 97 4| 36 97 7| 1 97 10| 2 9H 2 3 98 h\ 4 98 Sji 5 98 in 99 2| 6 7 99 52 99 8| 8 9 100 10 100 3| 11 100 6j{ 37 100 9^ 1 101 \ •> mm 101 3^ 3 101 6} 4 101 10 5 102 U 6 102 4 7 102 1\ 102 10{ 8 9 103 1 103 4| 10 11 Area. FeH. 855.301 859.624 863.961 868..'J09 872.665 877.035 881.415 885.804 890.206 894.619 899.041 903.476 907.922 912.377 916.844 921.323 925.810 930.311 934.822 939.342 943.875 948.419 952.972 957.538 962.115 966.770 971.299 975.908 980.526 985.158 989.803 994.451 999.115 1003.79 1008.473 1013.170 1017.878 1022.594 1027.324 1032.064 1036.813 1041. .576 1046..349 10.)1.130 1055.926 1060.731 10f>5.546 1070.374 1075.2126 10SO.059 1084.920 10S0.791 1094.671 109:>.564 1104.469 1109.3S1 1114.307 1119.244 1124.1M9 1129.148 Oirron. 109 %\ 109 11^ 110 28 110 h\ 110 8^ 111 111 3J 111 6 111 9 112 112 117 117 117 117 118 118 118 \ 3ii 112 6| 112 10 113 1| 113 4| 113 78 113 lOf 114 15 114 4^ 114 8 114 in 115 2\ 115 5# 115 9 115 11 116 2| 116 6 116 9 J ■t ^\ 6i 9| 4 74 118 10J 119 \i Ihan Ft. 38 1 2 3 4 5 6 7 8 9 10 11 39 1 2 3 4 5 6 7 8 9 10 11 40 1 2 3 4 • 5 6 7 8 9 10 11 41 1 2 3 4 5 6 7 8 9 10 11 42 1 2 3 4 5 6 I 8 9 10 11 Area. Feet. 1134.118 1139.095 1144.087 1149.089 1154.110 1159.124 1164.159 1169.202 1174.259 1179.327 1184.403 1189.493 1194.593 1199.719 1204.824 1-209.958 1-215.099 1-2-20.254 1-225.420 12.30.594 1-235.782 1*240.981 1-246.188 1-251 .408 1-256.64 1-261.879 1267.i:i3 1-272.397 1277.669 1282.955 1288.252 1293.557 1298.876 1.304.206 1309.543 1314.895 1320.267 1.325.6-28 1331.012 1.3.36.407 1341.810 1347.-2-27 1352.6.)5 1358.001 130.3.541 1369.001 1374.47 1379.952 13S5.446 1.390.-247 1396.462 1401.988 1407.522 1413.07 1418.6-29 1424.195 14-29.776 1435.367 1440.967 1446.580 Cirfnm. Ft. In. I 119 4i 119 7j : 119 105 120 2 I 120 5| i 1-20 Hi 1 120 Ui 1-21 2A 121 5J 121 8^ 121 in 122 31 122 61 1-22 9^ 123 i 123 3ji 123 6J 1*23 9| 124 IJ 1-24 4i 1 124 7H i 1-24 KU \ 1-25 if 1-2.) ^ 125 7| 1-25 11 1*26 2\ 126 bi 1-26 S4 1-26 1l| 1*27 25 127 5,' 1*27 9 1*28 i 1-28 3g 1-28 6j 1-28 9| 1*29 I 1*29 3j 1*29 7 1*29 101 130 U 130 4i 130 7| 130 lOS 131 n 1 131 5 i 131 8^ I 1.31 Hi 132 -2^ 132 51 132 HI 132 111 133 3 133 OH 133 91 134 i 134 ^ 134 63 134 9| 54 MENSURATION. -CIRCULAR ARCS. Areas and Circumferences of Circles (Feet etnd Inches). Diam. Ft. 43 1 2 I 5 6 7 8 9 10 11 44 1 2 3 4 ' 5 6 7 8 9 10 11 45 1 2 3 4 5 6 7 8 9 10 11 Area. Feet. 1452.205 1457.836 1463.483 1469.14 1474.804 1480.48.} 1486.173 1491.870 1497.532 1503..^) 1509.035 1514.779 1520.534 1526.297 1532.074 1537.862 1543.058 1549.478 1555.288 1561.116 1566.959 1572.812 1578.673 1584.549 1590.435 1596.:V29 1602.237 160S.155 1614.0S2 1620.023 1625.974 1631.9.33 1637.907 1643.891 1649.883 1655.889 CircBm. Diam. Ft. In. Ft. 135 1 46 135 4 135 1, 1 2 135 10 3 136 1 4 136 4i 5 136 7| 6 136 11 7 137 2i 137 5j 137 83 8 9 10 137 lit 11 138 2^ 47 138 5| 1 138 9 2 139 3 139 31 4 139 6 5 139 9 6 140 7 140 3 8 140 7 9 141 10, 10 141 l| 11 141 43 48 141 74 1 141 105 142 l| 2 3 142 5 4 142 8i 5 142 11 6 143 21 7 143 5 8 143 8^ 9 143 11 10 144 3 11 Area. Feet. 1661.906 1667.931 1673 97 1680.02 1686.077 1692.148 1698.231 1704.321 1710.425 1716.641 1722.663 1728.801 1734.947 1741.104 1747.274 1753.455 1759.643 1765.845 1772.059 1778.28 1784.515 1790.761 1797.015 1803.283 1809..562 1815.848 1822.149 1828.460 1834.779 1841.173 1847.457 1853.809 1860.175 1866.552 1872.937 1879.335 Cireum. Diam. Ft. In. Ft. 144 6 49 144 9, 1 145 2 145 3i 3 145 6: 4 145 9| 5 146 1 6 146 4 7 146 7 8 146 10 9 147 U 10 147 4 11 147 73 50 147 11 148 2 148 5 148 8 148 11 149 2 149 5 149 82 150 150 3 150 6 150 9i 151 151 3| 151 6 151 10 152 1 152 4: 152 Ik 152 10 153 13 153 4i 153 8| Area. Feet. 1885.745 1892.172 1898.504 1905.037 1911.497 1917.961 19-24.426 1930.919 1937.316 1943.914 1950.439 1956.969 1963.5 Cireia. Ft. In. 153 llj 154 2| 154 5) 154 8j 154 llj 155 2| 155 6 155 9J 156 1.56 156 61 156 9^ 157 I Circular Arcs. To find the length of a circular arc when its chord and height, or versed sine is given; by the following table. Rule. — Divide the height by the chord; find in the column of heights the number equal to tills quotient. Take out the corre- sponding number from the colunm of lengths. Multiply this number by the given chord. Example. — The chord of an arc is 80 and Its versed 6ine is 30, what is the length of the arc ? Ans. 30 -r 80 = 0.875. The lenglh of an arc for a height of 0.375 we find from table to be 1.840t«. 80 X 1.34063 = 107.2504 =? length of arc. MENSUKATION. — CIHCULAR ARCS. 55 TABLE OP CIRCULAR ARCS. Hght8. Lengths. Hghts. Lengths. Hghts. Lengths. Hghts. Lengths. Hghts. Lengths. .001 1.00001 .062 1.01021 .1-23 1.03987 .184 1.08797 .245 1.15.308 .002 1.00001 .063 1.01054 .124 1.04051 .185 1.08890 .-246 1.154-28 .00:J 1.00002 .064 1.01088 .125 1.04116 .186 1.08984 .247 1.15,U9 .001 1.00004 .065 1.01123 .126 1.04181 .187 1.09079 .248 1.15670 .005 1.00007 .066 1.01158 .127 1.04247 .188 1.09174 .249 1.15791 .oo-^ 1.00010 .067 1.01193 .128 1.04313 .189 1.09269 .250 1.15912 . .0J7 1.00013 .068 1.01228 .129 1.04380 .190 1.09365 .251 1.16034 .OOS 1.00017 .069 1.01264 .130 1.04447 .191 1.09461 .252 1.16156 .OO.J 1.00022 .070 1.01301 .131 1.04515 .192 1.09557 .253 1.16279 .010 1.00027 .071 1.01338 .132 1.04584 .193 1.09654 .254 1.16402 .Oil 1.00032 .072 1.01376 .133 1.04662 .194 1.09752 .255 1.16526 .012 1.00038 .073 1.01414 .134 1.047-22 .196 1.09850 .256 1.16650 .013 1.00045 .074 1. 01453 .135 1.04792 .196 1.09949 .-257 1.16774 .014 1.00053 .075 1.01493 .136 1.04862 .197 1.10048 .258 1.16899 .01.5 1.00061 .076 1.01533 .137 1.04932 .198 1.10147 .259 1.170-24 .016 1.00060 .077 1.01673 .138 1.05003 .199 1.10247 .260 1.17150 .017 1.00078 .078 1.01614 .1:39 1.05075 .200 l.ia347 .261 1.17-276 .018 1.00087 .079 1.01656 .140 1.05147 .-201 1.10447 .262 1.17403 .019 1.00097 .080 1.01698 .141 1.05-2-20 .'202 1.10548 .26:3 1.17530 .020 1.00107 .081 1.01741 .142 1.05293 .203 1.10650 .264 1.176.57 .021 1.00117 .082 1.01784 .143 1.05367 .204 1.10752 .265 1.17784 .022 1.00128 .083 1.01828 .144 1.05441 .-205 1.10855 .266 1.17912 .023 1.00140 .084 1.01872 .145 1.05516 .206 1.10958 .267 1.18040 .024 1.00153 .085 1.01916 .146 1.05591 .207 1.11062 .268 1.18169 .025 1.00167 .086 1.01961 .147 1.0566? .208 1.11165 .269 1.18-299 .026 1.00182 .087 1.02006 .148 1.05743 .209 1.11-269 .270 1.184-29 .027 1.00196 .088 1.02052 .149 1.05819 .210 1.11374 .271 1.18559 .028 1.00210 .089 1.02098 .150 1.0.")S96 .211 1.11479 .272 1.18689 .029 1.00-225 .090 1.02145 .151 1.0)973 .212 1.11584 .273 1.188-20 .030 1.00240 .091 1.02192 .152 1.06051 .213 1.11690 .274 1.18951 .031 1.00256 .092 1.02240 .153 1.06130 .214 1.11796 .275 1.19082 .032 1.00272 .093 1.02289 .154 1.06-209 .215 1.11904 .276 1.19-214 .033 1.00289 .094 1.02339 .155 1.06-288 .216 1.1-2011 .277 1.19346 .034 1.00307 .095 1.02389 .156 1.06:368 .217 1.12118 .278 1.19479 .035 1.00327 .096 1.02440 .157 1.06449 .218 1.1-2-225 .279 1.19612 .036 1.00345 .097 1.02491 .158 1.06530 .219 1.1-23:34 .280 1.19746 .037 1.00361 .098 1.02542 .159 1.06611 .2-20 1.1*2444 .-281 1.198S0 .038 1.0a384 .039 1.02593 .160 1.06693 .221 1.12554 .282 1. -20014 .039 1.00405 .10) 1.02645 .161 1.06775 .-2-22 1.1-2664 .283 1.20149 .040 1. 00426 .101 1.0289S .162 1.068.58 .2-23 1.1-2774 .284 1.20284 .(Wtl 1.00447 .102 1.02752 .163 1.06941 .2-24 1.1-2885 .285 1.20419 .042 1.00469 .10 J 1.02S06 .164 1.070-25 .-2-25 1.12997 .286 1.20555 .043 1.00492 .104 1.02860 .165 1.07109 .-2*26 1.13108 .287 1.20691 .044 1.00->15 .105 1.02914 .166 1.07194 .227 1.1.3-219 .288 1. -20827 1 .0*'> 1.00 ).39 .103 1.0-2970 .167 1.07279 .228 1.13331 .289 1.-20^)64 1 .046 1.0056:3 .107 1.03026 .168 1.07365 .2-29 1.13444 .290 1.21102 .047 1.00587 .108 1.03082 .169 1.07451 .-230 1.13557 .291 1.21-2.'39 .048 1.00612 .103 1.03139 .170 1.07537 .'231 1.13671 .292 1.21377 .049 1.0033S .110 1.03198 .171 1.076-24 .232 1.13785 .293 1.-21015 .050 1.00665 .111 1.03254 .172 1.07711 .233 1.13900 .294 1.-21654 .051 1.00692 .112 1.03312 .173 1.07799 .'234 1.14015 .295 1.21794 .(►.-.2 1.00720 .113 1.0:3371 .174 1.07888 .235 1.14131 .-296 1.219:33 .05:J 1.00748 .114 1.03430 .175 1.07977 .236 1.14247 .297 1.2-2073 .0.>4 1.00776 .115 1.03 J90 .176 1.08066 .-237 1.14363 .298 1.2-2-213 .055 1.00805 .116 1.03551 .177 1. OS 1 56 .-238 1.14480 .299 1.22:354 .050 1.00834 .117 1.03611 .178 1.0S246 .-230 1.14597 .300 1 .-2-2495 .057 1.00864 .118 1.03672 .179 1.083;J7 .-240 1.14714 .301 1. -226:36 .058 1.00895 .119 1.03734 .180 1.0S42S .2 41 1.148.32 .302 1.-2-2778 .059 1.00928 .120 1.03797 .181 1.08519 .242 1.14951 .30:j 1.-2-2920 .060 1.00957 .121 1.03860 .182 1.08611 .243 1.15070 .304 1.'2:]063 .061 1.00989 .122 1.03923 .183 1.08704 .-244 1.15189 .:305 1.23206 56 MENSURATION. — CIRCULAR ARCS. Table of Circular Aros {conciuded)], Hghts. .306 lengths. Hghts. Ungths. lights. lengths. Ughts. lengths. Hghts. Leigths. 1.23349 .345 1.29*209 .384 1.35575 .423 1.42402 .462 1.49651 .307 1.23492 .346 1.29.366 .385 1.3.5744 .424 1.42583 .463 1.49842 .308 1.23636 .347 1. -29523 .386 1.. 35014 .425 1.42764 .464 1.50033 ..309 1.2.3781 .348 1.29681 .387 1.. 36084 .426 1.42945 .465 1.:iO-224 , .310 1.23926 .349 1.29839 .388 1.36254 .427 1.43127 .466 1.50416 .311 1.24070 .350 1.29997 .389 1.36425 .428 1.4.3309 .467 1.50608 I .312 1.24216 .351 1.30156 .390 1.. 36596 .429 1.4.3491 .468 1.50800 ; .313 1.24361 .352 1.30315 .391 1.36767 .430 1.43673 .469 1.50992 1 .314 1.24507 .353 1.30474 .392 1.30939 .431 1.43856 .470 1.51185 ! .315 1.24654 .354 1.306.34 .393 1.37111 .432 1.44039 .471 1.51378 .316 1.24801 .355 1.30794 .394 1.37283 .433 1.44222 .472 1.51571 .317 1.24948 .356 l.:50954 .395 1.. 37455 .434 1.44405 .473 1.51764 .318 1.25095 .357 1.31115 .398 l.:37628 .435 1.44589 .474 1.51958 .319 1.25243 .358 1.31276 .397 1.37801 .436 1.44773 .475 1.52152 .320 1.25391 .359 1.314:37 .398 1.37974 .437 1.44957 .476 1.52346 .321 1.25540 .360 1.31599 .399 1.38148 .438 1.45142 .477 1.52541 .322 1.25689 ..361 l.:31761 .400 1.38.3-22 .439 1.45327 .478 1.527:36 .323 1.25838 .362 1.31923 .401 l.:38496 .440 1.45512 .479 1.52931 .324 1.25988 .363 1.. 3-2086 .402 1.38671 .441 1.45697 .480 1.53126 .325 1.26138 .364 l.:j-2-249 .403 l.:38846 .442 1.45883 .481 1.53322 .326 1.26288 .365 1.32413 .404 1.39021 .443 1.46069 .482 1.53518 .327 1.26437 ..366 1.32577 .405 1.. 391 96 .444 1.46255 .483 1.53714 .328 1.2(5)88 .367 1.32741 .406 l.:i9372 .445 1.46441 .484 1.53910 .329 1.23740 ..36S 1.32905 .407 l.:39548 .446 1.46628 .485 1.54106 .330 1.26892 ..383 1.33069 .408 1.397-24 .447 1.46815 .486 1.54302 :.m 1.270 U .3'<0 1.33-2:34 .409 1.39900 .448 1.47002 .487 1.54499 .332 1.2719 J .371 1.. 33399 .410 1.40077 .449 1.47189 .488 1.54696 .3:i3 1.27349 .372 1.. 3:3564 .411 1.40254 .450 1.47377 .489 1.54893 .:«4 1.27502 .373 l.a3730 .412 1.404:J2 .461 1.47565 .490 1.55091 .33> 1.27656 .374 l.:3:3896 .413 1.40610 .452 1.47753 .491 1.55289 .336 1.27810 .375 1.34063 .414 1.40788 .453 1.47942 .492 1.55487 .337 1.27964 ..376 1.342-29 .415 1.40966 .454 1.48131 .493 1.55685 .338 1.28118 .377 1.34:396 .416 1.4H45 .455 1.48320 .494 1.55884 .339 1.28273 .378 1.34583 .417 1.413-24 .456 1.48509 .495 1.56063 .340 1.28428 .379 1.. 34731 .418 1.41503 .457 1.48699 .496 1.56292 .341 1.28583 .380 1.:J4899 .419 1.41682 .458 1.48889 .497 1.56481 .342 1.28739 ..381 1.3506S .4-20 1.41861 .459 1.49079 .498 1.56681 .343 1.28895 .382 1.35-237 .421 1.4-2041 .460 1.49269 .499 1.56881 .344 1.29052 .38:3 1.35406 .422 1.42-221 .461 1.49460 .500 1.57<M0 Table of Leiig^ths of Circular Arcs whose Radius is 1. Rule. — Knowing the measure of the circle and the measure of the arc in degrees, minutes, and seconds; take from the table the lengths opposite the number of degrees, minutes, and seconds in the arc, and multiply their sum by the radius of the circle. Example. — What is the length of an arc subtending an angle of 13° 27' 8", with a radius of 8 fe<»t. Ana. Length for 13° = 0.2268928 27'= 0.0078540 8"= 0.0000388 1.30 27' 8"= 0.2:J47850 8 Length of arc = 1.8782848 feeL \l MENSURATION. —CIRCULAU ARCS. IjeugthB of Circular Arcs ; BadiuB = 1. I ziT^T^ri^rj.:* — -.i:sr:a> -if chords. ' • .:f Vir ■■••i-f ,T I,, f- r7»-M '.'§•* rft'tr^i '^f k'llf th€ arf^ and ." '*'r^9t"L it/i-i C7"i [jirtfji. (The vprswl ^^.J-,^ "ri^L* - ■Lit; ^itr^eniii.'alakT N/. Fig. 31.) , ^- A •-..'.. — J-'im :Iie siTau** of tli«* clionl of " B-^ -, ;;i-r -ir lt: ?;iijcru!!: "I^e itjo^ire of the versed iini-. lali ~jL£ii T3¥'jx zjifi si^iujre root of the ; i I -.:... i:' .J. — T'lr -iiirr. .t u-r' ^Iie ir: is •5i>, and the versed '■ ^ i.t. •'I- — J'- = =;X4, and \'J3kM = 48, ; iii -Ifr < z = \ft5^ the chord. ••i"j-( ir f:i f:-: r//*;;! :,itf diiMtter and versed »ine ! I. ji ■.-:." I" jt* '-rse-i sill' v- i. xai sobtncc the product from | .:•• li.i^it*. -r -j.-;i -jaijui-ur^ "-i- iiiiarn ot the reiuaiuder from .^' -r. ;:i.~- .r -.Air. lia.nrL-^r. i:ii ::4Ju uhe ft^aare root of that re- > ■•. 1-1- 3:. 4 ' ..:. — Titt riiOiHC-r }f a '."jrcfe is !♦». and the vereed ?»j.r .1 ui i.^- .i\ vjiL :r :L.e iIikc i»ik the iTC ? -i ... .._*_^ = -;;. : i: - Ti = 2S. lOUfS — 28* = 0216. \ -ii: f = f**. :iif ■-•xiodi oif the are. r- .1. ' I- I- ••■'■ r ■/ ' - 'I ■«••: jc&ea tAe cAord of the arc .::. — TLi^ vre <»; iir^ r-x-c or the sum of the squares of the ■■.>«■•: ?..:•' i:: . ;r '-« ' -ir .'iLOri of the arc. Y;> • \ • ..-.. — 7iH .-li.ri :r iji an: b ^ and the versed shie 30, V '..i^ > . It- :i:i.ri .c '■'■«^"*' "lit; at:? '"' " - >" • /■ •■: ■"' V.:""* T.1 'JT}: irA«i the diameter and rerW •. . ' . '-.1. "..•.— Xi ::i~ :j:h i^^i^iiecer by the versed sine, and take tlie "■. .■■-'■ '■.■«. C ."C l^*z.? V'"^.\1".JJ-. — V" ■'"■ •-■: *->.*f 5i;ujLre oi the chonl of half the arc by Kv.Y i. — A:.: :Jir? <i;ujLrv of half the chord of the arc to the o. ar*. . : :L*n: -■ rs •: <!-: . i:l<.I divide this sum by the versed sine. MENSURATION. —ARCS AND VERSED SINES. 59 Example. — What is the radius of an arc whose chord is 96, and whose versed sine is 36 ? Ans. 482 + 362 = o^qqq^ 3(500 -^ 36 = lOO, the diameter, and radius = 50. To compute the versed sine. Rule. — Divide the square of tlie chord of half the arc by the diameter. To compute the versed sine ivhen the chord of the arc and the diameter are given, ' Rule. — From the square of the diameter subtract the square of the chord, and extract the square root of tlie remainder; sul>- tract this root from the diameter, and halve the remainder. To compute the length of an arc of a circle when the number of degrees and the radius are given. Rule 1. — Multiply the number of degrees in the arc by 3.1416 multiplied by the radius, and divide by 180. The result will be the length of the arc in the same unit as the radius. Rule 2. — Multiply the radius of the circle by 0.01745, and the product by the degrees in the arc. Example. — The number of degrees in an arc is 60, and the radius is 10 inches, what is the length of the arc in inches ? Ans. 10 X 3.1416 X 00 = 1884.96 -f 180 = 10.47 inches; or, 10 X 0.01745 X 60 = 10.47 inches. To compute the length of the arc of a circle when the length is given in degreesj minutes, and seconds. Rule 1.^ Multiply the number of degrees by 0.01745329, and the product by the radius. Rule 2. — Multiply the number of minutes by 0.00029, and that product by the radius. Rule 3. — Multiply the number of seconds by 0.00000448 times the radius. Add together these three results for the length of the arc. See also table, p. 57. Example. —What is the length of an arc of 60° 10' 5", the radius being 4 feet ? Ans. 1. 60° X 0.01745329 X 4 = 4.188789 feet. 2. 10' X 0.00029 X 4 = 0.0116 feet. 3. 5" X 0.0000048 X 4 = 0.000W)6 feet. 4.200485 feet. MFNSVRATION.--C1RCULAR SEGMENTS, ETC. 7 -:'.:•:-:■: ■:/" -i a^rtf^r of circle ichen the degreea of the ^ ^ .:'•: 'An'! th^ rodvis are given (Fig. 82). F 5-32 ^,.— -^=^-— ^^^ RvLii. — Multiply the number of degrees in J ' _____r:?i.i ...^ _^ ; ^. .j^^ area of the whole circle, anddi- Ex-OiPLE. — Wliat is the area of a sector of A :.r.'-. ^* Lose radius is 5, and the length of the \ .y . ■--.■» .: >. A:rA •:: c'.role = 10 X 10 X 0.7854 = 78.54. 78.0 X 00 _ ^^^ TLrr. infa of sector = — ^^ — — 13.09. " . .•.-•■-.-,. .„ •-.-. :•;- .j.-ijrf'fs and mini(tes, reduce it >. v. :v.v..:.v V \ :':.v Arxra of the whole circle, and divide « I - i. ,--• .: rir^le irhen the length of the ' m * - — '^. .. ■•:'-• T-r:":. •"»: the arc by half the length of the X «' -.«h aw «*,«« ■»« ^« X % I . \ - ■* --ir.'/^ irhen the chord and . :\: -.' 'r'.'j* or diatui'ter of the circle "» -VN :'•. tw a xcmicircle). — Ascer >v.:,7 "..A* '.^ :"::o same arc as the segment, . .-\.. .' \ :-..%".-'-: fonuea l>v the chord of the ■."..-: s-:v:^'r. and late the difference of ■V V < ■•--••-— rV:n a ^mirirrle). — As- ^ -.. . : . : .ir-.,* i*: :he le>5er iH)rtion of the , Ar.A . : :"..t ul-.ole «.-ii\-le, and the remain- \ vv \ .:' . • :*-.; ::n.ujnfen»nce, and the . V . \ >:::■:* *^v of a sphere of 10 inches •- > V.:4li^= S1.416 inches,* - > : . . : :.t: >U7ftic« of sphere MENSURATION. — SPHERES AND SPHEROIDS. 61 To compute the surface of a segment of a sphei'e. Rule. — Multiply the height (be, Fig. 38) by the circumference of the sphere, and add the product to the area of the hase. To find the area of the base, we have the diameter of the sphere and the length of the versed sine of the arc abdy and we can find the length of the chord ad by the nde on p. 56. Having, then, the length of the chord ad for the diameter of the base, we can easily Fig. 33 find the area. Example. — The height, be, of a segment abd, is 36 inches, and the diameter of the sphere is 100 inches. What is the convex sur- face, and what the whole surface? Ans. 100 X 8.1416 = 314.16 inches, the circumference of sphere. 36 X 814.16 = 11309.76, th e conv ex surface. The length of ad = 100 — 30 x 2 = 28. V1OO2 — 28-^ = 96, the chord cwi. 962 X 0.7854 = 7238.2464, the area of base. 11309.76 + 7238.2464 = 18548.0064, the total area. To compute the surface of a spherical zone. Rule. — Multiply the height (cd, Fig. 34) ^ by the circumference of the sphere for the convex surface, and add to it the area of the two ends for the whole area. Fig.34 Spheroids, or Ellipsoids. Definition. — Spheroids, or ellipsoids, are figures generated by the revolution of a semi-ellipse about one of its diameters. When the revolution is about the short diameter, they are pro- late ; and, when it is about the long diameter, they are oblate. To compute the surface of a spheroid when the apheroid is prolate. Rule. — Square the diameters, ami nmltiply the square root of half their sum by 3.1416, anil this procluct by the short diamettn*. Example. — A prolate spheroid has diameters of 10 and 14 Uiches, what is its surface ? Ans. 10=2 = 100, and 142 = 19n._ Tlieirsum = 296, andi/-^ = 12.1655. 12.1655 x 3.1416 X 10 = 382.191 square inches. 62 MENSURATION.- CONES AND PYRAMIDS. To compute the mirface of n ipheroid when the fipheroid is obVite. KuLK. — Square the diameters, aud multiply the square root of lulf their smu by :i.l4ie, and tlila product by the long diameter. To tumipute thf mir/uM iff n ryllndm: liiiLK. — Multiply tlie ittngth by tiie circumference for the cod :X sarface, and add to the product the ares o> e two ends for the whole sm-face. I compute the HeetiontU urea of a circwtoi ring (Kg. 35). Ri'Mf. —Find the area of liotli circles, and subtract the area of t1ie sinaller from the area of tlie larger: the remainder will be the area of Fig.3S the ring. To i:im\\mti: the Hurfare of a eone. }{<T^E■ — Multiply the perimeter or circumference of the base by one-lialf the slant height, or side of the cone, for the convex area. Add (o this tlie ai'ea of the base, for ilie whole area. Example. —The diameter of the base of a cone ie 3 inches, and the slant height 15 inches, what Is the area of the cone f Ans. 3 X 3,141(i = 8.4248 = circumference of tmte. 6.4*248 X 7i ~ Hi.mi squai-e inches, the convex stu^ace. 3 X 3 X 0.TS54 = T.CI68 3(|iiare inches, the ares of baae. Area of cone = 77.7.J4 square Inches. PI jg To enmpute the itiea nf the surfneeof thefiru*- RULii. — MiUtipty tlie sum of the perinietets of the two cnils by the sinjit height of tlie fois- tnm, and <iivide l>y '2, fur the convex surface. Add tlie area of the lop and bottom surfaces. To rompiile the nurface ufa pyramid. Rule, — Multiply the perimeter of llie base by one-half the slant height, aud add to Uie product the area of the base. To i^nmpiite the nvrface of the fruttum <tf It pyrcmi.l. lti:i.K. — Multiply the sum of the perimeters of the two ends by the slant height of the frustiuu, lialve the pnxluct, aud add lo Uie result the area of the two euds. MENSURATION. - PIUSMS. 63 BfENSURATION OF SOLIDa To compute the volvme of a prism, RiJi.K. — Multiply tlie area of tli^ base by the height. This rule applies to any prism of any shape on the base, as long as the top and bottom surfaces are parallel. To compute the volume qf a prismoid. Definition. — A prismoid is a solid having parallel ends or bases dissimilar in sliape with qiuidri lateral sides. KuLK. — To the sum of the are^s of the two ends add four times the area of the middle section pai*allel to them, and ^ nmltiply this sum by one-sixth of the perpendicular height. Example. — What is the vol- ume of a quadrangular prismoid, as in Fig. 37, in which ah = 0", C(i = 4", ac = he = 10", ce = 8", ^ = 8", and //* = 6" ? Ans. Area of top Area of bottom Area of middle section 6jfJ 2 8 + 6 2 « + ($ X JO = 50. X 10 = 70. X 10 = 60. |50 + 70 + (4 X 60)1 X J^ = 600 cubic inches. Note. — The length of the end of the middle section, as mn in Fig. 37 = To find the volume of a prism truncated obliquely. Rule. — Multiply the area of the base by the average height of the edges. Example. — What is the volume of a truncated prism, as in Fig. 38, where (f = 6 inches, y7i = 10 inches, ea = 10, ft = 12, (?// = 8, an(l/^ = 8? Ans, Area of base = 6X10 =60 square inches. 10+12 + 8 + 8 Fig. 38 Average height of edges = = 9i inches. 60 X 9i = 970 cubic inches. 66 MEiNSUUATlON. — SPHEROIDS, PAUAB0L0ID3, ETC. the square of the radius of the base phis the square of the lieight 10:3 X 4 X 0.5236 = 341.3872 cubic inches vol- ume. Second Solution. — By the rule for fin«l- ing the diameter of a circle when a chord and its versed sine are given, we find that the diameter of tlie sphere in this case is 16.2o inches; then, by Rule 2, (3 X 16.25) — (2 X 4) = 40.75, and '!0.75 x 4^ X 0.5236 = 341.3872 Fig. 41. cubic inclies, the volume of the segment. To cowpiite the volume of a spherical zone. Definition. — The part of a sphere in- cluded between two parallel planes (Fig. 42). Rule. — To the sum of the squares of the radii of the two ends add one-third of the square of the height of the zone; nndtiply this sum by the height, and that Fig. 42. pi*oduct by 1.5708. To compute the volume of a nphei'ohh Rule. — Multiply the square of the revolv- ing axis by the fixed axis, and this product by 0.5236. To compute the volume of a parafjolold of revo- lution (Fig. 43). Rule. —Multiply the area of the base by half rifl.43 the altitude. To compute the volume of a hjperholoid of revolution (Fig. 44). Rule. — To the s(|Uare of the I'adius of the base add the square of the middle diameter; nmltiply tliis sum by the height, and the pix>tl- uct by 0.5236. To compute the volume of any Jiyure ^f revo- ^'^'^ lution. Rule. — Multiply the area of the generating surface by the clr- cuniference described by its centre of gi-avity. To compute the volume of an excavation, where the ground uf irrey- ular, and the bottom of the excavation is level (Fig. 45). Rule. — Divide the surface of the ground to be excavated Into equal squares of about 10 feet on a side, and ascertain by ineuu MENSURATION. — EXCAVATIONS. 67 a a A d a d d b Fig.45 d a a a of a level the height of each comer, a, a, a, ft, 6, &, etc., abo\e the level to which the ground is to be excavated. Then add togcllier the heights of all the corneis that only come into one scjuare. Next take twice the sum of the heights of all the corners that come in two squares, as 6, h, b ; next three times the sum of the lieiglits of all the corners that come in three squares, as r, c, c ; and then four timies the sum of the heights of all the j^ corners that belong to foiu* squares, as d, r2, d, etc. Add togetlwr all these ^ quantities, and multiply their sum by one-foiuth the ai-ea of one of the squares. The result will be the volume of the excavation. Example. — Let the plan of the excavation for a cellar be as in the figure, and the heights of each corner above the proposed bot- tom of the cellar be as given by the numbers in the figure, then the volume of the cellar would he as follows, the area of each square being 10 X 10 = 100 sqHai*e feet: — Volume = i of 100 (a's + 2 b's 4- 8 c's + 4 tZ's). The a's in this case = 4 + « + :J + 2+1 + 7 + 4 = 27 2 X the siun of the 6's = 2 X (3 + ($ + 1 + 4 + :{ + 4 )= 42 3 X the sum of the c's = 3 x ( 1 + ;^ + 4) =24 4 X the siuu of the *rs = 4 X (2 + 3 + + 2) =52 145 Volume = 25 X 145 = 3625 cubic feet, tlui <iUiintity of eailh to be exjavatetL 68 GEOMETKICAL PROBLEMS. OEOMETRICAL PROBLEMS. Problem 1 . — To bisect j or flimde into equal partSy a (/hen Ihu'.ah (Fig. 46). ^ From a and ft, with any radius greater tlian half of aft, describe ares intersecting in c and d. The line cd, connecting these intersections, will bisect a)), and be perpen- Fig.46 X^ l> diciilar to it. Pkoblkm 2. — To draw a perpendicular to a given straiyht line from a point witJf out it, 1st Method (Fig. 47). —From the point a describe an arc with sufficient radius that it will cut the line he « V X in two places, as e and /. From e and / describe two arcs, with the same radius, intersecting in g; then a line drawn from a to fj will be pei-peudicular to the line ftc. 2d Method (Fig. 48). — From any two » ^ • •s.-w —r^ points, d and c, at some distance apart in the given line, and with radii da and c« respectively, describe arcs cut- ting at a and e. Di-aw ae, and it wili l)c the I)erpendicular required. This method is useful where the given point is opposite the end of the line, or nearly so. Problem 3. — To draw a perpendicular to a straiyht line from a given point, a, in that line. >.i a Fig.49 1st Method (Fig. 49).— With any radius, from the given point a in the line, describe arcs cutting the line in the points ft and c. Then with b and c as centres, and with any radius greater than ab or ac, describe arcs cutting each other at d. The line Ja will be the perpendicular desireiL GEOMETRICAL PROBLEMS. 69 2d Method (Fig. 50, when the given point is at the end of the line). — From any point, 6, outside of the line, and with a radius ba, describe a semi- circle passing through a, and cutting the given line at rL Through b and d draw a straight line intersecting the semicircle at 6. The line ea will then be perpendicular to the line uc at the point a, 3d Method (Fig. 51) or the 3, 4, and 5 Method. — From the point a on the given line measure off 4 inches, ot-4 feet, or 4 of any other unit, and with the same unit of measure describe an arc, with a as a centre and 3 units as a radius. Then from b describe an arc, with a radius of 5 units, cutting the first arc in c. Then ca will be the perpen- dicular. This method is particularly useful in laying out a right angle on the ground, or framing a house where the foot is used as the unit, and the lines laid off by straight edges. In laying out a right angle on the ground, the proportions of the triangle may be 30, 40, and 50, or any other multiple of 3, 4, and 5; and it can best be laid out with the tape. Thus, first measure off, say 40 feet from (c on the given line, then let one person hold the end of the tape at b, another hold the tape at the 80-foot mark at a, and a third person take hold of the tape at the 50-foot mark, with his thumb and finger, and pull the tape taut. The 50-foot mark will then be at the point c in the line of the pei*pendicular. Problem 4. — To draw a strali/ht line parallel to a given line at a given distance apart (Fig. 52). i B d ( » Fig.52 I > From any two points near the ends of the given line describe two arcs about opposite the line. Draw the line cd tangent to these arcs, and it will be parallel to ab. 70 GEOMETRICAL PROBLEMS. Problem 5. — To eonstriici an (vngle equal to a given angie. With the point ^4, at the apex of the given angle, as a centre, and any radius, describe the arc BC, Then witli the point <r, at the vertex of tlie new angle, as a centre, and with the same radius as before, describe an arc like BC, Then with JiC as a radius, and h as a centre, describe an arc cutting the other at c. Then will cab b*» equal to the given angle CAB. Problem 6. — From a point on a given line to draw a line making an angle qf 6(P with. tJie (jiven line (Fig. 54). Take any distance, as ab, as a radius, and, with a as a centre, de- ^crilie the arc 6c. Then with 6 as a centre, and the same radius, describe an arc cutting the first one at c. Draw from a a line through (', and it will luake with ab an angle of 60^. Fig.54 Fig.55 Problem 7. — From a given point, A, on a given line, AE, to draw a line making an angle of 4^^ with the given line (Fig. 55). Measure off from A, on AE, any distance, -46, and at 6 draw a line perpendicular to AE. Measure off on this perpendicular be equal to Ab, and draw a line from A through c, and it will make an angle with AE of 45^. Problem 8. — From any point, A, on a given line, to draw a line which shall make any desired angle with the given line (Fig. 56). To perform this problem we must have a table of chords at hand (such as is found on pp. 85-'.)3), which we use as follows. Find in the table the length of chord to a radius 1, for the given angle. Then take any ra- ^ dius, as large as convenient, describe an arc of a circle be with A as a centre. Mul- tiply the chord of the angle, found in the table, by the length of the radius Ab^ and with the product as a new radius, and 6 as a centre, describe a short arc cutting be in d. Draw a line from A throngl: &, and it will make the desired aaglc with DE, Fig.56 GEOMETRICAL PROBLEMS 71 Example. — Draw a line from A on DE^ making an angle of 440 40' with DE. Solution. — We find that tlie largest convenient radius for our arc is 8 inches: so with ^ as a centre, and 8 inches as a radius, we describe the arc be. Then, looking in the table of chords, we find the chord for an angle or arc of 44° 40' to a radius 1 is 0.76. Mul- tiplying this by 8 inches, we have, for the length of our new radius, 6.08 inches, and with this as a radius, and 6 as a centre, we describe an arc cutting be in d. Ad will then be the line desired. Problem 9. — To biseet a given angle, as BAG (Fig. 57). With ^ as a centre, and any radius, descrl an arc, as eb. With c and b as centres, and any radius greater than one-half of eb, describe two arcs inter- secting in d. Draw from A a line through d, and it will bisect the angle BAC, Problem 10. — To biseet the anyle contained between two linen^ (IS A B and CI), when the vertex of the angle is not on the drawing (Fig. 58) Draw fe parallel to AB, and cd parallel to CD, so that the two lines will intersect each other, as at i. Bisect the angle cidy as in the preceding problem, and draw a line through i and o which will bisect the angle between the two given lines. Problem 11. — Through two given points, B and C, to describe an arc of a circle with a given radius (Fig. 59). With B and C as centres, and a radius equal to the given radius, describe two arcs intersecting at A» With ^ as a centre, and the same radius/ describe the ait; be, which Fig. 59 will be found to pass through the given points, B and C 72 GEOMETRICAL PROBLEMS. Problem 12. — To find the centre of a given circle (Pig: W)). Draw any chord in the circle, as ah, and bisect this chord by the perpendicu/ar cd. This line will pass through the centre of the circle, and ef will be a diameter of the circle. Bisect ^, and the centre o will be the centre of the circle. T*R0BLEM 13. — To draw a circular arc through three gii>en pointH, as A, B, and C (Fig. 61). Draw a line from ^ to J5 and from B to C. Bisect AB and BC by the lines aa and cc, and prolong these lines until they intersect at 0, which will be the centre for the arc sought. With o as a centre, and Ao as a radius, describe the arc ABC, Problem 14. — To describe a circular arc parsing through three given points^ when the centre is not availaJjle, by means of a tri- angle (Fig. 62). B^ Let il, JB, and C be the given points. Insert two stiff pins or nails at A and C. Place two strips of wood, SS, as shown in the figure; one against A, the other against C, and in clined so that tlieir intersection shall come at the third point, B. Fasten the strips together at their intersection, and nail a third strip, T, to their other ends, so as to make a firm trian^e. Place the pencil-point at B, and, keeping the edges of the trian^ against A and B, move the triangle to the left and right, and tbv l>eneil will describe the arc sought. OEOMETHICAL PROBLEMS. 73 X %/ / 4 ra Fig. 63 V When the points A and C are at the same distance from B^ if a strip of wood be nailed to the triangle, so tliat its edge de shall be at right angles to a line joining A and C as the triangle is moved one way or the other, the edge de will always point to the centre of the circle. This principle is used in the perspective linear cZ. PuoBLEM 15. .— To find a circular arc which shall be ianfjent to a f/iven point, A, on a straiyht lincj and ^ pass through a given point, C, ouUnde the line (Fig. 63). Draw from A a line perpendicular to the given line. Connect A and C by a straight line, and bisect it by the perpen- dicular ac. The point whei-e these two perpendiculars intersect will be the centre of the circle. Pkoblbm 16. — To connect two parallel linen by a reversed curve composed qf two circular arcs of equal radius, and tangent to the lines at given points, a« A and B (Fig. 64). Join A and B, and di- vide the line into two equal parts at C. Bisect CA and CB by perpen- diculars. At A and B erect i)erpendicu]ars to the given lines, and the intersections a and b will be the centres of the arcs composing the required curve. Pboblbm 17. ^On a given line, as AB, to construct a com- pound curve qf three arcs of circles, the radii of the two siue. ones being equal and qf a given length, atid their centres in the given line; the central arc to pans through a given point, C, on the perpen- dicular bisecting the^ given line, and tangent to Uie other two arcs (Fig. 66). Draw tlie pttpendlc- nlar CIX Lftj off Aa^ Bbf aud CCf eiudi equal to the given radius of the side arcs; join Fig.64 \:: / / I / Fig. 65 74 GEOMETRICAL PROBLEMS. ac; bisect ac by a perpendicular. The intersection of this line with tlie perpendicular CD will be the required centre of the central arc. Through n and h draw the lines De and De' ; from a and b, with the given radius, equal to Aa, Bby describe the arcs Ae'sind lie; from D as a centre, and CD as a radius, describe the arc eCef which completes the ciu^e required. Phoblem 18. — To conairuct a triangle upon a given straight line or bane, the length of the two tildes being given (Fig. 66). First (an equilateral triangle. Fig. 66a). — With the extremities A and B of the given line as centres, and AB sasi radius, descril)e arcs cutting each other at C Joiu AC and BC, Fig.GGa Fig. 60 b Second (when the sides are unequal, Fig. 66b). — Let ADh^ tt.e given base, and the other two sides be equal to C and B. With /) as a centre, and a radius equal to C, describe an indefinite arc With ^ as a centre, and B as a radius, describe an arc cutting the first at E. Join E ^dth A and 2>, and it will give the required triangle. Problem 19. — To describe a circle about a triangle (Fig. 67). Bisect two of the sides, us AC and CB, of the triangle, and at their centres erect perpendicular lines, as ae and />e, intersecting at e. With e as a centre, and eC as a radius, descril)e a circle, aud U will be found to pass through A and B. Fig. 67 Problem 20. — To inscribe a circle in a triangle (Fig. tSB), Bisect two of the angles, A and B, of the triangle by lines cntting each other at o. With o as a centre, aud oe as a radius, dMeribe » circle, which will be found to just touch the other two sideiu GEOMETRICAL rUOBLKMS. 40 PnoBLEM 21. — To inscribe a square in a circle^ and to describe a circle about a square (Fig. 69). To inscribe the square. Draw two diameters. AB and CDy r.t rigliL angles to each other. Johi the points A, 1), B, C, and we liavc the inscribed square. To describe the circle. Draw the diagonals as before, intersecting at E, and, with ^ as a centre and AE as a radius, describe the circle. PROBLKM 22. — To inscribe a circle in a fiqvare, and to deticrihe a square about a circle (Fig. 70). To inscribe the circle. Draw the diagonals AB and C7>,. inter- secting at E. Draw the i)eipendicular EG to one of the sides. Tlien with J^ as a centre, and EG as a radius, describe a circle, which will be found to touch all four sides of the square. To describe the square. Draw two diameters, AB and CD, at right angles to each other, and prolonged beyond the circumference. Draw the diameter GF, bisecting the angle CEA or BED. Drnw lines through G and JF* perpendicular to GF, and terminating in the diagonals. Draw AD and CB to complete the square. Pkoulem 23. — To inscribe a penta- gon in a circle (Fig. 71). Draw two diametei*s, AB and CD, at ^ right angles to each other. Bisect AG ,^ at E. AVith ^ as a centre, and EC as a A| « radius, cut OB at F. AVith C as a centre, and CF as a radius, cut the circle at G and U. With these points as centres, and the same radius, cut the circle at I and J. Join /, J, ff, G, and C, and we then have inscribed in tlie circle a regular pentagon. pROBt.KM 24. — 7b i»w(!rl6« B rflffntei SoLUTioH. — Lay off on tiie dm circle six times, and connoct the p Prohi.em 25. — To coniitmct a re^ KlriiiuUt line, AB (Fig. TA). From A and II, wiih a nullus equal at O. With aa a centre, aiid a rM circle, and from A and JS lay o'J tU fci'ciice of tlie circle, and join tbe result will be a regular liexagon. Pkoblem 26, — To coimtnict n re, atraUjkt line, AB (Fig. 74). Produce the line AB both vaya, an and Bb, of iudeflnite lei^^' Kaect B, and niake the length of the Itiiea e draw lines parallel to Aa, and eijual in centres G and I) describe arcs, witli i peiidiculara Aa and Bb In Fand E. ? E- Y ' \ A Flg.74 I'noni.KM 27.—Toniakeurrj/ultirct Dian- the diagonals .<1D and BC, C, and D, with a nuUin cqnal to A GEOMETRICAL PROBLEMS. 77 sides of the square in a, ft, c, d, c, /, h, and L Join these points to complete the octa^gon. Problem 28. — To inscribe a regular octagon in a circle (Fig. 76). Draw two diameters, AB and CD, at right angles to each other. Bisect the angles AOB and AOC by the diameters EF and Gfl. Join Af Ey I), 11 J B, etc., for the inscribed figure. a Fig.ZS /ig.77 PiiODLEM 29. — To inHcrihe a circle within a regular poh/f/on. Fimt (when the polygon has an even number of sides, as in Fig. T7). — Bisect two opposite sides at -4 and /?, and drawylZ?, and bisect it at C by a diagonal, DE, drawn between two opposite angles. With the radius CA describe the circle as required. Second (when the number of sides is odd, as in Fig. 78). — Bisect two of the sides at A and By and draw lines, AE and BD^ to the opposite angles, intersecting at C With C as a centre, and (J A as a radius, describe the circle as required. Pkoblem 30. — To deacribe a circle without a regular polygon. When the mmil)er of the sides is even, draw two diagonals from opposite angles, as ED and 67/ (Fig. 77), intersecting at C; and from C\ with CD as a radius, describe the circle required. When the number of sides is odd, find the centre, C, as in last pi-oblem; and with C as a centre, and CD (Fig. 78) as a radius, describe the cii'cle required. Fi8.78 GEOMETRICAL PKUBLEH8. Plioni.EM 31. —To describe an ellipse, the lengtli and hrei the Uro iiiex, behiji iiieeii. iH-hlg gh On All a as aiamcte from the centre, 0, ri A the circles. and CLDK nZberTf on the cir. eiicc of th. circle, aa 6", etc, an tlieiii dra» G to tlie cen FiB.79 cutting th, circle at tht a, a', a", elc, respectively. l,„.ji the points h, b', eU'., dra parAllel to the shorter axi 3; ami from the points n, a*, etc t^ . 1 lines parallel to tht — .,,,.^^ axis, and inlerset^ \, first set of lines i \ c", etc These last \ will be points in lipse, anil, h; obta K v\ ^ I the ellipse can ea y^ 2n Method (P ^^^ — Take the stmigl c of a stiff piece of FiB.eo canlboani, or woe Smm sor lie point, as ii, lu.ii rk off „b eqwal to half the sharle) / GEOMETRICAL PROBLEMS. 79 eter, and ac equal to half the longer diameter. Place the straight eilge so that tlie point h shall l>e on the longer diameter, and tlie point c on the shorter: then will the point a be over a point in the ellipse. Make on the paper a dot at a, and move the slip around, always keeping the points b and c over the major and minor axes. In this way any number of points in the ellipse may be obtained, which may be connected by a curve drawn freehand. 3d Method (Fig. 81, given the two axes AB and CD.) — FroTM the point Z> as a centre, and a radius A O, equal to D one-half of AB, describe an arc cutting AB at F andF'. These two points are called the foci of the ellipse. jOne property of the ellipse is, that the sum of the distances of any two points on the circumference from the foci is the same. Thus F'D + DF= F'E -f EF or F'G + GF.] Fix a couple of pins into the axis A B at F and F\ and loop a thread or cord upon them equal in length, when fastened to the pins, to AB, so as, when stretched as per dotted line FDF\ just to reach the extremity D of the short axis. Place a pencil-point inside the chord, as at E, and move the pencil along, always keeping the cord stretched tight. In this way the pencil will trace the outline of the ellipse. Problem 32. — To draw a tangent to an ellipse at a given point on the curve (Fig. 82). Let it be re- quired to draw a tangent at the point E on the ellipse shown in Fig. 82, First find the foci F and F'j as in the third method for describing an el- lipse, Hnuk from 1*^' 80 GKOMETRICAL PUOBl.EMS. E (li-aw lines EF and EF\ Prolong EF' to a, so that Ea shall equal EF. Bisect the angle uEF as iii 6, and through 6 draw a line touching the ciu-ve at E. Tliis line will be the tangent required. If It were tlt?sii*ed to draw a line normal to the ciu've at E, as, for instance, the joint of an elliptical arch, bisect the angle FEF\ and draw the bisecting line through E, and it will be the normal to the curve, and the proper line for the joint of an elliptical arch at that point. Problem 33. — To dmto a tarty ent to an ellipse from a yiven point without the curve (Fig. 83). Fig.83 From the point T as a centre, and a radius equal to Uio distance to the nearer focus F, describe a circle. From F' as a centre, and a radius equal to the length of the longer axis, describe arcs cutting the circle just described at a and b. Draw lines from F' to a and /;, cutting the circumference of the ellipse at E and G, Draw lines from T through E and G, and they will be the tangents reqiiired. PitOBLEM 34. — To describe an ellipse approximately, by means of circular arcs. First (with arcs of two radii, Fig. 84). —Take half the difference of the two axes AH and CD, and set it off fiom the centre O to (f and c on OA and OC ; draw ac, and set off half ac tx) d; draw dl parallel to ac; set off Oc equal to Od; join c /, and draw em and dm parallels to di and ic. On nt as a centre, with a i-adlus mC, describe an arc through C, terminating in 1 and 2; and with i as a'oentre, and id as a radius, describe an arc tlu'ough X>, terminating in points 3 and 4. On d and e as centres describe arcs through A and JS, connecting the points 1 and 4, 2 and 3. The four arcs' Urns de- GEOMETRICAL PROBLEMS. 81 smhod form approxiuiately an ellipse. Tliis methotl does not apply satisfaciov^^ when the conjugate axis is less than two-thirds of the li-ansvei-se axfs; Rg.04 C Second (willi arcs of three radii, Fig. 85). — On the ti-ansverse r.xis AB draw the rectangle AGEB, equal in height to 0C\ half the conjitgatc axis. Di-aw GD perpendicular to AC. Set off OK eqnal to OC^ and on AK as a diameter describe the semicircle 82 GEOMETlllCAL PROBLEMS. ANK, Draw a radiiis parallel to OC, intersecting the semicircle at N, and the line GE at P. Extend OC to L and to D. Set off OM equal to PJV, and on D as a centre, with a radius DM, descrilKj an arc. From A and B as centres, with a radius OX, intersect this arc at a and h. The i^oints //, a, 2), 6, //', are the centres of the arcs required. Produce tlie lines a/T, Da, Dh, hW, and the spaces enclosed determine the lengths of each arc. This process works well for nearly all ellipses. It is employed in striking: out vaults, stone arches, and bridges. Note. — In this example the point IT happens to coincide with the point K^ but this need not nccesuariiy be the case. The Parabola* PjtoiJT.KM 35. — To construct a parabola token the vertex A, the axis AB, and a jjoint, 21, of the curve, are given (Fig. 86). Construct the rectangle ABMC, Divide MC into any nmnbor of equal parts, four for instance. Divide ^C in like manner. Con- nect Al, A2, and ^13. Through 1', 2', 3', draw parallels to the axis. The intersections I, II, and III, of these lines, are i)olnt8 in the required ciure. Pkoblem 36. — To draw a tangent to a given points II, €f Hie parabola (Fig. 86). From the given point II let fall a perpendicular on the axis at 6. JCxteml the axis to the left of A, Make Aa equal to Ah, Draw (dl, and it is the tangent required. The lines perpendicular to the tangent are called normals. To find the, normal to any point 1, harhif/ the tangent to any oUier point, 11. Draw the normal lie. From I let fall a perpendicular Id, on the axis AB, Lay off de equal to be. Connect Ic, and we have the nonnal required. The tangent may be drawn at I bf iaying off a perpendicular to the uonnal le at L OKOMKTUICAL PIIOBLKMS. 83 Hie Hyperbola. The hyi>erbola possesses the characteristic that if, from any point, P, two sti-aiglit lines be drawn to two fixed points, F and jF", the foci, their difference shall always be the same. Phobi.em 37. — To ddHcrihe an hyperbola throvffh a </iven vertex, a, icith the (jwcu difference ahy and one of the foci, F (Fig 87). Draw the axis of the hyperbola AB, with the giveji distance ah and the focus F marked on it. From b lay off bFx equal to aF for the other focus. Take any point, as 1 on AB, and with a\ as a radius, and F as a centre, describe two short arcs above and below the axis. With 61 as a radius, and F' as a centre, describe arcs cutting those just described at P and P'. Take several points, as 2, :^, and 4, and obtain the corresponding points P.^, P;,, and P4 in the same way. Join these points with a curved line, am) it will be an hyperbola. To draw a tant/ent to any point of an hyperbola, draw linos from the givi'Ji point to each of the foci, and bisect the angle thus formed. The bisecting line will be the tangent recpiircd. 84 GEOMETlllCAL PROBLEMS. The Cycloid. __^^^_^QQ The cycloid Is the curve descrribed by a x>oint hi the circumference of a circle rolling in a straight line. Problem ;^. — To deacrihc a cy- cloid {Fi^. m. Draw the straight line AB slz the base. Describe the generating circle tangent to this line at the centre, and through the centre of Uic circle, C, draw the line EE parallel t<: the base. Let fall a perpendicular from C upon the base. Divide the semi-circumfer- ence into any number of equal parts, for instance, six. Lay off on A B and . CE distances 0*1', J '2', etc., equal to Q« the divisions of the circiunferencc. 5» Draw the chords Dl, D2, etc. From the points 1', 2', 3', on the line CE, with radii equal to the generating circle, describe arcs. From the points 1', 2^, 3', 4', 5', on the line BA, and with radii equal respectively to the chords 2)1, 7)2, D3, D4y 2)5, describe arcs cutting the preceding, and the inter- sections will be points of the curve required. GROMF-TRICAl. ritOBLEMS. 8B TABLE OF CHORDS ; Badios = 1.0000. KG GEOMETRICAL PROBLEMS. Table of Chords; Radius = l.OCXX) {continued). M. 1 IV 1J8- 13- 14- 1 .1917 .2091 .2264 .2437 I .l'>20 .2093 .22*57 .•2440 '2 .1923 .2096 .2270 .'2443 3 .1926 .2099 .2273 .2446 4 .192S .2102 .2276 .•2449 f» .19:J1 .2105 .2279 .•2452 1 .1931 .2108 .2281 .'2465 1 7 .1937 .2111 .2284 .•2458 ! 8 .1^0 .2114 .2287 .•24t>0 J « .1943 .2117 .2290 .•24<5:i ■10 .1946 .2119 .2293 .•2466 11 .1949 .2122 .22JK> .•2469 12 .1952 .2125 .22i>9 .•2472 13 .I9.'i5 .2128 .2:102 .•2475 14 .1M7 .2131 .2305 .•2478 15 .1960 .2134 .2307 .•2481 16 .1W>3 .2137 .2310 .2*84 17 .1^)66 .2140 .2313 .•2486 18 .1969 .214.-1 .2316 ••2489 19 .1972 .2146 .2319 .•2492 2U .1975 .2148 .2322 .*^95 21 .197S .2151 .2:J25 .•2498 22 .1981 .2154 .2328 .•2501 23 .198.1 .2157 .•2331 .•2504 24 .198«> .2UiO .2:133 .•2507 2.') .1989 .216:$ .2:1:16 ••2.510 21'. .IW2 .2106 .2339 .2512 27 .llW:') .2169 .2342 .•2515 2S .1998 .2172 .2:146 .•2518 2V .2001 .2174 ^148 .2521 30 .2004 .2177 .2:151 .•2524 31 .2007 .2180 .2354 .2527 '.)■> .2010 .2183 .2367 .25:10 3-i .2012 .2186 .2:159 .253:1 34 .2015 .2189 .2:162 .2636 3.-) .2018 .2192 .2365 .2.v:i8 36 .2021 .2195 .2368 .2.>H 37 .2024 .2198 .•2371 .2544 3K .2t)27 .2200 .2:174 .2547 3H .20:iO .220:1 .2377 .2660 40 .20: UJ .2206 .2380 .2653 41 .20;i6 .2209 .•2383 .2656 42 .2038 .2212 .2385 .•2559 4ii .2041 .2215 .'2388 .'2561 44 .2044 .2218 .•2:191 .2664 4i> .2047 .2221 .2:194 .2567 46 .2t).'>0 .2224 .2397 .•2570 J 47 .2a'»;) .222ti .2400 .•2573 .2a'HJ .2229 .2401 .2f>76 4H .20.')9 .22;i2 .240«) .•2679 (K) .2WJ2 .22:15 .2409 .•2^.82 Til .20t)5 .2238 .2411 .2585 ."•2 .2067 .2241 .2414 .•2587 61 .2070 .2244 .SM17 .2590 fi4 .2073 .2247 .2420 .25it:i r.5 •>076 .2260 .2423 .2596 f)6 .2079 .2253 .2426 .2599 •u .2082 .2256 .24'29 .2t)02 r>s .2085 .2258 .•24:12 .2605 M .2088 .2261 .24:14 .2»M)8 6tl .2091 .2264 .'2437 .•2611 16' .2611 .2613 .•2616 .2619 .2<)'25 .•2ti28 .2631 .•2»>W .'2636 .'26:19 .♦2642 .'2645 .'2648 .2651 .•2654 .'2657 .'2660 .•2662 .'2605 .•2668 .2671 .'2674 .2677 .'2680 .2683 .'2685 .2688 .2691 .2694 .2697 .2700 .270:1 .2706 .2709 .2711 .'2714 .'2717 .'27'20 .•27'2:j .'2726 .27^29 .27:12 .27:14 .2737 .2740 .274:1 .2746 .•2749 .2752 .2755 .2758 .•27(50 .•276:i .2766 .27459 .2772 .'2775 .•2778 .•2781 .•2783 16* .2783 .2786 .2789 .2792 .'2795 .2798 .•2801 .'2804 .'2807 .'2809 .•2812 .'2815 .•2818 .•2821 .•28-24 .'2827 .2830 .'2832 .'2835 .'28:18 .2841 .•2844 .'2847 .'2850 .'2853 .2855 .2858 .'2861 .2864 .2867 .•2870 .'2873 .•2876 i .'2878 . .•2vS81 , .•2S84 ' .•2887 ' .2890 I .•2893 ; .'2896 .'2899 j .•2902! .2904 1 .2907 ! .•2910 .•2D13 .•2916 .•2919 ! .29^22 .•2925 .'2927 .'2930 .'29:i:i .•29:m .-29:19 .2942 .-2945 .•2948 .•2950 .295:1 .'2956 17' .2966 .'2959 .'2962 .'2966 .'2968 .2971 .2973 .2976 .'2979 .2982 .2986 .2988 .2991 .2994 .2996 .'2999 .3002 .3005 .3008 .3011 .3014 .3017 .3019 .3022 .3026 .30'28 .30:11 .3034 .3037 .3040 .3042 .3046 .:1048 .3051 .:1054 .:1057 .3060 .3063 .:1065 .3068 .3071 .3074 .:J077 .:1080 .3083 .3086 .:i088 .3091 .3094 .3097 .3100 .310:1 .3106 .3109 .3111 .3114 .3117 .3120 .312:1 .31-26 .31-29 .3129 .3132 .31^4 .3137 .3140 .3143 .3146 .3149 .3162 .3155 .3167 .3160 .3163 .3166 .3169 .3172 .3176 .3178 .3180 .3183 .3186 .3189 .3192 .3195 .3198 .3*200 .3-203 .3'206 .3'209 .3212 .3216 .3218 .3221 .3*223 .3*226 .3-229 .3'232 .3-235 .3238 .3241 .3'244 .3-246 j .3-249 .3-252 .3255 .3-258 .3261 .3264 .3267 .3269 .3272 .3275 .3278 .3*281 .3-284 .3287 .3-289 .3-292 .3-295 .3-298 .3301 .3801 .3304 .3307 .3310 .3312 .3316 .3318 .3321 .3324 .33*27 .3330 .3333 .3335 .3338 .3541 .3au .3347 .3350 .3363 .3356 .3358 .3361 .2i\M .3367 .3370 .3373 .3376 .3378 .3381 .3384 .3387 .3390 .3393 .3306 .3398 .^1401 .3404 .3407 .3410 .3413 .3416 .:1419 .:1421 .3424 .34-27 :M80 .a433 .3436 .:1439 .:i441 oil 1 .Ol'l 1 .:1447 .3450 .3463 .3466 .3469 .:14<J2 .3467 .3470 .3473 «©• ai* .3473 ..1645 .3476 .3(U8 .3479 .3660 .3482 .3663 .8484 .3656 .3487 .3659 .3490 ^1662 .3493 .3665 .3406 .3668 .S409 .3670 .8502 .3673 .3504 .3676 .3507 .3679 .3510 .3682 .3513 .3686 .3516 .3688 M^'klO .3690 .3522 .3603 .3626 .3606 .3527 .3600 .3530 .3702 .3533 JNO& .3636 .3708 .3530 .3710 .3542 .3713 .3645 .3716 .3547 .3719 .3650 .3722 .3663 .3726 .3666 .3728 .3660 .3730 ..3662 .3733 .3665 .3736 .3567 .3730 .3670 .3742 .3573 .3745 .3676 .3748 .3670 .3750 .3682 .3753 .3685 .3766 .3687 .3760 .3600 .3762 .3603 .3765 .3606 .3768 mIuOO .3770 .8602 .3773 .3605 .8776 .3608 .3770 .3610 .8782 .3613 .3786 .3616 .3788 .3619 .3700 .36*22 .3703 .3626 .8706 .3628 .3709 .3030 jsaoi' .3633 .3805 .36:16 .3808 .3630 U»10 .3642 .3813 .8046 .3816 6 7 8 10 11 12 13 U 15 16 17 18 10 20 21 22 23 24 25 26 27 28 20 30 81 32 33 34 35 30 37 38 30 40 41 42 43 44 45 40 47 48 40 50 51 52 53 &4 55 56 57 58 50 «0 GEOMETRICAL PROBLEMS. 87 Table of Chords; Radius = 1.0000 (continued). M. aa* 88' «4- »5' «6' «?• 28* »9' 30* 3V 32' M. 0' (K .3816 .3987 .4158 .4329 .4499 .4669 .48.38 .5008 .5176 .5345 .5513 1 .3819 .3990 .4161 .4332 .4502 .4672 .4841 .5010 .5179 .5348 .5516 1 2 .3822 .3993 .4164 .4334 .4505 .4675 .4844 .5013 .5182 .5350 .5518 2 S .3825 •oVvD .4167 .4337 .4508 .4677 .4847 .5016 .5185 .5353 .5521 3 4 ..3828 •«J«I<T(I .4170 .4340 .4510 .4680 .4850 .5019 .5188 .5356 .5524 4 5 .3830 .4002 .4172 .4343 .4513 .4683 .4853 .5022 .5190 .5359 .5527 5 6 .3833 .4004 .4175 .4346 .4516 .4686 .4855 .5024 .5193 .5362 .5530 6 m I .3836 .4007 .4178 .4349 .4519 .4689 .4858 .5027 .5196 .5364 .5532 7 1 8 .3839 .4010 .4181 .4352 .4522 .4692 .4861 .5030 .5199 .5367 .5535 8 9 .3842 .4013 .4184 .4354 .4525 .4694 .4864 .5033 .5202 .5370 .5538 9 lu .3845 .4016 .4187 .4357 .4527 .4697 .4867 .5036 .5204 .5373 .5541 10 11 .3848 .4019 .4190 .4360 .4530 .4700 .4869 .5039 .5207 .5376 .5543 11 12 .3850 .4022 .4192 .4363 .4533 .4703 .4872 .5041 .5210 .5378 .5546 12 13 .3853 .4024 .4195 .4366 .4536 .4706 .4875 .5044 .5213 .5381 .5549 13 14 .:3856 .4027 .4198 .4369 .4539 .4708 .4878 .5047 .5216 .5384 .5552 14 15 .3859 .4030 .4201 .4371 .4542 .4711 .4881 .5050 .5219 .5387 .5555 15 16 .3862 .4033 .4204 .4374 .4544 .4714 .4884 .5053 .5221 .5390 .6557 16 17 .3865 .4036 .4207 .4377 .4547 .4717 .4886 .5055 .5224 .5392 .5560 17 18 .3868 .4039 .4209 .4380 .4550 .4720 .4889 .5058 .5227 .5395 .5563 18 19 .3870 .4042 .4212 .4383 .4553 .4723 .4892 .5061 .5230 .5398 .5566 19 20 .3873 .4044 .4215 .4:i86 .4556 .4725 .4895 .5064 ,5233 .5401 .5569 20 21 .3876 .4047 .4218 .4388 .4559 .4728 .4898 .5067 .5235 .5404 .5571 21 22 .3879 .4050 .4221 .4391 .4561 .4731 .4901 .5070 .5238 .5406 .5574 22 23 .3882 .4053 .4224 .4394 .4564 .4734 .4903 .5072 .5241 .5409 .5577 23 24 .3885 .4056 .4226 .4397 .4567 .4737 .4906 .5075 .5244 .5412 .5580 24 25 .3888 .4059 .4229 .4400 .4570 .4740 .4909 .5078 .5247 .5415 .5583 25 26 .3890 .4061 .4232 .4403 .4573 .4742 .4912 .5081 .5249 .5418 .5585 26 27 .3893 .4064 .4235 .4405 .4576 .4745 .4915 .5084 .5252 .5420 .5588 27 28 .3896 .4067 .42.38 .4408 .4578 .4748 .4917 .5086 .5255 .5423 .5591 28 29 .3899 .4070 .4241 .4411 .4581 .4751 .4920 .5089 .5258 .5426 .5594 29 30 .3902 .4073 .4244 .4414 .4584 .4754 .4923 .5092 .5261 .5429 .5597 30 31 .3905 .4076 .4246 .4417 .4587 .4757 .4926 .5095 .5263 .5432 .5599 31 32 .3908 .4079 .4249 .4420 .4590 .4759 .4929 .5098 .5266 .5434 .5602 32 33 .3910 .4081 .4252 .4422 .4593 .4762 .4932 .5100 .5269 .5437 .5605 33 34 .3913 .4084 .4255 .4425 .4595 .4765 .4934 .5103 .5272 .5440 .5608 34 35 .3916 .4087 .4258 .4428 .4598 .4768 .4937 .5106 .5275 .5443 .5611 35 36 .3919 .4090 .4261 .4431 .4601 .4771 .4940 .5109 .5277 .5446 .5613 36 37 .3922 .4093 .4263 .4434 .4604 .4773 .4943 .5112 .5280 .5448 .5616 37 38 .3925 .4096 .4266 .4437 .4607 .4776 .4946 .5115 .5283 .5451 .5619 38 39 .3927 .4098 .4269 .4439 .4609 .4779 .4948 .5117 .5286 .5454 .5622 39 40 .3930 .4101 .4272 .4442 .4612 .4782 .4951 .5120 .5289 .5457 .5625 40 '.41 .3933 .4104 .4275 .4445 .4615 .4785 .4954 .5123 ,5291 .5460 .5627 41 142 .3936 .4107 .4278 .4448 .4618 .4788 .4957 .5126 .5294 .5462 .5630 42 43 .3939 .4110 .4280 .4451 .4621 .4790 .4960 .5129 .5297 .5465 .5633 43 44 .3942 .4113 .4283 .4454 .4624 .4793 .4963 .5131 .5300 .5468 .5636 44 45 .3945 .4116 .4288 .4456 .4626 .4796 .4965 .5134 .5303 .5471 .5638 45 46 .3947 .4118 .4289 .4459 .4629 .4799 .4968 .5137 .5306 .5474 .5641 46 47 .3950 .4121 .4292 .4462 .4632 .4802 .4971 .5140 .5308 .5476 .5644 47, 48 .3953 .4124 .4295 .4465 .4635 .4805 .4974 .5143 .5311 .5479 .5647 48; 49 .3956 .4127 .4298 .4468 .4638 .4807 .4977 .5145 .5314 .5482 .5650 49 50 .3959 .41.30 .4300 .4471 .4641 .4810 .4979 .5148 .5317 .5485 .5652 50 bl .3962 .4133 .4303 .4474 .4643 .4813 .4982 .5151 .5320 .5488 .5655 51 i 52 .3965 .4135 .4306 .4476 .4646 .4816 .4985 .5154 .5322 .5490 .5658 52 53 .3967 .4138 .4309 .4479 .4649 .4819 .4988 .5157 .5325 .5493 .5661 53 54 .3970 .4141 .4312 .4482 .4652 .4822 .4991 .5160 .5328 .5496 .5664 54 55 .3973 .4144 .4315 .4485 .4655 .4824 .4994 .5162 .5331 .5499 .5666 55 56 .3076 .4147 .4317 .4488 .4658 .4827 .4996 .5165 .5334 .5502 .5669 56 57 .3979 AlbO .4320 .4491 .4660 .4830 .4999 .5168 .5336 .5504 .5672 57 58 .3982 .4153 .4323 .4493 .4663 .4S:J3 .5002 .5171 .5339 .5507 .5075 58 59 .3085 .4155 .4326 .4496 .4666 .48:j6 .5005 .5174 .5342 .5510 .5678 59 60 .3987 .4158 .4329 .4499 .4669 .4838 .5008 .5176 .5345 .5513 .5080 60 88 GEOMETRICAL PROBLEMS. Table of Chords ; Radius = 1.0000 {continued) , 0' 1 2 3 4 5 6 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 21 25 26 27 2S 29 30 31 32 •M 34 35 .'it; 37 :is 39 40 I 41 42 43 44 j45 '46 i47 '48 I 49 ■50 51 ,52 ;53 I 54 :6:. ' 5'> I 57 : 5S I 59 I 30 5680 5683 56S6 5689 5691 5694 5697 5700 5703 57a5 5708 5711 5714 5717 5719 5722 5725 5728 57:U) 57;j:j 5 'TM 5739 5742 5744 5747 5750 5753 5756 575S 5701 5704 5767 5769 5772 :> M O 577S 5781 5783 5786 5789 5792 5795 5797 5800 5803 5806 5S0S 5811 58 14 5817 5820 5SJJ 5h28 58:;i :.s:j4 5s:;r» .'i.s:j9 .■•s4-.; 58 J.') 5S47 .5847 .5850 .5853 .5866 .5859 .5861 .5864 .5867 .5870 .5872 .5875 .5878 .5881 .5884 .5886 .5889 .58;»2 .5895 .5897 .5900 .5'.Ktt ..5903 .5909 .5911 .5914 .5917 .5920 .5922 .5925 .5928 .5931 .5931 .5936 .59:19 .5942 .5945 .5947 .5950 .5953 .5956 .5959 .5961 .5964 .5967 .5970 ..5972 .5975 .5978 .5981 .5984 ..5980 .59S0 .5992 .5995 .59'.)7 .iUMH) .004 »6 .0011 .0014 .6014 .6017 .6(r20 .6022 .6025 .6028 .6a-a .6034 .etm .6039 .6042 .6045 .mil .605i) .605:^ .60r»6 .6058 .60«)1 jMm .6067 .6»)70 .0072 .6075 .6;»7S .6t)Sl .608:) .6;)8:'. .6089 .6(K»2 .60{»5 .0097 .61 on .6103 .6106 .6108 .6111 .6114 .6117 .6119 .6122 .6125 .6128 .6130 .61:^:) .6i:Mi .6139 .6142 .6144 .6147 .6150 .6153 .6155 .0158 .0101 .61('>4 .010<\ .010!) .0172 .0175 .iil7S .01 SO I .6180 .6183; .6186 .6189 .6191 .6194 .6197 .6200 .6202 .6205 .6208 .6211 .6214 .6216 .6219 .6222 .6225 .6227 .6230 .6233 .6-236 .6238 .6241 .6244 .6247 .6249 .6252 .6255 .6258 .6260 .6263 .6266 .6269 .6272 .6274 .6277 .6280 .6283 .6285 .6288 .6291 .Ui94 .62<H5 .6299 .6302 .6305 .6307 .6,310 .6313 .6316 .(»18 .6321 .6:J24 .r>327 I .O:i:io I .<^i:>2 .»):j;j5 .(Viil .o:n:i .0:> l«i r I 37' 38' .6511 39» 40* 4V 4«' 43* M. 1 .6346 .6676 .6840 .7004 .7167 .7330 0'. .6349 .6514 .6679 .6843 .7007 .7170 .7333 i; .6.%>2 .6517 .6682 .6846 .7010 .n73 .7335 2 .63.54 .6520 .6684 .6849 .7012 .n76 .7338 3 .6:j57 .6522 .6687 .6851 .7015 .n78 .7341 4 .62)60 6.525 .6«J90 .68.54 .7018 .nsi .7344 5 XuViH .6528 .6693 .68.57 .7020 .7181 .7346 «■ .631^5 i .6531 .6695 .6860 .7023 .7186 .7340 4 .6:J6S , .65:w .6098 .6862 .7026 .n89 .7352 8i .6371 1 .6536 .6701 .6865 .7029 .7192 .7354 »i .6374 .6.5.39 .6704 .6868 .7081 .7195 .7357 101 .a376 .6542 .6706 .6870 .7034 .7197 .7360 11 .6379 .6544 .6709 .6873 .7Cte7 .7200 .73G2 12 .6:182 .6547 .6712 .6876 .7040 .7203 .73«5 13 .0385 .6550 .6715 .6879 .7042 .7205 .7368 14 .6387 .6553 .6717 .6881 .7045 .7208 .7371 15 .6390 .6555 .6720 .6884 .7048 .7211 .7373 16 .6393 .6558 .6723 .6887 .7050 .7214 .7376 17 .0:»6 .6561 .6?25 .6890 .7053 .7216 .7370 18 .6398 .6564 .6728 .6892 .7096 .7210 .7381 19 .«U01 .6566 .6731 .6895 .7059 .7222 .7384 20 .6404 .6569 .6734 .6898 .7061 .7224 .7387 21 .6407 .6572 .6736 .6901 .7064 .7227 .7300 22 .MIO .6575 .67.39 .6903 .7067 .7230 .7302 23 .6412 .6577 .6742 .6906 .7069 .7232 .7395 24 .6415 .6580 .6715 .6909 .7072 .7235 .7308 25 .6418 .6583 .6747 .6911 .7075 .7238 .7400 26 .6421 .6586 .6750 .0914 .7078 .7241 .7408 27 .6423 .6588 .675:$ .6917 .7080 .7213 .7406 28 .6426 .6591 .6756 .692«) .7083 .7246 .7408 SO .0)429 .6594 .6758 .6922 .7086 .7240 .7411 30 .6432 .6597 .6761 .6925 .7089 .7251 .7414 31 .6434 .6599 .6764 .6928 .7091 .7254 .7417 32 .6437 .6602 .6767 .6931 .7094 .7257 .7410 38 .6440 .6605 .6769 .6933 .7097 .7260 .7^2 84 .6443 .6608 .6772 .6936 .7099 .7202 .7425 35 .6445 .6610 .6775 .6039 .7102 .7285 .7427 ao .6448 .6613 .6777 .6941 .7105 .7268 .7480 37 .6451 .6616 .6780 .6944 .7108 .7270 .7433 38 .6454 .6619 .67X3 .6947 .7110 .7278 .7433 39 .♦U56 .6621 .6786 .6950 .7118 .7276 .7438 40 .6459 .6624 .(.788 .6952 .7116 .7270 .7441 41! .0462 .6<)27 .6791 .6955 .7118 .7281 .7443 42- .6465 .60;M) .0794 .6958 .7121 .7284 .7446 43: .0407 .66:J2 .6797 .6961 .7124 .7287 .7440 44 .6470 .66:)5 .6799 .6963 .7127 .7280 .7468 45 .6473 .663S .6802 .6966 .7129 .7292 .74U 46 .6470 .6640 .6805 .61HJ9 .7132 .7205 .7467 47 .W78 .6<U3 .0S08 .6971 .7135 .7298 .7400 48 .6481 .(•)640 .6810 .6974 .7137 .7900 .7402 49> .6484 Am\) .0813 .0977 .7140 .7308 .7405 50> .6487 ' .0051 .0810 .0981 .714:5 .7306 .7468 51 .6tSt| .60 "i 4 .r.8l9 .0982 .7146 .7308 .7471 52' .fi4l»2 ; .0<r)7 .«>X21 .0985 .7148 .7311 .7478 53, .r>4y5 1 .ooiKJ .0824 .6!>88 .7151 .7814 .7476 54 .o49*< 1 sm-i .r>.S27 .0901 .7154 .7316 .7470 55 .»>;'i<M) .fiOiV) AW1\) .6'.>1»3 .7150 .7310 .7481 50; ,0511:; .Oi'is .0k:12 .0LKH5 .7159 .7322 .7484 57 i .O'ltMi .OiTl .ov;5 .6i>.t'.i .7Hi2 .7325 .7487 68 .0V»'.» .007:') .ris;;^ .7001 .7165 .7527 .7480 M .o-.ii .(•►«>7r» .0840 , .7(M»4 .7107 .7330 .7402 00 .ble of Chords ; Radius = 1.0000 (continued) • 4' 46* 46' 47' 48' 49' SO" 51- 68- 1 54* M. 0' 192 .7654 .7815 .7975 .8135 .8-294 .8452 .8610 .8767 .8924 .9080 m .7656- .7817 .797:^ .8137 .8297 .8455 .8613 .8770 .8927 .9082 1 m .7659 .7820 .7981) .8140 .8299 .8458 .8615 .8773 .8929 .9085 2 wo .7662 .7823 .7983 .8143 .8302 .8460 .8618 .8775 .8932 .9088 3 i03 .7664 .7825 .7986 .8145 .8304 .8463 .8621 , .8778 .8934 .9090 4 m .7667 .7828 .7988 .8148 .8307 .8466 .8623 .8780 .8937 .9093 5 m .7670 .7831 .7991 .8151 .8310 .8468 .8626 .8783 .8940 .9096 6 »ii .7672 .7833 .7994 .8153 .8312 .8471 .8629 .8786 .8942 .9098 7 il4 .7675 .7836 .7996 .8156 .8315 .8473 .8631 .8788 .8945 .9101 8 >16 .7678 .7839 .7999 .8159 .8318 .8476 .8634 .8791 .8947 .9103 9 »19 .7681 .7841 .8002 .8161 .8320 .8479 .8636 .8794 .8950 .9106 10 >22 .7683 .7844 .8004 .8164 .8323 .8481 .8639 .8796 .8953 .9108 11 »24 .7686 .7847 .8007 .8167 .8826 .8484 .8642 .8799 .8955 .9111 12 »27 .7689 .7849 .8010 .8169 .8328 .8487 .8644 .8801 .8958 .9113 13 30 .7691 .7852 .8012 .8172 .8331 .8489 .8647 .8804 .8960 .9116 14 33 .7694 .7855 .8015 .8175 .8334 .8492 .8650 .8807 .8963 .9119 15 35 .7697 .7857 .8018 .8177 .8336 .8495 .8652 .8809 .8966 .9121 16 38 .7699 .7860 .8020 .8180 .8339 .8497 .8655 .8812 .8968 .9124 17 41 .7702 .7863 .8023 .8183 .8341 .8500 .8657 .8814 .8971 .9126 18 •43 .7705 .7865 .8026 .8185 .8344 .8502 .8660 .8817 .8973 .9129 19 •46 .7707 .7868 .8028 .8188 .8347 .8505 .8663 .8820 .8976 .9132 20 49 .7710 .7871 .8031 .8190 .8349 .8508 .8665 .8822 .8979 .9134 21 •51 .7713 .7873 .8034 .8193 .8:J52 .8510 .8668 .8825 .8981 .9187 22 •54 .7715 .7876 .8036 .8196 .8355 .8513 .8671 .8828 .8984 .9139 23 .57 .7718 .7879 .8039 .8198 .8357 .8516 .8673 .8830 .8986 .9142 24 •60 .7721 .7882 .8042 .8201 .8360 .8518 .8676 .8833 .8989 .9145 25 •62 .7723 .7884 .8044 .8204 .8363 .8521 .8678 .8835 .8992 .9147 26 •65 .7726 .7887 .8047 .8206 .8365 .8523 .8681 .8838 .8994 .9150 27 •68 .7729 .7890 .80 JO .8209 .8368 .8526 .8684 .8841 .8997 .9152 28 .70 .7731 .7892 .8052 .8212 .8371 .8529 .8686 .8843 .8999 .9155 29 •73 .7734 .7895 .8055 .8214 .8373 .8531 .8689 .8846 .9002 .9157 30 •76 .7737 .7898 .8058 .8217 .8376 .8534 .8692 .8848 .9005 .9160 31 .78 .7740 .7900 .8060 .8220 .8378 .8537 .8694 .8851 .9007 .9163 32 •81 .7742 .7903 .8063 .8222 .8381 .8539 .8697 .8854 .9010 .9165 33 •84 .7745 .7906 .8066 .8225 .8384 .8542 .8699 .8856 .9012 .9168 34 86 .7748 .7908 .8068 .8228 .8386 .8545 .8702 .8859 .9015 .9170 35 .89 .7750 .7911 .8071 .8230 .8389 .8547 .8705 .8861 .9018 .9173 36 .92 .7753 .7914 .8074 .8233 .8392 .8550 .8707 .8864 .9020 .9176 37 .9.) .7756 .7916 .8076 .8236 .8394 .8552 .8710 .8867 .9023 .9178 38 .97 .7758 .7919 .8079 .8238 .8397 .8555 .8712 .8869 .9025 .9181 39 iOO .7761 .7922 .8082 .8241 .8400 .8558 .8715 .8872 .9028 .9183 40 m .7764 .7924 .8084 .8244 .8402 .8560 .8718 .8874 .9031 .9186 41 105 .7766 .7927 .8087 .8246 .8405 .8563 .8720 .8877 .9033 .9188 42 i08 .7769 .7930 .8090 .8249 .8408 .8566 .8723 .8880 .9036 .9191 43 ill .7772 .7932 .8092 .8251 .8410 .8568 .8726 .8882 .9038 .9194 44 il3 .7774 .7935 .8095 .8254 .8413 .8571 .8728 .8885 .9041 .9190 45 1 •16 .7777 .7938 .8098 .8257 .8415 .8573 .8731 .8887 .9044 .9199 40. •19 .7780 .7940 .8100 .8259 .8418 .8576 .8734 .8890 .9046 .9201 47, 121 .7782 .7943 .8103 .8262 .8421 .8579 .8736 .8893 .9049 .9204 48 , 124 .7785 .7946 .sio,-) .826) .8423 .8581 .8739 .8895 .9051 .9207 49, 127 .7788 .7948 .8108 .8267 .8426 .8584 .8741 .8898 .9054 .9209 50' i-29 .7791 .7951 .8111 .8270 .8429 .8.587 .8744 .8900 .9056 .9212 51 •32 .7793 .7954 .S113 .8273 .8431 .8589 .8747 .8903 .9059 .9214 52 35 .7796 7956 .8110 .8275 .S434 .8)92 .8749 .8906 .9002 .9217 53 1 i38 .7799 .7959 .8119 .S27S .8437 .8594 .8752 .8908 .9064 .9219 54; m .7801 .7962 .8121 .8281 .8439 .8597 .8754 .8911 .9067 .9222 55 43 .7804 .7964 .8124 .S283 .8442 .8000 .8757 .8914 .9069 .9225 56 46 .7807 .7967 .8127 .8286 .8444 .8602 .8760 .8916 .9072 .9227 57 48 .7809 .7970 .8129 .8289 .8447 .8605 .8762 .8919 .9075 .9230 5S >51 .7812 .7972 .8132 .8291 .8450 .8608 .8765 .8921 .9077 .92:32 59 1 154 .7815 .7975 .8135 .8294 .8452 .8610 .8767 .8924 .9080 .9235 60 90 GEOMETllICAL PROBLEMS. Table of Chords ; Radius = = 1.0000 1 [continued) • M. 55" 66' 67* 68" 50" eo* or 62' «8' 64* M. .9235 .9389 .9543 .9696 .9848 1.0000 1.0161 1.0301 1.0460 1.0698 1 .9238 .9392 .9546 .9699 .9861 1.0003 1.0163 1.0303 li)462 1.0601 1 2 .9240 .9395 .9548 .9701 .9854 1.0005 1.0166 1.0306 1.0466 1.0603 2 3 .9243 .9397 .9551 .9704 .9856 1.0008 1.0168 1.0308 1.0467 1.0606 3 4 .9245 .9400 .9553 .9706 .9859 1.0010 1.0161 1.0311 1.0460 1.0608 4 T) .9248 .9402 .9566 .9709 .9861 1.0013 1.0163 1.0313 1.0462 1.0611 5 6 .9250 .9405 .9559 .9711 .9864 1.0015 1.0166 1.0316 1.0466 1.0613 6 7 .9253 .9407 .9561 .9714 .9866 1.0018 1.0168 1.0318 1.0467 1.0616 7 8 .9256 .9410 .9564 .9717 .9869 1.0020 1.0171 1.0321 1.0470 1.0618 8 9 .9258 .9413 ,9566 .9719 .9871 1.0023 1.0173 1.0323 1.0472 1.0621 9 10 .9261 .9415 .9569 .9722 .9874 1.00-26 1.0176 1.0326 1.0476 1.0623 10 11 .926:$ .9418 .9571 .9724 .9876 1.0028 1.0178 1.0328 1.0477 1.0626 11 12 .9266 .9420 .9574 .9727 .9879 1.0030 1.0181 1.0331 1.0480 1.0028 12 13 .9268 .94-23 .9576 .9729 .9881 i.oa33 1.0183 1.0333 1.0482 1.0690 13 14 .9271 .9425 .9579 .9732 .9884 1.0035 1.0186 1.0336 1.0486 1.0633 14 15 .9274 .9428 .9581 .9734 .9886 1.0038 1.0188 1.0338 1.0487 1.0636 15 16 .9276 .9430 .9584 .9737 .9889 1.0040 1.0191 1.0341 1.0490 1.0638 16 17 .9279 .9433 .9587 .9739 .9891 1.0043 1.0193 1.0343 1.0492 1.0640 17 18 .9281 .9436 .9589 .9742 .9894 1.0045 1.0196 1.0346 1.0406 1.0043 18 19 .9284 .9438 .9592 .9744 .9897 1.0048 1.0198 1.0348 1.0407 1.0646 19 20 .9287 .9441 .9594 .9747 .9899 1.0050 1.0201 1.0361 1.0600 1.0648 20 21 .9289 .9443 .9597 .9750 .9902 1.0053 1.0203 1.0363 1.0602 1.0660 21 22 .9292 .9446 .9699 .9752 .9904 1.0055 1.0206 1.0366 1.0604 1.0668 22 23 .9294 .9448 .9602 .9755 .9907 1.0058 1.0208 1.0368 1.0607 1.0666 23 24 .9297 .9451 .9604 .9757 .9909 1.0060 1.0-211 1.0361 1.0600 1.0668 24 25 .9299 .9464 .9607 .9760 .9912 1.006;i 1.0213 1.0363 1.0612 1.0660 2ft 26 .9302 .9456 .9610 .9762 .9914 1.0060 1.0-216 1.0366 1.0514 1.0602 26 27 .9305 .9459 .9612 .9765 .9917 1.00(58 1.0218 1.0368 1.0517 1.0005 27 28 .9307 .9461 .9615 .9767 .9919 1.0070 1.0-221 1.0370 1.0519 1.0007 28 29 .9310 • J7^rO*T .9617 .9770 .9922 1.0073 1.0-2-23 1.0373 1.0622 1.0070 29 30 .9312 .9466 .9620 .9772 .9924 1.0075 1.0226 1.0376 1.0624 1U)072 30 31 .9315 .9469 .96-22 .9775 .9927 1.0078 1.0228 1.0378 1.0627 1.0076 31 32 .9317 .9472 .9625 .9778 .9929 1.0080 1.0231 1.0380 1.0629 1.0077 32 33 .9320 .9474 .9627 .9780 .9932 1.0083 1.0233 1.0.383 1.0632 1.0080 33 34 .9323 .9477 .9630 .9783 .9934 1.0086 1.0-236 1.0386 1.0534 1.0082 34 35 .9325 .9479 .9633 .9785 .9937 1.0088 1.0-238 1.0388 1.0537 1.0086 36 36 .9328 .9482 .96;}5 .9788 .9939 1.0091 1.0-241 1.0390 1.0539 1.0087 36 37 .9330 .9484 .96:JS .9790 .9942 1.0093 1.0-243 1.0393 1.0W2 1.0090 37 38 .9333 .9487 .9640 .9793 .9945 1.0096 1.0-246 1.0396 1.0544 1.0092 38 39 .93;t5 .9489 .9643 .9795 .9947 1.0098 1.0248 1.0398 1.0547 1.0004 39 40 .9338 .9492 .9645 .9798 .9950 1.0101 1.0251 1.0400 1.0649 1.0097 40 41 .9341 .9495 .9648 .9800 .9952 1.0103 1.0253 1.0403 1.0661 1.0099 41 42 .934:i .9497 .9650 .9803 .9955 1.0106 1.0-256 1.0406 1.0664 1.0702 42 43 .9346 .9500 .9653 .9805 .9957 1.0108 1.0258 1.0408 1.0566 1.0704 43 44 .9348 .9502 .9665 .9808 .9960 1.0111 1.0261 1.0410 1.0669 1.0707 44 45 .93.')! .9505 .9658 .9810 .9962 1.0113 1.026:} 1.0413 1.0661 1J0700 46 4t3 .9353 .9507 .9661 .9813 .9965 1.0116 1.0-266 1.0415 1.0564 1.0712 40 47 .9351) .9510 .9663 .9816 .99<>7 1.0118 1.0-268 1.0418 1.0566 1.0714 47 48 .9359 j .9512 .9660 .9818 .9970 1.01-21 1.0-271 1.0420 1.0560 1.0717 48 49 .93<)1 i .9515 .9668 .9821 .9972 1.01-23 1.0273 1.04-23 1.0571 1.0719 49 50 .9364 1 .9518 .9671 .9823 .9975 1.01-26 1.0-276 1.0425 1.0674 1.0721 50 51 .93(56 .9520 .9673 .0«2») .9977 1.01-28 1.0278 1.0428 1.0576 1.0724 61 52 .93()J .9523 Mid .9828 .9980 1.0131 1.0281 1.04:50 1.0^79 1.0726 52 53 .9371 .9525 .9678 .9831 .9982 1.0133 1.0-283 1.0433 1.0681 1.0729 63 54 .9374 .9528 .9681 .08:5^$ .99S5 1.0136 1.0286 1.04:15 1.0684 1.0731 64 55 .9377 .9530 .9(^3 .9836 .99S7 1 1.013S 1.0-2S8 1.0438 1.0686 1.0784 65 56 .9379 .9533 .9086 .9S3S .9990 j 1.U141 1.0-291 1.0440 1.0589 1.0730 66 57 .9382 .95.36 .9689 .9841 .9992; 1.0143 1.0-293 1.0443 1.0591 1.0730 57 58 .9384 .9538 .9691 .9843 .9i>95 1.0146 1.0-296 1.0446 1.0603 1.0741 68 59 .9387 .9541 .9694 .9846 .9998 1.0148 1.0-298 1.0447 1.0696 1.0744 50 60 .9389 .9543 .9696 .9848 1.0000 1.0151 1.0301 1.0460 1.0508 iun4o 00 GEOMETRICAL PROBLEMS. 91 Table of Chords ; Radius = 1.00O0 (continued] 1. M. 65* 66" 67' es" 69* 70- w 78* 73' M. 0' 1.0746 1.0893 1.1039 1.1184 1.1328 1.1472 1.1614 1.1766 1.1896 1 1.0748 1.0895 1.1041 1.1186 1.1331 1.1474 1.1616 1.1758 1.1899 1 2 1.0751 1.0898 I.IOU 1.1189 1.1333 1.1476 1.1619 1.1760 1.1901 2 8 1.0753 1.0900 1.1046 1.1191 1.1335 1.1479 1.16-21 1.1763 1.1903 3 4 1.0756 1.0903 1.1048 1.1194 1.1338 1.1481 1.1624 1.1765 1.1906 4 5 1.0758 1.0905 1.1051 1.1196 1.1340 1.1483 1.1626 1.1767 1.1908 5 6 1.0761 1.0907 1.1053 1.1198 1.1342 1.1486 1.1628 1.1770 1.1910 6 7 1.0763 1.0910 1.1056 1.1201 1.1345 1.1488 1.1631 1.1772 1.1913 7 8 1.0766 1.0912 1.1058 1.1203 1.1347 1.1491 1.1633 1.1775 1.1915 8 9 1.0768 1.0915 1.1061 1.1206 1.1350 1.1493 1.1635 1.1777 1.1917 9 10 1.0771 1.0917 1.1063 1.1208 1.1352 1.1495 1.1638 1.1779 1.1920 10 11 1.0773 1.0920 1.1065 1.1210 1.1354 1.1498 1.1640 1.1782 1.1922 11 12 1.0775 1.0922 1.1068 1.1213 1.1357 1.1500 1.1642 1.1784 1.1924 12 18 1.0778 1.0924 1.1070 1.1215 1.1359 1.1502 1.1645 1.1786 1.1927 13 14 1.0780 1.0927 1.1073 1.1218 1.1362 1.1505 1.1647 1.1789 1.1929 14 15 1.0783 1.0929 1.1075 1.1220 1.1364 1.1507 1.1650 1.1791 1.1931 15 16 1.0785 1.0932 1.1078 1.1222 1.1366 1.1510 1.1652 1.1793 1.1934 16 17 1.0788 1.0934 1.1080 1.12-25 1.1369 1.1512 1.1654 1.1796 1.1936 17 18 1.0790 1.0937 1.1082 1.1227 1.1371 1.1514 1.1657 1.1798 1.1938 18 19 1.0793 1.0939 1.1085 1.1230 1.1374 1.1517 1.1659 1.1800 1.1941 19 20 1.0795 1.0942 1.1087 1.1232 1.1376 1.1519 1.1661 1.1803 1.1943 20 21 1.0797 1.0944 1.1090 1.123+ 1.1378 1.1522 1.1664 1.1805 1.1946 21 22 1.0800 1.0946 1.1092 1.1237 1.1381 1.1524 1.1666 1.1807 1.1948 22 28 1.0802 1.0949 1.1094 1.1239 1.1383 1.1526 1.1668 1.1810 1.1950 23 24 1.0805 1.0951 1.1097 1.1242 1.1386 1.1529 1.1671 1.1812 1.1952 24 25 1.0807 1.0954 1.1099 1.1244 1.1388 1.1531 1.1673 1.1814 1.1955 25 26 1.0810 1.0956 1.1102 1.1246 1.1390 1.1533 1.1676 1.1817 1.1957 26 27 1.0812 1.0959 1.1104 1.1249 1.1393 1.1536 1.1678 1.1819 1.1959 27 28 1.0815 1.0961 1.1107 1.1251 1.1395 1.1538 1.1680 1.1821 1.1962 28 29 1.0817 1.0963 1.1109 1.1254 1.1398 1.1541 1.1683 1.1824 1.1964 29 80 1.0820 1.0966 1.1111 1.1256 1.1400 1.1543 1.1685 1.1826 1.1966 30 31 1.0822 1.0968 1.1114 1.1258 1.1402 1.1545 1.1687 1.1829 1.1969 31 32 1.0824 1.0971 1.1116 1.1261 1.1405 1.1548 1.1690 1.1831 1.1971 32 38 1.0827 1.0973 1.1119 1.1263 1.1407 1.1550 1.1692 1.1833 1.1973 33 34 1.0829 1.0976 1.1121 1.1266 1.1409 1.1552 1.1694 1.1836 1.1976 34 35 1.0832 1.0978 1.1123 1.1268 1.1412 1.1555 1.1697 1.1838 1.1978 35 36 1.0834 1.0980 1.1126 1.1271 1.1414 1.1557 1.1699 1.1840 1.1980 36 37 1.0837 1.0983 1.1128 1.1273 1.1417 1.1560 1.1702 1.1843 1.1983 37 38 1.0839 1.0985 1.1131 1.1275 1.1419 1.1562 1.1704 1.1845 1.1985 38 39 1.0841 1.0988 1.1133 1.1278 1.1421 1.1564 1.1706 1.1847 1.1987 39 40 1.0844 1.0990 1.1136 1.1280 1.1424 1.1567 1.1709 1.1850 1.1990 40 41 1.0846 1.0993 1.1138 1.1283 1.1426 1.1569 1.1711 1.1852 1.1992 41 42 1.0S49 1.0995 1.1140 1.1285 1.1429 1.1571 1.1713 1.1854 1.1994 42 43 i.oajvi 1.0997 1.1143 1.1287 1.1431 1.1574 1.1716 1.1857 1.1997 43 44 1.0854 1.1000 1.1145 1.1290 1.1433 1.1576 1.1718 1.1859 1.1999 44 45 1.0856 1.1002 1.1148 1.1292 1.1436 1.1579 1.1720 1.1861 1.2001 45 46 1.0859 1.1005 1.1150 1.1295 1.1438 1.1581 1.1723 1.1864 1.2004 46 47 1.0861 1.1007 1.1152 1.1297 1.1441 1.1583 1.1725 1.1866 1.2006 47 48 1.0863 1.1010 1.1155 1.1299 1.1443 1.1 58() 1.1727 1.186S 1.-2008 48 49 1.0866 1.1012 1.1157 1.1302 1.1 44r) 1.1 5S8 1.1730 1.1871 1.-2011 49 50 1.0868 1.1014 1.1160 1.1304 1.1448 l.ir)90 1.1732 1.1 S73 1.2013 50 51 1.0871 1.1017 1.1162 1.1307 1.1450 1.1.51)3 1.1735 1.1875 1.2015 51 52 1.0873 1.1019 1.1105 1.13()« 1.14r)2 1.1.505 1.1737 1.1878 1.2018 52 I 53 1.0876 1.1022 1.1167 1.1311 i.i4r)r) 1.150S 1.1730 1.1880 1.2020 53 ! .-4 1.0S78 1.1024 1.1109 1.1314 1.1457 1.1000 1.1742 1.1882 1.20-22 54 1 55 1.0881 1.1027 1.1172 1.1316 1.1460 1.1002 1.1744 1.1885 1.2025 55 1 56 1.0883 1.1029 1.1174 1.1319 1.1402 1.1005 1.1740 1.1887 1.20*27 56 , 57 1.0885 1.1031 1.1177 1.1321 1.1404 1.1607 1.1749 1.1889 1.2029 57 58 1.0888 1.1084 1.1179 1.13-23 1.1467 1.1609 1.1751 1.1892 1.2032 58 50 1.0890 1.1036 1.1181 1.1326 1.1469 1.1612 1.1753 1.1894 1.2034 59 60 iMm 1.1089 1.1184 1.1328 1.1472 1.1614 1.1756 1.1896 1.2036 60 92 GEOMETRICAL PROBLEMS. Table of Chords; Radius = 1,0000 (continued). M. 740 76- 76* 77* 78- 70' 80' 81* S^" M. 1.2036 1.2175 1.2313 1.2450 1.2586 1.2722 1.2866 1.2989 1.3121 O' 1 1.2039 1.2178 1.2316 1.2453 1.2689 1.2724 1.2868 1.2991 1.3123 1 2 1.2041 1.2180 1.2318 1.2455 1.2591 1.2726 1.2860 1.2993 1.3126 2 3 1.2043 1.2182 1.2320 1.2457 1.2593 1.2728 1.2862 1.2996 1.3128 3 4 1.2046 1.2184 1.2322 1.2459 1.2595 1.2731 1.2866 1.2998 1^130 4 5 1.2048 1.2187 1.2325 1.2462 1.2598 1.2733 1.2867 1.8000 1.3132 5 6 1.2050 1.2189 1.2327 1.2464 1.2600 1.2735 1.2869 1.3002 1.3134 6 7 1.2053 1.2191 1.23-29 1.2466 1.2602 1.2737 1.2871 1.3004 1.3137 7 8 1.2055 1.2194 1.2332 1.2468 1.2604 1.2740 1.2874 1.3007 1.3130 8 9 1.2057 1.2196 1.2334 1.2471 1.2607 1.2742 1.2876 1.3009 1^41 10 1.2060 1.2198 1.2336 1.2473 1.2609 1.2744 1.2878 1.8011 1.3143 10 11 1.2062 1.2201 1.2338 1.2475 1.2611 1.2746 1.2880 1.3013 1.3146 11 12 1.2064 1.2203 1.2341 1.2478 1.2614 1.2748 1.2882 1.3016 1.3147 .12 13 1.2066 1.2205 1.2343 1.2480 1.2616 1.2751 1.2886 1.3018 1.3150 13 14 1.2069 1.2208 1.2345 1.2482 1.2618 1.2763 1.2887 1.3020 1.8152 li 16 1.2071 1.2210 1.2348 1.2484 1.2020 1.2755 1.2889 1.8022 1.3154 15 16 1.2073 1.2212 1.2350 1.2487 1.2623 1.2767 1.2891 1.3024 1.3156 16 17 1.2076 1.2214 1.2352 1.2489 1.2625 1.2760 1.2894 1.3027 1.3158 17 18 1.2078 1.2217 1.2354 1.2491 1.2627 1.2762 1.2896 1.3029 1.3161 18 19 1.2080 1.2219 1.2357 1.2493 1.2629 1.2764 1.2898 1J»81 1.3163 10 20 1.2083 1.2221 1.2359 1.2496 1.2632 1.2766 1.2900 1.8038 1.3165 20 21 1.2085 1.2224 1.2361 1.2498 1.2634 1.2769 1.2903 1.3085 1.3167 21 22 1.2087 1.2226 1.2364 1.2500 1.2636 1.2771 1.2905 1.3088 1.3169 22 23 1.2090 1.2228 1.2366 1.2503 1.2638 1.2773 1.2907 1.3040 1.3172 28 24 1.2092 1.2231 1.2368 1.2505 1.2641 1.2776 1.2909 1.8042 1.3174 24 25 1.2094 1.2233 1.2370 1.2507 1.^2643 1.2778 1.2911 1.3044 1.3176 25 26 1.2097 1.2235 1.2373 1.2509 1.2646 1.2780 1.2914 1.8046 1.8178 26 27 1.2099 1.2237 1.2375 1.2512 1.2648 1.2782 1.2916 1.8040 1.3180 27 28 1.2101 1.2240 1.2377 1.2514 1.2650 1.2784 1.2918 1.3061 1.3183 28 29 1.2104 1.2242 1'.2380 1.2516 1.2652 1.2787 1.2920 1.8068 1.3185 20 30 1.2106 1.2244 1.2382 1.2518 1.2664 1.2789 1.2922 1.9056 1.8187 30 31 1.2108 1.2247 1.2384 1.2521 1.2656 1.2791 1.2925 1.8057 1.3180 31 32 1.2111 1.2249 1.2386 1.2523 1.2659 1.2793 1.2927 1.8060 IJSlOl 82 33 1.2113 1.2251 1.2389 1.2525 1.2661 1.2796 1.2929 1.3062 1^08 38 34 1.2115 1.2254 1.2391 1.2528 1.2663 1.2798 1.2931 1.8064 1.8196 84 35 1.2117 1.22.56 1.2393 1.2530 1.2665 1.2800 1.2934 1.8066 IJSIW 85 36 1.2120 1.2258 1.2396 1.2532 1.2668 1.2802 1.2936 1.8068 1.82U0 36 37 1.2122 1.2260 1.2398 1.2534 1.2670 1.2804 1.2938 1.8071 1.8902 37 38 1.2124 1.2263 1.2400 1.2537 1.2672 1.2807 1.2940 1.8073 1.8204 38 39 1.2127 1.2265 1.2402 1.2539 1.2674 1.2809 1.2942 1.8075 1.3207 30 40 1.2129 1.2267 1.2405 1.2541 1.2677 1.2811 1.2946 1.3077 1.32U0 40 41 1.2131 1.2270 1.2407 1.2543 1.2679 1.2813 1.2947 1.8070 1.3211 41 42 1.2134 1.2272 1.2409 1.2546 1.2681 1.2816 1.2949 1.3082 1.8218 42 43 1.2136 1.2274 1.2412 1.2548 1.2683 1.2818 1.2961 1.8084 1.8315 43 44 1.2138 1.2277 1.2414 1.2550 1.2686 1.2820 1.2954 l.'UKUt 1.8318 44 45 1.2141 1.2279 1.2416 1.2552 1.2688 1.2822 1.2956 1.8088 1.3220 45 46 1.2143 1.2281 1.2418 1.2555 1.2690 1.2825 1.2958 1.3090 1.3222 46 47 1.2145 1.2283 1.2421 1.2557 1.2692 1.2827 1.2960 1.8003 1JI224 47 4S 1.2148 1.2286 1.2423 1.2559 1.2695 1.2829 1.2962 1.3005 1.8226 48 49 1.2150 1.2288 1.2425 1.2562 1.2697 1.2831 1.2965 1.3007 1.8228 40 50 1.2152 1.2290 1.2428 1.2564 1.2699 1.2833 1.2967 1.8000 1.8231 SO 51 1.21.54 1.2293 1.2430 1.2566 1.2701 1.2836 1.2969 1.3101 1.3288 51 52 1.2157 1.2295 1.2432 1.2568 1.2704 1.28.18 1.2971 1.3104 1.8285 52 53 1.2159 1.2297 1.2434 1.2.571 1.2706 1.2840 1.2973 1.3106 1.3237 53 54 1.2161 1.2299 1.2437 1.2573 1.2708 1.2842 1.2976 1.3108 1.3280 54 55 1.2164 1.2302 1.2439 1.2575 1.2710 1.2845 1.2978 1.3110 1.3242 55 50 1.2166 1.2304 1.2441 1.2577 1.2713 1.2847 1.2980 1.3112 1.3244 56 57 1.2168 1.2306 1.2443 1.2580 1.2715 1.2849 1.2982 1.8115 1.8246 67 58 1.2171 1.2309 1.2446 1.2582 1.2717 1.2851 1.2985 1.8117 1.8M8 68 59 1.2173 1.2311 1.2448 1.2584 1.2719 1.2864 1.2987 1.3110 1.8860 50 60 1.2175 1.2313 1.2450 1.2586 1.27-22 1.2856 1.2989 1.3121 1.8259 00 OBOHBTRICAI. PROBLEMS. 93 94 HIP AND JACK RAFTERS. Lengrtlis and Bevels of Hip and Jack Rafters. The lines ab and be in Fig. 89 represent the walls at the angle of a building; be is the seat of the hip-rafter, and (jf of a jack-rafter. Draw eh at right angles to be, and make it equal to the rise of the roof; join b and 7^, and hb will be the length of the hip-rafter. Through e draw di at right angles to be. Upon b, with the radius bh, describe the arc hiy cutting di in L Join b and i, and extend nf to meet bi in.; ; then r/j will be the length of the jack-rafter. The length of each jack-rafter is found in the same manner, — by ex- tending its seat to cut the line ht. From/ draw yik at right angles to /r/, also fl at right angles to be. Makefk equal to fl by the arc Ik, or make u^' equal to (ij by the arc./A-V then the angle at J will be the top bevel of the jack-rafters, and the one at h- the down bevel. Backhu/ of the hip-rnftoy. At any conv(Miient place in be (Fig. 8i)), as o, draw mn at right angles to be. From o describe a circle, tangent to bh, cutting be in s. Join m and h and n and b ; then these lines will form at s the proper angle for bevelling the top of the hip-rafter. TRIGONOMETRY. 95 TRIGONOMETR7. ot the purpose of the author to teach the use of trigonom- ^hat it is; but, for the benefit of those readers who have icquired a knowledge of this science, the following con- formulas, and tables of natural sines and tangents, have erted. To those who know how to apply these trigono- iinctions, they will often be found of great convenience ty. tables are taken from Searle's "Field Engineering," John Sons, publishers, by permission. 96 T&IGONUMETRIC i'X>UMUlJLS. Tkioosomktbic FtTscnonL ljetA(Fig. lOT) = BJoglo BAC = mre Br^ajid let the radius Af— AB = We then hATe dii.f = DC eos^ = AC tan^ = DF txAA ^HO wocA ^AD eosee A = AG Tenia -4 = CF= BE covers^ = i;;.: = i.x exsec A = i?Z> cuerstx! -4 = BG chord -1 ^BF ^kOx^^A z=Zl=2LC FicKK. Ie tbe liglitnan.eltxl triangle ABC iTi|:. 107) ' L-et AB = r, -4C? = ft, end ^C = o j We then have : L sin.4 2. eo8.4 S. tan.f 4. col .4 \ 8ec.4 6t = — =cosi? c c a b b u f li = cotB = IoxlB .4 = - = p«*o B a c - h 7. Ters -4 = = ch^v^ts B c c - h R. cxaeo .4 =t js ('MeTStv B ^ 0, coT««r«^ r - rt :- vorsin B 10. omtxftHi.l .. - «»\mhW? u «U iiriM^ It a =-.ctinA = hUnA li. b =: ccosA = acot^ ah a.a .4 c*XiA H, o =ccos^ = 6cc>t^ i:v 6 =3 c sin J7 r= a tan ^ -- ah cvuj B t^u B 17. a =3 «' ^c -r 6» kc — ~6r TRIGONOMETRIC FORMULAS. 9: Boixmov OF Oiiu^uB Trumo: Fio. 10& GIVEN. 23 A,B,a 23 84 as 2G tit 28 29 81 as -A. a, 6 C,a,6 a,b,c souoar. C, 6, c -B, C, c -<<,;&, O.a Foiann.«. ' Bin ^ c = -r - - sin (^ 4- B) Rill ^ = - . 6, a O=180«»-U4-P), -T . sin C. area area sin A tanHU-J3)="-^^tanHU + P) K^y^abelnC. cos be «^=/n7^'«-H^yvs .mA = . — ; be vers A = 2 Cf - fc) (a —^c) 6c J: = ♦'a (« - a> (a — b) (j — c) a* sin B.tdn C K:^ »B1U ^ 98 TRIGONOMETRIC FORMULAS. GENERAL PORHULA. 34 sin ^ = = 4/ 1 — cos^ A = tan A cos A comic A 35 sin ^ = 2 sin J^ A cos l^A = vers ^ cot J4 -4 36 sin^ = |/ levers 2 -4 = f/j^d'— co8'2\4) 1 37 cos ^ = = V 1 — sina A = cot ^ sin A BOO ^ as cos ^ = 1 - vers ^ = 2 cos^ Y^A — l = 1—2 sin« ^ ^ cos^ = cos» 14 ^ — Bina 14 ^ = i^ 34"+>.i co8"2^ 40 t;in^l = -;- ^ ?^" ^ = ^"i^c^'A—l cot ^ cos A y cos-* ^ cos^ l+cos2^ ^ - . . 1 — cos 2 ^ vers 2 A ^ .. w ^ 42 tan -4 = - . = — ^ — -— - = exsec -4. cot JiS jl sin 2 ^ sm 2 -4 '^ ^« i. J 1 cos A , r— : T « '^'^ = tSn = Bn-3 = ♦'c«»ec'^-l 44 cot u4 = - -as ss ' - . 1 — COS 2 A vers 2 ^ sin 2 ^ 45 cot ^ = — , 40 vers -4 = 1— cos A t= sin -4 tan ^ ^4 = 2 sin* ^ j1 47 vers A — e::r*c A cos A 48 40 exsec A = sec -4 — 1 = tan A tan X^A — — — .- ^* cos A . .. . /l — cos A / smH^ = i/ 2 = i/- vers -4 2 BO Kin 2 A — 2r.Iny(cos-4 kt 1 y ^ /l + COS -<^ Bl cosj.^^ =1 i/' 2 ' 53 cos 2 ^ = 2 cos« A — 1 = ccs'^ A — Bin* >i m 1 ^tMn*^ TRIGONOMETRIC FORMULAS. 09 1 General Fobmula. tan A J u A 1 — cos^_^ /l — cos ^2 2^ = 2 tan A 1 — tan»-4 . _ sin A l_-f coSj4 1 ^ ~ vers A ~ sin -4 "~ cosec A — cot ^ « ^ = — :; lH^ = 2cot^ J<^ vers ^ 1 — cos A 1+*^1 — ^vers^ 2+ V2(l4-cos']4) i2A=:2 Bin* -4 ,, . 1 — cos^ (1 + cos ^) + V;si (1 4- cos ^) 2 tana ^ 3C2 A = 1 - tan« A iA ± B) = ^nA. cos P ± sin P . cos ji (-4 ± P) = cos A . cos J? 7 sin ^ . sin ^ 4 4- sin P = 2sin J^(4 + P)cos^(^ — B) 4 — sin B = 2 cos ^ M + B) sin ^ (^ — S) ^ -f cos B = 2 cos Ji^ (-4 H- 5) cos JiS (^ — -B) jB — cos ^ = 2 sin H (^ + J?) sin Ji^ U — B) A — sin« P = cos» B — cos« A = sin (^ + B) sin (^ — B) ' ^ — 8in« J5 = cos (^ 4- B) cos (-4 — B) ' COS ^ . COS B COS^.COSB J NATURAL SINES AND COSINES. 101 m 6 !. 1 6«» 7 8* 1 9 1 9 9 Sine Cosin Sine Cosin 71045?'. 99462 Sine Cosin Sine Cosin Sine Cosin "o ToKTior.owioi .12187 .99255 7l3J)17 .99027 715643 ".iW760 60 1 !.0874'> .996171!. 104831 .99440 .12216 .99251 .13946 .99023 .15(572 .98764' 59 2J.0H774 .99014 l.ia511 .99446! .12245 .90218 .13075 .90019 '.15701 .987(50: 5M 8 ! .0H80.-) .99C12 1.10540 .9944:3 .12274 .99214; .14004 .99015 1.157:301.98755; 57 4'.0«831 .99609 1.10560 .99410 .12302 .99210 .14033 .99011 i.l57.':8 .98751: 50 5;.0K8C0 .99607 .10597 .99437 .12331 .992371 .14001 .99000 .15787 .98740. So 6 .08889 .99604 .10626 .99434 1. 12360 1.90283 .14000 .99002 ; .15816 .98741 1 64 7 .08918 99602 .10055 .99431 .123891.992:30 .14119 .08908 1.15845 .987371 53 8 .0K&17 .99599 .10G84 .99428 .121181.99220 .14143 .98991 1.15873 .98732: 52 9 ; .0»^C .99596 .10718 .90124 .12447' .992221 .14177 .989CU ;. 15002 .987281 51 10 - .09005 .99594 1 .10742 .99421 .12176 .99219 .14205 .98980 .15931 .98?23; 60 11 .09034 .90591 .10771 .99418' .12501 .90215 .14234 .98982 1 .15959 .98718! 49 lSi.0(K)&3 .99588' .10800 .99415; .125331.99211! .14i>63 .98078 .is'jHy .98714' 48 18 ; .00092 .99586 .10829 .994121 .12662 .902081 .14292 .98973 .16017 .98700, 47 14 .09121 .99583 1.10858 .994091 .12591 .90204! .14320 .98900 . .16040 .98701140 15 .09150 .99580 1.10887 .994061 .12620 .99200 .14349 .98965 .16074 .98700) 45 16 .09179 .99578 1.10916 .994021 .12649 .99107 .14378 .08961 .16103 .98695! 44 17 .09308 .99575 1.10945 .09309. .12(}78 .90103 .14407 .98957 : .16132 .96690 43 16 ' .09287 .905?2 1.10973 .99306: .12706 .99180! .144:36 .98953 ' .16160 .986801 42 19 : .09306 .99570 ;. 11003 .993JW 1 .12735 .99186 .144(54 .96948 • .10189 .98681 1 41 20 .09295 .99567 .11031 .99390 .12764 .99182 .14493 .98911 j .16218 .98676 40 tl .09821 .99564 .11060 .99386 .12TJ)3 .99178 .14522 .08940 '.16246 .96671 80 22!. 09353 .<K)5G2 .110S9 .99ai3i .12822 .90175 .14551 .989:iG .16275 .98607: 38 28 .09382 .99559 .11118 .993801 .12C)1 .99171 .145801.98931 .16304 .98602 87 24 .09411 .99556 .11117 .99:^771 .12880 .90167 .14(508 .98927 .16333 .98657' 30 261.09140 .99553 .11176 .90374, .12008 .99163 .14637 .98923 .16361 .986521 ») 29 .09409 .99551 .11205 .99370, .12037 .99160 .14006 .98910 .16390 .98048: 84 27 .09496 .99518 .11234 .99:W7 .12066 .99156 .14C05 .98914 .16419 .98(543 8:3 26 .09027 .99545 .11263 .993:{1 .12005 .99152 ,14723 .98910 .16447 .98038 82 29 .00556 .99542 .11201 .99»iO .1:3024 .99148 .14': 52 .98000 1 .16476 .986:3:3: 81 80 .09566 .99540 .11320 .99357; .13053 .99144 .14781 .98902 .16505 .98629 80 81 .09614 .99537 .11349 .99354' .13081 .99141 .14810 '.98897 .16533 .98024 20 82 .09612 .995*4 .11378 .99a>l ; .13110 .99187 .14KJH .98803 .165(52 .98610, 28 88 .09671 .99531 .11407 .99:347 .18i:J0 .9013:r .14807!. 98880 1 .ia591 .98(514 27 84 .09700 .99528 .11436 .99:M41 .1:31(58 .90120 , .14800 .98881 1 .16020 .98600, 20 85 .09729 .99526 .11465 .993111 .13107 .90125 .14025 .98880 1.16048 .98004 25 86 .09758 .90523 .11491 .993:J7 .13226 .9912«> .1405.1 .98876 .10677 .98000,24 87 .09787 .9'J520 .11523 .99*i4 .13254 .901181 .14082 .988n .16706 .98595: 2:3 88 ! .09816 .99517 .11552 .993:31 .1328:3 .99114; .15011 .98867 .167:i4 .985001 22 | 88 .09845 .99514 .11580 .99327 .1^312 .991101 .15010 .98863 .107(53 .98585; 21 40 .00674 .99511 .11609 .99324! .18341 .99100 .16069 .98858 .16792 .98580,20 41 .00006 .09508 .116^ .99390' .ia370 .99102 .16097 .96854 .16820 .98575 10 42 .00932 .99506 .11667 .99:317 .13:jy0 .99008 .15120 .98W0 ;.16W0 .98570 18 48 .09961 .99503 .11606 .09314 .1^427 .99091 .15155 .98845 .16878 .98565 17 44 .00990 .99500 .11725 .99310' .l&4.-iC .990911 .16184 .96841 .16006 .98501, 10 45 .10019 .99497 .11751 •993071 .18485 .99087 .15212 .988:20 .160351.98556, 15 46 .10048 .99494 .11783 .99333' .ia514 .90083 .15241 .988:^2 ;. 16964 !.9&'}5ll 14 47 .10077 .9(M91 .11812 .09300 .135:3 .O'joro .15270 .98827 .10002 :.98,>161 13 46 .10106 .99488 .11840 .99207; .ia')72 .90075 .152001.98823 .170211.985-111 1:J 49 .iai33 .99485 .11860 .992a3 .I3c;)0 .90071 .15327 .98818 i.i7a"'.0j.985;:(> 11 50 .li)lti4 .99482 .11898 .09290. .13029 .91H)G7 .15350 .98814 .17078 .98531 10 51 .10192 .9MTD .11927 .99286' .136^9 .900^3 .in3R5 .98809 .17107 .98526 9 52 .loe-Ji .9^170 .HOW .99283' .130 17 .oixno; .ir>iii .9W'()5 .171 :m .98521 S 53 .lieso .99473 .li985 .992791 .1:3710 .OO(XV) .15412 .98800 ; .17101 .98510 r* 54 .i.WTO .99470 !. 12014 .99276' .13711 .9<K)51 .15-171 .98706 : .17103 .9a')11 6 55 .10908 .99467 1.12013 .992^2 .137?:^ .90017 .15.'i<-)0'. 98701 .17222 .98500: 5 66 .10337 .994^^1 ..12071 .99209: .13802 .00013: .15.V>0 .98787 .172.-)0 .98501 4 57 1086C .9!»W1 .12100 .99265 .i:38:u .9;)i):iol .15:.57 .98782 . 17279 i. 9849(5 3 56 .10695 .99158 ; .12129 .992(J2 .138<X) .aKlT)! .\X,:m .98778 :. 17308'. 98101 2 50 .10121 .99455 .12158 .99258 .13KS0 .95K):Jl 1 .l.':(515'.9877;J .17a-«5!. 98480 1 60 .10168 .9M52 i .12187 ;Co6ln 1 .99255 Bine ; .13017 Cosin .90027 "Sine" .1501):. 98709 ,.17305 1.98481 1 1 — . . . JO / Oorin Bine Cosin a: Sine 1 Cosin Sino 84* 1 88* 8 2» V* 8( NATURAL SINES AND COSINES. 103 — 1 I 16« 1 2 3 4 5 G 7 8 U 10 11 12 i:) 14 15 10 17 18 19 20 21 22 23 21 25 20 27 28 29 30 31 a2 33 31 35 I 36 I 37 3^ • :iJ - 40 : '» . 42 . 43 - 41 ! -15 ' 47 48 t.) Tit) I ie« J7« 18< Sine SlnQ j Cmnn Slne^ i Cosln .ai882^ 9659:3 '."27l)« .9612fi ; .29237 .2W10.98585^ .27592 .96118 .29^*fi5 .25938.90578 .27620 .9(5110 .29293 . 25966 . W r)70 . 27648 . JWl 02 ' . 29321 .25991 .f^'j«'^ .27676 .960!)4 .298-18 1-26022 .SMK65 .2770t .96086 .at)376 .26050 .96547: .27731 .96078; .29404 .26079 .96510 .27759 •96070. .29432 ' .26107 .96532, .27787 .96062' .29460 .aJ135 .{MW24 '1.27815 .96054:' .29487 ;. 26163 .96517 1.. 27813 .96016j .29515 ' .26191 .96.509 ' .27871 .96037 1 .29543 26219 .96502 .27899 .96029' .29571 : .26247 .9(^94 .27937' .96021 1 .29599 .9r)519; .31261 .26275 .96186 .27955 .96013 1 . 29620;. 93511 j .31289 .26303 ■.96179, .27983.96005 .29651 .95502 .81316 .29682.9.>493i .31344 .29710 .95485 1 .81372 .29737 '.95476; .31399 Cosin ' Sine .95(»j' ^30902 .95623 .301)29 .95613 .80957 .956(X; .:^^)^H5 .9559(> .81013 .(»5588 .31040 .95579 .310(58 .95571, .31095 .9.")562! .81123 .95554 .81151 .95545 .95536 .31178 .31206 . 9-3528 ! .81233 . .26331 ; .96471 1 .28011 ' .95997 .26359. 96163 1.280391.95989 .263871. 9(hl56 i .28067 1.95981 i .26415 .96148 .26443 1.9(»40 .26471 1. 96188 .26500 .9<U25 .26528 '.9(>117 .26556' 96410 .265&tl.96402 .266681.96:379 .26696,. 96371 .26?^:. 96363 .26752'. 96353 .20780 1.96*47 .26808 >. 96340, .2G836;.06332i .26861 .96321' .20892 1. 96:316 .26990,. 96308 .2G948'.9'J:301 .»J976 .9<5293 .27001;'. 96283 .2T03S'.9fl27? .2ro(K)'.iMW69, .270681.06261' .2nl6i. 96253; .271441.96216; .271721.962381 .25200. 