La 367 34 UC-NRLF SEM L. P. SHIDY FORMULAS AND TABLES ARCHITECTS AND ENGINEERS CALCULATING THE STRAINS AND CAPACITY OP STRUCTUKES IN IRON AND WOOD, BY F. SCHUMANN, C. E. ILLUSTRATED WITH MORE THAN THREE HUNDRED DIAGRAMS, DESiGSEL .uSi> ENGRAVED ESPECIALLY FOR THIS WORK BY J. C. LYONS. WASHINGTON CITY: WARREN CHOATE & CO. 1873. Entered according to Act of Congress, in the year 1873, by F. SCHUMANN, In the Office of the Librarian of Congress, at Washington. IN ME?/!Of?IAM KL STEREOTYPED BY tt OILL A WITHEROW, WASHINGTON, D. C. THIS VOLUME is RESPECTFULLY DEDICATED TO A. B. MULLETT, gUPERVISING ARCHITECT OF THE U. 8. TREASURY DEPARTMENT, BY THE AUTHOR. (iii) CONTENTS. EEEATA. On page 4, 10th line from bottom, read - - instead of 30. On page 4, 10th line from bottom, read 10.0036 instead of 10.036. a On page 4, 14th, 15th, and 16th lines from bottom, read - instead of a. On page 32, Fig. 70, insert I = distance between supports. On page 34, Fiq. 72, insert I = distance between supports. On page 34, Fig. 74, insert I = length of beam. On pages 38 and 39 w = total weight of beam between supports. On page 39, 5th line from top, read 1099000 instead of 1000000. On page 39, 5th line from top, read 1754 instead of 1757. On pages 144 and 145, in formulas for H n , change places of last minus sign with foregoing plus sign. (See 13th line from top.) Page 145, lines 1 to 7 from bottom, ~| Change places of Cand T Page 146, lines 1 to 3 from top, > under strains in Figs. Page 146, lines 13 to 22 from top, J 225, 226, 227, and 228. On page 149, 1st line from bottom, read -- instead //, of- N On page 197, 7th line from bottom, read 3.14159 instead of 1.14159. On page 204, 1st line from bottom, read A-\- A, instead ofAA,. Static and moving loads on bridges of wrought iron.,,... 192, 193 (vii) CONTENTS. PAGES. Summary of definitions and general principles 1-5 ^Moments of inertia and resistance of various sections.... 5-25 Strength of materials, &c 26-29 Resistance to cross-breaking and shearing 29 Capacity and strength of beams 29-99 Pressure on supports 100-102 Compressive strains and pressure on supports 102 Sloping beams, rafters, &c 102-103 Resistance to crushing 103 Strength of columns, pillars, and struts 103-110 Parallelogram of forces Ill Strains in frames 112-114 Strains in boom derricks 114-115 Strains in trusses.... 115-121 Strains in trussed beams 122-125 Strains in trusses with parallel booms 126-146 Strains in parabolic curved trusses 147 "Bow-string girders" 147-153 Capacity and strength of parabolic arched beams or ribs originally curved 153, 154 Strains in a polygonal frame 154, 155 Strains in roof trusses 156-178 Pressure of wind on roofs 178, 179 Pressure of snow on roofs 180 Tie rods and bars 181, 183 Joints or connections in iron constructions 184-186 Dimensions of bolts and nuts 187, 188 Compound strain in horizontal and sloping beams 188-190 Weight of moving loads 191 Static and moving loads on bridges of wrought iron 192, 193 (vii) Vlll CONTENTS. MISCELLANEOUS. PAGES. Geometry 197-201 Center of gravity of planes 202-204 Trigonometrical formulas 205 Trigonometrical functions 206-217 Circumference, area, and cubic contents of circles 218-223 Specific gravities of materials 224-226 Weight of a superficial inch of wrought and cast iron... 227 Weight per square foot of metals 228 Weight of a lineal foot of flat and square bar iron 229-233 Weight of a lineal foot of rolled round iron 234 Weight of bolts, nuts, and heads 235-237 Weight of materials used in building 238 1 Divisions of a foot expressed in equivalent decimals 239 Table for comparing measures and weights of different countries 240, 241 To cut the strongest and stiffest beam from a log 242 FORMULAS AND TABLES FOR ARCHITECTS AND ENGINEERS. Summary of Definitions and General Principles. EXTERNAL FORCES are those forces (loads, &c.) which cause or tend to cause the rupture of a structure. INTERNAL FORCES are those forces which resist the external forces; when one balances the other, the structure is said to pos sess Stability EXTERNAL FOKCKS. INTERNAL FORCES. Compressive strain. Resistance to Compression. Tensional strain. Resistance to Tension. Shearing strain. Resistance to Shearing. Cross-breaking strain. Resistance to Cross-breaking. COMPRESSION causes the material to fail by crushing or buck ling, or both. RESISTANCE to direct Crushing: In case pillars, blocks, struts, or rods, along which the strains act, are not so long in propor tion to their diameter as to have a