9C333: .27228 .9(5222, .2723(5 .1K;->14 .2<)»lj. 96206 •M 1.27812'. 96198 -■> . 28095 ' . 95973 . 297(5.-) ' . 95-107 .28123,. 95964 .28I50L959.56 .28178 .95948 .28206 -.95940 .282:31 .959:31 .28263 .95923 .28390 .95915 .28:318;. 95907 .283461.95898 .28374 .95890 .28402 05882 .28429 '.05874 .29793 ,.95459 .29821 '.95450 .29849 .95441 .2J)876 .0.5433 .29904 .95421 ; .209:33 .95415 . .29960; .95107 ' .31630 1.2993/1.953031;. 31 648 .31437 .81454 ' .81482 '■ .31510 I .81537 I .31505 i .81593 .30015 I .80048 ; .30071 .9.');i89i.. 31675 .0.->:380; .8170) .95872, .81730 .28485;. 95857 .28513 ,.95849 .28511 .ft5841 .28569 !.9.'y5S3 .285971.95821 .28625 '.95816 .28652 ■.95807' .SO.*^ .80348 ii .28457 '.95805, .80120 1.80164 :. 80182 1.80209 I.80C37 ! .80305 ■.8C292 .80098 .95363,;. 81758 .28680,. 95799 .287081.95791 .28786 .95783 .287Wi. 95774 .95766 .96757 .95749 .95740 .95782 1.80376 i. 30403 ,'.30181 1 .80159 .80486 .28792 ■7 .28875 .28903 .28931 .95724; .80597 .9535^4 ; .31760 .9o345 .81813 .95337 1 1.31841 .95328; '.31808 .95319 1;. 31890 .95310'!. 31 033 .95301 1 '.81951 .9539:3 ; .81079 .05281 Ij. 83006 .9.'5275' .82034 .95.3^3 1 .83001 .95357 ■.33089 .95218 ;,. 33116 .952:0 ;.83144 .80514 .OrrSA ' .33171 .8a'>12 .80570 .28959 .95n5 .289871.95707 .80025 .95333,;. £3100 .95313 N. 33337 .95304'i.833.>4 .95195, ,'.33282 I, .306r)3 .ailRO .32809 2r310 .Wil90 .3:*0iu;.9r)0{)8l .:3(M>«{)i 95177 i.S.':]:)7 .290i2 .950W| .80708 .95:08 ' .338(>4 . 29070 . it'yCm ' . 3or;J0 1 . 951 r.O . ;33:J0:5 .21)098 .9.757:3' . 30708 ;.Ori 150 .;33t19 . 2912(5 . 9500 4 . 3( )791 , . iul 13 . ;334 17 .29154.95650, .3(W19l.9'"31.8:5 .;33t74 .V3 .'SilV:^ .9»51K3 54 .373JK5 .9(5174 55 .27424 .Wl&i 56 , .27452 .9(5158 57 ' .27480 .96160 .'58 . .27508 .96142 .VJ .27580 .00181 tiU j .27561 .96126 j'Cosin; Sine i .2SJ182 .9rm7\ .30p«10.95l3i .:33.";03 .2lr309 .95639 . 20287 !. 95(530 Cosia|sine 78^ .30874,. 951 15 .3-i.->3i) .3(KXW Osin 9510(5 .33557 Sine : Cosin r2o Oosin .9.5106 .95(X»7 .9.')0^«8 .9507!» .9r,070 .9s)<)(51 .95053 .95043 .950:3:5 .9.')0::4 .95015 .95006 .94997 .94988 .949r9 .94970 .94961 .91953 .94943 .949:3:3 .04924 .04915 .94906 .94897 .94888 .94878 .948(59 .94800 .9-1851 .948l.'3 .04832 .94823 .94814 .94805 .9-1795 .94780 .94777 .94708 .94758 .94749 .94740 .94730 .94731 .94713 .04703 .94093 .04084 .94074 .94005 .940-ly .94037 .94037 .94018 .9-40t')0| .94509' .9I.7M);' .9:3.5801 .9-1.571 .91501 .94553 Siup W IV Sine .82557 .33.584 .33(513 .83089 .83f5<57 ..83()<M .83733 .83749 .33777 .83Hi)4 .338.33 60 Cosin .945.53 .94543, 59 .945:3:31 .58 .91.53:3 .94511 .*.).4504 .94195 .94-1K5 .91470 57 56 55 54 .5:3 53 .91400! 51 .9145^ .83H.59;. 94-4.47 .33HS7l.944:3;< .82914;. 94 438 .33{)43 .94418 .339(59 .94409 .94:399 .94300 .948S0 .94370 .94301 .94351 .33097 .38031 .83051 .3:3079 .83106 .831.%1 .83101 .3:3189 .33316 .83244 .83371 .83308 .3a']3(5 .3:3:55:3 .33381 .88408 .38130 .a8108 .83400 .a*35l8 .8:3545 .a3573 ' .83000 .3:3037 .33055 .33083 .3:3710 .a8787 .3:3704 .83793 .88J!19 .33816 .3:3874 .3:}:k)1 .S3J39 .330.50 .3:3:i-v;,.91()jo .aio 11!. 01050 .8ii»;ii.ni(h>o .84005.. 91010 .Siir.i;j;.94(K)0 .8413<) .93009 .84117;.980H0 .3 1175!. 93970 .3 1303 j. 93000 Cosin Siuo 50 49 48 47 46 45 44 43 42 41 40 39 .94251 .04245 .94235 .94335 .94215 .9430(5 .94190 .9418(5 .9417(5 .94107 .94313' 38 .94:3:331 37 .043331 86 .948131 85 .94.8a8l 34 .943o:r 3:3 .943^1; 33 .913741 31 .94304 30 29 28 27 26 25 24 2:3 23 21 20 10 1!5 37 10 15 14 38 13 11 10 9 8 7 6 5 4 3 2 1 .94157 .94147 .94187 .94137 .94118 .94108 .oioor. .9108H .91078 .91008 .04058 70^ NATURAL SINES AND COSINBS, 0° 1 Sine ICoain Sine -5 OOdOO Ono. .01741> » a s Mild Ono. 1 .01B03 Ono. .01801 One, O03W Ono. •"igiS IXe38 One. wim One. 10 aaai One. !0!0S6 a 00890 .99901 .OlOSS a .oau IS I '.osaa .oaiai IS .di9ii H 17 '.OMt !eoo9e .0224C IS IS B ioOUK :02M .00683 .(e^ 10 21 n ioooii » 23 .OOKIi! n *4 .006ft K K '.mg! B M '.wai .00907 U .OKK .09901 n .OOftll n SO '.max n » :008T! .99996 :<H618 » ,99096 » 3S .00906 :o£(i7( IB 33 .OOOO .99e-5 .0370; tr .OOM .990% K as .UlOll llBTOI IB 86 Inoos jOBnt H ST '.oimi .gODH .taw >S 38 .Olios .ooou .QBSX a .OllSl .OOQH .0^71 n 40 .01101, OOBW .0SW8 ID ,01103' .99B9S .OM38 » .OJOO IS 43 :oiiaii:TO3aa .om 17 44 .01880.90903 .0303 It 49 .01300 .wool .030^ U 46 .01S33'. 90091 .03033 47 .01837.90001 ^11! 48 .01396.99090 .0314: It 4« .tU4ra .00000 BO -OMM;. 90089 ^OUO! 51 .m4H3.9S9l« .eua 9 m .01313 .ou:mo 8 .oi(na.9onm '.mi 9 u .oian.oooiw .03310 9 cm .oiodoI.eooK!' .OSMl 8 s» 4 Mivii '.mm .OICRT .OOOfiB :0S4.'!; .01719 .Booes 1 ]0340( ~ Cosln"|l5r Codn BJf ft X^ %.^>^AA« .A^ftk^S ri6 174 )03 m )18 M7 we .mi9 .09617 .99614 .99612 .99600 .99607 .99604 99602 .99399 .99596 )05 .99594 )84'. 90591 )63 .99588 )92. .09586 121 '.99583 l50 .99580 179 .99578 J08 .00575 S87i.9g3?2 !66 .99570: 295 .99567 &4 )53 .99564 .90562 82 .99359 Ul MO 109 106 )86 n4 )42 571 roo 789 r87 ?16 374 333 361 990 )19 >18 377 106 133 l(» 192 221 230 279 308 W7 366 )95 m .90556 .99553 .90551 .00548 .99546 .00542 .99i^ .99537 tVuiyn .99531 .99528 .99526 .99523 .99520 .99317 .90514 .90511 .99506 .90503 .00500 .99497 .99494 .99401 .09488 .00485 .09482 .90479 .09-170 .99473 .90170 .90107 .9n4<J4 .00101 .90458 .99455 168..9S452 iin|siiie ^6« Sine 71045? .10482 .10511 .10540 .10360 .10597 .10626 .10655 .10084 .10713 .10742 .10771 .10800 .10829 .10a38 .10387 .10916 .10045 .10973 .110021 .11031 .11060 .110S9 .11118 .11147 .11176 .11205 .11234 .11283 .11291 .11320 .11349 .11378 .11407 .11436 .11465 .11494 .11523 .11652 .11580 .11609 .116® .11667 .11696 .11725 .11734 .11783 .11812 .11840 .11869 .11898 .11027 : .11030 .li985 1.12014 .12013 .12071 .12100 .121'J9 .12138 .12187 Cosin Cosin, .90452: .99440 .99446' .99443 .99410: .99437 .99434 .99431 i .99428 .99424 .99421 .99418 .994151 . 99412 i .99409, .99406 .99402, .09a')9i .99306; .90303! .993001 .99386; .99333^ .993H0 .993771 .99374 .99370 .99367 .99334 .99360 .99357, .99334 .99331! .90347 .90:»4| .99311 .993:^7 .99334 .99:i31 .99J27 .99324 .99320 .99:317 .99314, .993101 •99307 1 .99333' .99300 .99297- .99203 .99290 .99286' .90283 .992791 .99276 .99272 .99209 .90205 .00262 .00238' .00255, Sine 2187 2216 2245 2274 2302 23;n| 2360: 2380! 2418' JW47 12476 2504 2533 2562 2501 2620 2&49i 2(578 2706 i 2735J 2764 2703 2822 2S51 2880 2006 2037 2966 2005 3024 3053 3081 3110 31.'}9 31(i8 3107 3226 32.34 328:3 3312 3341 8370 a309 3427 34.36 3485 3314 35 J3 a372 3G29 3638 30 7 3710 371 1 3773 3802 ;38:U 3860 3KS9 8017 Sine Cosin Sino 88< i| • I Cosin T9^i35 .992511 .9JfcM8 .99-^Ui .99:^ .99237. .99233 .99230 .99226 .992221 .09219: .90215 .00211 .90208 .90204 .99200 .99197 .90103 .00180' .00186 .99182, .99178' .99175 .99171 .99167 .99163 .99160 .99156 .99162 .99148 .991^ .99141 .99187 .9913:3, .99120 ; .99123 .99122 .90118; .99114! .99110 .99106 .99102 .99008 .99004 .00091 .99087 .90083 .0:X)70 .99075 .90071 .00007 .90063 .000.50 .00(V)3 .00031 .90047 .00013 .00030 .900.^3 .90031 .00027 82< 8^^ Sine Cosin .99027 .99023 .90019 .99015 .99011 .99000 .09002 .98908 .98994 .98900 .98086 13017 13046 13073 14004 14033 140G1 14aJ0 14110 14143 11177 14205 14234 14263 14292 14320 14349 14378 14407 14436 14464 14493 14622 14351 14580: 14608 14637 14666 14005 14723 14^32 14781 14810 14fc3« 14807 14806 14923 14034 14082 16011 15040 16069 15097 16126 16155 16184 16212 15JM1 16270 15290 15327 15336 15385 13-n I 13113 16171 15,'M)0 13.'):>0 15."i7 in.")«6 inoi5 15613 1 Cosin j ?* ^ine j Cosin 15672} 15701 1 157:301 i57r.y 157871 15816; 158451 15873 15002 15931 .98962 .98978 .98973 .98969 .98965 .98961 .98957 .08953 .96948 .98044 .98940 .98936!! .989311 .98927ii .98923 .98919 I .98914 .98910 .98906 .989002 .98897 .98803 .98880 .98884 .98880 .98876 .988n .96867 .98863 .98868 .96854 .98840 .98845 .96841 .98836 .98832 .98827 .96823 .98818 .98814 .98809 .98803 .98800 .98706 .98701 .08787 .08782 .98778 .98773 .98769 Sine 01* 15959 15C88 16017 16046 16074 16103 16182 16160 16180 16218 16246 16275 16304 16333 16361 16300 16410 1&447 16476 16605 16538 16362 16301 16620 16&18 16677 16706 16734 10763 16792 16820 16649 16878 16006 16033 160&1 16002 17021- 17030' 17078 17107 17i:36, 1710-11 171031 1 r'.).>k> I 17230! 172701 17308! 173:36 i 17863 .98769 60 .987&4:50 .06700 5^.1 .08755, 57 .08751 ! 60 .98746 5.3 .96741164 .98737,63 .98732 62 .98728 51 .987^160 45 41 43 42 .98718: 49 .98714" 48 .98709, 47 .987011 46 .98700 .98695 .96690 .96686 .96681:41 .06676| 40 .9667l'89 .98667 38 .96662! 87 .98657 86 .96662185 .98648' 84 .986431 83 .98638! 82 .98633 .98629 81 80 .96624 29 .98619 28 .986141 27 .98609 26 .98604^25 .98600 24 .06695 .98590 .98585 .98580 23 22 21 20 19 18 17 16 15 14 Cosin .98576 .98570 .98565 .96561 .96556! .96551 .98546! 13 .96541: V2 .miZa 11 .98531 1 10 .98526' 9 .985211 8 .985161 7 .98311 .9a500 .9^301 .98496 .98401 .96486 ^^3481 Sine 80* NATURAL SINES AND COSINES. , 1 10° IP i| 12 l; • II K- line CMin SlDS COBlni Kna CoBln! Bii» Ccnbtl' OMtn ■ol.]786!i:«»81 .1008: -K^i' Tmrao «> I ;.IT393 .<)H4T( 91023 2 .lT4aa.lWI71 llMBt :ai553 97431 0701B 3:.lTJ51.0K16a .loie; 9T7W, .aasHO .97417 97l» ^|. 17419, .OSlOl 97701 .aaous 07001 5« i !llM3i iosissj 96091 0l:i7B87|:«il50 .19253 .88189 :B3a85 «7S08 OflOW 7 .iraw.twm .l!«a .981« .83008 97801 98980 s!.]<391 .lkU4l) .1030 .wtiia .(awa 97384 00073 as .1033 .SSTW 90906 Gl i 10 1 1™: :iKH3CI .lOSM ;»8I07 07871 00950. » ,0«425 .fflSIM 077*8 .a»»7 !lft4S3 07743 oooS ^ IS iiTTS; !S8414 .loite 08037:47 Hi.rra .W109 07720 oraiB oooaoiM .BSHM 97733, .33930 07888 lii': ?fB !]»S8 .32948 B^lti .IBSOO 9mi .83977 96000 43 .liBOS osoai 9770S .asocB S7B18 oeoce 42 .19683 OSMfl 97008 .23038 9731 SosmUi so : Tsa? .isesa 07003 OTKH 9«a87|« M .17« B9M4I OTBSe' .83090 07296 00880 30 SHOW SiWO .23118 97-JO 90878:18 2a 03038 ITS73 JU14D JW17S 9affi8 >0 sttsa eran 08861 s .2U31 9TS64 06814 » .lain .sssoo 9683; S3 W .1B1« Jseao ftWU .23388 IB !» .1B1« !o9isi Ams Bra» )ro3fl 071M4 B68aS SOl.ltKM .OBsa 97003 .^ 975187 06815 30 .98330 07^!! 23Brm BTSBO 06807 (0 .MOM 07.a8 BG80O a 0670E a a> !i!aa7D )7j93 97:3)3 19 30 ilKSM :OSiM oraoa: 37 .IMM ,98a« 975KI mw 06704 38 .lUSI .uttss iTsn » .vserr 07010 'naim « ■wiiiaaot .D&tTS 0WS4I iraao 33827 OTia ;»sw 41 .18SB8 ,98OT .2HS0 oT%ie 07500 cmos » a'.iesm .IM»1 a 07933, OIMS .S308J onis IS 4a!.185M 07918 17547 28718 07148 iS«i IT wi.isea .iwaa .0X00 BTOIOI 97141 10 45l.lHll3li .ma .tanu jmt. 40 .U<UBI .vsau msa .WBM ''^' H Ksi: .fti-UO 9mr ffiR .U«i3 .aun .ma-A^m 1 a) iionc .turn .imb«».9W!b:!Sb' 10 M .IIBU .<H«13' .aosi!i'.inn»'!.ii»fl)' U .IBSSJ .ftt«7> .xa» .omn .aiWK 230Rt .iiH«n; .aunwiLKtw .awj? )7WJ 23095 M Iwrni .wiwi .axtanl.BjMBi : .aaaa JT47B 24033 (B .IUBSHHW .3MIB..DTK1S ;.3:!Xn 0«..I«H7 .WIMJ .ann? .07«k)' .a^m: 34079 S7 ,11WBS .BMT9; .90?» .imwt .awiii 07437 «10S W..lilUSi .(B174 .»!nt..m>ST .fom J7450; 91180 .raBj Rt -i!io5a .mm .annSj.nHBi. .»H7 M104 :«mi <W 1 .IDMn .361A3 37487' 21103 CoSS|-Bb.e- Ocali -ffiaiciSSi 5^ Oosta aST <MB(ak. ~ 79- "l .- 11' 77 7fl 73- wl WTW .aSSST irmr,^ "mbstI itai Stool JoBlDJ Sine 1 Oslul 61na iCoalnl Sio' [;< • ll 106 NATURAL SINES AND COSINES. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 80» Sine Cosin .50000 .50025 .50050 .5007G .50101 .50126 .5()151 .50176 .50201 ..^)0>37 .50252 .50277 .G0a>7 .50352 .50377 .50403 .5042S .60453 .6W78 20 .60503 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 11 12 43 41 45 46 47 48 49 50 51 ryi 5:J 54 55 56 57 58 59 60 sr Sine ' Cosin 82« 88« .60528 .50563 .60578 .60603 .60628 .50651 .60679 .60704 .50729 .50754 .50779 .60804 .50629 .50864 .60879 .50904 .50929 .50954 .50979 .51004 .51029 .51054 .510?J .51104 .51129 .51164 .61179 .61204' .61229 .61;354i .86588 .80573 .80559 .86544 .80530 .80515 .86501 .86486 .P6471 .m^7 .80442 .80-127 .8(J413 .86393 .86384 .8(>369 .86354 .86340 .8(J325 .86810 86205 .86281 .86266 .8(5251 .86237 .80222 .86207 .86192 .80178 .86163 .86148 .80133 .86119 .86104 .860891 .86074' .86059 .8(5045 .8(5030 .80015, .86000' .K-)1)S5 .K->970 .K5956 .K>D41 .8")926 .85911 .K'>896 .F.")S81 .83806 .61279 .5l;)04 .51829 .51854 .51879 .51404 .51429 .514r>4 .61479, .51504 .85851 .r...iW .a">so6 .R'>792 )l M . .a")762 .K-)747 .K">732 .K->n7 51504 51529 51554 51579 51604 51628 51653 51678 51703 6172.i 61753 61778 51803 5182.i 5185^ 51877 51902 61927 61952 6i9rr 62002 82020 52051 52076 52101 621C6 62151 52175 52200 52225 52275 522LJ 52324 62349 62374 52399 62423 52448 52473 62498 82522 52547 625?^ 62597 52621 52646 5:^71 52096 52720 52745 52770 52794 52819 528'14 62869 52893 62J)18 62i)43 52967 62992 i .85717 .85702 .85687 .85672 .8r^(5.-)7 .85<>42 .a->027 .a")612 .^55^7 .KV)S2 .85567 .85551 .a5536 .85521 .85506 .85491 .85476 .85461 .85431 .85416 .85401 .avivS .85;j70 .85355 .avjio .a-)325 .a"),jio .a">294 .K-)279 .85264 .ft^>249 .a-i218' .85203! .a5188, .85173, .851571 .851421 .a-)1271 .85112 1 .85006' .85081 .a-iOOO .a-)051 : .aj035 .a^oos .81<)89i .81974 .a4U59 .84948 .8Ji)28 .8J913 .81897 .81882 .84866 .a4851| .81836 .84820 .84805 Sine 529!)2 63017 53041: 63066 63091 53115 63140 63104 531i:9; 5^14 63263 63312 53337 53.361 &3:»6 6.3411 5^435 531(W 63484 ssm 63.>34 53558 63583 63607 63032 63081 63705 53730 63754 53779 53*^)4 63f^ istm 63877 63{)02 63926 63a')l 53975 64000 54024 540191 W()73 64097 64122 64146 64171 541 a'} 64220 64244 642: ;9 5421W 5l:J17 64:J42 54306 &1391 M415 &4410 &44I54 Cosin .&4805 .a4789 .&4774 .&4759 .&4743 .&4?28 .a4712 .84697 .84681 .a4606 .84650 .o4635'! .&1019,' .&4604l| .84588 I .84573 ' .&4557 .84542 .84526 .84511; .84495 .84480' .84464 .84448 .84433 .8W17 .84 402 .84386 .84370, .84:J55 .84339 .84324 .84;yiw .84292 .84277 .84J61: .81245 .84^J0 .81214 .84198 .8U82 .84167 .84151 .84135 .84120 .841(V4: .84088 .84072 .84057 .84041 .84025 .84009 .a-5:^>4 .88978,1 .83%2;! .839461 .&ia*JO! .83915 .83899 .888K;J .88K67 .83676 .83660 .83045 .83629 .83613 .83397 .83581 .83.365 .88549 Sine 'Oosin 544<^ .83867 64488 .88851 54513 .83835 545371.88819 645611.88804 64586 = .83788 fr4610 .83772 64635 .83756 546,39 .83740 546835.83724 54708 .83708 54732 54756 &4781 M805 54829 54a>4 64878 64902 M927 54951 64975 54999! 6501^4! 53048 53072 53097 55121 65145 65109 55194 56218 55242 55206 652J1 53:}15 65:};39 65:}03 65388 66-412 55436 65460 56484 65509 6(>533 65657 65581 65605 65630 53654 55678 65702 65?^5 66750 65775 65799 66823 65847 558n 53895 65919 84« Sine ICk>sln' — / ' .65919 .66013 .559(53 .53992 .66016 ! .66040 .56064 .66088 .66112 .56186 .60160 .66184 .66308 .66282 .66256 .66280 .66805 .66329 .66853 .56377 .66401 .88583 .83517 ; .88501 ! .83485 •' .83469 , .88458 .88437. .83421! .834051 .83389! .88873' .88356 .88;^ .83324 .83806, .882921 .832761 .83260' .8:5«4 .83;iS8. Cosin I Sine ; Cosin bine fiS** 53« Cosin I Slue Cosiu Sine .88318 .83105 .83179 .88103: .88147, .88181 ! .88115 .88098 .880^2 .83066 .88050 .880171! .88001! i .82{)85; .82960'! .821K38 • .82936 I .82920 I .82901 ; .60125 .66449 .66473 .66497 .66521 .66546 .66600 .66593 .66617 .66041 .66665 .66689 .66713 .66736 .66760 .66784 .66608 .66882 .66856 .66880 '.66004 .668S8 .66052 .66976 .vrooo .670S4 .67047 .6TO71 .57003 .67119 67« 6e» .6T148 .5no7 .W191 .67i:i5 .57388 .67862 .57^280 .C?7810 .573U 0)7858 Cosin Bino .83904 60 .82887 60 .82871 68 .82855 57 .82889 66 .83823 65 .83806 51 .83790 68 .837r3 63 .K757i 61 .83741 60 .837^' 49 .83708 48 .83093 47 .82075 46 .83659 45 .82043 44 .82026 48 .83610 43 .82593 41 .83677 40 .82S6l'fl9 .83544 88 .63538 37 .83511 86 .63495 86 .83478 81 .83463 88 .83446 as .8^439 81 .8(9413 86 .88806' SO .83880 38 .88868 37 .88847 30 .68880 89 .88814' 84 .83897; 38 .8838r83 .88304 31 .81^48 30 .88881' 10 .88814; 18 .68198 17 .68181 16 .68165 15 .68148. 14 .63183 18 .83116 13 .88098 11 .88063 10 .880651 .63048- .83033 .68015, .819991 .81983 .61965. 8 .61949 $ .81983: 1 .81915 NATURAL SINES AND COSINES. 107 85< Sine Cosin 86< .673811 .67405! .67453 .57477; .67501; .575iJii 8 .67548. 1 2 3 4 6 G 9 .57^-'>" 572' 10 . .5759G 11 . .57619 12 .57043 13 .576C7 14 .67091 15 . .57715 10 .57738 17 .577(52 18 .57786 19 .57810 20 .57833^ 21 22 23 24 25' 26 27' 28' 29 80 biJ)i5 81899 81882 81805 81848 81832 81815 81798 81782 817G5 81748 I 81 S2 83 ai 85 86 = 87 J 88 89 40 41 42 43 44 45 46 47 48 49 60 .578571 .578811 .679041 .5TD28' .57052 .57976 .57999 .58023 .58047 .58070 .5S094! .C3118: .CJ141 .58165; .CS189! .582121 .58236 .58260 5«383 .58307 .58830' .58354 .58378. .58401 .584^ .5f^9 .58472 .68496: .68519) .58643 51 .5656;' 62 .6a'i00 63 .58014 M .68037 55 .68661 6«; .686&4 57 .68708 58 .68731 59 .58755- W .68779' ,C0Bin| .81731 I .8m4 .81098 .81081 .81604'; .81647,' .81031 .81614 .01597 .81580 .81563= .81546 .815::0 .81513 .81406 .81479. .814C3 .81443 .r.l4C3 .81413 .81395 .81.078 .81361 .81&44 .81327; .81310 .81293 .81276 .81C59 .81242 .81225- .81208 .81191 .81174 .81157 .81140 .81123 .81106 .810H9 .81072 Sine 'CJosin r5sr;9 .6SSa2 .68826 n. .a.^19 .5c^^r3 .668961 .63920! .58943 .58907! .68990 .69014 .59037 .69061 .59084 .50108 .50131 .59154 .59178 .59201 ' .59225 j .59248 .59272 '.59295 ' .59318 .50342 .59305 ■.59389 .694121 .594361 .59159' .594831 .8iWS5 .80SC7 .WS'jO .8i>S33 .r<fcjil6 .fX)?J9 .80782 .80765 .60748 .80730 .80718' .80096' .80679 .80002 .800^4 .80027 .80010 87* 88< '■1 .59506' ' .5a"'>29. .69552 .69576 .59599 .59022' .690461 .no(;93 .69716 .59739 .6JTG3' .wm .5I,'**09 .6JS32, .6'.>S56 .5:)S79 .6iHK>2 ..'):i!»26 .69949 .81055 .59072 .81(»38 .6;»(K>5 .81021 .0«i!l9 .81004 .&x>i-« .8IKI87 .600«J5 .80;»70 .eoosG .80953 .60112 .809'36 .001:35 .80919 .601.58 .809(» .60182 Sine Cosin .sa^ro .8055S .8a>ii .805:w4 .805U7 .SO4H0 .80472 .8<>455 .804.-i3 .804::X) .80403 .80386 .8^ .803.:4 .80316 .80203 .80Z-2 .80204 .80247 .80230 .80212 .80195 .80178 .80160 .8.)143 .&.)125 .80108 .8*K)r3 .8iwi5(; .800;id .80021 .TlO-0 .7;mk« ,.7IKi.")l .70910 .79J-:»9 .7fKsl Bine Sine Cosin 7G0182 /40.-'O4 .GtKN)5 .(i0228 .00251 .aK>74 .7 .G(VJii6 .7 Sine 'Cosin 89< 7' r9846 rosu 07ii3 51776 I . .6(ui44 , .00307, .0a390 .60414: .60437 .60400 .60483 .00506' .C(K529 .60553 .C057C .cav.:9 .G0G22 .60645 .60008 .t<>>91 .C0714 .ror:>M .CUTOl .607H4 .00807 00.^)3 1 .60876 .60809 .60028 .60945 .C09C8' .ccooi i C1015I OIOCS' ClOCl ' 6iaS4' 011C7 .79758 ,79741 .79723 .70700 .79688 .7%n .79053 .70035 .70018 .70000 .70583 .70505 .70.>I7 .70512 .79i04 "•f • — .7W.')0 .70441 .70iC4 .70-:c;o . < " »yO .79^35. .70318' .7w:a) .7120.1 .7r.~Mr .7f211 .7010.'} .7oi';o .70156 .oiyi4 .6l:«7 .613001 .01 nw; .ou-,>o .01 t.>i .01 574 .CI 107 : .01520 .6l.'>43" .70105 .70(:-7 .70<.»f;o .79a"i! .700:W .70010 .7M('. S .TbOJS) .78962 .7'-0-!l .7vr:(j .7^^:>i .7.'^^73 .7>*-.')5 .7»>:i7 .7^19 .7*W01 .615<;0 .61580. .61612" .616:i5 .61058 .61681 .61704 .61720 .01740 .617?J .61705 .61818 .61&41 .01804 .61887 .61900 .61052 .01055 .01078 .020CJ1 .02024 .02040 .02(X;0 .G2(H0 .02115 .62138 .02100 .0218:3 .02200 .02251 .62274 .02207 . CCS 05 .62.':88 .02411 .624:« .02-150 .62479 .61130 '.70140 .62502 .6115:3 .7I.1C2 .61170 .61 ICO .01-^22 .61 '.'-45 1 .C1C(;8! .01201 .02524 .62547 .62570 .62502 .62015 .6vaS .C20<X) .78^■;ll .78783 .7870o .78747 .78720 .78711 .78004 .78070 .78058 .780i0 .78622 .786^-' .78580 '■■ .78508 .78550 .7K532 - .78514 .78490 .7^478 : .7t>400 .■:^>442 . 784^4 .784a") .;-8.387 .78:>;0 .78351 .7833:3 .78315 .78297 .78279 .78261 .78243 .7^5225 .78C06 .78188 .78170 .78152 .7«134 .78110 .78008 .78U79 .78061 .7h04:3 .7H)25 .78007' .77988 '"""070 . 1 ( .77052 .779:34 G20S:J .7701'; 62700 .77697 64" 63- Cosin ; Sine 62« .65728 . J.. . I I .02?'J«j .C2«10 .02S-I-2 .o-.i^;i .62KS7 .020.32 Cobiu .77879 .77h;i .'.-7X13 .7:.'<01 . < < I T . < I <0.l . t ^ i.^1 .(it •/•! .III 10 Sine Sine Cosin .62055 .62077 .om'O .6:3(n:2 .0C0-i5 .63000 .63113 .0:3125 .63158 .77715, eo .7701.0 £0 . 770781 £8 57 50 .63180 .0:3203 .C3225 .C3248 .03271 .63203 .63310 , .63338 .63001 .63383 .63400 .634-8 .63451 .0*473 .03400 .03518 .035-10 .0350:i .63585 .03a'8 .63630' .0:3053 .03075 .G30i!8 .0:3720 .63742 .63705 .C3787 .CO.'^IO .63832 .essw' .0:3677 .C3809 .03022 .0::044 , .0:3900 .C:3G60 .(■4011 .(.10:33 .04050 .64078 .04100 .61123 .041*5 .64167 . 6419*3 .64212 .642:31 .642.V) .61270 Cosin .'iTOOO' .77041 ' . 77023 i .77005' .77586 .775i'»8 .77550 .77531 .77513 .77494 .77476 .77458 .77439 .77421144 , .77402 1 43 ' .77384' 42 .77306 41 .77347: 40 65 54 .•il 60 40 48 47 40 45 .77329 .77310 .77202 .77273 .77255 .77230 .77218 .77100 .77181 .77162 .7714-4 .77125 .77107 .77070 .77051 .77a0:3 .77014 .7CO-0 .70077 .7CO,'50 .70040 .7-0021; .7000:3 .':os>4 .70^:1} .70.^1:. .7'<>e8 .r-.^i.j .70702 I -, S^ I . ((:i <•; r . ■""• i . I Jt •' I .7()735 .7()?17 .7(''50.S .■;';<;:o .70001; .70042' .7002:J .70(Xi4 Sine 39 33 37 i 30 3-3 i 34 33 O.) 31 i 30 ' 29 I 28 : 20 ! 25 : 24 23 21 . SiO : 19 IS 17 10 !.-> 11 1:3 li 1! 5 4: 1 51' 60° NATURAL TANGENTS AND COTANGENTS. 4- 8- 8° 7- Ta^ rang Cotang Tflng_ Cotang Tang lOo J B749 iIImis :iS IS :!SI s* .07051 .123B8 b! s iiisiea a'.issa 8.1 :o7iio ; )S8«« 11.3789' 0.40904 ;]£G97 B.I 0.88307 .IZ^ iiiaois .is;:8 7 ; 0718 8 06983 ii!iaifl :i3515 t': fl .0^56 K»18 ; 0J76 ?.■ .10606 oiassw ;i267* 11 own 11. 0837 9.S80I6 .13808 7. IS ,07314 WlOl 10.0882 B.OAiie .12i:C3 13 .07273 10.CM9 9.1S028 9.16651 ! 12692 7" !o74ai 09189 B.18093 .isras 7. IG .07 Ul KSIS 10.6183 .0981 9,IOW0 .12^1 17 .07 30 10.8139 . 1011 !07.>J8 0!03370 !i^o 7. sa .07 a 1 09336 10:711B : 1090 0.00063 .12889 ai .07(107 ; (^65 J0.6T8S .11138 .98698 .12809 . 2as9 :07effi 1 lOloilB :ilJ8T .2908 7- SI .07005 I0.6;S9 .11217 .20t8 7. 25 .OTTi* 10.6462 . 8017 T. :o77aa I0:4U13 illEfflS ;81551 .'SOTO 7.- sa .07818 0057U ]0,4i3I .11335 .83^53 . 3106 7. so .OTSII ocaoo . 8136 7.' 09620 lOisSM ;ilB94 :77«W .ISWB SI .07800 09668 '*'-^ .11423 .nta . 8196 7. CJ X!038 . 8324 J1482 :a ioToaz K)T-I6 loiaooa isSTOl E5 .OS017 M7r« 10.2294 .86483 . SS13 7. K6 .OSOlfl 10.1963 . 8848 7. Kl 05S04 loiinai :il829 m .osiai T.^ IO.108O .iie.'-.o .67718 . W33 .08103 .nm .3461 7. 19952 lo.was .117I8 43 .08321 .tiacB .08251 .49188 :i8560 7. 10010 OioOOOT :iIS06 .18680 lOOCO 9.03101 .IIBSO :44Foa 40 .4£;95 .')8«3a 47 .40705 .18689 T.. IS 101 ns 0!k14S3 .SEcas .13808 7.. 4» !o81CT ioii;7 B.t.in4i ;iir54 .SKM .ISiSB 00 .08406 10218 B.;8S17 .11963 8.34496 .1K58 .08485 I0SJ8 .laoj!. B.aE446 .I3JB7 .OKU 9:rji7 .13)42 B.B&:06 63 9-71_H41 S.atffiTB losses 7.: 10363 ! 13131 T. '.mm 03!a oioL-aw .laico BisSftM .'l808S .aim 0422 B.RllOO s.soura .18008 68 8.18370 .18906 7. 69 e.I«3B8 .14004 7.' 60 !oer40 Cotang TmiE | C 0310 otang B:ai-i38 :i8S78 8.14486 .14064 7. TanB c«»¥ Tang Cotang ~T 8S' 1. 84° 88' .W NATURAL TANGENTS AND COTANGENTS. ni 8» 90 10*» 11"» 60 'smg Cotanff Tang .158:38 Cotang Tang .17633 Cotang Tang .19438 Cotang 14034 7.11537 0.31375 5.67128 5.14455 14061 7.10038 .15803 6.30189 .17003 5.66165 .19468 5.13658 59 14118 7.08540 .1589.J 6.29007 .17093 6.65205 .19493 5.12862 58 :414d 7.07059 .15928 6.27829 .17723 5.64248 .19329 5.12069 57 4173 7.03379 .15950 6.26655 .17753 6.63295 .19559 5.11279 56 43(Xb 7.04105 .15938 6.25486 .irr83 5.62344 .19589 5.10490 55 4232 7.02a'J7 , .16C17 6.24321 .17813 5.61397 .19619 5.09704 54 4262 0.91174 .16047 0.23160 .17843 5.60452 .19649 5.08921 53 4291 6.99718 i .icor7 0.22003 .17873 5.59511 .19080 5.08139 52 4321 6.98368 .10107 C.20&51 .17903 5.58573 .19710 5.07360 51 4351 0.96823 .16137 6.19703 .17933 5.57638 .19740 5.06584 50 4881 6.93385 .161C7 6.18559 .17963 5.56706 .19770 5.05809 49 4-110 6.93953 •IClCj 6.17419 .17993 5.55r<7 .l9;;oi 5.03037 148 4440 C.925?o .16220 6.1G283 .18023 5.51831 .19831 5.042G7 47 4170 6.91104 .10250 0.13151 1 .18053 5.53927 .19vS01 5.03499 46 4499 6.89688 .16230 0.14023 .18033 5.53007 .19891 5.02731 45 4529 6.83278 .16316 6.12309 .18113 5.52090 .19921 5.01971 44 4539 6.8G3T4 .ICOiG 6.11779 .18143 5.51176 .19932 5.01210 43 1588 6.83475 1 .iGc;o 6.10004 .18173 5.60264 .199^13 5.00151 42 1618 0.84083 .16435 6.095:2 .18203 5.49356 .20012 4.99695 41 1648 6.82694 .10435 6.08114 .18233 5.4W51 .20042 4.98940 40 1678 0.81312 .16465 6.07340 ' .182G3 5.47548 .20073 4.98188 39 1707 6.7JK)36 .16495 6.0GC10 .18233 5.400:8 .20103 4.97438 38 1787 6.re5G4 .16523 6.C5143 .18323 5.45751 .20133 4.96G90 37 1767 6.77199 .10355 6.04031 1 .18333 5.44837 .20104 4.95045 36 1796 6.73838 .16585 6.02332 .18384 5.439G6 .20194 4.95201 35 1826 6.74403 .16615 6.01878 .18414 5.43077 .20224 4.94460 34 1856 6.73183 .16645 6.00797 1 .18444 5.42192 .20254 4.93rai 33 1886 6.71789 .10074 5.99720 ! .18474 5.41309 .20285 4.92084 32 1915 6.70450 .1G7J4 5.98G40 .18504 5.40429 .20315 4.92219 31 1945 6.69116 .16731 5.97576 : 1 .18334 5.89552 .20345 4.91516 30 1975 6.67787 .16764 5.9C310 .18564 5.38677 .20376 4.90785 29 Kxe 6.60463 .10794 5.95413 .18504 5.37805 .20406 4.90056 28 Km 6.65144 .16824 5.94000 .18021 5.36936 .20436 4.89330 27 3064 6.6,ia31 .16834 5.93355 .18a51 5.36070 .20406 4.88005 26 S094 6.62523 .16834 5.92283 .180iil 5.35206 .20497 4.87882 25 S124 6.61219 .10914 5.91236 .18714 5.34315 .20527 4.87162 24 S153 6.59921 .16944 5.90101 .18743 5.33487 .20557 4.86444 23 3183 6.53627 .16974 5.89151 . 18775 5.32G31 .20588 4.85727 22 3218 6.57339 .170D4 5.83111 .18805 5.31778 .20018 4.85013 21 3243 6.56055 .17033 5.87080 .18835 5.30928 .20648 4.81300 20 3272 6.54777 .17063 5.86051 .18865 5.30080 .20679 4.83590 19 3802 6.53503 .17033 5.85021 .1«895 5.2oe:5 .20709 4.82882 18 5332 6.52234 .17123 5.84001 .18925 5.28CJ3 .20739 4.82175 17 5362 6.50970 .17153 5.82982 .18935 5.27553 1 .20770 4.81471 16 5391 6.49710 .17183 5.819G6- .18930 5.20715 .20800 4.80769 15 >121 6.48456 .17213 5.80953 .19016 5.25880 .208':0 4.80068 14 5451 6.47208 .17243 5.79944 .19013 5.25048 .208G1 4.79870 13 5481 6.45961 1 .17273 5.7f;938 .19070 5.24218 .20891 4.78673 12 5511 6.44720 .17303 5.779:36 .19106 5.23301 .20921 4.77978 11 j&lO 6.43484 .17333 5.7G937 .19130 5.22566 .20932 4.ri28C 10 5570 0.42253 .17363 5.75941 .19166 5.21744 .20982 4.76595 9 5600 6.11023 .17393 5.71019 .101'.i7 5.200-35 .21013 4.75900 8 TAW G.:}0y04 .1742:3 5.7:W0O 1 .19227 5.20107 .21013 4.75210 7 5060 G.3S3H7 .17453 5.70074 .19257 5.19203 .21073 4. 745:) 4 6 3r>.S9 6.37374 .17433 5.71992 .19287 5.18480 .21104 4.73851 5719 6.3G165 .17513 5.71013 .19317 5.17671 .21134 4.73170 4 5749 6.34961 .17343 5.700:37 ! .19347 5.16863 .21104 4.72490 31 3779 6.83761 .17573 5.69004 .19378 5.16058 .21195 4.71813 2| 5:^09 6.32566 i .17603 5.68094 .19408 5.15256 .21225 4.71137 1 ' 5838 6.81373 .17033 Cotang 1 5.G7128 j .10138 Cotang 5.141.55 .21236_ Cotang 4.70463 1\ tang Tang Tang Tang Tang 1 / sv i 1 SO** 1 79° 1 7 B° NATURAL TANOEMTS AND COTANGENTS. NATURAL TANGENTS AND COTANOKNTe. 113 le- l7» 1 18» 19« 1 / 60 Tang .30578 Cotang Tang '• Cotang .83493 1 8.07768 Tang .84433 Cotang nb 3.48741 3.27085 2.90421 w 3.^8850 ' .30606 8.26745 .32524 8.07464 .84466 2.90147 59 ■as 3.4'«Tn7 ! .30037 8.26406 .82556 3.07160 .84496 2.89678 58 -m i 3.47596 .80GC9 3.20067 .32588 3.06857 .84530 2.89600 57 100 8.47216 .30700 3.25729 .82621 3.06554 .84563 2.89827 56 138 3.46837 .80783 3.253P3 .82653 3.06253 .84596 2.89055 55 (64 3.46458 .30764 3.25065 .82G85 3.05950 .84628 2.88788 54 m 3.46060 .80796 3.24719 .32717 3.05649 .84601 2.88511 68 w 3.45703 .80828 3.24383 .82749 8.05^49 .84093 2.88240 63 158 3.45327 .80800 3.24049 .82782 3.05049 .84726 3.87970 51 190 3.44951 .80891 3.23714 .82814 3.04749 .84758 2.87700 50 61 3.44578 .80038 3.23881 .82846 3.04450 .84791 2.87480 49 33 3.44202 .80055 3.2:J048 .82878 3.04152 .84824 2.87161 48 84 3.43829 .30987 3.22715 .32911 3.03854 .84856 2.86893 47 16 3.43456 .31019 3.22384 .32043 3.03566 .84889 3.80624 40 47 3.43084 .81051 3.22053 .32973 3.03260 .84922 2.80366 45 79 3.42713 .81083 3 21723 .83007 3.02903 .84a>4 2.86069 44 10 8.42313 .31115 3.21393 .83040 3.02007 .84987 3.85823 43 42 3.41073 .31147 3.21063 .83072 3.02372 .35020 3.85555 42 74 3.41604 .81178 3.20734 .83104 1 3.02077 .35052 3.85289 41 OS 3.41236 .81210 3.20406 .83180 3.01783 i .35085 2.85028 40 37 8.40R69 .81243 3.20079 .88169 3.01489 ' .35118 3.84758 39 68 3.4Uj02 .31274 3.10752 .83201 3.01196 .85150 2.81494 38 00 3.4J136 .81300 3.10426 .83233 3.00CC3 .35183 2.81229 137 32 3.39771 .31338 3.10100 .832(30 3.00011 : .35216 3.83965 i33 08 8.30406 .81370 3.18775 .33298 3.00319 .85218 2.83702 33 95 8.39012 .31402 3.ia461 .33330 8.00028 .2&:in 2.83139 34 S6 8.38679 .81434 3.10127 .33303 2.09738 .35314 2.83176 33 68 3.3:]317 .81466 3.17804. .38395 2.99447 .35,^6 2.82914 32 90 3.C7Ga3 .81493 3.17181 .83427 2.99158 .85379 2.82653 31 fn 3.S7594 .31530 3.17159 .83400 2.98868 .85412 2.82301 30 58 8.371M4 .31663 3.10838 .83492 2.98580 .85445 2.82130 !20| 86 8.CJj75 .31594 3 10517 .3S5C4 2.98292 .35477 2.81870 28 16 8.3G516 .31026 3.10197 .Zr^iii 2.98004 ' .35510 2.81610 27 48 8.33158 .81058 3.15877 .33569 2.9m7 1 .35548 2.81360 '26 80 3.r,-;800 .01090 3.ir;,-,58 .33621 2.97430 .85576 2.81091 25 11 8.avi4d .31722 3.15240 .330,>4 2.97144 .35008 2.60638 24 43 8.33087 .31754 3.11923 .3;^cso 2.96858 .35041 2.80574 23 75 8.ai733 .31786 3.14605 .83718 2.CG5?3 .35074 2.80316 22 06 3.31377 .31818 8.14288 .33751 2.90288 .857V7 2.80050 121 38 8.34028 .31860 . 3.18973 .33783 2.96004 1 .85740 3.79802 20 70 8.33670 .31883 3.18656 .83810 2.95721 .85772 3.79545 10 01 3.33317 .31014 3.13341 .33^48 2.95437 .35805 3.79289 18 33 8.8::365 .31046 3.13027 .33881 2.95155 .35838 2.79033 17 65 8.3r:614 .31978 3.12718 .33913 2.94872 .35871 2.78r;8 10 97 8.3^264 .32010 8.12400 .33045 2.94591 .35004 2.78623 15 38 3.31914 .32043 8.12087 .33978 2.94309 .35a37 2.78269 14 60 3.31565 .32074 3.11775 .34010 2.94028 .35009 2.78014 13 IM 3.31216 .32106 8.11104 .3-1013 2.93748 .30002 2.VY761 12 34 .S.CIXSS .32139 3.11153 .34(m'5 2.0.'W(W .300:« 2.77507 11 56 8.U)o81 .32171 3.10843 .34108 2.93189 .30008 2.77254 10 87 8.30174 .325^3 3.10533 .34140 2.92910 .30101 2.77002 9 19 3.:J.Ki29 .32235 3.10223 .34173 2.92a*J2 .miu 2.707;.0 8 51 3.;::) 183 .3':.:07 3.n:)014 .3 12: '5 2.92.%! .30107 2.701J,8 7 83 3.'JJ139 .3i,VJ9 a. mm ' .312;]^ 2.92070 .3<;i09 2.70217 6 14 3.2S795 ,ii::m 3.()ik>98 .34270 2.«17J)9 .;i02:J2 2.75{«)0 5 46 8.38453 .32303 3.CS991 , .34:^3 2.91.':C:J .3<;2(r) 2.75740 4 78 8.2:^100 .32396 3.0S685 . .343:« 2.012-10 .30208 2.75400 3 00 8.27767 .3'>l28 3.OS.379 ,34Ci.8 ^:.{:c:.7i .303:31 2.76246 2 41 8.27436 .awx) 3.0S073 1 .3+400 2.90C96 .30304 2.74997 1 .78 S.2T0R6 .32493 Cotang 3.07708 .344.'i3 Cotang 2.00421 1 .3o;m>7 Cotang 2.74718 _0 ng Tiuig Tang Ta:ig Tang / 7 y TS'* 71° ,1 70« ! 114 NATURAL TANGENTS AND COTANGENTS. 20<= 2V Ji 3! 4. 5 6' 7; 8 9 10 11 la 13 14 15 IG 17 18 19 20 21 22 23 24 23 26 27 2^ 23 80 81 82 83 34 85 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 5:»| 53, 51 55 56 57 5ii 59 GO Tanff 36430 86-163 36106 3G529 36562 86593 36628 36661 36694 36727 36760 :56793 3r>823 30859 30'M)2 3G925 36958 30991 37024 37057 870?0 37123 37157 S7190 3?^^ 3?256 87289 37:522 3ra55 87388 87422 37453 37483 37521 37554 37588 37621 37654 37687 37720 37754 37787 37820 37a53 37{}37 37920 37953 3r9«6 3 '1.30 38053 3«»0W] 3S120 'JS153 3^1NJ 3S220 38253 382:46 3rvj20 38*r>3 383S8_ Cotarr: Cotang ~2.74r48 2.74499 2.74251 2.74004 2.73756 2.73509 2.73i263 2.?3017 2.73771 2.7.1526 2.7^2281 2.7.'3036 2.71792 2.71548 2.71305 2.71003 I 2 70.^19 i 2.70577 2.70333 2.7't034 2.03858 2.00612 2.0J371 2.03131 2.0:\392 2.CJ053 2.CM14 2.0:175 2.0V.:'57 2.orroo 2.67462 67225 069S9 GG752 66516 66281 2.66046 2.a5811 2.0.5376 2.0.5342 2.05109 261875 2.01 ; 13 2 01410 2.01177 2 &3315 2.03714 2.C3483 2.01-152 2.0^21 2.02791 2.62361 2.(;ii32 2.02103 2.01M74 2.01046 2.G1418 2.01190 2.00063 2.(50736 2.0)509 G0« Tan?r .38386 C->tanfr 2.a;o09 .88420 2.60283 .8&453 2.60057 .88487 2.53331 .88520 2.53006 .38553 2.59381 .38587 2.59156 .38020 2.C3933 .3"^54 2.58708 .38687 2.58484 .88721 2.68261 .88754 2.68088 .38^7 2.57815 .83821 2.57593 .33854 2.57871 .SS888 2.5n50 .889:1 2.56928 .33955 2.56707 .83983 2.66487 .39023 2.50206 .89056 2.56046 .89089 2.55827 .33123 2.53CL8 .89158 2.55389 .39190 2.r5170 .33?-?3 2.W052 .S3C57 2.51734 .83200 2.5-;516 .333:4 2.54209 .33357 2.51003 .89391 2.53805 .imvi 2.53^48 .33453 2.';3433 .33192 2.:i3217 .33523 2.53001 .33539 2.5:;:8 .33393 2.;"::571 .39023 2.r,:357 .80000 2.fJ142 .30094 2.:;19i:9 .39^7 2.51715 .89761 2.51502 .::3795 2.512S9 .::3H29 2.51076 .83803 2.50304 ..^'3896 2.50r;3 .339::o 2.5a«0 .33963 2.:;0e->9 .33997 2.rHWH ..rx)3i 2.10807 .40005 2. :3597 .40008 2.193'-<0 .40133 I 2.l.;i77 .4«)106 , 2.-I ■:);)7 ..;;)■»(« ! 2.! wrvS .4tU;n ' 2..J043 . 40:^07 2 'l".04O .40301 2.18183 .40335 2.-17924 .40309 2.47716 .40403 2. 17500 Cotanf? '1 -.r.i,' Cotaiicj Tang 22» Tangr CotanfjT .40408 2.47509 .40436 8.47303 .40470 2.47095 .40504 2.46888 .40538 2.4668S .40573 2.46476 .40606 2.46270 .40640 2.46065 .40074 8.45860 .40707 2.45655 .40741 8.46461 .40775 2.4S946 .40809 2.45048 .40843 8.44839 .403?r 8.44686 .40911 8.44488 .40945 8.44280 .40379 8.44027 .<1018 2.43::25 .41017 2.43023 .11081 8.43428 .41115 8.48220 .41149 8.43019 .41183 2.42319 .41217 2.42618 .41251 8.43418 .4121^5 8.42218 .41319 2.42019 .4ia53 2.41819 .413;?7 2.41020 .41421 8.41421 .41455 8.41223 .414'J0 2.410J3 .41524 2.4ai27 .4m58 8.406C9 .41693 8.40133 .416C6 8.40233 .41660 2.40033 .41694 2.89841 .41T-:3 2.890-13 .41763 2.89449 .41797 2.89258 .'118:U 2.29053 ..!lC<i5 2.88863 41099 2.88668 i .41933 2.88173 i ..-inr.s 2.88279 .4J003 2.38084 .42086 2.87891 .'!'2070 2.37097 ..42105 2.37J04 .42189 r:.37811 .12173 2.37118 , .'422t)7 2.86925 ' .42-213 2.86733 .422'. 6 2.86541 .42310 2.86349 .42845 2.86158 .42379 2.86967 i .42418 2.86776 ! .42447 2.a'S5W 23< Tang «M47 48488 42516 42551 42585 42619 42651 42033 42723 42757 42791 42r28 42800 42694 420S9 48968 42996 48088 48067 48101 4S186 4S170 48205 48383 48274 43308 43a43 43378 46418 43447 46181 48516 43550 48585 48634 48689 43734 48758 48703 48888 48868 48897 48038 48006 44001 44033 44071 44103 44140 44173 44210 44214 442rr9 44314 44349 44884 44418 44468 4(488 44388 Cotang 8.85686 8.85335 8.83205 8.85016 8.84825 8.64686 8.84147 8.81858 8.84069 8 83881 8.83608 8.83606 8.83817 8.88180 8.8891S 8.88756 8.S35TO 8.88888 8.88197 8.83018 8.81886 8.81641 8.81466 8.81871 8.81086 8.80008 8.80718 8.80631 8.80861 2.80167 8.89981 8.80601 8.89619 8.89487 8.89851 8.80078 8.88891 8.88710 8.88588 8.83318 8.88107 8.8T067 887806 8.87686 8.87447 8.87867 8.87068 8.80009 8.86780 8.866R8 8.36374 8.86106 8.86018 8.80&1O 8.85063 8.25186 8.85800 8.86188 8.81056 8 8mo 8.81001 GV QV Cotang Tung NATURAL TANGENTS ANE > COTANGENTS. 1 15 24«» 25° ' £8° 27^ 60 Tangr .44523 Cotang Tang .40631 C'otang 1 Tang .484 73 Cotang Tang .50953 Cotang 2.24004 2.14^^31 2.05C30 1.96;t01 1 .44558 2.24428 .46666 2.14288 .48809 2.04879 .509ii9 1.90120 59 2 .44593 2.24252 .40702 2.14125 .4^345 2.04728 .61020 1.95979 58 8 .44627 2.24077 .46737 2.13963 .4LS881 2.04577 .51003 1.95838 57 4 .44662 2.23902 .46778 2.1SS01 .48017 2.04426 .510CD 1.95698 56 6 .44097 2.23727 .46808 2.18639 .48953 2.04276 .51136 1.95557 55 6 .44738 2.23553 .46843 2.13477 .48989 ; 2.04125 i .51173 1.95417 :54 7 .41767 2.23378 .46879 2.1.3G16 .49026 2.0C975 .51209 1.05277 i53 8 .44802 2.23204 .4C014 2.181M .49CG2 2.03P25 .51246 1.95137 ;52 9 .41837 2.23030 .40950 2.12093 .49098 2.0CC:a .51283 1.94C97 51 10 .44872 2.22857 .40985 2.12832 .49134 2.a3526 .51319 1.94858 50 11 .44907 2.23683 .47021 2.12671 .49170 2.03376 .51356 1.94718 49 12 .44942 2.22510 .47056 2.12511 .49^06 2.0S227 .51393 1.94579 ,'48 13 .44977 2.22337 .47092 2.12350 .492:2 2.03078 .61430 1.94440 47 14 .45012 2.^164 .47128 2.12190 .492^8 2.02023 ..01467 1.94301 46 15 .45047 2.21992 .47163 2.12030 .49315 2.02780 i .51503 1.94162 45 IG .45082 2.21819 .47199 2.11871 .49351 ! 2.02031 .51540 1.94023 44 17 .45117 2.21047 i A?23i 2.11711 .49387 2.02483 .51577 1.93885 43 18 .45152 2.21475 .47270 2.11552 .494::3 2.02335 .51614 1.93746 42 19 .45187 2.21804 .47005 2.11C92 .494:9 2.021B7 .C1651 1.03608 '41 20 .45222 2.21132 .47341 2.11233 .49495 2.02039 .51688 1.93470 40 21 .45257 2.20961 .47377 2.11075 .49532 2.01891 .51724 1.03332 39 22 .45292 2.20790 .47412 2.1GJI6 .40.:l8 2.01743 .61761 1.93195 38 23 .45327 2.20019 .47448 2.10758 .49604 2.01596 .51798 1.93057 37 24 .45S6J 2.204i9 .47483 2.10600 .49640 2.01449 .£1835 1.92920 36 25 .45897 2.20278 .47519 2.10442 .49077 2.01302 .51872 1.92782 35 26 .45432 2.20108 .47555 2.10284 .49713 2.01155 .51909 1.92645 34 27 .45407 2.100C8 .47590 2.10126 .49749 2.01008 .51946 1.92508 33 28 .45502 2.197G9 .47626 2.09969 .49786 2.00862 .51983 1.92371 32 29 .45538 2.195C9 .47062 2.09011 .49822 2.00715 .52020 1.92235 31 80 .45573 2.19430 .47698 2.09654 .49858 2.00569 .52057 1.92098 30 81 .45608 2.19261 .47733 2.09498 .49894 2.00423 .52094 1.91962 29 82 .45643 2.10092 .47769 2.09341 .49931 2.00277 .52131 1.91826 28 83 .45078 2.18923 .47805 2.03184 .49967 2.00131 .52168 1.91690 27 84 .45718 2.18755 .47840 2.09028 .50004 1.99986 .52205 1.91554 26 86 .45748 2.18587 .47876 2.08872 .SOOiO 1.99841 .52242 1.91418 25 86 .45784 2.18419 .47912 2.08716 .50076 1.99695 .52279 1.91282 24 87 .45819 2.18251 .47948 2.08500 .50113 1.C3550 .52316 1.91147 23 88 .45854 2.18064 .47984 2.08405 .50149 1.C3406 .52353 1.91012 22 89 .45889 2.17916 .43019 2.03250 .60185 1.C9261 .52390 1.90876 21 40 .45024 2.17749 .43055 2.08094 .50222 1.99116 .52487 1.90741 20 41 .45960 2.17582 i48091 2.07939 .50258 1.98972 .52464 1.90607 19 42 .45995 2.17416 .43127 2.07785 .50295 1.98828 .52501 1.90472 18 43 .46080 2.17249 .48163 2.07630 .50331 1.98684 .52538 1.90337 17 4( .40065 2.17083 .48198 2.07476 .50308 1.98540 .52575 1.90203 10 45 .40101 2.10917 .48234 2.07321 .50404 1.98396 .62613 1.90009 15 46 .46186 2.10751 .48270 2.07167 .50141 1.98253 .52650 1.89935 14 47 .46171 1 2.10585 .48306 2.07014 .50477 1.93110 .52687 1.89801 13 48 .40208 2.10^120 .48342 2.06860 .50514 1.97966 .62724 1.89607 12 49! .4G242 2.1C55 ' .48.378 2.06706 .50550 1.97823 .52761 J. 895;^ 11 £0 .46277 2.16090 i .48414 2.06553 .50587 1.97681 .52798 1.89400 10 51 .46312 2.15925 .48450 2.06400 .50623 1.97538 .52836 1.8926b 9 1.2 .40348 2.157C0 .4-3486 2.00247 .50060 1.97395 .52873 1.89133 8 53 .46383 2.15596 .43521 2.00094 .50096 1.97253 .52910 1.89O00 7 54 .46418 2.15482 .48557 2.05942 .50733 1.97111 .52947 1.88r.G7 6 55 .40454 2.15268 .48593 2.05790 .50769 1.96969 .62985 1.88734 5 56 .40489 2.15101 .48629 2.05037 .50806 1.96827 .53022 1.88602 4 57 .40525 2.14940 .48665 2.05485 .50843 1.96685 .53059 1.884C9 3 58 .46560 2.147r7 .48701 2.05383 .50879 1.96544 .58096 1.88337 2 59 .46595 8.14614 .48787 2.05182 i .50916 1.96402 .53134 1.88205 1 1 / .46681 Cotang 2.14451 .48773 2.05030 .50953 1.962G1 .53171 1.88073 _0 Tang Cotangi 'I'ang Cotang Tang Cotang Tang f 66» e4« 6 30 6! 2» NATURAL TANGENTS AND COTANGENTS. NATURAL TANGENTS AND COTANGENTS. 1 17 * 82* ': 88<» 84» 1! 85» (io m Tangf I Ck>taiig: ' .62487 1.600:43 , Tang .04941 Cotong Tang j Cotang 1 .67451 . 1.4H2r)6 , Tang .70021 Cotang 1.539S6 1.42815 ll .62527 1.50980 .61982 1.53888; .67493 1.48103 .70UG4 1.42720 50 2| .625U8 1.59826 . .65024 1.53791 .67536 1.48070 .70107 1.42(:g:} \:a 3 .6:%0d 1.537e:j .65005 1.53093 .67578 1.47JJ77 .70151 1.42550 57 4 .62649 1.59da0 1 .65106 1.53595 : .67C20 1.47R85 .70194 1.42462 i56 5 .62689 1.59517 .^148 1.53197 . .07ca 1.47792 .70238 1.42^^74 ;55 6 .G2r^ 1.59414 .65189 1.53100 : .677l'5 1.47G99 .70281 1.42286 54 1.42198 53 1.42110 52 7 .68770 1.59311 .652:31 1.63302 .Ci i <8 1.47007 .70325 8 .62811 1.5020^3 .65272 1.5305 .6771:0 1.47514 .70:jf;.8 9 .62852 1.59105 .6>314 1.53107 .67S:i2 1.47422 .70412 1.42022 ;51 1 10 .628S» 1.50002 .05355 1.53010 .67875 1.47330 .70455 1.41931 50 11 .62933 1.53900 .6.5307 1.B2913 .67917 1.47238 . .70199 1.41817 49 12 .C;^r3 1.5J7J7 .0>4:38 1.52816 .679f» 1.47146 1.47053 ' .70:>42 1.41759 48 13 .63014 1.5S695 .05480 1.52719 .08002 .705S6 1.41C72 47 14 .CSOxS 1.53593 .05521 1.52C22 .68045 1.4C002 =: .70029 1.415K1 40 15 .030;» 1.58190 .C5563 1.51S25 .68088 1.40S70 :.' .70073 1.41497 45 16 .63136 1.583^38 .65004 1.52129 1 .68130 1.40778 .; .70717 1.41409 44 17! .63177 1.58286 .65646 1.62332 : .68173 1.4GC86 :■ .70700 1.41322 ^^ \^\ .fSSXl 1.581W .6"5C'« 1.52235 .68215 1.4C595 .70804 1.41£:« :42 19- .63258 1.58«3 1 .or^^ \XIV^ .6<*2.:S . 1.40503 .1 .70«4S 1.411 LS 41 2U .C32D9 1.579bl .OjiVI 1.52013 .68301 1.40411 ;| .70801 1.41001 40 21 .63810 ' 1.57879 .R-seia 1.5194C .68313 1.40320 '' .70035 1.40974 39 22 .63S30 1.577:1 .C^oCl ' 1.51850 .CK->6 1.4UJJ9 1 .70.,r9 1.40'-S7 3:^ 23 .63121 1.57G.0 .C^S>5 1.51754 .6S4:w;» 1.40137 .710a I.41.UK) 37 24 .63162 1.57675 .65033 1.51C.-i8 . .68471 1.40046 ; .7Ki<:6 1.40714 30 25 .635C3 1.574T4 : .65SS0 1.515C2 .0S511 1.450.'i5 ; .71110 1.40»;27 35 25 .63514 1.573T2 .66021 1.6l4r» . .08557 1.45:01 ! .71151 1.4U"10 •:4 27 .63581 \.h"i:i\ .660C3 1.51870 .68000 1.45ii'3 ' .7ir.Pi 1.40:.-4 3:1 2) .63625 1.5n70 .66105 1.51275 .68012 1.450-^2 .7ic:}2 1.40:.:;7 32 29 .63006 1.670C9 .66147 1.51179 .m>h 1.45.v:r2 .712-5 1.4^2^1 '31 ao| .63707 1.56969 .661S9 1.510c4 ; .^I'A 1.455U1 .71329 1.40105 "30 1 81 .63718 1.56868 .66230 i.soaas .68771 ' 1.4*^110 \ .71373 1.40109 20 32 .63789 1.5STC7 .00272 1.50SI*3 .f-^U : 1.45:,.'0 ! .71417 1.4'i:r« 'O'i 83 .63830 1.5o007 .00314 l.D£r797 : .OS857 ; 1.4.-; J ;9 . .71401 1.80036 27 81 .63071 1.56566 .CC3::6 1.50702 .CS900 . 1.45!:;3 : .7151)5 l.S0!OC :2*^. 85 .63912 1.56406 .60308 1.5«3C-/7 .0=^342 1 1.45049 .715 J9 1.80701 25 86 .63L53 1.5C3G6 .66140 1.50:^12 ; .CS0S5 ' 1.4;'.:.S 1 .71503 1.89C79 .21 87 .C39'J1 1.56265 .661S2 1.50417! .C9028 1.4'l=';S ; .71637 1.89:03 23 8rt .61085 1.56105 .60524 1.50322 .«'>j71 1.44773! .71081 ' 1.805"7 22 89 .61076 1.500G5 .60506 1.60223 ; .rolU 1.44C^S .n725 1.89 ni 21 40 .6ai7 1.559C6 .66608 1.50133 i .(^157 1.44598 : .71769 1.393:;6 :-^ 41 .61158 1.55866 .666.')0 1.50038 ' .09200' 1.44508 .71P13 ; 1.39250 'l9 \Ai\ .e4JiH> 1.557W .60092 1.4JJ44 .«.j0;3 i 1.44413 .71?.-.7 1.8..: ;5 1 ; 43 .61:^0 1.55CW . Xf-jt '-4 1.4aS49 .Uj2o6 1.44:^:9 1.49^55 .Cj3i9 1.4i2:^ 1.49601 .69372 1.4!Ii9 ' .710 :l 1.30 :9 17 44 .6J281 1.55567 .66776 .':i:.46 1.3. 04 10 45 .6i:jii 1.SM67 .66818 .71000 1.3S0U0 1.- 46 .OldGS 1.5986^ . .Oi-CO 1.49566 = .09116 \AV/A .7-a:.M 1.3->.^4 11 47 .64404 1.5SSG3 .6akl2 1.4:^72 . .ei^;59 i.4p%ro .7JJ78 l..'i:7:;S IC 43 .64446 1 1.55irj .6C:^ i 1.4&G73 «! .005.-2 1.43& 1 .72122 l.?J^.-J 1: 49 .61487 1.55071 .er;:;j5 ! 1.49171 : .6X-i5 . 1.4.'?n'2 .72107 1.3'.-. S 11 50 .04538 1.54972 .6r0t23 1.49190 '• .695>)8 X.A^r,^ .72211 1. ;:>;>! u 51 .61509 1 1.54873 .67071 1.49097 | .69^31 1.4CC11 .7225.5 1.3S.r9 9 52 .64010 1.&4774 .67113 1.4>yj3 .€:..; 5 1.4:i:.:5 .72i:/J 1.3SJ11 f= 53 .64052 1 5JC75 .671.5 1.4:i;9 .607:8 i.4;;;-5 .72311 1.3---. J 7 ;>i .61003 l.r^J576 .671 W 1.48.3:6 .60; 01 i.4.>i;7 .72;--J 1.3^145 6 ns .647^ 1.514:3 .67;,i9 1.4-722 - .Cf^- 4 1.4oi.S . .721':2 i 1.3SC-» 5 56 • .647^5 i.5ij:9 .67::r3 \A:^?J!i 1 .c:.;7 1.4;: .3 .721:7 ; l.:i7:.:'3 4 57 ' .64817 : 1.51iSl .6r.J4 1.4S536 ' .e: 1 1.43 ■) .72: .1 ' 1.37fe:i 3 .^e \ .61858 1.541S3 .67J36 1.4S«2 .f/y^i i.4;<>2 .?i: 5 i.37¥«>r 2 .72C:0 : 1.37722 1 .72?;-.! 1 .37cr;s Cotacg] Ting ES » .64SII0 1.51035 .67416 1.4«'^9 .Uw 1 7 1 .-x^-- -3 '6C » .64941 1.589B6 .67451 1.4^j6 .7aei 1.4715 1 f Cotaogt Tang 1 Cotcn^ 1 Tacg C^L-a.;: T^EiJ w Vf : 65' \ 34- _J 118 N.-iTURAl, TAXOKNTS AND COTANGENTS. 36« Tang I C()t:irijf 1 2 3 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 .3522-4 .35142 .35060 .;MU78 .34896 .34814 .S-1732 .34650 .34568 .^4487 .34405 .34323 .a4242 .34160 .a4079 .33998 .33916 .33835 .33754 .3:JG73 .335:12 .33511 .33430 .333-49 .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 .78a-J5 .78082 .78129 '^'^521 ,«r608 .32704 ..')24(U .:J2384 ..•12:W4 .:)2224 .321-44 ..*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 .79-196 .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 l-TTT ueu .1-708 ttOOB .103S 1 15(67 .lOUB :i-S61 j^ :S ;jK!K .nsi jMW .l-WB »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 wrought-iron columns have lately been y changed by some engineers; but as the question of the th of wrought-iron columns has not yet been satisfactorily I, and as the formulas herein given undoubtedly err on the ide if at all, we have thought best not to change them, espe- as they are still used by many bridge engineers. question of the wind-pressure on roofs has not been taken as thorough manner as would be needed for pitch roofs of Teat span ; but for ordinary wooden roofs, and iron roofs not ling one hundred feet span, the method given in Chap. I. is sufficiently accurate. r one wishing to study the most accurate method of obtaining feet of the wind-pressure on roofs will find it in Professor 's excellent work on " Graphical Analysis of Roof Trusses." ©nclusion, the author recommends these chapters as present- icurate and modern rules, especially adapted to the require- of American practice. EXPLANATION OF SIGNS AND TERMS USED IN THE FOLLOTVING FORMULAS. Besides the usual arithmetical signs and characters in general use, the following characters and abbreviations will frequently be used : — The sign y^ means square root of number behind. ^ means cube root of number beliind. ( ) means that all the numbers between are to be taken as one quantity, means decimal parts; 2.5 = 2t^, or .46 = iVo. The letter A denotes the co-efficient of strength for beams one inch square, and one foot between the supports. C denotes resistance, in pounds, of a block of any material to crushing, per square inch of section. E denotes the modulus of elasticity of any material, in pounds per square inch, e denotes constant for stiffness of beams. F denotes resistance of any material to shearing, per square inch. B denotes the modulus of rupture of any material. aS denotes a factor of safety. T denotes resistance of any material to being pulled apart, in pounds, per square inch of cross-section. Breadth is used to denote the least side of a rectangular piece, and is always measured in inches. Depth denotes the vertical height of a beam or girder, and is always to be taken in inches, unless expressly stated otherwise. LetKjth denotes the distance between supports in feetf unless otlu*rwis(» specified. Abbreviations. — In order to shorten the formulas, it has ()ft(Mi been found necessary to use cerUin abbreviations; such as bet. Tor Ix'twiMjn, hot. for bottom, dist. for distance, diam. for diaimtcr, lior. tor horizontal, scj. for square, etc., which, however, can in no cast' Wiul to uncertainty as to their meaning. Wli( IV tlie word "ton" is used in this volume, it always means 2(M)0 pounds. CHAPTER T. DEFINITIONS OF TERMS USED IN MECHANICS. The following terms frequently occur in treating of mechanical construction, and it is essential that their meaning be well under- stood. Mechanics is the science which treats of the action of forces. Applied Mechanics treats of the laws of mechanics which relate to works of human art ; such as beams, trusses, arches, etc. Rest is the relation between two points, when the straight line joining them does not change in length or direction. A body is at rest relatively to a point, when any point in the body is at rest relatively to the first-mentioned point. Motion is the relation between two points, when the straight line Joining them changes in length or direction, or in both. A body moves relatively to a point, when any point in the body moves relatively to the point first mentioned. Force is that which changes, or tends to change, the state of a body in reference to rest or motion. It is a cause regarding the essential nature of which we are ignorant. We cannot deal with forces properly, but only with the laws of their action. Kqiiilibrium is that condition of a body in which the forces acting upon it balance or neutralize each other. Statics is that part of Applied Mechanics which treats of the conditions of equilibrium, and is divided into: — a. Statics of rigid bodies. 6. Hydrostatics. In building we have to deal only with the former. Structures are artificial constnictions in which all the parts are intended to be in«equilibrium and at rest, as in the case of a bridge or roof-truss. They consist of two or more solid bodies, called pieces, which are connected at portions of their surfaces called joints. There are three conditions of equilibrium in a structure; viz. : — I. The forces exerted on the whole structure must balance each other. These forces are: — a. The weight of the structure. h. The load it carries. 126 DEFINITIONS OF TERMS c. The supporting pressures, or resistance of the foundation?, called external forces. II. The forces exerted on each piece must balance each other. These forces are: — rt. The weight of the piece. b. The load it carries. c. The resistance of its joints. III. The forces exerted on each of the parts into which any piece may be supposed to be divided must balance each other. Stability consists in the fulfilment of conditions I. and II., that is, the ability of the structure to resist displacement of its parts. Streng'th consists in the fulfilment of condition III., that is, the ability of a piece to resist breaking. Stiffness consists in the ability of a piece to resist bending. The theory of structures is divided into two parts; viz. : — I. That which treats of strength and stiffness, dealing only with single pieces, and generally known as strength of liiaterialH* II. That which treats of stability, dealing with structures. Stress. — The load or system of forces acting on any piece of material is often denoted by the term " stress,'* and the word will be so used in the following pages. The i)} tensity of the stress per square inch on any normal sur- face of a solid is the total stress divided by the area of the section in square inches. Thus, if we had a bar ten feet long and two inches square, with a load of 8000 poimds pulling in the direction of its length, the stress on any normal section of the rod would be 8000 pounds ; and the intensity of the stress per square inch would be 80{K) -f 4, or 2000 pounds. Strain. — When a solid body is subjected to any kind of stress, an alteration is produced in the volume and figure of the body, and this alteration is called the ** strain." In the case of the bar given al)ovo, the strain would be the amount that the bar would stretch under its load. The Ultimate Stronprth, or Breaking: Load, of a body is the load riHiuircd to prothK-e fracture in some specified way. The Safe Load is the load that a piece can support without impairing: its strciii^tii. Factors of Safety. — When not otherwise specified, & factor of safety means the ratio in which the breaking load exceeds the safe load. In designing a i)i{^ce of material to sustain a certain load, it is required that it shall be perfectly safe under all circum- stances; and henc(^ ii. is necessary to make an allowance for any defects in the material, workmanship, etc. It is obviona, that, for USED IN MECHANICS. 127 Is of different composition, different factors of safety will ired. Thus, iron being more homogeneous than wood, and »le to defects, it does not require so great a factor of safety, ^in, different kinds of strains require difiPerent factors of Thus, a long wooden column or strut requires a greater »f safety than a wooden beam. As the factors thus vary irent kinds of strains and materials, we will give the proper of safety for the different strains when considering the ce of the material to those strains. iiiction between Dead and Live liOad. — The dead load," as used in mechanics, means a load that is ap- j imperceptible degrees, and that remains steady; such as 3;ht of the structure itself. ive load '' is one that is applied suddenly, or accompanied .brations; such as swift trains travelling over a railway- or a force exerted in a moving machine. \ been found by experience, that the effect of a live load on or other piece of material is twice as severe as that of a id of the same weight: hence a piece of material designed r a live load should have a factor of safety twice as large lesigned to carry a dead load. load produced by a crowd of people walking on a floor is considered to produce an effect which is a mean between a dead and live load, and a factor of safety is adopted modulus of Rupture is a constant quantity found in aulas for strength of iron beams, and is eighteen times the : the constant " A." ulus of Elasticity. — If we take a bar of any elastic 1, one inch square, and of any length, secured at one end, he other apply a force pulling in the direction of its length, i find by careful measurement that the bar has been stretched ;ated by the action of the force. if we divide the total elongation in inches by the original )f the bar in inches, we shall have the elongation of the bar b of length; and, if we divide the pulling-forre per square this latter quantity, we shall have what is known as the s of elasticity. e we may define the hkkIhIiis of fUintirUij an the pullinfj or uiing force per unit of .'section divided by the elongation iresnion i)er unit of Unfjth. \ example of the method of determining the modulus of y of any, material, we v^ill take the following: — )8e we have a bar of wroiight-iron, two inches square and ten feet long, securely fastened at one end, and to the other end we apply a pulling-force of 40,000 pounds. This force causes the bar to stretch, and by careful measurement we find the elongation to be 0.0414 of an inch. Now, as the bar is ten feet, or 120 inches, long, if we divide 0.0414 by 120, we shall have the elongation of the bar per unit of length. Perfonning this operation, we have as the result 0.00034 of an inch. As the bar is two inches square, the area of cross-section is four s(iuare inches, and hence the pulling-force per square inch is 10,000 pounds. Then, dividing 10,000 by 0.00084, we have as the modulus of elasticity of the bar 29,400,000 pounds. This is the method generally employed to determine the modulus of elasticity of iron ties; but it can also be obtained from the deflection of beams, and it is in that way that the values of the modulus for most woods have been foiuid. Another definition of the modulus of elasticity, and which is a natural consequence of the one just given, is the number of pounds that would be required to stretch or shorten a bar one inch square by an amount equal to its length, provided that the law of pei-fect elasticity would hold good for so great a range. The mod- uhis of elasticity is generally denoted by E, and is used in the detomiination of the stiffness of beams. Moment. — If we take any solid body, and pivot it at any point, and apply a force to the body, acting in any direction except in a line with the pivot, we shall produce rotation of the body, provided the force is sufficiently strong. This rotation is produced by what is called the moment of the force; and the moment of a force about any given point or pivot is the product of the force into the perpendicular distance from the pivot to the lin(i of action of the force, or,an common phrase, the product cf the force into the arm with which it acta. The Centre of Gravity of a body is the point through which tlie resultant of the weight of the body always acts, no mat- ter in what, position the body be. If a body be suspended at its centre of tjjravity, and revolved In any direction, it will always be in e<iuilihriinn. (For centre of gravity of surfaces, lines, and soliils, see Chap. IV.) CLASSIFJCATION OF STRAINS. 120 CI.A88IFICATION OF STRAINS WHICH MAT BE PRODUCED IN A SOLID BOD7. The dififerent strains to which building-materials may be exposed are: — I. Tension, as in the case of a weight suspended from one end of a rod, rope, tie-bar, eta; the other end being fixed, tending to stretch or lengthen the fibres. II. Shearing Strain^ as in the case of treenails, pins in bridges, etc., where equal forces are applied on opposite sides in such a manner as to tend to force one part over the adjacent one. III. Conipressiony as in the case of a weight resting on top of a column or post, tending to compress the fibres. IV. Transversa or Cross Strain, as in the case of a load on a beam, tending to bend it. V. Torsion, a twisting strain, which seldom occurs in build- ing-construction, though quite frequently in machinery. 130 FOUNDATIONS. CHAPTER n. FOUNDATIONS. The following chapter on Foundations is intended to furnish the reader with only a general knowledge of the subject, and to enable him to be sure that he is within the limits of safety if he follows what is here given. For foundations of large works, or buildings upon soil of questionable firmness, the compressibility of the soil should be determined by experiments. The term ^'foundation" is used to designate all that portion of any structure which serves only as a basis on which to erect the superstructure. This term is sometimes applied to that portion of the solid mate- rial of the earth upon which the structure rests, and also to the artificial arrangements which may be made to support the base. In the following pages these will be designated by the term " foundation-bed." Object of Foundations.— The object to be obtained in the construction of any foundation is to form such a solid base for the superstructure that no movement shall take place after its erection. But all structures built of coarse masonry, whether of stone, or brick, will settle to a certain extent; and, with a few exceptions, all soils will become compressed under the weight of almost any building. Our main object, therefore, is not to prevent settlement entirely, but to insure that it shall be uniform ; so that, after the structure is finished, it will have no cnacks or flaws, however irregularly it may be disposed over the aroa of its site. Foundations Classed. — Foundations maybe divided into two classes : — Class I. — Foundations constructed in situations where the natural soil is sufficienthj flnn to bear the weight of the intended structure. Class II. — Foundations in situations where an artyicicU bear^ ing-stratum must be formed, in consequence of the 9rftne89 or looseness of the soil. FOUNDATIONS. 131 Each of these two great classes may be subdivided into two divisions: — a. Foundations in situations wliere water offers no impediment to the execution of the work. 6. Foundations under water. It is seldom that architects design buildings whose foundations are under water; and, as this division of the subject enters rather deeply into the science of engineering, we shall not discuss it here. Boringf. — Before we can decide wliat kind of foundation it will be necessary to build, we must know the nature of the subsoil. If not already known, this is deterininetl,* ordinarily, by digging a trench, or making a pit, close to the site of the proposed works, to a depth sufficient to allow the different strata to be seen. For important structures, the nature of the subsoil is often de- termined by boring with the tools usually employed for this pur- pose. When this method is employed, the different kinds and thickness of the strata are determined by examining the speci- mens brought up by the auger used in boring. Foundations of tlie First Class.— -The foundations in- cluded under this class may be divided into two cases, according to the different kinds of soil on which the foundation is to be built : —% Case I. — Foundations on soil composed of mateiHals whose stability is not aff^cteA by saturation with water, and which are firm enough to support the weight of the structure. Under this case belong, — Foundations on Rock. — To prepare a rock foundation for being bfuilt upon, all that is generally required is to cut away the loose and decayed portions of the rock, and to dress the rock to a plane surfsice as nearly perpendicular to the direction of the pressure as is practicable; or, if the rock forms an inclined plane, to cut a series of plane surfaces, like those of steps, for the wall to rest on. If there are any fissures in the rock, they should be filled with con- crete or rubble masonry. Concrete is better for this purpose, as, when once set, it is nearly incompressible under any thing short of a crushing-force; so that it forms a base almost as solid as the rock itself, while the compression of the mortar joints of the masonry is certain to cause some irregular settlement. If it is unavoidably necessary that some parts of the foundation shall start from a lower level than others, care should be taken to keep the mortar Joints as close as possible, or to execute the lower portions of the work in cement, or some hard-setting mortar: other- wise the foundations will settle unequally, and thus cause much injury to the superstructure. The load placed on the rock should at no time exceed one-eighth of that necessary to crush it. Pro' 132 FOUNDATIONS. fessor Rankine gives the following examples of the actual intensity of the pressure per square foot on some existing rock founda- tions: — Average of ordinary cases, the rock being at least as strong as the strongest reil bricks 2000(; Pressures at tlie base of St. KoIIox chimney (450 feet below the summit) On a layer of strong concrete or beton, 6 feet deep .... 0070 On sandstone below the beton, so soft that it crumbles in the hand 4000 The last example sliows the pressure which is safely borne in practice by one of the weakest substances to which the name of rock can be applied. M. Jules Graudard, C.E., states, that, on a rocky ground, the Roquefavour aqueduct exerts a pressure of 26,800 pounds to the square foot. A bed of solid rock is unyielding, and appears at first sight to offer all the advantages of a secure foundation. It is gen- erally found in practice, however, that, in lai^ge buildings^ part of the fowidations will not rest on the rock, but on the adjacent soil; and as the soil, of whatever material it may be composed, is sure to be compressed somewhat, irregular settlement will almost invariably take place, and give much trouble. The only remedy in such a case is to make the bed for the foundation resting on the soil as firm as possible, and lay the wall, to the level of the rock, in cement or hard-setting mortar. Foundation on Compact Stony Earths, such as Graieel or Sand. — Strong gravel may be considered as one of the best soils to build upon ; as it is almost incompressible, is not affected by exposure to the atmosphere, and is easily levelled. Sand is also almost incompressible, and forms an excellent foun- dation as long as it can be kept from escaping; but as it has no cohesion, and acts like a fluid when exposed to running water, it should be treated with great caution. The foundation bed in soils of this kind is prepared by digging a trench from four to six feet deep, so that the foundation may be started below the reacli of the disintegrating effects of frost. The bottom of the trench is levelled ; and, if parts of it are required to be at different levels, it is broken into steps. Care shoulil l)e taken to keep the surface-water from running into the trench; and, if necessary, drains should be made at the bottom to carry away the water. The weight resting on the bottom of the trench should be pro* portional to the resistance of the material forming the bed. FOUNDATIONS. 133 Mr. Gaudard says that a load of 10,500 to 18,300 pounds per square foot has been put upon close sand in tlie foundations of Gorai Bridge, and on gravel in the Lock Ken Viaduct at Bordeaux. In the bridge at Nantes, there is a load of 15,200 pounds to the square foot on sand; but some settlement has already taken place. Ilankine gives the greatest intensity of pressure on foundations in firm earth at from 2500 to 3500 pounds per square foot In order to distribute the pressure arising from the weight of the structure over a greater surface, it is usual to give additional breadth to the foundation courses: this increase of breadth is called the spread. In compact, strong earth, the spread is made one and a half times the thickness of the wall, and, in ordinary earth or sand, twice that thickness. Case II. — Foundations on soils firm enough to support the weight of the strtictiire, but whose fttaMUty l8 affected by water. The principal soil imder this class, with which we have to do, is a clay soil. In this soil the bed is prepared by digging a trench, as in rocky soils; and the foundation must be sure to start below the frost-line, for the effect of frost in clay soils is very great. The soil is also much affected by the action of water; and hence the ground should be well drained before the work is begun, and the trenches so arranged that the water shall not remain in them. And, in general, the less a soil of this kind is exposed to the air and weather, and the sooner it is protected from exposure, the better for the work. In building on a clay bank, great caution should be used to secure thorough drainage, that the clay may not have a tendency to slide daring wet weather. The safe load for stiff clay and marl is given by Mr. Gaudard at from 5500 to 11,000 pounds per square foot. Under the cylindrical piers of the Szegedin Bridge in Hungary, the soil, consisting of clay intermixed with fine sand, bears a load of 13,300 pounds to the square foot; but it was deemed expedient to increase its sup- porting power by driving some piles in the interior of the cylinder, and also to protect the cylinder by sheeting outside. Mr. McAlpine, M. Inst. C.E., in building a high wall at Albany, N.Y., succeeded in safely loading a wet clay soil with two tons to the square foot, but with a settlement depending on the depth of the excavation. In order to prevent a great influx of water, and consequent softening of the soil, he surrounded the excavation with a puddle trench ten feet high and four feet wide, and he also spread a layer of coarse gravel on the bottom. Foundations in Soft Eurths. — There are three materials in gen- eral use for forming an aitificial bearing-stratum in soft soils. 134 FOUNDATIONS. Whichever material is employed, the bed is first prepared by ezca^ vating a trench sufficiently deep to place the foundation-courses below the action of frost and rain. Great caution should be used in cases of this kind to prevent unequal settling. The bottom of the trench is made level, and covered with a bed of stones, sand, or concrete. Stones. — When stone is used, the bottom of the trench should In; paved with rubble or cobble stones, well settled in place by ramming. On this paving, a bed of concrete is then laid. Sand. — In all situations where the ground, although soft, is of sufficient consistency to confine the sand, this material may be used with many advantages as regards both the cost and the stability of the work. The quality which sand possesses, of distributing the pressure put upon it, in both a horizontal and vertical direction, makes it especially valuable for a foundation bed in this kind of soil; as the lateral pressure exerted against the sides of the founda- tion pit greatly relieves the bottom. There are two methods of using sand; viz., in layers and as piles. In fonning a stratum of sand, it is spread in layers of about nine inches in thickness, and each layer well rammed before the next one is spread. The total depth of sand used should be sufficient to admit of the pressiu^ on the upper surface of the sand being distributed over the entire bottom of the trench. Sand-piling is a very economical and efficient method of forming a foundation under some circumstances. It would not, however, be effective in very loose, wet soils; as the sand would work into the surrounding ground. Sand-piling is executed by making holes in the soil, or in the bottom of the trench, about six or seven inches in diameter, and about six feet deep, and filling them with damp sand, well rammed so as to force it into every cavity. In situations where the stability of piles arises from the pressure of the ground around them, sand-piles are found of more service than timber ones, for the reason that the timber-pile transmits pressure only in a vertical direction, while the sand-pile transmits it over the whole surface of the hole it fills, thus acting on a large area of bearing-surface. The ground above the piles should be covered with planking, concrete, or masonry, to prevent its being forced up by the lateral pressure exerted by the piles: and, on the stratum thus formed, the fomidation walls may be built in the usual manner. Fouiidatious on Piles. — Where the soil upon which we wish to build is not firm enough to support the foundation, one of the most common metliods of fonnhig a solid foundation bed is FOUNDATIONS. 136 by driving wooden piles into tlie soil, ami placing the foundation wails upon these. The piles are generally round, and have a length of ahout twenty times iheir mean diameter of cross-section. The diameter of the hcjid varies from nine to eighteen inches. The piles should be straight grained, and free from knots and ring strokes. Fir, beach, oak, anil Florida yellow-pine are the best woods for piles; though spruce and hemlock are very commonly used. Where piles are exposed to tide-water, they are generally driven with their bark on. In other cases, it is not essential. Piles which are driven through hard ground, generally require to have an iron hoop fixed tightly on their heads to prevent them from splitting, and also to be shod with iron shoes, either of cast or wrought iron. Long piles may be divided into two classes, — those which trans- mit the load to a firm soil, thus acting as pillars; and those where the pile and its load are wholly supported by the friction of the earth on the sides of the pile. In order to ascertain the safe load which it will do to put upon a pile of the first class, it is only necessary to calculate the safe crushing-strength of the wood; but, for piles of the second and more common class, it is not so easy to determine the maximum load which they will safely support. Many writers have endeavored to give rules for calculating the effect of a given blow in sinking a pile; but investigations of this kind are of little practical value, because we can never be in pos- session of sufficient data to obtain even an approximate result. The effect of each blow on the pile will depend on the momentum of the blow, the velocity of the ram, the relative weights of the ram and the pile, the elasticity of the pile-head, and the resistance offered by the ground through which the pile is passing; and, as the last-named conditions cannot well be ascertained, any calcula- tions in which they are only assumed must of necessity Ikj mere guesses. I^ad on Piles. — Professor Rankine gives the limits of the safe load on pilesy based upon practical examples, as follows : — For piles driveil till they reach the firm ground, 1000 pomids per square inch of ar^ of head. For piles standing in soft ground by friction, 200 pounds per square inch of area of head. But as, in the latter case, so much depends upon the character of the soil in which, the piles are driven, such a gcneml rule as the above is hardly to be reconunended. Several rules for the bearing-load on piles have been given, Perhaps tho nile most commonly given is that of Major Sanders, United-States En«jint;er. He experimented largely at Fort Dela' ware, and in 1851 gave the following rule as reliable for ordinary pikMlriving. Sanders's Rule for determining the load for a common wooden pile, driven until it sinks through only small and nearly equal distances under successive blows : — ,, , , , . „ weight of liammer in lbs. X fall in inches Safe load m lbs. = SXslnkin^t iS^blo^v^ Mr. John C. Trautwine, C.E., in his pocket-book for engineers, gives a rul(i which appears to agree very well with actual results. His rule is expressed as follows: — cube root of weight of x O^.*! Extreme load in _ fall in feet ha m mer in Ib:^. "'"^^ tons of 2240 lbs. ~ Last sinking in inches -h 1 For the safe load he recommends that one-half the extreme load should be taken for i)iles thoroughly driven in firm soils, and one- fourth when driven in river-mud or marsh. According to Mr. Trautwine, the French engineers consider a pile safe for a load of 25 tons when it refuses to sink under a liam- mer of 1344 pounds falling 4 feet. The test of a pile having been sufficiently driven, acconling to the best authorities, is that it shall not sink more than one-fifth of an incli under thirty blows of a ram weighing 800 pounds, falling 5 f(H>t at each blow. A more common rule is to consider the pile fully driven wlien it does not sink more than one-fourth of an inch at the last blow of a ram weighing 2500 pounds, falling 80 feet. In ordinary pile-driving for buildings, however, the piles often sink more than this at the last blow; but, as the piles are seldom loaded to their full capacity, it is not necessary to be so i)articular as in tlie foundations of engineering structures. A common practice witli :ircliitects is to specify the lengih of the piles to be usi»d, and the ])iles ;in» driven imtil their heads are juat al)Ove ground, and then left to he levelled off afterwards. Kxamplo of I^ile Foundation. — As an example of the ni"thf/!l of di'termining the necessary numl>er of plh»8 to 8up]K)rt a i:iv«'n building, we will determine tho numlKT of piles nM|ulr«Ml to MUi)port the sidivwalls of a warehouse (of which a vertical sec- tlon is shown in Fig. 1). The walls aro of brick, and the weight may be taken at 110 pounds per cubic foot of masonry. The piles are to be driven in two rows, two feet on centres; and It is found that a pile 20 feet long and 10 inches at the top will sink Fig. 1. one inch under a 1200-pound hammer falling 20 feet after the pile has been entirely driven into the soil. What distance should the piles ba on centres lengthwise of the wall ? 4 138 FOUNDATIONS. Hy calculation wc find tliat the wall contains 157i^ cubic feet of masonry per running foot, and hence weiglis 17,306 pounds. The load from the floors which comes upon the wall is: — From the first floor 1500 lbs. From the second floor loSO ll>s. From the third floor 1380 lbs. From the fourth floor 790 lbs. From the fifth floor 720 lbs. From the sixth floor 720 lbs. From the roof 240 lbs. Total 6730 lbs. Hence the total Ayeight of the wall and its load per running foot is 24,0:56 pounds. Tlie load which one of the piles will support is, by Sanders's rule, 1200 X 240 — ^^"^7~f — — 36000 pounds. By Trautwine's rule, using a factor of safety of 2.5, the safe load would be (^20 X 1200 X 0.023 — *j 5 X (14-1 ) ~ ^'^ ^^^^ ^^^ ^^'^ pounds), or 33600 pounds. Then one pair of piles would support 72,000, or 67,200 pounds, according to which rule we take. Dividing these numbers by the weight of one foot of the wall and its load, we find, that, by Sandei-s's rule, one pair of piles will support 3 feet of the wall, and, by Trautwine's rule, 2.8 feet of wall: hence the piles should be placed 2 feet 9 inches or 3 feet on centres. In very heavy buildings, heavy timbers are sometimes bolted to the tops of the piles, and the foundation walls built on these. In Boston, Mass., a large part of the city is built upon made land, and hence the buildings have to be supported by pile founda- tions. The Building Laws of the city require that all buildings "exceeding thirty-five feet in height (with pile foundation) shAll have not less than two rows of piles under all external and party walls, and the piles shall be spaced not over three feet on centres in the direction of the length of the wall." ^l.s (m example of the load which ordinary piles in the made land of Boston will support, it may be stated that the piles under Trinity ('hurch in Boston support two tons each, approKimately. For engineering works, various kinds of iron piles are used; baft they are too rarely used for foundations of buildingB to come within the scope of this chapter. For a description of these FOUNDATIONS. 139 le reader should consult some standard work on engineering, good description of iron piles is given in "Wheeler's Civil jering," and also in " Trautwine's Handbook." icrete Foundation Beds. — Concrete is largely used ndation beds in soft soil, and is a very valuable material for rpose; as it affords a firm solid bed, and can be spread out > distribute the pressure over a large area. ;rete is an artificial compound, generally made by mixing cement with sand, water, and some hard material, as bi*oken slag, bits of brick, earthenware, burnt clay, shingle, etc. e is any choice of the materials forming the base of the :e, the preference should be given to fragments of a some- K>rous nature, such as pieces of brick or limestone, rather > those with smooth surfaces. {See page liSa.) broken material used in the concrete is sometimes, for con- 2e, called the agrjregate, and the mortar in which it is incased, sitrix. The aggregate is generally broken so as to pass b a li or 2 inch mesh. imp ground or imder water, hydraulic lime should of course I in mixing the concrete. ingr Concrete. — A very common practice in laying con- 1 to tip the concrete, after mixing, from a height of six or 3et into the trench where it is to be deposited. This process ;ted to by the best authorities, on the ground that the heavy :ht portions separate while falling, and that the concrete is •re not uniform throughout its mass. best method is to wheel the concrete in barrows, immedi- fter mixing, to the place where it is to be laid, gently tipping position, and carefully ramming into layers about twelve thick. After each layer has been allowed to set, it should pt clean, wetted, and made rough, by means of a pick, for the yer. ; contractors make the concrete courses the exact width d, keeping up the sides with boards, if the trench is too This is a bad practice; for when the sides of the founcla- ;s are carefully trimmed, and tlie concrete rammed up solidly them, the concrete is less liable to ha crushed and broken it has entirely consolidated. It is therefore desirable that K:ifications for concrete work should require that the whole of the excavation be filled, and that, if the trenches are ted too wide, the extra amount of concrete be furnished at itractor's expense. Tete made with hydraulic lime is sometimes designated as 140 FOUNDATIONS. The pressure allowed on a concrete bed should not exceed one> tenth part of its resistance to crushing. Trautwine gives as the average crushing-strength of concrete forty tons per square foot. Foiiudations in Compressible SoiL— The great diffi- cully mot with in fonuing a iinu bed in compressible soils arises from the nature of the soil, and its yielding in all directions under pressure. (See page 144.) There are several methods which have been successfully em- ployed in soils of this kind. I. When the compressible material is of a moderate depth, the excavation is made to extend to the firm soil beneath, and the fomulation put in, as in firm soils. The principal objection to this method is the expense, which would often be very grea.t. II. A second method is to drive piles through the soft soil into the tlrm soil beneath. The piles are then cut oif at a given level and a timber platform laid upon the top of the piles, which serves as a support for the foundation, and also ties the tops of the piles together. III. A modification of the latter method is to use shorter piles^ which are only driven in the compressible soil. The platform is made to extend over so large an area that the intensity of the press- ure per square foot is within the safe limits for this particular soil. lY. Another modification of the second method consists in using piles of only five or six inches in diameter, and only five or six feet long, and placing them as near together as they can be driven. A platform of timber is tlien placed on the piles, as in the second metho<l. Tht^ object of the short piles is to compress the soil, and make it tirmor. ''This practice is one not to be recommended; its effect bein<i^ usually to pound up the soil, and to bring it into a state which can best Xh', described by comparing it to batter-pudding." * V. Still another method is to surround the site of the work with shccL-piling (flat piles driven close together, so as to fonn a sheet), to prcvi>nt the esca^Mi of the soil, which is then consolidated by driving ]>iles into it at short distunires from each otlier. The piles are then sawn oft' level, and the ground excavated between them for two or three feet, and filled up with concrete: the whole is tlien planked ovt;r to re(!eive the superstructure. The great point to be attended to in building foundations in soils of this kind is to distribute the weight of the structui'e equally 1 Dobeon on Fouiidatloiirt. FOUNDATIONS, 141 over the foundation, wtilcfa will then seLlle In a vertical direction, and cause little Injui-y; wh^'eas any irregular aettlement would rend the work from top to bottom. Planking for Poiinaation Beds.— In erecting buildings □n soft groimd. where a large briiring-siirface ia required, planking may be resorted to with great advantage, provided tbo timber can lie kept from decay. If the ground is wet ami the timber good, there ia little to fear in thia respect; but in a dry aituatlon, or one expoaed to alternations of wet and dry, no dependence can be placed on unprepared timber. There are several methods cm- ployed for the presei-vation of timber, such as kyanlzing oi' creo- Mting: and the timber used for fouiidatlona should be trcaleil by one of these methods. The advantage of timber Is, tliat it will resist a great cross-strain with very triOing flexure; and therefore a wide fooling may l>e ob- tained without any excessive spreailing of the bottom courses of tbe masonry. The best method of employing planking under walls is to cut the stuff into short lengths, which should be placctd acroKS Uie foundation, and tied longitudinally by planking laid to the width of the bottom course of masonry in tlie direction of the length of the wail, and firmly spiked to the bottom planking. Another good method of using planking ia to lay down sleepers on the ground, and fill to their top with cement, and then place tlie planking on the level surface thus formed. For the cross'timbers, four-Inch by six-inch timber, laid flatwise, will answer in ordinary FouiKlations for Cliimiteys. — As examples of tlie foun- dations i'ci|uired for very high chimneys, we quote the following front a treatise on foundations, in the latter part of a work on "Foundations and Foundation Walls," i»y George T. PowelL Fig. 2 represents the l>ase of a cliimiiey erected in IfS Chicago Refining Company, 1.51 feet high, and 12 feet aqm 142 FOUNDATIONS. SooL Tlic bnse, merely two courses of lieavy dlmeiuloD stone, lu shown, is bedded upon the aurface^ravel near the mouth of the rivet, there recently deposited by the lake. The inorUr employeil In the joint between thu stone Is rooting-gr&vel in cement. The an'a of the base is '£>!; square feet, the woight of chimney, iDcluslvu of bnse, 025 tons, giving a pressure of 34 pounds to the square inirh. This foundation provei! to \x; perfecL Fig. 3 represents the base of a chimney ereelcil in 1872 for tliii Hcl'orniick Iteaper Works, Clilcago, which is 160 feet liigh, 14 feet square at the foot, with a round flue of (t feet 8 inches diameter. FiB- 3. The base covers 025 square feet; the weight of the chimney and base is approximately 1100 tons; the pressure upon the ground (liry liard clay) ia therefore 24^ |>ouniis to the square inch. This foundation also proved to be perfect in every respect. Bftitrinif Power of Soils. {Added to A'Mli JtlditioH.j In u imjier publislied in tJiu Ameritmn Arehiteet and BuHdinf JVVjuw, November 3. 188«, hy J'rof. Ira O. Baker, C.R.. on the Hearing I'owcr of Soils, iio sums up the resulta of his discussion in tho following liibli', which t;ivus values which seom to the writer to be both praclieal anil I'vliablu. The remiirks ((blowing the tBl>lc should al.so bo cart^fully cnnsidifred. FOUNDATIONS. r4b Kind of Matbbial. Rock— the hardest— in thick lay^ers, in native bed Hock equal to best ashlar masonry Kock equal to best brick masonrj' Kock equal to poor brick masonry Clay on thick beds, always dry Clay on thick beds, moderately dry m Clay, soft Gravel and course tiand, well cemented Sand, compact and well cctmeuted Sand, clean, dry Quicksund, alluvial noils, etc Bearing power in tons per square foot. Min. Max. 200 25 30 15 20 5 10 4 6 2 4 1 2 8 10 4 6 2 1 4 0.5 1 " Conclusion. — It is well to notice that there are some practical considerations which modjiy the pressure which may safely be put upon the soil. For example, the pressure on the foundation of a tall chimney should be considerably less than that of the low mas- sive foundation of a fireproof vault. In the former case a slight inequality of bearing power, and consequent unequal settling, might endanger the stability of the structure; while in the latter no serious harm would result. The pressure per unit of area should be less for a light structure subject to the passage of heavy loads— as, for example, a railroad viaduct — than for a heavy struct- ure, subject only to a quiescent load, since the shock and jar of the moving load are far more serious than the heavier quiescent load." The following list of actual known weight on different soils will give a very good idea of what has been done in actual practice. Rock. — St. Rollox chimney, poorest kind of sandstone, 2 tons per square foot. Clay. — Chimney, McCormick Reaper Works, Chicago, 1^ tons per square foot on dry, hard clay. Capitol at Albany, N. Y., rests on blue clay containing from GO to 90 per cent, of alumina, the remainder being fino sand, and con- taining 40 per cent, of water on an average. The safe load was taken at 2 tons per square foot. In the case of the Congressional Library at Washington, which rests on "yellow clay mixed with sand," 2^ tons per square foot was taken Tor the safe load, ** Experience in Central Illinois shows that if the foundation is carried down below the action of the frost the clay subsoil will bear 1^ to 2 tons per square foot without ap- preciable settling. " * * In O. Baker, Amerkan Architect, November 8, 1888. 144 FOUNDATIONS. Sand and Gravel. — "In an experiment in Finance, eiean river sand, compacted in a trench, supported 100 tons per sqaare foot. ** The p'.ers of the Cincinnati suspension bridge are founded on a bed of coarse gravel 12 feet below water; the maximum pressure on the gravel is 4 tons per square foot. *'Thc piers of the Brooklyn suspension bridge are founded 44 feet below the bed of the river, upon a layer of sand 2 feet thick resting upon bed-rock ; the maximum pressure is about 5^ tons per square foot. ** At Chicago, sand and gra,vel about 15 feet below the sarfaoe are successfully loaded with 2 to 2.V tons per square foot. '* At Berlin the safe load for sandy soil is generally taken at 2 to 2^ tons per square foot. " The Washington Monument, Washington, D. C, rests upon a bed of very fine sand 2 feet thick. The ordinary pressure on cer- tain parts of the foundation i^eing not far from 11 tons per square foot, which the wind may increase to nearly 14 tons per square foot."* Foundations on Soft, Yielding Soil, BuUt of Steel Seams and Concrete. — On page 141 is described the method of planking for foundations, wliich does very well where the timber is sure to bo always wot, but, if there is any chance of its ever becoming dry, iron or steel beams should be used instead. Steel rails were first used embedded in concrete, but they oflfer, however, comparatively little resistance to deflection, and for this reason, if allowed to project beyond the masonry to any considerable length, the concrete filling is liable to crack, and thus the strength of the foundation become impaired. Steel I-beams, more recently used for this purpose, are found to be superior in every respect. A greater depth can be adopted, the deflection thus reduced to a minimum and a sufficient saving effected to more than compensate for their additional cost per pound. The foundation should be prepart-d (see illustration, p. 146) by first laying ji bed of concrete to a depth of from 4 to 1*3 inches and then placing upon this a row of I-beams at right angles to the face of the wall. In the case of heavy ])iei's, the beams may be crossed in two directions. Their distances apart, from centre to centre, may vary from 9 to 24 inches according to circumstances, i.e,, length of their projection beyond the masonry, thickness of concrete, esti- mated pressure per square foot, etc. They should be plaoed at least far enough apart to permit the introduction of the oonczeto * Ira O. Baker, American Architect, Novonber 8, 18B8. FOUNDATIONa 145 filling and its proper tamping between the beams. Unless the concrete is of unusual thickness, it will not be adyisable to exceed 20 inches spacing, since otherwise the concrete may not be of suffi- cient strength to properly transmit the upward pressure to the beams. The most useful application of this method of founding is in localities where a thin and comparatively compact stratum overlies another of a more yielding nature. By using steel beams in such cases, the requisite spread at the base may be obtained without either penetrating the firm upper stratum or carrying the footing-courses to such a height as to encroach unduly upon the basement-room . MBTHOD OF OALCULATINa THB 8IZI3 AND LENGTH OF THE BEAMS.^ Let L — Weight of wall per lineal foot, in tons. and h = Assumed bearing capacity of ground, per square foot (usually from 1 to 3 tons). Thei;i -r = IF =? Required width of foundation, in feet. w = Width of lowest course of footing stones. p = Projection of beams beyond masonry, in feet. 8 = Spacing of beams centre to centre, in feet. Evidently the size of beams required will depend upon their strength as cantilevers of a lengthy, sustaining the upward reaction, which may be regarded as a uniformly distributed load. Thus ^ & = uniformly distributed load (in tons) on cantilevers, per lineal foot of wall, and ph8 = uniform load in tons, on each beam. The table on the following page gives the safe lengths p for the various sizes and weights of steel beams, for s--l foot and 6 rang- ing from 1 to 5 tons per square foot. For other values of 8 say 15 inches, i. «., 1| i^^t, the table may be used by simply considering b increased in the same ratio as 8 (see example below). As regards the weight of beams, it is advantageous to assign to 8 as great a value as is warranted by the other considerations which obtain. EXAMPLE SHOWING APPLICATION OF TABLE. The weight of a brick wall, together with the load it must sup- port, is 40 tons per lineal foot. The width of the lowest footing- course of masonry is 6 feet. Allowing a pressure of 2 tops per * This and the next page are taken by permiBsion from Carnegie, Phipps & Co.*8 Pocket-book. FOUNDATIONS. Bquare foot od tho foundation, what dse ftnd length of steet I-bemu 18 inches dcnCre to centre will be required ? Am : L -40 ;6 = 2;w = C;a = U. Therefore ir = 40 -^ 3 = 20 feet, the required lei^h ol beams. The projection jj = HSl* - 8) = 7 feet. In order to apply tho table (calculated for « = 1 fool) wc must consider 6 increased in tho same ratio as «, t'.e., 6 = 3 x 1^ =S In the eolumn for 3 tons, we find the length 7 feet to agree with 30 inches I-lieams G4.0 pounds per foot. TABLE OIVINQ SAFE LENGTHS OF FROJECTIOKS p IN FEBT (BSB ILLl'STRATION). FOK I - 1 FOOT AND VALUES OF ft BAITQING FROM 1 TO 5 TONS. Depth lw«tght >. Tos 7i Foot). u 11 7i SO SO \i %\f,i !o:o a 15 7S 15 flO IS 1 « 11 5 , 10.5 S,B ■m Wt li 40 10 ! ai. 6 V li ""a g ' « 1 ! h.o m FOUNDATIONS. 147 The foregoing table applies to sied beams. Values given leased on extreme fibre strains of 16,000 pounds per square inch. Chicago Foundations*'" — The architects and builders of Chicago probabijT have to deal with the most unfavorable condi- tions for securing a good ^foundation for their heavy buildings of any people in the world. 1 he soil under the central part of the city consists of a black loamy clay, which is tolerably firm at the surface, and will sustain a load of from one to three tons per foot, depending upon locality. A few feet below the natural surface of the ground the soil becomes quite soft, growing more and more so the deeper the excavation is carried, and at a depth of from 12 to 18 feet it is so yielding that nothing can be placed upon it with any reliance. Nor is this all. It has been discovered, by many failures in buildings, that there is a broad subterranean layer of soft mud which lies directly across the most heavily built portion of the city, extending under the Post- office, and reaching from the lake to the river, a distance of three- quarters of a mile. The first of the larger structures were built with continuous foundation walls, with wide footings, the width being proportioned tx) the loads bearing upon them. This method, however, did not prove successful, as it was foimd that the wall will settle more than a pier, and the comers of the wall will settle less than the centre. After experiments of one kind and another, it has come to be the accepted practice in Chicago of dividing the foundation into iso- lated piers, the footing of each pier being carefully proportioned according to the load upon it, its position in the building, char- acter of the superstructure, etc., so that all shall settle at exactly the same rate without any crackings or detriment to the super structure. The footings of the piers are built of steel beams and concrete, as described on page 145, except that the beams are often crossed three and four times ; in this way a great spreading is obtained in a small height. In determining the area of the footings, the ground is assumed to be capable of sustaining a safe load of from 1| to 2^ tons per square foot. The loads on the piers of the Board of Trade building vary from 2| to S^ tons per square foot. The size of the footings under the piers and the corners is made less than under the walls, to offset the difference in settlement of the different portions of the building. •^-.^ * 0. H. BlMkall, in American Architect, p. 147, Vol. XXUI. 148 FOUNDATIONS. It is found that a heavy pier will sink proportionally more than a light one, so that the area under the larger piers is made relatively greater than under the smaller ones. Again, it is necessary to take into account the material of which the superstructure is to be built. Thu?, a footing under a brick wall i^ made larger than a footing under a line of iron columns, so that if both footings aro loaded with the same weight, thiit under the columns will settle the most, to allow for the compression in the joints of the mason- work. It is impossible to build heavy buildings on the Chicago w)il without settlement, and the architect must therefore plan his build- ing so that all parts shall settle equally, and this has been success- fully done in many of the largest buildings. In a building where the footings aro proportioned to give a bear- ing weight on the ground of 2+ tons per square foot, it is esti- mated that the building will settle about 4 inches altogether. Piling has been successfully used under several buildings in Chicago, and there seems to bo no reason why it should not be more extensively resorted to. In the construction of the large grain elevators which are seat> tercd through the city the loads are so excessive, reaching as high as six tons per foot, that it would be impracticable to support them on ordinary footings, and piling has been resorted to. The piles are driven a distance of twenty to forty feet down to hard-pan, cap[)ed by wooden sleepers, with heavy wooden cross-beams and solid planking to receive the masonry. CONCBETE FOOTING FOB FOUNDATIONS. 148a OONORSTB FOQ!nNQ> FOR FOUNDATIONS. For the footings of foundations in nearly all kinds of soil where piles are not used, the writer believes a good concrete to be prefer- able to even the best dimension stone, for the reason that it acts as one piece of masonry and not as individual blocks of stone, and if made of sufficient thickness it will possess sufficient transverse strength to span any weak place in the soil beneath, if not of large area. When the best brands of Portland cement are used, the propor- tions may be as follows : One part Portland cement ; 3 parts clean sharp sand ; 5 parts chip stone, in sizes not exceeding 2 x 1^ x 3 inches. Using these pro- portions, one barrel of cement will make from 22 to 26 cubic feet of concrete. The above proportions were used in the concrete for the founda- tions of the Mutual Life Insurance Company's Building, New York City When the cement is not of the best quality, or other cement than Portland cement is used, more cement should be used with the other material. Using a cement made in the West, the author specifies that one part of cement to two of sand and four of broken stone should be used, and the result has been very satisfactory. It will generally be found wise to keep an inspector constantly on the ground while the concrete is being put in, as the temptation to the contractor to economize on the cement is very great. In mixing the concrete, the stone, sand, and cement should be thrown into the mortar box in the order named , and while one man turns on the water two or more men should rapidly and thoroughly work the material back and forth with shovels, when it should be imiuediatelv carried to the trenches. The concrete should be deposited in layers not over six inches thick, and each layer \\ell rammed. If one layer dries before the next is deposited it should be well wet on top, just before depositing the next layer. Care should be exercised to see that the trenches are not dug wider than the desired width of the footings ; and also in mixing the concrete, not to use more water than is necessary to bring the mass to a pudding-like consistency, as otherwise the cement may be washed away. 148^ COST OF CONCRETE. COST OF OONORSTB. The cost of labor in mixing concrete, when the proper facilities are provided, need not exceed three cents a cubic foot, and four cents is a liberal allowance, with wages at two dollars a day. The vunount of materials required to make 100 cubic feet of concrete may be taken as follows : proportion of 1 to 6, 5 bbls. cement (original package) and 4.4 yards of stone and sand ; proportion of 1 to 8, 3.9 bbls. of cement and 4i yards of aggregates. The cost of concrete at the present time in Denver is about thirty cents per cubic foot. The weight of concrete varies from 130 to 140 lbs. per cubic foot, according to the material used, granite aggregates making nat- urally the heaviest concrete. MASONRY WALLS. 149 CHAPTER III. MASONRT TV ALLS. Footingr Courses. — In commencing the foundation walls of a building, it is customary to spread the bottom courses or the masonry considerably beyond the face of the wall, whatever be the character of the foundation bed, unless, perhaps, it be a solid rock bed, in which case the spreading of the walls would be useless. These spread courses are technically known as " footing courses." They answer two important purposes : — :ist, By distributing the weight of the structure over a larger area of bearing-surface, tlie Uability to vertical settlement from the compression of the ground is greatly diminished. 2d, By increasing the area of the base of the wall, they add to its stability, and form a protection against the danger of the work being thrown out of "plumb" by any forces that may act against it. ... Footings, to have any useful effect, must be securely bonded into the body of the work, and have sufficient strength to resist the violent cross-strains to which they are exposed. Footings of Stone Foundations. — As, the lower any stone is placed in a building, the greater the weight it has to sup- port and the risk arising from any defects in the laying and dress- ing of the stone, the footing courses should be of strong stone laid on bed^ with the upper and lower faces dressed true. By laying on. bed is me^nt laying the stone the same way that it lay before quarryin{]j. In la3^ng the footing courses, no back joints should be allowed beyond the face of the upper work, except where the footings are in double courses; and every stone should bond into the body of the work several inches at least. Unless this is attended to, the footings will not receive the weight of the superstructure, and will be useless, as is shown in Fig. 1. In proportion to the weight of the superstructure, the projection of each footing course beyond the one above it must be reduced, or the cross-strain thrown on the projecting portion of the masonry will rend ft from top to bottom^ as shown in Fig. 2. la- boildllig 1st)9e mlMses of work, such as the abutments of 150 MASONRY WALLS. bridges and the like, the proportionate increase of bearing-surface obtained by the footings is very slight, and there is generally great risk of the latter being broken off by the settlement of the body f A' ^ P ] ^,/ EEL ^IL 1 Fig. 1. Fig. 2. of the work, as in Fig. 2. It is therefore usual in these cases to give very little projection to the footing courses, and to bring up the work with a battering-face, or with a succession of very slight offsets, as in Fig. 3. A 'r* /^''//x' Hl"l/ -^ Fig. 3. Footings of undressed rubble built in common mortar should never be used for buildings of any importance, as the compression of the mortar is sure to cause movements in the superstructun*. Jf rubble must be used, it should be laid with cement mortar, £o that the whole will form a solid mass; in which case the size aiul shape of the stone are of little consequence. In general, footing stones should be at least two by three feet on the bottom, and eight inches thick. The Building Laws of the city of New York require that ttie footing under all foundation walls, and under all plejs, columns, posts, or pillars resting on the earth, shall be of stone or concrete. Under a foundation wall the footing must be at least twelve inches wider 1 aan the bottom width of the wall, and under pler% wrtnmnUi MASONRY WALLS. 151 its, or pillars, at least twelve inches wider on all sides than the ;tom width of the piers, columns, posts, or pillars, and not less m eighteen inches in thickness; and, if huilt of stone, the stones ill not he less than two by three feet, and at least eight inches ck. Vll base-stones shall be well bedded, and laid edge to edge; and, Lhe walls are built of isolated piers, then there must be inverted hes, at least twelve inches thick, turned under and between the rs, or two footing courses of large stone, at least ten inches ck in each course. The Boston Building Laws require that the bottom course for all indation walls resting upon the ground shall be at least twelve hes wider than the thickness given for the foundation walls. footings of Brick Foundations. — In building with ck, the special point to be attended to in the footing courses is 1 BRICK ^^. T^^ <5s. -^=i A M BRICK <>^, 'y'yy^' ^^ H%v. %\ E3. y>- 'ssr ''^^ Fig. 4. Fig. 5. keep the back joints as far as possible from the face of tht: rk; and, in ordinary cases, the best plan is to lay the footings in 2 BRICKS 'W^Tm '////"/" =^ y/yy '''/// \\\ll'^ ^^-*-^ '// v/. \\\\ ''//// ;ll///^'^ I //////,, =g'.' y/ '^^///„ Fee. ;le courses; the outside of the work being laid all headers, and course pix>jecting more than one-fourth brick beyond the one >?e ity exo^ in. the case of an eight-inch wail 154 MASONRY WALLS. inches thick below the top floor, and stone walls not less than six- teen inches. The thickness of the walls required by the laws of the cities of Boston, New York, and Denver, Colo., are shown by the tables on pp. 155-157. The Boston Law also contains the following provisions, which form an excellent guide to architects in other localities : Section 38. Vaulted walls shall contain, exclusive of withes, the same amount of material v.a is required for solid walls, and the walls on cither side of the air-space shall be not less than eight inches thick, and shall be securely tied together with ties not more than two feet apart. Section? 39. In reckoning the thickness of walls, no allowance shall be made for ashlar, unless it is eight inches or more thick, in which case the excess over four inches shall be reckoned as part of the thickness of the wall. Ashlar shall be at least four inches thick, and properly held by metal clamps to the backing, or prop- erly bonded to the same. Section 40. External walls may be built in part of iron or steel, and when so built may be of less thickness than is above required for external walls, provided such walls meet the requirements of this act as to strength, and provided that all constructional parts are wholly protected from heat by brick or terra-cotta, or by plastering three-quarters of an inch thick, with iron furring and wiring. First and Second Class Buildings. Section 45. First and second class buildings hereafter bnilt shall have floor bearing supports not over thirty feet apart. These supports may be brick walls, trusses or columns and girders. Such brick walls may be four inches less in thickness than is required by this act for external and party walls of the same height, pro- vided they comply with the provisions of this act as to the strength of materials, but in no case less than twelve inches thick. When trusses are used, the walls upon which they rest shall be at least four inches thicker than is otherwise required by sections thirty-siz and thirty-seven, for every addition of twenty-five feet or part thereof to the length of the truss over thirty feet. Section 46. Second class buildings hereafter buHt shall be so divided by brick partition walls of (ho thickness prescribed for bearing partition walls and carried twelve inches above the roof, that no space inside any such building shall exceed in area tea thousand square feet, and no existing wall in any aeoond MASONRY WALLS. 165 building shall be removed so as to leave an area not so enclosed, of more than ten thousand square feet. Section 47. All walls of a first or second class building meet- ing at an angle shall be united every ten feet of their height, by- anchors made of at least two inches by half an inch wrought iron securely built in to the side or partition walls not less than thirty- six inches, and into the front and rear walls at least one-half the thickness of such walls. The New York Law also provides that the bearing walls of all buildings exceeding one hundred and five feet in depth without a cross wall, or piers or buttresses, shall be increased four inches in thickness for each additional one hundred and five feet in depth or part thereof; also, in case the walls of any building are less than twenty feet apart and less than forty feet in depth, or there are cross walls, or piers or buttresses, which serve to strengthen the walls, the thickness of the interior walls may be reduced in thickness at the judgment of the superintendent of buildings. In comparing the thickness of brick walls in the eastern and western portions of the country, it should be taken into consideration that the eastern brick arc much harder and stronger than those in the west, and that an eight-inch wall in Boston is probably as strong (to resist crushing) as a thirteen-inch wall in Denver, Colo. THIOKNBS8 OF WAIX8 REQUIRZSD IN DENVER, OOI.O. FOR DWELLINGS, TENEMENTS, OR LODGING HOUSES. Outside and Party Walls. Onestorjr, ,. Two stories.. Three stories. Four stories. . Five stories.. Six stories . . . *» , 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 « 2--S "2 a 2 S g 2 2 ^ a 9, 2-2 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 H-l (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 a-o 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 arrow-head, as in Fig. 3. The length of the line, if drawn to a scale of pounds, shows the value of the force in pounds; the direc- tion of the line indicates the direction of the force; the arrow-head shows which way it acts; and the point A denotes the point of application. Thus we have the direction, magnitude, and point of application of the force represented, which is all that we need lo know. Parallelog^ram of Forces, — II. Jf two forces applied at one point, and actiny in the same plane, be represented by two straight lines inclined to each other, their resultant loill be equal to the diagonal qf tlie parallelogram formed on these lines. Thus, if the Hues AB and AC (Fig. 4) represent two forces act- ing on one point. A, and in the same plane, then, to obtain the force which would have the same effect as the two forces, we complete the parallelogram ABDC, and draw the diagonal AD» This line will then represent the result- ant of the two forces. When the two given forces are at right angles to each other, the resultant will, by geometry, be equal to the square root of the sum of the squares of the other two forces. The Triaui^le of Forces. — III. If three forces acting on a point be repre- sented in magnitude and direction by the aides of a triangle taken in order, they icill keep the point in equilibrium. Thus let P, Q, and R (Fig. 5) represent thi"ee forces acting on the point O. Now, if we can draw a triangle like that shown at the right of Fig. 5, whose sides shall be respectively {Murallel to the forces, and shall have thfl^ same relation to each other as do the forces, then the Fig. 4. 160 COMPOSITION OF FORCES. forces will keep the point in equilibrium. If such a triangle cannot be drawn, the forces will be unbalanced, and the point will not be in equilibrium. The Polygon of Forces. — IV. If any nwnher qf forcen actiny at a point can be represented in magnitude xmd direction by the aides of a polygon taken in order, they will be in equilibriwn. This proposition is only the preceding one carried to a greatei extent. Moments* — In considering the stability of structures and the strength of materials, we are often obligexl to take into considera- tion the moments of the forces acting on the structure or piece; and a knowledge of what a moment is, and the properties of moments, is essential to the praper understanding of these subjects. When we speak of the moment of a force, we must have in mind some fixed point about which the moment is taken. The moment of a force about any given point may be defined as the product of the force into the perpendicular distance from the point to the line of action of the force; or, in other words, the moment of a force is the product of the force by the arm with lohich it acVi. Thus if we have a force F (Fig. G), and wish to determine its moment about a point P, we determine the perpen- dicular distance Pa, between the point and the line of action of the force, and multiply it by the force in pounds. For example, if the force F were equal to a weight of 500 pounds, and the distance Pa were 2 inches, then the moment of the force about the point P would be 1000 inch-pounds. The following important propositions relating to forces and moments should be borne in mind in calculating the strength or stability of structures. V. — If any number of parallel forces act on a 1>ody, that the body shall be in eqvilihrimn, the nmn P^ of the forces acting in one direction Fig. 6 P' Pi 4 4 Fig.7 Pi,Pj, and P3. must equal the sum of the forces actr D lug in the opposite direction. Thus if we have tlie parallel forces P\ P*, P®, and P*, acting on the rod AB (Fig. 7), in the opposite direction to the forces Pi, P„ P„ then, if the rod is in equilibrium, the sum of the forces P' , P*, P«, and P», must equal the sum of the loroet COMPOSITION OF FORCES. 161 Fa 1 Fs 4^ 1^ .. . . 1 ' *■ -2 — ^ ^ A ^ S ^ w .n ^. ^ « _J O^ \ Fig. 8 Fi . i. VI. If any nwnber of parallel forces act on a body in opposite directions, then, for the body to he in equilibrium, the sum of the moments tending to turn the body in one direction must equal the sum of the moments tending to turn the body in the opposite direction about any given point. Thus let Fig. 8 represent three parallel forces acting on a rod AB. Then, for the rod to be in equilibrium, the sum of the forces Ft and F3 must be equal to Ft. Also, if we take the end of the rod, A, for our axis, then must the moment of Fj be equal to the siun of the moments of F2 and Fi about that point, because the moment of Fi tends to turn the rod down to the right, and the moments of F^ and F^ tend to turn the rod up to the left, and there should be no more tendency to turn the rod one way than the other. For example, let the forces F^, F^, each be represented by 5, and let the distance ^a be represented by 2, and the distance Ac by 4. The force F, must equal the sum of the forces F3 and Ff, or 10; and its moment must equal the sum of the moments of F^ and Fs. If we take the moments around A, then the moment of F3 = 5 X 2 = 10, and of Fg = 5 X 4 = 20. Their simi equals 30: hence the moment of F| nmst be 30. Divid- ing the moment 30 by the force 10, we have for the arm 3; or the force Fi must act at a distance 3 from A to keep the rod in equilibrium. If we took our moments around b, then the force Fi would have no moment, not having any arm, and so the moment of F2 about 5 must equal the moment of F3 about the same point; or, as in this case the forces are equal, they must both be applied at the same distance from b, showing that b must be halfway between a and c, as was proved before. Tlie Principle of the Lever.— Tills principle is based upon the two pre- ceding prox>osltions, and Is of great im- portance and convenience. VII. Xf three parallel forces acting in one place balance each other, then each ^ force must Ije proportionaX to tJie distance jq between the other two. Thus, if we have a rod AB (Figs. 9a, Ob, and 9c), with three forces, P|, P^, F9, acting QU it» that the rod shall be balanced, we must have the 15 12 Fig. 9 a B Pi 162 COMPOSITION OF FORCES. following relation between the forces and their points of applica- tion; viz., — P, P2 P, or vn ' An ' AC Pi :P^ :Ps ::BC :AB : AC, This is the case of the common lever, anil gives the means of detennining how much a given lever will raise. p Pig.9 b B h Ftg.9o The proportion is also true for any arrangement of the forces (as shown in Figs, a, b, and c), provided, of course, the forces are lettered in the order sho^^Ti in the figures. Example. — Let the distance AC be 6 inches, and the distance CB be 12 inches. If a weight of 500 pounds is applied at the point B, how much will it raise at the other end, and what support will be required at C (Fig. 9b)? Ans, Applying the rule just given, we have the proportion: ^ P:, : P, :: AC : CB, or 500 : (P,) :: 6 : 12. Hence P, = 1000 poiuids; or 500 pounds applied at B will lift 1000 suspended at A. The supporting force at C must, by proposition v., be equal to the sum of the forces Pi and Pj, or 1500 ponnds in this case. Centre of Gravity. — The lines of action of the force of gravity converge towards the centre of the earth; but the distance of the centre of the earth from the bodies which we have occasion to consider, compared with the size of those bodies, is so great, that we may consider the lines of action of the forces as parallel. The number of tin? forces of gravity acting upon a body may be consicU ered as equal to the numbei' of particles composing the body. The centre of (jratlty of a body may be defined *a8 the point through which the resultant of the parallel forces of graTlty, actiiif upon the body, passes in eveiy position of the body. CENTRES OF GRAVITY. 163 If a iKxly be supported at its centre of gravity, and be turned about tliat point, it will remain in equilibrium in all positions. The resultant of the parallel forces of gravity acting upon a body is obviously equal to the weight of the body, and if an equal force be applied, acting in a line passing through the centre of gravity of the body, the body will be in equilibrium. Examples of Centres of Gravity. — Centre of Gravity of Lliies. StraiyfU Lines. — By a line is here meant a material line whose transverse section is veiy small, such as a very fine wire. The centre of gravity of a uniform straight line is at its middle point. This proposition is too evident to require demonstration. The centre of gravity of the perimeter of a triangle is at the centre of the circle inscribed in the lines joining the centres of the sides of the given triangle. Thus, let ABC (Fig. 10) be the given triangle. To find the centre of gravity of its perimeter, find the middle points, D, E, and F, and connect them by straight lines. The centre of the circle inscribed in the triangle formed by these lines will g- be the centre of gravity sought. Symmetrical Lines, — The centre of gravity of lines which are sjrmmetrical with reference to a point will be at that point. Thus the centre of gravity of the circumference of a circle or an ellipse is at the geometrical centre of those figures. The centre of gravity of the perimeter of an equilateral triangle, or of a regular polygon, is at the centre of the inscribed circle. The centre of gravity of the perimeter of a square, rectangle, or parallelogram, is at the intersection of the diagonals of those figures. Centre of Gravity of Surfaces, Definition. — A surface here means a very thin plate or shell. Symmetrical Surfaces, — If a surface can be divided into two symmetrical halves by a line, the centre of gravity will be on that line: if it can be divided by two lines, the centre of gravity will be at their intersection. The centre of gravity of the surface of a circle or an ellipse is at the geometrical centre of the figm-e ; of an equilateral triangle or a regular polygon, it is at the centre of the inscribed circle; of a parallelogram, at the intersection of the diagonals ; of the surface of a sphere, or an ellipsoid of revolution, at the geometrical centre of the body; of the convex surface of a right cylinder at the middle point of the axis of the cylinder. Irregular Figures, — 4^ny figure may be divided into rectangles 164 CENTRES OF GRAVITY. and triangles, and, the centre of gravity of each being found, the centre of gravity of the whole may be determined by treating the centres of gravity of the separate parts as particles whose weights are proportional to the areas of the parts they represent. Triangle, — To find the centre of gravity of a triangle, draw a line from each of two angles to the middle of the side opposite: the intersection of the two lines will give the centre of gravity. QuadrilateraL — To find the centre of gravity of any quadrilat- eral, draw diagonals, and, from the end of each farthest from their intersection, lay ofif, toward the intersection, its shorter segment: the two points thus formed with the point of intersection will form a triangle whose centre of gravity is that of the quadrilatenl. Thus, let Fig. 11 be a quadrilateral whose centre of gravity is sought. Draw the diagonals AD and BC, and from A lay ofif AF= ED, and from B lay off BH = EC. From E draw , P a line to the middle of FH, and from Fa line to the middle of EH. The point of intersection of these two lines is the centre of gravity of the quadri- lateral. This is a method commonly used for finding the centre of gravity of the voussoirs of an arch. Table qf Centres of Gravity. — Let a denote a line drawn f "om the vertex of a figure to the middle point of the base^ and D the distance from the vertex to the cen- tre of gravity. Then In an isosceles triangle D = fa chord* In a segment of a circle 2) = 12 X area 2 X chord m / \ V Segment. In a sector of a circle, the ver- ) 7^ _ « ^^ _ tex being at the centre J ' ^ In a semicircle, vertex being at ) r • * X arc D = 0.4S6R Sector. the centre In a quadrant of a circle D = IB In a semi-ellipse, vertex being ) /) = 426a at the centre ) * ' In a pai-^bola, vertex at intersection of I D=^hi. axis wi* \i curve) ' ' In a cone or pyramid D = }a In a frustum of a cone or pyramid, let h = hei^t of complete cone or pyramid, Ji' = height of f rustiun, and the vertex be at apei of complete cone or pyi*amid; then 1> = a/ku^jJ \ * GBNTRES OF GRAVITY. 165 The oommon centre of gi'avity of two figures or bodies external to esLob. other is found by the following rule: — Multiply the smaller ai'ea or weight by the distance between centres of gravity, and divide the product by the sum of the areas or weights: the quotient will be the distance of the common centre of gravity from the centre of gravity of the larger area. Example. — As an example under the above rule and tables, let us find the common centre of gravity of the semicircle and triangle shown in Fig, 12. We must first find the centres of gravity of the two parts. The centre of gravity of the semicircle is 0.425 R Fig. 12 from A, or 2.975. The centre of gravity of the triangle is i of 8", or 2.666^' from A ; and hence the distance between the centre of gravity is 2.975" + 2.666", or 5.641". 3|X49 The area of the semicircle is approximately — 5 — > ^^^*^ square inches. The area of the triangle is 7 X 8, or 56 square inches. The sum of the areas is 133 square inches. Then, by the above rule, the distance of the common centre of gravity from the centre 66 X 5.641 or* gravity of the semicircle is Too — = 2.37 , or 2.975 — 2.37 = 0.605 inches from A, Centre of Gravity of Heavy Particles. — Centre of Gravity of Two Particles. — Let P be the p^^ weight of a particle at A (Fig. 13), and W | that at C The centre of gravity will be at some point, B, on the line joining A and ^;0 e The point B must be so situated, that if p^ Flo, 13 W the two particles were held together by a stiflf wire, and were supported at 5 by a force equal to the sum of P and W, the two particles would be in equilibrium. The problem then comes under the principle of the lever, and hence we must have the proportion, P+W :P :: AC :BC, or PX^ ^^■" P + W If TT = P, then BC = AB, or the centre of gravity will be half- 166 CENTRES OF GRAVITY. way between the two particles. This problem is of great impor- tance, for it presents itself in many practical examples. Centre of Gravity of Several Heavy Particles. — Let Wj , We, TF3, W4 and Ws (Fig. 14) be the weights of the particles. Join W] and W2 by a straight line, and find their centre of gravity ^ , as in the preceding 'Ws problem. Join A with W3, and find the cen- tre of gravity By which will be the centre of gravity of the three weights W^ , Wfy and W^. Proceed in the same way with each weight, and the last centre of gravity found will be the centre of gravity of all the particles. In both of these cases the Unes joining the particles are supposed to be horizontal lines, or else the horizontal projection of the real straight line which would join the points. Ws Fig. 14 RETAINING WALLS. 1^'^ CHAPTER V. RETAINING VSTALLS. A Retaining^ Wall is a wall for sustaining a pressure of earth, sand, or other filling or backing deposited behind it after it is built, in distinction to a brest or face wall, which is a similar structure for preventing the fall of earth which is in its undis- turbed natural position, but in which a vertical or inclined face has been excavated. Fig. 1 gives an illustration of the two kinds of wall. Retaining* Walls. — A great deal has been written upon the theory of retaining walls, and many theories have been given for computing the thrust which a bank of earth exerts against a re- taining wall, and for determining the form of wall which affords the greatest resistance with the least amount of material. There are so many conditions, however, upon which the thrust exerted by the backing depends, — such as the cohesion of the earth, the dryness of the material, the mode of backing up tlic wall, etc., — that in practice it is impossible to determine tli(» exact thrust which will be exerted against a wall of a given heiji:ht. It is therefore necessary, in designing retaining walls, to be guided by experience rather than by theory. As the theory of retaining walls is so vague and unsatisfactory, wc shall not offer any in this article, but rather give such rules and cautions as have been estab- lished by practice and experience. In designing a retaining wall there are two things to be consid- ered, — the backing and the wall. The tendency <^ tAe hacking to slip is very much less when it is ^^^ BETAINING WALLS. in a dry state tlian when it is filled with wnter, and hence eve pi'M^aution shouliJ be taken to secure good drainage. Besides bi face drainage, tiiere should be openings left iii tlie waJI for Ike | water which may accumulate l:>e1iind it to escape aud run off. The manner in which the material is HUed agaiust the wftll also affects the stability of the baclcings. ff the ground be made irregu- lar, as in Fig. 1 , and the earth weil rammed in layers inclined jVom tlie uatl, tliit pressure will be very trifling, provided that attention be paid to drainage. If, on the other hand, the earth tie tipped, in ttie usual manner, in layers sloping toteardu the wall, the full pi urc of the earth will be exerted against II, and It must be made of correaponding strength. Fig.3 FiB.4 Fig.! The Wall.— lietainingWAlls are generally built with a batter- ing (sloping! face, as this Is the strongest wall tor a given amonnt of material ; and, if the courses are inclined towards the back. Ilia tendency to slide on each other will be overcome, and it will not bs necessary Ut depend upon the adhesion of the mortar. Fig-I FIg.a The importance of making tlie resistance independent of tiw ailhesion of the mortar Is obviously very great; as It WonU other- wise be necessary to delay backing up a n^l until tba iDortar WH 'horoughly set, which might require several uonllni RETAINING WALLS. it™ e Back of tlie Wall shonld bo left Roagli.— In ivork It would be well to let every third or fourth course ^t an inch or two. This increases the frietion of the earth 9t the back, and thus causes tlie resultant of the forces acting d the wall to become nion? nearly vertical, and to fall farther n the base, giving increased stability. Jt also conduces to ;tli not to make each course of uniform lielglit throughout the less of the wall, but to have some of the stones, especially near ick, sufiiciently high to reach up through two or liiree courses, is means the wliole masonry becomes more effectually inter- 1 or bonded tc^etlier as one mass, and less liable to bulge. ere deep freezing occurs, the back of the wall should be sloped rds for threeor four feet belowitstop, aa at OC (Fig. 2), which 1 be quite smooth, so aa to lessen the hold of the frost, and at displacement. i. 3, 4, 5, and 6 show the relative sectional areas of walls of snt shapes that would be required to resist the pressure of a of earth twelve feet high ("Art of Building," E. Dobson, The first three examples are calculated to resist the maxi- thnist of wet earth, while the last shows the modified form y adopted in practice. il's for tbe Tlilckness of tlie Wnll.— As has been . the only practical rules for retaining walls which we have nplrlcal rules based iiixin experience and practice- John C. Trautwiue, C.E., who is considered authority on ?ering subjects, gives the following table in his " Pocket-Book igineers," for the thickness at the base of vertical retaining with a sand-backing deposited In the usual manner. • first cohmm coulains the verLiea) Iieight CD (Pig. 7) of tht^ as compared willi the vertical lieiglil of the wall ; which lal fn' 170 KETAINING WALLS. is assumed to be 1, so that tlie table begins with backing of the same height as the wall. These vertical wails may be battered to any extent not exceeding an inch and a half to a foot, or 1 In 8, without affecting their stability, and without increasing the base. Proportion of Retaining: Walls. f Total height of the earth com- Wall of Good mortar, Wall of pared with the height of the cut Btone rubble, good, dry wall above grouud. in mortar. or brick. rubble. 1 0.35 0.40 0.50 1.1 0.42 0.47 0.57 1.2 0.46 0.51 0.61 1.3 0.40 0.54 0.64 1.4 0.51 0.56 0.66 1.5 0.52 0.67 0.67 1.6 0.54 0.59 0.68 1.7 0.55 0.60 0.70 1.8 0.56 0.61 0.71 2 0.58 0.63 0.78 2.5 0.60 0.65 0.75 3 0.62 0.67 o.n 4 0.63 0.68 0.78 6 0.64 0.69 0.79 Brest Walls (from Dobson's "Art of Building").— Where che ground to be supported is firm, and the strata are honzontal, the office of a brest wall is more to protect tlian to sustain the earth. [t should be borne in mind that a trifling force skilfully applied to onbroken ground will keep in its place a mass of material, which, if once allowed to move, would crush a heavy wall ; and therefore great care should be taken not to expose the newly opened ground to the influence of air and wet for a moment longer than is requisite for sound work, and to avoid leaving the smallest space for motion between the back of the wall and the ground. The strength of a brest wall nuist be projiortionately increase<1 when the strata to be supported inclines towards the wall: where they incline from it, the wall need be little more than a thin facing to protect the ground from disintegration. The preservation of the natural drainage is one of the most im- portant points to be attended to in the erection of brest walls, as upon this their stability in a groat measure depends. Xo rule can be given for the best manner of doing this: it must be a matter for attentive consideration in each particular case. STBEKGTH OF MASOKBY. 171 CHAPTER VI. STRBNGTH OF MASONRY. By the term "strength of masonry " we mean its resistance to a crushing-force, as that is the only force to which masonry should bo subjected. The strength of the different stones and materials used in masonry, as determined by experiment, is given in the following table. (For Architectural Terra-Cotta, see page 186a.) Crushing Resistance of Bricks Stone, and Concretes, {Pressure at right angles to bed.) Pounds per sq. inch. Brick : Common, Maspachnsetts. 1U,000 Common, St. Louis -. 6,417 Common, Wtibhington, D. C 7,870 Paving, Illinois .... 6,000 to 13,000 Granites : Bine, Fox Island, Me 14,875 Gray, Vinal Haven, Me 18,000 to 18,000 Westerly, R. I 15,000 Rockport and Quincy, Mass 17.750 Milford, Conn 22,600 Staten Island, N. Y 22,250 East St. Cloud, Minn 28,000 Gannison, Colo 18,000 Red, Platte Caflon. Colo 14,600 Limestones: Glens Falls, N. Y 11,475 Joliet,Ill •. 12,775 Bedford, Ind 6,000 to 10,000 Salem, Ind 8,625 Red Wing, Minn 23,000 Stillwater, Minn *. 10,750 Sandttones : T)OTche»ter^N.B. {hrovfii) 9,150 Mary's Point, N. B. (fine grain, dark brown) 7,700 Connecticut Brown Stone on lied '. 7,000 to 18,000 LoDgmeadow, Mass. (reddish brown) 7,000 to 14,000 '* " average, for good quality 12,000 Little Falls, N. Y 9,850 Medi na, N. Y 17,000 Potsdam. N. Y. (red) 18,000 to 42,000 Cleveland, Ohio 6,800 North Amherst, Ohio 6,212 Beren, Ohio 8,000 to 10,000 Ilnmmcltitown. I*a 12,810 Fond du Lac, Minn 8,750 Fond du Lac, Wis 6,237 Manitou, Colo, (light red) 6,000 to 11,000 St. Vrain, Colo, (hard laminated). 11,505 3Iarble8 : Lee, Mass 22.900 Rutland, Vt 10,746 Montgomery Co., Pa .' 10,000 Colton.Cal 17,783 Italy 12,156 Flagging : North River, N. Y 13,425 Concrete : Rosendale cement 1, pand and stone 7A, 46 months old 1,.544 Portland cement 1, sand and stone 9, 6 months 2,000 * This stone should not be set on edge. 173 STRENGTH OF MASONRY. The stones in this table are supposed to be on bed, and the height » to be not more than four times the least side. Of the strength of rubble masonry, Professor Rankine states that "the resistance of fjood coursed rubble masonry to crushing is about four-tenths of that of single blocks of the stone it is built with. The resistance of common rubble to crushing is not much greater than that of the mortar which it contains." Stones generally commence to crack or split under about one-half of their crushing-load. Crushing-Height of Brick and Stone. — If we assume the weight of brickwork to be 112 pounds per cubic foot, and that it would crush under 450 pounds per square inch, then a vertical unifonn column 580 feet high would crush at its base under its own weight. Average sandstones at 145 pounds per cubic foot would require a column 5950 feet high to crush it; and average granite at 165 pounds per cubic foot would require a column 10,470 feet high. The Merchants' shot-tower at Baltimore is 246 feet high, and its base sustams a pressure of six tons and a half (of 2240 pounds) per square foot. The base of the granite pier of Saltash Bridge (by Biiinel) of solid masonry to the height of 96 feet, and supporting the ends of two iron spans of 455 feet each, sustains nino tons and a half per squai-e f oot . The highest pier of Rocquef avonr stone aqueduct, Marseilles, is 305 feet, and sustains a pressure at the base of thirteen tons and a half jyar square foot. Worldng-Strengtli of Masonry.— The worlring-streiigth of masonry is generally taken at from one-sixth to one-tenth of the crushing-load for piei's, colunms, etc., and in the case of arches a factor of safety of twenty is often recommended for computing tbe resistance of the voiissoirs to crushing. Mr. Trautwine states that it cannot be considered safe to expose even first-class pressed brickwork in cement to more tlian thirteen or sixteen tons' pressure per square foot, or good hand-moulded brick to more than two-tliirds as nmch. {Seepage 181.) Sheet lead is sometinH^s plac(ul at the joints of stone columns with a view to equalize the pressure, and thus increase the strength of the cohnun. Exi)oriments, however, seem to show that the effect is directly the reverse, and that the column is materiaHy weakened thereby. ' Piers. — Masonry thai is so heavily loaded tliat it Is necessary to proporlion it in regard to its strength to resist crushing, is, as a general rule, in the form of piers, either of brick or Btoue. As 1 Trautwine's Pocket-book, p. 176. STRENGTH OF MASONRY. 1*^3 these pien are often in places where it is desirable tliat they should occupy as little space as possible, they are oflen loaded to the full limit of safety. The material generally used for building piers is brick: block or cut stone is sometimes used, for the sake of appearance; but rubble- work should never be used for piers which are to sustain posts, pillara, or columns. Brick piers more than six feet in height should never bo less than twelve inches square, and should have properly proportioned footing courses of stone not less than a foot thick. The brick in piers should be hard and well burned, and should be laid in cement, or cement mortar at least, and be well wet before being laid, as the strength of a pier depends very much upon the mortar or cement with which it is laid: those piei*s which have to sustain very heavy loads should be built up with the best of the Rosendale cements. The size of the pier should be determined by calculating the greatest lead which it may ever have to sustain, and dividing the load by the safe resistance of one square inch or foot of that kind of masonry to crushing. Example. — In a large storehouse the floors are supported by a girder running lengthwise through the centre of the building. The girders are supported every twelve feet by columns, and the lowest row of columns is supported on brick piers in the basement. The load which may possibly come upon one pier is found to be 65,000 pounds. What should be the size of the pier ? ^iM. The masonry being of good quality, and laid in cement mortar, we will a^ume 12 tons per square foot, or 166 lbs. per square inch (see p. 181), for the safe working load. Dividing 65,000 lbs. by 166, we have 891 square inches for the size of the pier. This would require a pier 20 x 20 inches. It is the custom with many architects to specify bond stones in brick piers (the full size of the section of the pier) every three or four feet in the height of the pier. These bond stones are gener- ally alx)ut foiu" inches thick. The object in using them is to distribute the pressure on the pier equally through the whole mass. Many first-class builders, however, consider that the piers are stronger without the bond stone; and it is the opinion of the writer that a pier will be just as strong if they are not used. Section 3 of the Building Laws of the city of New York requires that every isolated pier less tlian ten superficial feet at the base, and all piers supporting a wall built of rubble-stone or brick, or under any iron beam or arch-girder, or arch on which a wall rests, or lintel supporting a wall, shall, at intervals of not less than thirty inches in height, have built into it a bond stone not less than 174 STRENGTH OF MASONRY. four inches thick, of a diameter each way equal to the diametei of the pier, except that in piers on the street front, above the curb, the bond stone may be four inches less than the pier in diameter. Piers which support colmnns, posts, or pillars, shonld have the top covered by a plate of stone or iron, to distribute the pressure over the whole cross-section of the pier. In Boston, it is required that '*all piera shall be built of good, hard, well-burned brick, and laid in clear cement, and all bricks used in piers shall be of the hardest quality, and be well wet when laid. '* Isolated brick piers under all lintels, girders, iron or other col- umns, shall have a cap-iron at least two inches thick, or a granite cap-stone at least twelve inches thick, the full size of the pier. ^* Piers or columns supporting walls of masonry shall have for a footing course a broad leveller, or levellers, of block stone not less than sixteen inches thick, and with a bearing surface equal in area to the square of the width of the footing course pluB one foot required for a wall of the same thickness and extent as that borne by the pier or colunm." For the Strength of Manonry WallSj see Chap. UL The following tables give the results of some tests on bclckf brick piers, and stoue, made under the direction of the author, in behalf of the Massachusetts Charitable Mechanics Ajbso- ciation. The specimens were tested in the government testingi-macliliie at Watertown, Mass., and great care was exercised to make tlie te~sts as perfect as possible. As the parallel plates between which the brick and stone were crushed are fixed in one position, it is necessary that the specimen tested should have perfectly parallel faces. The bricks which were tested were rubbed on a reyolTing bed until the top and bottom faces were perfectly true and parallel. The preparation of the bricks in this way required a great deal of time and expense; and it was so difficult to prepare some of the hanler bi'ick, that they had to be broken, and only one-half if :he brick prepared at a time. STRENGTH OF MASONRY. 175 TABLE f^howing the UUimaJte and Cracking Strength of the Brick, the Size and Area of Face, Name of Bbiok. Philadelphia Face Brick . . . • • • 41 U Average . (« Cambridge Btiok (Eastern) . «< *( (( Average Boflton Terra-Ck>tU Co.'s Brick, l( CI (I (( (( « It Average New -England Pressed Brick . i( <t «( «« «i <i (« 11 (« Average Size. Whole brick Whole brick Whole brick Half brick . Whole brick Half brick . Half brick . Half brick . Whole brick Whole brick Half brick Half brick Half brick Half brick Area of face in Bq. ins. 33.7 32.2 34.03 10.89 25.77 12.67 13.43 11.46 25.60 28.88 12.95 13.2 13.30 13.45 Commenced to crack under Iba. per sq. inch. Net strength lbs. per sq. inch. 4,303 3,400 2,870 6,062 5,831 5,862 3,527 5,918 3,670 7,760 3,398 3,797 9,825 12,941 11,681 14,296 4,655 12,186 11,518 8,593 3,530 13,839 11,406 9,766 7,880 11,670 3,862 8,180 2,480 4,535 10,270 13,530 13,082 13,085 4,764 12,490 The Philadelphia Brick used in these tests were obtained from a Boston dealer, and were fair samples of what is known in Boston as Philadelphia Face Brick. They were a very soft brick. The Cambridge Brick were the common brick, such as is made around Boston. They are about the same as the Eastern Brick. The Boston Terra-Cotta Company^ a Brick were manufactured of a rather fine clay, and were such as are often used for face brick. The New-England Pressed Brick were hydraulic pressed brick, and were almost as hard as iron. From tests made on the same machine by the United States Gov- ernment in 1884, the average strength of three (M. W. Sands) Cam- bridge, Mass., face brick was 13,925 pounds, and of his common brick, 18,337 pounds per square inch, one brick developing the enor- mous strength of 22,351 pounds per square inch. This was a very bard-burnt brick. Three brick of the Bay State (Mass.) manufacture showed an average strength of 11,400 pounds per square inch. The New England brick are among the hardest and strongest brick in the oonntry, those in many parts of the West not having one-fourth of the strength given above, so that in heavy buildings, 176 STRENGTH OF MASONRY. where the strength of the brick to be used is not known by actaal tests, it is advisable to have the brick tested. Prof. Ira 0. Baker, of the University of Illinois, reported some tests on Illinois brick, made on the 100,000 pounds testing machine at the university, in 1888-89, which gives the crushing strength of soft brick at <574 pounds per square inch, average of three face brick, 3,070 pounds ; and of four paving brick, 9,775 pounds. In nearly all makes of brick it will be found that the face brick are not as strong as the common brick. Tests of the Streni^li of Brick Piers laid with Various Mortars/ — These tests were made for the purpose of testing the strength of brick piers laid up with different cement mortars, as compared with those laid up with ordinary mortar. The brick used in the piers were procured at M W. Sands's brick- yard, Cambridge, Mass., and were good ordinary brick. They were from the same lot as the samples of common brick tested as described. The piers were 8" by 12", and nine coui-ses, or about 224'' high, excepting the first, which was but eight courses high. They were built Nov. 29, 1881, in one of the storehouses at the United-States Arsenal in Watertown, Mass. In order to have the two ends of the piers perfectly parallel surfaces, a coat of about half an inch thick of pure I'ortland cement was put on the top of each pier, and the foot was grouted in the same cement. March 8, 1882, three months and five days later, the tops of the piers were dressed to plane surfaces at right angles to the sides of the piers. On attempting to dress the lower ends of the piers, the cement grout peeled off, and it was necessary to remove it entirely, and put on a layer of cement similar to that on the top of the piers. This was allowed to harden for one month and sixteen days, when the piers were tested. At that time the piers were four months and twenty-six days old. As the piers were built in cold weather, the bricks were not wet. The piers were built by a skilled brick-layer, and the mortars were mixed under his superintendence. ITie tests were made with the government testing-machine at the Arsenal. The following table is arranged so as to sbow the resalfc of these tests, and to afford a ready means of comparison of the strength of brickwork with different mortars. The piers generally failed by cracking longitudinally, and some of the brick were crushed. The 1 The report of these tests was first pablished in the AmBrican Aidiileel^ June 8, 1882. STRENGTH OF MASONBY. 17'( Portland cement used in these tests was known as Brooks, Shoo- bridge ft Co. 'a cement. As the aetaal strength of brick piers is a very important coneid- eration in bnildiog constmetion, the following tests, made by the United States Government at Watertown, Mass.. and contained in tbe rrport of the tests mode on the (iovcrnment testing machine for the year 18B4. are given, as being of much value. Three kinds of brick were reprasent«d in the conatruction of the piers, and mortars of different composition — ranging in strength from lime mortar to neat Portland cement. The piers ranged in cross-section dimensions from H' x 8" to 16" x IS", and in height from 16" to 10 '. The piers were tested at the age of from 18 to 24 months The following table gives the reaiUts obtained, and memoranda regarding the size and character of the piers. « SSS223SSS2 ■3 11 Jiiiiiiiiii ,IIWJI|I.I.I. 1 lii iiilll^ 5 J 1 1 nrl r;::il • il i I I I 180 STRENGTH OF MASONRY. Tests of Mortar Cubes. — The following tests of 6" oabesof mortar were made by the United States Gk)veniment at Watertown, Mass., in the year 1884. Ttie mortar cubes were allowed to season in the open air, a period of fourteen and a half months, whpn they were tested. The age of tlic plaster cube was four months. It should be noticed that, while the cube? of Rosendalc cement and Hme-mortar showed a greater strength than when sand alone was mixed with the cement, with the cubes of Portland cement and lim^-mortar the reverse was the case, differing from the result obtained by the author. This shows the necessity of a number and variety of tests. TABULATED RESULTS, 6" MORTAR CUBES. Crubhino Stbbngth. No. of test. Composition. First crack. Ultimate Btrength persq. in. Weight per CO. ft. 3a Sb Zc 1 part lime, 8 parts sand, ti 4( H lbs. Ibe. 185 119 118 lbs. 118 111 106 4a 4b 4c 1 part Portland cement, 2 parts sand, • • « • 11,600 660 606 888 116 180 115 6a bb 5c 1 part Rofiendale cement, 2 parts sand, (t It It tt tt it tt it 4,600 166 186 148 •111 100 107 6a 6b ec Neat Portland cement, kt it it ti • • • • • • • 96,000 2,678 8,548 4,887 196 189 185 7a lb 7c Neat Ro^endale cement, it it it it 11,000 19,000 19,900 481 615 686 94 90 vr 8a 8b 8c 1 part Portland cement, 2 parte lime-mortar, ^ it ti it i( It it ii it • • • • • • • 804 196 175 100 110 lOi 9a 9b 9c 1 part Rosendole cement, 2 parts lime-mortar,^ ti it ti ti it ti it it PlasttT-of-Paris. • • • • • • 194 198 16-2 1,981 105 1(« 106 74 Workings Stren^h of Masonry.— The faUowing table has been compiled as representing the practice of leading engineen, and the average requirements of recent building laws. The author believes that the values may be relied upon with eafetf , ftod with- 1 Lime-monar, 1 part lime, 8 parui STRENGTH OF MASOKBT. 181 out andae waste of materials. For the size of oast-iron bearing plates on masonry, see page 342&. For strength of architeotnral terra-cotta, see page 186a. SAFE WORKING LOADS FOR MASONRY. Briektoork in isalls or pier», TONS FBB SqUABS VOOT. Bastem. Western. Bed brick in lime mortar 7 6 ** hydraulic lime mortar 6 *' natural cement mortar, 1 to 3 10 8 Arch or pressed brick in lime mortar 8 6 ** •* " natural cement 13 9 ** ** ** Portland cement 15 12^ Piers exceeding in height six times their least dimensions should be increased 4 inches in size for each additional 6 feet. Stonework, (Tons per square foot.) Bubble walls, irregular stones 8 ** coursed, soft stone %^ ** hard stone 5 to 16 Dimension stone, squared in cement : Sandstone and limestone 10 to 20 Granite 20 to 40 Dressed stone, with |-inch dressed joints in cement : Granite 60 Marble or limestone, best 40 Sandstone 30 Height of columns not to exceed eight times least diameter. CoTicrete. Portland cement, 1 to 8 8 to 15 Rosendale cement. 1 to 6 6 to 10 Hydraulic lime, best, 1 to 6 5 HdUow Tile* (Safe loads per square inch of effective bearing parts.) Hard fire-clay tiles 80 lbs. *• ordinary clay tiles 60 ** Porous terra-cotta tiles 40 ** Mortars. (In 4-inch joints, 8 months old, tons per square foot.) Portland oement, 1 to 4 40 Rosendale cement» 1 to 8 18 Lime mort r, beet. . : 8 to 10 Best Portl d cement, 1 to 2. in 4-inch joints for bedding ixonp tea 70 182 8TBENGTH OF MASONRY. Actual Tests of the Crushingr-Stren^h of Sand- stones (made under the direction of the author for the Massachu- setts Charitable Mechanics' Association). — These tests were made with the Government testing mac^hine at the United States Arsenal, Watertown, Mass., and every precaution was taken to secure accu- rate results. Wood's Point (X.B.) Sandstone. — This stone is of about the same color as the Mary's Point stone, but it has a much coarser gmin, and is not very hard. Block No. 1 measured 4.03" x 4.03" X 8". Sectional area 16.2 square inches. Commenced to crack at 50,000 pounds, on the comers, and con- tinued cracking, along the edge^ and at the comers, until it was crushed under 80,000 lbs.' pressure, or 4932 lbs. per square inch. Block No, a measured 4" x 3.«8" X 7.25". SecUonal area 15.02 square inches. This stone commenced to crack under a pressure of 44,000 pounds, and failed under a pressure of 62,500 pounds, or 3976 pounds per square inch. Long MEADOW Stone. — The Bay of Fundy Qiiarryhig Com- pany also quarry a variety of the Longmeadow (Mass.) sandstone, which is a reddish-brown in color. Block No. 1 measured 3.S0" x 3.87" X 7.42". Sectional area 14.71 square inches. This stone showed no cracks whatever until the pressure bad reached 152,000 pounds, when it conmienced to crack at the cor- ners. When the pressure reachetl 200,000 pounds, Uie stone sud- denly flew from the machine in fragments, giving an ultluiato strength of 13,506 pounds per square inch. This stone did not fit into the machine vei7 perfectly. lilock No. f measured 3.30" x 3.07" X 7.5". Sectional area 15.6 square inches. The stone commenced to crack along the edges under a pressure of 47,000 pounds. Under 78,(KX) pouuils, a large piece of the stone split off from the bottom of the block, and under 142,300 pounds* pressure, the stone failed, cracking very badly. UUimale lUmngUi per aqiuirc inch 0121 jjtmnilfi. Bkown Sandstone fhom East Lon«meaj>ow, MAsa. — Quap- ried by Norcross Brothers tfe Taylor of East Longmeadow. This finii works several (juarries, the stone differing in the degree of hard- ness, and a little in color, which is a reddish brown. The different varieties take the name of the quarry from which they oome. Soft Saulsbubt Bbownstone. — This stone is one of the STRENGTH OF MASONRY. 183 softest varieties quarried by this firm, althougli it is about as liard as the ordinary brownstones. The specimens tested were selected by the foreman of the stone-yard without knowing tlie purpose for wliich they were to be used, and were ratlier below the average of this stone in quality. Block No. 1 measured 4" X 4" X 7.58". Area of cross-section 16 square inches. Ultimate strength 141,000 pouuci*, or 8812 j>oi/hc/« per square inch. Stone did not commence to crack until the pressure had reached 130,000 pounds. Block No. t measured 4" X 4" X 7.85". Area of cross-section 10 square inches. Ultimate strength 129,000 pounds, or 8062 pounds per square inch. There were no cracks in the specimen when it was under 100,000 pounds' pressure. Hard Saulsbury Brownstone. — This is one of the hardest and finest of the Longmeadow sandstones. Block No. 1 measured 4.16" x4.1(')" x 8". Sectional area 17.3 square inches. Ultimate strength 233,iKK) pounds, or 13,520 pounds per square inch. Stone did not commence to crack until the T?^*^sure had reachecl 220,000 pounds, almost the crushing-stronjrth. Block No. 2 measured 4.15" X 4.i:>" x S". Sectional area 17.5: square inches. Ultimate strength 2,b2,{M) pounds, or 14,650 i^ownd* per square inch. This specimen did not commence to crack until the pressure had reached 240,000 pounds, or 13,953 pounds to the square inch. The following table is ari-anged to show the sectional area and strength of each specimen, and the average for each variety of ^tone. The cracking-strength, so to speak, of the stone, is of con- sideitible unportance, for, after a stone has commenced to crack, its permanent strength is probably reached ; for, if the load which caused it to crack were allowed to remain on the stone, it would probably in time crush the stone. In testing the blocks, however, some in- equality in the faces of the block might cause one corner to ciack when the stone itself had not commenced to weaken. STKENGTH OF MASONRY. Cell. Q. A. Gillinore, a few yeura ago, tested tbe strength Uiauy vai'ielies of saii<latoiie by (.'I'lisliing Lwo-liiuli cubes. The r suits obtalnetl by bliii laiigtvl fi-otii 4:t50 pounds to 9830 poanda pi square inch. Coniparicig the Btrengtli of the stones lealed by tli author with these values, we find that tlie specimens of liar Sa»lsbiU7 sanilstone had a strengtli far aluve tlie average for smk stones, anil tlie oilier specimens have about the same value* i tliose obtained by Gen. Gllliuore. We should expect, liowever, smaller values from block) 4" X 4 X n" than fioni two-inch cubes; for, as a rule, small spednMnu ( almost any material show a greater strength than large speclmeiu It is interesting to note the mode of fractare of the btocki i browiistone, which was the same for each spechnen. The block fractui'ed by the sides bursting off; and, when takca fram tin ■! STRENGTH OF MASONRY. 185 shine, the specimens had the form of two pyramids, with their aj>exes meeting at the centre, and having for their bases the com- pressed ends of the block. The pyramids were more clearly shown in some specimens than in others, but it was evident that the mode of fracture was the same for all specimens. KruBK Sandstone. — In 1883 the writer superintended the testing of two six-inch cubes of the Kibbe variety of Longmeadow sandstone, quarried by Norcross Brothers. One block withstood a pressure of 12,590 pounds to the square inch before cracking, and the other did not commence to crack until the pressure had reached 12,185 pounds to the square inch. The ultimate strength of the first block was 12,619 pounds, and of the second 12,874 pounds, per square inch. Strength and H^eight of Colorado BalldiniT Stones. The following are the most reliable data obtainable of the strength and weight of the stones most extensively used for building in Colorado. * Med Ghranite from Platte Cafton, Crushing strength per square inch, 14,600 pounds. Weight per cubic foot, 164 pounds. Bed Sandstone from Pike's Peak Quarry, Manitou. Crushing strength, 6.000 pounds per square inch. ** Red Sandstone from Greenlee & Son's quarries, Manitou (adjacent to the Pike's Peak quarries). Crushing weight, 11,000 pounds per square inch on bed. Weight, 140 pounds per cubic foot. * Oray Sandstone from Trinidad, Crushing weight, 10,000 pounds per square inch. Weight, 145 pounds per cubic foot. t Ldva Stone, Curry's Quarry, Douglas County, Crushing {trength, 10,675 pounds per square inch. Weight, 119 pounds per ;abic foot. (Experience has shown that this stone is not suitable for piers, or where any great strength is required, as it cracks very saslly.) * Fort Collins, gray sandstone (laminated), much used for foun- dations. Crushing strength, bed 11,700 pounds, edge 10,700 pounds per square inch Weight, 140 pounds per cubic foot. (One ton of this stone measures just a perch in the wall.) * SI. Vrains, light red sandstone (laminated), excellent stone for foundations. Very hard. From tests made for the Board of Capitol Managers (of Colorado) by State BnglDeer E. 8. Nettleton, in 1885, on two-inch cnbes. t Floiii tests made by Denver Society of Civil Engineers, in 1884, also on two- ndi eobes. ♦• Tested at V. S. Arsenal, Watertown, Mass. 186 STBKNGTH OF MASONRY. Crushing strength, bed 11,505 pounds, edge 17,181 pounds per square inch. Weight, 150 pounds per cubic loot. Eft'ects of Freezing on Mortar.— Both cement and lime- mortar, mixed with salt, have been used in freezing weather with- out any bad clfcjts. (See American Architect. v«)l. xxi., p. 2>G.) Kule for the proportion of salt said to have been used in the works at Woolwich Arsenal: *' Dissolve one pound of rock-salt in eighteen gallons of water when the temperature is at 32 degrees Fahr., and add three ounces of salt for every three degrees of lower tempera- ture." durability of Hoop Iron Bond.— I believe that, embed- ded in liine-mortar at such depth &s to protect it from the air, hoop iron bond is indestructible*. In cement mortar containing salts of potash and soda, I doubt its lasting 1,500 years iinooRoded. — M. C. Meios, May 17, 1887. Grouting.* It is contended by persons having large experience In building that masonry carefully grouted, when the temperature is not lower than 40' Fahr., will give the most efficient result. The following buildings in New York City have grouted walls : Metropolitan Opera House. Produce and Cotton Exchanges. Mortimer and Mills Buildings. Equitable and Mutual Life Insurance Buildings. Standard Oil Building. Astor Building. The Eden Musee. The Navarro Buildings. Manhattan Bank Building. Tho Presbyterian, Gorman, St. Vincent, and Woman's Hospitals. etc ; also, the Mersey Docks and Warehouses at Liverpool, £ng: land, one of the greatest pieces of masonry in the world, have been grouted throughout. It should b(} stated, however, that there arj niiiny engineers and others who do not believe in grouting, claim- ing that there is a tendency of the materials to separate and fona lavers. * See American Architect, July 21, 1S87, p. 11. STRENGTH OF MASONRY. 186a Architectural Terra- Cotta— Weight and Strength. The lightness of terra-cotta, combined with its enormous resist- ing strength, and taken in connection also with its durability and absolute indestructibility by fire, water, frost, etc., renders it specially desirable for use in the construction of all large edifices. Terra-cotta for building purposes, whether plain or ornamental, is generally made of hollow blocks formed with webs inside, so as to give extra strength and keep the work true while drying. This is necessitated because good, well-burned terra-cotta cannot safely be made of more than about 1^ inches in thickness, whereas, when required to bond with brick-work, it must be at least four inches thick. When extra strength is needed, these hollow spaces are filled with concrete or brick- work, which greatly increases the crushing strength of terra-cotta, although alone it is able to bear a very heavy weight. *• A i'Olid block of terra-cotta of one foot cube has borne a crushing strain of 500 tons and over." Some exhaustive experiments, made by the Royal Institute of British Architects, give the following results as the crushing strength of terra-cotta blocks : Crushing wt. per en. ft. 1. Solid block of terra-cotta 523 tons. 2. Hollow block of terra-cotta, unfilled 186 *' 8. Hollow block of terra-cotta, slightly made and unfilled. 80 " Tests of terra-cotta manufactured by the New York Company, which were made at the Stevens Institute of Technology in April, 1888, gave the following results : Crushing wt. Crushing wt. per cu. in. per cu. ft. Terra-cotta block, 2-inch square, red 6,840 lbs. or 492 tons. Terra-cotta block, 2-inch square, buff 6,236 *' '* 449 Terra-cotta block, 2-inch square, gray 5,126 " " 369 ( ( (( Prom these results, the writer would i)lace the safe working strength of terra-cotta blocks in the wall at 5 tons per square foot when unfilled, and 10 tons per square foot when filled solid with brick-work or concrete. The weight of tem-ootta in solid blocks is 122 pounds. When 186* STBENGTH OP MASONRY. made in hollow blocks 1^ inches thick, the weight varies from 6f to 85 pounds per cubic foot, the smaller pieces weighing the most. For pieces 12" x 18" or larger on the face, 70 pounds per cubic fool will probably be a fair average. For the exterior facing of fire-proof buildings, terra-cotta is non considered as the most suitable material available. STABILITY OF PIERS ANP BUTTBESSES. 187 CHAPTER VIT. 8TABII1ITT OF PIERS AND BUTTRESSBS. A PI Kit or buttress may be cousMered stable when the forces acting upon it <lo not cause it to rotate or "tip over," or any course of stones or brick to slide on its bed. When a pier has to sustain only a vertical load, it is evident that the pier must be stable, although it may not liave sufficient strength. It is only when the pier receives a thrust such as tliat from a rafter or an arch, that its stability must be considered. In order to resist rotation, we must have the condition that the moment of the tluiist of the pier about any point in the outside of the pier shall not exceed the moment of the weight of the pier about the same point. To illustrate, let us take the pier shown in Fig. 1. Let us suppose that this pier receives the foot of a rafter, which exerts a thrust T in the direction AB» The tendency of this thrust will be to cause the pier to rotate about the outer edge b 1 ; and the moment of the thrust about this point will be T X a lb I, a lb i being the arm. Now, that the pier shall be just in equilibrium, the moment of the weight of the pier about the same edge must just equal T X a, 6,. The weight of the pier will, of course, act through the centre of gravity of the pier (which in this case is at the centre), and in a vertical direction; and its arm will be 6|<r, or one-half the thickness of the pier. Ilcncc, to liave equilibrium, we must have the equation, T X ttibi = W X bic. Ihit under this condition the least additional thrust, or the crush- ing off of the outer edge, would cause the pier to i-otate: hence, to have the pier in safe equilibrium, we must use some factor of safety. This is generally done by making the moment of the weight c(iual to that of the thiiist when referred to a point in the bottom of the pier, a certain distance in from the outer tnlge. This distance for piers or buttresses should not be less than one- fourtb of tbe thlcknesa of the pier. 18R STABILITY OF PIERS AND BUITBESSEI^. Rcpresontiiig this point in the figui*e by h, we have the neceasuj e(i nation for the safe stability of the pier, TX ab= W X it, t denoting the width of the pier. We cannot from this e<iuation detenuine the dimensions of a pier to resist a given thmst; becanse we have the distance ah, /, and W, all unknown quantities. Hence, we must first guess at i\w size of the pier, then find the length of the line a6, and sec if the moment of the pier is equal to that of the thrust. If it is not, we must guess again. Graphic Method of determining: the Stability of a Pier or Buttress. — When it is desii-ed to determine if a givon pier or buttress is capable of resisting a given thrust, the probleiu can easily be solved graphically in the following manner. TiCt ABCD (Fig. 2) represent a pier which sustains a given thnist T at B. To detennine whether the pier will safely sustain tliis thrust, we pi-oceed as follows. Draw the indefinite line liX in the direction of the thnisL Through the centre of gravity of the pier (which in this case Is at the centre of the pier) (h-aw a vortical line until it intersects tint line of the thrust at c. As a force may be considered to act any- where in its line of direction, we may consider the tlinut and Ih* weiixht to act at the point c: and the resultant of these two forces can l)e obtained by laying off the ihnist T from e on eX, and Ui« wcijrlit of the pier IT, from c on the line cY, lx)th to the same scale (pounds to the inch), completing the parallelogram, and dimw- ing the diagonal, if this diagonal prolonged cats the base of the pier at less than one-fourth of the width of the liase from the outer eilge, the pier will l>e unstable, and its dhneusious must beduuigiad. The stability of a pie7' may be increased by adding 10 U* ira%|l STABILITY OF PIERS AND BUTTRESSES. 186 (by placing some heavy mnterial on top), or by Increasing Its width at the base, by means of " set-offs," as in Fig. 3. Figs. 3 (A and B| show the method of determinit^ the stability ^f a buttress with offsets. The flrst step Is to find the vertical line paaslng throngli the centre of gravity of tlie whole pier. This is best done by dividing the bmtresa up into quadrilaterals, as ABCD, DEFG, and GIIIK (Fig. 3A), finding the centre of gruvity of each quadrilateral by the method of diagonals, anil then measuring the perpendicular distances A'g, A'„ X^, from the diSei^ent centres of gravity to the line KI. Multiply the area of each qitadrilateral by the distance of its centre of gravity from the line KT, and add together the areas and the products. Divide the sum of the latter by the sum of the former, and the result will be the distance of the centre of gravity of the whole buttress from KI. This distance we denote by X^. Example I. — Let the buttress shown in Fig. 3A have Ilia dimensions given l)etween llie cross-marks. Then the arv& of the quadrilaterals and the distances from their centres of gravity to KI would be as follows; 1st area = 35 sq. ft X, = (V.ft-> 1st area x X, - M.2.5 2d area = 23 sq. ft. X, = t'M 2d area x A\ = 67.85 3d ai-ea = 11 sq. ft. X-i = i^.OS 3d area X A',, = 54.45 Total a L, mi s<|. ft. Total t, 155.55 Tlie sum of the moments is IS.^..^; and, dividing this by the total area, we have 2.25 as the distance Xu- Measuiing tliis to the scale of the drawing froqj KI, we have a point through which the Tertic«l line fMlng through the centre of gravity moat pass. 190 STABILITY OF PIERS AND BUTTRESSES. After this line is found, the metho<l of dctemiining the stability of the pier is the same as that given for the pier in Fig. 2. Fig. 3B also illustrates the method. If tlie buttress is more than one foot thick (at right angles to the piano of the paper), the cubic contents of the buttress must be obtained to find the weight. It is easier. howeviT, to divide tlie real thrust by the thickness of the buttress, which i^ivi's the thrust per foot of buttress. J^ine of lleniiitsince, — Dcjinition, The line of resistance or of i>nvs.sures, of a pier or buttress, is a line drawn througli the centre of pressure of each joint. The centre of prenftitre of any joint is the point where the resultant of the forces acting on that portion of the pier above the joint cuts it. The line of pressures, or of resistance, when drawn in a pier, shows liow near the greatest stress on any joint comes to the edges of tliat joint. It can be drawn by tlie following method. Let AIU'I) (Fig. 4) be a pier whose line of I'esistance we wish to draw. First divide the pier in height, into portions two or three feet high, by drawing horizontal lines. It is more convenient to make the i)ortions all of the same size. Proloiii: the line of the thrust, and dr.'iw a verti<'al line through th(» centre of giiivity of the pier, intersertiiig tlu* line of thrust at tin' i)oint (I. From a lay off to a scah' the thrust T and the weights of the different ]M)rtions of the pier, eonnnencing with the w«Mght of the upper portion. Thus, ir, r(*pn»s<*nt8 the wi'ight of the porti(m alM)ve the ir-*t jiiinl : z'*^ represents the w«»ight of tin* .s.M'<)n:l iH>rtion; and so on. Tin* sum (if the /r's will <M|nal the whole \\«'ii:iil of the pier. Ilaviiii: iti-oeeeded thus far, etmipMi* a |)Hralleloffraiii, with 7* and w^ tor it> two sides. Dniw the diagonal, and prolong U. When- it eius iiii> first, joint will Im' a |N>iiil hi the line of mlitAnoe. Draw another parallehtgram, with 7' and Wi + lOg for lU iwotklML Draw the di;igonal intenMH^ting the second Joint at 8. rromud !■ Fi|.4. .Ji. ^.«kX:>- - ai2 -«■*■ •■rill.— :_. u- v' •■ ij- :i: *• uLk.*5i*-- ■•:««*•■■ »^- .I'-'i^ ii-.*««** — ;!. '.■;• — Lij;"'- li 'i ..:: r TIIMMT- f 5^»^ *^*» ^ iiftUii**-— *':i;; T*U* v _i'_ v v. -.y. hulki: V "U^-- uUTtt' »•:. i - ti 1 at. a:T '-• f «•'■" v::. ^ *^^ *■— wfcr- t« jsrjs^tf- * _*•■ *-L ill ;.'zr.?«' lui'-v- ll- i, ... ;^i»*»' - . ^■**^ nil - - * t^ — — - J/ = ^ -*- Z'. = *' ^ ZZ J< w : ji = ^..,: ^i* • » •>...■ 190 STABILITY OF PIERS AND BUTTEES8ES. After this line is found, the method of determining the stability the pier is tlie same as that given for the pier in Fig. 2. Fig. i also illustrates the method. If the buttress is more than one fc thick (at right angles to the plane of the paper), the cubic contei of the buttress must be obtained to find the weight. It is easii however, to divide the real thrust by the thickness of the buttre: which gives the thrust per foot of buttress. J^iiie of Kesistaiice. — Definition, The line of resistan or of pressures, of a pier or buttress, is a line drawn through t centre of pressure of each joint. The centre of pressure of any joint is the point where t resultant of the forces actmg on that portio.n of the pier abo the joint cuts it. The line of pressures, or of resistance, when drawn in a pi< shows how near the greatest stress on any joint comes to the edg of that joint. It can be drawn by the following method. Let ABCB (Fig. 4) be a pier whose line of resistance we wish to draw. First divide the pier in height, into portions two or three feet high, by drawing horizontal lines. It is more convenient to make the portions all of the same size. Prolong the line of the thnist, and draw a vertical line through the centre of gravity of the pier, intersecting the line of thrust at the point a. From a lay off to a scale the thrust T and the weights of the different portions of the pier, coiuniencing with the weight of the upper portion. Thus, to i represents the weight of the portion above the lirst joint; i02 represents the weight of the second portion; and so on. The sum of the to's will equal the whole weight of the pier. Having proceeded thus far, complete a parallelogram, with T u w I for its two sides. Draw the diagonal, and prolong it. Whfi it cuts the first joint will be a point in the line of resistanc Draw another parallelogram, with T and Wi+Wt for iU two aldf Draw the diagonal intersecting the second Joint at % Fit.4. STABILITY OF PIERS AND BUTTRESSES. 191 this way, when the last diagonal will intersect the base in 4. Join the points 1, 2, 3, and 4, and the resulting line will be the line of resistance. We have taken the simplest case as an example; but the same principle is true for any case. Should the line of resistance of a pier at any point approach the outside edge of the joint 'neai-er than one-quarter the width of the joint, the pier should be considered unsafe. As an example embracing all the principles given above, we will take the following case. Example II. — Let Fig. 5 represent the section of a side wall of a church, with a buttress against it. Opposite the buttress, on the inside of the. wall, is a hammer-beam truss, which we will sup- pose exerts an outward thrust on the walls of the church amount- ing to about 9600 pounds. We will further consider that the resultant of the thrust acts at P, and at an angle of 60° with a horizontal. The dimensions of the wall and buttress are given in Fig. 5 A, and the buttress is two feet thick. Question. — Is the buttress sufficient to enable the wall to withstand the thrust of the truss ? The first point to decide is if the line of resistance cuts the joint CD at a safe distance in from C To ascertain this, we must find the centre of gravity of the wall and buttress above the joint CD. We can find this easiest by the method of moments around KM (Fig. 5A), as already explained. The distance Xi is, of course, half the thickness of the wall, or one foot. We next find the centre of gravity of the portion CEFG (Fig. 5A), by the method of diagonals, and, scaling the distance X«, we find it to be 2.95 feet. The area of CEFG = ^g = 10 square feet; and of GIKL = Ax = 26 square feet. Then we have, X,-\ ^, =26 ^, X X, = 26 Xt = 2.95 ^2 = 10 A^X Xi- 29.5 36 36 ) 55.5 Xo = 1.5 Or the centre of gravity is at a distance 1.5 foot from the line ED (Fig. 5). Then on Fig. 5 measure the distance Xn = 1.5 foot, and through the point a dmw a vertical line intersecting the line of the thrust prolongisd at O. Now, if the thrust is 9600 pounds for a buttress two feet thick, it would be half that, or 4800 pounds, lor a buttrass one loot thick. We will call the weight of the IBS STABILITY OK PIEKS AND BUTTHB88BS. masonry of whicb the buttreea itDd wall la built IiJO ponnila pef ciibie foot. Then tbe Ihiiist is equivalenl lo 4800 -^ 150, or Hi cubic fctt of masonry. Laying tbls off lo a scale from O, in the illreotion of the Ihnist ami the area of the masonry, :tl> square feel from on tbe vertical line, completing the rectangle, anil (Irawjug ilin iliaguiial, we find it cnts ibe joint CD al ti, within tbe Uinlls of safety. We must next Qud where theliueof resistaoce cuts tlie base ^fi. First Hml the centre of gravity of tbe wtiole Ognre, wbUib I* fuiMiit by ascertaining the distances X,', X3', in fig. 6A, and making the following computation: 2'.98 A^< = 24 A,' ■K J,' = 11.62 4'.e5 A, '= 12 ^,'X ,lV = i»-« TO 70 1 imw T„' = 2.35 Then from the line EJi (Fig. 0| lay off the disUncv Xt' = 2'.2.'i, and ilraw through il a vi'rtlcal line iutcraeuliug tbe line of tlie tiirust at V. Un this vertical fi-uni O'jucasurc down the whole area 76, and from its extremity lay off tbe thniit T^ U at tl» STABILITY OF I'lKRS AND BUTTRESSES. 193 proper angle. Di*aw the line O'e intersecting the base at c. Tliis is the point where the line of resistance cuts the base; and, as it is at a safe distance in from A, the buttress has sufficient stability. If there were more offsets, we sliould i^roceed in the same way, finding where the line of resistance cuts the joint at the top of each offset. The reason for doing thisis because the line of resist- ance might cut the base at a safe distance from the outer edge, while higher up it might come outside of the buttress, so that the buttiess would be unstable. The method given iu these examples is applicable to piei's of any sliape or material. Should the line of resistance make an angle less tliau 30^ with any joiut, it might cause the stones above Uie joint to slide on their bed. This can be prevented either by dowelliug, or by incliu- lug the joint. It is very seldom in architectural coustruction that such a case would occur, however. 194 THE STABILITY OF ARCHES. CHAPrER vin. THE STABILITT OF ARCHB8. The arch is an arrangeimmt for spanning large openings by means of small blocks of stone, or other material, arranged in a par- ticular way. As a rule, the arch answers the same purpose as tbe beam, but it is widely different in its action and in tbe effect that it has upon tlie appearance of an edifice. A beam exerts merely a vertical force upon its supports, i>ut the arch exerts both a vertical load and an outward thrust. It is this thrust which requires that tho arch sliould be used with caution wliere the abutments are not abundantly large. Before taking up the principles of the • arch, we will define the many terms relating to It. The distance ec (Fig. 1) is called the ftpan of the arch; ai, its rise; b, its crown; its lower boundary Hue, eac, its 9(^t or intrados ; the outer boundary line, pi^l its back or extrados. The terms "soffit" and "back'' are also applied to the entire lower and upper curved surfaces of the whole arch. The ends of the arch, or the sides which are seen, are called its faces. The blocks of which the arch itself is composed are called voussoh'-s : the centre one, K, is called the keystone ; and the lowest ones, .S.S, the tfprintfei'H, In nf*/- weiital arches, or those whose intrados is not a complete semicircle, the springers generally rest upon two stones, as RR, which luive their upper surface cut to receive them: these stones are called skewhdcks. The line connecting the lower edges of the springers is called the sprinyhKj-Une ; the sides of the arcli are called the haunches ; and the load in the triangular space, between the haunches and a horizontal line drawn from the crown, is called the spandrel. The blocks of masonry, or other material, which support two sucrcssive arches, are called piers : the extreme blocks, which, in the Cease of stone bridges, generally support on one side emlMuak- ments of earth, arc calle<l ((hutments. A pier strong enough to withstand the thrust of ^ther areh, should the other fall down, is sometimes called an nhnUneni pier. Resides their own weight, arches usually support a pemnneiit kiad or surcharge of masonry or of earth. In using arches in architectural constructions! thit flom of fki THE STABILITY OF ARCHES. 195 arch is generally governed by the style of the edifice, or by a limited amount of space. The semicircular and segmental forms of arches are the best as regards stability, and ai-e the simplest to construct. Klliptical and three-centred arches are not as strong as circular arches, and should only be used where they can be given all the strength desirable. The strenytJi of an arch depends very much upon the care with which it is built and the quality of the work. In stone arches, special care should be taken to cut and lay the beds of the stones accurately, and to make the bed-joints thin and close, in order that the arch may be strained as little as possible in settling. To insure this, arches are sometimes built dry, grout or liquid mortar being aftei*wards nm into the joints; but the advantage of this method is doubtful. !Brick Arches may be built either of wedge-shaped bricks, moulded or rubbed so as to fit to the radius of the soffit, or of bricks of common shape. The former method is imdoubtedly the l>est, as it enables the bricks to be thoroughly bonded, as in a wall ; but, as it involves considerable expense to make the bricks of the proper shape, this method is very seldom employed. Where bricks of the ordinary shape are used, they are accommodated to the curved figiu-e of the arch by making the bed-joints thinner towards the intrados than towards the extrados; or, if the curvature is sharp, by driving thin pieces of slate into the outer edges of those joints; and different methods are followed for bonding them. The most common way is to build the arch in concentric rings, each lialf a brick thick; that is, to lay the bricks all stretchers, and to depend upon the tenacity of the mortar or cement for the connec- tion of the several rings. This method is deficient in strength, unless the bricks are laid in cement at least as tenacious as them- selves. Another way is to introduce courses of headers at intervals, so as to connect pairs of half-brick rings together. This may be done either by thickening the joints of the outer of a pair of half-brick rings with pieces of slate, so that there shall bo the same number of courses of stretchers in each ring between two courses of headers, or by placing the courses of headers at such distances apart, that between each pair of them there shall be one course of stretchers more in the outer than in the inner ring. The former method is best suited to arches of long radius ; the latter, to those of short radius. Hoop iron laid round the arch, between half-brick rings, as well as longitudinally and radially, is very useful for strengthening brick arches. The bands of hoop iron which traverse the arch radially may also be bent, and prolonged In tbe bed-Joints of the backing and spandrels. 196 THE STAlilLlTY OF ARCHES. By the aid of hoop-iron bond. Sir Marc-lsanibard Brunei half-arcli of bricks laid in strong cemtint, which stood, pr< from its abutment like a bracket, to tlie distance of sixty fe< it was destroyed by its foundation being undermined. The New- York City Building Laws make the following i ments regarding brick arches: — " All arches shall be at least four inches thick. Arches o"\ foot span shall be increased in thickness toward the hauu additions of four inches in thickness of brick. The first ad< thickness shall commence at two and a half feet from the c< tli(^ span ; the second addition, at six and one-lialf feet from I tre of the span ; and the thickness shall be increased then inches for every additional four feet of span towards the liai " The said brick arches shall be laid to a line on the centr a close joint, and the bricks shall be well wet, and the join with cement mortar in proyoitions of not more than two < to one of cement by measure. The arches shall be well | and pinned, or chinked with slate, and keyed." Hide for RadUis of Brick Archett. — A good nUe for the of segmental brick arches over windows, doors, and othe openings, is to make the radius equal to the width of the Oj This gives a good rise to the arch, and makes a pleas- ing proportion to the eye. It is often desirable to span openings in a wall by means of an arch, when there is not sufficient abut- ments to withstand the thrust or kick of the arch. In such a case, the arch can be formed on two cast-iron skewbacks, which are held in place by iron rods, as is shown in Fig. 2. AVhen this is done, it is necessai^ to proportion the size rods to the thrust of the arch. The horizontal thrust of the very nearly represented by the following formula: — load on arch x span Horizontal thrust = y x rise of arch in feet' If two tension rods are used, as is generally the case, the ter of each rod can be detennined by the following mie: — ^. . . , / total load on arch X span Diameter lu iiicl.es = y/ ^ x rise of aich in fee»^ THE STABILITY OF ARCIIES.y 107 If only one rod is used, 8 should be substituted in the place of 16, in the denominator of the above rule; and, if three rods are used, 24 should be used instead of 1(5. Centres for Arches. — A centre is a temporary stnicture, generally of timber, by which the voussoirs of an arch are sup- ported while the arch is being builU It consists of parallel frames or ribs, placed at convenient distances apart, cui'ved on the outside to a line parallel to that of the soffit of the arch, and supporting a series of ti-ansverse planks, upon which the arch stones rest. The most common kind of centre is one which can be lowered, or struck all in one piece, by driving out wedges from below it, so as to remove the support from every point of the arch at once. The centre of an arch should not be struck until the solid part of the backing has been built, and the moi*tar has had time to set and haixlen ; and, when an arch forms one of a series of arches with piers between them, no centre should be struck so as to leave a pier with an arch abutting against one side of it only, imless the pier has sufficient stability to act as an abutment. When possible, the centre of a large brick arch should not be struck for two or three months after the arch is built. Mechanical Principles of the Arch, — In designing an arch, the fii-st question to be settled is the form of the arch; and in regard to this there is generally but little choice. Where the abut- ments are abundantly large, the segmental arch is the strongest fonn ; but, where it is desired to make the abutments of the arch as light as possible, a pointed or semicircular arch should be used. Depth of Keystone. — Having decided upon the form of the arch, the depth of the arch-ring must next be decided. This is generally determined by computing the required depth of keystone, and making the whole ring of the same or a little larger depth. In considering the strength of an arch, the depth of the keystone is considered to be only the distance from the exti-ados to the intra- dos of the arch; and if the keystone projects above the arch-ring, as in Fig. 1, the projection is considered as a part of the load on the arch. There are several rules for determining the depth of the key- r.tone, but all are empirical; and they differ so greatly that it is <lifficidt to recommend any particular one. Professor Rankine's Itule is often quoted, and is probably true enough for most arches. It applies to both circular and elliptical arches, and is as follows: — Rankine's Rule. — For the depth of the keystone, take a mean proportional between the inside radius at the crown, and 0.12 of a foot for a single arch, and 0.17 of a foot for an arch form- ing one of a series. Or, if represented by a formula, •Mi) THE STABILITY OP AECHE8. Bnt, if we sliouM compute the stability of a •eraidreular ardi of 20 foot span, and 1.3 foot depth of keystone, we should find thai the arch was vei^ unstablp; hen^e, in this case, we must throw tlw rule aside, and go by our own judgment. In the opinion of the autlior, such an arcli should have at least 2i feet depth of ucb- Ttng, and we wiil try the stability of the arch with that thickness. In ali calculations on tlie arch, it is customary to conaltler tlie an'U to be one foot thick at rightangles toltsface; for it is evident, thai, if an arch one foot thick is stable, any utmiberof arches of the same fliiiiensioiis built alongside of it would be stable. Graplilc Solution of tlie Stalilllty of tlie Arcli.— Tlie most convenient luctbod of detennlning the stability of the arch is by the graphic mutliod, as it is called. 1st Stbi'. — Draw one-half the arch to as large a scale as con- venient, and divide it up Into voussoirs of i!qual size. In this exaniiile, shown In Fig. '-i. we have divided the arch-ring into ten equal voussolrs. (It is not necessary that these should be the actttal voussolrs of which the arch is built. ) The next step Is to And the area of each voussolr. Where the arch-rfi^ Is divided into voussoirs of equal size, this Is easiest done tiy computing th« ana of the arch-ring, and dividing by the number of voussoira. Fls.3 Ridi' for 'W'li of •iiif-hiiif vf urdi-rim; is as follows: — Area in square feet = 0.7854 X (outside radius squared — itaW.c radius squared). In this example the wholi' area equals 0.78-Vl X J12.5* — Id*) = 44.2 s<|iiare feet. As tiiere are ten equal voussoira, the area of «*ch vonssilir is 4.4 square feet. Having drawn out one-half of the arch-ring, we divide eack Joint into tliree equal parts; and from the point A (Fig. 8] we lay off to a scale the area of each voussoir, one below the ot' THE STABILITY OF ARCHES. 201 with the top voussoir. The whole length of the line AE will equal the whole area drawn to same scale. The next step is to find the yertical line passing through the centre of gravity of the whole arch-ring. To do this, it is first necessary to draw vertical lines through the centre of gravity of each voussoir. The centre of gravity of one voussoir may be found by the method of diagonals, as in the second voussoir from the top (Fig. 3). Having the centre of gravity of one voussoir, the centres of gravity of the others can easily be obtained from it. Next, from A and E (Fig. 3) draw lines at 4b^ with AE, inter- secting at O. Draw 01, 02, 03, etc. Then, where AO intersects the first vertical line at a, draw a line parallel to 01, intersecting the second vertical at b. Draw 6c parallel to 02, cd parallel to 03, and so on to kn parallel to OlO: prolong this line downward until it intersects AO, prolonged at D. Then a vertical line drawn through 1) will pass through the centre of gravity of the arch-ring. 2i) Step. — Draw a horizontal line through A (the upper part of the middle third), and a vertical line through D; the two lines intersecting at C (Fig. 3). Now, that the arch shall be stable, it is considered necessary that it shall be possible to draw a line of resistance of the arch within the middle third. We will, then, first assume that the line of resistance shall act at A, and come out at B'. Then draw the line CB, and a horizontal line opposite the point 10, between Q and P. This horizontal line represents the hori- zontal thrust at the crown. Draw AP equal to QP, and the lines PI, P2, P3, etc. Then, from the point where AC prolonged intersects the first vertical, draw a line to the second vertical, parallel to PI ; from this point a line to the third vertical, parallel to P2 ; and so on. The last line should pass through B. If these lines, which we will call the line of resistance, all lie within the middle third, the arch may bo considered to be stable. Should the line of resistance pass outside of the arch-ring, the arch should be considered unstable. In Fig. 3 this line does not all lie in the middle third, and we nuist see if a line of resistance can yet be drawn within that limit. 2i) Triai.. — The line of resistance in Fig. 3 passes farthest from the middle third at the seventh joint from the top; and we will next pass a line of resistance through A and where the lower line of the middle third cuts the seventh joint, or at B (Fig. 4). To do this, we must prolong the line <jh, parallel to 07 (Fig. 4), until it intersects AO. In this case it intersects it at O; but this 18 merely a coincidence; it would not always do so. Through O draw a vertical intersecting PA prolonged at C. Draw a line 303 THE STABILITY OF ARCHES. through C &nd D, and the horizontal line p^, oppoalte the point 7: this line represents the new horizonUkl thrust H,. Disw AP = pQ, and the lines PI, P2, etc.; then draw the line of resistaniK )is before. It should pass through D if drawn correctly. This lime we aee that the line of reslatance Ilea within the middle third, except jiist a short distance at the springing; and hence we nw} consider the arch stable. If it had gone outside the middle third this time, to any great extent, we should have considered the anHi unatable. The above Is the method of determining the stability of M unloaded semicircular arch. Such a case very seldom occurs In practice; but it is a good example to Illustrate the method, whidi applies to all other cases, with a little difference in the method of determining the centre of gravity of loadod arches. FiB.4 Example II. — Loaded or awcharf/ed semicircular areh. We will take the same arch as in Example L, and snppoM It to l>e loaded with a wall of masonry of the same thickness and welgbt per square foot as tliat of the arch-ring ; the horizontal snrtece of rhc wall being 3 feet C inches above the arch-ring at the crown. 1st Stei-. — Find centre qfgraHty, Commencing at Ibe crown, divide the load and aFch-rlng Into strips two feet wide, making the last strip the width of the areb- ring at tlie springing. Then draw the joints as shown In Hg, G. Measure with the scale the length of each vertical line, Aa, Bb, etc. ; then the area of Aalili Is equal to llie length of An + Bb, M the distance between them is Just two feet. The area of ffKk li, of course, FfX width of areh-ring. In this case, the areas of the slices are as shown by the Ognnt on their faces (Fig. 5}. Now <]lvlde the areh-ring into thirds, and from the top of tba middle thin<, at It, lay oS in succession, to a iHmla, tbe ntut td THE STABILITY OF ARCHB8. SOB iKcefl, commencing with the first slice (ram the crown, AaBb. m areas, when measured off, wilt be represented by the line 2, $ ... B (Pig. 5). From the extremities of this liile, if and 6, V lines at 45° with a vertical, intersecting at O. B>om O draw t to 1, 2, 3, 4, 5, and 6. Next, draw a vertical line through the re of each slice (these lines, in Fig. 5, are nnmlKred 1, 2, 3, I. From the point in which the line RO intersects vertical 1 , t a line paraJle) to 01, lo the line 2. From this point draw a to vertical 3, parallel to 02, and so on. The line parallel to will intersect vertical 6 at F. Then through F draw a line owards at 4^°, iniersecting OB at X. A vertical Hue drawn ngb X will pass through the ceutre of gravity of the arch-rlog its load. I Step. — To find the thnat at thecrojnnand at the i>pringing. ) find the thrust at the crown, draw a vertical line through .V, a horizontal line through B, intersecting at V, Now, the weight ■ch and load, and the resultant thrust of arch, must act throi^h point. We will also make the condition that the thrust shall through Q, the outer edge of the middle third. Then the at of the arch must act in the line VQ. Opposite 6, on the ical line throi^h B, draw a horizontal line IT, between KA' V<i. This horizontal tine represents a horizontal thrust at B, •h would cause the resultant thrust of the arch to pass through Now draw the horizontal line BP, equal in length to H, and I P draw lines 1, 2, 3 ... U. The line P6 represents the thrust be Mcb at Uie springing. lie amouut In cubic feet of masonry be detennined by measuring its length to the proper scale. 204 THE STABILITY OF ARCHES. 3d Step. — To draw the line of resistance. The lines PI, P2, P3, etc., represent the magnitude and dirae- tion of the thrust at each joint of the arch. Thus PI represents the thrust of the first voussoir and its load ; P2, that of the flret two voussoirs and their loads; and so on. Then from the point a', where the line BP, prolonged, intersects the vertical line 1, draw a line a7/ parallel to PI; from 6', on 2, draw a line 6V parallel to P2, and so on. The last line should pass through Q, and be parallel to P6. Now, if we connect the points where the lines a'6', 6V, etc., cnk the joints of the arch, we shall have a broken line, which is known as the line of resistance of the arch. If this line lies within the middle third of the arch, then we conclude that the arch is stable. If the line of resistance goes far outside of the middle, we must see if it be possible to draw another line' of resistance within the mid- dle third; and if, after a trial, we find that it is not possible, we must conclude that the arch is not safe, or unstable. In the example which we have just been discussing, the line of resistance goes a little outside of the middle third; but it is very probable that on a second trial we should find that a line of resist- ance passed through R and Q' would lie almost entirely within the middle third. . The method of drawing the second line of resistance was explaineil under Example I. ; and, as the same method applies to all cases, we will not repeat it. The method given for Example II. would apply equally well for a semi-elliptical arch. Example 111. — Segmental archy with load (Fi^ 6). 1st Step. — To determine the centre ofgravify. In this case we proceed, the same as in the latter, to divide the arch-ring and its load into vertical slices two feet wide, and compute the area of the slices by measuring the length of the vertical lines An, Bh, etc. Having computed the areas of the slices^ we lay them off in order from R, to a convenient scale, and then proceed exactly as in Example II., the remaining steps detenAinlng the tlirust; and the lines of resistance are also the same as given under Example 11. In a flat segmental arch, there is practically no need of dividing the arch-ring into voussoirs by joints radiating from a centre, but to consider the joints to be vertical. Of course, when built, they must be made to radiate. Fig. 6 shows the computation for an arch of 40-loot flpan, and with a load 13i feet high at the centre. The depth of the arch- ring is 2 feet inches. It will be seen, that the curve of pres as lies a iralj irlllifai - TIiE STABILITY OF ARCHES. 305 iddle third; uid hence the arch is abundaatlj safe, or stable, ■tild be remarked, that the line of resIstaDce in a segmental should be drawn through the toteer edge of the middle third springing. lii be noticed that the horizontal thrust, and ttie thrust T, springing, are very great as compared wiih those in a seml- Lr arch; and hence, aJthough the segmental arch Is the er of the two, it requires much heavier abutments, se three examples serve to show tlic method of determining tUlity and thrust of any arch sucli as is nseA In building. 20(1 RESISTANCE TO TENSION. CHAPTER IX. RESISTANCE TO TENSION. OR THE STRENGTH OF TIE-BOD8, BARS, ROPES, AND CHAINS. The resistance which any material offers to being pulled apart is due to the tenacity of its fibres, or the cohesion of the particles of which it is composed. It is evident that the amount of resistance to tension which any cross-section of a body will exert depends only upon the tenacity of its fibres, or the cohesion of its particles, and upon the number of fibres, or particles, in the cross-section. As the number of the fibres, or particles, in the section, is pro- portional to the area, the strength of any piece of material must be as the area of its cross-section; and hence, if we know the tenacity of the material per square inch of cross-section, we can obtain the total strength by multiplying it by the area of the section in inches. The tenacity of different building-materials per square inch hM been found by pulling apart a bar of the material of known dimen- sions, and dividing the breaking-force by the area of the croti- section of the bar. Table I. gives the average values for the tenacity of building- materials, as determined by the most reliable experiments. Knowing the tenacity of one square inch of the material, all that is necessary to determine the tenacity of a piece of any uniform size is to multiply the area of its cross-section, in square inches, by the number in the table opposite the name of the material. Tliii would give the weight that would just break the piece; but, as what we wish is the safe load, we must divide the result by a factor of safety. Most engineers advise using a factor of safety of five f6r a (lead load, although the New-York City and also the Boston Building Laws require a factor of six. Denoting the factor of safety by Sf and the tenacity by T, we iKive as a rule. For a rectangular bar, breadth x depth XT Safe load = ^;7-^^^ (1) RESISTANCE TO TENSIOlf. For a round bar, „ , , , 0.1854 X diameter squared x T Safe load = ~ -g — — (2J ExAMPLBl. — Wliat is the safe load for a tie-bar of wUite pine B b; 6 inches ? Ans. Here the breadtb and depth both equal G inches, T — 7000, and we will let tf = 5; then. 20tf X RESISTANCE TO TENSION. y e size of the bar is desired, we have, iS X load The breadth = g^^j^^^ (3) For a round bar, _. -S X load Diameter squared = q '^054 v T ^^' Example II. — It is desired to suspend 20,000 pounds from a round rod of wrought-iron : what shall be the diameter of the rod to carry the weight in safety ? Ans, In this case T = 50,000; and taking 8 at 5, we have 5X20000 Diameter squared = 0.7854 x 50000 = ^-M. The square root of this is 1.6 or 1§ inches nearly: therefore the diameter of the rod should be If inches. Tensile Strength and Qualities of SteeL The elastic limit of steel should not be less than 40,000 poonds per square inch for high grade steel, 36,000 pounds for medium steel, and 30,000 pounds for solt steel. The ultimate tensile strength of high grade steel should range between 70,000 and 80,000 pounds per square inch ; of mediom, between 00,000 and 70,000 ; and of soft steel, between 52.000 and 60,000 pounds per square inch. The elongation in a length of 8 inches should be not less than 18 per cent, for liigh grade steel, 23 per cent, for medium, and 25 per cent, for soft stcol. The reduction of area at point of fracture should be not less than 35 per cent, of tho original area. Jligh grade steel i85 per cent, carbon) should be used for com- pression, bolsters, bearing-plates, pins, and rollers. Medium steel (1j per cent, carbon) should be osed for tension members, floor system, laterals, bracing, and, unless high gnde steel is specified, should be used for all steel members except rivets. Soft steel (11 or 12 j)er cent, carbon) should be nsed in rivets only, and should bo tested by actually making up into rivets, riveting two plates together, and upon being nicked and cut cmt should show a good, tough, silky structure, with no crystalline appeannoe. Rivet steel should not have over 0.15 per cent, oaifoon. Steel made by the Bessemer process shonld not re over 0.06 per cent, of phosphorus, and open hearth steel e ow -^ cf 1 RESISTANCE TO TENSION. 209 per cent. The amount of phosphorus allowable should always be stated in the specitications, as this determines the price of the pig iron required to make the steel. About 0.04 per cent, of sulphur is allowable, and sometimes more.* The Working Streiig^h of steel in bi-idges is generally taken at 12,000 pounds per square inch, and in roof trusses, and struct- ures sustaining a steady load, at 15,000 pounds per square inch ; or, in a general way, the strength of steel is generally taken at 20 per cent, over that allowable for wrought iron under the same conditions. standard spxsoifioation, adopted by bridgb- buhiDErs, for material and workmanship of iron and steel structures. quality of materials. Wn OUGHT Ibon. Character and Finisli. — I. All wrought iron must be tough, ductile, fibrous, and of uniform quality for each class, straight, smooth, free from cinder pockets or injurious flaws, buckles, blisters, or gracks. As the thickness of bars approaches the maximum that the rolls will produce, the same perfection of finish will not be required as in thinner ones. 2. No specific process or provision of manufacture will be de- manded, provided the materia] fulfils the requirements of this specification. Standard Test Piece. — 3. The tensile strength, limit of elasticity and ductility, shall be determined from a standard test piece, not less than one quarter inch in thickness, cut from the full- size bar, and planed or turned parallel ; if the cross-section is reduced, the tangent between shoulders shall be at least twelve times its shortest dimension, and the area of minimum cross-sec- tion in either case shall be not less than one-quarter of a square inch and not more than one square inch. Whenever practicable, two opposite sides of the piece are to be left as they come from the roils, but the finish of opposite sides must be the same in this respect. A full-size bar, when not exceeding the above limitations, may be used as its own test piece. In determining the ductility the elongation shall be measured, after breaking, on an original length the nearest multiple of a qinirttT inch to ten times the shortest dimension of the test piece, in which length must occur the * JTaioeB BUobo, before the Civil Engineers' Club of Cleveland. 210 RESISTANCE TO TENSION. curve of reduction from stretch on both sides of the point o< frut- ure, but in no case on a shorter length than five inches. Tension Iron for Open Trusses. — 1. Ail iron to be used in the tensile members of open trusses, laterals, pins and bolts, ex- cept plate iron over eight inches wide and shaped iron, must show by the standard test piece a tensile strength in pounds per square inch of : f-rt rxr^n. 7,000 X arca of original bar , ,, . . , . 52,000 r^ — , i- ^.—r-a- (a^ ^ inches), circumference of onginal bar with an elastic limit not less ttian one-half the strength given by this formula, and an elongation of twenty per cent. Plate Iron. — .*). Plate iron 24 inches wide and under, and more than 8 inches wide, must show by the standard test pieces a tensile strength of 4d,C00 pounds per square inch, with an elastic limit not less than 26,000 pounds per square inch, and an elonga- tion of not less than 1 2 [)er cent. All plates over 24 inches in width must have a tensile strength not less than 46,0CO pounds per sqoue inch with an elastic limit not less than 26,000 pounds per sqoue inch. Plates from 24 inches to 86 inches in width must have An elongation of not less than 10 per cent. ; those from 86 inches to 4B inches in width, 8 per cent. ; over 48 inches in width, 5 per cent. Shaped Iron. — 6. All shaped iron and other iron not herein- before . specified must show by the standard test pieces a tensile strength in pounds per square inch of : 7.000 X area of original bar 50,000- circumference of original bar* with an elastic limit of not less than one-half the strength given by this formula, and an elongation of 15 per cent, for bars fifo- eighths of an inch and less in thickness, and of 12 per cent, lor bars of greater thickness. Hot Bending. — 7. All plates, angles, etc., which are to be bent hot, in th(> manufacture must, in addition to the above rs- quirements, be capable of bending sharply to a right angle at a working heat without sign of fracture. Rivet Iron.— 8. All rivet iron must be tough and soft^ and pieces of the full diameter of the rivet must be capable of bending cold until the sides are in close contact without sign of fracture on the convoif side of the curve. Bending Tests. — 9. All iron specified in claase 4 most bend cold, 180 degrees, without sign of fracture, to a oorve the innflr radius of which equals the thickness of the pieoa tested- - RESISTANCE TO TENSION. 211 10. Specimens of full thickness cut from plate iron, or from the flanges or webs of shaped iron, must stand bending cold, through 90 degrees, to a curve the inner radius of which is one and a half times its thickness, without sign of fracture. Niiiiiber of Test Pieces.— 1 1 . For each contract four stand- ard test pieces and one additional for each 50,000 pounds of wrought iron will, if required, be furnished and tested by the contractor without charge, and if any additional tests arc required by the pur- chas'ir, they will be made for him at the rate of $5J!0 each ; or, if the contractor desires additional tests, they shall be made at his own expense, under the supervision of tlie purchaser, the quality of the material to be determined by the result of all the tests in the manner set forth in the following clause. 12. The respective requirements stated are for an average of the tests for each, and the lot of bars or plates from which samples were selected shall be accepted if the tests give such average results ; but, if any test piece gives results more than 4 per cent, below said requirements, the particular bar from which it was taken may be rejected, but such tests shall be included in making the average. If any test piece has a manifest flaw, its test shall not be considered. For each bar thus giving results more than 4 per cent, belov/ the re- quirements, tests from two additional bars shall be fumishe<l by the contractor without charge, and if in a total of not more than ten tests, two bars (or, for a larger number of tests, a proportion- ately greater number of bars) show results more than 4 pier cent, below the requirements, it shall be cause for rejecting the lot from which the sample bars were taken. Such lots shall not exceed 20 tons in weight, and bars of a single pattern, plates rolled in univer- sal mill or in grooves, and sheared plates shall each constitute a separate lot. Time of Inspection. — 13. The inspection and tests of the material will be made promptly on its being rolled, and the quality determined before it leaves the rolling-mill. All necessary facili- ties for this purpose shall be afforded by the manufacturer ; but, if the inspector is not present to make the necessary tests, after due notice given him, then the contractor shall proceed to make such number of tests on the iron then being rolled as may have been agreed upon ; or, in the absence of any special agreement, the num- ber provided for in clause 11, and the quality of such material shall be determined thereby. Variation of Weiglit. — 14. A variation in cross-section or weight of rolled material of more than 2^ per cent, from that speci- fl€d nwy be catiBe for rejection. 212 liESISTAKOK TO TENSION. Steel. 15. No specific process or provision of manufMetare will be de- manded, ])rovided tl^c material fullils the regniremg^ts of this specitication. Test Bars.— IG. From three seiiarate ingots of each casta round sample bar, not less than three-quarters of ivn inch in diame- ter, and having a length not less than twelyo diameters between jaws of testing machine, shall be furnished and tested by the manu- facturer without charge. These bars are to be truly round, and shall be linished at a uniform heat, and arranged to cool onifonnljf, and fro:n these test pieces alone, the quality of the material ahaU be determined as follows : Tensile Tests.— 17. All the above described test baramut have a tensile strength within 4,000 pounds per square inch of that specified, an elastic limit not less than one-half of the tenaile strength of the test bar, a percentage of elongation not leas than 1,200,000 -f- the tensile strength in pounds per square inch, and a percentage of reduction of area not less than2,40O,O0O -f- thetensQe strength in pounds per square inch. In determining tbe ductUitj the elongation shall bo measured after breaking on an original length of ten times the shortest dimension of the test piece, ia wliicli lengt h must occur the curve of reduction from stretch on both sidn of the point of fracture. Finish and Reduction of Area on Finished Ban.^ IS. Finished bars must be free from injurious flaws or cracks and must have c workmanlike finii^h, and round or square test pieoee cut therefrom when pulled asunder shall have reduction of area at the point of fracture as above specified. [Number of Test Pieces.— 19. For each contract foor.snch tests respectively for reduction of area and for bending, and one additional of each for eax;h 5 J,()()0 pounds of steel will, if zeqoired, be made by the contractor witliout charge ; and if the porohaaeris not satisfied that the I'eduction of area test correctly indicates the effect of the heating and rolling, such additional tests for tenaik) strength, limit of elasticity, and ductility, as ho may desire, will bo made for him on test pieces confomiing to the provisions of daoso 8, at the rate of $5.00 each, or, if the contractor desires additional tests, he may make them at his own expense, under tho saperviuon of the purchaser, the quality of the material to be determined bj the result of all the tests in the manner set forth in the fbUowing clause. 20. Except for tensile strength, the respective : ijpiinaiinli BESLSTANCE TO TENSION. 213 stated &re for an average of the tests for each, and the lot of bars or plates from which samples were selected shall be accepted if the tests give such average results ; but, if any test piece gives results more than 4 per cent, below said requirements, the particuhir bar from which it was taken maybe rejected, but such tests shall be in- cluded in making the average. If any test piece has a manifest flaw, its U^st shall not be considered. For each bar thus giving results more than 4 per cent, below the requirements, tests from two additional bars chall be furnished by the contractor without charge, and if in a total of not more than ten tests, two bars (or. for a larger number of tests, a proportionately greater number of bars) show results more than 4 per cent, below the requirements, it shall bo cause for rejecting the lot from which the sample bars were taken. Such lot shall not exceed 20 tons in weight, and bars of a single pattern, plates rolled in universal mill or in grooves, and sheared plates shall each constitute a separate lot. Rivet Steel. — 2l. Rivet steel shall have a specified tensile strength of 60,000 pounds per square inch, nnd test bars must have a tensile strength within 4, 03 pounds per square inch of that spe- cified, and an elastic limit, elongation, and reduction o ' area at the point of fracture, as stated in clause 17, and be capable of bending double, flat, without sign of fracture on the convex surface of the bend. Time of Inspection. — 22. The inspection and tests of the material will be made promptly on its being rolled, and the quality determined before it leaves the rolling-mill. All necessary facili- ties for this purpose shall bo afforded by the manufacturer ; but, if the inspector is not present to mak(^ the necessary tests, alter due notice given him, then the contractor shall proceed to make such namber of tests on the steel then being rolled as may have been agreed upon, or, in the absence of any special agreement, the number provided for in clause IG or 10, and the (luality of such materia] shall be determined thereby. Variation of Weigrhts. — 23. A variation in cross-section or weight of rolled material of more than 2^ per cent, from that specified may be cause for rejection. CAhT Iron. 24. Except where chilled iron is specifie 1, all c;astings shall be of tough gray iron free from injurious cold ^huts or blow holes, true to pattern, and of a workmanlike finish. Sample pieces 1 inch aqiuune oast from the same heat of metal in sand moulds shall be 214 RESISTANCE TO TENSION. capable of sustaining on a clear span of 4 feet 6 inches a centnl load of 500 pounds when tested in the rough bar. Workiiiansjiip. Inspection. — 25. Inspection of the work shall be made as it progresses, and at as early a period as the nature of the work permits. 26. All workmanship must be first-class. All abutting surfaces of compression members, except flanges of plate girders where the joints are fully spliced, must be planed or turned to even bearings so that they shall bo in such contact throughout as may be obtained by such means. All finished surfaces must be protected by white lead and tallow. 27. The rivet-holes for splice plates of abutting members shall be so accurately spaced that when the members are brought into position the holes shall be truly opposite before the rivets are driven. 28. When members are connected by bolts whioh transmit shearing strains the holes must be reamed parallel, and the bolts turned to a driving fit. 29. Hollers must be finished perfectly round and roller-beds planed. Rivets. — 80. Rivets must completely fill the holes, have foil heads concentric with the rivet, of a height not less than ,0 the diameter of the rivet, and in full contact with the surface^ or be countersunk when so requiretl, and machine-driven wherever prM- ticabie. 31. Built members must, when finished, bo true and free from twists, kinks, buckles, or open joints between the component pieces. Eye Burs and Pin-hole, and Pilot Nuts.— 82. All pin-holes must be accurately bored at right angles to the axis of the members, unless otherwise shown in the drawings^ and in piec^es not adjustable for len^.th no variation of more than one- thirty-se3oncl of an inch will be allowed in the length between centres of pin-holes ; tlio diameter of the pin-holes shall not exceed that of the pins by more than one- thirty-second inch, nor by more than one-fiftietli inch for pins under three and one-half inobes diameter. Eye bars must Ic strai^^ht before boring; the holes must be in the centre of the heads, and on the centre line of the bars. Whenever links arc to be packe;! more tiian onemi^tli of an inch to the foot of their length out of parallel with the axis of the structure, they must bo bent with a gentle RESISTANCE TO TENSION. 215 the head stands at right angles to the pin in their intended position before being bored. All links belonging to the same panel, when placed in a pile, must allow the pin at each end to pass through at the same time without forcing. No welds will be allowed in the body of the bar of eye bars, laterals, or counters, except to form the loops of laterals, counters, and sway rods ; eyes of laterals, stirrups, sway rods, and counters, must be bored ; pins and lateral bolts must be finished perfectly round and straight, and the party contracting to erect the work must provide pilot nuts where neces- sary to preserve the threads while the pins are being driven. Thimbles or washers must be used whenever required to fill the vacant spaces on pins or bolts. Tests of Eyes on Full Size Bars.— 33. To determine the strength of the eyes, full size eye bars or rods with eyes may be tested to destruction, provided notice is given in advance of the number and size required for this purpose, so that the material can be rolled at the same time as that required for the structure, and any lot of iron bars from which full size samples are tested shall be accepted — 1st, if not more than one-third the bai*s tested break in the eye ; or, 2d, if more than one- third do break in the eye and the average of the tests of those which so break shows a tensile strength in pounds per square inch of original bar, given by the formula — g3 pQQ_7,000 X area of origjnal bar _ ,^^ ^ ^.^^^^ ^^ ^^^ ^j -^ Circumference of original bar inches), and not more than one-half of those which break in the eye fail at more than 5 percent, below the strength given by the formula. Any lot of steel bars from which full size samples are tested shall be accepted if the average of the tests shows a strength per square inch of original bar, in those which break in the eye, within 4,000 pounds of that specified, as in clause 17 ; but if one- half the full size samples break in the eye, it shall be cause for rejecting the lot from which the sample bars were taken. All full size sample bars which break in the eye at less than the strength here specified shall be at the expense of the contractor, unless he shall have made objection in writing to the form or dimension of the heads before making the eye bars. All others shall be at the expense of the purchaser. If the contractor desires additional tests thev shall be made at his own expense, under the supervision of the purchaser, the acceptance of the bars to be determined by the result of all the tests in the manner above set forth. A variation from the specified dimensions 216 RESISTANCE TO TENSION. of the heads will be allowed, in thickness of one-tblrty.second inch below and one-sixteenth above that specified, and in diameter of OD^ourth inch in either direction. Piincbiug' and Reaming. — 34. In iron work, the diameter of the punch shall not exceed by more than one-sixteenth inch the diameter of the livcts to be used. Rivet-holes must be accurately spaced ; the use of drift-pins will be allowed only for bringing together the several parts forming a member, and they must not be driven with such force as to disturb the metal about the holes ; if the hole must be enlarged to adnut the rivet, it must be remade; all rivet-holes in steel work, if punched, shall be made with a punch one-eighth inch in diameter less than the diameter of the rivet in- tended to be used, and shall be reamed to a dluneter one-sixteenth inch greater than the rivet. Annealing. — 35. In all cases where a steel piece iu which the Full strength is required has been partially heated, the wlM^e piece must be subsequently annealed. All bends in steel most be nade cold, or if the degree of curvature is so great as to require heatings the whole piece must be subsequently annealed. Painting. — 86. All surfaces inaccessible after assembling must be well painted or oiled before the parts are assembled. 37. The decision of the engineer shall control as to the interpre- tation of drawings and specifications during the execution of work thereunder, but this shall not deprive the contractor of his right to redress, after the completion of the work, for an improper decision. BESISTANOE TO TENSION. 217 TABLE II. Tables showing the Strength given by tJie Form'tUce of Sections 4, 6, and 88, for Iron Bars of Various Dimensions. 7,000 X area of original bar For Standard Te«t Pi, ce of Bars, 88,000 - i,- j-^SSiS^e ^^tTri^iiiA bif- For ey*i8 of Full Size Eye Bars, 7,000 X area of original bar ^^ . ,. . u * ia^u 62,000 ,- —i- c . _i 11- - 5 .0 lbs. per inch of width. ' circumference of original bar 7,000 X area of original bar For Standard Te^t Piece of Angles. 50,000 - ^jrcuiSference of original h^' Size of bar. 1 X 1 u xU u xli 2 X 2 2 X \ 2 X I 2 X 1 8 X i 8 X J 8 X 1 4 X \ 4 X 1 4 xli 5 X } 5 xl 5 xli 5 X li 5 x2 6 X ? 6 x 1 6 xll 6 X U G X 2 7 X 1 7 xli 7 X 2 Standard test piece. 50,250 49,8>0 49,380 48,500 50, GOO 50,090 49,670 ,50,510 49,91)0 49,:i80 49,790 49,200 48,070 49,720 49,090 48,500 47,9G0 47,010 49,670 49,000 48,390 47.800 46,750 48,940 47,680 46,560 Eyes of full size eye bars. 40,150 49,195 48.G:J0 47,500 49,600 49,090 48,670 49,010 48,400 47,880 47,790 -47,200 46,070 47,220 46,590 46.00* » 45,460 44,510 46.070 40,000 45,390 44,800 43,750 45.440 44,180 48,060 Size of angle. 6 X 6 X i 6 X 6 X J 4 X 4 X x^ 2 f 3.x 3 X i 3 X 8 X li 2 X 2 X i ii Standard test piece. 48,320 47.165 48,750 47,620 49,160 47,870 49,180 48,810 BESISTANCE TO TENSION. TABIjB UL Strength of Iron Bode.' Bxra Tehiilb STBBHaTsn or Round WBODsOT-lHoir Roia i to 4 Ik< IK DllKETBB, AND TH* WkioHTS PBn FOOT, TBI SaPK BrBBISTH B lAEEH AT 10.000 PoDNDe PIR SqUABE IhCE. Tensile Strength and Quality of Wrou^ht-Imm. The best American rolled iron has a. breaking tenatle itTength of from fifty thousand to sixty thousand pounds per sqaitre Inch tar epecimens not exceeding one square inch in section. Ordlnar7bM>- iron should not brealt under a less strain than fifty thouNnd pounds per square inch, and sliould not take a set under a Knu less than twenty-five thousand pounds per square inch. A bar one inch square and one foot long should stretch fifteen per cent of Ui length before breaking, and should be capable of being bent, coH, 00° over the edge of an anvil without sign of fracture, ud should show a fibrous lestnre when broken. Iron IliJit will not meet these re<[airements fs not suitable for structures; Imt notliinii is gained by speclfyii^ more severe tMts, because, in bars of the sizes and shapes usually required for tneb work, nothinp mon? can be atlaineil with certainty, and coniden- Eiuus milkers will be unwilling to agree to furnish that which ll la not practicable to produce. The aorking-iarerirjtb of wrought^iron ties hi trUM RESISTANCE TO TENSION. 219 taken at ten thousand pounds per square inch. In places where the load is perfectly steady and constant, twelve thousand pounds mav be used. The extension ofir&n, for all practical purposes, is as follows : — Wrought-iron, ru^no of its length per ton per square inch. Cast-iron, ^,,^01) of its length per ton per square inch. Appearance of the Fractured Surface of Wrouglit- Iron. At one time it was thought that a fibrous fracture was a sign of good tough wrought-iron, and that a crystalline fracture showed that the iron was bad, hard, and brittle. Mr. Kirkaldy's experi- ments, however, show conclusively, that, whenever wrought-iron breaks suddenly, it invariably presents a crystalline appearance; and, when it breaks gradually, it invariably presents a fibrous ap- pearance. From the same experiments it was also shown, that the appearance of the fractured surface of wrought-iron is, to a certain extent, an indication of its quality, provided it is known liow the stress was applied which produced I he fracture. Small, uniform crystals, of a uniform size and color, or fine, close, silky fibres, indicate a good iron. Coarse crystals, blotches of color caused by impurities, loose and open fibres, are signs of bad iron; and flaws in the fractured surface indicate that the piling and welding processes have been imper- fectly carried out. Kirkaldy^s Conclusious.^ Mr. David Kirkaldy of England, who made some of the most valuable experiments on record, on the strength of wrought-iron, came to some conclusions, many of which differed from what had previously been supposed to be true. The following are of special importance to the student of build- ing construction, and should be carefully studied : — " The breaking-strain does not indi(uite the quality, as hitlK'ito assumed. ** A hUjh breaking-strain may be due to the iron being of superior quality, dense, fine, and moderately soft, or simply to its being very hard and unyielding. ** A ioKJ breaking-strain may be due to looseness and coarsenc^ss in the texture; or to extreme softness, although very close and fine in quality. 1 Kirkaldy *B Ezperiraents on Wrought-iron iind Steel. 220 RESISTANCE TO TENSION. ** The contraction of area at fracture, previously oyerlooked, fo an essential element in estimating the quality of specimens. ** The respective merits of various specimens can be correctly as tained by comparing the breaking-strain y(9t/i% with the contraci of area. " Inferior qualities show a much greater variation in the breakj strain than superior. ^* Greater differences exist between small and lai*ge bars inooi than in fine varieties. ''The prevailing opinion of a rough bar being stronger thai turned one is erroneous. " Rolled bars are slightly hardened by being forged doii'n. '' The breaking-strain and contraction of area of iron plates greater in the direction in which they are rolled than in a tn verse direction. *^ Iron is less liable to snap, the more it is worked and rolled. '* The ratio of ultimate elongation may be greater in short tl in long bars, in some descriptions of iron; whilst in others then is not affected by difference in the length. '* Iron, like steel, is softened, and the breaking-strain reduced, being heated, and allowed to cool slowly. '* A great variation exists in the strength of iron bars which hi been cut and welded. Whilst some bear almost as much as uncut bar, the strength of others is reduced fully a third. " The welding of steel bars, owing to their being so easily bun by slightly overheating, is a difficult and uncertain operation. '^ Iron is injured by being brought to a white or welding heat not at the same time hanmiered or rolled. '^ The breaking-strain is considerably less when the strain is ai^ suddenly instead of gradually, though some have imagined that ' reverse is the case. '* The specific gravity is found generally to indicate pr^ty correc th<* quality of spiH'inieus. "' Till' doiisity of iron is decreased by the process of wire-draw and by the similar ])rocess of cold rolling,^ instead of increwted, previously imagined. *' The density of iron is decreased by being drawn out nude tensile strain, instead of increased, as believed by some. "" It must be abundantly evident, from the facts which have b * The couclusioii of Mr. Kirkaidy in renpect to cold rolllDg ia undoubtedly t when the rolling amonntrt to wirc-dniwini;: but, when tbe oomprenkm of Hurface by rolliiiK diminidheH the MH:tional area in greiUer proportion thtt cxtcndd the bar, the result, accordinfc to the experience of tho PlttsbnTj^ mi facturerH, ia a slight iucreaise in the density of the Iron. 1 [STANCE TO TENSION. 221 produced, that the breahing-strain, when taken alone, gives a false impression of, instead of indicating, the real quality of the iron, as the experiments which have been instituted reveal the somewhat tiarthng fact, that frequently the inferior kinds of iron actually yield & higher result than the superior. The reason of this diHer. enoe was shown to be due to the fact, that, whilst the one quality retained its original area only very slightly decreased by the strain, the other was reduced to less than one-half. Now, surely this variation, hitherto unaccountably completely overlookedj is of importance as indicating the relative hardness or softness of the material, and thus, it is submitted, forms an essential element in considering the safe load that can be practically applied in various structures. It must be borne in mind, that, although the softness of the material has the e£fect of lessening the amount of the breaking- strain, it has the very opposite effect as regards the workimj-Htrain. This holds good for two reasons: first, the softer the iron, the less liable it is to snap; and, second, fine or soft iron, being more uni- form in quality, can be more depended upon in practice. Hence the load which this description of iron can suspend with safety may approach much more nearly the limit of its breaking-strain than can be attempted with the liarder or coarser sorts, where a greater margin must necessarily be left. '* As a necessary corollary to what we have just endeavored to establish, the writer now submits, in addition, that the working- strain should be in proportion to the breaking-strain per square inch of fractured area, and not to the breaking-strain per square inch of original area, as heretofore. Some kinds of iron experi- mented on by the writer will sustain with safety more than double the load that others can cuspend, especially in circumstances where the load is unsteady, and the structure exposed to concussions, aa in a ship or railway bridge." Eye-Bars and Screw-Ends* Iron ties are generally of flat or round bars attached by eyes And pins, or by screw-ends. In either case, it is essential that the proportion of the eyes or screw-ends shall be such that the tie will not break at the end sooner than in the middle. In importaiit work, eyes are forged on the ends of flat or round bars, by hydraulic pressure, in suitably shaped dies; and, while the risk of a welded eye is thus avoided, a solid and well-formed eye is made from the iron of the bar itself. A similar process is adopted for enlarging the screw-ends of long 222 RESISTANCE TO TENSION. rods ; so that, when the screw is cut, the diameter of the screw il the root of the thread is left a little larger than the body of the rod. Frequent trials with saeh rods has proven that they will pull apart in tension anywhere else but in the screw ; the threads remaining perfect, and the nut turning freely after having been subjected to such a severe test. By this means the net section required in tension is made available with the least excess of material, and no more dead weight is put upon the structure than is actually needed to carry the loads imposed. T/ie diameter of the eye in flat bars, having the same thiokneBB throughout, should be 0.8 the width of the bar. The width of the metal on each side of the eye should be \ the width of the bar, and in front of the eye should be equal to the width of the bar. Wlien it becomes necessary to use a larger pin than here described (as when a bar takes hold of the same pin with bars of larger size), the amount of metal around the eye should be still further increased. The weight of an eye-bar, proportioned as here described, will be about equal to that of a plain bar of a length equal to the distaDce from centre to centre of the pins, plus twice the diameter of the pin multiplied by the width of bar, both in inches. The thickness of flat hara should be at least one-fourth of the width in order to secure a good bearing surface on the pin, and the metal at the eyes should be as thick as the bars on which they are upset. Table IV. gives the proportion for eye bars, sleeve nuts, and clevises, as manufactured by ttie ^ew Jersey Steel & Iron Co. Table VI. gives the proportion for upset screw-ends for dif- ferent sizes of rods, as adopted by the keystone Bridge Com' pany. Cast-iron has only about cno-thirJ the tensile strength of wroujj:! It-iron ; and as it is liabk* to air-holes, internal strains from uiH'ipial contraction in cooling, and other concealed defects, redu- cing its effective area for tension, it should never be used where it is subject to any great tensile stress. Tables. The following tables give the strength of iron rods, bars, steel and iron wire roi)es, nianila ropes, and dimensions of upset screw* ends. The diameter in Table III. is the least diameter of the rod; and, if the screw is cut into the rod without enlarging the end, the effective diameter between the tlu^ads of the icrew dumld be ised in calculating the strength of the rod. ' BBS:8TANCK TO TEN8IOH. TABLE IV. Aa WE1.DLES3, DIE-FOEGSD EYE BARS, 1* .SSKISSSSTSSSSSSTJSStESsS p is3=sai!==Sf=s2"S2"»'s""-*»— a- 11 ii.i,,.i,i,.„i,,,%„„i.f. I' The snulleM diameter iif i>[n given for each width <ir tiuris the xiandunl i 11m larger fliea given are Ih« iBivwt that ai-c nJlowatile with each head. SThe thlckneaa of the ban ahonld not he more than ) nor lesa than t their wi l]n>-ban an hored J, Inch larger than the diameecr of the pin. Other eizes befamlMhed- 224 RESISTANCE TO TENSIOK Table YIl. was compiled from data furnished by the John A. Roebling's Sons Company of New York. The ropes with nineteen wires to the strand are the most pliable, and are generally used for hoisting and running rope. The ropes with seven wires to the strand are stiffer, and are better adapted for standing rope, guys, and rigging. Table IX. is taken from Trautwine's " Pocket-Book for Engi- neers.*' Table X. gives the weight and proof, or safe strength, of ofaains manufactured by the New Jersey Steel and Iron Compuiy. RESISTANCE TO TENSION. TABLE V, Safe Strength of Plat Rolled Iron Bar». e. per gquare toob. 226 RESISTANCE TO TENSION. TABLE V. (concluded). Safe Strength of Flat Rolled Iron Bars, s ^ Width iu iucbeB. Thicknei in incbef 3J" 3 J" 4" ^" H" ^" 5" H" 6" 6i" IbB. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. Ibe. A- 2,190 2,340 2,500 2,660 2,810 2,970 3,130 3,440 3,750 4,060 i 4,380 4,690 5,000 5,310 5,630 5,940 6,250 6,880 7,500 8,130 A 6,560 7,030 7,500 7,970 8,440 8,910 9,380 10,300 11,300 12,200 i 8,750 9,380 10,000 10,600 11,300 11,900 12,500 13,800 15,000 16,300 ■h 10,900 11,700 12,500 13,300 14,100 14,800 15,600 17,200 18,800 20,300 i 13,100 14,100 15,000 15,900 16,900 17,800 18,800 20,600 22,500 24,400 iV 15,300 16,400 17,500 18,600 19,700 20,800 21,900 24,100 26,300 28,400 i 17,500 18,800 20,000 21,300 22,500 23,800 25,000 27,500 30,000 32,500 A 19,700 21,100 22,500 23,900 25,300 26,700 28,100 30,900 33,800 36,600 f 21,900 23,400 25,000 26,600 28,100 29,700 31,300 34,400 37,500 40,600 \i 24,100 25,800 27,500 29,200 30,900 32,700 34,400 37,800 41,300 44,700 i 26,300 28,100 30,000 31,900 33,800 35,600 37,500 41,300 45,000 48,800 +1 28,400 30,500 32,500 34,500 36,600 38,600 40,600 44,700 48,800 52,800 1 8 30,600 32,800 35,000 37,200 39,400 41,600 43,800 48,100 52,500 56,900 +? 32,800 35,200 37,500 39,800 42,200 44,500 46,900 51,600 56,300 60,900 1 35,000 37,500 40,000 42,500 45,000 47,500 50,000 55,000 60,000 65,000 We 37,200 39,800 42,*b00 45,200 47,800 50,500 53,100 58,400 63,800 69,100 n 39,400 42,200 45,000 47,800 50,600 53,400 56,300 61,900 67,500 73,100 lA 41,600 44,500 47,500 50,500 53,400 56,400 59,400 65,300 71,300 77,200 U 43,800 46,900 50,000 53,100 56,300 59,400 62,500 68,800 75,000 81,300 n 48,100 51,600 55,000 58,400 61,900 65,300 68,800 75,600 82,500 89,400 H 52,500 56,300 60,000 63,800 67,500 71,300 75,000 82,500 90,000 97,500 is- 56,900 60,900 65,000 69,100 73,100 77,200 81.300 89,400 97,500 105,600 1} 61,300 65,600 70,000 74,400 78,800 83,100 87,500 96,300 ia->,ooo 113,800 15 65,600 70,300 75,000 79,700 84,400 89,100 93,800 103,100 112,500 121,900 2 70,000 75,000 80,000 85,000 90,000 95,000 100,000 110,000 120,000 130,000 RESISTANCE TO TENSION. 2 TABLE Vi. Upset ScretB-End» fm- Round and Square Bars. StINDAHD PKOFORTIOm OP THE KETBTOKK BRIDGE COUPAKr. RESISTANCE TO TENSION. TABLE VI. (concluded). Upset Srrew-Enda. RE81STANCB TO TKN8I0M. TABLE Vn. Strength <tf Irott and Steel Wire Bopen, Mahutictdbed by thk Jobs A. Koeblikh'b Sons Co., New Tobk. In IIh. uf roio •i'-'liX. CastSte 230 RESISTANCE TO TENSION. Ropes, Hawsers, and Cables. (HASWKLL.) Ropes of hemp fibres are laid with three or four strands of twisted fibres, and run up to a circumference of twelve inches. Hawsers are laid with three strands of rope, or with four rope strands. C<(hles are laid with three strands of rope only. Tarred ropes, hawsers, etc., have twenty-five per cent less strength than white ropes: this is in consequence of the injury the fibres receive from the high temperature of the tar, — 290°. Tarred hemp and manila ropes are of about equal strength. Manila ropes have from twenty-five to thirty per cent less strength than white ropes. Hawsers and cables, from having a less pro- portionate number of fibres, and from the increased irregularity of the resistance of the fibres, have less strength than ropes; th^ diflference varying from thirty-five to forty-five per cent, being greatest with the least circumference. Ropes of four strands, up to eight inches, are fully sixteen i^er cent stronger than those having but three strands. Hawsers and cables of three strands, up to twelve inches, are fully ten per cent stronger than those having four strands. The absorption of tar in weight by the several ropes is as fol- lows : — Bolt-rope . . . .18 per cent Shrouding . . 15 to 18 per cent Cables 21 per cent Spun-yarn . . 25 to 30 per cent White ropes are more durable than tarred. The greater the degree of twisting given to the fibres of a rope, etc., the less its strength, as the exterior alone resists the greater portion of the strain. To compute the Strain that can be borne with Safety by New Ropes, Hawsers, and Cables, deduced from tlie Experiments of tlie Russian Government upon tlie Relative Strengtli of Different Circumferences of Ropes, Hawsers, etc. The United-States navy test is 4^00 pounds for a white rope, of three strands of best Ri(/a hemp, of one and three-fourths inches in cArcvmference (i.e., 17 ^000 pounds per sqxiare inch); but in thefol- lowing table 14^000 pounds is taken as the unit of strain that can be boime with safety. Rule. — Square the circumference of the rope, hawser, etc., and multiply it by the following units for ordinary ropes, etc EESI6TANCE TO TEN8I0W. 331 TABLE VIIL Showing the Unltx for compiitiny the Safe Strain that may be home by Eo/ipk, Ilftienem, nnd Cablea. WTien it is required to uncertain the vjeiylit or strain that can be borne by ropes, etc., in yeneral use, the above units sliould be redut^ed one-third, in order to meet tlie reduction of tlieir atrength by chafing, and exposure to ilie weather. TABLE IX. Streniilb and irpi(/At 0/ Manila Hope. m RESISTANCE TO TENSION. TABLE X. Weight and Proof Strength of Chain. HE KewJebbet Steel ahd Iroh ( StrCDl^rth of Old Iron. — A square link 12 inches broad, 1 incli (hick, and about 12 feet long was taken from the Kieff Bridge, then i ) years old. and tesl-od in comparison with a similar link which hiid been preserved in the slock-housc since the bridge was built. The following is a record iif a mean of four longitudinal test pieces, 1 >i IJ n 8 inehes, taken from each link. Old link from bridge. "•ss^ 21.8 n'.a (TlH Hwhaoiul Worid, London.) JtSSlBTASCS TO SUEAKINO, CHAPTER X. RESISTANCX! TO SHBAKINO. Bt shearing is meant the pushing of one part of a piece by the Other. Thos in Fig. 1, let abed be a, beam resting upon the sup- ports 8S, which are very near logclher. If a sufflcientl; heavf load were placed upon tlie beam, it nould cause the beam to break, not by. bending, but by pushing the whole central part of the beam thrai^b between tlie ends, as represented in the figure. This mode of fracture is called " shearing." The resistance of a body to shearing is, like its resistance lo tension, directly proportional to tbe area to lie sheared. Hence, if we denote the resistance of one square inch of tlie material to shearing by F, we shall have as ihe safe resistance to shearing, Safe shearing > _ area to be sheared X strength fc S ft denoting factor of safety, as before. A piece of timber may be sheared either longitudinally or trans- versely; and, as the resistance is not the same in both cases, the value of F will be different In the two cases. Hence, in substi- tuting values for F, we must distinguish whether the force tends to shear the piece longituilinally (lengthwise), or Iransyersely (across). Table I. gives the values of F, as determined by experiment, tor ) materials employed in architectural con uo- (1) JtEBlSTANCE TO SHEARING. Showini/ the Reninlnn'-.f of Materials to Shearing, hoUi Longtta- dlualljf and Traii^terxelf/, or the Values of f. MATsnr^tLs. VaiuMofr. It«. MO'l 470 d 640. 732* lb*. K.7(»i> as: si: 4!«)0c a,«uc 6.700 « ^000. !J3;i:^°« Tliere are but few cases in ai-fliEtectural construction in vrbicb tbe resistance to siiearing tms to lie provided for. The one moat frequently met witii is at the end of a tie-beam, as in Pig. S. Fifl. 2. Tlie I'afier U e\pits a iluiisl ivliicli teiKls to push or shear off the pifice A HVD, ami tli« area of the section at CD slioiild offer enough resiatanci' to kei^p tliu rafter In place. This area is eqnal to CD • Ranklnt^. bKlrkaldy. c Tcuulwtm. >1 Hntfield. o Uu)Ied.SUt« iSomtB- RESISTANCE TO SHEARING. 235 times the breadth of the tie-beam; and, as the breadth is fixed, we have to determine the length, CD. If we let // denote the hori- zontal thrust of the rafter, then, by ,a simple deduction from formula 1, we have the rule: — Length of CD in inches = b.^th o^beam x r <2) F, in this case, being the resistance to shearing longitudinally. Example I. — The horizontal thrust of a rafter is 20,000 pounds, the tie-beam is of Oregon pine, and is ten inches wide: how far should the beam extend beyond the point D f Ana. In this case H = 20,000 pounds, and from Table X. we find that jP = 840; aS we will take at 5. Then 5 X 20000 = 10 X 840 * ^^ nearly 12 inches. Practically a large part of the thrust is generally taken up by an iron bolt or strap passed through or over the foot of the rafter and tie-beam, as at A (Fig. 2). When this is done, the rod or strap should be as obliquely inclined to the beam as is possible; and, whenever it can be done, a sti-ap should be used in preference to a rod, as the rod cuts into the wood, and thus weakens it. The two principal cases in building construction where the shearing strength must be computed, are pins and rivets; for the latter see pages o57-565. Strength of Pins in Iron Bridge and Roof Trusses. — Iron and steel trusses are now so generally used that it is neces- sary for the architect who is at all advanced in his profession to know how to determine the strength of the joints, and especially of pin joints ; and to facilitate the calculation of the necessary size of pins, we give Table II , which shows the single shearing strength and bearing value of pins, and Table III., showing the maximum bending moment allowed in pins. Pins must be calculated for shearing, bending, and bearing strains, but one ol" the latter two only (in almost every case) deter- mines the size to be used. By bearing s( rain is meant the force required to crush the edges of the iron plales against, which the pin bears. The several strains usually allowed per square inch on pin con- nections in bridges are : shearing, 7,500 pounds; crushing, 12,000 pounds ; and bending, 15,000 pounds for iron, and 20,000 pounds for steel. The shearing strain is measured on the area of cross-section ; the 236 STRENGTH OF PINS. crushing strain, on the area measured by the product of the diame- ter of the pin, by the thickness of the plate or web on which it bears. The bending moment is determined by the same rules as given for determining the bendiug moment of beams. When gi'oupsof bars are connected to the same pin, as in the lower chords of trusses, the sizes of bars must be so chosen, and the bars so placed, that at no point on the pin will there be an exces- sive bending strain, on the presumption that all the bars are strained equally per square inch. The following example will show the method of determining the size of pin in a simple joint. Example.— Fig. 3. Determine the size of pin for the joint in the lower chord of a truss, shown in Fig. 3, the middle bar being a vertical suspension rod, merely to hold the chord in place. 40,000 IX 4' I -. ^ IX 4'^40,000 * IX 4'-40.000 ^ 40,000 1X4' 4- i Fig. 8. Ans, The shearing and crushing strain in this case is 40,000 pounds. The bending moment will be 40,0(iO x 1"; the distance between the centres of the two outer bars = 40,000 pounds. Prom Table III. , we find that to sustain a bending moment of 40,000 lbs., with a fibre strain of 15,000 lbs., will require a 3" or 3^" pin. From Table II., we find that the bearing value of a 3^" pin is but 37,500 lbs., and that we must increase the size of the pin to 8f inches. The shearing strength of a 3|" pin is, from Table II., 67,500 lbs., so that the size of pin we must use in this case is deter- mined by the bearing strain. To be sure of the correct size of the pin, one must make the calculation for all three of the strains. STBSNaTH OV FINB. 237 II Tin I 8 ^J 12 £ ?^£ iS2 ' o- fl * * * * • • • oooo JO » » ak aK ss fooo •b * tfte «% S to iO t-fr-»00 C3CD CO CPvD «D»Ol>i-l 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 §coapT-Joo H04Ol8OeO ©t^TH-ri 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 t-oa"^ CQCQosa i>cooaoo oooit* COOIA <OeOOt« lOCOrHO AOOt-CO gt> «>Qdo6 oJOfH— c^oo-Tjlio 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'- CQt-CO -^ ^ '^ 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 Hand-book. STRENGTH OF PINa 239 Bending Moment in Pins. The only difficult part of the process of calculating the sizes of pins will generally be found in determining the bending moment. In cases where the strains all act in the same plane, the bending moment can generally be determined by multiplying the outside force by the distance from its centre to the centre of the next bar, as in the foregoing example. When, however, the forces act in several planes, as is generally the case, the process of determining the bending moment is more difficult, and can be best determined by a graphic process, first published by Prof. Chase Green, and in- cluded in his lectures to the students in engineering at the Univer- sity of Michigan. As the pieces acting on any well-designed joint are symmetrically arranged, it is unnecessary to consider more than one-half of their number. Fig. 4 shows a sketch of one-half the members of a joint in the lower chord of a Howe truss. The pieces are parallel to the plane of the paper, and the pin is perpendicular to the same, but drawn in cabinet perspective, at an angle of 45° with a horizontal. The bars are assumed to be each one inch thick, and the channel to have one-half -inch web. The centre of the hanger is }" from the centre of the channel. The method of obtaining the bending moment is as follows : Draw the line A B at an angle of 45° with a horizontal, and, com- mencing with c, lay off the distances between the centres of the bars to a scale (1^" or 3" to the foot will be found most convenient) ; then draw the lines 1-3, 2-3, etc. , parallel to the pieces which they represent in the trass, to a scale of pounds. Resolve the oblique forces into their horizontal and vertical components (in this exam- ple there is but one oblique force). Next draw the stress diagram (Fig. 6) as follows : On a horizon- tal line lay off 1-2 equal to the first or outer force ; 2-3, equal to the next, 3-4 ; and 4-1, being the horizontal component of the brace, closes the figure. In the same way, lay off the vertical forces 15, 5 6, 61. If the forces are correct, the sum of the forces acting in one direction will always equal those acting in the opposite direction. From 1 draw the line 1 at 45", equal to the same scale of, say, 20,000 pounds, or any other convenient length. Draw 2, 3, 4, etc. Then, in Fig. 5, starting at the first horizontal force, draw c d parallel to 2, 6^ e parallel to 3, «/ parallel to 4, and/^ parallel to 1. In the same way, starting at the first vertical force, draw r 8 par- allel to 5> s ^ parallel to 6, and t 2 parallel to ' ' '^ 240 STREN(iTH OF PINS. line c d e fk will represent the boundary of the horizontal ordi. nates, and /• ,9 1 'O the boundary of the vertical ordinate?. And to find the resultant of these ordinates at any point on the pin, it is o ^ t only nooci^sary to draw tlic diagonal from the ends of the ordinates ut that ))<)ii)t. Thr.s. thi> resultant at X^ Fig. ft, will be i»-ii, uid it is evidtnt that this is the longest hypothenuM whk^ onn be BTRENGTH OF PINS. 241 dxawn ; and this hypotheause, multiplied by 0-1 (20,000 pounds), gives 62,600 pounds as the maximum bending moment on the pin. To obtain the maximum bending moment,, it is necessary to take the longest hypothenuse that can be drawn, no matter at what place it occurs. If one desires to try the effect of changing the order of the bars on the pin, it can readily be done. Suppose the diagonal tie to change places with the next chord bar. The horizontal stress dia- gram then becomes 1-2, 2-si, 3-4', 4-1. The equilibrium polygons A Fig. 11. will now be (Pig. l)cdef' k' and r' s' f w, and the longest hypoth- enuse, w a*, or 3J", which makes the bending moment 75,000 pounds, showing that the arrangement in Pig. 4 is the best. As a rule, in arranging the bars on a pin, those forces which counteract each other should be close (ogcthor. To further illustrate this method of dotcrniininp: the bending moment on pins, we will determine the bending moment for the pin at the joint A, Pig. 8. This is the some truss as worked out on page 686, the strains given in Pig. 8 being ^ of the strains at the joint, as all the pieces are doubled. Pig. 9 shows the size and •RBOgMDei of the ties and strat. It is assumed that the web of '242 STREN(4Tn OF PINS. the channel is reenforced to make it §" thick. Drawing the line AB, Fig 11, we lay off the outer force at a; then measaring off an incli. the distance between centres of the two outer bars, we lay off the next force {)arallel to the direction in which it acts ; and in the oame way, the other two forces. The three inclined forces must be resolved into their horizontal and vertical components. We next draw tlio stress diagram (Fig. 10) to the same scale of i)ounds, mak- ing 1 e(iual 20,000 pounds. The lines 4 and 6 ha[)pen, in this case, to coincide. Then, in Fig. 11, we draw a d parallel to 2, '> f parallel to 8, c d — 4, and d e parallel to 1. In the same way, we obtain i\w line hjk B. In this case, it will be seen tlial the longest horizontal ordinate is h by while at that point there is no vertical ordinate^ ; also, that no hypothenuse can be dra^Ti which will he as long as h b, so that we must take A 6 as the greati»st re- sultant : and this, multiplied by 20,000 pounds, gives 31,800 {xmnds as the inaxirnuni bending moment cm the pin. It will be seen that this is just the prmluct of the outer force by its arm to the centre of the next bar, so that the greatest bending moment is at that point. To determino the sizeof the pin, we find, from Table III., that for a steel j;iii to sustain this moment, allowing a flbro strain of 20,<MM) pounds, wc shall need a 25" pin. This pin has a bearing value of JU,5()) i)()un(ls for a bar an inch thick. The outer bar in this case is J thick, and has a strain of J31.800 [>ounds, equivalent to 42,4'K) pounds for a 1 bar. And we see, from Table II., that we shall need to u.'^e a lU' pin to meet this strain. The shearing streij;^^th of a ii\" pin is 36 tons, or nn)re than double the strain. Hence we must use a lU" pin. or. by increasing the thickness of the bars, we might reduce the pin to 3 inches. BEARIKQ-PLATES FOB GIBBEBS AKD COLUMKS. PROPORTIONS OF OAST-IRON BBARINGkPLAT FOR aiRDERS AND COLUMNS (1896). If a heavily loaded column or girder should rest directly up wall or pier of masonry, the weight would be distributed over a small area that in most cases there would be danger of cms the masonry, particularly if it were of brick or rubble work. Section • / ^ * \^ Pi an Fig I <£ n — n FiqZ ^P Fig 3 prevent this, it is customary to put a bearing-plate between end of the beam or column and the masonry, the size of the j being such that the load from the column or girder divided bj area of the piate shall not exceed the safe crushing- strength o masonry per unit of measurement. The load per square inch on different kinds of masonry not exceed the following limits : 242^ BEARIKG-PLAT£8 FOB GIBDfiBS AND COL17MK8. For granite 1,000 lbs. per sq. in. •• best grades of sandstone 700 ** " " ** ** soft sandstone 400 " *** '* ** '* extra hard brickwork in cement mor- tar 150 to 170" " " " ** good hard brickwork in lime mortar. . . 120 " ** ** ** ** good Portland cement concrete 150 ** " ** " '* sand or gravel 60 *« « " « Example 1. — The basement columns of a six-story warehouse support a possible load of 212,000 pounds each ; under the oolumn is a base-plate of cast-iron, resting on a bed of Portland cement concrete two feet thick : What should be the dimensions of the base-plate ? Answer. — ^As the plate rests on concrete, the bottom of the plate should have an area equal to 212,000 -h150= 1,413 square inches, or 37 inches square. The column should be about 10 inches in diameter and 1 inch thick. The shape of the base-plate should be as shown in Fig. 1. The height K should be equal to the projection P, and D should be equal to the diameter of the column. The thickness of all p<w- tions of the plate should be equal to that of the column above the base. This is not so much required for strength as to get a perfect casting, as such castings are liable to crack by unequal cooling when the parts are of different thicknesses. The projection of the flange G should be three inches, to permit of bolting the plate to the bottom of the column. It will be seen that in such a plate no transverse strain is developed in any portion of it. THICKNESS OF FLAT BASEPLATES. For small columns and wooden posts with light loads, plain flat iron plates are generally used. They may have a raised ring to fit inside the base of an iron column, or for a wooden post, a raised dowel, 1^ inches or 2 inches in diameter. If the plate is very thick, a saving in the weight of the plate may be made by bevel- ling the edge, as shown in Fig. 2, without loss of strength. The outer edge, however, should not be less than one inch thick. When such a plate is used, it is evident that if the plate is to distribute the load equally over its entire area, it must have suffi- cient transverse strength to resist bending or breaking, and this strength will depend upon the thickness of the plate. It is diffi- cult to make an exact formula for the thickness of such platu^ IKABiyG-PLkTES 50^ ^1212X5 JlTl/ COLrHTs. 34ic bat the writer ]u« -is-ricr-i -JiaE: *>_*: will be *lway? :c li* <^*-^ iLie strength: •r '-M«— ' ThickZKS Lt Z'J^Z>t -Z. Jl< Jlfr* I ' -^f in which r = TZc . *i c v.- : ^-.- ::~:j*:d '." *> u-»sk r. v/;az^ inches, azati P -j:± pr- ;.-r; • - f m.- ^i^- ^ -os s a-'.* >-- i-oi 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/ BEABIKG-PLATES FOB GIBDEB8 AliTD GOLUHKS. multiplied by 7,000, gives 42,000 pounds as the safe stiength of one bracket. The resistance to crushing may be found by multiplying the distance X by the thickness of the bracket and the product by 13,000. Thus, if X is four inches and the thickness one inch, the resistance to crushing would be 52,000 pounds. Such a bracket would support the end of a 20- inch light steel beam of 16 feet span under its full load ; for heavier beams, the thickness of the bracket and also the length D should be increased. ■v^ STHEMGTU ( POSTS, STRUTS, AND C0LUMN3. 243 CHAPTER XL STRBirGTH OF POSTS, STRUTS, AND COLUMNS. As the strength of a post, strut, or column, depends primarily upon the resistance of the given material to crushing, we must first determine the ultimate crushing-strength of all materials used for this purpose. The following table gives the strength for all materials used in building, excepting brick, stone, and masonry, which will be found in Chap. VI. TABLE I. Average Ultimate Crushing-Loads, in Pounds per Square Inch, for Building-Materials. ' . Crashing Crushing Material. weight, in lbs. Material. weight, in lbs. per sq. inch. per sq. inch. C. C. For Stone, Brick, Woods (continued). and Masonry, see Beech 9,300 » Chap. VI. Birch .... Cedar . . . 11,600 a 6,500 a Metals. Hemlock . . . 5,400 b Cast-iron .... 80,000 Locust . . . 11,720 b Wrought-iron . . . 36,000 Black walnut 5,690 Steel (cast) .... 225,000 a White oak . . Yellow pine . 3,150 to 7,000 4,400 to 6,000 Woods. Ash 8,600 a White pine . . Spruce . . , 2,800 to 4,500 The values given for wrought and cast iron are those generally Tised, although a great deal of iron is stronger than this. The values for white oak, yellow pine, and spi*uce, are derived from experiments on full-size posts, made with the government testing- machine at Watertown, Mass. ; the smaller value representing the strength of such timber as is usually found in the market, and the larger value, the strength of thoroughly seasoned straight- grained timber. For these woods a smaller factor of safety may be a Trautwine. b Hatfield. ii44 STRENGTH OF WOODEN POSTS AND COLUMNS. used than for the others, tlie strength of wlilch was derived from experiments on small pieces. The values for wood are for dry timher. Wet timlx»r is only about one-half as strong to resist compression as dry tindx*r, and this fact shouhl be taken into account when using gr«'en timlHT. TJk- sfrcntfth of <i ro/«////i, jwat, or Mrut depends, in a large nu'asun', uiK)n the pr(>j)ortion of the length to the diameter or least thickness. Up to a certain length, they bre^k simply by comi)ressi()n, and above that they break by first l>ending sideways, and then breaking. Wo<Mlen Columns. For wooden colunms, where the lengtli is not more than twelve times the least thickn(*6s, the strength of the column or strut may be computed by the nde, area of cross-section x C Safe load- - -factor of safety ~ . <1' where C* denotes the strength of tlie given material as given in Table I. The factor of safety to l)e used dei>end8 ujwn the plaoc where the cohiinn or strut is used, the load which comes ujion it, the <iuality of the material, and, in a large measure, ut>on the value takt'H for (\ Tims foi- white oak, yellow pine, and spruce, the value C is the actual cru.sliiiig-strength of full-size i)OSts of ordinar>' quality: hence wc need not allow a factor of safely for these greater tlian four. For the other wootls, we shouhl us«* a factor of safety of at least six. //■ //// ItKhJ ujto)! the rolfunu or iM)st is su«'h as conies upon the lloor of a iua<'hine-shoi», or where heavy machinei'y Is us«m1, or if the strut is for a railway-bridge, a larger factor of safety sliould be used in :ill ciises. If tin (judlitf/ of t/ir thntur is <>xce]>tionally goo4l, we may ust* the Imui'i- v;iln«'< f(»r the constant (\ in tb** cjise of (he last four WikhIs i:i\< n in <iic tabl(>. For (»rdinary bard pine or oiik imwIs, uudtiply lilt' ;iri;i n\ cross-section in inches by HMM); for >pru(v. by SiM», und t«»r wliite pine, by 7"*) pounds. V.\ \Mi'M. 1. —What is the siife load for a hanl-pine pust 10 by b) in. h. s, IJ ft'ct long? Ans. Ana of cross-section = 10 X 10 = 100 square Incliet; 100 X KNNI - lOO.IMM) i)ound». STRENGTH OF WOODEN POSTS AND OOLITMNS. 245 ExAMPm II. — What is the safe load for a spruce strut 8 feet long. G" X 8" ? An8. Area of cross-section = 48 ; 48 x 800 = 38,400 pounds. Stren^h of Wooden Posts over Twelve Diameters in Length. When the length of a post exceeds twelve times its least thick- ness or diameter, the post is liable to bend under the load, and hence to break under a less load than would a shorter column of the same cross- section. To deduce a formula which would make the proper allow- ance for the length of a column has been the aim of many engineers, but their formulse have not been verified by actual results. Until within two or three years the formulse of Mr. Lewis Gordon and Mr. C. Shaler Smith have been generally used by engineers, but the extensive series of tests made on the Gov- ernment testing machine at Watertown, Mass., on full-size col- umns, show that these formulae do not agree with the results there obtained. Mr. James H. Stanwood, Instructor in Civil Engineering, Mass. Institute Technology, in the year 1891 platted the values of all the tests made at the Watertown Arsenal up to that time on full- size posts From the drawing thus obtained he deduced the fol- lowing formula for yellow pine posts : Safe load pec square inch = 1,000 — 10 x . ' ^■,^. . — r-^ ^ ^ breadth m ins. The author has carefully compared this formula with the results of actual tests, and with other formulae, and believes that it meets the actual conditions more nearly than any other formula, and he has therefore discarded the tables of wooden posts given in the previous editions of this work and prepared the following tables for the strength of round and square posts of sizes coming within the range of actual practice For other sizes the loads can easily be computed by the formula. The loads for oak and white pine posts were computed b} the following formulse : 346 3TBENGTH OF WOODBN POSTS AND COLD For oak and Norway pine : For white pine and spmce posts : Safe load per sqnare inch = 635-6 X !!°^^° '^ . "^ ^ breudth id ius. in which the breadth is the le ast sid e of a rectangular stnit, or the diameter of a round post. The round posts were compnted for the half-inch, to allow tor being turned out of a square post, of the size next larger. The formuUe were onl^ used for posts exceeding ISdiameters for yellow pine, and ten diameters for other woods. For posts having bad knots, or other defects, or which are known to be eecentrically loaded, a deduction of from 10 to 35 per cent should be made from tbe values given iu the tables. 8APB LOAD IN POUNDS FOR YELLOW PINB POSTS <IU>ITND AND SQUARE). STRENGTH OF WOODEN POSTS AND COLCHNS. 347 248 STRKNGTH OF WOODEN POSTS AND OOLUMNa eccentric Loardrng. When the load on a post is applied in such a way that it is not distributed uniformly over the end of the post, the loading is called eccentric and the effect on the post is much more injurious than if the load were uniformly distributed. When a post supports a girder on one side only, or when the weight from one girder is much more than from the other, the load becomes eccentric, and an allowance must be made in the safe load varying from 10 to 25 per cent., according to the amount of eccentricity. The exact allowance cannot bj calculated, so that one must necessarily use his judgment in the matter, remembering that it is best to be on the safe side. Iron caps for timber pillars are often used in important con- structions, and are an excellent invention, as they serve to dis- tribute the thrust evenly through the pillar, and also form a bracket, which is often desirable, for supporting the ends of girders where a second post rests on top of the first. Fig. 1 shows the section of one of the simplest forms of caps. The Goetz and Duvinage caps, described at the end of Chapter XXIV., are the best shape for mill construction. STRENGTH OF CAST-IRON COLl \S, 249 Cast-Irou Columns. For cast-iron columns, where the length is not more than six or eight times the diameter or breadth of colunm, the safe load may be obtained by simply multiplying the metal area of cross-section by ()'i tons, which will give tons for the answer. Above this proportion, that is, where the length is more than eight times the breadth or diameter, the following formulas should be used. These formulas are known as Gordon's and Rankine's. Formulas — For solid cylindrical -cast-iron columns, Metal area x 13330 Safe load in lbs. = fi n — = — : — \ . (4) so. of length in inches ^ ' 1+ ^ sq. of diam. in inches X 266 For hollow cylindrical columns of cast-iron, O , . „ Metal area x 13330 Safe load in lbs. = sg. of length in inches ' <^) 400 X sq. of diam. in inches For hollow or solid rectangular pillars of cast-iron, Metal area X 13330 Safe load m lbs. = fi n — : — : — i • (6) sq. of length m mches ' ' 500 X sq. of least side in inches For cast-iron posts, the cross-section being a cross of equal arms, ^ , , , . , Metal area X 1.3330 Safe load m lbs. = sq. of length in inches ^^^ 133 X sq. of total breadth in inches Example I. -What is the safe load for a hollow cylindrical cast-iron column, 10 feet long, 6 inches external diameter, and 1'' thickness of shell ? Ans. We must first find the metal area of the cross-section of the column, which we obtain by subtracting the area of a circle of four inches in diameter from the area of one six inches in diameter. The remainder will be the area of the metal. The area of a six- inch circle is 28.27 square inches, and of a four-inch, 12.56 square inches; and the metal area of the column is 15.71 square inches. 250 STRENGTH OF CAST IKON COLUMNS. Then, substituting known values in fonnnla. 5, we liave 15.71 X 18830 Safe load = .^^^^^^ = 104700 pounds. ^"^40(rx"36 There is no use in carrying tlie result farther than the nearest hundred pounds, because the accuracy of our formulas will not warrant it. Example II. — What is the safe load for a cast-iron column 12 feet long, the cross-section being a cross with equal arms, one inch thick, the total breadth of two anns being 8" ? Ana, The area of cross-section would* be 8 + 7 = 15 square inches. Then, by formula 7, 15 X 13330 Safe load in lbs. = 20736 ~ 58300 pounds. ^■^ 133 X 04 Projectingr Caps. Hollow columns calculated by the foi-egoing formulas should not be cast with heavy projecting mouldings round the top or bottom, Fig. 2 as in Fig. 2, at a and 6. It is obvious that these are weak, and would break off under a load much less than would be requhredto STRENGTH OF CAST-IRON COLUMNS. 251 cnish the column. When such projecting ornaments are deemed necessary, they should be cast seimrately, and be attached to a pro- longation of the cohimn by iron pins or screws. Ordinarily it is better to adopt a more simple base and cap, which can be cast in one piece with tlie pillar, without weakening it, as in Fig. 3. In all the rules and formulas given for cast-iron colunms, it is supposed that the ends have bearings planed true, and at right angles to the axis of the column. When the columns are used in tiers, one above the other, the end connections of the columns should be made by projecting flanges, wide enough to received-inch bolts for bolting the columns together, as shown in Fig. 4, page 242^, and the entire ends and flanges should be turned true to the axis of the column. The end joints are generally placed just above the floor beams, for convenience in erecting the work. The basement columns should be bolted to cast-iron base plates, as shown in Fig. 1, page 242a. The author does not consider it advisable to use cast-iron columns with hinged ends, or in build- ings whose height exceeds twice their width. Tables of Cast-iron Columns. By an inspection of the foregohig fonnulas for cast-iron columns, it will be seen, that, all other conditions being the same, the strength per square inch of cross-section of any column varies only with the ratio of the length to the diameter or least thickness. Thus a column 15 feet long and 10 inches diameter would carry the same load per square inch as a similar column 9 feet long and 6 inches diameter, both having the ratio of length to diameter as 18 to 1. Owing to this fact, tables can be prepared giving the safe load per square inch for colunms having their ratio of length to diame- ter less than 40. On this principle Table IV. has been computed, giving the loads per square inch of cross-section for hollow cylindrical and rectangu- lar cast-iron colunms. To use this table, it is only necessary to divide the length of the column in inches by the least thickness or diameter, and opposite the number in column I. coming nearest to the quotient find the safe strength per square inch for the column. Multiply this load by the metal area in the cross-section of the column, and the result will be the safe load for the column. Example III. — Wliat is the safe load for a 10-inch cylindrical cast-iron column 15 feet long, the shell being 1 inch thick ? Ans. The length of the colunni divided by the diameter, botn in inches, is 18, and opposite 18 in Table lY. we find the safe load 252 STRENGTH .OF CAST-IRON COLUMlSrS. per square inch for a cylindrical column to be 7,360 pounds. The metal area of the column we find to be 28.27 inches ; and, multi- plying these two numbers together, we have for the safe load of the column 208,236 pounds, or about 104 tons. Besides this table, we have computed Table V. following, which gives at a glance the safe load for a cast-iron column coming within the limits of the table, and of a thickness thei*e shown. Thus, to find the safe load for the column given in the last example, we have only to look in the table for columns having a diameter of 10 inches and a thickness of shell of 1 inch, and oppo- site the length of the column we find the safe load to be 10^ tons, the same as found above. The safe load in both tables is one-^ixth of the breaking-load. TABLE IV. Strength of Hollow Cylindrical or Rectangular Cast-Iron Pillars, (Calculated bt Formulas 5 and 6.) Length Breaking-weight in pounds Safe load in i)ound8 divided by per square inch. per square inch. | external breadth or diameter. CyJindrical. Rectangular. Cylindrical. Rectangular. 5 75,294 76,190 12,549 12,698 6 73,395 74,627 . 12,232 12,438 7 71,269 72,859 11,878 12,143 8 68,965 70,922 11,494 11,820 9 66,528 68,846 11,088 11,474 10 64,000 66,666 10,666 11,111 11 61,420 64,412 10,236 10,735 12 58,823 62,111 9,804 10,352 13 56,239 59,790 9,373 9,965 14 53,859 57,471 8,976 9,578 15 51,200 55,172 8,533 9,195 16 48,780 52,910 8,130 8,817 17 46,444 50,697 7,741 8,440 18 44,198 48,543 7,366 8,090 19 42,050 46,457 7,008 7,748 20 40,000 44,444 6,666 7,407 21 38,050 42,508 6,341 7,085 22 36,200 40,650 6,033 6,776 23 34,455 38,872 5,742 6,479 24 32,787 37,174 5,464 6,195 25 31,219 35,555 6,203 5,926 26 29,741 34,014 4,957 5,660 27 28,343 32,547 4,724 5,423 28 27,027 31,152 4,504 5,192 29 25,785 29,828 4,297 4,971 30 24,615 25,571 4,102 4,761 31 23,512 27,310 3,918 4,818 32 22,472 26,246 3,745 4,374 33 21,491 25,172 3,581 4405 34 20,565 24,154 3,427 4,026 35 19,692 23,188 3,282 8,814 STRENGTH OF CAST-IRON COLUMNS. 253 TABLE V. Showing Scrfe Load in Tons for Cylindrical Cast-Iron Colvmns, Thickness of Shell | Inch. 1 Length Diameter of column (outside). of column. Gins. 7 ins. Sins. 9 ins. 10 ins. 11 ins. 12 ins. 13 ins. Feet. Tone. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 6 60.6 78.1 94.0 110.8 128.6 144.9 161.7 180.0 7 55.7 72.2 88.9 106.9 124.2 140.1 166.4 176.0 8 60.7 66.3 83.8 101.1 117.7 136.2 151.1 170.3 9 45.8 61.9 78.7 95.2 113.4 130.4 146.8 164.5 10 40.8 56.0 73.5 89.4 106.8 123.2 140.5 168.7 11 37.1 51.5 68.4 83.6 100.1 118.3 135.2 153.0 12 33.4 47.1 63.3 79.7 95.9 113.5 129.9 147.2 13 30.9 44.2 58.1 73.9 89.4 106.3 124.6 141.4 14 27.2 39.8 54.7 70.0 86.0 101.4 119.2 135.6 15 24.7 36.8 49.6 64.1 78.5 96.6 114.0 129.9 16 22.3 33.9 46.2 60.3 71.9 91.8 108.7 124.1 18 - 29.0 41.0 52.5 67.6 84.6 103.4 118.3 20 — 24.4 36.0 44.7 63.3 77.2 98.1 112.6 Metal area of croes-eection. sq. ins. sq. ins. 14.73 sq. ins. sq. ins. sq. ins. sq. ins. sq. ins. 26.51 sq. ins. 12.37 17.10 19.44 21.80 24.16 28.86 Thickness of Shell 1 Inch. Length Diameter of column (outside). of column. • 6inB. 7 ins. 8 ins. 9 ins. 10 ins. 11 ins. 12 ins. 13 ins. ! Feet. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 6 77 100 121 143 167 188 211 234 7 71 92 118 138 161 182 204 230 8 64 85 108 131 153 176 197 222 9 58 79 101 123 147 170 ' 190 215 10 52 72 95 116 138 161 183 207 11 47 66 88 108 130 154 175 200 12 42 60 81 102 124 147 169 192 13 39 57 75 95 116 138 162 184 14 35 52 69 90 110 1.32 155 177 15 31 47 64 83 104 126 148 170 16 28 43 69 78 j 96 119 142 162 18 25 39 53 68 ; 88 105 128 151 20 22 35 46 1 6S 1 79 94 114 136 Metal area of cross-section. sq. ins. sq. ins. sq. ins. sq. ins. sq. ins. sq. ins. sq. ins. sq. ins. 15.71 18.82 22.00 25.14 28.27 31.41 34.66 37.70 255a STBEl^GTH OF CAST-IRON COLUMNS. The principal disadvantage, as found in practice, is the difficulty, if not impossibility, of making rigid connections with the beams and girders. In buildings of not more than five or six stories, however, this is not of great importance. (Jast-iron is, of course, subject to flaws, and the columns are liable to be cast of uneven thickness of metal, but by careful inspec- tion these defects can be discovered, and any columns containing them rejected. For unprotected columns, c'ast-iron is unquestionably better than steel, as has been quite conclusively demonstrated by the experi- ments of Prof. Bauschinger, of Munich. Cast-iron, three quarters of an inch or more in thickness, is also practically uninjured by rust, while it is clnime<l that wrought-irOn or steel may be almost destroyed by it. Although cast iron columns may be made in a great variety of shapes, the hollow cylindrical and rectangular columns have thus far been the principal shapes used, and for interior unprotected columns the cylindrical column probably meets the usual require- ments better than any others. Every year, however, the require- ments of building regulations are being made more strict, so that at the present time it is required in most of our large cities that all vertical supports in buildings over five stories in height shall be protected by fireproof material, and for such buildings the author would call attention to the H-shaped column, as offering the following advantages : 1. Being entirely open, with both the interior and exterior sur- faces exposed, any inequalities in thickness can be readily discov- ered, and the thickness itself easily measured, thus obviating any necessity for boring, and rendering the inspection of the columns much less tedious. 2. The entire surface of the column can be protected by paint. 3. When built in brick walls, the masonry fills all voids, so that no open space is left, and if the column is placed as shown in Fig. 4, only the edge of the column comes near the face of the wall. 4. Lugs and brackets can be cast on such columns better than on circular columns, ■pjQ 4 especially for wide and heavy girders. 5. The end connections of the columns dp not require projecting rings, or flanges, which are often objection- able in circular columns. The cost of columns of this shape should not exceed that of cir- cular columns of the same strength. STBBKOTH 07 OABT-IBON COLDKKS. 2SBb As to the strength of such columns, the onl; experimental data which we have on the subject is that obtAined from the experiments of Mr. Eaton HodghinsoD, which give them about theaame strength as cyliudrieal columns of the same diameter, when the length does not exceed thirty diameters and the thicliness ia not less than three- quarters of an inch. When surrounded bj- masonry they would probably be stronger than the cylindrical column. The column may be flreprocfed in the same way as the Z-bar column, which it much resembles. The space occupied by the column 8lig}itly exceeds that ot both the cylindrical and Z-bar column. Fio. 6. PiQ. B. hut not enough to be of any serious consequence. Figs. 5 and 6 show details of end connections and brackets, and ot baseplate. The beams running at right angles to the web should be tied togeihcp by wrought-iron straps passing through holes in the web of the column. The following table has been calculated with the same stress per square inch of metal as allowed for the columns in Table V. STBBNGTH OF CA8T-1B0N OOLUHNS. TABLE V.a Siy% Loads in That of 3.000 Poand« fo B-*haped U 18 20 ISi a 87 48 30 an 48 S4 80 49 sst S8 58 SO a** 97 79 77 «4 1«! n 48 2a ra 88 72 3J m 88 «t S»! las 85 3** 124 114 107 ea 25 90 7S or 8? 92 m 138 129 120 9t 104 TO 811 ja 128 96 Slk 160 144 136 125 111 101 lar 138 98 u 841 1«8 4I( lie 101 40| 1S8 IBS 461 204 w 170 141 30 183 108 91 m IM m JSi 189 1B0 1S2 491 »T4 li! 207 232 ^ s IS 16G 1S2 125 lis vsi 205 1% IK7 IM 188 4Bi Mi 251 209 18> 300 286 200 280 2ia 4JI ess air %*■ 198 189 180 183 147 ; m 296 3^ 198 1T4 ! 801 830 aos 283 271) tat «n m «M 278 «) u« \ ?5| 399 353 337 .121 aot 277 STBBKGTH OF CAaX-IBO^ COUJMNB. 265d Hollow Rectangular Cast-iron Columns. The increasing use of hollow rectangular cast-iron columns in buildings, particularly when enclosed in brick walls, has led the author to compute Table V.6, which gives the safe loads for a large number of sizes and lengths, the application of the table being readily apparent. The loads correspond with and are based upon those given in the last column of Table IV. The author would recommend that the various sizes be not used for greater lengths than those given in the table. 266e STRENGTH OF CAST-IBON COLUHKa TABLE V.6 iSafe Loads in Tons of 2,000 Pounds far HoUovo Bertnngular Cast-iron Columns. LENGTH or COLUMN IN FBBT. W C * a. y. - x >:• = 6< H"^ ut^. h-y: U.?5i =-- w^"" C t- X 1 10 12 13 14 15 16 18 90 6x0 J 151 .^8 48 44 40 " 1 20 74 61 56 51 " li Si\ H7 7.3 66 61 6x8; J 18i ♦5<> .58 52 48 " ", 1 34 88 74 67 62 -, ij 2S} 106 88 80 74 6 X 10 J 21} m 67 61 56 51 it .. 1 < 28 u« 86 78 72 66 '• " 1 1} 33J 124 104 94 87 80 < X « , } 18S 78 67 62 58 58 t> Ik 1 24 100 86 80 74 68 7 X 1); J 21} 91 78 78 67 68 (i i(| 1 28 117 100 93 86 79 8x8' { 2U 100 87 81 76 71 65 ", 1 28 128 118 105 98 98 84 " li 33J 155 185 186 118 110 101 8 X 10 i 24} 113 90 92 m HO 74 Ik il 1 32 147 128 180 118 105 96 it 4k li mi 178 155 145 136 185 n« 8 X 12 J 21} 127 111 104 97 90 88 41 Ik -t 36 IJW 144 135 188 117 108 "It 43} 201 175 164 158 144 125 10 . 10 J Tt\ 14.3 19U 123 117 111 105 94 '•! 1 36 186 169 160 151 144 136 188 "1 li i 13} 220 ao5 194 1H4 175 166 148 "i 'i 1 •'>i •^^ 239 227 215 804 198 m 10 > 1'^ i ' 3()j 1.59 144 137 130 122 116 1(M "1 10 2(Ni 18S 17H 168 160 158 186 •• 1- 1 Hj 252 229 217 2<t5 195 1H5 I«^ •* n , r>r 2U3 267 253 240 828 316 198 11) u } :»| 174 I.5S 1.50 143 135 138 III 1 1 ••11 n 227 2i;6 196 1K5 176 167 119 111 ir, 1 1 IS 218 225 214 802 192 183 168 M l^ 1 ' .VJ 2ris 241 2:J1 219 808 197 ITli in '*\ I ' »'»! 3:jo 3<NI 2S-) 870 3:^*1 843 317 I'J . 1-J ; :W, 1^7 171 HkS 161 154 IH 186 184 •1 11 214 227 219 210 801 193 177 lil2 •• •• i; •'•••<; ■.flH 278 2»;7 2515 346 236 317 l\9i •' ]. k\:\ 349 32r) 812 800 889 8T7 8M 383 IJ 11 . :ic. 2«»;i 1S9 1S2 175 IfiH 161 14K I* 1 iK 2ti(> •J.IK 23J» 229 320 811 las l~ I-.' !•; I .v: 2SS 2GS 2.-is 848 8SH 88S 810 I'.tt l-j -.I 1 ♦^s 371 ."151 .338 335 812 899 874 <:>l u It 1 :.«•• .WO .31 >H 2tr7 2HK 8iK 86K 8BII 05 If. it'i 1 r»<) :{.-.! .^3li :») 324 S18 810 891 87S i«-. i" 1 i\\ 37; .35M 358 345 389 880 814 tBS l'^ J'' 1 ».s lit 401 391 3HI) 874 887 8ii W 1" -jj 1 HI |K.v< 472 460 44M 440 «8 408 888 STEEJi^QTH OF CAST-IRON COLUMNS. 255/* Wrought-Iron and Steel Columns and Struts (1891). Within the past three years wrought-iron and steel columns have been gradually taking the place of cast-iron columns in fire- proof buildings, and the time is probably not far distant when wrought- iron oi" steel columns will be used almost exclusively for the inte- rior supports of all largo buildings. In iron or steel trusses the struts are invariably made of the same material, though, of course, the strut bars are of a different section from that usod for ties. There are many contingencies which may arise in the manufact- ure of cast-iron columns which preclude anything approaching uniformity in the product. Among these are unevenness in the thickness of the metal, which has sometimes been found to be very different on one side of a round column from that on the opiK)site side. The presence of con- fined air, producing '* blow holes ** and *' honey-comb," and the col- 356 STRENGTH OF WROUGHT-IRON POSTS. lection of impurities at the bottom of the mould are aAso frequent sources of weakness in cast iron. The most critical condition, however, is that due to the unequal contraction of the metal during the process of cooling, thereby giving rise to initial strains, at times of sufficient force to produce rupture in the column or in its lugs on the slightest provocation. In many cases the trouble is due to faulty designing or careless- ness in the execution of the work ; yet, even under favorable condi- tions, it is so difficult to secure equal radiation from the moulds in all directions that castings entirely exempt from inherent shrink- age strains are probably seldom produced. As a protection against these contingencies, resort must be had cither to the uncertain expedient of a high factor of safety, or a material such as wrought iron or rolled steel must be adopted of a more uniform and reliable character than cast iron. Columns built up o* rollcl socLioiis alsj offer better facilities for fire-proof covering ; and for columns where extreme loads are to be supported, as in the lo.ver sLorieii oi' very high buildings, wrought- iron and steel columns wiU occupy less room than a cadt-iron column, and in many instances will be found to be cheaper. The forms of rolled columns now in general use in buildings are the ** Phoenix," '* Larimer," " Gray," and *• Z-bar" columns, illus- trated on pages 267-389A. For the strut bars of trusses two-channels bars, angle or T-bars, are generally used. In trusses with pin connectiotis the channel bar offers the best shape for the struts. I-beams are also often used. Streiigrtli of Wroiijjflit-iron Posts. The formulas most generally accepted by engineers of the present day for the strength of irre^^ular- shaped sections (such as nearly all these struts are) are as follows : Column — Square Bearing, Ultimate strength / _ 40,000 in lbs. per sq. inch i "~^ sq. of le ngth in inches ^ ' ' 36,000 X r* 1 + Column — Pin and Square Bearing^ Ultimate strength | _ 40,000 in lbs. per sq. inch ) ~"I sq. of le ngt h"lnlnches ^^ STRENGTH OF WROUGHT-IRON POSTa 257 Column — Pin Bearing, Ultimate strength ) _ 40,000 in lbs. per sq. inch ) ~' sq. of length in inches ' ' 18,U00 X r» in which r denotes th»j radius of gyration. A column is square hearing when it has square ends which butt against, or are firmly connected with, an immovable surface, such as the floor of a building, or riveted connections : it is pin and square hearing when one end only is square bearing, and the other end presses against a close-fitting pin ; and it is pin bearing when both ends are thus piti-jointed with the axis of the pins in parallel direc- tions (for example, the posts in pin -connected trusses). To shorten the process of computation by this formula, Table VI. has been computed, which gives the ultimate strength per square inch of cross-section for different proportions of the length in feet, divided by the radius of gyration. The radius of gyration of the principal patterns of rolled bars now on the market may be obtained from the tables given in Chapter XIII. To use these tables, it is only necessary to divide the length of the strut (in feet) by the least radius of gyration, if the strut is free to bend either way, and from the table find the load per square inch corresponding to this ratio. The area of the cross-section, multi- plied by the load, taken from the table, will give the ultimate strength of the strut or column. To find the safe load, divide by 4 for columns used in buildings, and 5 for trusses. Example 1. — What is the greatest safe load of a pair of Carnegie angles, 6" x 6", 33 pounds per foot, riveted together, 12 feet long, with square or fixed ends ; the angles being used as a strut bar in a truss ? Ans, The least radius of gyration is 1.85. which is contained in 12, 6.5 times. The strength for a column, with square ends, for this ratio of _ is, from Table VI., about 34,200 pounds per square r inch ; this, divided by 5, gives a safe strength of 6,840 pounds per scjuare inch, or a total safe load for the two angles of (6,840 x '.9.90) 136,116 pounds, or 68 tons. When two or more angles, channels, or I-beams are connected together by lattice work, the radius of gyration for the whole sec- tion should first be obtained, and then the method of calculation is the same as for a single bar. Channel bars are generally used in pairs, either connected by lat- tice work, or, where additional strength is required, by wrought-iron 258 STRENGTH OF WROUGHT-IRON POSTS. plates riveted to the flanges of the channels. In sach cases, the channels should be spaced far enough apart so that the colomu will be weakest in the direction of the web ; i.e., with neutral axis at right angles to the web, for which case the radius of gyration of the column is the same as that of a single channel. In Table VII. the quantities d and D show the distance that the channels should be separated to have the same radius of gyration about either axis. If the radius of gymtion is wanted for the neutral axis through the centre of section paraliei with web, it can readily be found, as the distance between the centre of grjivity of channel and centre of section with the aid of Column VI., in tables, pages 301-21. If two channels are connected by means of two plates, instead of lattice bars, it is necessary to obtain, fii*st, the moment of inertia of the section, whence the radius of gyration is found as the square root of the quotient of the moment of inertia divided by the area of the section. This moment of inertia, for a neutral axis, through centre of sec- tion perpendicular to the plates, is ecjual to the cube of the width of the plate, multiplied by ,'2 of tiie thickness of the two plates added, plus the combined area of the two channels multiplied by the square of the distance from their centres of gravity to the neu- tral axis. For a neutral axis in a direction parallel to the plates, it is equal to the moments of inertia of the channels as found in the tables, increased by the area of the two plates multiplied by the square of the distance between the centre of the plate and the centre of the section. The strength of such a strut may, however, be calculated with suflBcient accuracy for most purposes, by taking the radius of gyra- tion of a single channel, and getting the strength per square inch of cross-section, and then multiplying by the total area of the sec- tion. If the channels are s[)aced according to Table VII., or even greater, the true radius of gyration will be a little larger than that of the single channel, so that what error there is will be on the saf^ side. Table VII. has been computed on this basis, giving the strength of two channels, used as a strut. The heavy figures give the safe load (factor of saf('ty of 5) for the two channels latticed together, and the figures in italics give the safe load per square inch of sec- tion ; so that, in case the pair of channels alone do not give sufficient strength, one can readily tell how much additional area will be required. Table VIII. gives the safe load of Carnegie T-bara» used singly. STRENGTH OF WROUGHT-IRON POSTS. 259 Example ?. — A certain strut in a roof truss (18 feet Jong) has to withstand a stress of 50 tons, and it is desired to use two channels for the purpose ; what sized channels will be required, the strut baviiiij pin joints ? Ans. Looking down the column headed 18 (Table VII.), we find the nearest load under 50 tons is 40.8, for two 10" channels, pin bearing, and the safe strength per square inch is WA tons. As the load in the table lacks 9.2 tons of that required, the section of the channels 9 2 must be increased by -^, or 2.7 square inches, which is equivalent to 9 pounds per foot additional weight for the two channels ; so that we should use two 10" channels, weighing 24^ pounds per foot each, and the channels should be spaced 9.1" out to out, the flanges being turned in. In pin-connected trusses, two channels make the most practical form of strut bar. A common form of column or strut to be recommended for com- paratively light loads is that formed simply of two angles riveted together back to back, or four angles united either with a single course of lattice bars or a central web plate, as in Fig. 4, page 264. The radii of gyration for such struts are tabulated on pages 319-21. In cases where four angles are used, the two pairs should be spaced far enough apart to make the column weakest about a neu- tral axis parallel to the central web or latticing. The radius of gyration will then be the same as that given in the tables for a single pair of angles, since the moment of inertia of the web plate about such an axis is so small that it may be disregarded entirely. Example 3. — A strut 16 feet long, to be fixed rigidly at both ends, is needed for supporting a load of 80,000 pounds. It is to be com- posed of two pairs of angles, united with a single line of i" lattice bars along the central plane. What weight of angles will be re- quired, with a safety factor of 5 ? • Ans. We will assume four W x 4" angles, and determine the thick- ness of metal required. The angles must be spread ^" in order to admit the latticing. From the table on page 321, we find the radius of gyration of a pair of light 3" x 4" angles with the 3" logs par- l 16 alleland^"aparttobe 1 97 '. Hence the value of - = Y~Q7 — ^ 1» for#vhich the ultimate strength, as per Table VI. = 31,680 pounds. The allowable strain per square inch with a safety factor of 5 will therefore be 31,680 ^ 5 = 6.34 ) pounds, and the area of the re- quired cross-section 80,000 -t- 6,340 = 12.62 square inches, or 3.16 square inches for each angle. Hence the weight per foot of each 260 STKENGTU OF WROUGHT-IRON COLUMNS. TABLE VI. Ultivuite Strength of Wrought-iron Columns. For diflerent proportions of loiigtli in feet ( = O To leawt rudiua of L'yratiou in iiiclu'H ( — r). I'o obtain Safe lie^JHtance : P'or quics<>ent loudt*. an in hnil(lin<;H, divide by 4. For moving loads, as in bridges, divide by 5. I r 3.0 3. J 3.1 3.0 3.S 4.0 4.-J 4.4 4.(i 4.K 5.0 5.4 5.n 5.K 6.0 (i.L> 6.1 6.»i 6.S 7.0 7.2 7.4 7.H 7.H H.O S.I H.»l '.».•• '.I •-' '.'.I '.' '• '.« *» III i) lo.j lo I lO.ii It's I'ltimatc Ktrength in pounda per fr(jnare inch. ' Square. I P^" ai»<l • square. :3H,610 3s.4;m) 3s,2:^o 3sj);io 3r,--^u 3:.r.i»o 3r,3r^) 3T,I:A) 36.S71) 3li.«»-.'0 3«;,:«'.0 .'IC.OiiO .•i:..s*jo 3:),.-)40 3i,!»ro 3t,r.70 3 1.370 34.or.0 :i::,r:)0 •X\. \ 10 ••{:{, -.-{O 3-^*.'^10 3J. UK) :«.i;o 3I.*'.'»0 3I..VJ|> :'.l.!:«o : 3i.^7o :ii:..Mo 3 •.•.•!(» I 'I I :.:.' I '."» '.'.;( I •> !». IMn. •.'..'.fjll ^1 . • 1 1' I 37,J»r)0 37,«H<) 37,4(H) 37,110 :i0,sio 36,500 36, KO a").H40 ;i"),5oo 35.140 34,7SO :i4.4i>o :M,(K)0 :^3,670 3:j,-280 3--»,SflO I 3'J,o<H) 3v»,110 31.7111 31.310 30,910 3<»v'>10 80,110 iil>.710 :i9,3lO 2*<,'.I00 iN,5mi ;rr.70i» •*':.3io •J''..'^»o •.'•..110 •.»."•.:•■,(» •j:).3rn 'Jl.lKtO 'Ji, 60 87,210 86,!I70 36,610 3'i.a40 35,860 a5,«60 3.V50 31,(^0 33,770 33,3.30 :w,4n» 3I,«S0 31,5:^0 31,Ofio 3t).5'.)0 30.1.30 :2!),6ro 88,740 a8.-*>70 t>r,8:i0 'J7..KiO 2(S,<no !M,OM iJ.*>,.')7o •r..i:t» 'J4,7«iO v>l.-,»70 ■i-i.i;{i» •3.o:jii ^ « t ^^^* ' 'J 1. 140 *JI.INiO /• 11.0 11.2 11.4 11.6 11.8 12.0 12.2 12.4 12.6 12.8 13.0 1.3.2 13.5 13.8 14.0 14.2 14.5 14.8 15.0 1.V2 15.5 1.'>.8 16.0 16.2 16 5 16..S 17.0 17.2 17.5 17.S 18.0 IS.'J IK.-, Ivs r.».2 v.\ .-I l».8 21 Ml 2i».2 20 5 2l».K Ultimate str(>ntfth in jiounds I>cr square incli. I Sqnarc. 26,950 26,(>44) 26,.3iO , 26,(KM) I 25,6!)1i . 25,380 i 25,070 ' 24.770 ; 24.170 ' *1,170 j 23.870 : 28,570 23,114) 22,700 2.\ 120 22,l.-rf> 21.710 2 1, .320 21,050 20,7110 2l»,21W 20,<l20 19,7f*0 1!).510 I'M 50 iK.r.t) 18,.\'S«» 1S.320 I7!i8i» 17.120 I r. 21 Ml Iti.SNi) i6..'»:o 1 6.3:0 Hi. 1:0 i:i.87o I.V><i> l.\8sii I.VJI4I IJ/rJl) H.ftTiU Pin and nquart*. 23.170 22,S20 I 22,170 ! 22, i:*) ' 21. 8U) i 31.460 i 2i,iao : 20,810 20.4.10 . 20, ISO ! 19,860 ' 19.560 19.110 18,t>70 ! 18,:W) I lS,Hr0 K.H'.'O ' 17.290 j ir,02() 16.7tt> 16,3UI» > 16,010 I 15.77»J 15.M0 15.190 j 11.680 > 11.410 I 14.1i!0 I 13,790 i 13.5iiO i 13.:t!ii> 13.1«4) IJ.»'20 12.ii:ii) I 12. ni 12, 190 I ll.Wi) ii.;in 11.600 11,3110 1I.11W Pin. 20.2:» 19.960 19.6!0 19.2n> 1S,1«30 18.590 I8.2ii0 17.1M0 17.IU0 17,310 17.000 Itt.riO ia,2H0 15.KS0 1.V580 15.310 14.920 i4,&ao 14.290 14.<»40 l.-S.lil«t I3.3S0 13.120 12.910 12.590 12.2H0 I2.«W ll.K*« 11.590 ll.«IU 11.140 lO.'.itiO 10.700 10.4:10 10.290 lo.i:fl) 9.A20 8,in) STRENGTH OF CARN£6IE IRON CHAN2 S. 261 •-3 9 fej 2 H < a o OS o ^ O $ Eh 4S 1 J I Da 3Q •1; M § s a a Cfi 0L, GG £ Pui 02 P^ (/) Pi QQ A4 '/) p^ I a at a s »-■» ^ t>» «4 30 ?0 *^ ^stooo S^S2§^ *a«9»ac0»a S*=S8 • • • • • • fe*^«^S^S*'i;'^g*'^''SJ • • • ^ *'^'^§8*^^''8?''S''^^S5 •yjos »» 8 00 1^ c. <»^ «* 'O OS 55 aoao '^«o ?o g^g^g*ig*5^»jg^g»5g*:gj>5^*»^ •5g*» 00 t^ >.-iO «OQ0<t»o 5cS *»»-• -«© S» otSao 00 »^ • • • CO »» '-^ ^i ^ i.-^ » "^ » «--^ go oi <>» OOOi^ <0 00 ^ 00 OS <0 TO t^CC^Cl • • • • • • • 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* f-i f^ 91 »-" »^ Si'SS te s'jt ZJ $ii:?» $ -c '»x"^i ©•^x'^i ee ■*»o "'•t- ■'!« *» a: 7. '«v •^ "* »« 'ji *•> •»* y. 1/ ^ • '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^ »* i-M 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 T-BARS. 263 . V a> a 1 t • • •• •••• t'. •» « '-i 0» «« !.« «t • • • • • 00»<«O«) • ••• •••••• tO>-iiO«) »0 »-• SO »-i 00 »-< 1^ 1-^ S25 Sec ?S»»S^ o§?S^ 88IJ5S2 S§^j58S 00 »l «o '^ 9^ t' e« OS 9« CO «) >0»«I0») iO»-iOO»-«00*-« . 00 • • 00 «) • • • • •■•• •■•• #••• t-«) ri»)QO«) o«tt«*t e«4>0») • ••••• •• • • •••• •••• 00 «) e»^as«) o«)^«•) 1-1 1^ • • • • • ••••• .• 1-- ♦« ^ »< ^ •? 00 »-» to t^ «-* ^aoSoo QO^Sos «o 2os «>. S*3S^^5 « S 00 «N ©»5MO<M r^e>»00i>» 1-1 rH »-i I, <»1«{M 00 *»'«"•» ^ •» 93 VI I'. ^:^ R?2S^g^ 8:5SJ§ »^5S OS »i 00 "50 O »J 1-1 Tl ^*iOOCO ao ®» i- «» 00%tlO»)^«) ^%l <0 s^ S Sos ^» 38^Sa»ffi^ S?c 1-1 1-^ ^ *5i-i nn eo*50oc^ 1-1 T-1 11 00 00 to 0^ os>dto»)in«) "^•i « OS ».-i • • TH • • CO -iT) m • •••• •••• 1-^ 1^ 1"i v^ .... W *5 00 "50 10.82 S.ll 6.20 S.18 5.8 • • -■TJ< • • •••• •••• • • • • o "^s 00 "o 1-t • ••••• •• Least radins of gyration. • O 0.64 0.79 0.78 0.84 0.86 • • o o S ^ S 13 • • • • o o o o 08 • QC CO 11.9 3.57 15.2 ' 4.56 11.8 3.54 • • • • Gi Gi '9' OS S X « . • • « 1-1 rH T-1 4^ '3 I- • 0» 1-1 1" 1-1 • • CO OS I, eo OS* QC «C lO o o • • • • 1-1 CO CO «o 1-1 s 1 'a OD X to of X X X •^ eo X X "* CO X X oS" iS* 1* 00 ^ XXX •0 eo eo 00 X of WROUGHT-IRON AND STEEL <X)LU1IK8. angle vill be 3.16 -i- 0.8 = lO.S Ibe. This weight will be foood b agree with a thioknesa of 1 inch for a 4" x 8" aug-le. iTZ'SaV CDlumn *'Z:B>r OclWMl ^ of Ooliinuu bjr IVflootlon or Bnckllimr*— ri>nt,'tjt-in)ii l-oIuidds fail either by deflecting bodily out 111 line, or l>f the bui-kling of the metat botwaen rivata iLDli) uf supjiurt. Both actions maj take place at tfa* \ lOUGHT-raON AND STEEL COLUMNS. i!Ot> same time, bat if the Latter occurs alone, it maj be aa indication that the rivet spacing or the thickness ol the metal is losufficient. The niJe has been deduced from actual eiperiments upon wrought- fron columns, that the distance between centres of rivets should not exceed, in the line of strain, sixteen times the thioltness of metal of the parts joined, and that t]je distance Ijetween rivets or other points of support, at right angles to the lino of strain, should not exceed tbirt;-tiTO times the thickness of the metal. On page 244 sections are shown of some of the most common forms of steel and wrougbt-lron columns. Figs. 5 and 6, as well as the Ph<Bnix and Keystone Columns illustrated on piiges 267 and 377, belong to the type known as Cloeed Columns. As it is impractica- ble to repaint the inner surfaces of such columim, they should pref- erably be used only for interior work, where tlie clianggs in tem- perature are not considerable, and the air is comparatively dry. In places exposed to the extremes of temperature and unprotected from the rain, the paint on the inner surface of the column will, sooner or later, cease to be a protection to the iron, corrosion will set in, and, once begun, will continue as long as there is unoxidjzed metal left la the column. Figures 4 and 8 on page 264 represent types of }]=- columns with open sections, which readily admit -^-J of repainting, and are therefore suitable for out- Ik. door work. J 0( these, the latter, designed bj C. L, Strobel, C.B., and known as the Z-bar Column, is believed to oSer advantages equal, if not superior to those t any other steel or wrought-iron column in the f market. Bracing of Channels. — When chaqnels are i oonnected by hittioe work (as in Fig. 1). that there ^ may not be a tendency in the channels to bend be- £__jl tween the points of bracing, the distance I should jr be made to equal the total length of strut, mul- P,g i_ tiplied by the least radius of gyration of a single colunm, and the product divided by the least radius of gyration for the whole section ; or, I = „ where the letters have the following significance : / = length between bracing, L = total length of stmt. r = least radios of gyration for a single channel. B =■ leaat ntdiu of gyration for the whole section. i 266 STRENGTH OF STEEL COLUMNS. When the radius of gyration of channels, about an axis parallel with the web, is not ffiven, it will be sufficiently accurate to use for r tlie distance given in CoJumn VL in the tables on pages 801- 321. Example 4. — We will determine the distance l^ for the strut calcu- lated in Example 2. In this case 2/ = 18 feet, or 216 inches, R = 3.85 : and in Column VI., page 804, the distance d for a 20-pound channel is .70, for a 35-pound channel .75, so that we will assume .72 as the proper distance for a 24-pound channel ; or r = .72; 216 X .72 then I — 3.85 .- - = 40 inches. This same rule will also apply for angles, though with them the lattice work is generally doubled, in Fig. 2. ii 11 Steel Columns. '' Exi)oriments thus far made upon steel stmts indicate that fof Icnfifths up to 90 radii of gyration," (or 7.5 in Table VI„) '* their ulti- mate stnngth is alx)ut 20 |)or cent, higher than for iron. Beyond this )N)int. th<' excess of strength diminishes until it becomes zero at about 200 nulii. After passing this limit, the compressive re^ist- anco of <\y'v\ and iron seems to Iwcome practically equal.*** In Tables Vil and VIII. the loads to the left of heavy black line are for ratios less than 90 diameters. an<l those to the right for ratios alH)V(' that limit. Sp<M*ial Forms of AVr<MiKlit-iroii aiul Steel Coluiiiiui. 7'A/ 1* In mix Sifjincnfal ('obtmn\ has now been on the market fi>r a iiiiiiitMr of years, and is very extensively used in buildings, ami al.-^o lor posts in bridges. ^ .Mauiiiucturfd by the Phoenix lYoo Comptny, FliUaMphlik ROUGHT-IRON AND STEEL COLUMNS. 267 CO bages are : Economy of metal, simplicity of construc- bility to the requirements of building construction, and i. limns are made up of the rolled segments *'(','* which are riveted together, by rivets about six inches apart, by moans of flanges along their sides, as shown at * ' A " (Fig. 18). Between every two segments an iron bar is frequently inserted, through which the rivets pass. These bars, or '* flats *' as they are called, increase the area of the cross-section, and contribute much to the strength of the pillar. Table IX. gives the sizes of the columns rolled by the Phoenix Iron Company, as pub- lished in their book of sections. The interior surfaces of all Phoenix columns are thoroughly painted before riveting the segments together. After twenty years of service in exposed situ- " ations, columns have been cut open and l5 found uninjured by rust, and the paint still in good condition. The illustrations on pages 270 and 271 show methods of joining the several tiers of columns in a building, and the connections with girders, etc. Bearings for girders or beams at ir- regular heights are provided by project- ing brackets that are properly riveted to a segment, or by a plate passing transversely through the column be- tween the flanges, with seating angles alon^r its upper edge. For joining columns at the levels of different tiers, inside sleeves of wrought iron may be usod. They are riveted to the segments of the lower column, and cting tenon which is fastened by diagonal through bolts colamn when it is put in place. line the actual value of Phoenix columns under loads, have been made at different times and on various id especially that of the United States (Government at STRENGTH OF WRODQHT-lR< COLDMNB. TABLE VI. TJliiraate Strength of Wroiight-iron Cotumiu. Pordiaen-iit propunioiiaorkiigth in feet ( = I) SSS.l.Hl iS 21, AM) M,7«l acMSB ao.uao i 1S.7C0 RS isItbo H.KW i7.'jao 17.6W 14,W> li:H8U 1S.M0 18,380 IS.ICO li«70 15,5T0 ili is.sflo u'.fso 14,«N> II.TXI 10,190 un STRBNQTH OF CABNSQIB IRON CHANNELS. rn t . i .11 ^ If s ij 1 1 ? I 268 PHCBNIX WROUGHT-IKON COLUMNS. TABLE IX. Sizes of Phcenix Columns. One Segment. One Column. Least 1 radius of Mark. Thicknes? in inches. Weight in ponndB per yard. Area in sq. inches. Weight in pounds per foot. 12.6 iryration in inches. A A 9i 3.8 1.45 4 segment. i h 12 14i 4.8 5.8 16.0 19.8 1.50 1.56 3|" inter, diam. 8 17 16 6.8 22.6 1.59 i 6.4 21.3 1.92 B' h 19^ 7.8 26.0 1.96 23 9.2 30.6 2.02 4 segment. iV 26i 10.6 85.8 2.07 4|f" inter, diam. 1 1% 30 83i 12.0 13.4 40.0 44.6 2.11 2.16 i 87 14.8 49.8 2.20 m 7.4 24.6 2.84 B' A- 22i 9.0 30.0 2.89 I 26^ 10.6 35.8 2.48 4 segment. A- m 12.2 40.6 2.48 ^le.'/* j_ !• i 34i 13.8 46.0 2.62 5if inter, diam. ft 38i 15.4 51.8 2.57 s 8 i 42i 17.0 66.6 2.61 25 10.0 88.8 2.80 >'b 30 12.0 40.0 2.a5 2. 35 14.0 46.6 2.90 ,'b- 40 16.0 58.8 2 94 i 45 18.0 60.0 2.98 C ,\ 48 19 2 64.0 8.08 i 53 21 2 70.6 8.08 4 segment. li 58 23 2 77.8 8 12 7ft" inter, diam. i 63 25.2 84.0 8.16 iJ 68 27.2 90.6 8.21 1 » 73 29.2 97.8 8.26 1 83 33.2 110.6 8.84 H : 93 37.2 124.0 8.48 11 103 1 41.2 187.8 8.6d PH(BNIX WROUGHT-IRON COLUMNS. TABLE IX,— Concluded. Sizes of Pluenix Columns. >islied colnmoe. the VVatortown (Mass.) Arsenal. Prom these enjierimonts formu- las have been deduced Irom which the aeeompanyinfc tables have boun prepared, in which are shown the safe loads in net tocjs for each size and length of the several patteros made. 272 PHCENIX WROUGHT-IRON COLUMNS. columns are unequally loaded, then it will be adyisable 1 the tabular figures or use heavier sections for the case, a indicated hj the circumstances. Steel Columns. — These tables have been prepared columns. If it is desired to use steel, it will be proper to for loads from 15 to 20 per cent, more than those giv( tables, the greater value being for short, and the lesser columns. PHOENIX IRON COLUMNS. 273 SAFE LOADS IN TONS OP 2,000 POUNDS. PHOESNIX IRON COLUMNS. Square Ends. 4 Segment, A Column, 8f Inside Diameter. Length of ^" \" h" f" column hi 12.6 11)8. per ft. 8.8 D in. 16 lbs. per ft. 19.8 lbs. per ft. 22.6 lbs. per ft. 6.8 a m. feet. 4.8 □ in. 5.8 n in. 10 17.29 22.17 27. -W 32.36 12 16.87 21.65 26.57 31.63 14 15.99 20.54 25.23 30.05 16 15.08 19.30 23.84 28.48 18 14.17 18.24 22.45 26.79 20 13.29 17.12 21.10 25.21 22 12.39 15.99 19.73 23.61 24 11.57 14.95 18.47 22.13 4 Segment, B* Column, m" Inside Diameter. Length i" 21. 3 lbs. column per ft. in ft'ei. 6.4 Din. 10 30.30 12 29.45 14 28.49 16 27.46 18 28.40 20 25.28 22 24.14 24 23.00 26 21.88 Bibs 26 lbs. per ft. 7. Sain. 37.40 36.36 35.20 33.94 32.64 31.27 29.89 28.50 27.14 jf It 30.^ lbs. per ft. 9.2niii. 44.67 43.44 42.07 40.59 39.05 37.44 35.80 34.17 32.56 _7 T«.. 35.3 11)8. per ft. 10. Gain. 52.10 50.68 49.10 47.40 45.08 43.77 41.90 40.01 88.16 k" 40 lbs. per ft. 12 Din. 59.71 58.10 56.31 54.88 52.38 50.28 48.15 46.02 43.92 9 // 44.6 lbs. per f r,. 13.4nin. 67.47 65.68 63.69 61.53 59.29 56.95 54.57 52.19 49.84 49.3 lbs. per ft. 14.811 in. 70.41 73.43 71.28 68.84 66.37 63.78 61.16 58.53 55.94 \ Segment, B^ Column, 5^g" Inside Diameter. Lenj;th \" 30 lbs. \" J " IB k"" ■^b" \" of 24.6 lbs. 85.3 lbs. 40.6 lbs. 46 lbs. 51.3 lbs. 56.0 lbs. column per ft. per ft. per ft. per ft. per ft. per ft. per ft. in feet. 7.4 a in. 9 D in. 10.6 3 in. 12.2 Din. 13.8a in. 15.4 Din. 17Din. 10 a"), or 44.30 52.r9 61.14 60.85 78.72 87.75 12 85.25 43.33 51.56 59. 9d 68.51 77.23 8ii.l0 14 34.43 42.32 50.38 58.59 66.97 75.50 84.20 16 33.^3 41.23 49.09 57.12 65.30 73. H4 82.14 18 32.57 40.06 47.72 55.53 63.50 71.04 7^.93 20 31.55 38.83 46.26 53.86 61.61 69.52 77.60 22 80.48 87.58 44.73 52.09 59.61 67.29 75.14 34 29.41 3^.22 43.19 50.32 57.61 65.06 72.67 26 28.31 84.89 41.62 48.51 55.57 62.78 70.15 28 27.23 33.57 40.06 46.72 53.54 60.52 67.66 214 PHCENIX IRON COLUMNS. SAFE LOADS IN TONS OF 2,000 POUNDS. PHCSNIX IRON COLUMNS. Square Ends, 4 Sboment, C Column, 7^'' Inside Diaxbter. Length y of 33.3 lbs. column per ft. in feet. [ lOoin. 8 // per ft. 12 a in. 46.6 lbs. per ft. 14 Din. 7 // 53.3 lbs. per ft. 16 Gin. 60 lbs. per ft. 18 a in. 641b6. per ft. 19.2oin. 70.6 IbB. per ft. 21 .2 Din. 10 12 14 16 18 20 22 24 26 38 30 32 34 36 38 40 50.97 50.33 49.62 48.91 47.87 46.93 45.92 44.86 43.77 42.63 41.48 61.16 60.40 59.54 58.59 57.46 56.31 55.11 53.83 52.63 51.16 49.78 48.42 71.35 70.46 69.46 68.48 6;. 02 65.70 64.29 62.81 61.28 59.68 58.07 56.4!) 54.85 81.55 80.53 79.30 7H.2I) 76.t)0 75.08 73.48 71.78 70.04 68.21 06.37 64.56 02.69 60.88 91.74 90.60 89.31 88.04 86.17 84.47 82.66 80.75 78.79 76.74 74.67 72.63 70.53 68.43 66.37 97.86 96.64 95.87 08.91 91.92 90.10 88.17 86.14 84.04 81.85 79.65 77.47 75.23 7:^.00 70.80 68.61 106.05 10i».71 105.19 103.69 101.49 99.49 97.36 95.11 92.80 90.38 87.94 85.54 83.07 80.60 78.17 75.75 Lenffth TT.Slbs. per ft. 2:12 D in. 84 lbs. 90.0 lbs. 97.3 Ibe. 110.6 lbs. 124 IbB. 187.3 lbs. column in feet. per ft. 25.2cin. 128.45 per ft. 27.2Din. 138.65 per ft. 29. 2 Din. per ft. 33.2Din. per ft. Sf.Soin. per ft. 41. 2 a in. 10 118.2(5 148.84 169.23 189.QS 210.01 12 116. n 120.84 13»i.91 140.97 167.11 187.94 207.38 14 115.11 125.04 134.90 144.89 164.73 184.68 904.43 10 113.48 123.20 133.04 142.83 162.39 181.96 201.59 IS 111.07 120.04 130.22 139.79 158.94 178.00 197.94 20 108.87 118.20 127.04 137.03 155.80 174.67 193.85 22 100.54 115.73 124.91 134.10 152.47 170.84 189.91 24 104.08 113.U5 122 M 131.01 148.95 166.89 184.84 26 101.5.-) 110.31- 119.00 127.82 145.33 168.84 180.85 28 98.91 107.44 115.90 124.49 141.54 158.0O 175.65 30 90.24 104.54 112. S3 121.13 137.71 164.» iro.9i 32 93.01 101.08 109.75 117.82 133.91 160.10 166.94 34 90.90 98.74 10<).5S 114.42 130.00 146.78 161.44 36 RS.iJO 95.81 103.41 111.01 126.22 141.48 156.64 38 85.55 92.<>2 100.30 107.67 122.42 187.17 161.98 40 82.90 90.05 97.19 104.34 118.64 ias.96 147.23 PHGBNIX IKON OOLCJMNS. 275 SAFE LOADS IN TONS OF 2,000 POUNDS. PHCSNIX IRON COLUMNS. Square Ends, 6 Segment, E Column, 11'' Inside Diameter. Lens^th of 56 lbs. per ft. 16.8D in. 641 bs. 72 lbs. 801b-. 88 lbs. 9611)8. 1" 106 lbs. column per f I. I>er ft. per ft. per ft. per ft. P'^r ft. in feet. 19.2a in. 21. 6 Din. 24 Din. 26.4Din. 28.8Din. 31.8 a in. 10 86.94 99.36 111.78 124.20 186.62 149.04 164.56 12 86.41 98.76 111.11 123.45 135.80 148.14 163.57 14 85.79 98.06 110.31 122.56 134.82 147.08 162.40 16 85.09 97.24 109.40 121.56 13:3.71 145.87 161.06 18 84.30 96.34 108.88 120.48 132.47 144.51 159.66 20 83.44 95.36 107.28 119.20 131.12 143.04 157.95 22 82.52 94.81 106.09 117. ^s8 129.67 141.46 156.20 24 81.51 93.15 104.80 116.44 128.00 139. 7^ 154.29 26 80.47 91.96 103.46 114.^ 126.45 137.95 152.82 28 79.88 90.72 102.06 118.40 124.74 18tl.08 150.25 30 78.28 89.41 100.59 111.76 l-.?2.94 184.12 148.09 32 77.02 88.08 99.08 110.04 121.04 132.04 145.80 34 75.76 86.50 97.41 108.24 119.06 129.88 143.41 86 74.50 85.15 95.79 106.44 117.0R 127.72 141.03 38 73.21 as. 67 94.13 101.59 115.05 126.51 138.58 40 71.90 82.17 92.44 102.72 112.99 123.26 180.10 Leno^h 116 lbs. piT ft. 34.8 D in. 1" \l" 1" V H" U" 126 lbs. 186 lbs. 146 lbs. 166 lbs. 186 lbs. 206 lbs. column per ft. per ft. l)er ft. per ft. per ft. per ft. in feet. 87.8Din. 40.8Din. 43.8 Din. 49. 8 Din. 55.8Din. 61 .8 Din. 10 180.09 195.61. 211.14 226.66 257.71 288.76 819.81 12 179.01 194.44 209.87 225.30 256.17 287.03 :317.89 14 177.71 193.04 20S.3I) 2:23.68 254.32 284.97 315.61 16 176.26 191.45 206.65 2:31 .84 252.23 282.62 :313.01 18 174.63 189.68 204.73 219.78 249.89 280.00 310.10 20 l':2.85 187.75 202.6,'3 217.55 247.35 277.15 :^06.96 22 170.93 185.67 200.40 215.14 244.62 274.08 :30:3.50 24 168.84 18:3.40 197.96 212.51 241.62 2,0.74 i299.85 25 166.69 181.06 195.43 209.80 288.54 207.28 21)0.02 28 164.4:3 178.60 192. 7S 200.95 235.30 203.05 292.00 30 162.06 17'?. 08 190.00 203.97' 231.91 259.80 287.80 32 15^.55 17:3.31. 187. 0<5 200.82 22S.33 255.84 283.35 84 156.94 170.47 184.00 197.53 •.'24.59 251.05 278.71 86 154.88 107.64 180.94 194.25 220.86 247.47 274.08 88 151.H5 104.78 177.80 190.88 217.02 243.17 2^9.32 40 148.94 161.78 174.62 187.46 213.14 238.82 264.50 276 PHCENIX IRON COLUMNa SAFE LOADS IN TONS OP 2,000 POUNDS. PHCENIX IRON COLUMNS. Square Fnda. 8 Segment, G Column, 14j" Inside Diameter. Length 801be. 93.3 ll)s. iV 106. ti lbs. 120 lbs. 13:^.8 Ihs. 1" 146.6 lbs. 16Ulb8. column per ft. per ft. per ft. per ft. 36 Din. , per ft. per ft. 44 Dill. per ft. In feet. 24 Din. 28Din. 32 D in. 40 Din. 48 Din. 10 124.92 145.74 166. .56 187.38 208.20 229.02 240.84 1-2 124.44 145.18 165.92 186.66 207.40 228.14 ^48.88 14 123.5H 144.56 165.21 18'). 8() 206.. 52 227.17 247.82 16 123. 2.S 143.83 161.38 iai.98 205.48 226.02 'Zm.57 18 122.59 143.02 163.45 183.88 204.82 224.75 245.18 20 121.82 142.12 162.43 182.73 208.04 223.84 243.64 22 120.98 141.14 161.81 181.47 201.64 221.80 241.96 24 120.04 140.05 160.06 180.07 200.06 220.as 240.09 26 119.11 18S.96 158.81 178.66 198.52 218.87 288.22 28 118.08 137.76 157.44 177.12 196.80 216.48 23(>.16 80 117.00 13>i.50 156.(0 175.50 195.00 214.60 234.00 82 115.84 135.15 154.40 178.77 198.08 212.86 231.69 84 114.fi4 133.75 152.86 171.97 191.06 210.18 229.29 86 113.28 132.16 151.04 169.92 188.80 207.68 226.56 88 112.08 I30.7() 149.44 168.12 186.80 205.48 2^.16 40 110.80 129.27 147.74 166.21 184.68 908.14 221.61 Length of column in feet. 10 12 14 16 18 20 22 84 26 28 80 82 81 Si) 38 40 J/, 173.3 lbs. per ft. '^2 n in. 270.66 269.62 2i)8.47 267. 1-' 26.-). 61 26:^.95 262.13 2(K).10 :l'58.07 25.'>.84 'J53.50 :>51 .00 248.40 245.44 243. K4 240.08 186. «) lbs. por ft. 5(5 a in. 291.48 290.36 289.12 287.67 28H.04 2g4.25 2S2.29 280.11 277.92 275.52 273.00 2; 0.31 26r.51 264.32 2(n.52 258.55 200 lbs. per ft. 60nin. 312.30 311.10 309.78 808.22 30«).48 :iOI.56 302.46 300.12 297.78 295.20 292.50 289.(2 28.i.r)2 28{.:ii0 2S0.20 277.02 1" 226.6 lbs. 258.8 lbs. 980 lbs. 806.6 Ibfl. per ft. per ft. per ft. 84 Din. per ft. 68Din. (6 Din. 9iain. 358.94 895.58 487.29 478.86 3.V2.58 394.06 48.'^.54 477.09 351.08 892.88 488.69 474.99 349.81 80;). 41 481.60 472.60 :i47.84 388.90 4WJ.07 469.03 345.10 88.1.77 496.88 466.99 342.78 383.11 498.44 468.77 34). 13 380.15 420.16 460.18 387.48 877.18 416.89 4S6.m 334.56 373.98 418.28 452.64 381.50 370.50 409.60 448.60 328.28 866.85 40r>.46 444.08 324. t^a 883.06 401.96 489.48 :V>().96 35-).T8 896.48 4^.24 317.56 854.92 399.« 49>).64 313.95 850.89 887.89 424.78 STOl OCTA »K COLI F. 211 Keystone Octagon Column. Another special form of wrought-irou column is that known as the Keystone Octagon Column, manufactured by Carnegie, Phipps & Co. It is made of four rolled segments of wrought iron, riveted together as shown in Fig. 5. mmr///M Fie. 5. The table oo the following page giyes the diameters, areas, and weights of these columns as rolled. To compute the strength of these columns it is first necessary to find the radius of gyration (r), when the strength per square inch can then be determined from Table VI. The radius of gyration may be found by the following formulo : J= 7= A r 12 r = /4.' in which moment of inertia ; D area of column ; d radius of gyration. = outside diameter ; = inside diameter ; 278 KEYSTONE WROUGHT-IRON COLUMNS. 1- ll. tf u Q. CO 1- z m o ^ S hJ o J z o < o CO < ^ hJ o OC o < < o tH z o Q o z H Q. ^ CO llJ o q: Eh oc C/} tH o ^ M Q Z < CO hJ CO CO UJ Z ^ o •8saa3[3iqx "i «e-^ •e^fstnc ^■^ ^^ 1 i 1 •-» • js Si a • t* ^ S) ^ COM 5 «T.; O Ci • • too oct- • • 1 1 1 1 * ' i 1 » • J ■^ 1 o JS . eo 5<ll- cooa 1 a to I 5 ceo SS ^^ 1 1 1 1 , 1 1 le, Q t-* p) 1 < • • • 82 • • QCC) 1 1 1 1 • « as as js it a 3 -^o • • 00 1- • • 1-H COCO *ico 1 1 ;j • ,j *< o ja . r-x C50 ^•N 00 ^ O a • a J5 xr5 — ^^ 2^ tt t-i ^s 1 1 I 9> (O OQ 1 • • • • • • • • 1 1 -<{ j*^^ xo 1-1 CO -^CD »^ r-i 1^ 1-H l-H • % 1 QD x-»o • • Oi-i (NO • • CDQO 5K 1-1 1-1 1^ 1^ 1-H »-hO* \J »j *^ c •s. O coo x« C40 Oif i ^ 2 ;s 5 1 3^ - CO ^% • • (NO s^ t-t ■n a »>' o o -^ 7: c H- ^ ! 1 * 0; . «- 1 Sao ss ss ss < 1 ? ^ • • ^co T^ 1-1 • • i-« f-i ^5 ^d j 1 2 oi • c:x XX r-t- o« oo 1 ^ .2F 5 a 1 1 1 • • • • T-i 1-H 1-H(N tt t;^ • J3 -^co -*o Ofth- • ■» CO-N a • 5 13 i 1 1 • • • or- J^ig 3^ ood s > *^ •iH »H o X 101 'NX 38 So ^V. S!3i ^ < 1 1 • • • • X ^ 1=.^ tt t^ Sl^ 'Mtia 4.L ua .- ■c^"^ *- :>• 5^» Z-BAR COLUMNS. 279 Z-Bar Columns. Within the past three years, what is known as the Z-bar column has been introduced, and is now manufactured by all the leading iron mills. It is built up of four Z-bars, riveted together, as shown in Figs. 7 to 12, page 264. The dimensions of the different shapes manufactured will be found in the tables given in Chapter XIII. This column possesses so many advantages for building purposes that it is undoubtedly destined to be extensively used. Its claims for superiority are based mainly on the following qualities : 1. Cheapness, — The Z-bars are furnished at a lower price per pound than channels and I-beams, and only two rows of rivets are required, while four or more are used for any other column of an equal sectional area. 2. High Ultimate Resistance to Gompressioii. — Careful tests made upon fifteen full -sized (Carnegie) specimens, in which the web plates were replaced by lattice bars, showed an average ultimate resistance per square inch of 35,650 pounds for lengths ranging from 64 to 88 radii. These results are as favorable as have been obtained for closed cylindrical columns, and are more favorable than have been obtained for any other open columns. For detailed report of the tests referred to, see paper by C. L. Strobel, in Trans. Am. Soc. C. E., April, 1888. 3. Great Adaptability for Effecting Connections with I-heams. — When used in buildings, for supporting single floor beams, or double beam girders, this quality is of the greatest importance. The illustrations on pages 280 and 281 show different methods of making the connections, as employed by Carnegie, Phipps c: Co. This column may bo easily covered with terra-cotta blocks, for fireproofing, and finishing with plaster or cement, and the air-space between the tiling and the metal adds to the protection of the latter in the event of fire. The recesses in the columns may bo used to good advantage for conducting water and gas pipes, electric wires, etc. 4. Favorable Form for Inspection and Repairing. — This is a very desirable feature when used for out door work. When unusually heavy loads must be provided for, as in the case of columns for tho Iov,cr stories of very high buildings, the stand- ard sections of Z-bar columns may be reenforced to the required strength by using either a double central web plate, or by the addi- tion of outside cover plates, or, if need be, both, forming thus a Z-BAK COLUMNS. Connaotient ol IBomu is"in<I ij"I Bwu ni3fr'°."9">nd B" T'^nd S' I Tods. J] Too). ^ Buiw I Booh 17.6 Tom, B.I Tool Ccnnsotlonl d >dDubl« Som glnittt« Fluign otZB«r*< 88T0111. IBMioi IBhum IBaoM S3 Tons. 35 Tom. >7>< Ttm, n*Htimitre//eii(iidlcaitd,aftutttAibaJtfinimfirttmmttf girdtrtjir ahick Iht ennKiiimt arr prr^triiuud, BivtUamtBtlttHdta.—AUBtlUtui.vtirt,tMllta4t. Z-BAB OOLl rs. 281 DETAILS OP BTANDABD CONNECTIONS OPI^EAMSTO Z-BAR COLUMNS. leot Z-SarColun: Numitr »f riv4U rtf Hired /sr nntactiannef differ, c/l^^mi 10 »t*i o/Zbi<r>.-wiU be the !a.«t ni jEokph . 282 Z-BAR COLUMNa closed or box column. A form of column, offering advantages in some cases, especially if the column is to be finished circular in form, is shown by Fig. 3 on page 281. Pig. 8 on the same page shows the manner of splicing columns, whether of equal or unequal size. •* The standard connections for double I-beam girders and single iloor beams to Z-bar columns, detailed on pages 280 and 281, were designed to fairly cover the lunge of ordinaiy practice. When the maximum loads in tons indicate<l for each case are exceeded, the connections may be correspondingly strengthened by simply using longer vertical angles for the brackets and increasing the number of rivets. In proportioning these connections, the shearing strain on rivets was assumed of a maximum intensity of 10,000 pounds per square inch. For steel Z-bar columns, the maximum loads given for these ccmnections may be safely increased 15 per cent.'* The following tables give the safe load in tons for standard Z-bar columns of different lengths, as manufactured by Carnegie, Phipps &Co. The values for steel Z-bar columns should be used only for cases in which the loads are for the most part statical, and equal, or very nearly so, on opposite sides of the columns. When there is much eccentricity of loading, or the loads are subject to sudden changes, the tabulated values must be n^duced according to circumstances. The Carnegie Steel Co. has discontinued the manufacture of iron bars of all kinds, and their product is now confined entirely to steel, which has practically superseded iron in structural work, being sold at the same price per pound, while 20 per cent, stronger. (The steel here referred to is what is knovrn as "mild" steel, having an ultimate strength of about 60,000 pounds per square inch, and containing a comparatively low percentage of carbon.) Example. — What size of Z-bar column, 30 feet long, with square bearing ends, will be required to carry a load of 200 tons, using a safetv factor of 4 ? A7is. Referring to table of steel Z-bar columns, page 287. we find that for a length of 30 feet, a 12-inch column with |-inch metal, weighing 118 4 lbs. per foot, will support with safety 202.6 tons, which is slightly in excess of the load. * Carnegie, Phipps & Co.'t) Pocket Companion, 1890. E-BAB COLUHN DIUENSI0K8. Z-BAR COLUMN DIHBHeiONS. fOf J>- fOl M ^m - ^-y* % of Z-BsT columns in inches for mil mum thicknesaes. Note. — In columns A. B, C. D, E, and F, the thickness of the Z-bars iind web plates does not vary, the variations in the strength of the eoliimn being mode in the thickness of the side plates. Columns G. H, K, and L, have no side plates, and the variations are in the thickness of the bars snd web plate. All of Column B and part of A have four side plates, two on each side, the others have but one plate on eacli side. STBBL Z-BAK C0LDMN8. BAFB LOADS IN TONS OP 3,000 LBS. BTBBI. Z-BAR COLUMNS. Square Endt. ine per BQQflrB inch 1 18.000 Ibn., for length of TO rsdll ornode ilely fsclor 4 : ' ^ n.lOO-SI-^, for lenglhe over so ndU. 90" Z-BAR COLUHNS.-A. 4Z-B«riSi" 1". 1 Web Plate U" y I". Side Plata SO" w SO" Z-BAK COLUMNS.— B. Secllan: 4Z-Birsei" > 4". 1 Web Plate 14" > 1". 4 SMe Plata *0" wlda. BZBBL Z-BAB COLUHNB. BAS% LOADS IN TONS OF 1,0I» LBS. . STBEI. Z-BAR OOLnMNS. Sgttare Endt. Allows d«™in*per.quflretacl Bsfely factor 4 ; I,.iia,0001b8., fi jrlenellHofBOradiiornnder. .torlcni^thioterBOradU. BTEEL Z-BAB OOLUHNS. ' , . <, ll,«aO IbB., rar hnglbe oT 90 ndll 01 '"( 17,100-ST J. , tor li^iigUiB owr SO ni Section : 4 Z-Bara fll STEEL Z-BAR COLUHM& 8AFB LUADS IN TON» OP 3.000 LBS. STBBL Z-BAR OOI.nMN8. Square End*. w\ w! i|*,«. «,r 1SS.6 SOI ;iis- IBO.g m 4 194. STEEL ZBAK COLUHHS. SAPB LOAl>S IN TONS OP 1,0110 LBS. STBBI. Z-BAR OOIiUMNS. Square End*. ved slraina per square ttee\ -. HSfet; faclor Length tf( colnmn In r«l. mm ^ Ml |?!IS ...m,.,.n,k.r 47:t I wis ST.- «> st'.s te'.t si 4C.4 4i.a »4.0 IM.B Si 7B.7 i:l U.1 •JK .. 81 S:i lABIMEBS PATENT ALL STEEL COLtTMN. 389 i^matBBm patent aza steel oolomn. (MiaufactxKd by Jonea & LmghllnB, Pilteburgb.) This cotumn was patented Jutio 3, 1891. It is made by bending two I-beams at ngbt angles in tha middle of the web and riveting LARIMItn'e PlTEl theiu together as in tbe illustration. The porumn is very light and com pact, aad has but one row of rivets. The fallowing table gives tbe strength of tbe eolumn. 289a luAKIMKRS PATENT ALL STEKL COLUMN. s Hi o D Hi n n i I Hi Z h) f ^* n I <i ^ CO •= < 5 h] ^ of : 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» — i-i coc:= cere ■cc-so ^^ ■- ^^ -■* 2: t<>»i- I 1-1 ^ii — i ^ — i 3 c» »» c* ci ■?»' at«»aK wSrS w*^3* cj I? ?/ ?»' ?* *?* •:» c* c» r* 7» *:» 1 "" '" ! '""" ?J /. — .- at c *c •-* ~. i^X-j* — 1-^ r-^ ft r-m 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— ^«^ x-jiv 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 jifl-llliMiS 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:.i|s p.iAvo'nv Ml piiD'l .)ji-s "' ! -H'l 111 i-til '>iiii.jjs P'"''ll\ MilMJl I Jl. I'T.I.I.VV ' = "■■'11 I J'»"iS r- ■■« < - 51 "^ r ■- — r. x SCI - — £ — — 15 SsTir "r;; =-rJ— S:-*?? _ - _ 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 cover-plates riveted to their faces, when necessary to increase the strength of the column. Any bridge shop can make these columns by paying a small royalty to the patentee. As angles are the cheapest shape of rolled steel that is manufact- ured, this should be an economical column. The special advantages claimed for this column are : 1. A strong, economical section. 2. Provides continuous pipe space from basement to rool 3. Has four flat sides for connections. 4. Size of column does not vary when section is iucreased or diminished. 5. Does away with ** cap-plates," and joins sections of colamns firmly together, making a continuous column. Tests made in the hydraulic machine of the Keystone Bridge Works on 14-inch columns, 11 feet long, developed a resistance to crushing of from 38,000 to 40,000 pounds per square inch of section, and a modulus of elasticity of from 24.030,000 to 27,750.000 pounds. The tables on pages 2SQe-2S9h give the safe loads of several siaea of square, wall, and corner columns as computed by Mr. Gray. By varying the thickness of angles and adding cover-plates, the strength of the column can be greatly increased. Tables of wall and corner columns, and further particulars, maj be obtained by addressing Mr. J. U. Gray, C. E., Chicago. TBB GRAY STEEL COLUMKS. 269e THE GRAY STEEL COLUMNS, SAFE LOADS IN TONS OF 2,000 LBS. BY FORMULA 17,100 LBS. - 57 -. SQUARE OOIiUMNS WITHOUT OOVBR PI1ATB& 10" COLUMN. No. Pieces. 8 8 8 8 8 8 8 2k" X 2i" Lb. i 9.52 44 it A 11.76 (t »» I 13.84 ti tt 16.00 H ti k 18.00 2k" X 3" La. k 20.00 ti t» A 22.24 r. 13 ft. S.16 69.0 8.15 85.2 3.13 100.0 3.12 116.0 3.11 130.1 3.00 143.4 2.98 159.5 16 ft. 90 ft. 64.1 60.8 80.1 75.0 94.2 88.1 108.8 101.7 122.2 114.3 134.5 125.4 149.3 139.1 89 ft. 55.8 68.8 78.0 84.2 94.5 102.6 118.6 12" COLUMN. 8 8" X 8" Ls. i 11.52 3.81 86.1 81.9 77.8 67.6 8 (t ii A 14.24 3.79 106.3 101.2 96.1 83.2 8 ti ii f 16.88 3.77 125.9 119.9 118.7 96.4 8 3" X 4" Ls. » 19.84 3.57 149.2 141.6 138.7 114.4 8 ii ii ^ 22.96 3.55 169.8 160.9 158.1 129.9 8 8" X 5" Ls. /« 26.48 3.36 194.1 183.2 178.5 145.5 8 it ii i 30.00 8.34 219.0 207.8 195.1 164.8 8 it ii A 33.44 3.32 244.6 230.8 217.6 188.8 8 ii ii i 36.88 3.30 ♦J69.5 254.2 S:«.9 800.7 8 ii ii H 40.24 3.28 293.7 276.9 260.1 818.8 8 ti ii f 43.52 3.26 317.3 299.0 280.7 285.0 8 ii ti H 46.72 3.24 340.3 320.6 800.8 851.5 14'' COLUMN. 8 4" X 8" Ls. ^ 16.72 4.63 128.2 123.5 118.4 105.8 8 ii ii f 19.84 4.61 152.0 146.1 140.2 185.4 8 4" X 3i" Ls. t 21.36 4.50 163.1 156.9 150.2 188.9 8 4" x4" Ls. X 22.88 4.40 174.3 167.2 160.1 142.8 8 ii ii 26.48 4.39 201.7 193.4 185.2 164.5 8 4" X 5" Ls. ^ 30.00 4.12 226.6 216.7 206.7 181.8 8 (i ti i 34.00 4.10 256.7 245.3 834.0 205.6 8 4" X 6" Ls. k 38.00 3.93 285.2 272.0 258.7 1^.5 8 . ii A 42.48 3.92 321.7 806.7 891.8 854.8 8 ii it f 46.88 3.91 851.6 a35.2 318.8 877.6 8 (i ti H 51.29 3.89 384.4 866.3 84S.4 808.3 8 it i 55.52 3.88 416.0 396.3 876.8 887.9 8 it tt \l 59.76 3.87 447.6 426.4 405.8 808.5 8 if i 63.92 3.86 478.5 455.9 488.8 876.6 THE GllAY STEEL COLUMNS, 289/ SAFE LOADS IN TONS OP 2,000 LBS. BY FORMULA 17,100 LBS. - 57 -. r SQUARE OOX.UMNS WITHOUT OOVZSR PLATBS. 16" COLUMN. No. Pieces. Dimensions. Thick. Area Sq. In. r. 12 ft. 16 ft. 20 ft. 30 ft. 8 5" X 3" Ls. 1 22.88 5.45 178.4 172.7 166.9 152.6 8 5" X 34" Ls. f 24.40 5.85 190.8 184.6 178.3 162.6 8 5" X 4" Ls. f 25.84 5.24 200.7 194.0 187.2 170.4 8 it (I iV 30.00 5.21 232.8 225.0 217.1 197.4 8 5" X 6" Ls. ^ 33.44 5.01 258.5 249.4 240.2 217.5 8 ii (t * 88.00 5.00 293.7 283.4 272.8 246.9 8 It (t A 42.44 4.98 338.2 316.5 804.9 275.7 8 it it f 46.88 4.96 362.1 349.2 336.2 303.9 8 It t( H 51.36 4.94 396.4 382.3 868.0 832.5 8 ti tt i 55.52 4.93 428.5 413.1 897.7 359.2 8 ti tt H 59.68 4.92 460.5 443.9 427.3 385.9 18" COLUMN. 8 6" X 8i"L8. f 27.86 6.15 215.7 209.6 208.5 188.3 8 6" X 4" Lb. k 28.88 6.07 227.4 220.9 214.4 198.1 8 tt tt 38.44 6.05 268.2 2.55.7 248.1 229.3 8 tt tt i 38.00 6.03 299.0 290.4 281.8 260.2 8 6" X 6" Ls. ^n 40.48 5.64 316.6 806.8 297.0 272.5 8 tt tt i 46.00 5.63 359.8 348.6 837.4 309.6 8 tt tt ^ 51.44 5.62 402.5 389.7 8n.2 346.9 8 t. tt f 56.88 5.60 444.6 480.7 416.8 332.1 8 tt (t H 62.24 5.59 486.5 471.3 456.1 417.9 8 tt tt i 67.52 5.57 527.3 511.0 494.4 452.9 8 tt tt H 72.72 5.55 56S.0 550.0 632.0 487.3 8 tk It i 77.92 5.54 608.5 589.2 569.9 521.9 22" COLUMN. 8 8" X 6" Ls. i 54.00 7.30 431.4 421.3 411.1 385.8 8 tt T?B 60.48 7.29 483.1 471.8 460.4 4a2.0 8 it f 66.88 7.27 534.2 521.5 508.9 477.5 8 <t \h 73.38 7.26 585.2 571.3 557.5 5-23.0 8 tt f 79.52 7.24 634.8 619.8 604.8 507.3 8 tt H • 85.76 7.23 684.6 638.4 652.1 611.6 8 tt i 91.92 7.22 783.6 716.3 698.8 655.3 8 tt H 98.06 7.21 782.7 764.2 745.6 699.0 8 tt 1 104.16 7.20 881.2 812.4 791.6 742.2 289iir THE GKAY STEEL COLUMNS. SAFE LOADS IN TONS OF 2,000 LBS. BY FORMULA 17,100 LBS -.1. WAIiZi COLUMNS WITHOX7T OOVBR PZJLTB& 10" COLUMN. No. Pieces. • Dimensions. Thick. Area sq. in. r. 12 ft. 16 ft. 20 ft. 80 ft. 6 2i" X 2i"L8. i 7.14 2.25 48.0 48.7 89.3 26.5 6 it (t A 8.82 2.25 59.3 58.9 48.6 86.2 6 II it I 10.88 224 69.7 68.4 67.1 41.2 6 it It 12.00 2.24 80.0 78.8 65.0 47.6 6 It - it i 13.60 2.28 90.6 81.9 74.0 68.8 6 2k" X S" Ls. i 15.00 2.17 99.9 90.4 81.0 67.0 6 it it fk 16.68 2.16 110.9 100.8 89.8 68.8 12" COLUMN. 6 3" X 3" Ls. i 8.64 2.71 60.8 66.4 68.0 41.8 6 It it A 10.68 2.70 75.1 69.7 64.8 60.7 6 it it t 12.66 2.69 88.9 82.5 76.1 60.9 6 3" X 4" Ls. i 14.88 2.56 103.4 95.4 87.4 07.6 6 it it h 17.22 2.55 119.7 110.4 101.1 78.0 6 8" X 5" Ls. S 19.86 2.47 186.8 125.8 114.8 87.8 6 tt it k 22.50 2.47 155.0 142.5 180.0 98.0 6 t( it A 25.05 2.46 1^2 6 158.6 144.6 100.8 6 It it * 27.66 2.46 190.8 174.9 169.6 181.1 6 tt it H 30.18 2.45 207.4 190.6 178.8 181.6 6 tt it i 32 64 2 44 224.1 205.8 187.6 141.8 6 tt ti H 35.04 2.43 240.4 220.7 801.0 161.6 14" COLUMN. 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4" x8" Ls. ^ 12.54 it it 1 14.88 4" X sy Ls. f 1(5.02 4" X 4" Ls. k 17.16 ii ii 19.86 4" X 5" Ls. /e 22.50 it ii i 25.5U 4". X 6" Ls. i 28.50 it it A 31.86 tt it « 35.16 tt it ii 38 47 it it i 41.64 it tt H 44.82 tt ti i 47.94 3.83 3.31 3.25 3.19 3.18 8.06 8.05 2.97 2.96 2.95 2.95 2.94 2.94 2.93 91.8 86.7 81.6 108.8 102.7 06.6 116.7 110.0 108.8 124.6 iir.3 100.9 144.2 135.6 127.1 ll;6.2 152.1 142.1 183.7 172.8 160.9 204.3 191.2 178.1 228.2 213.5 198.7 251.7 235.4 819.1 275.3 257.6 289.7 297.9 278.6 259.8 820.7 299.7 878.7 342.7 320.4 298.1 74.8 88.6. 86.4 01.4 106.6 110.0 188.8 146.8 101.9 178.8 196.1 810.7 886.8 848.0 IH£ QBAY STEEL COLUHMS. SAFE LOADS IN TONS OF 8,000 LBS. BY FORMULA 17,1«» - BT -■ CORNER OOIiUMNS WITHODT COVER PLATBa 11' COLUMN HEDUCKD FROM H" COLUMN, 15" COLUMN KBDUCED E 1 IB" COLUUH. 1 it; 1 Vi i;iS i f li i iflois' 1*S i:».4 i5« III 290 BENDING-MOMENTS. CHAPTER Xir. BENDINGMOMENXa Tmk bonding-niomont of a beam or tnws represents the destnic- live energy of the load on the l)eani or truss at any point for which tlie ]>en(hng-nionicnt is computed. The moment of a force around any given axis is the product of the force into the pen^^ndicular distance between the line of action of the force and the axis, or the product of the force into its arm. In a I^eam the forces or loads are all vertical and the arms hori- zontal. The bending-moment at any cross-section of a beam is the alge- braic siun of the moments of the forces tending to turn the beam ai*ouud the horizontal axis passing through the ceuti'e of gravity of the section. Example. — Suppose we have a beam with one end securely fixed into a wall, and the other end projecting from it, as in Fig. I. ]jet us now 8upix>se wc liave a weight, which, if placed at tlie end of the beam, will cause it to break at the point of support. /^^v, ^ Then, if we were to place the ^\^ ^^^ weight on the Ix^am at a point ^^v, ^\^ ^-x near the wall, the beam would ^>.^ ^^V/ support the weight easily; but, as ^\^ we move the weight towards the outer end of the beam, the beam bends more and more; and, wh<»n ^' the weight is at the end, the beam breaks, as shown by the dotted lines. Fig. 1. Now, it is evident that the destructive eneigy of the weight la greater, the farther tlie weight is removed from tlie wall-end of the beam, thouixh the weight itself remains the same all the time. Tlie reason for this is, that the moment of the weight tends to turn the beam alwut the point A, and thus producer a pull on the ui>i)er fibres of the beam, and compresses the lower fibres. As the weight is moved out on the beam, its moment becomes greater, and hence also the pull and compression on the fibres; and, when tlie ^rm ^^ BENDING-MOMENTS. 291 moment of the weight produces a greater tension or compression on the fibres tlian they are capable of resisting, they fail, and the beam breaks. Before the fibres break, however, they commence to striitch, and this allows the beam to bend: hence the name "Ixmd- ing^nionient" h«s been given to the moment which causes a beam to bend, and perhaps idtimately to break. There may, of course, be several loads on a beam, and each one having a d liferent monvent, tending to bend tlie beam; and it may ilso occur that some of the weights may tend to turn the beam in different directions: the algebraic sum of their moments (calling those tending to turn the beam to the right +, and the others — ) would be the bending-moment of the beam. Knowing the bending-moment of a beam, we have only to find the section of the beam that is capable of resisting it, as is shown in the general theory of beams. Chap. XIV. To determine the bending-moments of beams mathematically, requires considerable training in mechanics and niathematics; but, as most beams may be placed under son\e one of the following cases, we shall give the bending-moment for these cases, and then show how the bending-moment for any other methods of loading may be easily obtained by a scale diagram. Examples of Bencliugr-Momeuts. Case I. Beam fixed at one end^ and loaded with concentrated load W. Bending-moment = W X L. {L may, or may not, be the whole length of the beam, according to where the weight is located. ) Case II. Beam fixod at one end, loaded with ^^^ u dt-'itribntt'd load \V. ^-'^ Bending moment = W x - • Note. — The length L mast always* he taken In the same unit of measurement «>« is listed for the breadth and depth : thus, if B and D are in inches, L must be in inches. 292 BBNDING-MOMENTS. Case III. Jionm fixpd at one end, loaded with both a concentrated and a distributed load. /., Bending-moment = P X Lj + JK x -^ Casr IV. licam supported at both ends, loaded with concentrated load lU centre, W J Bcnding-raoment Case V. Beam s^iipported at both ends, loaded with a distributed load W. V -, - ■■: 'm% <?; 'n Fig.6 Bending-moint;nt Cask VI. livam supported at both tnids, loaded with concentrated load nol at ('('litre Bending-moment = Wx m X n BKNDING-MOMENTS. 293 Cask VTI. Beam supported at both encU, loaded rcith two equal concen- trated loadSy equally distant from the centre. Bending-moment = W X nu m-^ v^m <rm Flg.8 iiW From these examples it will be seen that all the quantities which enter into the bending-moment aro the W?ight, the span, and the distance of point of application of concenti-ated load from each end. The hendin{i-moment for any case other than the above may easily be obtained by the graphic method, which will now b« explained. Graphic Method of Determining Benclin^- Moments. The bending-moment of a l)eam supported at both ends, and loader! with one concentrated load, may be shown graphically, as follows : — Let W be the weight applied, as shown. Then, by rule under Case VI., thebeuding- niomeut directly under [<« f^ J^ jjj W = IF X in X n Draw the beam, with ^ the given span, accu- rately to scale, and then measure down the line AB equal to the bending - mo- iiuMit. Connect B with each end of the beam. If, then, we wished to find the bending- moment at any other point of the beam, as at o, draw the vertical line y to BC ; and its length, measured to the same scale as ABf will give the bending-moment at o. Beam with two concentrated loads. To draw tlie bending-moment for a beam with two concentrated loads, first draw the dotted Hues ABl) and ACD, giving the outline 294 BENDINO-MOMENTS. of Ihe bending-nioment for each loful separately; KB heing eqiwl toWx It^^ „„(, rr pqjial to P X '-^ Fi9.IO Now, Ihe beiuting-momnnt at Uie pclnt E equals RJi, doe to tha loud ir, anil Kb, clue to tlie load P: liunce the l)Uiiilii^(-nioiuent at i'slioHld lie drawn ainul to En+ Kh — Kll, ; anil at Fthe beiMl- Iiig iiioiiienl shoHkt equal H,'+ Fc= FC,. The otOUne lor the bendlng-niomunt due to both loaits, tlieu, would be the Uiie AIl^C'iD, anil the greatest bending-iuoment would In this parUe- ular tasu be FC'i- Jleam with three concentrated load*. Fiy.tl Pmcpficl as in the laat ease, and drawthp hending-moment for eaoli load separately. Then make AD = A\ + A2 + AS, BB = m + m + /J:t, and i:F= (I + r2 + f73. The line IIDEFI urill then Ih- the c)ut1hie for the Ix'iiding-ii'ouiunl due to all tl)e wt^lghts. The iH>iidln(>-nioiuent for a lieani loadeil tvith nnj- number of «w <«ntiiited weiglita uiav be drawn In tlie oanie way. BENDING -HOHENT& Beam with untformly distrVnUed load. Draw the beam with the given spaii. accurately to a scale, m before, and at the middle of the beam draw the vertical line AH I equal to If x gi W representing the whole distributed load. Then connect the points C, it, D by a parabola, and It will ^ve the outline of the bendlng-moraents. if, now, we wanteil the bendlng-Qiomeat at the point a, we have only to draw the vertical line ab, and measure It to th<! same scale as ^ B, and it will be the moment deatred. Hethoda for drawing the parabola may be found in " Geometrical Problems," Part I. Beam loaded viith both diatritruted and concenlrated loads. To determine the bendlng-momcnt in this case, we have only to combine the methods for concentrated loads and for the distributed load, as shown in „ the accompanying figure. The bend- ing-moment at any point on the beam will then be lim- ited by the line ABC on top, and CHEFA on the bottom ; and the gii'atesi bendfiig- moraent will be the longest verti- cal tine that can be drawn between Ha.ia these two bounding lines. For example, the tiending-momeiit at X would be BE. The posi- tion of the greatest ben ding-moment will depend upon the position. of the concentrated loads, and it may aud may not occur at tlie 296 BENDING-MOMENTS. Example. —What is the greatest bending-ittmneilt In a hektk of 20 feet span, loaded with a distributed load of 800 pounds and a concentrated load of 500 pounds 6 feet from one end, and a con- centrated load of 600 pounds 7 feet from the other end ? L Ans. 1st, The moment due to the distributed load is W X ^* 800 X 20 or y = 2000 pounds. We therefore lay off to a scale, say 4000 pounds to the inch, Bl = 2000 pounds, and draw a parabola between the points Af B, and C. 2d, The bend- ing-moment fbr the concentrated load of 500 pounds is 50 X 6 X 14 20 , or 2100 pounds. Hence we draw E2 = 2100 pounds, to the same scale as Bly and then draw the lines AE and CE, 3d, The bending-monient for the concentrated load of 600 pounds 600 X 7 X 13 — , or 2730 pounds; and we draw i)8 = 2780 pounds, IS 20 and connect D with A and C. 4th, Make EII = 2 — 4, and DG = 3 — 5, and connect O and H with C and A and with each other. The greatest bending-nioment will be represented by the longest vertical line which can be drawn between the parabola ABC and tlu* broken line AHGC. In this example we find the longest veitl- cal line which can be drawn is xy ; and by scaling it we find the greatest bending-nionient to be 5550 pounds, applied 10 feet 11 inches from the point A. In this case, the position of the line Xy was determined by drawing the line TT\ parallel to IIG, and tangent to ABC, The line Xy is drawn through the point of tangency. Note. — As the measurements ased for determining the bending-momeiit \ in feet, we must multiply the moment by 12. to get it into inch poands; otfaar- wise, in working out the dimenaione of the beam, they would be in feot Inntfiad of inches. MOMENTS OF INERTIA AND RESISTANCE. 297 CHAPTER Xm. MOMENTS OF INERTIA AND RESISTANCE, AND RADIUS OF GYRATION. Moment of Inertia. The strength of sections to resist sti-ains, either as girders or as posts, depends not only on the area, but also on the form of the cross-section. The property of the section which represents the effect of the form upon the strength of a beam or post is its mo- ment of inertia, usually denoted by I. The moment of inertia for any cross-section is the sum of the products obtained by multiply- ing the area of each particle in the cross-section by the square of its distance from the neutral axis. Note. — The ueutral axis of a beam is the line on which there is neither tension nor compression; and, for wooden or wronght-iron beams or posts, it may, for all practical purposes, be considered as passing through the centre of gravity of the cross-section. For most forms of cross-section the moment of inertia is best found by the aid of the calculus; though it may be obtained by dividing the figure into squares or triangles, and multiplying their areas by the squares of the distance of their centres of gravity from the neutral axis. Moment op Resistance. The resistance of a beam to bending and cross^breaking at any given* cross-section is the moment of the two equal and opposite forces, consisting of the thrust along the longitudinally compressed layers, and the tension along the longitudinally stretched layers. This moment, called "the moment of resistance," is, for any given cross-section of a beam, equal to • moment of inertia extreme distance from axis* In the general formula for strength of columns, given on p. 281, the effect of the form of the column is expressed by the square of the radius of gfyration, which is the moment of inertia of the sectiou divided by its area; or -r = r^. The moments of inertia of the principal elementary sections, and a few common 206 MOMENTS OF INERTIA AND RESISTANCE. forms, are given below, which will enable the moment about any given neutral axis for any other section to be readily calculated by merely adding together the moments about the given axis of the elementary sections of which it is composed. In the case of hollow or re-entering sections, the moment of the hollow portion is to be subtracted from that of the enclosing area. Moments of Inertia and Resistance, and Radii of Gyration. I = Moment of inertia. R — Moment of resistance. G = Radius of gyration. • A = Area of the section. Position of neutral axis represented by broken line. 1 1 . --1 — rf w~— • i ui — Y-h--* I bcP ■" 12* R b(P = 6' & -12' I 6(i» = 3' <P 3 1 « — i- — » z 1 1 1 1 ? » 1 1 T I E 6 — \ r / = h(p - bii^ 12 i 21 I— ^ = ;/ ' 2X (^ = bd - b,d. I-Ream (another fonnula). Let a denote area of one flange, a' denotes area of w(»b, cT = effective depth between centres of gravity of flanges; then -v'+6;2 This is the formula generally used by the engineers for the iioiir companies. MOMENTS OF INERTIA AND RESISTANCE. 299 y-h'i Ie--6~-^ T- 1 — T 1 1 fy Ik _^ J._- ,. . 1 ^S 10 L_li 1 n. ihi ». — h- <b •t ■t J !■ h O- I 6# "■ 3 " -M. <!2 4 ~ 12' Gf2 I / 6d« = 36' It 3/ 24' G^ i d^ 18 I 6d« -12- G^ = 6' I 6d« = 4' C2 ~ 2* / _ bd^ 4- 6,(Z,« j__ {b,-b)dj 3 /? J G2 / " A' I = 0.7854)-*. R = 0.7854r3. (?2= -r. 7 = 0.7854 ()•*- i? G2 = 0.7854 U'S-^J r 1 r* - r* BOO TABLES OF INERTIA AND GYRATION. Moments of Inertia and Radii of Gyration of Mercliant Sliapes of Iron and Steel. For the sections of rolled iron beams and bars to be found in the tnarket, the moments of inertia are given in the '* Book of Sections " published by the manufacturers. The following tables give the moments of inertia and radii of gyration for the principal sections manufactured by ( amegie, Phipps & Co., the New Jersey Steel and Iron Company, and the Phoenix Iron Company (revised to October 1, 1891). The Pencoyd Iron Works have recently made changes in a number of their sections, and some of the old seotioDS of iron beams and channels have been abandoned, and they are not at present prepared to furnish the revised data. The tables give the least weight for each section of iron beam, and the minimum and maximum weights for channels, deck beams, and angle irons. These shapes can be rolled for any weight between the two given, while the weight of the beams can also be greatly increased. With the quantities given in these tables, one can find all the data required in usual calculations. The tables on pages 322-24 will be found very oonTenie&t in computing the strength of struts formed of two or four angle bart. TABLES OF INERTIA AND GYRATION. 301 MOMENTS OF INERTIA AND RADII OF GYRATION OF CARNEGIE BEAMS— IRON. V u IB A \1 -Oi I. n. • III. IV. .V. Size, in Weight per lw)t, in lbs. Area of cross- bection, Moments of inertia. Radii of j gyration. inches. in sq. in 24.0 Axis A B. Axis C D. Axis A B. Axis CD. 15 80 813.7 38.8 5.82 1.27 15 60 18.0 625.5 23.0 5.90 1.13 15 50 15.0 522.6 15.5 5.90 1.02 12 56.5 17.0 348.5 17.4 4.53 1 01 12 42 12.6 274.8 11.0 4.67 0.94 10^ 40 12.0 201.7 12.0 4.10 1.00 m 31.5 1^.5 165.0 8.01 4.17 0.92 10 42 12.6 198.8 13.74 3.97 1.04 10 36 10.8 170.6 10.02 3.97 0.96 10 30 9.0 145.8 7.43 4.03 0.91 9 38.5 11.6 150.1 12.84 3.61 1.05 9 28.5 8.6 110.3 6.79 3.59 0.89 9 2J.5 7.1 92.3 4.64 3.62 0.81 8 34 10.2 102.0 10.2 3.16 0.99 8 27 8.1 82.5 6.30 3.19 88 8 21.5 6.5 66.2 3.95 3.20 0.78 7 22 6.6 51.9 4.58 2.80 0.83 7 18 5.4 44.2 3.28 2.86 0.78 6 16 4.8 29.0 2.87 2.46 0.77 6 13.5 4.1 24 4 2.00 2.46 70 5 12 3.6 14.4 1.46 2.00 0.64 5 10 3.0 12.5 1.15 2.04 0.62 4 7 2.1 5.7 0.67 1.65 0.57 4 6 1.8 4.6 0.36 1.61 0.45 3 9 2.7 3 5 0.85 1.15 0.56 3 5.5 1.7 2.5 0.44 1.24 0.52 80a MOMENTS OF INERTIA MOMENTS OF INERTIA AND RADII OF aYRATION OF CARNEGIE BEAMS-STEEL. U 71 ~~i — J IB Size, in inches. 24 20 20 15 15 15 15 12 12 10 10 9 9 8 8 7 7 6 6 5 5 4 4 1 n m. Weight per foot, in lbs. 80 80 64 75 60 50 41 40 32 83 25.5 27 21 22 18 20 15.5 16 13 13 10 10 7.5 Area of Momcntfl of inertia cross- sec lion, ill sq. in. 23.2 23.5 18.8 22.1 17.6 14.7 12.0 11 9 9 7, 7 4 7 5 9 Rsdii of gyradon. 2,059.3 1.449.2 1,146.0 75r 7 644.0 529 . 7 424.1 281.3 222.3 161.3 12:^.7 110.6 6.2 84.3 6.5 71 9 5.3 57.8 5.9 49.7 4.6 38.6 4.7 28.6 3.8 23.5 8.8 15.7 3.0 12.4 2.9 7.7 2.2 5.9 sis CD. Axis A B. 41.6 9.42 45.6 7.86 27.8 7.80 40 1 5.86 80.4 6.04 21.0 6.00 14.0 6.94 16.8 4.90 10.8 4.85 11.8 4.08 7.32 4.06 9.10 8.72 5.56 8.70 6.62 8.38 4 35 8.80 5.52 2.91 8.47 2.91 3.24 2.47 2.27 2.48 1.99 2.08 1.29 2.06 1.22 1.62 0.75 1.68 Axis CD. 1.34 1.89 1.20 1.85 1.82 1.20 1.08 1.20 1.04 1.10 0.99 1.07 0.96 1.01 0.91 0.97 0.87 0.83 0.77 0.72 0.6(S 0.66 0.58 AND RADII OF GYRATION. 803 MOMENTS OF INERTIA AND RADII OP GYRATION OF CARNEGIE DECK BEAMS-IRON. [I _J : /^ D d-*\^ I. II. in IV. V. Size, in Weight per foot, in lbs. Area of cross- section, in sq. in. Moments of inertia. Radii of gyration. inches. Axis A B. Axis C D. Axis A B. Axis CD. 10 26.9 8.1 118.4 6.12 3.83 - 0.87 10 85 2 10.6 139.9 7.41 8.64 0.84 9 28.2 7.0 77.6 2.45 3.34 0.59 9 29.8 8.9 01.0 3.15 3.19 0.59 8 21.4 6.4 52.1 2.23 2.85 0.59 8 28.0 8.4 63.2 2.96 2.74 0.59 7 17.0 54 34.4 1.81 2.60 0.59 7 22.8 6.9 41.8 2 34 2.47 0.58 Deck Beams— Steel. 9 26 7.6 85.2 4.61 3.35 0.76 9 30 8.8 93.2 5.18 3.25 0.75 8 20 5.9 57.3 4.45 3.12 0.82 8 23.8 7.0 63.5 5.21 8.01 0.82 7 • 20 5.9 42.2 4.50 2.67 0.82 7 . 23.5 6.9 46.6 4.87 2.60 0.82 304 MOMKNTS OF INEBTIA MOMENTS OF INERTIA AND RADII OF GYRATION OF CARNEGIE OHANNEL-BARS—IRON. n IB ^ I. n. Moments IV. VI. R»dii of Distance of Siz<', in inches. Weight per foot, in 11)8. Area of cross-section, in sq. in. of inertia. gyraticm. centre «'f gravity fhxn oatdde of Axis A B. 473.1 AxIr a B. web. 15 60 18 5.12 0.88 15 40 li 360.6 5.48 0.82 12 50 15 247.3 4.10 0.88 12 30 9 17.]. 7 4.40 0.76 12 20 6 120.2 4 48 0.70 10 35 10.5 126.3 8.47 0.75 10 20 6.0 88.8 8.85 0.70 10 16 4.8 62.8 8 62 0.55 9 30 9.0 87.8 8.12 0.73 9 18 5.4 63.5 8.48 0.67 8 28 8.4 63.9 2.76 0.78 8 20 6.0 45.5 2.75 0.69 8 16 4.8 39.1 2.85 0.57 8 10 3.0 28. :J 8.07 0.50 7 20 6.0 37.7 . 2.51 0.67 7 18i 4.0 25.5 2.51 0.53 7 8^ 2.5 19.0 2.73 0.49 6 16 4.8 ?2.3 2.16 0.08 6 10 3.0 16.9 2.R8 0.62 6 7i 2.2 12 2 a 84 0.48 5 14 4.2 13.10 1.77 0.61 5 8^ 2.5 8.72 1.85 0.49 4 9 2.7 5.75 1.46 0.56 4 5 1.5 3.69 1 57 0.45 3i 8.1 2.4 3.82 1.25 0.52 3 6 1.8 2.23 1.15 0.51 AND RADII OF GYRATION. 305 FOMENTS OF INERTIA AND RADII OF GYRATION OF CARNEGIE CHANNEL-BARS— STEEL. ;b I. II. IV. VI. Moments Radii of Distance of Size, in Weight per foot, in lbs. Area of cross-section, in bq. iu. of inertia. gyration. centre of Lravily from oatside of incbes. Axis A B. Axis A B. web. 15 82 9.4 284.5 5.53 0.75 15 51 15.0 390.0 5.13 0.77 12 20 5.9 117.9 4.49 0.62 12 80i 8.9 153.9 4.17 0.62 iio 15i 4.5 63.8 3.80 0.63 10 23i 12| 20i 7.0 84.6 3.50 0.61 9 8.7 43.3 3.42 0.58 9 6 58.5 3.14 0.56 8 lOi 3.0 28.2 3.05 0.53 8 17i 5.0 38.9 2.78 0.52 7 Sk 2.5 17.4 2.67 0^49 7 m 4.3 24 6 2.42 0.48 6 7 2.1 11.1 2.31 0.48 6 12 8.6 15.6 2.09 0.47 5 6 1.7 6.5 1.94 48 5 lOi 3.0 9.1 1.75 0.47 4 5 1.4 3.5 1.57 0.48 4 Si 2.4 4.8 1.81 0.48 Deck Beams — Steel. 9 26 7.6 85.2 3.85 9 30 8.8 93.2 8.25 8 20 5.9 57.3 3.12 8 28.8 7.0 63.5 3 01 7 20 5.9 43.2 2.67 7 28.5 6.9 46.6 2.60 306 MOMENTS OF INERTIA MOMENTS OP INERTIA AND RADII OP GYRATION OF CARNEGIE ANGLE-BARS. For minimum and maximum thickneeses and weight. ANGLES WITH EQUAL LEGS — IRON OR STEEL. Weights in Table are for Iron; for Steely add 2 per cent. I. VI. n. IV. V. Distance Sizi*. in inches . Weight, per foot. Area of crosp- pection. of centre of gravity from out- Hide of Moments of inertia. Raclli of gyntioii. in sQ. in. flange, in inc.lietit. Axis A B. 17.68 Axis A B. AxIbOD. 6 xG J16.0 5.06 1.66 1.87 1.19 (33.1 9.95 1.85 34.09 1.85 1.17 5 x5 J12.0 3.61 1.39 8.74 1.56 0.99 127.0 8.28 1.61 20.00 1.56 1.00 4 x4 j 9.5 120.1 2.86 1.14 4.36 1.28 0.79 6.03 1.33 9.00 1.22 0.88 3ix3^ j 8.3 (17.4 2.48 l.Ol 2.87 1.07 0.68 5.22 1.20 5.90 1.06 0.72 3 x3 4.8 1.44 0.84 1.24 0.98 0.68 ^11.7 3.50 1.01 3.00 0.93 0.62 2^x2^ j 4.4 ) 9.0 1.31 0.78 0.98 0.86 0.64 2.69 0.95 2.22 0.91 0.06 2i X 2A \ 4.0 1.19 0.72 0.70 0.77 0.50 \ 7.9 2.37 83 1.44 0.78 0.60 2i X 21 j 3.5 ) 7.0 1.06 2.11 0.66 0.78 0.51 1.04 0.69 0.70 0.46 0.49 2 x2 \ 2 4 (».71 0.57 28 0.62 040 '( 5.5 1.65 0.60 0.06 0.68 0.64 1^x1! j 2.1 0.6i . 51 0.18 0.54 0.22 4 9 1.47 0.64 0.44 0.56 0.40 l^xli 1.8 0.53 0.44 0.11 0.46 0.29 \ 3.6 1 06 . r,4 0.24 48 0.88 li X \{ j 1.0 0.80 35 0.044 0.38 0.22 1.9 0.56 0.40 0.077 0.3V 0.24 HxU S 0.0 / 1.9 0.27 0.32 0.032 0.84 0.19 0.55 0.40 0.077 0.37 0.25- 1 xl j 0.8 \ 1.5 0.23 0.30 0.022 0.81 0.21 0.44 34 0.037 0.29 0.18 i x} J 0.6 ( 0.8 0.17 0.23 0.009 0.28 0.14 0.25 : 0.26 0.012 0.22 0.16 AND RADII OP GYKATION. 307 MOMENTS OP INERTIA AND RADII OF GYRATION OP CARNEGIE ANGLE- BARS. Forminimam and maximnm thicknesses and weight. UNEVEN LEGS — IRON OR STEEL, Wei ghts in TcUde are for Iron; . for Steel, add 2 per cent I. II. Mom€ inei III. mts of rtia. IV. V. VI. VI. Size, in Weight, per foot. Area of c roes- section, Radii of gyration. Distance from hate to neutral axis. inches. 1 int'q.in. Axis Axis Axis Axis Axis d. /. AB. CD. AB CD. 1.17 EP. 6 x4 J12.0 3.61 13.47 4.90 1.93 .88 1.94 0.94 (27.3 8.18 29.58 10.68 1.90 1.14 .88 2.15 1.16 6 x8i jll.4 3.42 12.86 3.34 1.94 0.99 .77 2.04 0.79 ^25.8 7.75 28.20 7.25 1.91 97 .78 2.25 1.00 5 x4 jlO.8 3.23 8.14 4.67 1.59 1.20 .86 1.53 1.03 (22.8 6.83 16.75 9.57 1.57 1.19 .88 1 72 1.22 5 x3i jlO.2 3. Oh 7.78 3.18 1.60 1.02 .76 1.61 86 I2I.4 6.42 15.99 6.52 1.58 l.Ol .77 1.80 1.05 5 x3 j 9.5 2.86 ► 7.37 2.04 1.61 0.85 .66 1.70 0.70 (20.1 6.02 15.19 4.18 1.59 0.83 .66. 1.89 0.89 4ix3 j 8.9 2.67 5.50 1.98 1.44 0.86 .66 1.49 0.74 (18.7 5.62 11.26 4.06 1.42 0.85 .67 1.08 0.98 4 x3i j 8.9 2 67 4.18 2 99 1.25 1.06 .73 1.21 0.96 (18.7 5.61 8 53 6.10 1.23 1.04 .74 1.39 1.14 4 x3 j 7.0 2.09 3.38 1.G5 1.27 0.89 .65 1.26 0.76 (17.4 5.21 8.09 3.92 1.25' 0.87 .66 1.47 0.97 3ix3 i 6.5 1.93 2.33 1.58 1.10 0.90 .63 1.06 0.81 (16.0 4.80 5.54 3.76 1.07 0.89 .65 1.27 1.02 3ix2i ( 4.8 1.44 1.80 0.78 1.12 0.74 .55 1.11 0.61 1 9.8 2.92 4.0s 1.81 1.17 0.78 .58 1.27^ 0.77 3ix2 i 4.2 1.25 1.36 0.40 1.04 0.57, .44! 1.09 0.48 ) 8,3 2.48 2.70 0.81 1.04 0.57 .45i 1.22 0.59 3 x2i i 4.4 1.31 1.17 0.74 O.Ooi 0.75; .53 0.91 66 } 8.7 2.60 2.34 1.49 0.951 0.70 .54 1 03 0.78 3 x2 j 4.0 1.19 1.09 0.39 0.90' 0.57 .44 0.99 0.49 ( 8.0 2.31 2 27 0.84 0.1)9 0.60 .47 1.12 0.63 2ix2 j 2.7 81 51 0.29 0.79, ('.60 .43 76 51 ( 7.2 2.18 1.38 0.80 80; 0.61 .44 0.87 0.67 2 xll j 2.6 ( 4.6 0.78 0.37 0.12 0.63 0.39 .30 0.69 0.37 1.^9 0.56 0.*^2 0.63 0.40 .31 0.79 0.47 l}xl 0.9 0.28 05 0.02 0.44 0.29 .22 0.44 0.26 308 MOMENTS OF INEBTIA MOMENTS OF INERTIA AND RADII OF GYRATION OF CARNEGIE T-BARS— IRON OR STEEL. c 8 Weights in Table are for Iron ; for Steely add 2 per cent. ni. Moments of inertia. Azi» : Axis A B. CD. 5 x3 5 x2i 4ix3i 4 x5 x5 x4i x4" x3 x2i x2 3.i X 4 3iLx4 3ix3* ^x^ iJixS Ux% 3x4 x3i x3 x3 x2i x2i 2Ax3 2i X 2i 2ix2i 2 x2 2 xH If'x 1} Uxli 1 xl 5.5 4.9 3.7 2 8 2.1 2.8 2.1 2.5 2.1 1.8 1 8 1.89 1.42 1.89 1.42 1.88 1.18 1.21 1.20 1.20 0.75 0.89 0.':5 0.44 0.44 0.2o 0.18 0.18 VI 0.08 0.02 IV. ^ I V. RadUof gyratifin. AzIh AB. 0.76 0.64 1.04 1.54 1.56 1.87 1.88 1.20 0.86 0.70 0.51 1.21 1.22 1.04 1.05 0.87 0.89 1.23 1.06 0.88 0.90 0.72 0.7;? 0.94 0.74 0.67 60 0.42 0.51 0.49 0.29 Axis CD. 1.21 1.26 90 0.79 0.78 0.81 0.80 0.88 0.88 0.91 96 0.72 0.70 0.74 0.78 0.77 0.76 0.59 0.62 0.64 62 66 0.65 0.51 0.52 0.47 0.42 0.45 0.37 0.84 0.21 VL Distance f/from ba.<te to neatnl azia. 0.67 0.87 t.ll 1.06 1.61 1.87 1.81 1.15 0.78 0.00 0.51 1.25 1.19 1.06 1.01 0.88 0.78 1.88 1.18 0.98 0.86 0.71 0.68 0.92 0.74 0.66 0.60 0.42 0.64 0.42 0. AND BAUII OF GYRATION. Weighh in Tabh are for Iron ; for Steel, add 2 per eent. 310 MOMENTS OF INERTIA. MOMENTS OF INERTIA AND RADII OP GYRATION OP TRENTON BEAMS— IRON. 7 I* .JL. B \J Weight per foot, in IViM I. Area of n. m. IV. V. Size, in IncheH. Moments ( >f inertia. Radii of gyration. IIL J vO • section, in sq. in. 90.6 27.20 Axis A B. Axis C I). Axis A B. AxiaCD. 20 1,650.3 46.50 7.79 1.30 20 66.6 19.97 1,238.0 26.62 7 88 1.15 15 66.6 20.02 707.1 27.46 5.94 1.17 15 50 15.04 523.5 15 29 5.90 1.01 15 41.6 12.36 434.5 11.64 5.98 1.02 12i 56.6 16.77 391.2 25.41 4.88 1.28 12i 41.6 12.33 288.0 11.54 4.80 .Vt 13 40 11.73 281.3 16.76 4.90 1.20 12 32 9.46 2-29.2 11.66 4.92 1.11 m 45 13.36 23:J . 7 15.80 4.18 1.10 loi ;J5 10.44 185.6 9.43 4.22 .96 lOA 30 8.90 164.0 8.09 4.29 .95 9' 41.6 12.33 150.8 11.28 3.47 .95 9 28.3 8.50 111.9 7.35 8.63 .98 9 23. :J 7.00 93.9 4.92 8.66 .84 8 26.0 H.03 83.9 7.55 3.28 . vO 8 21.6 6.37 67.4 4.55 3.24 .85 7 18.3 5.50 44.3 3.90 2.84 .84 6 40 11.84 64.9 18.59 2.86 1.25 6 30 8.70 49.8 ' 10.78 2.39 1.11 C 16.6 4.97 29 . 2 2.86 2.42 .70 6 18.3 3.98 23.5 1.61 2.48 .64 5 13.3 3.90 15.4 1.68 1.94 .66 5 10 2.99 12.1 1.04 1.99 .59 4 12.3 3 6() 9.2 1.74 1.59 .69 4 10 2.91 7.5 1.11 1 1.60 .62 4 6 1.77 4.5 .31 1.60 .48 AKP RADII OF GYRATION. 311 )MENTS OF INERTIA AND RADII OF GYRATION OF TRENTON BEAMS— STEEL. ►G- 7 r iB 4 I. II. III. rv. V. Size, in Weight per foot, in lbs. Area of cross- section, Moments of inertia. Eadii of gyration. inches. in sq. in. Axis A B. Axis CD. Axis A B. Axis CD. 15 50 14.70 529.7 20.96 6.00 1.19 15 41 12.02 424.4 13.94 5.94 1.07 13 40 11.73 281.3 16.76 4.89 1.19 12 82 9.46 229.2 11.64 4.93 1.10 10 45 13.14 216.1 17.94 4.05 1.17 10 33 9.67 1(J1.3 11.81 4.08 1.10 10 25.3 7.50 123.6 7. 82 4.06 .98 9 27 7.98 110 6 9.13 3.73 1.07 9 21 6.15 84.3 5.56 3.70 .95 8 22 6.47 71.9 6.62 3.34 1.01 8 18 5.28 57.7 4.36 3 30 .91 7 20 5.87 49.7 5 51 2.91 ,97 7 15.5 4.55 38.6 3.47 2.91 .87 6 16.6 4 97 29.2 2.86 2.42 .76 6 13.3 3.97 23.4 1.63 2.42 .64 5 13 3.80 15.7 1.98 2.03 .73 5 10 2.96 12.4 1.30 2 04 .67 4 10 2.94 7.7 1.22 1.62 .04 4 7.3 2.21 5.9 .75 1.63 .59 .l4i. _:.^_i-.- 312 MOMENTS OF IKERTIA MOMENTS OF INERTIA AND RADII OP GYRATION OP TRENTON CHANNEL AND DECK BEAMS— IRON. w IC d-r^^B I. II. m. IV. V. VI. Size, in inches. Weight per Area of crosg- Moments of inertia. Radii of gyration. Distanced of centre of gravity foot, lbs. sectioR, 8q. iu. Axis AB. 1 Axis ', Axis C D. A B. AxlH CD. from oat- aide of web. Channel Bars. 15 63.3 18.85 15 40 12.00 12i 40.6 14.10 12i 23.3 7.00 lOi 20 6.00 10 16 4.77 9 23.3 7.0-2 9 16.6 5.08 8 15 4.48 8 11 3 30 7 12 3.60 7 8.5 2.54 6 15 4 82 6 11 3.20 6 7.5 2 . eo 5 6.3 1.92 4 5.5 1.65 3 5 1.45 586.0 32.25 5.57 1.31 1.26 376.0 14.47 5.60 1.10 0.25 291.6 17.87 4.65 1.12 1.120 153.2 5.04 4.68 .86 0.755 88.4 3.84 3.84 .80 0.628 64.0 2.20 3.68 .68 0.666 82.1 5.35 8 42 .87 0.86 58.8 2.53 3.40 .70 0.08 44.5 2.54 8.15 .75 0.76 32 9 1.44 3.16 .66 0.68 27.1 1.96 2.74 .88 0.716 17.3 .8;^ 2.61 .67 0.611 21.7 2.12 2.24 .70 0.725 17.2 1.30 2 32 .64 0.68 12.6 .70 2.37 .66 0.64 7.2 .44 1.98 .48 0.464 3.9 .32 1.54 .44 46 2.0 .29 1.17 .45 0.61 Deck Beams. 8 7 21.6 18.3 6.25 5.35 1 54.7 35.1 8.7 3.6 2.96 2.56 .76 .82 • AND RADII OF GYRATION. 313 MOMENTS OF INERTIA OF TRENTON ANGLE-BARS. Size, in inches. Weight per foot, in J be. I. Area of erosB- section, iu sq.ins. II. Moment of inertia. VI. Distance d from base to neutral axis, in inches. EVEN LEGS. 6 in. 4i " 4 ** 3} " 3 25 2i 2i 2 13 li 1 1 << X X X X X X X X X X X X X X X 6 in 4i " 4 3i 3 2J 2i 2i 2 13 H li 1 J I a (( (( 19 to 32i 12i to 20^ 9i to 18 8} to 14i 4.8 to 12i 5.4 to 9i 3.9 to 7i 3^ to si to 2 to IJ to 1 to 3 to 0.6 to T%tO 6 4i 3i 2i li 1 0.8 5.75 19.910 1.685 Axis A 3.75 7.200 1.286 2.86 4.360 1.138 2.48 2.860 1.013 1.44 1.240 , 0.842 1.62 1.150 0.802 1.19 0.700 0.717 1.06 0.500 0.654 0.94 0.350 0.592 0.62 0.180 0.507 0.53 0.110 0.444 0.30 0.044 0.358 0.23 0.022 0.296 0.20 0.014 0.264 0.17 0.009 0.233 B UNEVEN -=- LEGS. 6 in. X 4 in. i 5 " X 3i " ; 4i " X 3 " i( X 3 (< 3^ " X H " 3 " X 2> " 8 " X 2 " 14 to 23 4.18 j 15.460 ] 5.600 1.964 0.964 10.2 to 19i 3.05 j 7.780 ] 3.190 1.610 0.8(K) 9 to U\ 2.67 S 5.490 \ 1.980 1.490 0.740 7 to 14^ 2.09 j 3.370 1 1.640 1.260 0.760 4.0 1.19 j 1.500 ] 0.170 1.320 0.320 4i to 9i 1.31 j 1.170 1 0.740 0.910 0.660 4 to 7i 1.19 ( 1.090 ] 0.390 0.990 0.490 Axis C " A 16 ii il a a n n il (t c A C A C A C A C A D B I) B D B D B D B D B C D A B 3U MOMENTS OF INERTIA MOMENTS OF INERTIA OP TKBNTON T-BABSL c B ^fe B Size, in inches. 4*x4 3^x31 3 x3 2ix2i 2 x2 6 x2i 3 x2 2 xli aixli 2 xl I. Weight per foot, in lbs. Area of croBs- Bection, in sq. in. m 3.75 9.6 2.87 7 2.11 5 1.46 3i 0.94 11.7 3.50 4.8 1.45 3.00 0.91 2.40 0.74 2.15 0.65 1.86 0.56 IL Moment of inertia. IV. i 5.560 ^2.620 j 3.260 <1&30 i 1.760 ^0.970 (0.850 (0.400 j 0.350 10.160 a.500 ^5.090 i 0.470 'i 0.680 (0.170 "(0.180 (0.060 ) 0.180 tS: 040 140 0.040 0.070 Radii of gyration. VI. 1.22) .84j: 1.06) .73 f .91) .62 f .76) .52 y .60^ .43 f .65) 1.20f .571 .68 f .43/ .45 f .29) .49 1) .26 .46 .26 .35 [ Distance d from base to nentral axis, In ineheB. 1.180 1.030 0.890 0.740 0.590 0.610 0.520 0.500 0.290 0.260 0.280 jAxisAB. i Axis CD. < AxisAB. ) Axis CD. jAxisAB. (Axis CD. jAxisAA (Axis CD. \ Axis AB. Axis CD. jAxisAB. ] Axis CD. j Axis A B. (Axis CD. ( Axis A B. (Axis CD. t AxisAB. CD. jAxIs AK (Axis CD. I AB. Axis CD. * The flret dimension Ib the width. AND BADII OF GTKATIOK. TRENTON IRON OE STEEL Z-BARS. PHCENIX IRON Z-BARS. 316 MOMENTS OF IKEBTIA. MOMENTS OP INERTIA AND RADII OP GYRATION OP JONES & LAUGHLIN'S, LIMITED, STEEL BEAMS. 17 IB 4 I. II. III. IV. V. !4i7P Weight Area of cross- Moments ol r Inertia. Radii of GyratioiL in inches. per foot, in lbs. section, in sq. in. Axis A B. Axis C D. Axis A B. AxiHCB. 15 70 20.6 731.1 37.8 5.95 1.85 15 59 17.3 640.9 30.3 6.08 IM 15 48 14.1 495.9 19.2 5.98 1.16 15 39 11.5 403.3 13.1 5 92 1.06 12 50 14.7 302.0 18.1 4.53 1.11 12 38 11.2 265 4 15.6 4.86 1.18 12 30 9.1 211.7 10.2 4.82 1.05 10 32 9 4 152.6 10.8 4.02 1.07 10 28.8 7.0 117.7 7.09 8.88 .05 9 24.5 7.2 101.1 7.80 8.74 1.04 9 19.75 5.8 79.8 5.03 8.71 0.03 8 25 7.3 71.8 6.66 8.18 95 8 18 5.3 57.3 4.27 3.28 0.89 7 18.8 5.4 40.4 5.02 2.98 0.96 7 15.25 4.5 37.9 3.38 2.89 0.86 6 16.6 4.9 2S.4 3.39 2.40 0.88 6 12.75 3.7 23.1 2.22 2.49 0.77 5 13 3.8 15.7 1.83 2.02 0.00 5 10 2.9 13.5 1.40 2.16 0.60 4 10.2 3.0 7.7 1.20 1.42 O.W 4 6.85 2.0 5.8 0.71 1.70 0.60 8 7 2.0 3 1 65 1 24 0.50 8 5.1 15 2.3 0.;;5 1.28 0.47 AND BADII OF GYRATION. • 317 IfOMENTS OP INERTIA AND RADII OP GYRATION OP PHGSNIX BEAMS— STEEL. fl U ^ B 71 \1 --0. Siase, tn inches , 15 15 15 15 12 12 m m 9 9 8 8 7 7 6 6 5 5 4 Weight I. per foot, Area of in lbs. cross- section, in sq. in. 75 22.05 60 17.64 50 14.70 41 12 05 40 11.76 82 9.41 83 9.70 m 7.47 27 7.93 21 6.17 22 6.47 18 5.29 20 5.88 15i 4.55 16 4.70 13 3.82 13 3.82 10 2.94 10 2.94 II. Moments of inertia. Axis A B. 757.7 644.0 529.7 424.1 281.3 222.3 179.6 137.3 110.6 84.3 71.9 57.8 49.7 38.6 28.6 23.5 15.7 12.4 7.7 Radii of gyration. Axis A B. Axi?CD. 5.86 1.35 6.04 1.32 6.00 1.20 5.94 1.08 4.90 1.20 4.85 1.04 4.54 1.10 4.52 0.99 3.7SI 1.07 8.70 0.95 8. as 1.01 8.30 0.91 2.91 0.97 2.91 0.87 2.47 0.83 2.48 0.77 2.03 0.72 2.05 0.66 1.62 0.66 31- M03fENTS OF DTEHTIA MOi£E>'TS OF lyERTlA ASD RADU OP GYRATION OP PHCELNIX DECK-BEAMS AND T-BARS. c c ^ =o^ IIL IV. V. VI. "T.'iini A4 »a^ ..C"?. n L>» u V-jnufSLtB jf SM. Bihlii of gyration. I>isUnee d from IWMtO neatnl «{. n. A^z^JlB. ^Ti*CD. AtjwJlR AxJgCD. * s: s I.I ^« s 4 s ^ ^ j' "• * ^ i '♦ V i .K.S 3£^ i05L T'S-'K ?raxs — Iho». k * 5 :*i "5 5 IT 4*1 0.74 4.37 > 5 .5: ^ 5 H 4£ O.TO 8.77 < * -^j -» 4 >4 3 27 0.84 2.96 < .1 TO ;• 3 -jS 2.90 0.81) 2.» ■« ■^^ 3 >H 2.53 0.77 2.96 'fc i :i « ^35 2.17 •.75 1.88 1 ^ ^ :i» 1.7» 0.51 2.41 -4 il ^ *» 4 41 9.n •.« 4.06 > ^ ^ -7 4 Z« 2.« 0.73 8.«8 « * - " i? 54 11 ? U 2.« • 85 2.85 5 3S ? i i*? \i i *? a. IS •.76 2.89 i '"A * . . :* m i 15 d.ltf •.73 2.78 ^-v :i > « « m m J 24 V-» aV 1.23 0.77 • * 4 » : * : 5» i M f.6» 1.17 0.66 «* ' ' « X ^ ^ % -^ d J8» # * •.•8 0.78 * ^ : > < . V. : i^ « m 1-01 ; 0.57 * ** • < • • » 4 s4 • a 0.84 '»■. s •'" . * N- I ft : ** I « •.76 . 1.03 ^ N '• ^ ^ "S ' 5» 4 •» •.•9 0.86 * - * X > -e ' 4t « :s •.S 1 0.75 • * • « i :t *.» ••"i 0.68 *^ O*. w:)m«aK«'a > AND RADII OF GYRATION. 319 MOMENTS OF INERTIA AND RADII OF GYRATION OF PHOENIX CHANNEL-BARS— IRON. A Ia n ^ IB Weight I. II. in. IV. V. VI. Size, in inches. per root, Area of Moments of inertia. Badii of gyration. Distance d in lbs. cropD- section, in sq. in. from base to neutral Axis A B. AxifiCD. Axis A B. Axis CD. axis. 15 66.6 20 554.57 23.61 5.27 1.09 1.08 15 60 15 421.87 12.39 5.30 0.91 0.86 15 38.3 11.5 351.56 10.01 5.53 0.93 0.83 12 50 15.0 235.73 8.44 3.96 0.75 0.80 13- 29.8 8.8 159.44 4.19 4.26 0.69 0.82 12 20 6.0 123.50 8.01 4.54 0.71 0.86 10 37 11.1 128.61 5.26 3.40 0.69 0.76 10 25 7.5 97.36 8.51 3.60 0.69 0.66 10 16 ^•^^ 63.67 2.21 3.64 0.68 0.56 9 38.8 10.0^ 94.27 5.24 3.07 0.73 0.76 9 23.8 7.0 75.29 8.69 8.28 0.73 0.70 9 15 4.5 61.01 2.36 8.49 0.69 0.70 8 19 5.7 43.99 2.14 2.76 0.61 0.66 8 10 8.0 26.20 0.85 2.96 0.53 0.45. 7 19 5.7 32 69 2.00 2.40 0.59 59 7 8.8 2.5 17.62 0.75 2.66 0.55 0.47 6 15.6 4.7 23.12 2.5 2.22 0.73 0.73 6 7 8 2.2 10.42 0.62 2.18 0.53 0.40 5 9 2.7 9.52 0.84 1.88 0.56 0.55 5 5.6 1.7 6.35 0.43 1.9:3 0.51 0.47 4 8 2.4 5.53 0.79 1.52 0.57 0.60 4 5 1.5 8.74 0.4 1.58 0.52 0.62 8 6 1.8 2.26 0.86 1.12 0.45 0.53 8 5 1.5 1.98 0.29 1.15 0.44 0.50 320 MOMKNTS OF INERTIA MOMENTS OF INERTIA AND RADII OP GYRATION OF PHCENIX ANGLE-BARS— IRON. ANGLES WITH EQUAL LEGS. I. n. • TTT. IV. V. VI. Wefght per foot, in lbs. Area of croBS- t^ection, sq. in. Moments of inertia. Badii of gyratiou. Distance dtTOOk Size, in inches. Axis AB. Axis CD. Axis AB. Axis CD. base to ikentral axis. 6 x6 6 x6 5 x5 5 x5 4 x4 4 x4 3ix3i 3ix3i 8 x3 3 x3 2|x2} 2Jx2i 2ix2i 2i X 2i 2J X 21 2i X 2i- 2 x2 2 x2 l}xlj lixH 33.3 16.8 20.6 12.3 17.2 9.4 13.6 6.8 9.4 5 8.6 4.5 7.9 3.5 6.1 2.6 4.6 2.5 2.0 1.5 10 5.03 6.2 3.7 5.16 2.81 4.1 2.05 2.81 1.5 2.58 1.34 2.36 1.05 1.83 0.8 1.4 0.75 0.61 0.44 35.17 17.22 14.70 9.35 7.18 4.39 4.35 2.30 2.23 1.33 1.65 1.01 1.22 0.62 0.82 0.40 0.49 0.29 0.18 0.9 13.98 6.77 6.07 8.77 3.01 1.71 1.84 0.95 0.95 0.54 0.62 0.41 52 0.25 0.35 0.17 0.20 0.12 0.07 0.04 1.87 1.85 1.54 1.59 1.18 1:25 1.08 1.06 0.89 0.94 0.80 0.87 0.72 0.77 0.67 0.71 0.59 0.62 0.55 0.46 1.18 1.16 0.99 1.01 0.76 0.78 0.67 0.68 0.58 0.6 0.49 0.55 0.47 0.49 0.44 0.46 0.88 0.40 0.85 0.29 r.84 1.08 1.55 1.46 1.22 1.16 1.06 0.96 0.98 0.87 088 0.8S 0.77 0.7 0.74 069 0.08 0.0 0.08 0.44 AND BADII OK GYBATION. ,r Nora.— E P b parallel to Uds tluongh ends ut aides. 322 MOMENTS OP INERTIA RADII OF GYRATION FOR A PAIR OP CARNEGIB ANGLES PLACED BACK TO BACK. ANGLES WITH EQUAL LEGS. n ymMmm \x ^^^^ ^vummm n Hadii of Gyration given, correspond to directions indicated by arrow-heads. Size, in inches. 6 x6 6 x6 5 x5 5 x5 4 x4 5.72 4 x4 12.04 ^x^ 4.96 8^x3^ 10.44 8 x8 2.88 3 x3 7.00 2^x2J 2.62 23x2J 5.88 2ix2i 2.88 2ix2i 4.74 2i X 2i 2.12 ^4,x2i 4.22 *Area of croBS- section, in inches. 10.12 16.56 Weitjbt per foot of 8ingle Hadii of gyration. angle, in lbs. ^0. 1.87 n. »•«• r». 16.9 2.50 2.67 2.76 33.1 1.85 2.62 2.80 2.89 12.0 1.56 % 09 2.20 2.85 27.6 1.55 2.24 2.42 2.62 9.5 1.28 1.68 1.86 l.d5 20.1 1.22 1.81 2.0(> 2.10 8.8 1.07 1.47 1.66 1.76 17.4 1.06 1.60 1.80 1.00 4.8 0.9] 1.25 1.43 1.58 11.7 0.98 1.37 1.56 1.66 4.4 0.85 1.15 1.84 1.44 9.0 0.91 1.31 1.60 1.61 4.0 0.77 1.05 1.24 1.84 7.9 0.78 1.14 1.38 1.48 8.5 0.69 0.96 1.14 1.84 7.0 0.70 1.05 1.24 1.85 AND RADII OF GYRATION. 323 RADII OF GYRATION FOR A PAIR OF CARNEGIE ANGLES PLACED BACK TO BACK. ANGLES WITH UNEQUAL LEGS. '2 M * V r.fV Radii of Gyration given, correspond to directions indicated by arrow-heads. Size, in incties. 6 6 6 6 5 5 5 5 x4 x4 x3i x3i x4 x4 x3i x3i 5 x3 5 x3 4^x3 4ix3 4 x3i 4 x3i 4 x3 4 x3 3^x3 3^x3 3ix2i 3^x2^ 3ix2 3ix2 3 x2i 3 x2i 3 x2 8 x2 2^x2 2ix2 ♦Area of cross- section, in inches. 7.22 16.36 6.84 14.50 6.46 13.66 6.10 12.84 5.72 12.04 5.34 11.24 5.34 11.22 4.18 10.42 3.86 9.60 2.88 5.94 2.50 4.96 2.62 5.20 2.38 4.62 1.62 4.36 Weight per foot of single angle, in Ids. 12.0 27.3 11.4 25.8 10.8 22.8 10 2 21.4 9.5 20.1 8.9 18.7 8 9 18.7 7.0 17.4 6.5 16.0 4.8 9.8 4.2 8 3 4.4 8.7 4.0 8.0 2.7 7.2 Radii of gyration. ro. 1.93 1.90 1.94 1.91 1.59 1 57 1 60 1.58 • 1.61 1.59 1.44 1 42 1.25 1.23 1.27 1.25 1 10 1.07 1.12 1.17 1.04 1.04 .95 .95 .96 .99 .79 .SO n- r,' 1.50 1.67 1.62 1.80 1.26 1 43 1.39 1.58 1.58 1.75 1.70 1.89 1.33 1.51 1.45 1.61 1.10 1.27 1.22 1.41 1.13 1.31 1.26 1.45 1.43 1.60 1.54 1.74 1.17 1.35 1.30 1.50 1.22 1.40 1.35 1.55 0.96 1.13 1.10 1.28 0.74 0.92 0.82 1.02 1.00 1.18 1.09 1.28 0.75 0.93 0.87 1.06 0.79 0.97 0.90 1.10 1.76 1.90 1.53 1.68 1.85 1.98 1.60 1.74 1.37 1.51 1.41 1.56 1.70 1.84 1.44 1.60 1.49 1.05 1.23 1.39 1.02 1.12 1.28 1.38 1.03 1.17 1.07 1.21 * 1 > flfmrpR in this column give the area of both angles. 324 MOMENTS OF INERTI4 RADII OP GYRATION FOR A PAIR OP GARNEGIB ANGLES PLACED BACK TO BACK. 4 .e-il ANGLES WITH 'UNEQUAL LEGS, %5;%:^^^;5^^ S$5SSSS55S5JSSS: Radii of Oyration given, correspond to directions indicated by arrow-headt. Size, in inches. ♦Area of cross- pectiou, ' in inches. 6 6 6 5 5 5 5 x4 x4 x3^ x3i x4 x4 x3^ x3^ 5 x8 5 x3 4^x8 4^x3 4 x3i 4 x3i 4 x3 4 x3 Six 2 3ixa 8 x2^ 8 x2^ 8 3 x2 x2 2ix2 2^x2 7.22 16.36 6.84 14.50 6.46 13.66 6.10 12.84 5 72 12.04 5.34 11.24 5.84 11.22 4.18 10.42 2.50 4.96 2.62 5.20 2.88 4.62 1.62 4.86 Weight per foot of single anffle, in lbs. 8^x3 3.8J 34x3 9.60 3ix2^ 2.88 8Jix2^ 5.91 12.0 27.3 11.4 25.8 10.8 22.8 10.2 21.4 9.5 20.1 8.9 18.7 8.9 18.7 7.0 17.4 6.5 16.0 4.8 9.8 4.2 8.3 4.4 8.7 4.0 8.0 2.7 7.2 ''o- 1.17 1.14 0.99 0.97 1.20 1 19 1.02 1.01 0.86 0.83 0.86 0.85 1.06 1.04 0.89 0.87 ' 0.90 : 89 i 0.74 \ 0.78 0.57 0.57 O.7.-) 0.76 0.57 60 0.60 0.61 Badii of gynttion. n- ri. 2.74 2.87 2.81 2.95 2.92 8.06 8.00 8.14 2.20 2.88 2.27 2.39 2.88 2.52 2.45 2.59 2.35 2.47 2.07 2.20 2 52 2.66 2.25 2.89 1.74 1.86 1.79 1.98 1.92 2.05 1.97 2.12 1.52 1.66 1.58 1.72 1.71 1.H6 1.76 1 91 1.51 1.00 1.31 1.40 1.70 1.80 1.60 1.59 1.88 1.49 1.10 1.18 1.57 1.69 1.28 1.87 8.01 8.16 8.10 8.24 2.48 2.62 2.65 2.69 2.62 2.77 2.86 2.40 2.08 2.15 2.07 2.22 1.80 1.96 1.86 2.01 1.80 1.91 1.69 1.69 i.er 1.79 1.89 1.48 * The figures in this colnmn give the area of both anslea. AND RADII OF GYRATION. 325 For compound sections made up of two or more beams or bars, the moments of inertia are found by combining those of the several shapes as given in the preceding tables. Thus : ~ / = G2 = Twice the moment of inertia for l)eam a (col. II.) + that for beam 6 (col. III.). I sum of areas of beams a and b (col. I.) 1^^^ a. rtS^^ I = B Twice area of beam a (col. I.) x d^ + twice moment of inertia for beam a (col. III. ) + that for beam b (col. II.). I d + i width flange of beam a L sum of areas of beams a and b (col. I.) ^ a i ^„ r%^iii t^--*- A^ a I = G^- Twice area of channels (col. I. ) y. d^ + moment of inertia (col. III. ), in which d = distance of centre of gravity of the channel from centre line of the combi- nation. 7 area of the two channels (col. I. ) t ^s-^ J fOJ..n.imm.m.<.i^ ■I 11 111 L mm XatHce I = Twice the moment in col. II. G'^ = Same as for single channel. When a section is employed alone, either as girder or post, the neutral axis passes through its centre of gravity. When rigidly connected with other sections forming part of a compound section, the neutral axis passes through the centre of gravity of the com- pound section; and therefore the moment of inertia of the elemen- tary section will not be that around its own centre of gravity, but around an axis at a distance from that point. The moment tjif inertia of a section aitout an axis other than that thronyh its mi- tre of (jravitjj is (Mjuai to the moment aixmt a i>arallel axis iHitt^ing tliruu^h its coiilrt' of ^Tavity plus the area of the section mulli plied by tlif square of tlu* distance Ix^tween the axes. The first step, then, in findinji; the moment of inertia, is to find the position of the »'entre of i^ravity of the se<^tion. For all sym- nirtrieal sections, this, of course, lies at the middle of the depth. For triani^lrs. it is found on a line j)arallel with the hjiso, and tlis- tant ont'-third the heii;ht of the triangle above the bast». For other sections, it is found by supposing the area divided up into elemen- tary sections, and nndti])lying the area of each such section by the distance of its centn* of gnivity from any convenient line. The sum of these products ilivided by the total area of the si»ction will give tile distance of the centre of gravity from the line from which the distances w<'re measured. KxAMPLK. — Find the neutral axis of a X siH'tion having the fo)l(>wini; dimensions : wi<lth, 8 inches ; depth. 10 inches : thick- ncs'< of metal, 2 inches. The area of the vertical flange, considering It as running through to the l)ottom of the section, would lx> 10 X :>, or 2n scjuare inches; and the distam'C of its centre of gravity alcove the l)ottoin line, 5 inches. The product of thest' quantities, lhen»- forc, is 1(H). The area of the bottom flange, not included in tlie Vertical Mange as above taken, is (> times 2, or 12 squan* inches: the distance ut' its centre of gravity above the l)ottom line. 1 inch; and the product of the two, therefore, 12. The sum of thesk* pHnUicls 112 divided l>v the total area is .,.r« or .'}.r> Inches, which is the distance •»- ot" till- centre of i^ravitv ab(»ve the lK>ttom line of the MVtion. 11a villi: tound the neutral axis of this .section, its moment of iiwiiia I-- readilv fomid bv the fornnda before given. Thus, in the la- i'l^-' >Upl»n«sed, f/ WOldd be 10 ~ :»..">= «i.."i. (/^ = .*{.5; »/.^ = l.o; ami tiif iiiouieiit WOldd be (see ]i. 2!hM, (2 X r...v«) -}- (S X ;l..v«) — (ti X 1.5-*) / - .J = 2t»«»l. The iiioMieiit «)f resistance of this sei'tion as a ginler would b(> . . . ■•! tl : and it* a Mrain on the tibres of the iron of 12,UM) poiimU i« i- sijiiure inch be allowed, then, sinet> the moment of n-^i^iani-f lit the ginler multiplied by hirain |ier s«|uaro iuoh musi AND BADII OF GYRATION. 327 «qiMl the bcndlng^moment of the load, it will be able to support a kmd whose bending-moment is 44^ times 12,000 pounds, or 536,000; Le., if used as a girder secured rigidly at one end, and loaded at the other, it would support a load, in pounds, of 536000 length in inches Or if supported at both ends, and the load uniformly distributed over the span, It would support a load eight times as great; the bending-moment in such case being one-eighth that in the former case (see pp. 291, 292). NoTs.— The formulas and fignree on pp. 296, 299, and 325. are taken, by per* iniP<)ion of The New-Jersey Steel and l9xm Company ^ from a hand- book which they pnblieh, entitled ** Usefal Information Tor Engineers and Architects,'^ and containiog fall information pertaining to the forms of iron which they mannfac- tnie. Radius of Gyration of Compound Shapes. {Ninth Edition.) In the case of a pair of any shape without a web the value of B can always be readily found without considering the tiioment of inertia. The radius of gyration for any section around an axis parallel to another axis passing through its centre of gravity, is found as follows : Let r = radius of gyration around axis through centre of grav- ity ; B = radius of gyration around another axis parallel to above ; d = distance between axes. When r is small, E may be taken as equal to d without mate- rial error. Thus, in the case of a pair of channels latticed to- gether, or a similar construction. Example 1 —Two 9-inch, 15-pound PhoB- c nix channel bars are placed 4.6 inches apart, ^^ ! i"^"1 K-44J-- as in the figure ; required the radius of gyra tion around axis C D for combined section. j Ans. Find r, in Column V., p. 819= ^ |-H— -i — H— B 0.69; and r^ = . 4761. Distance from base of channel to neutral axis. Column VI.. is .7. One-half of 4.6 = 2.3-1- .7 = 8, the distance l)etween neutral ^ axis of single channel and of combined section ; henoe, B = y9 + .4761 = 3.077 ; or, for all practical purposes, R = d, 328 RADIUS OF GYRATION OP COMPOUND SHAPES Example 2.-*Four 8x8 inches, 5-pound Phooniz angles as shown form a column 10 inches square ; find the rad gyration. Ans. From Column IV., p. 830, we find r = 0.94 aiu A .8886. The distance from base of angle to neutral axis, O VI.. is .87 ; hence, c? = 5 - .87 = 4.18 ; or, (f = 17.0609 .8= /i7.0509 + .8836 = 4.28. PRINCIPLES OF THE STRENGTH OF BEAMS. 829 CHAPTER XIV. GZSNERAL PRINCIPIiES OF THE STRENGTH OF BEAMS, AND STRENGTH OF IRON BEAMS. By the term "beam" is meant any piece of material which supports a load whose tendency is to break the piece across, or at right angles to, the fibres, and which also causes the piece to bend before breaking. When a load of any kind is applied to any beam, it will cause it to bend by a certain amount; and as it is impossible to bend a piece of any material without stretching the- fibres on the outer side, and compressing the fibres on the inner side, the bending of the beam will produce tension in its lower fibres, and compression in its upper ones. This tension and compression are also greatest in those fibres which are the farthest from the neutral axis of the beam. The neutral axis is the line along which the fibres of the beam are neither lengthened nor shortened by the bend- ing of the beam. For beams of wrought-iron and wood the neutral axis practically passes through the centre of gravity of the cross- section of the beam. To determine the strength of any beam to resist the effects of any load, or series of loads, we must determine two things: first, the destructive force tending to bend and break the beam, which is called the " bending-moment ; " and, second, the combined resist- anceof all the fibres of the beam to being broken, which is called the **ni oment of resistance ." The methods for finding the bending-moments for any load, or series of loads, have been given in Chap. XII. ; and rules for finding the moment of resistance, which is equal to the moment of inerlia divided by the distance of the most extended or compressed fibres from the neutral axis, and the quotient multiplied by the strength of the material, have been given in Chap. XIII., together with tables of the moment of mertia for rolled iron sections of the usual patterns. Now, that a beam shall just be able to resist the load, and not break, we must have a condition where the bending-moinent in the beam is equal to the moment of resistance multiplied by the strength of the material. That the beam may be abundantly safe Uy resist Ibe given load, the moment of resistance multiplied by y / 3150 rillNCIPLKS OF THE STRKNOTIl OF BKAMS. Atrcimtli of material must be several times as cjeat as the bendiiif;- nioinent; and the ratio in which this pnKlnet exe<'o<ls I lie ImmhI- ini:-iiH)inrnt, or in whirh the breaking- load exceeds the safe load, is known as the "factor" of sjifcty. r.y "ih*' strength of the material" is meant a certain constant (]iiantity, whiiji is dctermim^l by exiM'rinicni. and wliicli is known a> thf *• Mo:lu]us of Rupture." Of course this value isdifTerent for each ditlVn'nt mat<Tial. The following table contains the values of (I I is constant divided by the factor of safety, for most of the nianriais used in buildini;-const ruction. The moment of n^sistanee nmuiplied by these values will give the sttfe reiiiti(imj-\)OweT of the beasii. MoiU'Lrs OF lU'PTURE FOR SaFE STRENGTH. Vahu' <>r M.itt-rijil. f{. in \hj*. (':i-t Iron :..%« \Vrnii"|ii Iron l;!.i)i)0 si.r; Ki.iXK) Ain«Ti«;Mi :i*h '.».(KK) .\ni'-rir:iii r-cl IhtcIi l..H<)J) Am i<- III \i!lM\v Itirrh i.ii-a) ■\iiifiii-ni u lii'i" rciljir . l.(MN) Aii\i' ii in ili'i I.KI) Niu Kii. "iinil lir \.m) II'IIli'"' i» l.',1K» \mii : 1 .III \\ hill- o.ik !.:«► Material. !v,:. I lie of ill Urn. l.Osil AiiuTiraii wliitr pine.. .Anierimn yollnw |iim> l.sm Anii-rican fpnirr i l.'J»iil Om';;<)Ii pine I l.ViO niu(*Hti<iii> ll:ii;i;ii)K (liiiil-, son Rivni ' (•i:iniii>. avi'ntire Liine>ti>nf Marbli- Saiii|>toii4> Slatr I .TTR art) »IN) 'ri;.- ;ili..\.' valiH's I if R for wi*ou«:hi inui and st«H'l are one-fmirrh th.ii f"" t'l.- bri'akiFiir-l'»ads ; lor ca.st- iron. «me-sixth : for wt mm I, une- tliirl : jii'l i'"'!' >iniii-, <ine-sixih. Th*' constant'^ lor wimmI an- bJl^4•«l wj.-m ill- f.iiiii I'-t-i iiiaili" al the Massacliusetis Institute of Teeli- i,<i|<i-\ -Ml' :i l'!iil-^i/i- liiidH-rs of the usual i|uality found in build- ■n.r-. ! 'i-' tu'iir*-* ;;i\'ii "ii lh«' above labb- are bellev(»il to )>e amply sil".- ■■:■ i":rii> in ll<»nr-. ni" ilwrlliuirs. public balls. n»of.«*, etc.: but, fiir tl'X'i- III iiiiiN arnl warehouse- lb hm's, the niitbor ret -on in lends that iHit !ii>:i- than two-tbinls nf ilu- aUivr values Ih- iiM-d. The -:\U' ioj'l I T iIm' 'rn-ntiMi. iMio'nix. and »*arne«;ie s4><-tions. ustii as ih-ar::-. :ii'-.ill cDiiipiiIrd wjlii l'2.l>(M) pnuiids bir the >:ife value of /,'. ..■■ w"'- r.*.« OD |HHii)d>> libri' '^train, as it is •rcnerally calleti. :.-: 1-.::. iii| H'i.CMKI piiiinti> for >tcii. '!'•.'■ .■■ i-i riain ( a^« -« of be.ini> which most fri'ipieiitly occur :ti i<ii;l-i.:ij •••ii^t nn-i inn. f«ir which ftirmulas can In- given by wtiich tin -at I'ad^ fur llie bi-aiMS ma\ In- determiuetl ilirectlv ; hut U fieri liapp- u> ihal we may have either a iX'gularly nliapud bttun « • FBINCIPLES OF THE STRENGTH OF BEAMS. ^JSl Inregiilarly loaded, or a beam of irregular sectioh, but with a com- mou method of loading, or both ; and in such cases it is necessary to determine the bending-moment, or moment of resistance, and find the beam whose moment of resistance multiplied by R is equal to this bending-moment, or wliat load will give a bendinp:- moment equal to the moment of resistance of a beam nuiltipliod by R. For ezainplej suppose we have a rectangular beam of yellow pine loaded at irregular pomts with irregular loads: what dhnon- sions shall the beam be to carry these loads ? We will suppose that we have found the bending-moment caused by these loads to be 480,000 inch pounds. Then, as bending-moment equals moment of resistance multiplied by li, 480,000 pounds = —\^ x 1800 = J? x 2>« x 800 ; _ • , 4H00()0 or B X D^= " SOO ~ If we assume i> = 12 inches, then B = ^ =11 inches ; or, 144 the beam should be 11 inches by 12 inches. If, instep of a hard-pine beam, we should wish to use an iron beam to carry our loads in the above example, we must find a beam whose moment of resistance nuiltiplied by 12,000 equals 480, OOC) inch pounds. We can only do this by trial, and for the first trial we will take the Trenton I2:t-inch 125-pound beam. Tlie moment of inertia of this beam is given as 2S8; and its moment of resistance is one-sixth of this, or 48. Multiplying this by 12,000, we have 576,(X)0 pounds as the resisting-force of this beam, or 96,000 pounds over the bending-moment. Hence we should prob- ably use this beam, as the next lightest beam would probably not be strong enough. Fn this way we can find the strength of a beam of any cross-section to carry any load, however irregularly disposed it may be. Strength ol' Wrouglit-Iron Beams, Clianiiels, Aiijyle and T Bars. It is very seldom that one needs to compute the strength of wrought-iron beams, channels, etc. ; because, if he uses one of the regular sections to be found in the market, the computations have already been made by the manufacturers, and are given in their handbook. There might, however, be cases where it would be necessary to make the calculations for any particular beam; and to tneel such^cascs we give the following formulas. 332 PRINCIPLES OF THE STRENGTH OF BEAMS. Beams fixed at one end, and loaded at the other (Fig, 1). Safe load in pounds = 1000 X moment of inertia length in feet x y . (1) Beams fixed at one end, loaded with vniformly distributed load (Fig. 2). Safe load in pounds = 2000 X moment of inert ia length in feet X y . (2) Fig. 2. Beams supported at both ends, loaded at middle (Fig. 3). W Safe load in pounds = Fig. 3. 4000 X moment of inertia (81 span in feet x y Beams supported at both ends, load uniformly distributed (Fig. 4). Safe load in pounds = Fig. 4. 8000 X moment of inertia span in feet X y w PBINGIPUSS OF THE STRENGTH OF BEAMS. 333 Beaifis supported at both ends, loaded with concentrated load not at centre (Fig. 5). Safe load in pounds Fig. 6. 1000 X moment of inertia X span in feet ^(5) m X nX y Beams supported at both ends, loaded with W pounds, at a dis- tance m from each end (Fig. 6). Fig. 6. Safe load W, in pounds at each point = 1000 X moment of inertia (6) m in feet x y The letter y in the above formulas is used to denote the distance of the farthest fibre from the neutral axis; and, in beams of sym- metrical section, y would be one-half the height of the beam in inches. These formulas apply to iron beams of any form of cross- section, from an I-beam to an angle or T bar. For steel beams, increase the value of W one-third. Weight of Beam to be subtracted from its Safe Load. As the weight of iron beams often amounts to a considemble proportion of the load which they can carry, the weight should always be subtracted from the maximum safe load : for beams with concentrated loads, and for beams with distributed loads, one-half the weight of the beam should be subtracted. Example 1. — What is the safe load for a Trenton 12i-inch light I-beam, 125 pounds per yard, having a clear span of 20 feet, the load being concentrated at a point 5 feet from one end ? 1000 XIX span 1000 X 288 ;^ ^0 Ans. Safe load (For. 5) = - 12,500 pounds. mX nX y 5 X 15 X 6i 334 STRENGTH OF IRON AND STEKL BKAXB.' Example 2. — A 12-inch Carnegie iron channel-bar, wdgliing 90 pounds per yard, and having a clear span of 24 feet, supportBA concentrated load at two points, 6 feet from each end. Wliat is the maxiinuin load that can be supported at each point consistent with safety ? Avs. Safe load at each point = ;;-— ^ 6x6 4825 pounds. The moment of inertia for channels and an^e-bars, and other sections, will be found in Chap. XIII. Deepest Beam always most EconomicaL Whenever we have a large load to carry with a given span, It will be found that it can be carried with the least amount of iron by using the deepest beams, provided the beams are not too strong for the load. Thus, suppose we wish to support a load of tons with a span of 20 feet, by means of Trenton beams. We oould do this either by one 12i-inch beam at 125 pounds per yard, or by two 9-inch beams at 85 pounds per yard. But the 12Hnch beam, 21 feet long, would weigh only 875 pounds, while the two 9-incfa beams would weigh 1190 pounds; so that, by using the deeper beam, we save 315 pounds of iron, worth from three to five cents per pound. C The following table, under the heading |F?, gives the relative strength of Trenton beams in proportion to their weighty thns exhibiting the greater economy of the deeper patterns. Trenton Rolled I-Beams. Strength of each Beam in Proportion to its Weight. c c Bbam. w Bbam. W 15 inch, heavy .... 37.41 8 inch, light ».75 15 '♦ light . . . 36.76 7 " 55 pounds . 19017 124 " heavy . . . 12 " light . . . 28.41 6 •' 120 «• 14^ 30.61 6 " 90 " 44.07 loX ♦* heavy . . . 26.64 6 " heavy . . Mjas 10 1 •• light . . . 27.':0 6 • light . . 16.05 10| '* extra light . 27.78 5 ** heavy . . 18.S7 9 " extra heavy . 21.44 6 ' ' light . . 1S.90 P •• heavy . . . 23.41 4 • ' heavy . . . - . Mi " - • light 2:5.86 4 • ' light . . lOM " heavy .... 20.99 4 •' extra light . . IOjQO STRENGTH OF IRON AND STEEL BEAMS. 335 Another important advanta^ in the use of deeper beams is their greater stiffiiess. By referring to the tables, it will be seen tiiat a beam twenty feet long, under its safe load, if 6 inches deep will deflect 0.95 inch ; 9 inches deep, will deflect 0.63 inch ; 12 i^ inches deep, will deflect 0.46 inch ; and 15 inches deep, will de- flect only 0.38 inch. A floor or structure formed of deep beams will therefore be much more rigid than one of the same strength formed of smaller sections. There are, of course, cases where the use of deep beams would be inconvenient, either from increasing the depth of the floor, or from the fact that, with a light load and short span, they would have to be placed too far apart for convenience. In general, however, it will be best to employ the deep beams. Inclined Beams, — The strength of beams inclined to the horizon may be computed, with suflBcient accuracy for most purposes, by using the formulas given for horizontal beams, taking the horizon- tal projection of the beam as its span. Steel and Iron Seams. — The relative efficiencies of steel and iron beams depend upon the conditions under which they are used. The transverse strength of beams of the same length and section is proportional to the tensile strength of the material, or beams made of steel, of 65,000 pounds tenacity, will possess an ultimate stren^h about 80 per cent, greater than similar beams made of iron of 50,000 pounds tenacity. But the steel beam will not be stiffer than the iron beam — that is, it will deflect under working loads as much as the iron beam of the same length and section ; the steel beam merely bending farther than the iron beam without injuiT to its elasticity. Therefore, if strength without regard to stiffness is sought, the steel beam is the best ; but if stiffness without regard to ultimate strength is desired, beams of either material would probably prove of equal utility. Steel beams should not be used for their full load when the span in feet exceeds tivice the depth of the beam in inches. Note.— Since 1893 the Carnegie Steel Company has discontinued the manufacture of iron beams and bars for structural work, and now manufacture all their shapes in steel only. As steel beams, angles, etc., are sold at the same price per pound, and are about 20 per cent, stronger than iron, steel has naturally almost entirely superseded iron in rolled sections. Strengrth of Trenton, Pencoyd, Phoenix, and Car- negrie Rolled Beams, Channels, Angle and T-Bars — Iron and Steel. The foUowing tables ^ve the strength and weight of the various sections to be found m the market, together with the general dimensions of the I-beams. The tables are in all cases made up from data published by the 386 STBElfGTH OF IB017 AND 8TEBL BBAMB. respectiye manufacturers. The deflection of the beams under their maximum safe distributed load is also given in some of the tables. The tables on pages 849 to 363 will to found very convenient, for they can be used for the spans indicated, without any computations whatever. In these tables, the loads to the nght of and below the heavy line will crack plastered ceilings. When 12- to 24-inch beams are used to their full capacity for spans less than 10 feet^ the web should be stiffened at the ends. STRENGTH OF IBO^ AND STEEL BEAMa 887 tENGTH, WEIGHT, AND DIMENSIONS OP TRENTON ROLLED I-BEAMS— IRON. Blgnation of beam. ;h, heavy light heavy light light heavy light heavy light heavy light extra light . extra heavy heavy light heavy light 55 lbs 120 " 90 " heavy light heavy light heavy light extra light.. Weight per yard, in lbs. 872 200 200 150 135 170 125 120 96 195 105 90 125 85 70 80 65 55 120 90 50 40 40 30 37 30 18 n. Safe distributed load for one footof span, in lbs.* 1,320,000 990,000 748,000 551,000 460,000 511,000 877,000 875,000 806,000 360,000 286,000 250,000 268,000 199,000 167,000 168,000 185,000 101,000 172,000 132,000 76,800 62,600 49,100 38,700 36,800 30,100 18,000 m. Moment of inertia. IV. Neutral uxi8 perpen- dicular to web. Width of flange, ill ins. V. 707.1 523.5 434.5 891.2 288.0 281.3 229.2 283.7 185.6 164.0 150.8 111.9 93.9 83.9 67.4 44.3 64,9 49.8 29.0 23.5 15.4 12.1 9.2 7.5 4.6 6.75 6.00 5.75 5.00 5.00 5.50 4.79 5.50 5.25 5.00 4.50 4.60 4.50 4.50 4.00 4.50 4.00 8.75 5.2r 5.00 3.50 3.00 3.00 2.75 3.00 2.75 2.00 Area of cross- section, ininii. 27.20 20.00 20.02 16.04 12.86 16.77 12.33 11.78 9.46 13.36 10.44 8.90 12.88 8.60 7.00 8.08 6.87 5.50 11.84 8.70 4.91 4.01 3.90 2.S9 3.66 2.91 i.rr * For any other span divide this coefficient by span in feet. 838 STRENGTH OF lAON AND STEEL BICAMGL STRENGTH, WEIGHT, AND DIMENSIONS OF TRENTON ROLLED I-BEAMS— STEEL. I. n. ra. Moment of inertia. IV. V. Deeijifnation Weight per yard, in lbs. Safe distribated load for one foot of span in, lbs. Fibre strain of 16,000 lbs.* Width of Hange, in incnes. Aieaof of beam, in inches. Neatral axis perpendicu- lar to web. cnjoB- iectioii, iniuchet. 15 150 753,000 529.7 5.75 14.70 15 123 603,000 424.4 5.5 12.02 12 120 500,000 281.3 5.5 11.78 12 96 407,000 229.2 5.26 9.48 10 135 461.000 216.1 5.25 18.14 10 99 344,000 161.8 5.0 967 10 76 264,000 123.6 4.75 7.50 9 81 262,000 110.6 4.75 7.98 9 68 200.000 84.8 4.5 6.16 8 66 192,000 71.0 4.5 6.47 8 54 154,000 57.7 4.d5 5.28 7 60 151,000 49.7 4.25 5.87 7 46.5 118,000 38.6 4.0 4.55 6 50 104,000 29.2 8.5 4.07 6 40 83,300 23.4 8.0 8.27 5 39 67,000 15.7 8.18 8.80 5 30 52,900 12.4 8.0 2.90 4 30 41,200 7.7 2.75 2.24 4 22.5 31,400 5.9 2.62 2;2i * For any other span divide this coefficient by STRENOTH OP IRON AND STEEL REAMS. 339 lENUTH, WEIGHT, AND DIMENSIONS OF TRENTON CHANNEL-BARS AND DECK-BEAMS— IRON. esi^ation of bar. I. Weight per yard, in IbB. II. Safe distributed load, in lbs., for one foot of span.* III. Moment of inertia I. IV. Width of flange, in ins. V. Area of cross- section, in ins. Channel-Bars. ch, heavy light heavy light light heavy heavy light light extra light. light extra light heavy li^t extra light extra light extnt light extra light 190 120 140 70 60 48 70 50 45 33 36 25i 45 33 19 16i 15 625,000 401,000 381,000 200,100 134,750 102,500 146,000 104,000 88,950 65,800 62,000 89,500 68,300 45,700 aS,680 22,800 15,700 10,500 586.0 376.0 291.6 153.2 88.4 64.0 82.1 58.8 44.5 32.9 27.1 17.3 21.7 17.2 12.6 7.2 3.9 2.0 4f 4 4 8 2f 2i 2.2 2 2i 2i n H u 18.85 12.00 14.10 7.00 6.00 4.77 7.02 5.06 4.48 3.30 3.60 2.54 4.32 3.20 2.25 1.92 1.65 1.45 Deck-Beams. ch 65 55 91,800 63,500 54.7 35.1 4i 4* 6.29 I 5.35 * For coefficient of steel bars add one-third. 340 STRENGTH OF IRON" BEAMS. STRENGTH, WEIGHT, AND DIMENSIONS OF TRENTON ANGLE AND T BARS. I. n. I. n. Designation of bar. Weight per foot, in ibs. Safe diBtributed load for one foot of span, in lbs. Designation of bar. Weight per foot, inlba. Safe distriboted load for onej footofspao, Inlbf. ANGLEf ) Even Li EGS. Anolbs 1 [Jme^ual LBG8. 6 in. X 6 in. 44 " X 44 " 19.00 124 36,900 18,000 6 in. X 4 in. 14.00 ( 80»680 14,7S0 4 " X 4 " 34 •• X 34 " 94 81 12,184 9,200 6 " X 84 " 10.20 3 " X 3 " 2| " X 2| " 4.80 5.40 4,611 4,710 44 «» X8 " 9.00 f 14.680 ( T.oao 24 " X 24 " 2J " X 2i " 3.90 3.50 8,156 2,530 4 " X 8 ** 7.00 ( •,860 ( ».8n 2 " X 2 " 11 " X 1| « 3.13 2.00 1,970 1,150 34 " X 14 «« 4.00 r 6.616 ( 1,148 14 " X 14 « IJ " X l| " 1.75 1.00 832 393 3 " X 24 " ua ( M90 \ S,S88 1 "XI " 1 " X J « 0.75 0.60 246 186 3 " X 2 " 4.00 I 4.884 1 8,080 J " X g « 0.56 133 T-B. kR8. 4 in. X 4 in. 12.50 15,800 3 in. X 2 in. 4.80 2.640 34 " X 34 " 9.60 10,550 2 " X 14 " 8.00 1.866 3 " X 3 ♦• 7.00 6,680 2\ " X IJ " 2.40 604 24 " X 24 " 5.00 3,850 2 X 1 " 2.15 467 2 •♦ X 2 ♦• 3.13 1,970 14 •• XI •♦ 1.86 421 5 •• X 24 " 11.70 6,044 ** 7or coeflicient of steel barn add one-third. For any other tfma dMdo tilli foeiBcient by span. SISENGTH OF IBON AND STEEL BBAMS. 341 TRENGTH, WEIGHT, AND DIMENSIONS OF CARNEGIE I-BEAMS— STEEL. Depth of beam, in inches. Weight per foot, in lbs. Thickness of web, in inches. Width of flange. in inches. Safe dis- tributed load for one foot of span, in lbs. 16,000 lbs. fibre strain for buildings.* Safe dis- tributed load for one foot of span, in lbs. 12,500 lbs. fibre strain for bridges.* 24 100 .75 7.20 2,086,600 1,670,000 24 80 .50 6.95 1,830,500 1,486,000 ao 80 .60 7.00 1,545,600 1,207,500 ao 64 .50 6.25 1,222,400 955,000 16 75 .67 6.31 1,077,800 841,700 16 60 .54 6.04 916,800 715,800 15 50 .45 5.75 7.v3,aoo 588,500 15 41 .40 5.50 603,200 471,800 12 40 .39 5.50 500,100 390.700 12 S2 .85 5.25 395,200 3083)0 10 33 .37 5.00 344,000 268,800 10 25.6 .32 4.75 263,800 206,100 9 27 .31 4.'?5 262,200 204,900 9 21 .27 4.50 199,900 156,100 8 22 .27 4.50 191,600 149,700 S 18 .25 4.25 154,000 120,300 7 SO .27 4.25 151,400 118,300 7 15.5 .23 4.00 117,600 91,900 6 16 .26 3.fi3 101,800 79,500 6 13 .23 3.50 83,500 65.300 5 13 .26 3.13 67,000 52,400 5 10 .22 3.00 52,900 41,800 4 10 .24 2.75 41,200 32,200 4 7.6 .20 0.63 31,400 24,600 * For any other span divide tliis coefficient by span. 343 STRBNGTU OF IBOM AXSi 8TBEL STRENGTH, WEIGHT, AND DIMENSIONS OP CAENEGIl CHANNEL-BABS— IRON. STRENGTH OF IRON AND STEEL BEAMS. 34a STRENGTH, WEIGHT, AND DIMENSIONS OF CARNEGIE CHANNEL-B ARS- STEEL. Safe dis- tribured load Safe dis- tributed load Depth of cbaDnel, in inches. Weight per foot, in lbs. Thickness of web, in inches. Width of flange, in iuches. for one loot of span, in lbs. 16,000 lis. fibre strain for buildings.* , for one foot of span, in lbs. 12,500 lbs. fibre strain for bridges.* 15 32 .40 3.40 464,700 316,200 15 51 .775 3.775 554,700 433,400 12 20 .30 2.90 209,600 163,800 12 30i .55 3.15 273,600 213,800 10 15i .26 2.66 136,100 106,300 10 23} .51 2.91 180,500 141,000 9 m .24 2.44 102,700 80,200 9 2(H .49 2.69 138,700 108,400 8 lOi .22 2.22 75,n00 58,800 8 17i .47 2.47 103,700 81.000 7 8i .20 2.00 53,100 41,500 7 m .45 2.25 75,000 58,600 6 7 .19 1.<S5 39,400 80,800 6 12 .44 2.14 55,400 43,300 5 6 .18 1.78 27,900 21,800 5 lOi .43 2.03 39,000 30,500 4 5 .17 1.G7 18,700 14,600 4 8i .42 1.92 25,700 20,100 * For any other span divide this coefficient by span. 344 STBENGTH OF IBON AlSfD STBEL BEAHS. STRENGTH, WEIGHT. AND DIMENSIONS OP JONES ft LAUGHLIN'S, LIMITED, STEEL BEAMS. Safe dis- Safedis- tributed load tribnted load Depth of beam, In inches. Weight per foot, in lbs. Thickness of web, in inches. Width of flange, in inches. for one foot of span, in Iba. ltf,000 Ibe. fibre strain for buildings.* for one foot of span, in lbs. 12,000 lbs. fibre strain for bridges.* 15 70 0.64 6.366 1,089,700 810,700 15 59 0.468 5 968 910.000 710.900 15 48 0.406 5.726 705,200 650,900 15 39 0.375 5.475 673,600 448,000 12 50 0.598 5.723 536,800 419,400 12 38 0.343 5.468 471,800 868.600 12 30 0.312 5.218 876,400 294,100 10 32 0.3125 4.937 826.500 254,800 10 23.8 0.281 4.72 251,100 196.200 9 24.5 0.296 4.671 239.700 187,800 9 19.75 0.266 4.39 189.100 147,700 8 25 0.287 4.537 101,600 149,600 8 18 0.25 4.25 15i,800 119.400 7 18.3 0.2G6 4.266 141,400 110,600 7 15.25 0.25 4.0 115,500 90,200 6 16.6 0.265 3.765 100.900 78.800 6 12.75 0.25 3 5 8-^.100 64,100 5 13 0.31 3.06 07,000 62,800 5 10 0.22 2 845 67,600 46.000 4 10.2 0.28 2.78 41,100 82,100 4 7.9 0.25 2.(59 32.000 26,000 4 6.85 0.19 2.56 31.000 24,200 8 7 0.19 2.152 22,000 njioo 3 5.1 0.156 2.03 16,800 12,700 * For any other span divide this coefficient by spaa. STRENGTH OF IBON AND STEEL BEAMa 345 STRENGTH, WEIGHT, AND DIMENSIONS OF PHCENIX I-BEAMS— STEEL. Depth of beam. In inches. Weight per yard, in lbs. Thickness of web, . in inches. Width of flange, in inches. Safe dis- tributed load for one foot of span, in lbs. 16,000 lbs. fibre strain for buildings.* Safe dis- tribnted load for one foot of span, in lbs. 12,500 lbs. fibre strain for bridges.* 15 225 .62 6.375 1,076,000 840,600 15 180 .50 6.125 920,000 718,750 15 150 .45 5.75 752.000 587,500 15 123 .40 5.50 602,000 470,300 12 120 .39 5.50 500,000 390,600 12 98 .35 5.25 394,000 307,800 10^ 99 .35 5.00 368.000 287,500 lOi 764 .30 4.75 284,000 221,800 9 81 .31 4.75 262,000 204,600 9 63 .27 4.50 200.000 156,200 8 66 .27 4.50 190,000 148,400 8 54 .25 4.25. 154,000 120,300 7 60 .27 4.25 142,000 110,900 7 m .28 4.00 114,000 89.060 6 48 .26 3.625 100,000 78,120 6 39 .23 3.50 82,000 64,060 5 39 .26 3.125 66,000 51,560 5 30 .22 3.00 52,000 40,620 4 30 .24 2.75 40,000 31,'250 * For any other span divide this coefficient by span. 346 STB^GTH OF IRON AND STBBL BBAM& Peucoyd Beams and Cliaiinels* The coefficient for strength of the Pencoyd sections has been calculated for a fibre strain of 14,000 lbs. for iron, and 16.500 lbs. for steel. These tables also contain the maxim am load that should be placed on the beam, whatever the length, unless the web is stiffened at the points of support. Example. — What should be the maximum distributed load for a 15-inch 145-lb. iron beam of 10 feet span ? Ans. The coefficient of this beam is 648,600 lbs. Dividing by 10, we have 04.860 lbs., or 32.4 tons as the safe load ; but we see, by the last column, that it will not be safe to put more than 22.1 tons on the beam without stiffening the web. Hence, the safe load for that span is 22.1 tons. It is only for very short beams that this condition will apply. STRENGTH, WEIGHT, AND DIMENSIONS OF PENCOYD I-BEAMS— STEEL. Depth of beam, in inches. Weight per yard, in lbs. Thickness of web, in inches. Width of flange, in inches. Safe dia- tribated load for one foot of span, in lbs. 14,000 lbs. fibre strain for baildlnfiB.* Maxlmnm ]oad in tons, witlioat atlffeniiif welK 10 70.1 * .30 4.50 248,260 18.06 9 GO.l .28 4.80 198,010 10.44 8 51.7 .26 4.00 146,360 &g8 7 48.4 .24 3.75 106,840 7.60 6 34.9 .22 3.40 76,160 6.18 5 27.3 .20 3.00 49,000 4.04 4 •25.0 .22 2.6 &5,860 6.05 4 18.6 .16 2.8 27.180 8.16 * 3 20.5 .22 2.4 21,480 8.77 8 15.9 .16 2.2 17,880 %.7% ' For any other span divide this coefllcient by span. The load, be greater than that in next column, unless the web is stiflenad aft aoppoita STRENGTH OP IRON AND STEEL BEAMa 347 STRENGTH, WEIGHT, AND DIMENSIONS OP PENCOYD I-BEAMS— IRON. Depth of beam. In inches. • Weight per yard, in lbs. Thickness of web, in inches. Width of flange, in inches. Safe dis- tributed load for one foot ofspan,inlb8. 14,000 lbs. fibre strain for buildings. Maximam load in tons, without stiffening web. 15 190.0 .562 6.687 844,560 89.57 15 145.0 .437 5.125 648,600 22.10 16 124.1 .406 5.609 541,980 18.59 12. 1680 .656 5.5 578,640 88.63 12 120.0 .453 4.80 424,440 22.22 12 89.5 .343 5.0 817,440 13.60 10* 134.4 .468 5.25 429,560 22.13 10* 108.3 .406 4.87 347,420 17.71 10* 89.3 .343 4.5 288,460 13.35 10 111.7 .5 4.625 324,0^ 23.68 10 90.4 .343 4.375 276,860 13 18 9 90.0 .406 4.75 246,420 16.53 9 70,6 .312 4.25 195,880 9.94 8 80.0 .406 4.375 188.840 13.88 8 61.0 .297 4.0 161,400 10.46 7 65.8 .437 3.20 132,760 15.69 7 51.4 .234 3.61 114,880 6.17 6 115.5 .625 5.25 196,740 21.19 6 90.1 .5 4.87 160,000 16.42 6 55.5 .281 3.84 103,480 7.75 6 40.0 .218 3.47 76,500 5.25 5 29.7 .26 3.0 46,560 4.91 4 24.6 .22 2.6 30,000 4.33 4 18.2 .16 2.3 23,000 2.71 8 20.1 .22 2.4 19,340 3.23 8 16.6 .16 2.2 14,740 2.33 n.-.n .-n-KF-N-irrEi <if runs anti stkei, beamp. II) II II 111 i- ai a : ..;a;.iin4i..ioi!>.«i6.«ris."iw.:<iia.i7 ii .%!i!>«(i)<!inir<:mi3!H;Njniii!!ii ■j'.-m v . ■ii.ai !■ riio.'ri m.-n ?.!«i r.m li.i- :■ > j!is am i!bs| i!: : .;« »:« ■:'■ «' i.!jH i.nir ■■•.-■1 snimfiTB OF irok and stkkl beaiis. :iol li.wriftniod. E\ts 10 U ■< » su ss.rf M.-o «is n.c: ^.m m.m, si.«> -^.m 4>.nj m. lb i3> *■.;* ii,i3 «.M is,«s i«,w is.n-I i».:i w.-v u.iJ ui. 348 STRENGTH OF IRON AND STEEL BEAH8. STRENGTH, WEIGHT, AND DIMENSIONS OP PENCOY] CHANNELS. For Steel. Depth of channel, in inches. 1 ! Weight per vard. in lbs. Thickness of web, in inches. Width of flange, in inches. Safedis- tribnted load for one foot of span, in lbs*. 14,000 IbH. fibre strtiin for bnildini^. 1 1 1 Maximnm load ill ton*" u-iiiiout St ffeiilug web. 8 81.8 .22 1 i 2.27 79,0S0 6 55 7 26.6 .21 2.11 79,080 6.91 6 22.2 .20 1.95 42,600 6.25 6 18.1 .19 1.79 29,360 4.65 4 14.7 .18 1.P8 19,800 8.79 Foil IltON. 15 139.0 .562 8.94 539,940 84.84 15 106.0 .375 8.87 437,600 16.88 12 88.5 .406 2.94 284,280 18.49 12 60.0 • .281 2.61 192,440 9.14 12 61.5 .2.S1 8.09 206,460 9.06 10 59.7 .328 2.75 164,740 18.67 10 47.5 .25 2.5 133,660 8.46 9 52.7 .812 2.69 125,740 18.90 9 37.2 .234 2.36 92,640 7.17 8 43.0 .281 2.28 96,83» 8.77 8 39.5 .25 2.50 80,800 7.66 8 30.7 .218 2.28 68,940 4.66 7 41.0 .297 2.30 78,700 9.07 7 25.0 .171 1.95 49,aaao 8.42 6 81.9 .25 2.25 67,160 6.60 6 22.7 .20 1.7S 86,820 5M 5 28.9 .23 2.06 34,120 5.14 4 21.5 .25 1.69 24,060 6.19 4 16.5 .19 1.26 19,800 4.99 8 15.2 .22 1.68 12,640 8.49 8V 11.8 .25 1.87 0,660 8.90 1 8.8 .22 1.09 4,600 9.49 SAFE DBTRIBUTED LOADS AND DEFLECTIONS OP PENCOYD BBAMS-mON. .1 »«■ d.'llprll<>1» In Inc onwiMjndins i ~.uia markwr* ta-T be idled in ^Ic^il, ivlieii i\«- wplghis will be Incpeat otr cent. -faff loail aboul ao pvr cent. Detleclluii (.rarilcslly Ihe BUoe w Etn vKb aQiwl loadt. STRENGTH -OF TRENTON STEEL I <TI.— The flenreH tn Italic arc thndefli'cClnnH. In InchM. -ds above. For the dcllMHono or graileM nafB loiula In i DofttleUbiilarflgDrealii iUlliM. STREIfOTH OF OAHNEGIE IKON BEAHB. E DISTRIBUTED LOADS OP CARNEGIE IRON BEAMS. e loadi' In net tons In mlddlii, m In Weldrt 1 1^. Length of »pB n, in feet. IB 3W B U ISO Ml u tn Q u Mat V m »( uo M rlgbt and below bcavy lii STRBVOTH OF CARNEGIE BTEEL O 5 1 8 1 s g'l i| 1 1^. I Hi Hi ej- S 1 i 6. S ^^ 1 1*1 ° t i-^ J =?l,- CO '. ■5-g 1 £ M ^ Mil t !!l ss ; r| § Z^-^M — c i § 1 " s-IC^ 2 ill lli^ C = 3 „ siiin 2 S ^-^ ' !r = 1 ^ . "f s '1 ^ ■; ■§ -1 i ^■ ||S J?|j|| BTBBNaTH OF CABNEOIE STEEL BKAU8. 8S7 1 i ' " "I 1 s s s ^Is S u \ s ^ s STBENOTH or OABKEQIE 8TEBL ] 8TBENGTH OF OAKNBGIli: STEKL BKAMS. STBENQTH OF IRON BEAJK. STRENGTH OF IRON BEAMS. test safe load in IbB. iiDlforoil; dtsttibated. Including weight a ir 13,000 Ibe. fibre stnitn. lonceDtTBled load in middle of beam allow one-bair o[ (hat givu In Inches. «t'B*-l STRENGTH OF IBON BBAHB. Angles with UnegwU Legs — Long Leg VerHeal. vatceceafe In&d Id Ibn. untfomily distributed, inclndliis welifht of aiula- . For K.noo \bf. fliire atr^n. For coucuntnled IobU Iu middlB of b««m »&m STI :QTH or IRON ] JJ8. Attglet ailh Uaequal Leg* — Short Leg TeHietd. GraateBtrsfe ^•"•/> in ih- ..nif-^iiy diatribnled, Inclodine welabt of aosle- iron, f.ir 18,000 1 oonceolouad iod in mtadle of bom allow 364 BEAMS SUPPORTING BRICK WALLS. Beams Supportingr Brick Walls. In the case of iron beams supporting brick walls having no openings, and in wliich the bricks are laid with the UBual bond, the prism of wall that the beam sustains will be of a triangular shape, tlie height being one-fourth of the span. Owing to freqaenft iirogn- larities in the bonding, it is best to consider the height as one4hinl of the span. Fig. 7. The greatest bending-stress at the centre of the beam, mulling from a brick wall of the above shape, is the same as that caused by a load one-sixth less, concentrated at the centre of the beam, or two-thirds more, evenly distributed. The weight of brickwork is very nearly ten pounds per square foot for one inch in thickness ; and from tlds data we find that the bending-stress on the beams would be the same as that caused by a uniformly distributed load equal to 25 X square of span in feet X thickness In inches - 1» J) Having ascertained this load, we have merely to determine from the proper tables the size of beams required to carry a distrfbuted loail of this amount. £xAMi>LK. — It is proposed to support a solid brick wall IS inches thick, over an opening 12 feet wide, on rolled Iron beams: *. should be the size and weight of 1)eams ? x. Hy the rule given alH>ve, the unifonnly distributed load FRAMING AND CONNECTING IRON BEAMS. 365 which would produce the same bending-stress on the beam as the wail, equals 25 X 144 X 12 9 = 4800 pounds. As the wall is twelve inches thick, it would be best to use two beams placed side by side to support it, as they would give a greater area to build the brick on ; then the load on each beam would be 2400 pounds, or 1.2 tons. From the preceding tables for safe distributed loads on beams, we find that a 4-inch heavy beam would just about support this load; but as a 5-inch light beam would not weigh any more, and would be nmch stiffer, it would be better for us to use two 5-inch light beams to support om- wall. If a wall has openings, such as windows, etc., the imposed weight On the beam may be greater than if the wall is solid. For such a case consider the outline of the brick which the beam sustains to pass from the points of support diagonally to the out- side comers of the nearest openings, then vertically up the outer line of the jambs, and so on, if other openings occur above. If there should he no other openings, consider the line of imposed brickwork to extend diagonally up from each upper comer of the jambs, the intersection forming a triangle whose height is one-third of its base, as described above. When beams are vsed to support a wall entirely (that is, the beams run under the whole length of the wall), and the wall is more than sixteen or eighteen feet long, the whole weight of the wall should be taken as coming upon the beams ; for, if the beams should bend, the wall would settle, and might push out the supports, and thus cause the whole structure to fall. Framingr and Connecting Iron Beams. When beams are used to support walls, or as girders to carry floor-beams, they are often placed side by side, and should in such Fig. 8. Rg. 9. Fig. 10. Fig. 11. cases be furnished with cast-iron separators fitting between the flanges, so as to firmly combine the two beams. These separators "may be placed from four to six feet apart. Such an arrangement iB shown by Figs. 8 and 10, Figs. 9 and 11 showing fonus of sepa- ooo rnAJ»i:>ij ainu uuin w liu i ixn u ittuiM t5iSAM». rators usually employed; that with two bolt-boles being iimmI the 15-ineh and ]2i-inch beams, and that with a single hole smaller sizes. Fig. 12. Fig. 13. When beams are required to be framed together, it is usu done as shown by the accompanying cuts, in which Fig. 12 sli two beams of the same size fitted together. Fig. 13 shows a b fitted flush with the bottom flange of a beam of larger size. 14 shows a smaller beam fitted to the stem of a larger beam, al the lower flange. Fig. 14. Fig. 16. Wooden heanis may be secured to an iron girder in the si manner as an iron beam, by framing the end, and securing it b] ^-bracket; or an angle-iron may be riveted to the web of 3n eirder to afiford a flat bearine on which the wooden faeun i FRAMING AND CONNECTING IRON BEAMS. 367 The different rolling mills have standard connection? for con- necting iron beams with each other. The standard connection angles for all sizes and weights of steel and iron I-beams manufactured by Carnegie, Phipps & Co., Limited, are illustrated on page 3(58. These connections were designed on the basis of an allowable shearing strain of 10,00;) lbs. per square inch, and a bearing strain of v*(),000 lbs. per square inch on rivets or bolts, corresponding with extreme fibre strains in the I-beams of 16,000 and 12,00') lbs. per square inch, for steel and iron respectively. The number of rivets or bolts required was found to be dependent, in most instances, on their bearing values. The connections have been proportioned with a view to covering most cases occurring in ordinary practice, with the usual relations of depth of beam to length of span. In extreme instances, how- ever, where beams of short relative span lengths are loaded to their full capacity, it may be found necessary to make provision for additional strength in the connections. The limiting span lengths, at and above which the standard connection angles may be used with perfect safety, are given in the foUowing table : TABLE OP MINIMUM SPANS, FOR CARNEGIE I-BEAMS, WHERE STANDARD COxVNECTION ANGLES MAY BE SAFELY USED, WITH BEAMS LOADED TO THEIR FULL CAPACITY. Stbbl I-Beams. Iron I-Bbams. ^« I S$ S3 a> at V rS. (U cc a li gj K a> Designation Su Designation Designation ^y Depignation VT. of §.H of of §.£ of S.h: beam. Is beam. C OB •- P. 9.5 beam. 10. beam. '= 5 20" -80. lbs. 17.0 9"— 27. lbs. 15"-^. Ibp. 9"— 2S.5 lbs. 8.0 *' 64. 16.0 ♦' 21. 8"-22. 8.C •' 60. " 13. ( *' 23.5 " 8.0 15"-75. 12. 0' 8.0 " 50. " 13. ( 8"— 34. '' 7.0 " 60. 11.5 " IS. 7.0 12"-56.5 " 9.( " 27. " 7.0 " 50. 11. C 7"— 20. 6.0 " 42. " 8.0 " 21.5 " 6.5 ♦• 41. 10.5 " 15.5 5.5 lOi'MO. '' 9.( 7"— 22. '' 5.0 12"- 40. 8.5 6"— 16. 6.5^ " 31.5 " 10.01 '' 18. " 6.5 " 38. 7.5 " 13. 6.d 10''-42. " 10.5 6"— 16. •' 5.0 10" 88. lO.S 5" 18. 4.0^ '• 36. " 10.5 " 13.5 " 4.5 "- 25.6 9.0 " 10. 4.0 " 30. " 9"— 38;5 " 10.5 6.5 5"-12. " '♦ 10. ♦' 3.0 3.0 i STANDABU CONNECTION AH6LB8 KOE I-BEAJfS. % (H Ha ten* _rm ■III d++l + + + 4 ^4&t-l.-»-.»tf n. «x nt<x''-~°- 'rf''t- H-'iiil- fi SSPAaATOBS FOR CAKNEGIE STEEL BEAMS. 3<{9 SIZES AND WEIGHTS OF SEPARATORS FOR CARXECilE STREL BEAMS. Separators for 20" lieains arc maile nf I" nii'IHl. WITH TWO BOLTS, IS s SEPARATORS WITH 870 SEPARATORS FOR CASNBaiE IRON SEPARATORS WITH TWO BOLTfl. HEI-AR^TORS WITH OKB BOVt. la 36 1 66i lOJ S 12 8a 43 9? n lOi 4A 40 10,'„- 5 101 4/1 3n Bt 6 10 7 42 10 6i 10 56 ;i8 9i 5 10 5f» :iO 9,'r 4 9 6c 381 10 5 9 t» 28 85 4 9 »a a3i Si 4 3! 8 Se 31 91 e 8 86 37 84 ^ 8 8.1 311 8 5 7 96 2i 8rV 4J 7 9a 18 7i 4 e 105 IS !^'- 4 s - 1 6 10a 13J lit 1 116 13 «.) Si 1 6 llu 10 61 8i K i 13 7 6t 8 1 STRENGTH OF CAST IKON BEAMS. 371 CHAPTER XV. STRENGTH OF CAST-IRON. T7700DEN, AND STONE BEAMS — SOLID BUILT BEAMS Cast-iron Beams. — Most of our knowledge of the strength of oast-iron beams is denved from the experiments of Mr. Eaton Hodgkinson. From these experiments he found that the form of cross-section of a beam which will resist the greatest transverse strain is that shown in Fig. 1, in which the bottom flange con- tains six times as much metal as the top flange. When cast-iron be^ms are subjected to very light strains, the are^s of the two flanges ought to be nearly equal. As in practice;* it is usual to submit beams to strains less than the ultimate load, and yet beyond a slight strain, it is found, that when the flilnges are as 1 to 4, we have a proportion which approximates very nearly the requirements of practice. The thick- ness of the three parts — web, top flange, and bottom flange — may with advantage be made in proportion as 5, 6, and 8. If made in this proportion, the width of the top flange will be equal to one-third of that of the bottom flange. As the lesull of his experiments, Mr. Hodgkinson gives the following rul(» for the breaking-weight at the centre for a cast-iron beam of the above form :—- Fig 1 Breaking-load in tons = Area of hot. flange ^ depth ^ o 426 in square inches in ins. clear span in feet (1) Cast-iron beams should always be tested by a load equal to that which they are designed to carry. Wooden Beams, — Wooden beams are almost invariably square or rectangular shaped timbers, and we shall therefore con- sider only that shape in the following niles and fonnulas. 372 STRENGTH OF WOODEN IJEAMS, For beams willi a rectangular cross-secticHi, wo can simplify our formulas for strength by substituting for the moment of inertia }, X ip its value, viz., ~r:>~~ , where h = breadth of beam, and d its depth. Then, substituting this value in the genenil formulas for beams, W(> have for rectangular beams of any material the following foniiulas : — B V an LS fixed at one end, and loaded at the other (Fig. 2). Fig. 2, W or Safe load in pounds = Iheadth in inches = breadth x square of deptli X A 4 X length in feet 4 X load X length in feet s<|uarH of depth X A ' (2| (3) ficatns fired at one end, and loaded with uniformly dUdrihuled load (Fig. ;5). ■'^^y^y Fig 3 breadth x snuan^of depth X A Safe luad in j>ounds = ., ^ , . -.--; — \--i » * 2 X lengrh in feet or 2 X Icmjrih in fivt X loocl Iheailth in inches = — ^ .. ,. e~\r:zr\r^ — i — . 8(|uare of deplli X ^1 14) (&( STRENGTH OF WOODEN BEAMS. §73 Beams supported at both ends, loaded at middle (Fig. 4). W Safe load in pounds = Fig 4. __ broadlli X square of dopth x A^ span in feet or Breadth in inches _ span in feel x load (6) (7) square of <leptli X A' Beams supported at hoik andsj had. uniformly distributed (Fig. 5). Fig. 5. 2 X breadth x square of depth x A Safe load in pounds = span in feet ' ^^^ or Breadth in inches = : span in feet x load 2 X square of depth X A' (0) Beams supported at hidh ends, loaded with concentrated load yOT AT CENTRE (Fi^. (>). K-n—> m »w . / . / Fig 6 breadth x sf|. of depth X span X A Safe load in pounds — 4 X //< A // or BreaiUh in inches — 4 X load X /;/ X )i square of dcptli x span x A' (101 (11 374 STRKNGTH OF WOODEN BKAMS. Beams supported at both ends, and loaded wiUi W pounds at a distance m /row. each end (Fig. 7). ^■■^ :.:■■■■■ ' ■m-*- WM <rW ^W ^ Fig. 7. Safe load M' in pounds _ breadth X sciuare of depth X A ^ or at each point Breadth in inches = 4 X m 4 X load at one point X m (12) (13) scj. of depth X A Ndte. — Iti the lUKt two c-aflCH the ieiigthB denoted by tn and n should b« takeu in feet, the Huinc us the ripiiUH. Valuks of the Constant A, The letter A denotes the safe load for a unit beam one inch scpiare and oik! foot si>an, loaded at the eentre. This is also one- eii^hteenth of the modulus of rupture for safe loads. The follow- ing are tlie values of .1, which are obtained by dividing the moduli of rupture in Chap. XIV. by 18. TABLE I. Values of .4.— Co-kfficient for Beams. MuteriMl. .1 lbs. ;W8 888 KM) «) TO Matcrhil. .4 \\m. ( "nst iron Pino, white. Wentem '• Texue yellow S^)ruce \N hltewocKi (poplar) ; Rluostoiie tlagiiiii!; iHudvoii 1 Kiver) 05 Wrou'hi-iroM 90 Steel TO Aineriean wood.-* : (M:,.-fmit 05 Ilt'iiilix'k ti <);ik. \^ hiu- < Jr.inite, averaire Limestone 17 I'iiii-. • itor-'iM vellow 15 ( )1CMI11 Marble 17 iid or NDrway \\ hitc. Ka^te^n Sail' stone M 60 'lIu'M- v:ilnes for the ccwnieionl .1 are one-thinl of tlio hn»aking- u<iL:iii ot tiiiilMTs of the same si/.«> and i|U:tlity as that iisi'd in flrst- rla*-- Inii Minus. Tlii'< i»< a siiMirirnt allo'vanc** for timlM»rs in roof trii^^*'^, and lM'ani«« wliirh do not have to carry a nion* w^ven* Umd than that on a dwrllini: hou<(> floor, and small halls, etc. Wliori' tJMTi' i^ likely to \w M>ry much vihraiion, as in the lloor <if a mill, or a L;\niiiaNium tloor. or tlitoi-s of lari;*' public hail^i. llii* uiillitir r<rnmm«iid- I hat oidv foiu'-tifth.s of the :i1n»vc values of .1 In* usmmI. RELATIVE STRENGTH OF BEAMS. 376 « ExAMPLV 1. — What load will a hard-pine beam, 8 inches by 18 inches, securely fastened into a brick wall at one end, sustain with safety, 6 feet out from the wall ? Ans. Safe load in pounds (Formula 2) equals 8 X 144 X 100 4x6 = 4,800 lbs. EXA.MPLE 2. — It is desired to suspend two loads of 10,000 pounds each, 4 feet from each end of an oak beam 20 feet long. What should be the size of the beam ? Ans. Assume depth of beam to be 14 inches ; then (Formula 13\ breadth .— ^ * — ==^ — = 11 inches, nearly ; therefore the beam should be 11 x 14 inches. Helative Streng:th of Rectang:iilar Beams. From an inspection of the foregoing forniulas, it will be found tliat the relative strength of rectangular beams in different cases is as follows : — Beam supported at both ends, and loaded with a uniformly distributed load 1 Beam supported at both ends, and loaded at the centre ... i Beam fixed at one end, and loaded with a uniformly distributed load . . . . ; \ Beam fixed at one end, and loaded at the other | Also the following can be shown to be true : — Beam firmly fixed at both ends, and loaded at the centre . . 1 Beam fixed at both ends, and loaded with distributed load . . li These facts are also true of a uniform beam of any form of cross- section. When (I Hqiiare beam is supported on Us ethje^ instead of on its side, — that is, has its diagonal vertical, — it will bear about seven- tenths as great a breaking-load. The stronfjest beam which can be cut out of a e^ "^^ & round log is one in which the breadth is to the / ^ depth as 5 to 7, very nearly, and can be found / \ / yd / r^ \ \ I / / graphically, as shown in margin. Draw any [ diagonal, as ah, and divide it into three equal \ parts by the points c and d ; from these points draw perpendicular lines, and connect the points " #? and/ with a and h, as shown. ^' 1 CYLiNl>UI<''Af. Bkam.s. — A cylindrical beam is oidy .^ as 1 • I SIQ STRENGTH OF WOODEN BEAMS. • strong as a square beam whose side is equal to the diameter of the cirolo. [lonco, to find the load for a cylindrical beam, Hrst finil tlio propter load for the corresponding square beam, and then divide it by 1.7. 77/ r hcnrUiri of the ends of a 1>eam on a wall beyond a certain amount does not strengthen the beam any. In general, a beam slioulil have a bearing of four inches, though, if the beam be very short, the bearing may be less. Wv'ujUt of the Benin itHelf to be taken into Account. — The for- nuilas we have given for tlie strength of beams do not take into account the weight of the beam itself, and hence the safe load of tli(>. formulas includes both the external load and the weight of the material in the beam. In small wooden beams, the weight of th(i beam is generally so small, compared with the external load, that it need not be taken into account. But in larger wooden beams, and in metal and stone beams, the weight of the beam should be subtracted from the safe load if the load is distributed ; and if the load is applied at the centre, one-half the weight of the beam should be subtracted. The weight per cubic foot for different kinds of timber may be found in the table giving the Weight of Substances, Part III. Tables for the stren^li of yellow aud wliite pine» spruce, aud oak beauis, are given below, for beams one inoh wide. To find the strength of a given beam of any .other breadth, it is only necessary to multiply the strength given in the table by the breadth of the given beam Example. — What is the safe distributed load for a yellow-pine beam, supported at both ends, 8 inches by 12 inches, 20 feet clear span ? Alls. From Table II., safe load for one inch thickness is 1,440 pounds. 1,440 x 8 = 11,520 pounds, safe load for beam. Far a concentrated load at centre, divide these figures by 2. To find the size of a beam that will support a given load with a given span, find the safe load for a beam of an assumed depth .one inch wide, and divide the givcm load by this strength. KxAMPLK.— Wh.it size spruce beam will be required to carry a distributed load of S,64() pounds for a clear span of 18 feet ? Ann. From the table, we find that a beam 14 inches deep and 1 inch thick, 18 feet span, will support 1.524 pounds ; and diridiiig the load, 8.640 pounds, by 1.524, we have 5) for the breadth of the *t in inches : hence the V>eam should be 6 by 14 inohea, to oany ibuted load of 8,640 pounds with a span of 18 feei. *■■ STRENGTH OF HABD-PINE BEAMS. 311 GQ n 111 I n I 8 S) s a 03 I i •SI o C 'a 5 Si "Q o Jg 2 9i OQ $ OD «1 CO OC J5 1^ 0< IS t-t Ol 0» ^ rH rH of e* ^ o ^ rH O f- O 1-1 W TH K5 *> *k •» *> ^ ri T-i e< e< ot g s i § » «> f-i 11 e» e< Ol ^ s S^ s n iQ Q a o et OD -« rH tH T-l 01 kO OD Xi no o 09 Bi < ^5 r^ Ito ^ yf Id CO CO ^ lO t> I S I P § s i-H VI In 09 00 eo <M OD :0 00 s s s o» t- <?*. 00 eo -* OD 3 S QO g p « i-i" o» o» 00 -^ *o OB a s u > V c > o eS 09 O h3 X Xi © o rfS^ «o .-T i-T of of cc i >A CO ? S5 8 «o i-< c* 00 < '^^^©lOIOO^Ol-OO jog <Eto{^aoa»oo<'^*0(0 fl »M »-< »H n »-< 372 STRENGTH OF WOODEN 11EAM8, For beams wilh a rectangular cross-section, wo can simplify onr fonmilas for strength by substituting for the moment of inertia its vahie, viz., — t^", where h = breadth of beam, and il its deptli. Then, substituting tliis value in the general formulas for 1)eaMis, wo liave for recUingular beams of any material the following forniulas : — Beams fixed at one end, and loaded at the other (Pig. 2). or Safe load in pounds = Breadth in inches = Fig. 2, breadth x square of depth X A 4 X length in feet 4 X lo ad X length in feet scpiare of depth X A ' (2| (8J lieams fixed at one end, and loaded with nn{foTuHy dUArihiuUd load (Fig. 3). or Safe load in pounds = Breadth in inches = Fig 3. breadth X square of depth X A ~2'^lengMi in feet 2 X h'ugth in ft?et X load 8(juare of depth X A ' U) m STRENGTH OF WOODEN BEAMS. §73 Beams supported at hoik evds^ loaded at middle (Fig. 4). W Safe load in pounds = FI9 4. breadth x square of depth X A^ span in feet span in feet x load Breadth in inches = s,,„are of depTh x~7r- (6) (7) Beams supported at both ends, had uniformly distributed Fig. 5). Safe load in pounds = Fig. 6. _ 2 X breadth x STfuare of depth x A span in feet _ span in feet x load Breadth in inches = .> ^ . ..^..^^r .1 ^*u v < » 2 X •Kjiiare of depth x A (8) (») Beams sujtported at both ends, loaded with concentrated load ^OT AT CENTEE (Fiir. «)- y////,///y''* Safe load hi poancb = >x Brpailtli m \Tif\\c^. ~ Fig 6 breadth x vf. of depth x span X A 4 X ;<» X H 4 y Uy\t] y />/ y u ft'inar** of *\<'\A\\ / .s|>vif» y A' (101 (11 !•■ 374 STRENGTH OF WOODEN BEAMS. Beams supported at both ends, and foaded with W pounds (U a distance m from each end (Fig. 7). or FIfl. 7. Safe load H' in pounds __ breadth X square of depth X A ^ at each point 4 X m ^ 4 X load at one point X m Breadth in inches = sq. of depth X A * (12) (18) Note. — In the last two cascft the leugthf) denoted by m and n sliould tn taken in feet, the same as the spans. Values of the Constant A. The letter A denotes the safe load for a unit beam one inch square and one foot span, loaded at the centre. This is also one- eighteenth of the modulus of rupture for safe loads. The follow- ing are the values of A, which are obtained by dividing the moduli of rupture in Chap. XIV. ..by 18. TABLE I. Values of J.— Co-efficient fob Beams. Material. .4 lbs. Cast-iron 308 Wrou«;ht-iron ()0(5 Steel 888 American woods : Cies^tnnt 60 Hemlock 55 Oak, while 75 Pinu. (iCorL'ia yellow " Oreiron 100 90 " red or Norwav 70 " white, Eastern 00 Material. Pine, white. Western . . . •• Texas yellow .... Spruce I W hi te wood (poplar) . . . . , { Bhicstoiie flagging (Hndson j River) I Granite, average i Limestone Marble ' Sannstouc I Slaie AWm, 00 90 n 17 16 17 8 BO Tlu\s<; values for the co-oflicient A are one-third of the breaking- weiixht of timbers of th(> same si/.e and quality as that used in firat- class buildings. This is a sutticient allo'.vance for timbers in roof trusses, an<l beams whi<:h do not have to carry a more severe load than that on a dwelling-liousc floor, and small halLs, etc. Where there is likely to be very much vibration, as in the floor of a mill, "* gymnasium-floor, or floors of larg(> public halU, the author uenils that only four-tifths of the above values of ^ be used. BELATIVE STRENGTH OF BEAMS. 375 Example 1. — What load will a hard-pine beam, 8 inches by 12 inches, securely fastened into a brick wall at one end, sustain with safety, 6 feet out from the wall ? Ans. Safe load in pounds (Formula 2) equals 8 X 144 X 100 4x6 = 4,800 lbs. Example 2. — It is desired to suspend two loads of 10,000 pounds each, 4 feet from each end of an oak beam 20 feet long. What should be the size of the beam ? Ans. Assume depth of beam to be 14 inches ; then (Formula 13^ breadth — ' — -^ — = 11 inches, nearly ; therefore the beam should be 11 X 14 inches. Relative Strengrth of Rectangular Beams. From an inspection of the foregoing formulas, it will be found that the relative strength of rectangular beams in different cases is as follows: — Beam supported at both ends, and loaded with a uniformly distributed load 1 Beam supported at both ends, and loaded at the centre ... ^ Beam fixed at one end, and loaded with a uniformly distributed load . . . . ; 4 Beam fixed at one end, and loaded at the other i Also the following can be shown to be true : — Beam firmly fixed at both ends, and loaded at the centre . . 1 Beam fixed at both ends, and loaded with distributed load . . li These facts are also true of a uniform beam of any form of cross- section. When a square beam is supported on its edffe, instead of on its side, — that is, has its diagonal vertical, — it will bear about seven- tenths as great a breaking-load. The sironf/est beam which can be cut out of a e^ "^^.^ round log is one in which the breadth is to the / ^x / depth as 5 to 7, very nearly, and can be found / graphically, as shown in margin. Draw any | diagonal, as ab. and divide it into three equal \ yd / /^N \ \ \ / / parts by the points c and d ; from these points \ J/ \ draw perpendicular lines, and connect the points " "^^ — -^/ f; and/ with (t and ?>, as shown. '^' 1 Cylindrical Bkams. — A cylindrical beam is only .^ as 1*1 382 »ULll> BUILT WOOUBN BBAH8. When a beam is built of several pieces la lengtb afl well i (Icptb, tbej sliould break joints with each other, Tlie layen b the neutral a^U should be lengthened by tlie scarf or Rati y iiseJ for resisting tension; and the npperoues should have the abut against each other, using plain buU joints. 'I* Si i'l UM Many builders prefer using a hiiiU heam of selected Umber single solid oni', on acisjunt of the great dlfticutty of getting latter, whi'ii very lai^i', frep fi'orn defeotsr moreover, the atrei of the former is to l>e relieil u[>on, althougli it caimol be stro: than the corresponding solid one, if perfectly sound. STIFFNESS AND DEFLECTION OF BEAMS. 383 CHAPTER XVI. STIFFNESS AND DEFLECTION OF B£AMS. In Chaps. XIV. and XV. we have considered the strength of beams to resist breaking only ; but in all first-class buildings it is desii-ed that those beams which show, or which support a ceiling, should not only have sufficient strength to carry the load with safety, but should do so without bending enough to present a bad appearance to the eye, or to crack the ceiling : hence, in calcu- lating the dimensions of such beams, we should not only calculate them with regard to their resistance to breaking, but also to bend- ing. Unfortunately, we have at present no method of combining the two calculations in one operation. A beam apportioned by the rules for strength will not bend so as to strain the fibres beyond their elastic limit, but will, in many cases, bend more than a due regard for appearance will justify. The amount which a beam bends under a given load is called its deflection, and its resistance to bending Is caUed its stiffness: hence the stiffness is inversely as the deflection. The rales for the stiffness of beams are derived from those for the deflection of beams; and the latter are derived partly from mathematical reasoning, and partly from experiments. We can find the deflection at the centre, of any beam not strained beyond the elastic limit, by the following formula: — _ load in lbs. X cube of span in inches X c Def. in inches - ^duius of elasticity X moment of inertia* ^^^ The values of c are as follows : — Beam supported at both ends, loaded at centre . .0.021 " " *' uniformly loaded . . 0.01:3 ** fixed at one end, loaded at the other .... O.-^Vi " ♦* *' unifonnly loaded .... 0.125 By wM^lgi"e the proper substitutions . in Formula 1, we derive the 384 STIFFNESS AND DEFLECTION OF BEAMS. following formnla for a rectangular beam ^supported at bath ends, and loaded at the centre : — . _ load X cabe of span X 1728 Def. in inches - 4 x breadth X cube of depth X E^ *^' the span being taken in feet. From this fommla the value of the modulus of elasticity, E, for different materials, has been circu- lated. Thus beams of known dimensions are supported at each end, and a known weight applied at the centre of the beam. The deflection of the beam is then carefully measured; and, substituting these known quantities in Formula 2, the value of £ is easily obtained. 1728 Formula 2 may be simplified somewhat by representing a^e ^ ■^, which gives us the formula WX L^ Def. in inches = j^ x I^x F^ ^^^ For a distributed load the deflection will be five-eighths of this. Note. — The constant i'^ correBponds to Hatfield's F, in Us Tnuisreiae Stimiiu. If we wish to find the load which shall cause a given deflection, we can transpose Formula 2 so that the load shall fdrm the left- hand member. Thus : — Load at centre _ 4 X breadth X cube of depth X def. in ins. X E in pounds ~ cube of span X 1728 ' * ' Now, that this formula may be of use in determining the load tb put upon a beam, the value of the deflection must in some way be fixed. This is generally done by making it a certain proportion of the span. Thus Tredgold and many other authorities say, that, if a flooi> beam deflects more than one-fortieth of an inch for every foot of span, it is liable to crack the ceiling on the under side; and henoe this is the limit which is generally given to the deflection of beams in first-class buildings. Then, if we substitute for ** deflection" the value, length in feet -r 40, in the above fornmla, we have, breadth X cube of depth X e Load at centre = ^^— ^ ,-^jj , (5) E letting e = p=^- y engineers and architects think that one-thiriieUk qfan inch )t of span is not too much to allow for the defleetton of floor STIFFNESS AND DEFLECTION OF BEAMS 385 beams, as a floor is seldom subjected to its full estimated load, and then only for a short time. If we adopt this ratio, we shall have as our constant for deflec- _ E tion, €i - J2900- In either of the above cases, it is evident that the values used for Ef F, e, or Ci, should be derived from tests on timbers of the same size and quahty as those to be used. It has only been within the last three or four years that we have had any accurate tests on the strength and elasticity of large timbers, although there had been several made on small pieces of various woods. The values of the vaiious constants for the fii*st three woods in the following table have been derived from tests made by Professor Lanza and his students at the Massachusetts Institute of Tech- nology, and the values for the other woods are about six-sevenths of the values derived from Mi*. Hatfield's experiments. The author believes tliat the values given in this table may be relied upon for timber such as is used in first-class construction. TABLE I. Values of Constantn for Stiffness or Deflection of Beams, E = Modulus of elasticity, pounds per square inch. F = Constant for deflection of beam, supported at both ends, and loaded at the centre. € = Constant, allowing a deflection of one-fortieth of an inch per foot of span, e, = Constant, allowing a deflection of one-thirtieth of an inch per foot of span. Material. Cast iron . . Wrought-iron Steel . . . Yellow pine . Spruce . . . While oak . White pine . Hemlock . . Whilewood . CheHtaut . . A«h. . . . Muple . . . E. 15,700,000 26,000,000 31 ,000,0.00 1,780,000 1,294,000 1,240,000 1,073,000 1,045,000 1,278,000 944,000 1 ,48-.\000 1,902,000 F^ E 432" 36,300 60,000 71,760 4,120 3,000 2,S70 2,480 2,420 2,960 2,180 3,430 4,400 E 17280 907 1500 1794 103 75 72 62 60 74 54 86 no E ^1 " 12960' 1210 20:k» 23o8 137 100 95 82 80 98 72 114 146 394 CONTINUOUS GIRDBR8. Contimtons Girder of Three Equal Spans, Concentrated Load <^ W Poitnda at Centre of Each Span. Re-action of either abulment, R,=R, = i\W; (7) Re-action of either centi-al support, B, = A'j = U yV; (81 r or the re-action of the end supports is lessened three-tenths, and that of the central supports increaseil three-twentieths, of that which they would have been, had three separate girders of the samp cross-section been used, instead of one continuous girder. D Fig.2 Continuous Girder of Three Equal Sjmns uniformly loaded with w Pounda per Unit of Lenyth. Re-action of either end support, R,=R, = Uol; m Re-action of either central support, R^ = R, = \htol; (10) hence the re-actions of the end supports are one-fifth less, and of tlie central supports one-tenth more, than if the girder were not continuous. Strength of ContiuHous Girders, — Uviymg determined the re- action of the supports, we will now consider the strength of the girder. Tlu; strength of a beam depends upon the material and shape of the l)eain, jind upon the external conditions impose<l upon the beam. The latter j;ive rise to the bemling-moment of the beani, or tlu> amount by which the external forces (such as the load and supporting forces) tend to bend and break the beam. It is Ibis bonding-moment which causes the difference In the Ijoaring-strength of continuous and non-continuous girders of the same cross-section. Continuoua Girdtrs of Tico .s>«».s. — When a rectangular beam is at the point of breaking, we have the following conditions :^ Bendim;- _ Mod, of rupture x breadth X sq. of depth . moment "~ 6 ' '"' :hat the lieam may carry its load with perfect safety^ the load by a proper fac^tor of safety. CONTINUOUS GIRDERS. 395 Hence, if we can determine the bending-moment of a beam under any conditions, we can easily determine the required dimensions of the beam from Formula 11. The greatest bending-moment for a continuous girder of two spans is almost always over the middle support, and is of the oppo- site kind to that which tends to break an ordinary beam. Distributed Load. — The greatest bending-moment in a continu- ous girder of two spans, / and /i , loaded with a unifonuly distributed load of w pounds per unit of length, is Bending-moment = o /# , > » (12) V/hen i = f , , or both spans are equal, Bendmg-moment = -g-, (12a) which is the same as the bending-moment of a beam supported at both ends, and uniformly loaded over its whole length: hence a continuous yirder of two tfpans uniformly loaded is no stronyer than if non-continuous. Concentrated Load, — The greatest bending-moment in a con- tinuous girder of two equal spans, each of length /, loaded with W pounds at centre of one span, and with W^ pounds at the centre of the other span, is Bending-monaent •=^ h^(W+Wx). (13) When W = W\^ov the two loads are equal, this becomes Bending-moment = ^WU (13a) or one-fourth less than what it would be were the beam cut at the middle support. Continuous Girder of Three Spans^ Distributed Load. — The greatest bending-moment in a continuous girder of three spans loaded with a uniformly distributed load of w pounds per unit of length, the length of each end span being /, and of the middle span Ij is at either of the central supports, and is represented by the formula, Bendmg-moment = .,.>. , ^. v . (14) When the three spans are equal, this becomes Bending-moment = 7a» (14a) or one-fifth less than what it would be were the beam not con^^ tinuous. 388 STIFFNESS AND DEFLECTION OF BEAMa a as O u o c o a ^ o r if -2 c *- Is C ^ g«OQoaoo««io«o -• s s g £2 Ai( S s % s i ^ ^ § § R i s 00 i = C8 'J E-i 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 1-1 00 •- Ti Oi G) 01 00 1-H 00 IS 8 S ^ S 3 •-• ^ O t- *-< fi5 »• ^ r- • _ ^ ^ 1-1 1-1 ©I <ji 00 eo O P t- Q O ©» 1-i Wt « r-1^ -^ l-^ O r-i T-l »-i 01 CC 00 •^ i f2 f2 T-t 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 gcOXOiOM-TfOCO STIFFNESS AND DEFLECTION OF BE A J. Eh o a CD c 83 'O e3 O ti-i 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 ^ '^ T-t 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 » 1-1 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) i-I 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 yellow-pine beam of 10 foot span, to support a concentrated load of 4250 pounds, without deflecting more than ^ of an inch at the centre V Ans. A deflection of i of an inch in a span of 10 feet is in the proportion of y?, of an inch per foot of span; and as the load is concentrated, and applied at the centre, we should use Fomiula 7, employing for e the value given in the fourth column, opposite yellow pine. Formula 7 gives the dimensions of the breadth, and to obtain it we must assume a value for the depth. For this we will first try b inches. Substituting in Formula 7, we have, 4250 X 100 Breadth = 512 x l'j7 ~ ^ inches, nearly. This would give us a beam 6 by 8 inches. Example 8. —What is the largest load that an inclined spruce beam 8 by 12 inches, 12 feet long between supports, will cari7 at the centre, consistent with stiffness, the horizontal distance between the supports being 10 feet ? An9. Formula 12 is the one to be employed, and we will use the value of e given in the third column, opposite spruce. Making the proper substitutions, we have, ^ , . 8 X 1728 X 75 Safe load = — r2 x"To — ~ ^^^^ pounds. Cylindrical Beams. For cylindrical beams the same fonnulas may be employed as for rectangular beams, only, instead of #■, use 1.7 X e, : that is, a cylindrical beam bends 1.7 times as much as the circumscribing rectangle. Deflection of Iron Beams. For rolled-iron beams the deflection is most ac(;urately obtained by Fonimla 1. The following ap])roximate formula gives the de- flections quite accurately for the maximum safe loads, s^iuar(» of si)an n\ feet Deflection in inches = 70 x iht: depth of bv.am The deflections for tlie PJuvnix, Pencoyd, Trenton, and Car- negie beams, arc given in the tables for strength of beams, in Chap. XIV. In using iron beams, it should be n*membered that the deepest btatu is aJways the most economical; and the stiffness of a floor is almtys gi!Qftt!er wheu a suitable number of deep beams are used. 302 CONTINUOUS GIRDERS, CHAPTER XVII. STRENGTH AND STIFFNESS OF CONTINUOUS GIRDERS Girders resting upon throe or more snpiwrts arc of quite fi-e- qiieiiL octurreiioi* in buiUling construction; anti a great variety of oi)inions is held as to the relative strength and stiffness of continu- ous and iion-continnous girders: very few i)ersons, probably, having any coiic'Ct knowledge of tin* subject. In almost every building of importance, it is necessary to employ girdtrs iisiing ui)on jiiers or columns placed from eight to fifteen feet ai)ari ; and in many cases gndcrs can conveniently bo ubtaiueil wliitli will span two and even three of the spaces l)etween the piera or columns. When this is tlie case, the question arises, whether it will be heller construction to use a long continuous girtler, or to have each ii:irdcr of only one span. Most aiehitects an? probably aware tliat a girder of two or more sj)aii> is sirougi'i and stifTer than a gn-der of the same section, of only one s])an. but just htnn much stronger and stiffer is a question they are unable to answer. As it i> -eldoin ihai a iiirderof more than three spans ih employed in (Utlmaiy bni Idlings, we shall c<»nsid(»r only these two caM*.s. hi all struelures, the first point which slumld Ih» considennl is the n'sistaiiee require<| of (Im* su])poris, and we will first cimsider the resistance offered by the siq)iH)r(> of a continuous ginler. In this elia)>ter we shall iioi go into the mathematical flisciission of the "subject, but leier any readi'is inter«»sle<l in the derivation of the toiniulas for (-ontinnous girders to an article on thai sulijt*<-l, b\ the author, in the .Inly (J8^;ij number of Van NostrandV " Knmneeiiiig Magazine.'' Supporting- Forces. (iinhrs <tj Two spuuM, lotuh'tl al lUf. Centre qf Baeh Span. —If a ginler ot two si>ans, / and /,, is loaded at tho centre of the tpui I CONTINUOUS GIRDERS. 393 with W pounds, and at the centre of ^ with Wi pounds, the re-action of the support Ri will be represented by the fonnula R =: -* 32 (i) the re-action of the support R.^ ^Y «2 = j^(ir + ^r,), and the re-action of the support R^hy the formula 13 IK, -:]W (2) A»» = 32 (3) If H^ = IF,, then each of the end supports would have to sustain 1^ of one of the loads, and the centre support V- of W, Were the girder cut so as to make two girders of one span each, then the end supports would carry ^ or tb W', and the centre support |g ]V: hence we see, that, by having the girder continuous, we do not require so much resistance from the end supports, but more from the central support. ABC m m m R2 Fig 1 Girder of Two Spans, uniformly Distributed Load over Each Span, — Load over each span equals lo pounds per unit of length. Re-action of left support, 10 r ^« + / « -| -2L' 4/(f-h/.)J- Re-action of central support, R, = w{l-hl,)-R, - /?3. Re-action of right support, ^^ - 2U' 4/, (/ + /.)]• (4) (5) (6) When both spans are equal to /, the re-action of each end support is i Kj/i, and of the central support t '«' ' hence the girder, by being contuuious, reduces the re-action of the end supports, and increases thai of the central support by one-fourth, or twenty -five per cent. 394 CONTINUOUS GIRDERS. Continuous Girder of Three Equal Spans, Concentrated Load of W Pounds at Centre of Each Span. Re-action of either abutment, R,=R, = ;\,}V; (7) Ro-action of either central support, liz = H, = U ^V; («) or the re-action of the end supports is lessened three-tenths, and lliat of the central supports increased three-twentieths, of that which they would have been, had three separate girders of the sam^ cross-section been used, instead of one continuous girder. D Continuous Girder of Three Equal Spans uniformly loaded with w Pounds per Unit of Lent/ th. Re-action of either end support, /r = /?4 = i tot; (9) Re-action of either central supi>ort, /?, = /^^ = ^,( ,o/; (10) hcnco the re-actions of the end supports arc one-fifth less, and of the central supports one-tenth more, than if the ginier were not continuous. str'iKjtli of' ('(nitiuuous Girders. — Having detemiineil the re- action of the supports, we will now consider the strength of the irinler. The streiij^th of a beam depends upon the material and shai)e of the beam, and ii]K)n the external conditions imiH)s<Hl ii{H>n the beam. Tlie lattei ijive rise to the ben<ling-moment of the l)eani, or the amount by wbieh the external forces (such as the load and support iiiu forces) tend to ben<l ami break the l)eam. It is tliis bendini^-monient which causes the ^liflTerenee in the l>eaiiiiLj-^tiemitb of continuous and non-continuous ginlers of tie* >ame cross-section. ('(mfiiiii'iiis (iirdti's o/* Tii'n spiois. — When a rectangular heam is at the point of breakiuir, we have the following (^mditions : — Hendini:- _ ^lod. of rupture X bre mllh X s<|. of depth , moment ~ " «" ' *"* and. that the beam may carry its load with perfect safety, wemiut divide the load b> a proper factor of safety. CONTINUOUS GIRDERS. 395 Hence, if we can determine the bencUng-moment of a beam under any conditions, we can easily determine tlie required dimensions of tlie beam from Formula 11. The greatest bending-moment for a continuous girder of two spans is almost always over the middle support, and is of the oppo- site kind to tliat which tends to break an ordinary beam. DiMtrlbuted Load. — The greatest l)ending-moment in a continu- ous girder of two spans, / and /, , loaded with a uniformly distributed load of w pounds per unit of length, is Bending-moment = o /^ ^ > » (12) V/hen Z = i I , or both spans are equal, top Bending-moment = -g-, (12a) which is the same as the bending-moment of a beam supported at both ends, and uniformly loaded over its whole length: hence a continuous girder of two ifpams uniformly loaded is no stromjer than if non-continuous. Concentrated Load, — The greatest bending-moment in a ron- tinuous girder of two equal spans, each of length if, loaded with \V pounds at centre of one span, and with Wi pounds at the centre of the other span, is Bending-monaent ^ ^iHW+Wy), (13) When W = ITi , or the two loads are equal, this becomes Bending-moment = A IT/, (13a) or one-fourth less than what it would be were the beam cut at the middle support. Continuous Girder of Three Spans^ Distributed Load. — The greatest bending-moment in a continuous girder of three spans loaded with a uniformly distributed load of w pounds per unit of length, the length of each end span being /, and of the middle span I, is at either of the central supports, and is represented by the formula, wl^ 4- wli^ Bending-moment = .,... , ^i y (14) When the three spans are equal, this becomes xol^ Bending-moment = Tqi (14a) or one-fifth less than what it would be were the beam not con-^ tinuous. 396 CONTINUOUS GIRDERS. Conconfrated Loads. —The greatest bending-moment in a con- tinuous girder of three equal spans, each of a length 2, and each loaded at the centre with [V pounds, is Bending-moment = ^,^ Wl, (15/ or two- fifths less than that of a non-continuous girder. Deflection of Continuous Girders. CoutiniiOHs Girder of Two Eqvdl Spniis. — The greatest deflec- tion of a continuous girder of two equal spans, loaded with a uniformly distributed load of w pounds per unit of length, is id* Deflection = 0.005416 ^. (16) {E donotos modulus of elasticity; /, moment of inertia.) The deflection of a similar beam supported at both ends, and uniformly loaded, is Deflection = 0.01:3020 ^. ITencc the deflection of the continuous girder is only about two- fifths that of a non-continuous girder. The greatest deflection in a continuous girder is also not at the centre of either span, but betweer. the centre and the abutments. The greatest deflection of a continuous girder of two equal spans, loaded iit the centre of one span with a load of IV pounds, and at the centre of the other span with IV i pounds, is, for the span with load \V, (28n'-0lF,)/« Deflection = - 153^.^^ ' <"> for the si)an with load ir,, (2:ur, ~oir)/« Deflecti(m = — I'l'A' EI * (Ha) When ImMIi si)ans have the same load, 7 ir/« Deflection = >.^■^ ^^ • (17'>) T]i<> drilci tion of a beam su])ported at l>oth ends, and loaded at tin* (•»'iiin- with \y pounds, is Deflection = .^ j^-,-j\ or tlir ditlrction of the continuous girder is only seven-tlsteenUit of the nun-eontinuous one. CONTINUOUS GIRDERS. 397 Continuous Girder of T/tree Eqiial Spans, — Uniformly distrib- uted load of 10 pounds per unit of length, Deflection at centre of middle span = 0.00052 ^ ( 18) Greatest deflection in end spans = 0.006884 j^ (10) or the greatest deflection in the girder is only about one-half that of a ncn-continuous girder. Concentrated load of W pounds at centre of each span, I \Vl^ Deflection at centre of middle span = t^ -^t- (20) II Wl» Deflection at centre of end spans = kqk -^j- (21) or only eleven-twentieths of the non-continuous girder. Several Observations and Formulas for Designing: Continuous Girders. From the foregoing we can draw many observations and conclu- sions, which will be of great use in deciding whether it will be best in any gi\^n case to use a continuous or non-continuous girder. First as to the Su2)ports* — We see from the formulas given for the i*e-action of the supporting forces in the different cases, that in all cases the end supports do not have as much load brought upon them when the girder is continuous as when it is not; but of course the difference must be made up by the other supports. This might often be desirable In buildings where the girders run across the building, the ends resting on the side walls, and the girders being supported at intermediate points by columns or piers. In such a case, by using a continuous girder, part of the load could be taken from the walls, and transferred to the columns or piers. But there is another question to be considered in such a case, and that is vibration. Should the building be a mill or factory in which the girders had to support machines, then any vibration givea to the middle span of the beam would be carried to the side walls if the beam were continuous, while if separate girders were used, with their ends an inch or so apart, but little if any vibration would be can-ied to the side walls from the middle span. In all cases of important construction, the supporting forces should be carefully looked after. Strength, — As the relative strength of continuous and non- oontinuouB girders of the same size and span, and loaded in the •aoie wny, is as their bending-moments, we can easily calculate the .^9S CONTINUOUS GIRDERS. strongth of a continuous girder, knowing the formulas for its bend* ing-nioni(Mit. From the values given for the bending-nioments of the various cases considered, we see that the portion of the girder most strained is tliat which conies over the middle supports; also that, except in llie single case of a girder of two spans uniforndy loaded, tlie strength of a girder is greater if it is continuous than if it is not. But tlie gain in strength in some instances is not very great, altliough it is generally enough to pay for making the girder continuous. Stijrnc's^i. — The stiffness of a girder is indirectly proportional to its deflection; that is, the less the deflection under a given load, the stiffen the girder. Xow, from the values given for the deflection of continuous girders, we see that a girder is rendered very much stiffer by being made (continuous ; and this may be considered as the principal advantage in the use of such girders. It is often the case in building-construction, that it is necessary to usi^ beams of nmch greater strength than is required to carry the superimposed load, because the deflections would be too great if i\ui beam were made smaller. But, if we can use continuous girders, we may make the beams of just the size required for strength; as the deflections will be lessened by the fact of the gird- ers being (continuous. It should therefore be remembered, that, wh(>re great stiffness is required, continuous beams or girders should be used if possible. Foriuulas for Strciigtli and Stiffhess. For eonvenienee we will give the proper formulas for calculating the streni;tli and stiffness of continuous ginlers of rectangidar cross-s(»etion. The fonnulas for strength are deduced from the fornuda, Bending-moment = ;. * (22) where 1i is a (constant known as the modulus of rupture, and la ei^litecu times what is generally known as the co-efticicnl of stn'nijth. SiKKNJ.Tn. — (.'ontinnoits tjirder of two equal Hpana^ loadtd nnij'nnnhj oi'cr ((ir/i span^ 2x nx U^x A lirealving-weight = i ' (23) where li d«>ri()tes the breadth of the ginler, D the depth of the girder (botli in inches), and L the length of one span, in/eef. The CONTINUOUS G1KDEB8. 399 values of the oonstant A are three times the values given in Table L, p. 874. For yellow pine, 800 pounds ; for spruce, 210 pounds ; and for white pine, 180 pounds, — may be taken as reliable values for A. Continuous girder of two equal spans, loaded equally at the centre of each span, 4 B X D^x A Breaking-weight = 3 X r • (24) Continuous girder of three equal spans, loaded uniformly over each span, « , . . , f) Bx D2x A Breaking- weight = 9 ^ L * ^^^ Continuous girder of three equal spans, loaded equally at the centre of each span, 5 B X D^x A Breaking-weight = 3 x j • (26) Stiffness. — The following formulas give the loads which the beams will support without deflecting more than one-thirtieth of an inch per foot of span. Continuous girder qf two equal spans, loaded uniformly over each span, Bx l>^x e Load on one span = q 26 x L-^ ' '^'^^ Continuous girder of two equal spans, loaded equally at centre of each span, 16 B X D^x e Load on one span = "7" x j-^ • (28) Continuous girder* of three equal spans, loaded uniformly over each span, B X Z)'^ X e Load on one span = q 33 x L^ ' ^^^ Continuous girder of three equal spans, loaded equally at the centre of each span, 20 B X D^x e Load on one span = TT ^ jo • (oO) The value of the constant e is obtained by dividing the modulus of elasticity by 12,1)(50 ; and, for the three woods most commonly used as beams, the following values may be taken : — Tellow pine, 187 ; white pine, 82 ; spruce, 100. 400 CONTINUOUS GIRDERS. For iron beams we may find the l)ending-inoinent by the for- mulas given, and, from tahles saving the sti-ength and sections of rolled beams, find the beam whose moment of inertia = bending-m oment X depth of beam 2000 •.vhen tli«* beuilinsj moment Is in foot pounds. For (^xjunphs we have a continuous l-lwam of three equal spans, loaded ovtM- each span, with 2(KM) pounds per foot, distributeil. Each span being 10 feet, then, from fonnula 14(r, we have 2(KX) X 100 Bemling-moment = rr^ = 20000. 2(XNK) Moment of inertia = ~:^^^ x depth of beam; 20,(XM) -^ 2(MM) = 10, and we must find a beam whose depth multl- plic 1 hy ten will c(jUJil its moment of inertia. If \\v try a ten-inch lK*am, we should have 10 X 10 = 100; and we sec from Tal)lcs, i)p. 2(50-272, that no ten-inch beam lias a moment of Inertia as small as KM): so we will take a nine-inch beam. W X 10 ~ INK and the lightest nine-inch beam has a moment of inertia of \Y,\: so we will use that beam. In tluj case of continuous I-i)eams of three e(|nal spans, (upially load(>d with a distributed ItKid. wi* may take four-fifths of the load on one siKin, and find the iron beam which would support that load if with only one span. KN.VMri.i:. — if we have an I-beam of three equal siNins of 10 feet each loadcil with 20,000 pounds over each span, wliat Hize beam should we use? Ans. -! of 20.<MM)= 10,000. Tlie ecpiivalent load for a span of oui- foot would be 10,000 X 10= UMMMM). rrnin Tables, Chap. XIV., we find that the beam whose eo-efll- ciint is nearest to this is the nine-inch light lM*ani, — the s;inie beam wbicb we found to carry the same load in the prt*(*e«Iing c\aiii|>lc. Tor iK'anis of two equal spans loiidtMl uniformly, the >nciii:ib <»t the beam is the same as though the beam were not colli iinioiis. rin- t'oi iinila^ ui\en for tbe re-actioiis of the sup]M)rts ami for the (l<-tli-iri«>ii oi (-v)iitiiiMoiis Lcirders with eoneentnili'd NhmIsi, were vnitii-l bv Mm- aulboi- b> means of careful experiments on small sr«'-! bai->> IIm- other forinulas have Inn'ii veriH«Ml hy <>oni]iAri9un witli iitbi-r iiiilboi'it ies, wliei'i* it was |His.sible to do so; though one or iwo ot tbf l-a^e^ uJMMi, tli(* auliiur has never seen dlicuaa»d in ail) woiU on tbe .subject. FLITCH PLATk GIRDEttS. 40 J CHAPTER XVITT. FLITCH PLATE GIRDERS. In framing large buildings, it often occurs that the floors must be supported upon girders, which themselves rest upon columns ; and it is required that the columns shall be spaced farther apart than would be allowable if wooden girders were used. In such cases the Flitch Plate girder may be iron flate used, oftentimes with advan- tage. A section and elevation of a Flitch Plate girder is shown in Fig. 1. Fig. 1. The different pieces are bolted together every two feet by three- fouiths-inch bolts, as shown in elevation. It has been found by practice that the thickness of the iron plate should be about one- twelfth of the whole thickness of the beam, or the thickness of the wood should be eleven times the thickness of the iron. As the elas- ticity of iron is so much greater than that of wood, we must propor- tion the load on the wood so that it shall bend the same amount as the iron plate: otherwise the whole strain might be thrown on the iron plate. The modulus of elasticity of wrought-iron is about thir- teen times that of hard pine; or a beam of hard pine one inch wide would bend thirteen times as much as a plate of iron of the same size under the same load. Hence, if we want the hard-pine beam to bend the same as the iron plate, we must put only one-thirteenth as much load on it. If the wooden beam is eleven times as thick as the iron one, we should put eleven-thirteenths of its safe load on it, or, what amounts to the same tiling, use a constant only eleven- thirteenths of the strength of the wood. On this basis the follow- ing formulas have been made up for the strength of Flit(;h Plate girders/ in which the thickness of the iron is one-twelfth of the braidth oi the beam, approximately : — 40-2 FLITCH PLATE GlRDEES. Let 1) = Depth of beam. B = Total thickness of wood. L = Clear span in feet. i = Thickness of iron plate. f __ i 1^^> pounds for hard pine. f 7o pountls for spruce. W = Total load on girder. Then y for beams supported at both ends, Saf<» load at centre, in pounds = j- (/B-\-*JnOt), (11 22)2 Safe distributed load, in pounds = —f— (/B + 7500. (21 For distributed load, D = \/ 2/7i-f Kiitbt ' I irZ For load at centre D = \/ >^"j_7^' (3) (4) As an example of the use of this kind of girder, we will take tl«(* case of a railway-station in which the second story is devoted to offices, and where we must use girders to support the second floor, of twonty-liyc feet span, and not less than twelve feet on centres. If we can avoid it. This would give us a floor area to be supported by the girder of 12 X 25 = :300 square feet; and, allowing 105 i>ounds p«T s()iiare foot as the weight of the suiKjrimposed load and of the floor itself, we have ol,r>00 pounds as the load to be supported by the ginlcr. Now we find, by computation, that if we were to us«» a M)li(l girder of hard pine, it would re<iuirea8eventeen-lncli by four- teen-inch beam. If we were to use an iron Ix'ani, we find tliat a fifteen-inch ln^iivy iron beam would not have the requisiti^ strength for this span, and that we should be obliged to use twotwelve-4nch beams. We will now see what size of Flitch Plate ginler we would recpiire, sliould we decide to use such a girder. We will assume tlie total breadth of both beams to be twelve inches, so that we can use two six -inch tind)ers, whi<'h we will have hanl pine. The thick- ness of the iron will he one inch and one-eighth. Then, substi- tuting in Formula JJ, wt* have. / :{!.')( M) X 25 . — ^' = V- X KM) X 12 + I;V)T7^rHt = VIW, or 14 inches. Hence we sliall require a twelve-inc4i by fourteen-incb girder. NoVt FLITCH-PLATE GIRDERS. 40;^ for a comparison of the cost of the three girders we have considered in this example. The seventeen-inch by fourteen-inch hard-pine girder would contain 515 feet, board measure, which, at five cents a foot, would amount to $25.75. Two twelve-inch iron beams 25 feet 8 inches long will weigh 2083 pounds; and, at four cents a pound, they would cost $83.82. The Flitch-Plate girder would contain 364 feet, board measure, which would cost $18.20. The iron plate would weigh 1312i poimds, which would cost $52.50; making the total cost of the girder $70.70, or $13 less than the iron beams, and $45 more than the solid hard-pine beams. Flitch-Plate beams also possess the advantage that the wood almost entirely protects the iron; so that, in case of a fire, the heat would not probably affect the iron until the wooden beams were badly burned. 404 TRUSSED BEAMS. CHAPTER XIX. TRUSSED BEAMS. AVhexkveti wo. wish to support a floor upon ginlers having a span of more than thirty feet, we must use eitlier a trussed ginler, a riveted iron-phite fjinU^r, or two or more iron beams. The clieap- esi and most convenient way is, probably, lo use a large woo<leu girder, and truss it, either as in P'igs. 1 and 2, or Figs. JJ ami 4. In all these forms, it is desirable to give the girders as much <!epth as the conditions of the case will permit; as, the deei)er the ginler, the less strain there is in the pieces. In the belly-rod truss we either have two beams, and one rod which runs up between them at the ends, or three beams, and two rods runnini^ up between the beams in the same way. The beams should be in one continuous length for the whole span of the ginler, if they can be obtained that length. The requisite dimensions of the Me-rod, struts, and beam, in any given case, must be deter- mined by lirsi tindiui^ the stresses which come ui>on these picH»t»s, and then the area of cross-section required to resist these sti-esses. Foi: sixciu: srui t iielly-kod tkisses, sucli as is represented by FJLi. 1, the strain ni)on the pieces may be obtained by the foUow- ini: formulas : — For DisTiniu'iEi) LOAD ir over whole (jiriJeTf '1 'ension in r ^ o 10 w X ( oinpression in r ^^ s' w. ( 'ompH'ssion in li zz 10 ir X length of T length of C le ngth of B length of Cf (1) (2) m TRUSSED BEAMS. 405 For CONCENTRATED LOAD W 09€r C, ,«.,«, ^ length of T Tension in T = y x ,^„g,i, ^^ ^T Compression in C = W. . . „ H^ length of B Compression in B = g- x ^^^^^^ ^^ ^ W (5) For girder trussed as represented in Fig. t under a distributed LOAD W over whole girder, 3 length of S Compression in S = j^ »' x lengthof C" (6) Tension in R - ^w. Tension in B _ 3 length of B 10 "^ length of C (7) For CONCENTRATED LOAD, W at centre, , ^ W length of S Compression in S = ^ X i^ng^j^^f^- Tension in 1? ^ W, W length of B Tension in B = y x j^^^pT^f^. (8) (9) For double strut belly-rod truss (Fig. 3), with distributed i,OAD W over whole girder. B Tension in T Fig.3 length of T = 0.307 W x 7- '^ length of C Compression in C = 0.367 W. ^ length of B Comp. in iJ or D = 0.367 H^ X i,„^„ ^f p - (101 (11) 406 TRUSSED BEAMS. Fo7' coNCKNTRATEi) LOAD W over cQch of the HtruU C, leneLli of T , , Conipression in C = W, leiigtli of Ji Coiiip. in B or tension in /) = \V x ip,iiwj7Qf7"'' (**^) For (jinlcr trusffvd, as in Fly. 4, under a distkiuuted load H' over whole (jlrder, r^ ^Jp -v ^^-TU Fig.4 lon^tli of .S , , , , = 0.307 irx,^.^g,^-^jr-,^. (14) = 0.307 1 r. ConipR'ssion in S Trnsion in R lonslh of W .,,. Tension in li or conip. in D = ().:>07ir X \7r{^(u~^i~fy '*^' igtli Under ('ON<'KNTnATKi) loads W applied (H 9 and 3. len^h of iS ('oinpivssion in S Tension in H = W X = W. len<;tli of H (16) lon^li of Ji Tension in /; or conip. in I) = M' X i^^^^jT^fT;- (17) Trusses sneh as shown in Figs. 3 and 4 should Iw divided so that the rnds li, (»r I lie struts (', shall divide the lont^th of Iho ginler into three (M|Mal oi* n*'arly e<|ual parts. The len<;ths of the pi«»ci»s T", (\ li, li, >, rt<'.. should he measured on the <'entrt»s of the pleees. Tiius iIk* lrui;th of li should he taken from the eeiitre of llie lie- heaui r» lo the <-eutre of the strut I) : and the leii«;tli of Cshoiilil Im inraviiiTil from the eentre of the rod to the ivntre of the strut- IXMMI li. After dt'terminiui: the strains in the pieees hy these formulas, we may compute the areii of the eross-s(>eti(>ns hy (he folluwliig rules ; — eonip. in strut Area of cross-section of strut = — -r, • (18) <. . . . , , /tension In rod Dianu'ter of smjjle th^nMl » = \/ i^^i . {\9) ^ Al:<>^^ inL' 1'J.(MNi iioiiiidrt Hufo ifiiHiuii iN*r Miii«rc tiieh In Ibo rod. TRUSSED BEAMS. 407 ^* . . , , . , /tension in rod Diameter of each of two tie-rods = a/ T^gso * (20) For the beam B we must compute its necessary area of cross-, section as a tie or strut (according to which truss we use), and also the area of cross-section required to support its load acting as a beam, and give a section to the beam equal to the sum of the two sections thus obtained. Area of cross-section of B to / tension comp. resist tension or compression j T C ' ^ ' In trusses 1 and 2, Wx L Breadth of iJ (as a beam) = o x Z>=^ ~x~A' ^^^ In trusses 3 and 4, -^ ^'■' ^/'/^■-* - 2 X If X L Breadth of B (as a beam) = 7 ^ n^ x A ' ^^^ Id these formulas, C — 1000 pounds per square inch for hard pine and oak, 800 pounds per square inch for spruce, 700 pounds per square inch for white pine, 13,000 pounds per square inch for cast-iron. T = 2000 pounds per square inch for hard pine, 1800 pounds per square inch for spruce, 1500 pounds per square inch for white pine, 10,000 pounds iDei' square inch for wrought-iron. A = 100 pounds per square inch for hard pine, 76 pounds per square inch for oak and Oregon pine, 70 pounds per square inch for spruce, 60 pounds per square inch for white pine. Examples. — To illustrate the method of computing the dimen- sions of the different parts of girders of this kind, we will take two examples. 1. — Computation for a (jlrder snch as is shown in Fig. 7, for a span of 30 feet, the truss to be 12 feet on centres, and carrying a floor for which we should allow 100 pounds pi^r sc^uare foot. The girder will consist of three strut-beams and two rods. We van allow the belly-rod T to come two feet below the beams B, and we will assume that the depth of the beams B will be 12 inches; then the length of C (which is measured from the centre of the beam) would be 80 Inches. The length of B would, of course, be 15 feet, and by computation, or by scaling, we find the length of T to be 15 feet 2i inches. 408 TRUSSED BEAMS. The total load on the girder equals the span multiplied by the distance of girdei*s on centres, times 100 pounds = 90 X 12 X 100 = 3(KX)0 pounds. Then we find, from Fonnula 1, Tension in nxl = f», of 30000 X g^V"^^ = 65664 pounds; and, from Fornuda 20, /6y064 Diameter of each rod = x/jM^g = Ij inches, nearly. The striit-heams we will make of spruce. Tlie compression in the two strut-beams = i% of 36000 X '/,P = 64800 pounds, or 21600 pounds for each strut. To resist this compression would require 21600 -^g^ , or 27 square inches of cross-section, which corresponds to a beam 2^ inches by 12 inches. The load on B = i of 36000. or 18000 pounds; and, as there are three beams, this gives but 6000 pounds' load on each beam. Then, from Formula 22, 6000 X 1.5 _ . ^ . 1^ ^ ~ 2 X 144 X 70 " • incbea^ and, adding to this the 2} inches already obtained for compression, we have for the strut-beams three 65-inch by 12-incli spruce beams. The load on C= ^ Fl', or 22500 pounds. If we are to bave a num- ber of trusses all alike, it would be well to have a strut of cast-iron; but, if we are to build but one, we might make the strut of oak. If 22500 of cast-iron, the strut should have ^.w^q , or 1.8 square inches of cross-section at its smallest section, or al)out 1 inch by 2 inches. If 22500 of oak, IL would require a section equal to "Tqqq • or 22i square inohos, = 4^ inches by 5 inches, at its smallest section. Thus we hav(> found, thai for our truss we shall require three stmt-l)eanis 7 inclu's by 12 inches (of spruce), about 31 feet long, two belly-rods U inches diameter, and a cast-iron strut 1 inch by 2 inches at the smallest end, or else an oak strut 4i inches by 5 inches. 2. — It is desired to support a floor over a lecture-room forty feet wide, by means of a trussed girder; and, as the room above is to be used foi- electrical i)uri>oscs, it is desiretl to have a truss with very little iron in it, and hence we use a truss such as is shown in Fig. 4. re the girders rest on the wall, there will be brick pilasters g a projection of six inches, which will make the span of the 10 feet ; ^nd we will space the rods /if /^ so as to diTldeUieUe- into thi-ee equal spans of 13 feet each. The tie-taun will •TUUSSED BEAMS 409 consist of two hard-pine beams, with the struts cominjGf between them. We will have two rods, instead of one, at i?, coming down each side of the strut, and passing through an iron casting below the hoanis, forming supports for them. The height of truss from centre to centre of timbers we must limit to 18 inches, and we will s})ace the trusses S feet on centres. Then the total floor-area sup- ported by one girder equals 8 feet by 89 feet, equal to .*U2 square feet. Tin; heaviest load to which the floor will be subjected wiii be the weight of students, for which V) pounds per square foot will be ample allowance; and the weight of the flooi* itself will be about 25 pounds; so that the total weight of the floor and load will be UK) pounds per square foot. This makes the total weight liable to come on one girder 81,200 pounds. Then we find, Formula 14, 157 ins. Compression in struts = 0.;^>7 W x .o. ,., = 106800 pounds. 156 ins. Tension in both tie-l)eams = 0.867 ir X ^^ .^^.^ = 106000 pounds. Tension in both rods i? = 0.807 W = 1 1450 pounds. The timber in the tniss wdl l>e hard pine, and hence we must have 10(>8(X) —TTwTTT-, or 107 square inches, area of cross-section m the strut, which is equivalent to a 9-inch by 12-inch timl)er . or, as that is not a merchantable size, we will use a 10-inch by 12-inch strut. The tie-beams will each have to carry one-half of 106000, or 58000 5800()__ pounds ; and the area of cross-section to resist this equals ^j^ — 27 inches, or 2^ inches by 12 inches. The distributed load on one section of each tie-beam coming from the floor-joist equals i:J X 8 X 100 = 10400 pounds; and from Formula 28 wo have ^ = ^ 7T. 7 = ^ — ^Mj ^/w> = 3? inches. Then the breadth 5 X JJ- X A 5x 144x 100 of each tie-beam must be 84^ inches + 21 inches = 6 inclies : hence the tie-beams will be 6 inches by 12 inches. Kach rod will have to /57/i5 ^..^ = } inch, nearly. Thus we have found, for the dimensions of the various pieces of the girder: — Two tie-beams 6 inches by 12 inches; two rods at each joint, J inch diameter i and strut-pieces 10 inches by 12 inches. A\0 BIVETEU PLATKIHON GIKDKHS. CHAPTER XX. RIVETED PLATE-IRON GIRDZSR8, Whenever the load upon a girder or the span is too great to admit of using an iron beam, aiul the use of a tmssed wooden girder is impi-acticable, we must employ a riveted iron-plate girder. Ginlers of this kind are quito commonly used at the present day ; as they can easily be made of any strength, and adapted to any span. They are not generally used in buildings for a greater span than sixty feet. These girders are usually made either like Fig. i tlWW n'A'AMyVitf.wj Fig. 2. or Fig. 2, in section, with vertical stiifeners riyeted to the web- plates (»very few feet along their length. The vertical plates, called '' web-plates/" are made of a single plate of wronght-iron, rarely less than ont^-fourth, or more than five-eighths, of an inch thick, and geiKM-aliy tliive-i>ightlis of an inch thick. Under a distributed load, the web of three-eighths of an inch thick is generally snfll- ciently sti-ong to resist tlu^ shearing-stress Ln the girder without ng, provided that two vertical pieces of angle-lroD ; r ivebed >^eb, near each end of the girder. Tliese ve ii i !«■ of >n or T-iron, whichever is used, are c "; ten the girder is loaded at the centre, ana : • If K4":- . RIVETED PLATE-IRON GIRDERS. 411 under a distributed load, it is necessary to use the stiffeners for tlie whole length of the girder, placing them a distance apart equal to the height of the girder. The web is only assumed to resist the shearing-stress in the girder. The top and bottom plates of tlie girder, wliich have to be proportioned to the loads, span, and lieiglit, are fastened to the web by means of angle-irons. It has been found, that in nearly all cases the best proportions for the angle-irons is :i indies by 3 inches by .J inch, which gives the sectional area of two angles five and a half square inches. The two angles and the plate taken together form the flange; the upper ones being called the '* upper flange," and the lower ones the ** lower flange." RiVKTs. — The rivets with which the plates and angle-irons are joined together should ho, three-fourths of an inch in diameter, unless the girder is light, when five-eighths of an inch may l)e sutti- cient. The spacing ought not to exceed six inciies, and should be closer for heavy flanges : and in all cases It should not be more than three inches for a distance of eighteen inches or two feet from the end. Rivets should also not be spaced closer than two and a half times their diameter. Rules for the Strength of Riveted Girders. In calculating the strength of a riveted girder, it is customary to consider that the flanges resist the transverse strain In the girder, and that the web resists the shearing-strain. To calculate the strength of a riveted girder very accurately, we should allow for tilt* rivet-holes in the flanges and angle-irons ; but we can com- pute the strength of the girder with sufficient accuracy by taking the strength of the iron at ten thousand pounds per square inch, instead of twelve thousand pounds, which is used for rolled beams, and disregardnig the rivet-holes. Proceeding on this consideration, we have the following rule for the strength of the girder : — 10 X area of one flange x height Safe load in tons = :] x span in feet ' ^ ^ ) Area of one flange I _ 3 x load X span in f eet in square inches ) 10 X height of web in inches' ' The height of the girder is measured in inches, and is the height of the web-plate, or the distance between the flange-plates. The w(^b we may make either one-lialf or three-eighths of an inch thick ; anil, if the girder is loaded with a concentrated load at llie centre or any other point, we should use vertical stiffeners the whole length of the girder, spaced the height of the girder apart. 412 RIVETED PLATE-IRON GIRDERS. If the load is distribvted^ divide one-fourth of the whole load on the girder, in tons, by the vertical sectional area of the web-plate: and if the quotient thus obtained exceeds the figure given in the following table, under the number nearest that wlifcli wouhl 1.4 X height of ginler bo obtained by the following expression, " thickness of wci7 ' then stiffening pieces will l)e required up to within one-eighth of tho span from the middle of the girder. c/ X 1 .4 t 31) 3.08 35 2.84 40 2.61 45 2.39 50 2.18 55 1.99 60 1.82 65 1.60 70 1.52 75 1.40 80 1.28 85 1.17 90 1.08 9;) 1.00 100 0.92 Example. —A brick wall 20 feet in length, and weighing 40 tons, is to be supported by a riveted plate-girtler with one web. Tho girder will be <24 inches high. What should be the area of each flange, and the thickness of the web ? 3 X 40 X 20 4ns. Area of one flange = — m x 2^ ~ ^^ square inches. Subticicting 5 squai*e inches for the area of two 8-inch by 8-inch angle-irons, we have 5 s(iuare inches as the area of the plate. If we make tho plate 8 inches wide, then it slK>uId be5-r8,orfofan Inch thick. The web we will make J of an inch thick, and put two stiffonors at each end of the girder. To find if it will be necessary to use more stiffeners, wo divide J of 40 tons, equal to 10 tons, by the area of the vortical section of the web, which eqimls f of an inch X 24 inches = sciuaro inches, and we obtain 1.11. The exin^esslou 1.4 X lioii^ht of girder thioknoss of \vA) — ' *" ^^^'^^ **^**^' ^^^^^^"^ ^'^' ^* number near- est this in the table is 00, and the flgure under it is 1.06, which is a little less than 1.11 ; showing that we nnist use vertical stiffeners uj) to within i\ feet of tho centre of the girder. These vertical stiff- eners we will make of 2i-ineh by 2j-inch angle-irons. From tlie fonnuhl for th(> area of flanges, the following table has been coni- piilei). wliiel) greatly faeilitato.s the process of finding the necessary area of flanges for any given girder. RIVETED PLATE-IRON GIRDERS. Co-efficient for deLenninin;; Ihe area required in flanges, allowing 10,00IJ pouiiils ]wr siiuare incb of cross-section fibre strain ; — 1U:lk. — Mnlliply Use load, in tons ot -JOOIl i)Oiinii» unffomily ilistribiitetl, by tlie co-?fbcient, and dividu by 1000 pounds. Tlie quotient will be the gross area, in square inches, required for each llan^. I im ms. ExAMl'l.E. — l.ol iLS take the same giriler that we have jiisl c'0]iipiite<l. Here llie a]>an was 20 feet, and the depth of girder 24 iuehes. From the table we find the eo-efli<!ient to \»: 2-~)0 ; and multiplying this by the loail, 40 tons, and ilividlng by 1000, we have lU square inches as the area of oue Sange, being the same result as thai obtained before. 4U RIVETED PLATE-IRON GIRDERS. Girders intended to carry plastering should be limited in depth (out to out of web) to one-twenty-fourth of the span-length, or half an inrh per foot of span: otherwise the deflection is liable to eau<e the plastering to crack. In heavy girders, a saving of iron may often i)e made by nMlucing the thickness of the flanges towanls the ends of th(^ i^irder, where t\w strain is h'ss. The bendinir- moment at a number of points in the length of the girder may Ix' detJMiiiined, and the area of the flange at the different i)oints nia<h' propoilional to the bending-moments at those points. The thick- ness of the llanges is easily varied, as required by forming them of a sutticiiMit numlu'r of plates to give the greatest thickness, and allowing them to extend on each side of the centre, only to such distanc'es as may be nt'cessary to give the required thi<:kness at each point. The deflection of girders so formed will be greater than those of uniform cross-sectiou throughout. TABLES OF SAVE LOADS FOR RIVETED PLATE- IROX GIRDERS. The tables given on pp. 414 and 415 have I)een computed ac- cording to the fonnula on p. 411, to give an idea of the siz«* of girder that will be reiiuired for a given load, of the heights and siKin^ inlieiited. If i; i- r(nuinl)ered that the strength of a girder depends tUrectly as tlh- ;i!( a of its llanges and its height, the width and thickness of the tl,in.r<- pi ite may be changed, inttrided the area rcniahis the .sn,in . witlnni* altering its strength. Thus a girder ii(5" liigh, with tlaiu.- tni.i,.- i of 4.r' X 4f' X ^" angles, and f X 24" plate, would be as vT,.ni:, as one with th«' same aniilos and 1" X 12" plate, pro- vi«l.' I iIm' u»'!> plates are ])r()perly stitTened, as described on p. ;i47. In eompuiiiv,' li:e weight of the ninlcrs in the tables, no allow- ancf b.l«^ h«M'ii made for siitT<»ners. In computing the stn^ngth of rivet*'. 1 uiidrr'*, it will be convenient to know that — The ana of two :V' x ;}" X |" angle-irons = ').iy stpiare niches. :U X ;)f' X f *' =({.4 *• 4' X 4" X f *' =7.4 " 4f' X 4f' X f ** =v{.4 «-• RirKTBD PLATR-IRON QIBDKR8, 41 PSPSSBsasss I SBSSESSSS38K SSSSSSSSSSgg SSSKSSSSKSaS ' gS3SS3SS5 !SS Si!5 S SSSS*5:;«S3(3; as 55 ' g'-SSSSSSSS ^gv s:i^sssgs;3ss$$$:is RlVbrrKD PI^TE-IBON UlKDEHa 1^ t I' ' n I a ■' sisssas^psisus > SSiSSiSSESESS I gSgSSSSfiSSSSSSS SSS£«SS$$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 I-beams with top and bottom flange plates. Such girders are commonly used for supporting interior walls in buildings. The following tables give the safe loads for ** Carnegie " beams, with different thicknesses of plates. They were prepared for steel girders on account of the advantages possessed by steel beams over beams of iron. The former are more economical of section and permit the use of a higher unit strain than the latter. The values given in the tables are founded upon the moments of inertia of the various sections. Deductions were made fgr the rivet holes in both flanges. The maximum strain in extreme fibres was limited to 13,000 lbs. per square inch, while in the tables on- rolled steel beams a fibre strain of 16,000 lbs. was used. This reduction was made in order to amply compensate for the deterioration of the metal around the rivet holes from punching. Box girders should not be used in damp or exposed places, since the interior surfaces do not readily admit of repainting. ExajIPle. — A 13' brick wall, lo feet high, is to be built over an opening of 24 feet. What will be the section of the girder re- quired ? Ans. — Assuming 25 feet as the distance, centre to centre of bearings, the weight of the wall will be 25 x 15 x 121 = 45,375 lbs., or 22.68 tons. On page 420 we find that a girder composed of two 12" steel beams, each weighing 32.0 lbs. per foot, and two 14" x i" flange plates will carry safely, for a span of 25 feet, a uniformly dis- tributed load of 23.23 tons, including its own weight. Deducting the latter, 1.42 tons, given in tho next column, we find 21.81 tons for the value of the safe net load, which is 1 . 07 tons less than re- quired. From the following column we find that by increasing the thickness of the flange plates ,^j" we may add 1.52 tons to the allowable load. This will more than cover the difference. Hence the required section will be two 12" steel beams 32.0 lbs. per foot, and two 14 ' x ■^%" steel cover plates. 27 418 bterl beam qibderb. stki':l beam girdbbb. safe loads in tons, uniformlt disnubhtbel S-X" eti.«l (Caiiiogle) I-beama and 3 aleel platw 18" x J" it liii" lit in.'iu lb*, prraq. !■. STEBL BEAM OIRDEBS. 419 STKBL BEAM QIRDBRg. SATB LOADS IN TONS, UNIFORMLT DI8TRIBUTXIO. X-IS" Bieel (Carnegie) I-beams and 9 utee] plates 14" > |" 420 STEEL BEAM GIRDERS. STEEL BEAM GIRDERS. SAFE LOADS IN TONS, UNIFORMLY DISTRIBUTBIX si-lS" steel (Carnegie) I-beams and 2 steel plates 14" x i" « tt c /> jf — 6 — >r«, ^ 1 ^^ a) ^ b=^'„„ . , Si Vi" steel . . ^ r, uJ X> steel I-beamn. ^^ *^-» " 12" steel ^Ui O plates, 40.0 lbs. 2 steel I-beams, £ 14" X J" per foot. .JA*^?',- 1 83.0 lbs. •§« centi eet. 4 ^r-^ 14 Xf- I ^1I^ per foot. •a'5 0<4- ti S **^ .i^ 1 1 I JS 9 «.s 1 «k-«M «- ♦-» '-a 1 o • » ( j:«i-i*-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 5-3 r — "S Ti Ti •S.5 si .5.S 10 , (U.IM ' <).r.5 3.':.') 58.08 0.57 i 8.81 0.06 11 5.>.(W 0.71 3.40 32. SO 0.63 i 3.45 0.08 12 r>4.1.> O.T". 3.12 4S.40 0.68 3.17 0.08 13 1 41). IC) 0.S4 2.8S 44. (W 0.74 2.W . 0.04 14 4()..i'.) n.!)l 2.«S 41.48 0.80 2.7« 0.04 ir. 13. vM) U.')7 2..-0 38.72 0.85 2.58 0.04 K 4 ).:.!) 1.04 2.34 3<)..30 0.91 2.88 0.U5 K ; ;«.-.>() 1.10 2.21 i^.m 0.97 2.34 0.05 is 3(».i>S 1.17 2.08 82.27 1.03 3.11 0.06 IS) 31. 1« 1.2.3 1.97 30.57 1.08 > 3.00 i 0.05 iM 3i.47 1.3«) 1.87 29.04 1.14 : 1.90 U.06 iil 3). '.13 1.3«; 1.7S 27.r.« 1.20 ' l.Hl 0.06 .>.) -v^ •>I!)..V2 1.43 1.70 2(5.40 1.25 1.78 0.06 i>.3 ^s.\>:j 1.1'.) i.r»3 25.2.'> 1.31 1.65 0.07 i»l '.■:■.<)»» !..')<; i.r><» 21.20 1.37 1.5H 0.07 U.-) .::>.i»s i.r.i \.rii\ 23.2.3 1.42 1.52 U.07 •,M) :i.'.ts 1. »•.'.» 1.14 2-2. :« 1.4S 1.46 O.08 •^>r !!.'•.') 1.7.") i.:is 21.r.l 1.54 1.41 0.06 '.»S ,'3. 1) 1.S2 1.34 2.1.74 1.1,0 1.86 0.08 •^.t •,»•,». 31) l.SH 1.21) 20.03 1.K5 1.31 0.08 ;j' 'JI.JJ-, l.li.-. 1.2.". 11). :W l.;i 1.27 0.(« ;n •J ».!).'i 2.111 1.21 18.7:1 1.77 1.23 0.09 3j •JO.-JI ■J. OS 1.17 IS. 15 1.82 1.19 O.OO ;{:$ I'l.'N 2.M 1.14 17.J»o l.SM 1.15 . 0.10 Ml l!».l.t 2.21 1.10 i;.«>8 1.1»4 1.13 0.10 .-i.-) IS.-.:) 2.-,: l.i>7 10..-..) 1.19 1.09 0.10 .*i'i IS. .11 2.::4 1.04 in. 13 2.05 i.r6 0.10 3: 17..-.-. 2.4" 1.01 15. :o 2.11 1.08 0.11 i^ K.iM 2.4-: o.ii'J 1.').2S 2.17 l.ilO 0.11 i** i"i.»ir. .> r Vl o.w I 11. M) 3.32 0.98 0.11 .\)mi\i- \.tiu('>< .ire WaM* i nn nia.\iniuin lUirc i^lniins of 18,01)0 Ibi*. pvr M|. to. Uiv'-i iiii!i-< III l.i.ili ilHiit{«'?< deducteil. Weif^hti* uf KirUvn corKiipuiid tolcngtba I'fri re ft. rriiJn- i.r licurin^'!*. STEEL BEAU OIRDEBS. 431 STEEL BEAU GIRDERS. SAFE LOADS IN TONS, UNIFORMLT DISTBIBUTBDl S-IO" steel (Camegic) I-beama and 2 eteel plates IS" n )" Atvore TmlneH ore ba^cd on m'jximu Blvet holes In botb naDgee deducted. «mA» to emtae of beuinge. 422 CAST-IRON ARCH-GlRDKUa. CHAPTER XXT. STRENGTH OP CAST-IRON ARCH-GIRDERS, "WITH WROUGHT-IRON TENSION-RODS. Oast-iko.v jircli-girders are now (juite extensively enii)loyeil to support tlic front or rear walls of brick buildinfj^s. Fig. 1 shows the usual form of such a girder, the section of the casting and roil hv'uvj; shown in Fig. 2. — ii .' Fig. 1. Fig. 2. Th<^ casting is niad(» in one ])iece with l)ox ends, the latter having LCroov*'^ and scats to nMH'ivc the wrought-iron tie-rod. rii«' ti('-!().l is madi* from one-eighth to three-eighth8 of an Ineh sliorirr tlian the casting, and has scjnare ends fonning shoulilers so as t(» li! into the castings. The rod has usually one weld on its liimtli. and ixrcat <*arc shouh' he taken that this weld lie jXTftH't. Tlir r<> I is ('X])and«'d hy heat, and then pla<'etl in i)Osition in the (•as; in.:, and allowed to eontra<'t in cooling; thus tying the two enils ()!" iln' ca^'ing together to form abutments for receiving the hori- zontal iliiiist of tie* areji. If the rod is too long, it will not n*eeive till- full ]i'-oportion of the strain un'il the east-iron \iha so far dr- ll««t«i|. tliai its lower edge is >ubji'eted to a severe tensile strength, whirh cast-iron <'an feebly resist. If the ti(»-rod Is made too short, the ea^tinu is eambered up, and a sev(>re initial strain put Upon both the east and wrought iron, which enf(*4>hles lK>th for carryii|g CAST lEON ARCH-GIRDERS. 423 a load. The girders should have a rise of about two feet six inches on a length of twenty-five feet.i Rules for Calculating^ Dimensions of Girder and Rod. A cast-iron arch-girder is considered as a long column, subject to a certain amount of bending-strain ; and the resistance will be governed by the laws affecting the strength of beams, as well as by those relating to the strength of columns. Fig. 3. If we regard the arch as flexible, or as possessing no inherent stiffness, and the rod as a chord without weight, we can deduce the following formula for the horizontal thrust or strain : — Hor. thnist _ ^^^^ P^^ ^^^^ ^^ span x span in feet, squared^ or strain "" 8 x rise of girder in feet ^ ' From this rule we can calculate the required diameter of the tension-rod, which may be expressed thus : — Diameter in inches Vloail on girder X span in feet 8 X rise in feet X 7854 (2) The rule generally used, however, in proportioning the wrought- iron tie to the cast-iron arch is to alloiv one square inch of crosa- section of tie-rod for every ten net ton.^i of load impoaed upon the span of the arch. The following table, taken from Mr. Fryer's book on " Architec- I _ I — 1 Andiiteflliml Iron- Work for Buildings. — William J. Frter, Jun. Pp. 38. 4J4 OAST-IHON ARCII-GIRDERS. tiiral Iron- Work," shows ihe section of the cast-iron arch requirptl to supinrrt solid hrick icallsy and haciny a span of from 13 tot6 feet. I<it{lil of wall. Tliifkncss wall. of Di Top flange 4u ftM't. -III " I 12 iuchi'H 12 " 1 ir» 1 10 " 1 . 12" X 1" 12" X 1 y 12" X 1, " 10" X ij'' _ DiMEXHioxH OF Section. Centre web. 1-2" X 3'' i— '^ 8 12" X »•' 12" X 1" Bulb. ;i" X 2" 1// y ,>// X 2" 3i" 4^' I Substitute for Cast-iron Areli-Ciirder, In tlu* cast-inm arch-j;inior with wrou.uhl-iron tcnsioii-roil. the ca^tin;^ only serves to resist coinnn'ssion. Its place can as wi*!! be till«Ml l»y a l)rick arch foottMl on si \mv of cast-iron skewbacks, wliich an* thcnis(»lvcs held in ixwilion by a pair of tie-rods, as in In I Ills case, Fornniht 1 will still jjivc the horizontal pull to be resistci by the tie-rods ; i)Ut, as vvc nnist have two rotls instoad of ont , tlic diameter of eac.'h will bo obtained ))y the Ibnniila, Diameter o. each _ /Tj*!*.") )*>»''^"" arch X sjian ^.j rod in inclics \ Hi X rise of arch in fiM'l X 'tSTA N.r.. — TJu- rlH- Ik nie!i!>.i:r«Hl from thi* cent a' of the nnJ to the eentre of the :t:i-li. It will alM) ln' rem*>ini)<3n'd that the hpan iK tti \h.'. (i/irt/^jr taken In feet* mile--. DiliiTwi-if spti'iti-'d. Kx.vMPi.i-: I. — It is desired to siipiM)rt a 12-incb brick wall Ai) til-* liiuli «'\ci- an n]H'nin.Lr -'► l'e«»i wide, with a easi-iri>n an-li-iiinliT. ''.'Ii;t! -imidd lie ilic dim 'n-^inji-! of lln' u:ir<ler'.* I'nr !!•« riistin;;. we lind from the tabic that the eross-sei'tion of ;li.' llanv:"- hnnid be li: iin-b:" l>y 1 ineb : of tli«' web, TJ inebi's b\ : in-li : and of I lie bnlb. :! incbes by li inches. W'v will make llw ri'^e nt I'lic udrder *J feet and <> incbes. and fnon Forninla 2 Wf HniP \\ei<.;bt of wall X s]ian i)iim. **i I .kJ in im-bes - \ j*^ • ^ ... :.. «• , v -w?i"i \ s X riNc ot areli in feel X iK>4 Miio X -JO X Ml') X :io_ , — \ s X lM X 7s:h " ^ •■'•" = '-^* ^^ > I '••I:-! I>!i-;L; Ibai ihi* uinli-r wonlii o:ll^ Kup|N>rt atNiiit twcnly feet of Ihf ■\)k\\ in lii-iL'ht, thi> will! abiiM- l>i,tt ^uppnrtillK tlMfif. WOODEN FLOORS. 425 CHAPTER XXII STRENGTH AND STIFFNESS OF WOODEN FLOORS Strengrtli of Floors. — In calculating the strength of floor- beams, the first thing to be decided is the span of the beams, which is generally determined by the size of the opening to be covered ; and the second is the load which is to come ui)on the floor. Wooden floor-beams should not have a span of more than twenty- five feet (if it can be so arranged ) : for, if they are of a greater length than this, it is difficult to stiffen them sufficiently to prevent vibra- tion under a heavy or moving load When the distance between the l)earing- walls of a building is greater than the above limit, par- tition-walls should be built, or else the beams should be supported by iron or wooden girders resting upon iron or wooden columns. The Building Laws of the cities of New York and Boston require that m all buildings more than thirty feet in width, except churches, theatres, schoolhouses, car-stables, and other public buildings, the space between any two of the bearing- walls shall not be over twenty- five feet, unless ginlers are substituted in place of the partition- wall. Floor-beams, when supported at three or more points, should always be made continuous if possible, as the strength of each portion of the beam is thereby greatly increased. Superimposed Loads. — There is some difference of opinion among authorities as to what should be allowed for thc^ suprrim posed load upon the floor of a dwelling or upon the floors of public buildings. The New- York Building Law requires that in all build- ings every floor shall have sufficient strength to bear safely upon every superficial fool of its surface seventy-five pounds, and, if used as a place of public assembly, one hundred and tvv«^nty pounds. In dwelling-houses, where the maximum load consists of nothing but ordinary furniture and the weight of some ten or twelve people, it is not necessary to allow more than forty pounds per square foot for the superficial load ; and, in most cases, eighty pounds per s(|uare foot 18 ample allowance for the weight of an assemblage of peopl(^ Only in cases where people are liable to be jammed together during 426 WOODEN FLOORS. a jMinio or some unusual circumstance, is it possible to p;pt a weight on the tl<«>r of one humlriMl ami twenty pounds per Rr{uarp fool. Tlu' follt)\vin^ tablt* iiivcs tlie weight per squaiv foot which shouM l»e assume* I, in addition Lo the wciglit of tlir floor, for thcso various cases : — For stHM't l^ridges for general public traffic, S*t lbs. per s<{uarp foot. For tln»)r'5 of dwellings 4H lbs. {-ter s<|uaiiMont. Ft»r iliunh»'>. theatres, and ball-rooms, SO to V2i) lbs. i>er si|iian* foot. For s'hools ^<0 lbs. per sqiiar** foot. Fur hay-l«»tts S() lbs. per square loot. For si(>rai;e of gram HH) lbs. \n*r st{uare fool. For wan'houses anil general merchandise, '1')^) lbs. jHir siiuare foot. For fa<tories 1(M> to 4(X) lbs. per square foot. F'or oHirt- buildings liH> His. per square foot. Wan 'ho list '-floors are sometimes very heavily loaded, and for lhe>f a >iMMial compulation should bt? made in each case. Til- i.»ll.nvhig table, compiled by Mr. 0. J. H. Woodbury.' gi'<*« the tli>«>r areas, cubic space, and weights of merchandise, as usualty siori'd in warehouses. If the goods are piled two or luore cas^ hiu'h. I lie weight per square foo: of lloor will of course he increased in proportion. " The measuremenis were always taken to the outside ol case or package, and gross weights of such packages are given." Matkkial. W»>ol. Ha!.- K.i^- I- .li.i . . . . *• A;:*'.:. I i.i . . . . " S 11*. !; \iiiiTii"a . . I •* o ,_:n 1 I •• < '.I ;:■■;• li.i .... Kiii W ■'. >;.u^ . :" S. ■iind Wiml . Wiiolll'll (ifMXls. ' ';i"'f }• .% . x-t" •• K r ;:• -. hi':»\ y . . I >ri'-- « i■nlll^ . '• < ".i-- iii>-*i ■«.... •• 1"'. :• u r.ir .... •• li I A :» .... •• II - W .iiikii.-. . . Ci»!ttii». I'tr. i;.i- • -^-1 ii . . . ! •■ . i> « ■■•in:ni-«-i\l •• .1 -.- ..'... •• -1 :"■ I i-hiiiir-* ■' VI I i: I . . X -I I •^«il • • • I a MkaSI- HEME NTS. Floor ...0 .'•.s 7 7.5 7.1 .'i . .'» M..-» » ■! .•! 111.:; 4.0 N.l 4.1 •J. 4 •J.n ■• .t •*».S I • II Mo M. ■o. 3i). 1J.7 'J-J.rt ■JS.il •Jl.O 14.0 44.-2 ■« ,.• .1. 1 iii..'i :;4.7 17.11 Wbiuuts. (fPiM*. M. ft. 113 IVr cubic f :. ;mo 28 s-^ m lA lUiJO 14S 29 4S-2 70 lA .'mO 73 17 •M 40 7 — - 5 ■*jrt 40 17 .^•.0 40 22 , 4i*iO M 21 ! .vm ft2 •» ;;.-> 4)« 10 4.'H> 44 13 ■j..i» 63 18 M.*, 64 12 :»-.o 134 25 l-V. 100 40 :«k) 125 4:.i» 174 43 •>rt 88 « TOO Rl » 41 «) 7» M I Dif Kirt> PriM.u-tiitii ttf Mills, ii. lift WOODEN FLOORS. 427 Iatbrial. I €k>ods. leached Jeans . 3k vn Sheetings «hed Sheetings . t8 t Cloth. . . . ts ings >tton Yam . . ?*ng in Bales. nen tton )tton . . . . ivings . . . . td Book . . endered Book er . . ard . toard Bags Bulk « mean lour on side •• on end tags . . . in Barrels ags . . . lay . . . lerick Compressed « « Measurements. tiiflf'*, etc. I Bleaching Powder, Soda Ash . . . ?" rh ac oda in iron drum . arch >arl Alum .... act IvOgwood . . ime .'raent, American . " English . . aster Floor space. 4.0 1.1 3.6 4.8 7.2 4.0 4.5 3.3 1.4 8.5 9.2 7.6 7.5 16.0 7.5 2.8 4.2 4.1 3.1 3.6 3.7 3.3 5.0 1.75 1.75 1.75 1.75 11.8 10.8 3.0 4.0 1.H 4.3 3.0 3.0 1.06 3.G 3.8 3.8 3.7 Cubic feel. 12.5 2.3 10.1 11.4 19.0 9.3 13.4 8.8 5.3 39.5 40.0 30.0 34.0 65.0 30.0 11.1 4.2 5.4 7.1 3.6 5.9 3.6 20.0 5.25 5.25 5.25 5.25 39.2 29.2 9.0 3.3 4.1 0.8 10.5 10.5 .8 4.5 5.5 5.5 6.1 Weights. Gross. Per Per sq. ft. cubic ft. 300 72 24 75 68 33 235 65 23 330 60 30 296 41 16 175 44 19 420 93 31 325 99 37 — — 11 130 — 30 100 70 24 910 107 23 715 78 18 442 50 15 507 68 15 450 28 7 600 80 20 400 143 36 50 — — 69 ~ _ 38 _ 33 _ _ 59 _ _ 64 . _ 10 — - 37 165 39 39 _ 44 _ _ 39 — — 41 218 53 40 218 , 70 31 112 31 31 218 59 37 96 29 27 284 57 14 125 72 24 100 67 19 150 86 29 100 57 19 1200 102 31 1800 167 62 385 128 43 1.50 38 45 160 100 39 600 140 88 250 83 23 350 117 33 55 52 70 225 63 50 325 86 59 400 105 73 325 88 53 4L>h5 WOODEN FLDOKS. Matbbial. I I Dye RtnflDB, etc—OonVd. Barrel KuHiii •• LardOU Uope . Miftcellaneoos. Box Till •' GhL*^ C rate ( 'rockery (':i-k Crockery I>aie Li-ailier " (rnatr<kin8 " iiaw Hides " " '• compref8ed, '• Sole Leather . . . Pile S.ilf Leather . . . I>arrel Granulated K^ugar. Brown Sugar . . Cheese Measubexents. Floor space. 3.0 4.3 2.7 9.9 1U.4 7.3 11.2 rt.O 0.0 lli.tt 3.U 3.0 1 Cubic feet. 9.0 12.3 0.5 39.6 42.5 12.2 16.7 3<).0 30.0 s.y 7.5 Wbiohts. OroBB. 430 422 139 1600 600 190 300 44X) 700 200 317 340 Per I Per BQ. ft. cubic ft. 143 98 48 4A 09 278 . 60 102 40 52 14 26 16 27 18 67 13 117 23 22 16 — 17 106 42 113 45 - ao AV<Mj4:Iit of tlie Floor itself. — Having <lecided upon the span of the Moor boanl^ an. I upon the siiiH.Tinii)Oseil load, we must nt'xl consithi- the weiijjlit of th»? tlix^r itsrlf. WoodtMi floors in (hvellinjxs wiMirh. on thcavprago, from 8eveni.»H»n to twrnty two i>oiimls ]kt vS(|uai(' foot of floor, incluiling tht* weight of tin* plastt'rini: on the nmh'r sn\v. For onlinarj' spans tho Wiight may l)»' takrn at twontv pounds iH»r squan* fool. Jind, for lorn; spans, twnity two pounds por squan* f<K)t. For floors in public bulldins^, tin* \\«'ii:lit piM- sq nan' foot seldom oxcoeds twenty -five pounds, and it nia\ NMti'Iy ))e assumed at that amount. In wanliouse floors, whieh havi- to sustain ver\* hoavv loads, the w»'iu'lii iM'i- sqiian- fool may souH'times 1h» as gnMt as forty or fifty lHinnd>: and m Mieh ease*^ the a]»pro>Limate weight of the floor ^kt Miuan* \\n*l >hoiild l>e tirst caltulateil. FjU'tor of Safety to be used.— In eonsiderlni; tho load on a tliiiir. it siiould !>«> lememhtM-ed that the efTt>et of a load bud di-nlx applied uiK)n a Ix'am is twiei> as i:n-at as that of the Hanie ln:ii| i:i'.idnaliy applieii: and hrnrc the fa<'toi of s;ifety utH*4l for llu* fiiiimr »Li)Mld In* I win' a<« ijiral ax> that for the latter. The loail i-.iiio.l li\ a i-mwd of priipir i^ usually ron>ider«Ml to pn>ihii'«' an <th-<-i vxliii-ii i» a iiii-aii iH-iWft'ii thai nl llir sinH* ItKid wheli ;;ratlu- a)l\ and w Inii sitild<-nl\ a|tplled ; ami hmer a faelor of safKy IS iiii|>lii\fd wliirji i.s a mean lH*tw«'en that for a live and for a dead load. Tin- faihu-H of safi'ty for lltNir-iindM'rs adoptetl by the best engfn- citn \ar\ troni -i to 't. For short s|»ans hi onllnary dwelllngSi pnhlic Ituijilinu^. and Moivs, :{ is probably amply HUlHcieiil for I'JI ' - WOODEN FLOORS. 429 strength ; but. for long spans, and flooi*s in factories and machine- shops, a factor of safety of 5 should often be used.' Rules for the Strength of Floor-beams. — In consid- ering the strength of a floor, we assume it to be equally loaded over its whole surface, as this would be the severest strain to which the timbers could be subjected. Hence, in calculating the dimensions of the floor beams, we use the formula for a distributed load. That formula i^ for rectangular beams, 2 X bread th x depth squared X A Safe load - span in feet x S ^^^ *S being the factor of safety. For floor-beams the safe load is represented by the superimposed load and weight of floor supported by each beam. The areA of floor supported by each beam equals the length of beam multiplied by the distance between centres. If we. let f de- note the weight of the superimposed load per square foot of floor surface, and/' the weight of one square foot of the floor itself, then the total weight per square foot will be (/+/') pounds, and the total load on each beam will ecjual Length of beam X distance between centres x (/4-/'). Now, if we substitute this expression in place of the safe load in the above formula, and solve for the depth, we shall have, Square of __ S x dist. bet, cen tres x length squared x (/ + /') depth. - 2 x~ bread thlT^ ' ^^^ or, if we solve for the distance between centres, we shall have, Distance between _ 2 x breadth x depth squared x A centres in feet - ^sVlength"^ared x (/ + /') ° ^^ N. B.— The length and distance between centret* must be taken in feet' and the length meanB only the distance between sapports, or the clear span. The values of the constant A for the four woods in general use are as follows : / Spruce 210 Eard pme 300 Oak 225 White pine 180 Formulas 3 and 8 apply to all floors supported by rectangular beams, whatever be the factor of safety employed, the weight of > Until very recently It has been our custom to use factors of safety twice as great as these : bat, as we have had occasion to reduce the constants for strength to abont one-half of that formerly used, we have reduced the lactors of safety ■ecofdins^y. It will be found that the result is the same as that obtained by the n3M«f odborirMtefs. llic sii[«TiiiiiM)sf.l limil, or of the Htm- ilstll. To illustrete the ii|il)liculi(iii (if tlicsc tnrniulas. ve will i;ive two examples such as K\Ai;i'i.K 1.— IVIiiii sliiiiilil In- ilu- <1iiui'nsiniiB of thi- HpruLi' nonr-liiMins in !i liwlliii;:. Ilii' lu-aiuM lo limv ii 8)Ht]i of 13 reel, and tub,, j.lmoil llliiKOu-s. i.L-1': fis'l. iinci'ii1n.sV -l».. Iti lliisciisi' w,-ninililusi' a TiLcKir ..f siifet)- of -I : / sl.oiil.l In- iHki'ii III Id ix>iiiuis. / ,11 i I iHiiiiiils, ami .'I is 210 jKimids, A*. suizii' 3 iiiilics fur till' lirmilli. Tlieii, by Fiirmulu •. . eo_ 2 . 2 X aio •■''H.ti.-'V' r,:™'!0.5 Till' ilopfli A "■^.■' " 'il"c '"■'■■■ fl ifitlies. Tkiifo. to haw the miuisil.' s1n>ii},'lb. 1bi> IxMins sli.mlc! lit '> x 111 iTich.'s. KxAMi'i.i: a. -11 is lU-simi 1.. us,- 3 by 10 im-li VL-lIow.i.inr (•.■iuns in til,. n.«.r of ii ■■l.iircb. ill,- lu'U.iis to Imve a spin of IB ii','t. What ili-^tiini'i' sbiiiilil lli,-y Ix' s|m<ril ,>ii i:uln» i nils. / ■•', iioiuiilii, and A -■ 300 I"' „ls, 'n,.'ll, by F„ \ SIM) l.isri,n.v b,.,w.rTi ...jntivs -. ' 1 ' .;.;;- J,,": ^ ^ 0.73 ft.. ..rft i..». Ut-iic-i' III,' MiKir vrillbeMitli(-ii'ullyslrouj;if thclH'iiiuaiiru pluoottlt IlrUl^iiiK of l''t(>'>r-lM'anis. — lly "luidcina" i<* ini>nnt sy-d'iii of lirai-iiii; fl«Hir-l¥iiui«, i>iibi'r by iiu-iiiis of siuitll Htnil*. iis 111 Kif;. I, or lij- iiu-niH of siuitli- lii.i^'s of iHianlH at ri^lit aiiuh^ III llii- joists, mill titliii); jil Ih^ l«-<vn IlK'iii. Tl m-i-t of tliU l>ni<-in|; \» il.- iliMdlmti'J loiid. Tlii- ii: :ils« siiiTi-iis tU>' joints. I'M'iiis Mii-in fnoii tiiniiiii; .>'. It \* cMliIoiiiiirv III niuM of •-hisH-liridi:iii..: M iiiylivi- t<ii-lf<hl fii-liiiilic ' Iliiy -IkiiiM U- ill »tnii).'l>t I imiy iibul tllrwtly uiwd W UUUlfi JN I'L.UUKB. 4b 1 those adjacent to it. The method of bridging shown in Fig. 1, and known as "cross-bridging," is considered to be by far the l>est, as it allows the thrust to act parallel to the axis of the strut, and not across the grain, as must be the case where single pieces of board are used. The bridging should be of li-inch by 3-inch stock. Carriage-beams, Headers, and Tail-beams.— Fig. 2 represents the plan of the timbera of a floor, liaving a stairway opening on each side. The short beams, as KL, are called the ** tail-beams : " the beams jEF and O//, which support the tail- beams, are called the ** headers : " and the beams AB and ('D, the "carriage-beams," or "trimmers." The tail-beams are calculated in the same way as ordinary floor- joist; but it is evident that the headers and trimmers will require separate computations. It would be very difficult to give formulas that would serve for '•.vei'y case of trimmers and headers ; and the best way in any case is to find the load which the trimmer has to carry, and then, from the formulas already given, determine the required dimensions. In a floor such as is represented in Fig. 2, it is evident that the floor- area supported by EF or Gil = y X ^n. Multiplying this area by (f'\-f), we should have the load which each header would be required to support ; and then, by Formula 9, Chap. XV., we could determine its necessary dimensions. As the headers are wcakene.l by the tail-beams being mortised into them, a certain allowance should be made for mortising in calculating the dimensions, in ordinary cases it would probably be enough to make the breadth from one to two inches more than the calculated dimensions. 4:VJ WOODEN FLOORS The tritnmerft, A B and C'/A have to support one-lialf of the load rarrit'd l»y KF plus one-half the load carried by ^»7/, and also one' half ot tin* load su])p<)ried by the ordinary joist. The l)esl way in wliK-li to (-ali-iilalc siuh a triiiiiiirr i> to <'on>ider it to Im' made up iH two l»ain^ plact'd >ide by side, oiH' to earry the end of th«» he:ul ns KF \\\u\ (wll, and the second bein^ one-half the thi<-kness of th»» (H«linarv joinI The breadth of tlu» part carryinj: the ends of tlh' tiiiniiKi-; ruul.l then be calculated by Foruuda V-\, ("hap. XV., and the ti)ial breadth of the trimmers found by addiniz tot^*tht>r the bnadihs of the two })arts into which it is supixi^iiHi to Ik? divided. We have not the sjiace here to consider further the slun^ih of headtTS and irnnniei-s, but would lefer any readers dcsiriuL: further informatu>n on the subje<'t to IlatHehrs *• Trans- vei"se Strains,*' where they will tind the subji*<*l fully discussed. Fig. 3 Stirriip-Iroiis.— At the iM)int of eonmn'tion of the end of \\ir li«-.i«i« I with the trimmer, tlu' load on th«' trinun<T (^onun? fioiii tli«- ixadrr is a conrcntrated one : and all mortising at this |iniiii. In nrtlv*' ihr header. sh(»uld In-axoided. It is now tlie etis- tniti. Ill til -!-<j;is»< r»)ii>iiMetinii, !<► support the (>nds <if Inniders l>y nit,i:i- •'! »• ;rriip-iroMs, mn nIiowii in Fiu. '•'*. Tin* ISoston ami New- \ Ml k I'.ii ! '.Jul: Laws !»'i|nire tlia' '"I'verx trinnui'r or lieiuler nion» tli.iii titiii t'«->'t lolc^^ u^rd in any builiiiiii; e\e(>pk a dwi^lHnf;, shall bf liiiiiL.' Ml ^tin iip-iroii'^ of suitable tbiekness for the sixe of tlie t JndM I-.'" It 1^ i-vidi'iit that t'aeh vertical part of the stirnip will liave to WOODEN FLOORS. 438 carry one-fourth of the load on the header ; and we can easily deduce the rule, , . load borne by header Area of cross-section of stirrup = --- — Sfijoo * W The stirrup-irons are generally made of iron bars about two inches wide and three-eightlis or one-half inch thick. The headers are also generally bolted to the trimmer, as shown in the same figure; so that the trimmers shall not spread, and let the headers fall. Joist Hangers. — On page 437/ are shown two styles of patented joist hangers, which are intended to take the place of the stirrup iron, at less cost. . Oirders. — Formulas 2 and 3 will also apply to wooden girders supporting the floor-joist, neglecting the weight of the girder itself. In this case the distance between centres would, of course, mean the distance between the centres of the girders. The application of thijse formulas to girders being the same as for the floor-joist, it seeujB hardly necessary to illustrate by examples. • Solid or Mill Floors. By Solid or Mill Floors we mean a floor constructed of large beaniS spaced about eight feet on centres, and covered with plank of suitable thickness, and this, again, covered with maple or hard- plue flooring as desired. Such floors will be found fully described in Chap. XXIV. For calculating the large timbers, the best method is to compute the greatest load that the beam is ever liable to carry, and then determine the necessary size of timber by means of the proper formula, which may be found in Chap. XV. ; or if the beams are spaced a regular distance apart, and have only a uniformly dis- tributed load to carry, they may be computed by Formulas 2 and 3, given above. The floor-plank may be computed for their strength by the fol- lowing tonnula, supposing the load to be unifoi*mly distributed:^ V weight per square foot x X'^ x. 8 — ' yT x~l ' ^ ^ ^ They would, however, bend too much, when proportioned by this fommla, for use in mills, and in buildings where the under side of the plank must be plastered. For such buildings the thickness of the plank should be propor- tioned by the formula for stiffn(>ss, which is, 434 WOODEN FLOORS. Thickness of plank = ?/weight per squ are foot x U (gj y 19.2 X c e being the constant for deflectiou given in Chap. XVI. For s])riice, o — KM) pounds, and for hard pine 187 iwunds, for a defli'olion of on»'-tIiirti('th of an incli per foot of span. The \v»-ii;nt i)«'r sriuare foot should include the su|M»rfioial load on tlu* ll(>(u- and tin* wcii^ht of the ])lank and upper flooring. KxAMiM.K. — AVliai sliouhl be the thickness of the spnice plank in a mill where the ])eanis are spaced 8 feet on centres, and the superficial load may attain 12t) pounds ix»r square foot ? J//N. 'i'he weight of the plank and flooring, with deafening iM'tweeu. will weigh a])<)ut I.") i)()unds jM'r S(iuare foot, making tlie total load per scjuare foot 185 pounds. Then, from Formula 0, Thickness of plank = \/- ^{:,- u>^~ = ^ .>j • i i i ^ \ 1U.2 X 100 or :W-inch plank. Tlie ])iaiik would j)rol)ably come in two or three lengths, which would iiiakc the lloor considerably St iffer; but, as there nught oiMnir eases when the Ih^or wouhl have to sustain heavy conoentrate<l loads foi- a short lime, it would Ik^ hardly wise to use a less tlii(*k- ness of plank. The following table, taken from Mr. C. J. H. Woodbury*8 excel- lent work on "Tlie Fire Protection of Mills, and Construction of Mill-Floors,*' shows the dimensions of Ix'ams, and thickness of plank for waichouse-floors loaded with from fifty to three hundred pounds ]H*r s(|uarc foot, the ])eanis ])eing spaced eight feet on ci?ntres. The ])lank is supposed to b«* of spruce, and the beams of hard or 8outli- eni ]>iii('. Scv«'! a! si/.cs ui h<'ams are given ; so that a selection of those which will appl> m(»st convenieuily to any specific case may be made. WOODEN FLOORS. 435 STRENGTH OP SOLID TIMBER AND PLANK FLOORS. (By C. J. H. Woodbury.) Weight per Square Foot op Floor. Super- ficial load. 50 75 100 125 i50 ! 175 200 225 250 275 300 Weight of b^m, iu lbs. 3.00 4.08 5.33 3.00 4.08 5.33 3.00 4.08 5.33 3.00 4.08 5.33 3.00 4.08 5.33 3.00 4.08 5.33 3.00 4.08 5.33 3.00 4.08 5.33 3.00 4.08 5.33 3.00 4.08 5.33 3.00 4.08 5.33 Weight of floor- plank. 6.07 I 7.40 8.55 9.55 10.45 11.26 12.05 12.75 13.45 13.55 14.72 Total. 59.07 60.15 61.40 85.40 86.48 87.73 111.55 112.6:3 113.88 137.55 138.63 139.88 163.45 164.53 165.78 189.26 190.34 191.59 215.05 216.13 217.38 240.75 241.83 243.08 266.45 267.53 268.78 291.55 292.63 293.88 317.72 318.80 320.05 Dimensions oi Depth, 1 Breadth in in inches. inches. 12 6 14 7 16 8 12 6 14 7 16 8 12 6 14 7 16 8 12 6 14 7 16 8 12 6 14 7 16 8 12 6 14 7 16 8 12 6 14 7 16 8 12 6 14 7 16 8 12 6 14 7 16 8 12 6 14 7 16 8 12 6 14 7 16 8 Span, in feet. 20.95 26.16 31.63 17.42 21.82 26.46 15.25 19.12 23.23 13.73 17.23 20.96 12.59 15.82 19.25 11.71 14.70 17.91 10.98 13.80 16.81 10.38 13.06 15.90 9.86 12.40 15.08 9.43 11.86 14.46 9.03 11.36 13.85 Thickness of floor- plan k, in inches. 2.43 2.96 3.42 3.82 4.18 4.51 4.82 5.11 5.38 5.62 5.89 Stiffness of Wooden Floors. Floors in first-class buildings should possess something more than mere strength to resist fracture : they should have sufficient stiff- ness to prevent the floor from bending, under any load, enough to cause the ceiling to crack, or to present a bad appearance to the eye. To obtain this desired quality in floors, it is necessary to cal- colate the requisite dimensions of the beams by the formulas for nttffniyft ; and, if. the dimensions obtained are larger than those £j.. i'M) WOODKN KIXK)US. obtained by the. formulas for strength, they should be adopted, instead of those obtained by the hitter fonnulas. The only way in which we can b(^ sun^ tliat a beam is botli stronj^ <'non«^h and stiff enou^li to bear a ^iveii load is to calculate the re<]ulred dimen- sions liy both tlie formula for streni^th and th;* fornuda for stiff n»'ss, and take the lari^iM* dimensions obtained. Asagenenil rub;, thos** lieanis in vhicii the proportion of <lrpfh to Innjth is very mmiH siioiild !)(' calculated l)y the form uhis for .s/>y'//f/^//, and ricf rfi'sn. Foiinula 10, (hap. XVI., giv(?s the load which a given Ix'iun will carry without dellecting more than one-fortieth or one-thiilieth of an inch per foot of sj)an, according to tlu^ vahu^ of e which we use. Kornuda II, Chap. XVI., gives thii dimensions of the beam to carry a given load under the same conditions. In the cas«' of tloor-lx'ams, the load is given, and is represented, as wc saw under tin* Sfrcntfth of Flttors, by the expression. Distance between centres in feet X length in feet X ( /'+/'). Tlicn, if we substituti^ this expression in place of the load in Kormula 11, (hap. XVI., we shall have the fornmla, T) X (list, between centres X cube of length X (f+f) Jreadth — u \y , \ ^-.i .4i s^ (<) 8 X cuIm; oi depth X c ^** or S X breadth X cube of dei)th X e Dist. between centres = . ^ ^.^^j^^. ^^^. ,^^^^^^^,^ ^ {f^rT <*^^ The piojKT valu<'s for./' and/' have been given under the Stmnjth of ll<>oi> in the i)receding part of this cha])ter, and the value of f foi- any iriven casci may be found in Chai). XVI. ^ In oiiiinarv floors, when the values of /" used are thos4' n^-oin- im<iii|im1 al)o\e, a <ietlection of one-thirt i<>th of an inch {kt f(H>t of l>aM Mia> saftly be allowed, as the lloors would probably Ik» \ery raicJN loailed to their utmost ca])acity, and then but for a short tiiiH-: so ili.it it would have no injurious effects. As ail .".ample .showing the ai)plicjiti(»n of Fonnula 7, we will taivi- i!\aiiiplc I under the strength of wooden floors. Ill iiii> e\aMipIe, the l)eams were to have a sj)an of 15 fe«»t, and be , la.-.il I . t<et on centres: ./' was taken at -lo pounds, and./' at 'JUi I.oiiii.i-. What should be the dimensions of the In'anis. that thi'V iiia\ ^at- l\ cairy the Itiad upon them without deflecting more than ;. I'l a:i 'ii<|i |)cr foot of span ? > i >.■ -.1' .<- t')!' '-. fur r<|irii('(.*, hard pltic, iiiui ifuk, uns l>.f. jlnA. !>*/.- 4*0^ hM 76 11 :•> • l:i7 108 < '•*' M TS s WUODEJN FLOOtttt. 437 Ans, We have simply to substitute our known quantities in Formula 7, assuming the depth at 10 inches, and taking the value of e at 100 pounds, the beams being of spruce. Performing the operation, we have, ^ . ^ 5 X li X 15» X (40 + 20) Breadth = s X W x lOO =1.0^ inches. This gives us about the same dimensions that we obtained when considering the beam in regard to its strength only : hence a Ix^am two by ten inches would fulfil both the conditions of strength and stiffness. In the case of headers, stringers, etc., where the joist has to carry not only a distributed load, but also one or more concentrateci loads applied at different points of the beam, the required dimensions can best be obtained by considering the beam to be made up of a number of pieces of the same dei)th, placed side by side, and {;om- t)uting the required bre^adth of beams of that depth to carry each of the loads- singly, and then taking the sum of the breadths for the breadth required. The formula for stiffness of plank-floors has already been given on p. 484. Dimensions of Joists and Girders for Different Loads and Spans. To enable an architect to tell at a glance the size of joists and girders required for the ordinary classes of buildings, the author has computed the following tables, which give the dimensions required for spans from 10 to 24 feet for joists, and also the maxi- mum distance that the joists should bo sjmced on centers. Dimen- sions for girders are given for different spans and spacings. The beams and girdera in the first three classes were comjmted from the tables on pages 888, 389, and 390, and in class I) from the tables on pages 377 and 379. The application of the tables will doubtless be (wident to all. When the girders are not s[)Mce(l uniformly, or then^ is only one row of girders, take the width of flcK)r area supported by the girder, for the distance apart. In several cases two sizes an^ given, both of which have sufflcient strength, although one contains less lumber than the other. In most oases the deeper b(»am has a little excess of strength, but for convenience the shallower beam might be preferred. 437^ WOODKN FLOORS. TABLE L Dimensions of Floor Joists for Different Loads and Spans. [Note.— Tlu» iiiinibcr followin;^ the dash diMiotcs the distance aiMurt of joiBts in inches on centerf<.| A. FOU DWELLINGS. (Total Weight, 70 lbs. per Square Foot.) Timber. Yellow I 10 12 ( LE.\u Span in Fket. 14 10 IS ' t»0 •»•> 24 H Ui Whtto I I'liic. 1 ■.-• — ^ ., . io_o4 « 2 r> n 1 2 12 IS 2-M-:io : ,: r: j-ie-ie 2 12-18 ; 3, Jo 20 * ' ^* ■** >. « ifl 2 10 21 »2>10-12 „ ,, -. t 2 '12-11 8<1«-2J ,. , 1 n irt "'. "\ "9 '' 9-ii>-it; 9.19— sn 9'H--iii .'"'i*~'J ••U— 14 riii.'. ' ^ « 10 - "* (2 10 24 -* '^^ ' 3'1-S XO ^ IS 10 ,3,<,j_|g .!^U_-.o B FOR IIOTKLS, SCHOOL- HOOMS, LKJIIT OFFICES, ETC. (Total Wei^'lit, 1(H) lbs. per S(iuare Foot.) Ttmukk. CLKARSPAN is rKKT. 10 1-2 14 U\ IS 20 22 ! 24 Whltf » „ J ,r o 11 iJ I 2 ' 10 Hi 9 12-13- ,„ ,- ,^,, ,. , 1'1,„. > 2-H 10 2 -K' 1" ,2 12 2.. ."5 -12 19* '* '* SXll-H. | Spru'i-. •.' — \'J .„.,,. „o ...10 .>A 2 12 -Irt 8 -12- 17 -. , . ,i_,i» tAl4-14 i'lii'' ■* Jl 'i \i> 22 "J 12 24 2 A 1 .-, 2 • ID 20 I 8 • 14—19 t 2 • 10 12 IS* It Ifl 2-io-irt ;|.|;; ''2.12 154 3.11- !• .;-;;j_;j «.i4i» C. Foi: oKFKM-: I?riLl)IN(iS, A^SKMBLV ROOMS, AND LIGHT STORKS. (Total Wciu'ht, m) Wx. ptT Sciuare Foot.) 'llMUKI. Wt.lti- , ( ■.> I'ln." I -J Siirm-i' •.' Yi'liow - I'liii-. > 10 12 <'LKAK Sj'AN IN l""KICT II U\ IS 20 tt ! <« 2 10 H -• 12 1"' S 12 !.-> 3 14 1« j 2 • I- 17 2 12 Hi ]l'.]l ]l •''■12 IS 3-14- !»' j .., • 2 - 12 ., I lo 'J: ' :\ 12 12 3 It IS iM li 2 12 17 :;.\i_\; ,.i; ;; i a>i4- u it^u-ii WOODEN FLOORS. 437* D.-FOR STORES AND FACTORIES.* (Total Weight, 180 lbs. prr Square- Foot.) Clear Span in Fekt. 10 2X10-16 \tx 8-ia "(SXlO-17 SX 8-17 12 r2xio-ii (2XH-1 2x10-12 2x12-18 2X10-19 14 txit-ii 3X12-17 •^X12-18 3X12-20 *2X10-18 ( -^X 12-19 16 18 8X12- 8X14- 18 -18 8X12-1« 1X12-1) 8XM-I4i (8x12-12 « 8x14-16 ( 2<12-12 ■( 8x12-18 20 8X14-11 22 8x14-181 8X14-11 8X14-16 8x12-14 8X14-19 24 SX14-18I- * Calculated for strength only. TABLE II. A. Dimensions of Wooden Girders for Dwellings. (Total Weight, 70 lbs. per Square Foot.) SPRUCE. IN Distance apart on Centers in Feet. 10 i 6x10 ■« 8x 8 6x10 6x10 8x10 J 6x12 I 10 X 10 8x12 12 6x10 8x 8 6x10 8x10 \ 6x12 I 10x10 8x12 9x12 14 16 6x10 8x10 8x10 8x10 ) 6x12 / 9x10 8x12 10x10 8x12 8x12 10x12 10x12 10x12 1 10x14 (12x12 18 8x10 \ 8x12 ( 10x10 8x12 10 X 12 \ IOxhI "111x121 20 9x10 8x12 10x10 10x12 10x12 10x14 12x12 10x14 10x14 22 I 8x12 1 10x10 8x12 10x12 \ 10x14 ) 12 X 12 10x14 12x14 24 8x12 10x10 9x12 10x12 10x12 12x14 11x14 12x14 YELLOW PINE. IN r. Distance apart on Centers in Feet. 10 6x 8 12 6x 8 14 16 18 6x10 8x 8 t»0 22 24 \ 6x10 / 8x 8 6x10 8x 8 6x10 8x10 8x10 6x 8 \ 6^10 < 8x 8 6x10 6x10 8x10 8x10 8x10 9x10 6x10 6x10 6x10 8x10 8x10 S 6x12 1 10 X 10 8x12 10x10 8x12 6x10 8x10 8x10 1 6x 12 'i 10 X 10 8x12 10x10 8x12 8x12 10x12 8x10 S 6x12 ■/ 10 X 10 6x12 10x10 8x12 8x12 10x12 10x12 10x12 i 6x12 1 10 X 10 6 X 12 8x12 8x12 10x12 11x12 i 10x14 ) 11 x 12 10x14 8x12 8x12 10x12 10x12 < 10x14 ) 12 X 12 10x14 10x14 10x14 437c WOODEN FLOORS. B. DiMExsioxs OP Wooden Girders for Hotels, School-rooms, Light Offices, etc. (Total Weight. 100 lbs. per Square Foot.) SPKICE. bl'AN IN Dis 12 TANCE APAKT ON' 14 , 1« 1 , 1 Centei IS :s IN Fk 20 ET. Fkkt. 10 2« 24 10 11 (3. ^ 8. ( i\ > 8v / 10 . 10. 1 12 t id 8-10 8 . 10 ♦i !•.> 8- 12 \ 8 ^ 10 8 ^ 12 10- 10 10 X 12 8>12 10. 10, 8xli> h)>. 12. 8x12 10 X 12 \ 10 . 14 "( 12 X 12 10x12 10x12 10x14 12x12 10 X 12 10x14 '1 12 X 12 10x14 10x12 10^14 12 -IS 12' 14 v: 8 •-» 10 12 10 12 \ 10. 14 / 1- 12 10x11 12x14 12x14 14 '14 v.\ 10 X ■i 10 . 12 \ 10- 14' , 12 12 10 > 14 12x14 12x14 » 12x16 »14x 14 12 • IC 14 10-- 2 < 1" < 11 . 12- 12 10 - 14I 12 • 14 14 A 14 12x16 12x16 14*^16 i:> > 10 . . 1 '? • 1 2 10-14 12 .-14 _ 1 ( 12. It; ( 14 •• 14 1 12x11) 14x16 14x16 16^16 YKLLOW PINE. Sl'\N IS Ki;i T. 10 Distance ai-aiit on Tkntekh in Feet. 12 14 16 ' - l itt > () 10 \ )') 10 / '^ - s , S *< »; 10 8-10 H. 10 \ 8 ' 12 ti H- 12 ('» 10 iS . Id 10 10 (1 • 1(1 r> 111 »« .«• . \-i 10 - M ^ 12 8 . 10 » r, . 12 . 10. 10 8 10 8 12 10 10 N 12 in ^ I-.' I 10 • 12 10 . 10 10 . 10" H - 12 8- IJ 10- 12 li». 12 I I'l 1 J 1-.' 1-J V* \i 10 11 8 12 10 12 10 . 12 ; }i; ' \\ 10- 12 ' II!" \i 10. 14 10-11 11 -12 *ll]" I:* 10- 11 10x14 12*14 ( 12 - IJ I ''•■I* 111 11 i.>^i« to 11 »12*16 12x12 ^^''^* ^-''^* '-*^^ -,14.14 10- 11 :2- 11 l:j-14 )}J;{5 W"!" 24 s. 12 10- 1".» 10. 12 10 :t 12- 12 12-11 12*11 12. 1« 14-14 i2»ie WOODEN FLOORS. 43Vc? C. Dimensions OF Wooden Gibdebs fob Office Buildings, Assem- bly Rooms, and Light Stobes. (Total Weight, 130 lbs. per Square Foot.) SPRUCE. Span in • Distance apart on Centers in Feet. Feet. 10 12 14 16 18 20 22 24 9 10 11 12 18 14 8x10 j 8x12 110x10 8x12 10x12 (10x14 1 12 X 12 10x14 (8x12 1 10x10 8x12 10x12 (10x14 n2xl2 10x14 12x14 8x12 10x12 j 10 X 14 1 12x12 10x14 12x14 ( 12 X 16 "1 14 X 14 10x12 (10x14 ) 12x12 10x14 12x14 (12x16 1 14x14 12x16 10x12 10x14 10x14 12x14 12x16 14x16 (10x14 112x12 10x14 12x14 ( 12 X 10 1 14 X 14 18x16 15x16 10x14 12x12 12x14 18x14 12x16 14x16 10x14 12x14 14x14 14x16 YELLOW PINE. Span in Distance apart on Centers in Feet. Pket. 10 12 14 10 18 8x12 10x10 8x12 10x12 (10x14 ) 12 X 12 10x14 12x14 (12x16 1 14 X 14 20 22 8x12 10x12 10x14 12x12 12x14 14x14 12x16 14x16 24 9 10 11 12 13 14 15 6x10 8x10 8x10 8x12 8x12 10x12 (10x14 1 12 X 12 8x10 8x10 (8x12 / 10 X 10 0x12 10x12 (10x14 1 12 X 12 10x14 8x10 ( 8x12 / 10 X 10 8x12 10x12 ) 10x14 / 12 X 12 10x14 12x14 (6x12 ) 10 X 10 8x12 10x12 11x12 10x14 12x14 12x14 8x12 10x12 ( 10x11 'i 12 V 12 10x14 12x14 14x14 12x16 10x12 ( 10x14 ■/ 12x12 10x14 12x14 14x14 12x16 14x16 4376 WOODEN FLOORS. D. Dimensions of Wooden Girders for Stores and Factories. ^Total Weijrht, 180 lbs. per Square Foot.) SPRUCE. Stan iv Fkkt. 9 10 11 13 10 DljiTANC K APART ON CENTERS IN FbET. 12 14 16 IS SO 8vl> UK le 10x12 *]S^]:1 10x14 12x14 10 ,,y > 10x14 jn^i. 10^,4 10^ ti » 10x16 '"' -I'Jxl-J ^^^1-* 1-X14 1^X14 jj^^jj » 10 « 14 1-2 » 1-J 10 ^ 14 12x14 » 10 X 16 14x14 12 X 16 13 X 16 10 ^ 14 V2 > 14 14 X 14 12 x 16 14 x 16: l"2v'. I 14x14 12x16 14x16 ' 22 24 18x14 » 10 X 16 M4xl4 12 X 16 14 X 16 14x16 YKM.OW riNE. 10 l>i-r\\iK AivKT i»\ Crx-^KUs IN Fket. I "" 14 16 IS 20 22 24 ^ • 1 *- • i '.' 10 10 > 1 ...... • '-'^^ 1» 1- 1 J * '.I : 1 S X IC 114 < • 12 10 X 12 1-' 1 1 10 14 Iv! V II 12 ' : 1 :s - ] I li 14 12 ".i". :i- it» 1(1x12 10x14 12x14 10x14 11x14 12>14 12-14 13x14 11-14 I4xll' 12.16 ISxlA I 12. li. 13x16 15x16 1 1 • 16 13 « 16 WOODEN FLOORS. 437/ JOIST HANOBRa Fio. 4.— Ddflei Joibt Hanqeii. Fia, B.— Ooetz Joibt Haxbeb. Pig:s. 4 and S show two styles ot joist hangers that have been put on the mwkot within a few years. Both these aneliors are warranted to be stronger than the timber they support. 'I'hey are made in numeroiis sizes, and are inserted in holes bored la the sides of the girder, or trimmer. While these hangers themselves, however, have ample strength, they mu-^t weaken t« some extent the tim- ber into which the holes are bored, which is not the caso with the stirrup Fig. G shows a similar hanger made to support the wall end of floor joist. The writer believes this to be much superior to the method of building the p,^ 6.-Dtt«.e» BaitrK Wall joist into the wall, as it absolutely Hanoer, prevents dry rot. and permits the joist to fall in case of fire, without throwing the wall. It also gives the weight a good bearing on the wall. FiHK-PllOOF FLOUKS. CHAPTER XXUI. FIRE-FROOF FLOORS. TnE tPrin " fire-proot floor '' is hert unrlenstcXK] to mean a flool rfin^triii'liil of (irt-proof mnterial, RupiMirtcil on or betwe n iron iit 9ti'i-l ix-aiiiH or gmlcrs, or fire-proof wiiUs. anil entirely ]irot«cliij); tin- ironwork from tlie action of fire. The various materiais si |iri-." ril iiMil ill the <'on struct ion of absolately firu^proof floors lire bri'k. iinlliiu' [Mjruiis liJu, liullou' dense tlie, ibin pl&tea uf dense tile I ;ir..ilLii'i~ iif I'l.'iy: iiiiil (-iiiiiTi'te of Piirtlnnrl in^ment nnd i-itlii-r i'i'~. I'i'iikrii iil>'. ~i<iiii'. Ill' tirii'k; iiiui iiiac) eoin[H)iiilioiiii niiiile lila-iiT ol' I'^iri^ .-IS II (i-iiii-iitiiif MiiiliTint. The flr^l tlim' -ri;il~ III'" p'tirnilly ii-..-<l ill ill.' r<iv I iiR-lies net iHrtwiTn tiic II'.. 'I li.' iliiii |.liii'." -f i1>'iiM- till' iiri' iisi'd f.ir (oniiin^' vuiiltK ..i; uirl.T- »'..ii.h.|,'isuv.-.l.'iili,i'inllii'f"riiiof iin«rcli...t I.I -:.\-. n'riniiiL' <1<"<r ami .'I'iliii^'. »'illi liolliiw iiileriur : ill Die -. ir'ii liiir', i-x[iiini|i'<l nii'iiil. or wiiv lirs iin- iiiilii'Udiil. Inin \.: I. 1 111^ iin- <:>'n< rilly liiiil in tliK>rs nr uliiiwh in Fifr. I, Iho - ittl.,'1' jvsliiis nil 1..]. of llie ninlern. Kb in Kig. 3, or lioltei] to ~iili'^ <if ilii'i^rders. FIRE-PROOF FLOORS. ,430 Fig. 3 shows the detail of connection when the under sides arc made flush ; Fig. 4, the joint to bring tlie upper sides flush; and Fig. 5 shows the form usually adopted when the beams are of the same size, or the centre lines are brought together. Arrangements of this kind are also used to connect the trimmer-beams of hatch' ways, jambs, and stairways.^ P ][ Fi... \///m Fig. 6. The wall ends of the joists and girders should be provided with nhoes or beariny plates of iron or stone, as the brickwork is ant to crush under the ends of beams, unless the load is di^tribjuted by this means over a sufficient surface. Anchor-Htrapa should be bolted to the end of each r/irder and to the wall end of every alternate joist, binding the walls firmly from falling outwards in the event of fire or other accident. Several simple modes of anchorage are shown in Figs. 3, 4, and 5. When one beam docs not give sufficient strength for a girder, it is customary to bolt tosjcther two or more with cast separators between them, as shown in Fig. 6. *■ The details of the coiiuectioDs aud framing of iron beams kre more clearly shown on pp. 366, 366. KlllE-rUOOF KLOOKK. Itric-k Arc'li«s, I vviiy of iiiiikiiiu i> lln'-iU'oKf floor of hrir-k \s to fill .■-■11 111.' ji.isls«illi Lrii^k un-ln's. n■^Ii1l;,• <>ii llli- low.T ;.'iT;i-.'.>n:i <ir biit-k sk'-wliitrks. M'hi-ii tliis mctlio<l '^hoiiM lu' tiiki'ii Ilial llii- l<rli-ks of niiiHi tli.' .irvli.-s aiv of ^1111.1 slia|i.'. :iii.l v.'iy liiir.1. Tli.'V sIiomM I>' wiilL .■iLili oilirr, iiiiliuiLt liiir: i.ii.l iii: IW ji.iiit* trviU. mill Ih> kt-yiil n'itli lill.M ttUll 111.' IH-St i.T f..iir iii.h.^ llii.'k fi.r f]-^n> l«'tw.--H . llii.-k l'..i' -iMiis l..-i».'.-ii »:\ :iii.l .'k'lit III tl,.' -k.« l.:i.k- .,iri.- -..U.l.ima > .imn- I:, li-.-.'l tin :llvl,-h..|IMI.|-:l^..ll..■l.- ..; ^.i...tl. ai..| FIRB-PROOF FLOORS. 441 angle- bar or channel serving as a wall plate for distributing the strain produced by the thrust of the first arch (Fig. 7). The weight of n brick arch with cement filling is about seventy pounds per superficial foot of floor. Experience has shown that such a floor cannot be considered as fire-proof unless the lower flanges of the beam are protected by porous terra-cotta, fire-clay tile, or wire lathing, kept an inch away from the beam. Brick floor arches are largely going out of use, owing to the fact that a fire-proof floor may be more cheaply constructed of other material. Hollow Porous Terra-cotta and Hollow Dense Terra-cotta Floors.— For convenience, these materials will be referred to as Porous Tiling and Dense Tiling. A description of the materials, their nature and manufacture, will be found in Chapter XXV. They consist principally of clay, which is manu- factured into hollow blocks, generally with angles on side or ends, according to whether the arches of the floors are to be of end- method design or side method design. In some instances, to a limited extent, rectangular blocks have been successfully used, but this shape is not approved. The general practice in flat con- struction is to make bevel joints — radius joints are seldom used ; the best workmanship) and best results are found to be obtained with a bevel joint of about one inch to the foot. There are two general schemes of flat construction : one in which the tile blocks abut end to end continuously between the beams, and one in which they lie side by side, with broken joints, between the beams. In the end systems, it is not the practice to have the blocks in one row break joints with those in another, as it entails extra expense in setting. When this is done, however, the substantialness of the floors is increased. In some forms of flat construction a side-method skewback (or abutment) is used, with end-to-end interiors and keys, or end-tOr end interiors and side-method keys. Experience has shown that in the side method of flat construction the skewback, or abutment, was the weakest — in case of failure, sometimes collapsing, but gen- erally shearing off at the beam flange ; consequently, the side- method skewback is not approved in the end- method construction unless provided with partitions runninc^ at ris^ht anoflos to the beam. Keys should be end to end, or solid. The latter, when made very small, are preferable. A practice which has become somewhat general, especially In the East, is for the owner or general cqntractQr tp buy tjles, and the tnasofi ^{itraoter on the job to build them in plaee in the building. FIBE-FROOF FLOORS. 443 beams, and like centrepieces above, crosdng the beams. The ptanka on whieh tiles arc laid shfiuld be two-inch, dressed on one ode to uniform thickness, and should lie on lower centres, at right Angles to beams anil placed close together. J'he soffit tlto should be a separate key-shaped pieue. oC ei[iial width of beam, and laid directly under tbe beam on the planking, aftor whicb the eontring is tightened by screwing down tlii) nuts on the T-boits, until the sofflt tile are hard against the beams and the planking has a crowa ' not esc«&diag one-fourth oC an inch in spans of sis feet. This sys- tem gives what is very essential— a lirin and steady centre on which to construct the flat tile worlt. The tiles should be '■ shoved" in jilace with close joinb'. and keys should fit close. The centres should remain £n)m twelve to tliirty-six iiours, according to condi- tions of ireather, depth of tiling, and moj'tar used. When centres are "struck," the ceiling should be straight, even, free from open joints, creTices. and cracks, ready to receive the plastering. Figs. til 12 show types of flat constructions in use. Different tfianufacturers have various modifications of these. Pig. 9 is the most general design for dense tiling, although porous tiling, very similar in design, may be had from some manufacturers. The end- method design is preforahlo, however, for porous tiling. Fig. 10 is a light-weight dense-tile design, nol so gvinerally useil as fonncrjy. Figs. It and lli show the simplest end-methnd design for porous tiling, which has become known as iho ■'Leo end-method areh." It was first brought into general use by Mr. Thotnas A. I.ee, now of New York City. It was used by him in the tests conducted at Denver in Dceember, t"S(}, by Messrs. Andrews, Jaques & Ran- toul, architects In those tests the design ^ihowod superiority over the Dtberdesigns. It has the advantage of simplicity and economy, both Id mannfaoturo and construction. Tbe manufacttirer can FIRE PROOF FLOORS. 445 reduced and the stability of construction mcreased.- The reduo- tkm in price of all tiling makes the cost rather in favor of increas- ing the thickness of tiling and reducing the thickness of concrete. Among the advantages possessed by hollow tiles in their ap[)lica- bion to fire-proof floors, between steel or iron beams, are these : They are absolutely incombustible, because made of clay and laving withstood a white heat in the course of manufacture. They are sound-proof, from fact of being hollow. They are superior to any concrete material used for the same pur- jose, owing to their being free from shrinkage, thereby avoiding ihe unsightly cracks often seen in ceilings laid with concrete blocks. They are proof against rats and vermin. Floors made of them are forty per cent, lighter than by the old system of segmental solid brick arches levelled with concrete. They offer a flat surface on the bottom and top after being laid, Fig. 16.—** Austria " Arch, Patented by Pr. von Emperoer. ihereby giving a flat ceiling ready for plastering, and a flat founda- :ion for the floor strips. The flat arches should in all cases be capable of sustaining, with- )ut injurious deflection, after being set in place, an equally distrib- ited load of 500 pounds upon each superficial foot of surface. In laying the tile, a mortar composed of lime mixed up with joarse screened sand, in proportions of four to one, and richly tom- 3ered with hydraulic cement, should be used. This makes a strong nortar, and works well with the tile. In no case should a joint jxceeding one-half inch in thickness be permitted The laying of flat construction in winter weather without roof protection should not bo practised in climates where frequent tevere rain and snow storms are followed by hard freezing- jind ;h!iwii)g, as tho mortar joints arc liable to be weakened or ruptured, 'esulting in more or less deflection of the arches. The upper su rface of these arches is generally covered withcon- jrete of a sufficient depth to allow for bedding in it the wooden (trips to which the floor board-; are nailed. The concrete can be nade of light and cheap materials, such as lime or native cement knd clean rolling-mill cinders, coke screenings, broken flre-proo€ 14(5 FIRE-F»RO()F FLOORS. tiling, etc. The floor strips should be of sound and seasoned wood, 2 inches thi(»k by 2 inches wide on top. bevelled on each hide, to 4 inches wide on Iwttoni, paced about 1(> inches on ('i-n- tros 'rh(^ coiicrct(» should ix; firmly bodded beneath and ugiiin^t oMch <[(\l'. Instead ol' coiicr.'tc filling. tKt?., a filling is soinetiines made l)y layiii*; lidllow p.iitition bloc^lis on top of the arches. Tlicsc loiin excellent toundations tor marble or other linished tile liuoiin^^. Tlic j)i}icticc ol* puttini; in comparatively thin flat arcfh eonstruc- tiuij U) form ceiling's, then heavy wood strips from lx»am to btjani to carry the v. ciiriit ol' the floor, leaving a hollow s^iace between top of arclns and under side of wood flooring, ij« not approved. The amount of wood contained in such a floor is sufficient to produce u Very (lamairing lieal. The hollow space enables the wood to burn readily, and niakes a Are very difficult to fight. Such coDstruction, Fu.. 17. thereiore. i< danufcrous. and sIkjuM not be considere<l as first-class fire 1 1 root' \-. ork. '1 li«- VMi'iition in width of spans between beams is pn>vidi*ii for by ^u|i|'l\ iil: tiles of dilTerent sizes, both for interiors and keys, wihii'hy ): \arie»y of eond)i!iations can be sj'cured. When i!i-^ii'el to aitaeh iron oi" wood work to the soillts of the hoiioA 111. iloo- archer. sli>t holes are puiich-Ml in the tiles, and T- h. a-; (i i ol". ;ii-f inserted and secured a> >h«»wn in Fig. 17. '■ In-n ;..fiiie. terracotta tile are used, cleats nuiv Im' naile<l or S !• U. .| i'.l.ctiv to th.' tile. I:: ;■: ! :'ij' ii« n work, too «;r«'at can- cann<»t l)i' exercis^'tl that all 1 ai'i- ii.- |>I.i. -d paiallil. e»<pe«-ially '.her- (».ie or both emb «if 1' .:i:.- !■.-• •■•1 i-ri-kui-rk. rn'am-^ plaeed out of parallel make il \.n I \|.. !.-;\«' III M-i tile lire |«r-'i'li!i-..r. «»f'en nnpiirin:; cutting nf II!'-. aIi:< h -^ tiaiiiauin.;: ami injurious, and shoidd not U* tloue. \\ '■ I. -|..iN. - LTineni.!! hollow liie arches isee Figs. IS and IViarv >o!ii- t :...- - n-»-'i in wai'ehouscs, factories, ami fur mofs, in thick ne.^M.-^ of i.\ and eight inches. I'sually the tiles an* 0x6 inches, FIRE-PROOF FLOORSw 446a or 6 X 8 inches, and 12 to 16 or 18 inches long. Spans may be any width up to 20 feet, rise about one inch to foot of whole span, in some instances the joints are pointed after the centres have been removed, and the whole under side painted.- This form of hollow- tile work in wide spans from girder to girder is cheaper and lighter than flat construction with floor beams. 4" to U c)t^'<n,t«CU)b\ ^vck X!Uu^ 0^^>^v ^X» 5l>\lH. . 5vmi'«iUo'Iq^o' ac&oram% Xo »a.c o^ Fig. 18. Weights and Safe Spans for Dense-tile Arches.— The following table gives the weight and span of flat hollow dense- tile arches made bv the Raritan Hollow and Porous Brick Com- pany. This is about an average for spans given by different manu- facturers. The Pioneer Fire proof Construction Company, and some others, make a lighter grado of tile than this, but their heavy tiles correspond very closely with the table below. Dense tiles may also be had from Lorillard Brick Works Company and Henry Maurer's Son. New York ; the Empire Fire-proofing Company, Pittsburg ; Parker & Russell Company, St. Louis ; and others. WEIGHTS AND SPANS OF FLAT HOLLOW DENSE- TILE ARCHES. Depth of Arch. Span, between Beams. 3 ft. 6 in. to 4 ft. Weight per sq. ft. 6 in. 29 lbs. Tin. 4 ft. to 4 ft. Gin. 3? lbs. •8 in. 4 ft. 6 ill. to 5 fr. 6 in. 35 lbs. 9 in. 5 ft. to 5 ft. in. 87 lbs. 10 in. 5 ft. 1) in. to ft. () in. 41 lbs. 12 in. 6 ft. 6 in. to 7 fi. in. 48 lbs. The following table gives the weight and span of flat hollow porous-tile arches of the Lee end -method design, which may be FIRE-PROOF FLOORS. 446c olted together with f-inch tie rods, secured to the web of the cams near the bottom flanges, and drawn tightly to place by nut Ad thread. These tie rods should be set from five to seven feet bpart. The cost of hollow-tile arches of either kind, set in place ready br plastering, in lots of 20,(MM) square feet , ranges from 14 cents to •6 cents per square foot, according to size and weights of the tile, n Chicago the average price may be taken at 20 cents. Specifications for Transverse System of Elnd- Pressure Floor Arch. The following form of specification may be of assistance to rchitects in preparing their specifications for tile floors : Contractors submitting proposals for fire-proof floor arches shall, hen required, prepare detail drawings showing the sjrstem and ^plication of floor arch proposed to be used. The general require- lents of such design shall be as follows : 1st. Arches to be level top and bottom, filling space between the Bams from a point not less than seven eighths of an inch below le soffit of beam up to within one inch of the top of the beam. 2d. The abutment tile adjoining or resting upon the floor beams lall have its hollows run parallel with the beams, but the vous- )irs shall be laid transversely, with hollows running at right Dgles to the floor beams, so that the tile blocks forming the arch lay receive the pressure resulting from imposed load on their end .Kstion and distribute it lengthwise of their respective web members. 3d. Soffits of all beams shall be covered with tile slabs keyed 5curely in place, flushing with under surface of arch. Tests. Each arch shall be subjected to a test of a moving load consisting f a roller weighing 1 ,000 pounds to each lineal foot, and applied 3rty-eight hours after the centres have been struck and before the oncrete has been filled in. This roller to be rolled over the top of be tile wherever the supervising architect or his superintendent hall direct. In addition to such rolling test, the arches, after being set in lace seventy-two hours, shall be subjected to a dropping test made 1 the following manner : Before the concrete is applied on the rches, a bed of sand two inches thick shall be spread loosely over le top of the arches, Rud a wooden block or timber, weighing 200 mnds, shall be dropped thereon from a height of ten feet. If the 4iG</ FIKE-PUOOF FLOOUS. arclies withstand this impact for three c-ontiniious blows without breiikin<>: through, the test shall bo considered satisfactory, and the floor arches bo accei)ted. Should the floor arches break throu^rh under the blows, it >\v.i\\ be deemed (conclusive that the metliod of floor arch employed is faulty, and the contractor will Imj r(H|uired to remove same from the building and provide arches suitable to withsi;nnl the tests recjulred. Strt'ii^tii of Flat Hollow l>oiiso and Porous T<MTa-('ottJl AiM'lios. — Either of these materials, when prop- erly made an<l erected, should have a strcnjy^h of at least 5(:() lliS. pcrsijUMie foot. One of the most complete an<l practical tests oi" llo(»r arches I'ecorded was made in Denver, < ol., iindtT the direction of Messrs. Andrews, Ja(iues & Kantoul. architects, for the Dfiiver K.iuit.'blr P>iiilding (N)mpauy, Decendn'r r20-2o, 1890, oi" which a tuil reporl was ])ublislied in the A /."trioii' Architect and littiUUug \rirs, M.'injh "Js. IbiOl. Kight an-hes built of hollow bum«*d lin*- elay til«', and four of ])orous terra-iottu, were subjei^ted to four kinds of te-1s. under as nearly the same comlit ions as p(»ssible. Thraifrhes wri» earrie*! on 10-inch steel Mn^'ims, set- 5 feel apail on centres, and were built of 10- inch tile. The tcrra-eotta tile were manufactured by Mr. Thomas A. Lee, and were of the en< I -const met ion type shown in Kiirs. 11 and 11//. and it is dtaibtless owing tu thb fact that tliesj- arches (h'veloped the strength shown by the testti. The U'>ls were as follows : l>i. I>y still loatl. increased until the arches broke (h)wn. V.M. \'>\ -li<Mk>, repeatiMl until the arches nc re destroyed. :M T«-i- by lire and water, aliernaiiug until the an-hos were till I>\ •onlinnons tire of high heal, until the arches were Ill !<■ iin fii-!s| t:st.'iiii' ••; the llre-elay tile nivhe- bri»k«- at .">. U ; Il»s 1)1- "in 11'^. pn- s luari" foot, and the other at H.riTI lb'... nj- ]"J^ |i.>. |)iT s.|uar< f'ii>: ; brnh i»! theM* ar«'!ies liad but i-ne h<>ri/i»nial wi b, w hieh wa.- at t in- cent n- ul' tin- tilr. I'.oth of »ln>o ri'i'-c. ■■■iVi- • !iv sndd'iily. tin- wlmh- .-ir'-li iatlini;down. tbi' failun' i:i b !l i-.i-.'- lakinu: plaee m 1 1 r '-LiW baeks. t he remaill'^iT nf I hp ;i'. : !■ '■'.) . i.iiii.im.il '"li- p»i|i-M> le:-,;: i "fi ali'll. wl'i-h ad 1»0 ill 1 1. ■!■■ d wfli^ .-n.-t; iiMil a Iliad ol I."), ll.'i \\'< . i.il b>. per «•■ unre :...'. ;. : ! . . Imnr-* willn'Ut breikimr. ^*lll■n the l«ijid war di-imi- I i!i:i- •! 11. ■>("ImI .sriii-^ of tc^i.s wa^ madi- iiy dinppiiig a piece of tim- b.i \': MM iii> -^iiuare auii -i iett Inn" weighing 134 ]l)s . tnmi a hei>:lit •<] -.ix liei. u|Niii the inahile of the an-li. Ikjlh of thi* hoU FIRK-PROOF FLOORS. 447 low flre-olay tile arches broke at the first blow of the ram, the arches dropping from between the beams, the tile breaking *^ like a sheet of glass, indicatiu.u: extreme brittleness in the material/' The porous terra-cott:i arch withstood four blows from a height of six feet, and seven blows from a height of eight feet, the areii dropping at the last blow. Pieces of one or more of the tile, how- ever, dropped out at nearly every blow. Under the fire and water test, one of the fire-clay arches was destroyed by three ap[>lications of the water ; the other withstood fourteen applications of the water, alternating with extreme heat. The porous terra-cotta. arch withstood eleven applications of water, alternating with extreme heat, uninjured. The temperature of the tile at the time the water was applied varied from 1,300^ to 1,600° F. Under the continuous fire test, both fire-clay arches were destroyed after being subjected to a most intense heat for twenty- four hours. The porous tcrra-cotta arch, after having a continuous fire under it for twenty-four hours, was practically un- injured, as it afterward supported a weight of briyks of 12,5o0 lbs. on a space 8 feet wide, in the middle of the arch. These tests were conducted with perfect fairness, and unquestion- ably show the superiority of the [)orous terra-colta arches. The porous terra-cotta tile, new and dry. weighed 34 lbs. to tlio sfiuare foot ; the fire-clay tile which stood the tests the best weighed 40^ lbs. per square foot, and the other 32 lbs. per square foot. Other Tests. — During the construction of the Board of Trade building, in Chicago, in 1884, a 6-inch tile arch of 3 feet 8 inches span, made by the Wight Fire-proofing Company, of Chicago, was loaded up to 7o6 lbs. per square foot without injuring the arch. The arch was also severely tested by dropping heavy dry- gooils cases upon it from a height of 4 feet, without injury. When the large (l6-feet) sfwin arches were laid in the Commerce building, on Pacific Avenucj, in Chi(;ago, each arch was tt^sUnl by rolling an iron pulley, 6 feet in diameter and 14 inches wide. weighing 2,180 lbs., over each square foot, before the concrete had been filled in the haunches. This is a convenient method of test- ing the strength of a floor after it is laid, and its use is to be highly recommended. Streivsrth of Briek Arches.— Brick arches, properly built betwt»en iron beams, as described on j)age 440, are practically inde- stnictible, from any usage or load that could occur in a building. When the Western Union Telegrai)h building, in New York City, was being erected, Mr. P. C. Merry, the architect, made a series of tests on several forms of floor arches, supported by irou 448 FIRE- PROOF FLOORS. beams placed about five feet apart, by dropping a piece of granite, li5 inches s(juare and 4 feet lon^j:, with rounded edges, from a height of three f(»et. on lop of tlic arches : and. while ail of the other jin^lu'S wci*e destroyed, the brick urcli withstooil the nhock S(!veral times uninjured, and only after repeated |)oundings in the saiiK^ phicc one brick at a time was knocked out until the arch was finally hroken down. That l>ri(;k floor arch(\s will endure prreat distortion was sliown by tin' loiding of an arched fUK)r at the Watertown Arsenal, Mass. A flooi- \JI) feel square, was miule of five ir)-inch I-l)eam8, 20U lbs. per yard, carrying brick arches. The beams were 7 feet 4.8 inches apart on eenlres, and rested on l)ri(!k walls 28 feet inehe.<« apart. The rise of the brick arches was y.5 inches. ''Common, rather soft-burn(Ml ]>rick were us(mI, laid (m edge with lime mortar. The arches were i)acked with concrete, and planked over. The miixi- nuiin load carried by t his fl(K)r (when tlie Ijeams, and not the arches, failed) was 50:} lbs. per scpiare f(K)t. This load caused a gnulual and continuous yielding of the beams, winch was aHowed to con- tinue till till' ll(M)r was deflected a distance of 13.07 inulies, meas- ured at the centre of I he mi(hlle berims." '*The brickwork en- dured this great deflection. an<l apparently wouhl have stood much more without failur(>," had it been |K)ssi)>le to carry the test further.* FiiM'- Proof Floors with Tension Mem born (1805). — WitMJii a lew y»'ar-< several styles of nn?-proof floor construction havr li. en iiitroduce(l, of whicii there are two general olusi«os ; the first ela>s <'onsists of tension memiMM' floors, which in liicmselTei furnish tin ne<essary strength for sustaiidng the lhM)r from wall to wall, or wall to ginler, without the usi* of (hK)r l)eams; and the other ela^s consists of 1 U^ams iivt^ or six feel apart for sustaining the fli»)r. with rods or bai*s -usiK^niled or nesting upon the U>ani8, su]ip()rting win; cloth, netting, or expanded metal. whi<-li carries th<- concrete or plaster filling. I'rondiient among the first ilevici'S ineni iniii'.j :ire the II vat t riblnHl metal ties and Portland cement conerite ii«i,)r> built by 1*. 11. .ia<-ks<in. Sun {''rnncix-o ; tlh* con- crete an<l t w ist -d liar floors built bv the Ransome & Smitli Cciin- pany. ot" Cliieagt): and the Lee hollow tile and cabh- nwl fliior«, built l»y till' liCc Fii*e pniof Construction ('om|iiiny. of New York. ppiiiiiin lit among the l-)N>am and concn'te tiliiiig devices an* the sNv;i,Mi«^ nf t Iie .Metropolitan Kin»-I*r«H)Hiig Compiiny. of Tn»n- t'li. N. .1.: tliee\{ianiie«l metal con si r u ct I on com |ni nies of St. liiuiis * -I I-; n<>\\.iii|. ill .\itniii-a» An'hifn-f ttntl linihliittj .Vf/fA, Mttreb lU, I FIRE-PROOF FLOORS. 449 and New York ; and the New Jersey Wire Cloth Company, of Trenton, N. J. Hyatt and Jackson Concrete Floors.— Concrete com- posed of broken stone, fragments of brick, pottery, and gravel, held together by being mixed with lime, cement, asphaltum, or other binding substances, has been used in construction to resist compressive stress for many ages. With the introduction of Portland cement, concrete construction has taken a more important position among the various methods of building, so that now entire buildings are constructed of con- crete, such as the Hotel Ponce de Leon. fi.t St. Augustine, Florida; and in (Jalifornia. especially, concrete is largely used in the con- struction of floors, sidewalk arches, etc. The concrete is not used between iron beams, as are the brick and tile arches, but the concrete itself is made self-supporting from wall to wall by means of embedding iron in the bottom of the con- crete. Portland cement concrete has a great resistance to com- pression, but possesses little tensile strength. In 187G Mr. Thaddeus Hyatt, the inventor, while considering the matter of fire-proof floor construction, conceived the idea of forming concrete beams by embedding iron«in the bottom of the concrete to afford the necessary tensile strength which the concrete lacked. Mr. Hyatt made many experimental beams, with the iron introduced in a great variety of ways, as straight ties, with and without anchors and washers ; truss rods in various forms ; flat pieces of iron set vertically and laid flat, anchored at intervals along the entire length. These experimental beams were tested and broken by David Kirkaldy, of London, and the results pub- lished by Mr. Hyatt for private distribution, in the year 1877. By these tests Mr. Hyatt proved conclusively that iron could be perfectly united with concrete, and that it could be depended upon under all conditions for its full tensile strength. The method Mr. Hyatt adopted as the best for securing perfect unison of t'.ie iron and concrete was to use the iron as thin vertical blades placed near the bottom of the concrete beam or slab, extend- ing its entire length, and bearing on the supports at both ends ; Fig. 14. 450 FIRK-PKOOF FLOORS. tbcso vortical blades to be anchored at internals of a few inches by round win>s threaded through holes punched opposite each other in the vertical blades, thus forming a skeleton or gridiron, as shown in Fit;. 14 F^>r a perfect combination of these substances, it is essential that the one should 1)6 united with the other in such a maimer that the iron cannot stretch or draw without the concrete extending with it. The only person in this country to make practical application of the method devised by Mr. Hyatt, so far as the author is aware, is Mr. P. II Jackson, of San Francisco, Cal., who has used it quite extonisivoly in that city foj: covering sidewalk vaults, and for tl>e support of store lintels ; also, for self-supf)orting floors. Mr. Jack- son publislicd a pani[)hlot in 189.), entitled Impromment in BuUd^ ing ('onnfnfrfit^ny which gives a great amount of information on this sul).ject, and on concrete in general construction. To sliow the strength of this method of construction, Bfr. Jack- son, in Aug\L«*t. 1885, prepared a beam, 7 x 14 inches in section and 10 fiM't 6 inches long ; near the bottom were sitven vertical blades of iron extending the entire length ; three of these were i y \ inch, and four wore i x 1 inch, with i-inch wires threaded through overy 3 inches. Near the top were bedded two cast-iron rope moulding bars to assist the compressive strength of the con- crete, which, however, was siiown to IxMin necessary. The concrete at the top and bottom was one ])art cement to one of sand ; centre portion, oni' of cement to iwoof siind. Thi' Iwam was supfiorted by U-inch In-irings at both cn<ls. thus leaving it fed in the clear be- tween snp|»«»ris. Tlie beam was loaded with pig-irrm piletl lu-roes it, anil l)n)lv<' un:l(>r a lo.id of 5.>,(ii'>4 llw.. by Kcparating till the lon;;itu«Iin.-il bladi's on tlic line of on(> of the cn»ss-wires near the centre. .Inst lN>fore breaking, the deflection was measunnl, and foumi to Ih' \^_ in<'h. The breaking load of this lM>ani was aU>ut oMe-}):iir I hat which would have broken a hanl-pine beam of same dimensions and average ipiality. Tlu' Kaiisoiiu' and Kinllli Floor. \N hile Mr .Ia<'k^i>n was ex)M'iimenling with tlie Hyatt tics, Mr. Iv L lian-iiuf. a vrry >ui-'essrul workiT nf enn('ret«* in Sail Fran- ii-ii). iin.c.iMd the ide:> (it using siuan* b>irs i»f iron and .'^ti*«-l, twi^tni t|.,ii- entire leiurth. in place of the flat Uin* and win*s used )>y Mr .l.ick^on. as >)io\vn in Fig. 15. It was found that thest* bars Win- !■• Ill ill ihe fimerete i (lUiillv as well, if not UMIer than IIm ol liiM*. .ili'i lli'il (hey were niileh le*«s exiH'Msive. Nolle uf thtf in»D E-PBOO? FLOORa. in the ties is wasted, and it hae been demoastnted by careful ez- perirnents that the procesB of twisting the bars to the extent desired strengthens the rods instead of weakening them. Fig. IS. Mr. Bansome patented his improvement in 1884, and since that time it ha3i>een used quite extensively in San Franijisco. The bars, preferably made from the best quality nf rectangular iron, are twisted at an expense not exceeding from twenty-five to fifty cents per ton, which constitutes an inaigniflcant item of cost. The sizes so far used range from \ inch to 2 inches square. Concrete floors, as made by Mr. Eansome, are made in two forms — flat, and receesed or panelled. It can be and has been used for spans up to 34 feet, A section of a flat floor, in the California Academy of Science. 15 x S3 feet, teas tested in 1890 with a uniform load of 41,^ lbs, per square foot, and the load left on for one month. The deflection at the centre of the 23-feet space was only ; inch. It was estimated by the architects that the saving in this construction over the ordinary use of steel beams and hollow- tilo arches of the same strength, and with similar cement-finis I led floors on lop, amounted to tii< cents per square foot of floor. As a flre-i>roof construction, the concrete and iron con- struction above described is undoubtedly equal to any other con- Btruction in use. Oampotitum of the Concrete. — Regarding the concrete used tor these floors, the proportions are given for a cement of good average quality, that will develop a tensile strength of 350 lbs. per square inch in fourteen days. II a weaker cement is used, the quantity should be proportionately increased. The aggregates should be of any of the following solmtances, which are named about in the order of merit, the first being the best: Hard limestone rock, hard clinker brick, hard broken pottery, granite or basalt, hard clinker'^, broken flint or other hard rock. Care riiould be taken tj> use neither dirty nor soft clayey rock. The aggregates should be broken so as to pass through a two-inch FIHE-I'UOOF Fl.()i rinp, ami the fmo iliist, roitjoved by wiishing or screening (washing prcfL-rri'ih In mixing mid sufficient wnter to bring tbe mass into a fotl, |msly iiiitilitioii, itnil tam]i it. thoi'(>ii);hly iuto place. On (he lH)l.tc)in iif llie iiiiiulil ]>la(:c iilKiut one inch and a half of ctiniireic niiulw of ono jjiirl. cement lo two parts of agjrrcKitles vary- ing tnjiii ,',, to i ineli in diiinietur. I jiy Hio lower iron liars on this niixlure hikI tamp Ihuni Uunn into it ; tiien 1111 uji with a. conrre'" ci)ni|>o!<e(l of oni! [Nirl cement and six parts aggregates, making the final layer of double atrengtb. TIk> L<><- H<>IIow Tile and Cable Uoil Floor. Fig. 22 i» » Hk(>leh typic-al of thp Lee Hollow Tile anil Cable Goil FliHir. with a finislicil c-eineiil top. The flours are usiuily (li'signi'd cm a luisis of ^ inch in cIcpCh foreuuli foot of %paii. The spaiiH extend fioni wall to wall or from ginlcr to ginlur, no i- i terra-cotta tiles having siiuaru ends and a rod griKiri' aliiii;; iini^ hIiIi- near tliP Iwse, hfp iibi^. TIipm! lilea are sim- ilar t" IIh' l.i'i- i-nil arrb tiles, 'iVininimry focniH carried im honwa uri' iinividiil, and the lilei: are laid with i'ortland crinent niurtar in rows, curl in end, fioni wall In ginler. or fnitn ginli'r lo giixler. Into ibe ^'I'lKU-e of ca.-li row .if tiles soft eenient in phu'eil, >iiui iin<- or more nils, acc-oriiiin: lu stn-ngth rt<i)uirenii-nli>. iin- Imriwl in lllesoft e.-1ilel.t. The pilH'.-Bii is n-|>IMlvrl until 111.' wliol.- Hour iS fr.rMi.-.l Thr r.Hls iitop at emls of Ibe tili^s at wall lines Ani-liur lying the lloor lo tin- mipimrtii. tti'l f.>r ISy II LT 1 1ll to^n-lhel wliiih mar Ih' appli.'l lo lllling t: ,!<„< all Ihrusl i- lakrn ii|. bv the ealile HhI. and .-aeh iiid in ils i.l'K'e. CrMr'ks. deneetloDs. nii.l »lli.-r i|ef. <•(■ iidiii^' IImi jir<'lie> Hh- iivoi^le.1. Th.- Doors an- firm, rigitl. Tbu tloors are Iwwil n[ion the Inkn^vcnv strength uf FIRE- PROOF FLOORS. 452a beams. (Computations, verified by actual tests, are made, and the use of needless material and weight is thereby avoided. The cable rods used in the l«ee system are made of round drawn steel rods of about thirty onc-hundredths of an inch in diameter, ]aid spirally together, usually in two strands, as that form affords large gripping surface for the cement. Mr. Lee's patents cover a variety of forms, some containing several strands, with different shaped buttons, washers, etc., for affording great cement engaging surface. The rods being of drawn steel, they have high tensile strength, and are specially free from flaws or defects ; hence are found to make excellent tension members. The rods are spaced 8, 10, or 12 inches apart, according to width of tile used. The widths and shapes of tiles are varied to suit different spans and loads. Fig. 23 shows one design of roof for ten-foot spans. It is a f^25 fty^* special adaptation of the system, to cases requiring large protection to the metal from heat, as in dust chambers of smelters. Fig. 24 shows light design with finished wood top, suitable for dwellings, the wood top being more expensive than cement top. With a cement top the completed structure is but little more ex- pensive than a wood joist structure for the same purpose. The floors are absolutely incombustible, sound-proof, and vermin-proof. Strength and weight tables are furnished by the builders, giving various depths of floor structures for different spans and loads. The Metropolitan Company's Floors. Under this system, which has heretofore been known as the •'Manhattan*' system, and is protected by letters patent, fire- proof floors are made as follows : Cables, each composed of two galvanized wires, twisted, arc placed at given distances apart over the tops of the beams and transversely with them, as shown in Fig. 25. These cables i>ass under bars in the eenfre of the si)ans, and are thus ^iven a uni- form deflection between each pair of beams. The distance between the cables is varied with the loads to be provided for. Forms or centres are then placed under them, and a composition, made prin- cipally of plaster of Paris and wood chips, is poured on. This composition solidifies in a few minutes, after which the forms or iti-j/i FIKK-PliOOF FLOOKti. ceiitros are removed. 1'he rcsultini; lloor is anfitcientlr stroDf; to be iiM'il at once uiiilor tiie IoiuIh for which it has been calcukted, mill UM ids Kiirfni-'C is imiforin itiid bvul with llie tops of thu bnimx, a working Uimt U llius riiniishwl. 'I'LJs iii of (.'riini advarituKi- in fiicililaiiujr ihc tinnentt i-(iiislructiuu of builiiinKs. Fici. 3S. owM tbn urmnf^nient einplo;rod ill caiWR whura a flat ntijiiin^I. Id this iirniiij,'eraciit tlie nniler siili! of Ihe ■■■iiixhfs n coiliiiK Hortui-t' n'»ly lor |ilusb.'riu((. Thu iM iif thu bi'UtDH, |ti-ojcutiu^' ua thuy do bclciw tbo floor- -rr/ 1.1.'-: KhUl rry ilu- n«>r-[>li>t<-H. iirrnnjfi'iiM-iil fiii|>l(>ytfil whem a flat rciUiif; PIHE-PBOOF FLOORS. 452c id desired. In this case Ihe floor-plate i? the same aa in Pig. 26. Tha ceiling-plate is lormed as follows : Dars are placed upon the lower Annges of the beams, ami on these wire netting is laid. Centres are placed one incli below the beams, and the composition is poured thereon. The centres are then removed, and the ceiling thus made is readf for plastering. Whether a ceiling like that shown in Pig. 36, or a flat ceiling as shown in Fig. ■4'!. is osad, the webs of all beams are covered with about three inches in thiukness of the Metropolitan composition, which thoroughly pro- tects the beams from the etfeeta of heat It is claimed that this ma- terial is so remarkable a non-conductor of hat that a moderate thickness of it prevents the passage of nearly all warmth. " In.sETere Are tests the l)eams have rfmained cold, and conse- quentlj were unaffected. When exposed to flame for a long time, the Metropolitan composition \b attacked to a depth of from ,\- to A of an inch, the remainder being unaffected ; nnd when nater is thrown upon it, the mass (iocs not !ly or crack. When made thor- oughly wet, as would happen from water thrown into a building during a Are. the composition is nofdestroyod." In Paris a composition of plaster of Paris and broken brick, chips, etc , has liocn used for giineraiions f;.r fniiunig ceilings ijutwpon beams, so tliat the question of its durability is there fully settled. The strength of floor? made under the Metropolitan system has been accurately determinB<l for vitrious spans by 11 great number of carefully-made tests " The loads that so break up the oonpositioa of floors made 452^7 FIllE-l'ilOOF FLOOUS. under this system as to RMjuiro it to be replaced, vary from 1,100 to 2.00) j)<)un(ls ppT square foot on spans of from 4 to 6 feet. 'I'he W(M<^lit of ?i floor finislie:!, as shown in Fig. 26, when ready for the plaster underneath and the floor above, is about IS pounds I>er sfiujire toot ; and for a floor and ceiling such as is shown in Kig. 27, 24 pounds per sciuaro foot; the thickness of the floor plate is alxmt Ji'l iiu'lies. T1h» proprietors of this system reconmiend that the floor beams be spa('e<l about i> feet apart, as this distance appears to give the best results witli the greatest economy. P'or further information concerning tliis system, the reader is referred to the Metrojx)litan J^'ire Proofing Co., Trenton, X. J. There are several styles of floors constructed on the principle of the Metropolitan floor, although nearly all of the others use Port- land cement concrete instead of the plaster c<)mi)osition. Wire lathing, (■xj)anded metal, and various shaped bars are used for the t(?nsiun menil)ers. The jn'incipal advantage sought in these floors over the icrra-cotta file arches, is a reduction in the weight of the fl(K)r, thereby causing a saving in the steel construction. The floors themselves are also, as a rule, a littie cheaju'r than the tile floors. Another important characteri.stic of all floors constructed on this j)rincip]e is, any st^ttling of the anhes. or filling, will tend to draw tiie beams (»r girdei's together, instead of pushing them apart, as is the case wiih tile arches ; and tie rods are, therefore, unnecessary. The strains infl(M>rs of this kind are the same as in those of a beam, ilie e!V. c t of tlie load Ixnng to pull the tension members ai>art at ilu' 1k):im:ii. and to ciu>li the concrete on top. Wlien the eon- (I'ete i> of tlie proper thickness, and of g(XMl ([uality, the stn*ngthof th«' llonrwill bedetermined l»ythe strength of the tension n)(Mnl)cr>ii. Several ti'^is ot" beams made oi" .ortlaiid cenuMtt. eoncn'te. and wile neitiiiLr made by the NciW .lersey Win; Clnih ('om]»any. apjirur to show that only about one half the strength of the ten>ion ineni- lK'rs'\\h 11 of wire cloili) can be; d«;velo|H'd. In all floors van- strueted of coneiTie. plaster, or tile with steel tension nuMnU'rs, it is ^ii tiie iir^t imiMirtance that tiie two materials shall 1m* so elosi'lv united that the tension memU'r-? will not be dnitrn thror^h, or slip ill the eoiiepie : Inr the minute this (K'cui*s, the strength of the llniT. lis (I III (I III, is (lest roved. \\ lid« >'iiii" of the^r tension memU'r fl<M)i-s liave been ns«»d .sufTi- <-ieiiiI\ t(» :iill\ (h'lnonstiate their strength and praetieabiJiiy. yet th<- wr.'.ei i't'iieves that new arningements n devjees should lie u^>i| \M!li • At r-ine eaution and oidy after they havi) buuii t4S8ted an<l apprcVMl i>y experienced eagineers. FIRE-PROOF FLOORS. 4526 Concrete and Wire Netting Floors. '■ Pigs. 28, 29, 30, and 31, show two styles of fire-proof floors, devised by the New Jersey Wire Cloth C'ompany, and described, together with several other applications of concrete and wire net- ting, in a pamphlet published by them. The segmental arch shown ^^^r ^ ■;*: ' ^^ >— J — ''Vl m ^^v^^^ fiii^'^ m^ ^j Fig. 28. in Fig. 28 is constructed by forming a centre, made of small rods, cut the proper length to form the desired curve, and to just reach into the angles of the web and lower flange of the floor beams. These rods are inserted between the meshes of wire lathing, and the sheets, which would be three feet or more in width, are then Fig. 29. bent to the curve and sprung into place. A succession of these sheets placed side by side fill the entire space from wall to wall, and make a continuous network of iron wire and rods, upon which concrete can be spread from above without the use of any other support. The lower flanges of the beams are covered by wire lathing attached to a succession of rods hooked over the arch rods and held in place by the wedges which are inserted between the beams and the rods. The under side of the arches and the lathing around the beams is then plastered and finished in the usual way. It is claimed that with this construction the strength of the arch is only limited by the ability of the beams to carry the load. The weight of the concrete will vary from 30 to 40 pounds per square foot. FIllE-i'ROOF PIX)ORS. Fi^. BO and 31 ahov u flour coustnuitioi: designed on the com- pcjsiic Ikjuth principle. It is i'iiii[in.il by till! iiiaimfiiclurerB. that a load ol from 70 to HO n lie carried ■i of I>VIL1UB. i^-ljt of the corii^rute. uiru, anil rmls, For both Qoor aad ceilinj,', "ill vary troiu 8;Ho 4o jiounds |)iT wjnaru fool. a foot, with a fat-tor o( safety of si i»ii in spalls of isix feet bftwei'U oi - iif tilis HiNir ininKlriK'tion is n m-rJeH of rmIs honked over III tli<> lii'aiiiH. or iiUai-tx'rl ti> iIu'mi liy rliiw iltvitnHil for 'I- Till- rtxU Hre plwcl alxml Iwelvi- iiu-hiH Ui>«rt, Hnd in' spnixl sIhtIs of wiri' iHlhiiii; riLntiini; piimllcl with m-r ihc lii|i of till- iH-atiiN. Tlii' coiii-H'li- is tlii-n spn-ad icivK. ini>-iilli of Iwo tr.llirri'ini-lii-s Nn iTiitcrinK w - Ilii' iTirw iiii-slii'.-' of Ihi- liilliiiii; iin> mi oI<w loKutlior iK^iii;) iK'n-li' Hill •,!•< tiiroii^'li lo llrmly >iiii-hiir tb« ■V tl>- ('..tu-n>u- W M-l. iW unili-r sid.- >li<.iilij W iiIkmIiw) II ^r. i,s to .'lltill.ly >-!I|Ih'.1 till' win- >LII<l hHU. lis slioiilil In- |iriitii-liil liy win- IilIIimlk and pliiitti'rinR, KTilal ii'ilhif;, HUjiiHirliHl by linxioH riHla. inay U- hung b<>rs <a(i In- con^ttnietetl with eiimndiil iiieUl Iftthlog. FIRE-PROOF FLOORS. 462^ The Fawcett Ventilated Fire-Proof Floor. This is a style of floor construction differing almost entirely from any of the floors herein described. It has been used exten- sively in England, and to some extent in this country. In the construction of this fire proof floor, the special feature is a Tubnla/r Lintel^ or hollow tube, made of flre or red chimney pot clay, and burned mellow. Iron Beams (of sections to suit the spans and loads) are placed at two feet centres, and the lintels are fixed between, with their duig- onals at right angles to the beams ; the end of each bay is squared by cutting (during manufacture) an ordinary lintel, parallel to the diagonal ; the piece cut off when reversed goes on the other end. Thus the ends and sides of all lintels are open next the walls. These are called ** splits." The lintels being in position, specially prepared, cement concrete is filled in between and over them, which takes a direct bearing upon the bottom flange of the beams, thus relieving the lintels of the floor load, which is taken by the iron and concrete, the lintels forming a permanent fire-proof centering, reducing the dead weight of the floor twenty-five per cent, and saving about half the concrete. Cold Air is admitted (through air bricks in the external walls) into any of the open ends or sides of the lintels, and passes through them from bay to bay under the beams. Note, only two air bricks are absolutely necessary in each room, to insure a thorough current of air. The flat bottom of the lintel completely incases the bottom flange of the beam without being in contact with it, a clear half- inch space being left for the passage of cold air. It is claimed that the chief tire-resisting agent in this floor is not so much the terra-cotta or the concrete as the cold air, and that the circulation of air through the floor and around the beams will actually prevent the iron from ever getting hot at all. The Fawcett Company claims that their floors have never been injured by fire and water, Ixiyond what could be repaired bv replas- tering the ceiling and redeconiliug the walls. This floor needs no centering or any other support from below while in course of con- struction, and can be used as soon as finished. It is guaranteed to carry fmm 150 to 750 pounds to the square foot, according to the requirements of the building, with perfect safety. Although the author has never seen this floor put up, it appears FIKE-PltOOP FLOOitS. Ml m 11 1 i yi 1 FIBE-PKOOF FLOORS. 463 to him to be a very superior floor, although probably more expen- sive than the other styles herein described It requires more con- structional iron work than the systems generally in vogue in this country. The Guastavino Tile Arch System. Within a few years a method of constructing floors, partitions, staircases, etc., by means of thin tile cemented together so as to make one solid mass, has been introduced by R. Guastavino, of New York. The floors in this system are constructed by cover- ing the space between the girders by a single vault, constructed of tile about 6" x 8", and ^ inch thick, cemented together in three or more thicknesses, depending upon the size of the vault. The thick- ness is generally increased at the haunches. The strength of these floor vaults, considering their thickness, appears to the author very remarkable. This method of forming floors is especially desirable where a vaulted ceiling for decorative purposes is wanted, as the vault can be made the full size of the room. The iron- work used for posts and girders must bo piote:jted as in other methods of fire- proofing. The iron-work of the floors must be especially arranged for this system when it is desired to use it. As far as the author can judge from an inspection of the system, it possesses some ad- vantages over all other present methods of construction (and, pos- sibly, some disadvantages), and is likely to be largely used in the future. It has been employed in a number of buildings in New York and Boston, and a few other cities. The new Public Library Building in Boston has the Guastavino floor system, which is ar- ranged so as to give a fine effect of vaulting in the ceiling. Rules for Determining the Size of I-Beams, etc. The method of computing the size of the iron beams used in fire- proof floors is merely to determine the exact load they will have to support, and tlicn to find the required size of beam to carry tliat load. The weight of the floor itself should be determined for each par- ticular case, as it will vary with the kind and size of tile, the amount of concrete filling, kind of flooring, etc. The weight of the arch itself may be taken from the manufact- urer's catalogue, or from the table on page 445, and to this weight should be added about 5 pounds per square foot for mortar used in setting. For each inch in depth of concrete add 8 pounds; for plastered ceiling, 8 pounds ; for hard-wood flooring, 4 pounds ; for 454 FIUE- PROOF FIDO US. marble floor tiles, 1 inch thick. 1-1 pounds. The weight of the betims may bo taken at 5 pouncls per square foot for 9-inch bojuns. and () pounds lor 10 and 12-inch l)eains. Very few fire-proof floors will be found to woii^li less than 75 pounds per square foot, and where marbh^ tiles are used for the flooring? the weight of the (con- struction often reaches 1)5 pounds. The superimposed loads will, of course, be the sam(^ as those jLriven on page 426. The weight to be suj)p»)rted by the beams will be, w = distance between centers x span of beams x (/ -f /'); / representing the superimposed load, and /' the weight of the floor construction, including an allowance for the weight of the beams. Having obtained the value of this expression, the size of beam required to carry this load may be easily ol)tained from the tables in Chapter XIV.' To save the labor of making these calculations in the principal classes of buildings in which fire-proof floors are used, the follow- ing tal)les have been computed, which may be safely relied upon. Tables of Floor lioains. Tables showing the size and weight of Carnegie steel beams re- quired for dilT(»renl spans and sjiacings in different classes of build- ings, using hollow tile or terra-cot t a between the arches — the l)eams not to deflect so as to crack the phistering: TAIiLK I. -F()I{ b'LOOKS IN OKFICKS, IIOTKIjS, AND .\P.\kTMKXT IIOI SKS. 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 a-j \'i Si .. •jn 1m • •.'.". •• 1*1 .'.•J • 1;' :w IJ .ftj M *! •• •.'1 I'.' •• .. , . . :« • V2 •.\'i • • 1 I-i .-iV la .« " • >•> 1 .' ■• • i"* i>i :« » Vi '. '* 1 Vi :k ].". ■II " • »•► ^» 1 1 . • • I'l • 1 w • v: :i-3 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 '• FIRE-PROOF FLOORS. 455 TABLE II.— FOR FLOORS IN RETAIL STORES, THEATRES, AND PUBLIC BUILDINGS. (Superimposed load, from 125 to 180 ponnds per square foot.) Span, in Distances between Centres of Beams. feet. 4 feet. 4 feet 6 inch's. 5 feet. 5 feet 6 inch's. 6 feet. 10 6 in. -13 lbs. 6 In.— 13 lbs. 7 in.-15ilbs. 7 in.— 16*lb8. 7 in.— 15*lb8. 11 7 " - 15i " 7 " _15| " 7 " —15* " 8 "-18 " 8 "—18 " 12 7 " —15^ " 8 "—18 " 8 " —18 " 8 "—18 " 8 "—18 '* 13 7 " —15^ " 8 "—18 " 8 "—18 " 9 " -21 " 9 "-21 " 14 8 ''—18 " 9 "-21 " 9 "-21 " 9 " -21 " 10 " —25* »* 15 9 "-21 '' 9 "-21 " 9 " -21 " 10 " -25* " 10 "-25* " 16 9 "-21 *' 10 " —25* " 10 " -25* " 10 " -25* " 18 "-82 " 17 10 " — 25J " 10 " -25* " 12 "-32 " 12 " -32 " 12 " -82 " 18 10 " -25* " 12 " -32 " 12 "-32 " 12 " -32 " - 12 "-82 " 19 12 "—32 " 12 "-32 " 12 "-82 " 12 "-32 " 12 " -82 " 20 12 *• —32 " 12 "—32 " 12 "-32 " 12 "-82 " 12 " -40 " TABLE III.—FOR FLOORS IN WAREHOUSES. (Superimposed load, from 200 to 210 ponnds per square foot.) Span, Distances between Centres of Beams. in feet. 1 4 feet. 4 feet 6 inch's. 5 feet. 5 feet 6 inch's. 6 feet. 10 6 ill.- 13 lbs. 6 ill. —13 lbs. 6 in.— 13 lbs. 7 in.— 15Alb8. 7 in.— 15* lbs. 11 7 " -15* *' 1 7 " —15* " 7 •• —15* •• 7 " -15* •' H "—18 " 12 7 " -15* ' 8 " —18 " 8 " —18 " 8 " 18 " 9 " -21 " 18 8 " -18 " 9 " —21 '' 9 " -21 " 9 " -21 " 10 '*-25* " 14 9 " -21 " 9 " -21 " 10 " —25* " 10 " -25* " 10 " -251 •* 15 10 '• - 2.5* " 10 " — 25i " 12 " -32 " 12 " -m " 12 •• 32 " 16 10 " -25i " 12 " -32 " 12 " -32 " 12 " 32 '' 12 " 40 " 17 12 " - 32 " 12 " -32 " ,12 "-32 " 12 " -40 " 15 " 41 " 18 12 " -32 " 15 " -41 " 15 " -41 " 15 ' -41 " 15 "—41 " 19 15 " 41 " 15 " -41 " 15 "—41 " 15 " .'50 •' 15 " -50 " 20 15 •' 41 " 15 " ^11 " 15 "—50 '' 15 " -50 " 15 " —50 " It will bo seen from these tables that it is more coonomioAl to space the l^eains farther apart, and use as short spans as the condi- tions of the building will {)errait. For example, if we have an office floor 48 feet square, to support with iron beams and tile arches, wc; may eith(T use one girder down the centre, with 12-inch beams, spaced 4 feet apart ; or two girders, and lO-inch beams spiU3ed 6 feet apart. In the former case we should require 11 beams the full width of the building, weighing 455a FIRE-PROOF FLOORS. 16,896 pounds, and in the latter 7 beams weighing 8,568 pounds, a saving of nearly 50 per cent, in the steel. From this, however, will have to be deducted something for extra girders and columns, but tho total saving would probably equal '^5 \)er cent. In rcganl to the columns, it will not make much difference in the amount ol' iron used, whether there are one or two rows, as the total weight to 1)6 supported is the same in either case, and if one row of girders is used the columns will be closer and heavier than if two rows are used. l)<»rtecti(>ii of Rolled I-Reaiiis. — The deflection of rolled iron I-beams can be computed by Formula 1, under the Stiffne^ts of BcatHs, Chap. XVI. Accordiiiir to the calculations of Mr C. L. Strobel, C.E., tho beams in the foregoing tables will not deflect over one-thirtieth of an iucli for every foot of span, under the load which they have been calcnlatcd to support. T!(»-r<)(ls. — Tie-rods from fivt^-eighths to (me inch in diameter are ordinarily em])loyed to take the thrust of the bri(rk arches, and to add to th(^ security of the floor. These may l>e spaced from eight to ten times the depth of the beams ajiart, and the holes for them should always be punched at the centre of tlio depth of the l>eam. The formula for the diamet^^r of the tie-rod for any floor is, . W X span of arch, in feet Diameter 8«iuared .^., . - i • .» i ; ' 6J8;j2 X rise of an-h, m fi»et * irdenotin;: weight of IKh)!*. and superimjH)srd load nesting on the arch ii.iU'-way between the tic^-nxls on each side. i^x.\.MiM.K. What shoul'l 1h* the diameter of the tii^-HKl to take the thni>i of a 1 brick arch, bctwt'en 10 ' iM'ams, spac«*<l 5 fwt npart : i he an li having a rise of iJ , and the tic-nxls to Im» sitiicod 7 U'c\ apart ? I'lie su]M'riinpo.«<ed Imul to U' taken at 100 llw. Anf<. in thisca.>^c the span 5 ftn't, nearly ; W 170 x 5 x 7 5050; and r . fn(.|. Then />- '*".*' "' , 0, <»r /> 1 in4*h. t>ys."j ' k nearly. ( )f «M>ui->^r. where arches abut aL^•tinst each siih- of a iH'um. there is no n< I'll -!' iixis tn take thi' ihnist of the arches : but it is alwavs safer !•• u-<- •'HIM. as the nntsidt- bay of the thmr might \h' puslied •fV --ifiewise if the who!*' were not tie<l thniugh ; also, if one of the arehes >hi)(iid fail, or bn'ak through, the hmIs would keep the other arches in place. 456 ^ILL CONSTliUCTIOXi. CHAPTER XXIV. MILL CONSTRUCTION.! In this ('hai)t('r it is proposed to describe the principal oonstruc* tivc f«'atun*s of what, in tli«* Eastern States, is known as the " Mill Const ruction," or **Slow-]>nrnina; Construetion." It is a method of const rui't ion lirought al)(>nt largely throngh the influence of the factory imitiial insurance companies, and especially through the efforts of Mr. William i». Wliitinj^, whose mechanical judgment, experience, and skill as a manufacturer, have been «levote<lfor many years to tli* interests of the factory nnitual comiMinies and to the improv(>ni(>nt of factories of all kinds. Mr. K<lward Atkinson, presi<lent of tlie IJoston Manufacturers' Mutual Jnsumnce Coni- ])any. has also done a •xreat deal towards influencing the public In favor nt" tlilN mod*' of construction. Tlie /// xl'h I'lifmu in this mode of constniction is to have a build- inu: wli<»c nuoide walls shall he built of niiisonry (g<*nerallyof brick) con<'«ntraici in piers or buttresses, with only a thin wall i-ontaln- in^tlie windows l)ctween, and the floors and niof of which shall 1)e conMructf I of liiri^e tinduTS, covered with plank of a suitable tliickn«»: tin' ::ii'ders heiuL; supporte«l In'twi^'n the walls by W(M)deii ]M><ts. No t'lirrini^ or conc(>aled spa(*es an' allowed, and nothing is perniitteil which will allow of the accunndation of dirt, the con- cealni* nt of tire, or, in short, any thini^ that is not needed. Mr. i\ .1. II. Woodbury, ins]M*ct<»r Utr tin' factory mutual fln»- insur.iiH'i' conii>anies of Massachusetts, who has written a \iT>" able 1u)ok on the " Kire rroteetion of Mills" (publisheil by .lobn Wilev A- SoiiN o;' New York), has ;:iven such conci.se and «*N'ar stat<ini*nt^ oi what does ami what does not constitute safe iH>n- stniciion lor niills and warehouses, that with his iKTUiission we quote thrni nrfnitlin from his wurk. > Cuts I In t) in thi» rluiptrr an- inkcii fnmi WiMNlhiiryV Kirc lYolecttoil of MIIIh, iiiiil n-ilu< fil, fii rnnf«iriii lo tlio i-ixc of ilif imh^v MILL CONSTRUCTION. 457 r '^ Prevailiiig' Features of Bad Constrnetion of Mills and Storehouses. — The experience of the Factory Mutuals has shown that in mill and storehouse construction, where considerations of safety, convenience, and stability are es* sential, tlie following prevalent features of bad construction should be omitted : — " Bad roofs. " Rafters of plank, eighteen to twenty-four inches between centres, set edgewise. " Any roof-plank less than two inches thick (three inches pre- ferred) ; any covering which is not grooved and splined. " Any hollow space of an inch or more in a roof. '^ Any and every mode of sheathing on the inside of the roof so as to leave a hollow space. " Any and every kind of metal roof, except a tin or copper cover- ing on plank. " Boxed cornices of every kind. " Bad floors containing hollow spaces or unnecessary openings. " Thin or thick floors resting on plank set edgewise, eighteen to twenty-four inches between centres. "All sheathing nailed to the under side of plank or timber, making a hollow floor. ** Bad finish, leaving hollow spaces, or flues. "All inside finish which is furred off so as to leave a space between the finish and the wall. " Wooden dados, if furred off. " Open elevators. " Iron doors, iron shutters. " Any and all concealed spaces, wooden flues, or wooden ven- tilators of every kind, in which fire can lurk or spread, and be pro- tected from water. " Any and all openings from one floor to another, or from one department to another, except such as. are absolutely required for the conduct of the business (all necessary openings should be pro- tected by self-closing hatches or shutters, or by adequate wooden fire-doors covered with tin; automatic doors preferred in many places). " Ji^ssential Features for the Safe Construction of Mills an<l Storehouses. — Solid beams, or double beams bolted near together, eight to ten feet between centres. Not to be painted, varnished, or * filled' for at least three years, after the building is finished, lest dry-rot should ensue. Ends of timbers ventilated by an inch air-space each side in the masonry. " Roof nearly flat. Timbers laid across the tops of the walls to >• 458 MILL CONSTRUCrriON project eighteen to thirty-six inches, as may be desired, serving as brackets. Plank laid to the ends of the timbers. Neither gutters nor boxod cornices of any kind. Wooden ])osts of suitable size, not taperctl, unless wlu^n single posts turned from the trunks ot trees with tlu^ heart as a centre, following the natural ta]H?r. <.'ore.s })()re(l one and a half inches diameter ; two half-inch holes trans- ver.-.ely through tlu* post n(^ar top and bottom for ventilation. " Floor-i)lanks not less than three inches thick for eight-foot bays, three and a half to four for wider bays. In some cases, beams have b(?en i)laced twelve feet apart, witli four-inch plank for th<* floor ; but in such cases a careful computation of the strength should be madt*, based upon the load to be placed thereon, l)efore so wide a s})ace between beams is adopted, lest there sliould \ye. ex- Cisssive dellection. 'I'he better method, wliere tlie arrangement of the machin(»ry reipiires such wide l)ays, is to alter the plan of floor- timbers. Toj) lloor one and a quarter inch boards of Southern pin<', mjiplc, or some hard wood. The best construction requints this top I'ioor to Ih' laid over three-quarter inch mortar, or two thicknesses of rosin-sized sheath ing-paiR»r, certain grades of which are now made especially for this piU'i>ose. ••All rooms in which sp<'cial dangers exist, such as hot drj'ing, to be ])n)tcct«Ml overhead with jdastering on wire-lath, following the •inc of ceilin!^ and timlnT, thus avoiding any cavity in the ceilirg. In su'h rooms, the wooilen i)Osts should also be i)rott»cted with tin; car«' hciiii: taken to leave the half-inch holes through the ])us'.8 mar \h*' top and base uncovered, so that dr>-rot may not take r:acc. Kig. 1 re])»-(»s'nts th<' iM*op<T const met ion of one bay of a thie-"- siory ndll, e:it]i bay being like the others, and the building In-ii ^■ iorniid of auN number of su<'h bays pla.'eil one aft4*r the otluT. Such a buil ling cannot be <'onsidered as fire-pnH)f: but the im'- terial In in ^iuli a sha])«' that it would not reailily take tin*, aM-.- wonll l»ui n >Io\vly even ihen. Moreover, the construction is mm-I . that any jtari of the building can be easily reached by a stream •*: wale,- : so ijiat a lin* <"an be n'adilv extinguisheil In'fon* it ha. gaine I inueli headway. Ill a luiik Imildinii im 'jraiiHr shnnhl fir ?/«cf/. except f«)r sti'pr and nndi-rpiniiinLr. a^ it sjdits badly when ex)K)s<Mi to heat, an 1 i.* theretme nn -uitablf for <<ills or lintels, or any work liable to Ih» exiM)s('ii to an\ intense h«'at in case the building >hoiild In' on tlrt*. 'I'he hi^t i|Mahiies oi br«>\\n sandstone maybe u.sin! fur sills, aiiik for nihei- |)|aecs it would be blotter to use brick or ttTfaHHtttu. Mnnl ]•■ I liri<-ks ari> now manufactured in a gn*al variety of fornit. and are nell suii,.d for deconilive work. MILL CONSTRUCTION. 45\ The best factories and woollen mills Is Husachusetta are now generally bulll with the beams eiRht teel. apart from centres, end with a span of twpnty-flve or twcncy-tour feet, there being one or more rows of posts ai^oriling to the size of the mill. Fig. I repre- sents the section of a mill having two rows of posts. Fig. I. The floor-heams are iwually twelve inches by fourteen inches hanl-pinn tliiihers,' which n>st on twenty-inc'h brick piers in the basement, antl on wnwleii posts and the outside walls in the other Stories. The ends u'liicli rest on the outside wail are arranged so BB to iiave an air-spncit around tlie end of the timber, and are aTH'hon'd to the wall by a cast-iran plate on which the beam resls. Tills plate, shown in I'ig. 2, has a transverse projection on the lllHHT sui-fai-c, wlilc'h fit^i into a groove in the bottom of the beam, and is turned down alHiut six tuehes into the brickwork at the encl. The hrlekwiirk for about five courseji above the beam sliotild be laid dry, and the upper edge of the end of the beam slijtlitly rounded. In ease of tlie possible hiimlng of the beam, this would allow the beam to fall without throwing ont the wall. Tlif finnr an top of these iH'ams is ronstnicted, first, of three- Inch planks, not over nine Inches wide, planed both sides, and grooved on both cdgis, which are filled with splines of hard wood (generally haiil pini>) alM>ut lliri>i'-f<iiu'tlis of an inch by an inch im MILL CONSTRUCTION. [Ill A lialf. In ciailin^ the planks, it la l)etter to "blind nfUl" lipiii. !<t'U.'i tlie iiiann<^r of iiailitig miLtrhed floors in dwelling-IiouMt ml storfis ; tliat is, ilriviiij; tin- nails obliiiiiely LliroiiRli th« Rroove I't'oru t!ie a[itinu la put ill : lliis hIIouh Lite plank to sUrink or u't'll without tracking, and wilbout afliliin^ ttic s)ilin(>g. Fl(. 2. nalli'il In tlih way. rarli pinnk miul |iiit <liinii. Tills lakfH (-onstilfrahle ay a niiiiilxT of planks, Wiilj;!' Ilii'iu lliic-s fiiini oni'. cnil, &nd nail Uirw.'tly Fig. 3. Till' ujipcT tliMiiinir is p-iii'ially o( sonic liant wooil, an iiirli wml 'liuiriir rliU k. iin'ri-ly .joliiI.'<i. -I'll.' Ill »ir,-.^l.. mill ii.- ivn.irn-il ua(iT-(iylit l.y flim-folinlis i>( an .■liof iii..rkiilu.riv.-.-ri (hr III.].!-!' .111.1 lowri' (l.,..is. Til.' Iiiyi'r ot oiuir |.iv.,TV- III.' Iiiiiilirr from il.riiy. jiivM-iils iW ll.M.r fniin ':ii'ly III" pi'i'iif llian iiny ntli'T I'llii'llral iih'IIliiiI of iMnHlrm- liu. :; -lii.iis J. si-i'iinii ilirnii-li mi.1i a lliM.ra- v.>- liavi- iIi-wtMhiI. h- .■:;/ j. :;iii,-r:ill> iDiuuii of U-ii-iiic-li liy 1Hi-lvi'-iiii-li lianl-I'llw MILL CONSTKUCTIOM. 461 timbers placed the same as those below; and the outside end is allowed to project over the wall from eighteen inches to two feet, forming brackets to support the eaves. These timbers are covered with two and a half or three inch spruce plank, grooved and splined the same as for the floors. The plank extend to the end of the overhanging timbers, and form the eaves to the building, no boxed cornice being allowed. If the roof is flat, as is generally the case in mills and factories, the plank should be covered with tin, gravel, or duck. If tin is used, it should be the best " M. F." tin, painted on the under side with two coats of red-lead, and well dried before the sheets are laid. If a gravel roof is used, it should be equal to the best quality of tar-and-gravel roofing over four thicknesses of the best roofing-felt. Cotton duck is gradually coming into use as a roofing material, and has for a long time been used for covering parts of vessels. It is light, durable, does not leak, and is not readily inflammable. The material should be twelve-ounce duck, weighing sixteen ounces to the square yard, and should be thoroughly stretched, and tacked with seventeen-ounce tinned carpet-tacks, the edges being lapped about an inch. If the roof-planks are rough, or not of an even thickness, a layer of heavy roofing-paper should be laid before the duck is put down. After the duck is laid, it should be thoroughly wet, and then painted with white-lead and boiled linseed-oil before it becomes dry ; which makes it water-proof. To protect from fire, give it two more coats of white-lead, and over this a coat of iron- clad paint. Instead of the four coats of white-lead and oil, the duck may be saturated with a hot application of pine-tar thinned with boiled linseed-oil. This lias been found to work perfectly. The ironclad paint should be applied, whichever method is used. If the roof is pitched, it should be covered with shingles or slate laid over three-quarters of an inch of mortar; which protects the slate from the heat, should the building take fire, and rentiers the roof cooler in summer, and warmer in .wintei*, whether slate or shingles are used. Where there are no buildings near, shingles are recommended, as they are warmer than slate (thus saving in the cost of heating), and are also cooler in summer. If the shingles are painted, which is advisable, they should be dipped in paint before being laid, so as to be entirely covered on all sides with paint: otherwise, moisture Avill get into the shingle through tlie place not painted, and, being prevented from evaporating by the paint on the outside, will rot the shingle. The columns for such a mill are usually round columns, nine incbee diameter in the first story, eight in the second, and seven i& 4(12 Mll.r, CONSTttUCTION. tlie third; thpse l>eine Uie least diamet<'rs of the columns. Ifth< (■"luniris iirc tapered, t1»!)- may he half (in inch loss in diameter al the top, and oiu: itiHi [iiore at the bottom, making the taper on FI9, 4. hf I'dliiinn thrfc-foiiiths of an inch. They nhniild iiir'l-piiir or 'i'(k tiiiiliiT. tlmLims(lily seasoned, and ire.s Imr.'d one ami a half hirhw In illanietcr. with lii>li's Iransvi'rsiOy tlinMigh the [mst, npar top ami lit ilat ion mill to pri'vi'iit di-j-rot. Tlif tytlimins aro iM.<t-irim cuits. as shiiim in V\!i. 4. wliid) support •• Ilooi-beaiiis; ami, "hi'ii" there Is a vcrtlL-al line ul u:i1i iron itinllMi, wlileli ounnert of tiM-oIlii-r. ;in'V<'ntii)grhpKiii^ I by the ueigliC oil tin euluiimii MILL CONSTRUCTION. 463 above. The ends of the pintles and the iron plates against which they rest should be turned true, so that the contact will be uni- form. Fig. 5 represents a vertical section through the floor and the centre of the columns, and Fig. 6 shows a perspective view of a pintle with the base of the upper column coming down over the top. The brick piers in the basement supporting the columns should be capped with an iron plate twenty inches by twenty inches, an inch and three-fourths thick. The above is the most approved method of construction now in vogue for mills, factories, and storehouses; and the dimensions given for the various parts will answer for any cotton or woollen factory where the bays are not more than eight feet long from centres. Where the bays are more than this, or the loads on the floors are greater, as may be the case in storehouses, the floor-plank and timbers should be proportioned according to the rules for strength and stiffness given in Chap. XXII., and the columns proportioned according to the rule given in Chap. XI. ff partitions are desired in such a mill or store- house, they should be built of two-inch tongued and grooved plank placed together on end (forming a solid partition), and plastered both sides, either on wire, or on dovetailed iron lath. Such partitions have been found to work well after a trial of twelve years, and offer effectual resistance to fire. Mill doors and shutters should be built of two thicknesses of inch boards, covered on all sides with tin, as described in Chaj). XXVI. For a thorough description of the apparatus and appliances used for the fire protection of mills, and for a thorough discussion of the vibration of mills, the deflection of the floor-planks, and, in fact, every thing that refers to the construction and protection of mills and factories, the reader is referred to Mr. Woodbury's work on Ihe "Fire Protection of Mills," mentioned al)ove. The cost, per square foot of total floor area of mills and factories at the present time (1884), according to Mr. Edward Atkinson, is as follows : — Mill with three stories for machinery, and a base- ment for miscellaneous purposes 75 to 80 cts. Mill with two stories for machinery, and no l)a«ement 65 " Mill with one story, of about one acre of floor, with basement for heating and drainage only . . . about 85 ** The above is for the total area of floors in the building, above Fig. 6. MILL OONSTBUCTION. 466 ncrt eTen weakened by the sftace left in the wall, because the anchor remalnB, and the crashing strength of this cast-iroa box is much greater than that of the wall. No break or breach is made in the vail, and. the anchor that remains, securely held, forms a space for the easy repiaeoment of joist. The anchor provides a perfect and seoiire foundation for each joist. Fire from a defiictive flue cannot ignite a joist end, because it is protected by a rentllated east-iron box. The boxes, or anchors, also have air spaces in the sides, J inch wide, which permit a eircujation of air around the ends of the joist, effectually preventing dry rot in Che ends of tile timbers. If timber is wet or unseasoned it will have a ohanca to dry out after it is put in the buildiiin- Tliea; aur^hors are obviously greatly superior to the ordinary method of anchoring beams and girders to walls, and their use would, in case of fire, undoubtedly save much loss by the falling of the walls, which are almost invariably MILL CONSTHL-CTIOS. pulltnl ciown by tlio ordinary iron anchors. The avenifie wdght ol alinx liki; Fig. 7. Tor 2 x li joist, in l.j lu 17 lbs.; of Fig. 8, from woihI [Hists. Thw OB]) linlds . piuviik-s vuntiliiliim uIh>uI T..U..ri} ii'l I. I .laii «<iTilnl liinlior' tn fall. ■n,.-. ; li..r.. ,in<l .-111^ iir.- r.''-..]iirii.iiiii'il l.v Ih.- fiK-tiiry iiiiiriml iiiojr: ,.|„, i,.s <>r N<'»- l':ii:.'liiiL.). :itia <-Mli >'•' iiiii.li- ill aliv f..iii. :ry. t.y |.:.yiiii: ii mviilly ..C : r.[ a .vtil jkt |..iiii<l on »lt tliiit :ir.'i..;. i.-. '.- ihr ti.^-fx ll..\ Aii'Liir ('..in|.iiiiy. of N'lw AllKiny. 1ml. r It.iv ;. r,..;.' \- . ■,.. „f l-.n^iklvTi \. Y,. Ii;nv p:.1i-iil...| l|„>„i„ ti..rs iiiirl .'.'.|. -i,:x', i„ ri:r I ]. I (!.,.y |,„v.. lavn lis. > ;i sM.T !l)>'. <'\'< II'. Til.' '':it> itiiriT- rilllll 'hr <iiii'l^ .':i|l [.rill<-i[1lll> ill llli- Miii-ii'ii i f I'iii- for 111,- |1l^.j,.Iill-ril. I'liii'li li..|iU llu' liliil- r>i. ll i- I'i^ii'.i,.] i)i:it III.. |.iii...i.>iiMt ,7iii<i' Ilii- liiiiN'r. (■K'li.'ck Hiiil 111,, liiiilnr- t.i il ,',|, K.iili..|- „nli,..-. r,.riiis ..f I'lifsulKl 1111,'linrs is sii|ii.rii>i' til lliiiM' in i-niiiiiinii um', Tlu'y iiiiisl iiul lie u*sl, hu«r- evcr, witLuut u IIuvH-w fruiu tlic jmluaUHM. FIBE-PBOOF OONSTEUCTION FOB BUIIJ)INQS. 467 CHAPTER XXV. MATERIALS AND METHODS OF FIRE-PROOF CONSTRUCTION FOR BUIIiDINGS. The terai fire-proof is applied to various kinds of buildings, sometimes correctly, but more often incorrectly. The buildings most generally referred to by this term may be classed as follows : 1st. Those in which all the structural parts, both on the interior and exterior, are of non-combustible materials carefully protected from the action of fire by fire-resisting materials. (See also quota- tion from Chicago building ordinance, page 485.) 2d. Those built on the so-called •* mill principle," and protected by fire-proof material. 3d. Those built in the usual manner with wooden construction, and protected by fire-proof material. Of these classes the first is the only one that is considered by experts to be absolutely impreg- nable to the effects of fire. MATERIALS. Various materials have been introduced for the purpose of mak- ing incombustible buildings, and for the purpose of fire-proof pro- tection of other materials in structural parts of buildings, all more or less effective. Experience, however, has shown that the only materials upon which it is safe to rely are the products of clay, some concretes, and lime mortar under certain conditions. Plaster blocks have been found to be useless to withstand the effects of fire, moisture, and frost. The lime of Teil was for several years used in the manufacture of fire proof material, but to the best knowledge of the writer this has been discarded. All methods of fire-proofing by the use of exposed iron in any form are also acknowledged to be ineflicient. Of all materials, burnt clay has the most numerous applications in incomI)ustible building. It stands preeminently first as the most efiicieni fire-proof material in all departments of building, and especially so for interior filling of floors and parti- tions. For this it is used in hollow tiles of two general kinds. Tliey are known by several different names : the one by such as porous terra-cotta, terra cotta lumber, cellular pottery, porous til- 40^ FIKE-PROOF COXSTRrrTTOX FOR BriLDIX"08. in^, otr-. ; the other by fire-clay tile, Iiollow pottery, hard tile, terra- coUa, <lonso tiliiip:, etc For convenience, the first is herein referred to as porous tiling, and the second as di.*nsL' tiling Tht» terms " hollow tiling "' ahd "fireproof tiling" will Ikj usid when Ixiih are r< IVrred to in ii general way. They will 'oe descrilxnl in Ihoir order. l*or<Mis Tiliii$4:. — A substance formed by mixing sawdust with pun' clay and submitting it. to nn intense heat, by tho action oi which the siwdust is destroyed. leaving the material Jijjhl an>l poroiw. like pumice-stone. When prop«*rly mailc it will not cnkk or br. ak Irom unerjual heating, or from l»eing suildenly cooUmI l)y water wIitMi in a heated condition. It can also bo cut with a 9av or edire tools, and nails or screws may be ea>ily driven into it for si-.urin;_' interior finish, slates, tiles, etc. For the successful r»'si>t- ancc (M li< at, and as a non-c(mdu(-tor. thei*o is no building nmterial «<iua! to it. A"^ a casing, covering, or lining for the protection of (•tiicr material, it is to 1h.» preierrcfl alx)ve every oihor material. li shnul.l bi* manuiactun'd from touixh. plastic clays. A small jHTi-i-ntaLTf of lire-clay mixed in is«h'sirable but not essential. Till- {)i'oporlioii of sjiwdusl .should be from forty to sixty per cent., jic- onling to toughness of clay use<l. ('are is nH|uin>tl in m.-iiiur.-K-i ire that the work of idxing, drying, and burning be i)i<ii'<Mi:.-|il\ •joMi*. The bui'uing should be done in down-iiniuglit kii:i> ii\ I .iek process. Tin* prcxhiet should Ihj compact, tmigh, aii'i ii.!;- :. riniring when struck wiih metal. Pixjrly mixed, pn*s«!iHl, nr Ii .riie j lil--^, nr tiles from >luiri or sjindy clays, present a nigged, ^r)^:. .iifl eruiiibly apjiearaiicc, and mv nut desirable. .\ ;i:- hPHii fllliui; and protecting material should lie substantial a- \\\: .- iti(-<i]riliusiible. In a building made of alisolutely inctini- li!i-:iM-- i:i;iieria!< it isnf the first i!n|)ortance t!iat the firi»-pnH»flng If- .iM- i<- \vith<ia!id niiii^h usage, for, in th«' event of Hn». daniat:!' to til -tnietural parts will lie serious if thefire-))rix)fingisdisliMi>:iHl, Hi'ls . I jiMvt. .ir yields to the aetii;:i of fire, or of waliT when a fire i> in prii;^' :•■>■■;. or if it- cullajises under sudileii liwiiN, jars, or imfiiu->. .-ilih<<u.:li !'-i(> nriier<;<| ii<. If may not burn a! all. In siicti huiM- ii;_- iji.liiiiMu' '!ualiii'>:. b-ith <if the Hre-pr'Mif material and its (•••n- -;■ . rii'ii. :■.'•>• as vit.il aii'l import mt as the incombustibility «if the Hi. I* rial, in the eveu' t'f !i"e. the fir>l ilaiiLT'-r i< fii»m the (-olla{x«* of !-. m.i'-ri.ii and imi frum its cundiusiioii. l'!x[H'rii'ner has sh-i 'I •■;.!' I'r*- j»r«M»| till- if p'a-iii- i-lav'*, w iieu jMirou.s an* iiiore enduriii.' ih.-tn den>'e tilcN, i>\cii it I In* deiiNC ii|i-> Ih- tif (larl «ir ail fire>-!'.iy. iNinais tiles are tough and ila.<tie. Men.<«e tiles are hanl and uriitle The most esM-ntial reipti^itcs of a fire pnKif filliugand FIRE-PROOF CONSTRUCTION FOR BUILDINGS. 469 protecting material are these : It should be tough, not easily shat- tered by impact ; non-expansive, not easily cracked by heating or cooling ; slightly elastic, yielding gradually to excessive loads, but not breaking or collapsing ; compact and hard burned, but not dense ; strong enough, but not of excessive crushing strength. Blocks should bo light weight by being porous, but not by having thin shell and webs ; should be built in between beams by such metiiods as bring all parts of the tiles into position to do the great- est service, whereby n. structural eflBciency equal to the efficiency of the material is obtained. These requirements are very fully met by properly made and properly built-in porous tiling. Shells of porous tiles should be from seven -eighths to one inch thick, and webs from three-quarters to seven -eighths, according to size of hollows. Dense Tiling;, — Next to porous tiling as a fire-resisting mate- rial must be placed dense tiling, also a product of clay. It is made into hollow tiles of much the same shape and size as porous tiling. A variety of clays are used. Most manufacturers, though not all, use more or less fire-clay, and combine with it potter's clay, plastic clays, or tough brick clays. It is very dense, and possesses high crushing strength. In outer walls exposed to weather, required to be light, it is very desirable. Some manufacturers furnish it with a semi-glazed surface for outer walls of buildings. For such use it has great durability, and effectually stops moisture. In using dense tiling for fire- proof filling, care should be taken that the tiles are free from cracks, and sound and hard burnt. In the earlier days of fire proof construction dense tiling seemed to supply the wants very well, but in later years the improvements in the manufacture of porous tiling have resulted in the displace- ment of dense tiling to a considerable extent. Concrete. — Concrete made of Portland cement mixed with broken pieces of burnt fire-clay, broken bricks or tiles, burnt ballast or slag, and clear sand, is said to resist an intense heat suc- cessfully. It is recommended for fire-proof construction by English writers, and concrete construction has been largely used in Cali- fornia on account of its fire-proof qualities. Thaddeus Hyatt, who invented the process of combining iron and concrete so as to resist transverse strains, describes a remarka- bly severe test by both fire and water, of concrete construction, in a work published by him. entitled, Portland Cement Concrete Com- bined mth Iron as a Building Material. The concrete was heavily loaded and heated red-hot on the under side, when a stream of water was thrown against it for a period of fifteen minutes, and 47l» FiI{i:-JM:()OI' CON'STKrCTlOX FOR uriLDixos. the stren«^lli (jr (lumbilitv of tin* (Mincrote nunaiiKMl unuircottHniv tin- tl'>t. l*ias(or, or IjIIIU* Mortar, wIhmi dirccily appliei I in brick or lilr. will witlistaml llio acliiuior hotli lire iiiid "Mhr; ;.lso \»!nu «j»I>Iie«l to tlir suitncc of j)l!inks ami tii!il»i'r> l>y im-juis oi win.- lalli- illu^ ]ir(A'icl('(l ii lill.s all thr spaci* l«!t\\'('rii lin- wiir ainl ilu' tiiiiiicr. JMa>t«':- oil win- Jatli, appliiMl to a ci'ilin.r "ii tin umh-r >i«lf uf \v«>(».lcn joisi s])a('('(l 1:2 or 10 inches on renin's, will sue(vs.-riiily ri "-i.-xi an, <»r(!inarv fire, 1 ml is lial)h' to Ik- dania'Ti'il l>v \v;'.ii-r. PlaNi.'r Itlocks are not siiilal)le as a tire-proof material. In usiiiir linn- [»la>tir ri>r fire-ju'oof proti-etion. il slionM not tM)niain any j)l.i>ii'i' of Paris. l>ri<'k and St oiio.— Common brick will wiihtJtunW a u'nat anii'iirit of Ileal wiiliont malerial (lania«;e, tlion^'li mil in so j^real a (ie;;n'i- a- liir brick, jiorous terra-c<itta, ami lire clay tile. S«»nif >:iml>i<'nr> «io imi appear to be mncli aifccied by heal, csin'ciiJiy ijii.^f c' niaininLT «'«>ii>i«leral)le iron. Marl)le, limcsiono, an«l .i;raiiitr biiiiiiH- .•.»iiii)j.t»M\ desir«»ve<l under till- ai-lion «:f inien*^* heat and water, and .-liniiid luu be useil in pla<'es when I In- stability <il" the biiildiiiLr \\oiild Ix' endan*;* re<l by ilielr demoliiion. Terraroiia is undt'wbi.'dly the b«si iin-proof material I'ttr ilie i-xieri<»r <leeoratinn <»l biilldiii:,'^. MKTIIODS (H-' lOXSTUrCTloN. ]. niilMiii'^'s (*oiistrti(*t<Mi ol* liiroiiilMistihh* Mat€'- riaK projirrly l*rot ortod.— 'i'he mellnnis of construe: iin; iii< ir-'H I l-iiildJMu^ liavr been Lrr»'ally improved during: tiic past li-w . ■.!!>. alii: i-t •■•iinplilcly !e\ olul ionizJn;; t lie old inetlaNis o] build iiij. Tl.- id«al liri priM)f buildin;: should Im« con^tructid cuiinly *'\ iidii iir >i-'.l. drf>-M'd <in the oulsidc with lirick. sanil>t<»ni-. or lci:a <■■ ta. and j.roU;cle(l on the inside by Iiri'-pnK)f inatt-rials. Til- n.o-i apjirovj'd metiuxl of c«inslrui-iin^ hijrh buililini;s is lo ■■liiM tlif loundation (U the i>olal«-d pii-r >y^ti-m. and oii top f)f liicNi- pifis place >ieel nr w rouirlil-iron cobimi; cMendiui; t)in>UL;h (^■■i!itir< i- '-.dii lit till- bnihfiinr. bdilinn i he n .t-ide walls and in *ihi' iiiii ijiii" of ihe b^ildiiiir Al ca<!i lln-ir hvel ifoii Lrivdi-r> an* bolted I"'!,.- ('(.liimn-. anil llh* whnle sv«.tem braceil bv diair'Hial tie> in tin- ihn-kiii-N'* nf the Ihinr. 'rhu< i< f.irmed an iron «•!• ^teil ca;:c rt>liiiL' ' uiirely mi t! .- foundaliitn piei"s aiiu 'hich. sn l«»ii;: as ii can bi-kij-t irDiii lIn- a>-iioij of heal and iiMii-tiire, will endure fi»n-vi-r. 'I'll'- ••lii'.iile «;dl- are then biiili nf lirick. shine, nr lerni- cutla. cn<lo>iii.; il:e biiiliiiuLT and proiectiii^ lis contents fruin thv FIRE-PROOF CONSTRUCTION FOR BUILDINGS. 471 weather. Only sufficient, strength is required in this wall to with- stand its own weight, and if any of it should be destroyed it would not cause the destruction of the building. The interior columns should be encased by porous terra-cutta or fire-clay tiles, finished in plaster or Keene's cement, or Portland cement if preferred, and the floors should be constructed of iron beams filled in between with tile arches, the bottom and top of the beams being carefully pi'otected by the same material. All partitions for dividing the various floors into rooms, cor- ridoi-s, etc., should be built of fire- proof partition tile, or hollow bricks, and the roof and upper ceiling should also be constructed of the same material, supported by iron-work. In such a building it is impossible for the construction of the building to be en- dangered by either a local fire or by a conflagration, though the inside finish may be entirely consumed. It is possible, however, to finish the building in such a way that there will be but little wood to consume, which could be easily replaced ; also, by provid- ing fire-doors to the openings in the fire-proof partitions, any fire originating in the building can be confined to the part of the build- ing in which it started. DETAILS OF OONSTRUOTION. Floors. — The various approved methods of constructing fire- proof floors have been described in Chapter XXIII. Iron Columns. — The destruction of iron columns by in- cipient fires has been the common cause of the loss of vast amounts of property ever since iron columns have been useil. Their destruc- tion during fires, in buildings supposed to be flre-proof and in which incombustible materials of construction have been used, has shown the necessity for protecting them from the effects of intense heat under all circumstances. These disastrous effects have been intensified by the sudden throwing of cold water upon the heated columns, causing them to bend suddenly by contraction on the side upon which water is thrown, and consequently to break with ordinary loads. The expansion which occurs in iron columns before they have bec^n materially weakened by heat is another cle- ment of weakness. The first result in such cases is to raise the floors or walls ; and inasmuch as the strain required to raise them is much greater than that needed to hold them, the work to be done by the columns is much greater under such circumstances. The almost universal practice at the present day is to use wrought-iron and steel posts for the interior supports, and protect FIBE-PBOOP CONBTKUOTION FOE BUILDISOB. 4T3a the floors, the aams material vill generally be beat lor protecting tbo girders. Fig. 6e shows several wsjs in which this maj be Fib. Oa, -'Two-foot Coluhh CovuuHae ur tdb Fab«t BinLDDre. © Pio. ».— Section or CisT-raON Comm Fib. tc.—Vax-SBoor SountB Cots A'i'lh FIRE-PKOOF t'OXSTUUCTION FOU BUILDINGS. e:5.¥5".W VS* Toa\v\vo^'Yv\v-»«.\oaa****^ Partitious. The method at present most in favor for constructing fire-proof partitions apiieai-s to i)e by tlio use of hollow blocks or tiles, of either dense or |)or()U£> terracotta. Partitions arc sometiinct; built by using 4-inch isteel beams for studding, and fastening metal lathing on each side ; but this is not as practical a iMir- tition as one made of torra-cotta blocks. Par- titions constructed of terra-cotta blocks, either donso or porous, have many vMluable features other than their tire-proof qualities. They have the greatest degree of strength combined with light- ness. They are entirely vermin pro«jf, and do not reatlily transmit cold. heat, or sound. Wiien dense tile are used, courses of porous tile should be placed op|)osite the l^ase or any wood mouldings, as they will receive and hold the nails while the dense tile are apt to be ))rokon by the nails. Several styles of partition blocks are mana- factiired. of both dense and porous terra-cotta. some with grooved or (love tailed surfaces, and others with plain surfaces. Tiie weiu'ht of ))artition tile per square foot will average about as follows : WF.IGIIT PER SyrARK FOOT OP TERRA-COTTA PARTITION BLOCKS. Den-sf T«Tra-cotra. \Vt. per Ml flH>t, IbK. :j inches thick 13 4 '• •' j 17 't ..... «w r» •• ! -JG 7 *■ •• i w^ 8 •• •• ' :« i: Porous Ti'ira-cotta. Wt. per nq. foot, IImi. 3 inches thick 4 m h • • • I 5 " •• ' « •• •• ] 7 ** " ! « •• •• 12 17 21 26 82 38 Til ill rin»-pr<H>l* Partitions.- Tn a considoniblt* extent in finicf l)uililiiii;s. sonic hotels and apartment hi>uscs, iiartitioQS are n>i'v uscil which flni^ih fninioneatid thn*e-«|uarter inches to two and thri'c-<|iiar-tcr iiiche<« in total thicknesb. There are a number of dif- ferent dcvii-es and methods, all accomplishing substantially Um FIRE-PROOF CONSTRUCTION FOR BUILDINGS. 472c same results. Prominent among them are the expanded metal companies, using cliannel bars or flat bars and expanded metal lathing' ; the Lee Fire Proof Construction Company, using a core of one-inch tile, and burying Lee tension rods ^similar to those used in the flooi-s) in thv; plastering on each side ; the Doring Fire- proofing Company, using rods, bars or channels, and burlaps ; and the two-inch porous terra-cotta [)artition made by Henry Maurer & Son. The expanded metal system requires a scratch coat of plaster- ing on one side, the usual brown coat work on each side, and the usual finish coat on each side — altogether, five coats for the com- pleted partition. The Lee and Maurer systems require no scratch coat, but the usual brown coating on each side, as done with hard- setting mortar, and the finishing coats. The Doring requires a scratch coat on each side, and then the usual brown and finishing coats. An essential thing for all thin partiti<ms is that the plastering be of hard-setting mortar, such as Acme Cement, King's Windsor, Adamant, Rock Wall, and many others. The walls largely acquire their stififness from the solidity of the plastering ; hence the firmer and harder the plastering, the more substantial the walls. Roofs. — For mansard roofs the most economical method of constniction is by using I-beams, set 5 to 7 feet apart, and filled in between with 3-inch hollow partition tile, provision for nailing slate being made by attaching 1^ x 2 inch wood strips to the outer face of the tile, the strips being set at the proper distances a[)art to receive the slate, the spaces between the strips being then plastered flush and smooth with cement mortar. In case of a severe confla- gration the slate would probably be destroyed, and the wooden strips might be consumed, but the damage could go no farther. In place of partition tile porous terra-cotta bricks or blocks may be usi»d for filling bc^tween the I-beams. For roofs where the pitch is not over 45 defjrci^s, 8x3 inch T-irons, set 10 inches between cen- tres, and filled in with slabs of porous terra-cotta, makes a very desirable roof. If slales are used they may be nailed directly into the tiles, or if it is (h^sired to use hollow tile, strips of wood may l)e nailed to the tile for receiving the slate, and the spaces b'.^tween the strips filled in with cement. This method may also be used for flat roofs. The b(?st construction for flat roofs, however, is to build the roof like the floors, with tile arches between ircm beams. The arches should then be covered with Portland cement, or rock asphalt, flashed around the edges with copper, and then tiled with terra-cotta tile, about 0x8 inches, and | inch thick. This makes a durable and substantial roof, perfectly water-tight and absolutely FIRE-PKOOF CONSTRUCTION FOR BUILDINGS. 473 proof against fire. Composition, cement, and asphalt have a natural affinity for the tile, and adhere readily to it without the use of nails or fastenings. If the roof is exposed on the under side, it can be plastered and finished the same as the under side of a floor. TriisscM. — Where steel trusses are used to support the roof or several stories of a building, it is very important that they be pro- tected not only from heat sufficient to warp them, but so that they will not expand sufficient to affect the vertical position of the col- umns by which they are supported. The following description of the covering of the trusses in the new Tremont Temple, Boston, furnishes a good illustration, of the way in which this should be accomplished : *' The steel girders were first placed in terracotta blocks, on all sides and below, these blocks being then strapped with iron all around the girders, and upon this was stretched expanded metal lathing, covered with a heavy coat of Windsor cement ; over this comes iron furring, which receives a second layer of expanded metal lath, the latter, in turn, receiving the finished plaster. There is. consequently, in this arrangement for fire protection, first a dead air space, then a layer of terracotta, a Windsor cement covering, another dead air space, and finally the external Windsor cement." Ceilings. — In office buildings having a flat roof, there is gener- ally an air space, or attic, between the roof and ceiling of upper story, ranging from three to five feet in height. This space is often utilized for running pipes, wires, etc. Generally the ceiling is constructed in the same way as the floors, with the difference that lighter beams and filling are used. It sometimes occurs that a suspended ceiling is desirable under pitch roofs, to form a finish for the upper story, and protect the roof construction. If only the weight of the ceilmg itself is to be provided for, such a ceiling can be constructed at least expense by u.-^ing wire or expanded -metal lathing stretched over light T's or angles, suspended from the roof construction. The angles or T's may be plac^ed four or five feet apart, and tension rods fastened to and under them, to support the lathing ; such a ceiling would weigh only about twelve pounds per square foot. Plaster boards or thin porous terra cotta blocks, placed between T bars, also make a light ceiling, and a goo<l ground for the plaster. Walls.— If it is desired to further outside walls they should in DO case be strapped with wood, but should be furred or lined with porous terracotta or fire-clay linings, as shown in Fig. 6. on which the plastering may be applied. This not only affords a protection FIBB-PBOOF C NSTBUCTION FOB BtJILDINim 476 blocka, the same as described under Class 1. Id this method of buitding it is also neciessary to protect tiie upper side of ttiu floor planii. olticrwiKc tiiu fire would burn tliroug-li Cri>ui tlie top. This is best done either by laying; an inch uC mortar between it and the upper floor, or by using liollow tiln blocks laid on top of the plaitlt- ing, with strips between lor nailing the upper flooring to. The flrst method is much the cheapest, and as fire is very slow in attacking a floor, suuh a construction would probably resist the ac- tion of the fire as long as would the other portions of the bnildlng. The first point attacked by any Are is the ceiling of the room or story iu which it onginates, and every precaution must be taken to Pio. 7— Mill Cohhtbijction, Protkcteu by Plabtbr on Wire LiTHrao. make the ceiling imprepiable. Espoeial pains must be taken to see that all angles and junction of L-eilings with the walls and parti- tions are carefully protected, so that there may be no places in which the flre may work its way through the protection back of the plastering. Partitions. — The partitions in this class of buildings shonid be constfueted either of hollow tile partition blocks or bricks, as in Class 1, or they may bo built of 3-inch plank, tongued and grooved, and covered both skies with wire lathing from floor to ceibng, and back of the door jambs. The Walls should either be plastered directly on the brick- work, or furred with hollow tile blocks, as previously described. VTbea carefully built, a building of this kind will be practically PIBE-PROOF CONSTRUCTION FOR BUILDINGS. 477 Comigrated-wire Lathingr consists of flat sheets of double-twist warp-lath, with corrugations ^ of an inch deep running lengthwise at intervals of 6 inches. These sheets are made 8x8 feet in size, and applied directly to the under side of the floor timbers, to partitions, or brick walls, and fastened with staples. The object of the corrugation is to afford space for the mortar to clinch behind the lath, and at the same time do away with furring strips. The corrugations alto strengthen the lathing. This form of lathing, however, is not as desirable as those fol- lowing. Stiffened Wire Lathing. — The Clinton stiffened wire lath has corrugated steel furring strips attached every 8 inches cross- wise of the fabric, by means of metnl clips. These strips constitute the furring, and the lath is applied directly to the under side of the floors or to brick walls, etc. This lath is made in 32-inch and 36- inch widths, and comes in 100 yard rolls. The New Jersey Wire Cloth Co. also make a stiffened wire lathing by weaving into the ordinary 'wire cloth V-shaped strips of No. 24 sheet iron every 7.V inches. This is an excellent lath About the only difference between it and the Clinton cloth is that the bars in the latter are attached to the cloth instead of being woven in. Hammond's Metal Furring*. — A combination of shoet- metal bearings with steel furring rods, on which ordinary wire cloth is applied, makes one of tlio best fire-proof ceilings. By means of this furring the plaster may be kept an inch from the bottom of the timbers, thus allowing a free circulation of the air over the ceiling. It is claimed that t!iis is of importance in connection with fire- proofing, and is required by the building ordinance of the city of Chicago. The steel wires used for furring are fo small that the mortar entirely covers them, thus securely binding the cloth and rods together, greatly stiffening the ceiling. This method may be applied to any form of construction. Slieet-iron Latliing. — A number of styles of sheet-iron lathing have been invented and placed on the market, but they are objectionable from the fact that, in case of fire, the heat expands the iron and contnicls the mortar, so that the latter becomes sepa- rated and f:ills off. Even without considering its fire-proof quali- ties, sheet-iron latliing is not desirable, as it is difficult to get a good clinch on the mortar, so as to securely hold it in place. In the wire cloth, the amount of metal in the strands of wire is so sniall, and it also becomes so well l)edded in the mortar, that the action of intense heat does not affect it, and it has been practically demonstrated, both by actual fires in buildings and by fire tests, FIBE-PBOOF OOS&TRUUnoS FOR BUILDINGS. 4 79 elftborate decoration is to be applied, as it affords a much better sur- face than any other material. The upper surface of the floor must also be protected, either by putting an inch of mortar between the under and upper floor board- iug, or by filling in between the joist with fire-clay bridging tilo, or by brick nogging and covering with cement mortar, on top of which the upper floor is laid. As in the previous class, especial pains must be taken to see that all corners and angles are well protected. Roof. — If the building has a flat roof it should be protected the same as the floors, substituting for the upper floor boards, composi- tion roofing covered with flat tiles laid in cement. For steep roofs, efficient fire-proofing becomes a difficult problem. In the opinion of the author no building, five stories high or over, should be cov- ered with a pitch roof constructed of wood ; but if such a roof is used, it can be protected for a time by covering the roof boarding with porous tena-cotta blocks, aoout 15 inches square and 1^ inches thick, and nailing the slate directly to them, bedding the slate in cement as it is laid ; or the tile may be nailed to tiie rafters without boarding. For protection on the under side, if the attic space is finished, the under side of the rafters may be pro- tected as described for ceilings ; or, if the roof space is unfinished and more or less filled with trusses or other supports, a thoroughly fire-proof ceiling beneath, without any openings, would probably be as good a protection as could be obtained. The walls and partitions should be treated as in Class 2. Complete information regarding the particidar forms and sizes of the various fire-proof blocks inanufactur(Ml may be obtained by ad- dressing The Raritan Hollow and Porous Brick Co, , of New York City ; The Wight Fire-proofing Co., of Chicago or New York ; 2'he Pioneer Fire-proof Construction Co., of Chicago ; Henry }Iaurer d; Son, New Tork City ; 2 7ie Lee Fire-proof Construction Co . N(r.v York ; and llie t^taten Island Terra Cotta Lumber Co., New York ( ity. Details, Finish, etc. After tlie constructive portions of the building are completed and the building is plastered, there are yet many details to be arranged, so as to afford the least possible material for a fire, and also com- bine strength, durability, and often elegance. Stairs.— The most important of these are the stairs, which, owing to the necessity of their being located in a sort of well or shaft, are always fiercely attacke<l by a fire. To construct a thoroughly fire- proof stair is nither a difficult undertaking. Many architects con- tent themselves by merely making the strings and risers of wrought 4S() FTKK-PnOOF OOXSTKUCTION FOR BUILDINGS. or cMst-iron, and Wm'. treads of slate, marble, or wood. Siic-li stnir: ?!'(• iiTiil«.ul)tcdlv UiY better than the ordinarv wooden stairs, but liicy ni't' iiKMcly iiUM)mluisiibl('. In biiildin^i: such stairs \vro!i|Erht iron string- sii( uld hi' ust'd with slate tn-ads ami iron ris«TS I'wi iw in .h I lianiirl l>ars inak<' excellent strings, turning llie ilanL:«'> <'U.. anl i;«)ilinj,^ tlic n>eis to ih-. stem as shown in Fi^. U. Tile 1« <t >iair> lor a fir('-])ro()l" biiildiiiL; are tliose built of lirit-k DV Portland «i'nu'nt t'oncn-tt', witii at least one end sujijiortrd hy a Itii.i; Wall. If coinM-ctp >tairs an' constructed llic;* should b*- built :;'iuari- .-.nd xijid — thai i. , liaviiiLT the same sliapeon the bottom i\-nm ihi' loj,. II tin- stairs an- liuill Ix'twciMi two brick walls, as ^iioiilil alwa\s :•-■ ihc (•;i>.- in a thi-atn'. thrv will have sullieient strenu'th bv ixiiiidinj- ill. in 1 inc!:e< into the hrick wall. If only one enil i*: ^uj^ iM'iird l»y M wail, iin- other end can be su}>iM>rte(l by wrou,trhi-in>r =iriMj-- i.iiili int.. tiii- (M.neretf. ricT. 1 f ^liow> tw<» M-ciions of -i brick stairway. Stairways ^iI:l; I.-! :. : Fig. 9. SECTION OF WKO-^v.*H7 il.ON STAIRS. :■ i!i li- ill the ?i.w Pi-nsii'ii liniiilinj; at NVashincton A.i\ ■::.i ■. I ■.:--iiii'-.'il .1^ ;i.i>»«.lut» ly fin'-pniof Ni \f l« .1- Nt >"iii-^. ih-- ainiii'!' U"ulil n-i-ommcnd stairs ntu- I ■' v. . n ..'• ,•;.-.: ;!•■ !i "-iriiiL's. I'n'ifeti'd •*!» I !;t' uiiii' I .■ :• :'. - ;■.■:;.':■•.:.-;■ ni-io- !a lii . ami with >l.iU- ...■'. »; - _; -!■. \ : :;■ w :!i tili- r j! i-»Ti-.- ■.:! Ill^•.!d^ li' II I '1 ;;«!-' ha\<' Jw.M t'"UM'l u!nli-»ir- !"■ I".,. ":'■■ ^ \. •■". «.; w :,i :|"::i \ ;; -."iimtl: 1. :l III' !- ;! an- "-j'.i'-i- bridri'i r. «.o 'ii.-it i- I'll- .I'- \ .!'. «•■ .Id !•<■ far InMit ihiiH '.■ .■■'.- :■ . ! -•■ i: ■■. !h-. :_-!i xn h a ^-tairf. Thf •ilrinir* '.'.•■ ! i . : ;■-' . Mi'ha^ •■•■i It >>rnaiueutatU)n <ii';tMilr >lair>. li: ii..iiiv •: !i.> (i -vi rniiii-.ii buililin;r'< tbr FIEtE-PBOOF CONSTRUCTION FOR BQILDINOS. 481 stairs ftre constracted all of granite, a seotion throni^ the steps be- ing like that shown in Fig. 12. One end at the steps is boilt into wall, and the other depeods upon the edu^ol tlte steps for support Granite and most other natural euin stroyed b; the action of fire and water, s my be coiisidered as fire-proof. As to tho stair railing, if bnckstMrs are used, st - - . ' .tf - \^ :•.♦.»:•> ♦\ r.- :. •• iJ. :. . -' •* • B B k • M < ^ y Fie,, n. F.'-l '-'-ZZt :• : • : :.;• : « r.»- w,. l»y • •-. ' K'\ I'.i it • -.ii r. :i~ A > . '■• I y :>»•-: r - f. \ (lit il:i( ion :iii(| l|ol-:iir l'!iir«». 1''.— -".'i'"!!-! \\\ \\\\ '■•••■':••••; . !' ! ".■•'••- *• :!•' ••■ '.ikMn that I •:.:- <!..-" A .. :•• i;ia;f«i iiif.i-r bv slt-aiii or huC 1 il..- U>f »'M-tl.«t.U fi.r heiitiii;: •>nice> iit dvsicribed PIRK-PRX)P OONSTRUCnON FOR BUILDINGS. 483 In the article on Steam-Heating, under Direct-Indirect Radiation. If this method is employed, no hot-air flues will be needed, and it will only be necessary to provide for ventilation flues. In running iron and lead pipes, etc., in the walls and partitions, they should run in channels in the brickwork, and be covered with d Fig. 12. SECTION OF GRANITE STAIRS. (SELF SUPPORTING.) sheets of boiler iron about three-sixteenths of an inch thick, put up with screws, in an iron frame fastened to the brickwork. This can be painted as desired, and afford ready access to the pipes. No pipes should be carried in a wall or partition where they are not accessible. In finishing around elevator doorways, etc., where considerable ornamentation is required, cast-iron, painted in color, can be used with good results. Where there is no combustible material, there can of course be no fire. Cement Fig. 13. SECTION THRO' DOOR JAMB Stand-pipes. — A very important adjunct to every fire-proof building is a stand-pipe of 2-inch wrought iron, connected with the street main and running up above the roof (if flat), and provided on e^ch floor with suitable valves, hose, etc., ready for instant use. PIRB-PROOP CONSTRUCTION FOR BUILDINGS. 486 thousand square feet, without special permission, based upon un- usual and satisfactory precautions. 6. That every building to be erected, which shall be three stories high or more, except dwelling houses for one family, and which shall cover an area of more than twenty-five hundred square feet, should be provided with incombustible staircases, enclosed in brick walls, at the rate of one such staircase for every twenty-five hun- dred square feet in area of ground covered. 7. That wooden buildings, erected within eighteen inches of the line between the lot on which they stand and the adjoining prop- erty, should have the walls next the adjoining property of brick ; or when built within three feet of each other, should have the walls next to each other built of brick. 8. That the owner of an estate in which a fire originates should be responsible for damage caused by the spread of the fire beyond his own estate, if it should be proved that in his building the fore- going provisions were not complied with. A certificate from the Inspector of Buildings 4hall be considered sufficient evidence of such compliance, if the building shall not have been altered since the certificate was issued. In addition to these general propositions, another series of sug- gestions was adopted, providing for proper fire-stops between the stringers in wooden stairs, and between all studdings and furrings, in the thickness of the floors, and for six inches above ; for car- rying brick party-walls, and outside walls adjoining neighboring property, above the roof, and for anchoring* wooden floor-beams to brick walls in such a way as to prevent the overthrowing of the walls in case the beams should be burned oft and fall. Chicago Definition of Fire-proof Construction. "The term 'Fire-proof Constracfion ' shall apply to all bnildings in which all parts that carry weights or resist strains, and also all stairs and all elevator enclosnres and their contents, are made entirely of incombustible material, and in which all metallic structural members arc protected against the effects of lire by coverings of a material which must be entirely incombustible and a slow heat conductor. The materials which shall be considered as fulfilling the conditions of fire-proof covering are : First, brick ; second, hollow tiles of burnt clay applied to the metal in a bed of mortar and constructed in such manner that there ehall be two air sp ices of at lea-<t three- fourths of an inch each by the width of the metal surface to be covered, within the said clay covering ; third, porous terra-cotta which i^hall be at least two inches thick, and shall also be applied direcT to the metal in a bed of mortar ; foarih, two layers of plastering: on metal lath.*' JQA WOODEN ROOF-TRUSgKS. CHAPTER XXVI. WOODEN ROOF-TRUSSES, WITH DETAIIiS.! WnK.vKVKR it is rt^uired to roof a hall. room, or ImiMing. where the flt'ur ST»an is inon* than tweiitv-tive f*t*t. the roof should be siil»l»ortf 1 hy a truss of some fonn. Tlie various forms of trusses uslmI tnr tliis ]iuri)ose have e«*rtain ft-atures anil ]>rincip]es In fom- luon. (litfcring from those in bridge and floor trusses, which have ^ PlC;R seam CR CEiw So. J S' V^ — >r rLATE SPANS UP TC 2-4 -T II i»tl In L'lnr.i.iiii: tlu-m in on»» rlas'S. r:illi*»l " n^of-tnisses." Xeariy all II ■i!-Tn>^»'^ in r!nir«lif>. ;i3! ! li.ill-. n! like rhHRi"!er. an«l Ihe l.ii«'. r i:«i"."riiiiM ••! ini-»«>t-s usi- 1 in :ill kin.l'* of hiiildin;;^ an» itm- >tr;i<-t<l '.•riiii-ip.tlly iit wi^tid. ^\ith unly iron tii'-rods anil Uilts ; aii I. ;i«« \ii>. I li 11 iru>'*i"* an* nt nn»n- inifn'sl to \\\o an'^.it«^-t and liiiii !• r r-.ti; irnu tru*"'-'*. ilii-> hn\*' Ui-n nion' ittini^lctely d«*- Miii*.- i. .i:\ \ .1 Lin-jiTiT \:»rifT\ ni lorm'S an* irivon than for inm 1:. j.r .; t-> ^t. V. : ihi -.:i:-. -.:« lr<i'»'i-> »!.■ u: .m- .i ! iir:iu!i i>litfhlly iNil of u \: iM il.i \ .I't- ji'ii.ni i> .:>-ihiT. Till- tiii»'M-i> (huii kiok hmvy tai ..«.- : 'ilii :iii.'-%:. tul the rtUtii'ii «.>f lliv \.irio>i» WOODEN ROOF-TRUSSES. 487 roof-tmsses, which are. discussed in another chapter. In the Northern States and Canada, where there are often heavy snow- storms, experience has taught that the best form of roof for a building, except, perhaps, in large cities, is the A, or pitch roof. The inclinations of the roof may vary from twenty-six degrees, or six inches to the foot, to sixty degrees, or twenty-one inches to the foot, but should not be less than six inches to the foot for roofs covered with slate or shingles. For roofs covered with com- position roofing, tin, or copper, the inclination may be as little as five-eighths of an inch to the foot. PaiNOIPAL RAFTER ACK RAFTER CEIUNQ JOIST SPANS FhOM 2C TO 40 FT, The simplest form of pitch roof is that shown in Fig. 1. It con- sists simply of two by ten or two by twelve inch rafters, supported at their lower ends by the wall-plate, and holding themselves up at the top by their own stiffness and strength. A piece of board, called the "ridge-plate," is generally placed between the upper ends of the rafters, and the rafters ai-e nailed to it. In some locali- ties this ridge-piece is not used, but the upper ends of each pair of rafters are held together by a piece of board nailed to the side of the rafters before they are raised. The walls of the building are prevented from being puslu'd out- ward by the floor or ceiling beams, which are nailed to the i)late. The rafters are placed about two feet, or twenty inches, on centres, and the boarding is nailed directly on the rafters. The horizontal joists support the attic-floor and the ceiling of the room below. Such a roof can only be used, however, when the distance between the wall-plates is not more than twenty-four feet ; for with a greater span the rafters, unless made extremely heavy, will sag very coni^derably. i^- WOODKN KOOl'-TltrsShS Kin^ Post Truss. — WIumiovit wv wish to roof a hiiilrtin^i ill wliirh till- wall-iilat«'*< an* iiiort' tlian twniiy-four ftM-t aitait. wv iiiU'n; a-lojii sniiH" uM'tlunl for sui»]i«>niiiix tlu" rat'ti'i*s at tlic «"«'ntn'. TIm- iiii"li»».l ::«'ii«M\!lly fiiP'loNfl i^liDwii ill Kiir. -) is to iisi- Iras'"!- ilk.- ili.i! ^linwn ill ill'" liiruii". ^'i.n-!-! n'miir i\vrl»r t"«-ft apart in ih- li-ii:::); nt till- lniiMiiiLT. aiitl on ilii-M' iilar** larm* lifauis, i-alli'il ••imr : '. R n 2 « 8 CE-LIN3 J C o T Fiu.3 I.M'." \\i::'i. -:r.;i»i': iln- i-ooi". ur jark-raii«-rs. A^ tin* iIistani-<' from !■:. ■; : ' •; :.i :':••• :m\! i«. ni»l L't-m-rallN niun* than six or riirht fffi. i!.'- :.ii-..-:.:;:- r*- ni;i\ l-i- nia«li- a** >niall as iwn in'-hr*; hy six iih-hr". \\ .i»i! '!.■■ ^'.•:\\i 111 ill'- ti-n>s is innn- than tliirly-foiir fiH-t. t\vi» jijir- l;ii> ii.:«:.: in- ]-lai-r.l um «-a«Ii >iilf i»f ihi- iru^N. or at -I and .1. It i'. aiv.;.>- i" -:. hi'\\«'\ir. !•» I'hu-i- ihr pnrliiis only ov«'r tht* finl iif a LiM-'-. "! :o a jniii}, w ln-n it fan In- so arranu'fil. Tin" «"fihni: of tin ri...!M ii«\< !■. I !•> IJM' r«i'*!" i> franp-il witji li:;in juisis su|iiiortiil li\ W •* ■'!■ iM.-^. 1 ii' -• • < . ".1.^ 'ni-l" '»!ii'"!M ni»! hi* iiiiir • '■ . I-::' ^"i." . ! i- ■:••:. .1 ! ■-. ■»-;u<-li h\ funr-ini-li !.■■■ ; .' - ■• III! .1* "■.••v.!; \',. l.^. •■. • ■:.' ;::- - \- ■■:-::.::■. ;;v. I.. 1. ii i- ilitlimll ■:■ ■■:-!■■■.» •:.«■;_!; I--' : I.I r •.-!•■ -.u., uiihtiut spliriniZ. .1- t.".- I.- ;|m- I>i -• ii!i'!i«>-U ■■? liuilihiiL; up Iht' lii»- WOODEN BOOF-TBU88B8. 480 beam is lo make it of two-Inch plank bolted together, the pieces breaking joint, so that no two joints aha!! be oppoait* each other. This form ot truss is very rarely used where the timbers may be se«n from the room below, ami they are therefore generally left rough. If they were to be planed, and maile a part of the finish ot the room below, it would be necessary to use solid tie-beams spliced together, or else build the truss of hard pine, of which wood, timbers may be obtained fifty or sixty feet long. The form of truss sliown in Fig. 2 la the modem form of the old king post truss, shown in Fig. 4, which was made wholly of wood, excepting the iron straps used to connect the piece* at tlie joints. Queen Post Truss. — When the span to be roofed is between thirty-five and forty-five feet, a truss such as is shown in Fig. .i is pref- erable, for several rea- sons, to the king post It consists of a hori- zontal straining-beam, separating tlie upper ends of the principal rafters, and a rod at each end of the strain- ing-beam, leaving a large space in the cenr tre of the beam clear. This is a great advan- tage in many eases where it la desired to uliline the attic for This form of trass should not be used for a si>an of over forty feet. For spans from forty feet to fifty feet, anotlier form of the same truss, shown in Fig. 6, should be used. This ia a very strong form of truss, and leaves considerable clear apace in the centre. In tliis truss the principal rafter should be made of two pieces, — one running to the top, the other only to Qie Btralniag-beam. This gives the greatest economy in construe- *. -* .« •".. I. I •'■_ •■ \ I ..I i. :' I ' I : i..«. vi .1" ■.j?':ii.r.u i jVAit-r .omr it 3. It ^huiiiii V .. ..'i '.■.i\ ■,.«• -111*-;:-*:. •'" 1 'i"i.N-i iiv.iHini*; !iuir«'!v i|.,,n ^ . • ■- '• '« ^ I.'- .i/'.'n*'! "••ti"! .ii'!'. iiiti iiar I ■;"'i>- • V . , -.> -....-i: .: >■.•- • *,.«• Ill .■ii..i:i:ii:> n .'..mi :• •: '»■' • • -« «••' ^ ••■'.4i I \ - ■■••!! I ' • • • •■ I :: ' . 11" V I • 1 • > • I .. -I*... :• unci jiuiui til. WOODEN BOOF-TRnsSBS. 491 1 an enUrged detail of It is shown in Fig. 10. This tniM la m the Museum of Fine Arts, St. Louis, Ho., Hessn, Peabod; & ams, architects, Boston, Moss, )'0F Bpana of from forty to eighty feel, a truss such as is shown In ;. 11 ts one of thit best tonus to adopt, where a pitch I'oof is rhe strutB should be largest towards the centre, and ttie tie-rods a. rhe main rafter, on the contrary, and the tie-beam, have the stMt strain at the joint A. Figs. 12 and 13 show details of 192 WOODEN ROOF -TRUSSES, The tmsses which have thus far been given are the simplest forms of nioilern trusses for spanning ox>enings up to sixty or seventy-five feet in width, or even gn?ater, wliore it is <li»- sired to liave a pitcli roof. At the present <lay. how- ever, flat roofs are very ex- tensively used; and, when i; is desired to carry a flat ro«if. a different fomi of truss iiill be found more economical. WOODEN ROOF-TaU88B8. 498 The form of tnua generally employed for SM roofe is that shown in Pigs. 14 and 15. This truss may be adapted to any span from twenty to one Imndred feet, b; simply changing the height of the truss and the number of braces, and proportioning the various parta to the strains which they carry. Tlie hciglit of the truss be- tween the centres of the chords ought ^^ not to be less than one-eighth of the span, and, it possible, should he made oiie-seveiitli, as the higlier the truss, tlie less will be the strain on the chords.' It shoulil be noticeil, that in this truss the braces are Inclined in the opposite direction to that in which t);oy are placed in the •s r -: wrXjI/ZN R«X)r-TEV>SE5. F.>'. ]►'■ -:.■ \^- ::-►- l-<t iiiKhr"! of fomiinc th»* joints. A, A, A, Ji. /;. li. ••*'■. F;j. 1.' . A!*.ln»i:::ii in.'t '.rn" frt^iumily used in roof- •..- F".- -Tia::> ov»-r i"rty f»-»-:. th-^ ii».*-N*am should be made up or ]•. i.vk l"^;*.-'l :«.iL:»-tli»T. as sh'.»wii in Ki:;. ;\ unless it is possilile •'. }..t". - Ml- ?:t-]i*-aiii in nn*- j-i^-'-^-. This :s a ::iH"1 form of truss for *!.»-atr*-. ai-'i larj*- hall-? wli^-n- ih-r»- i> a hi.>rizontal ceiling. < 'oiiiif «T-llrjirrs, — li it i^iliNJrri! tn Iiku) thf tniMH at any iHiiiit u;lii-: liiaii tli*- rcMtn' uith a rtiui'i'iitralnl Ioa«l, — as, for in^tan<-(-. ^ii»pt-n<liii^ a :;all«'ry hy UK-ans uf rv*\a from the roof- K)D£N ROOF-TRUSSES. 495 trasses, — the trass should have additional braces, called ''counter- braces," slanting in the opposite direction to the braces shown. These counter-braces need only be used when the truss is unsym- metrically loaded. Wooden 'I russes with Iron Ties. — In all trusses whera the tie-beam of the truss is not horizontal, but higher in the centre than at the ends, it is better to substitute an iron tie for the wooden tie-beam. Fig. 17 shows a form of truss very well suited for the roofs of carriage-houses, stables, or any place where it is desired to have considerable height in the centre of the room, and a ceiling is not desired. The horizontal iron rod is fastened to the two struts at their ends, and the other two rods are fastened only at t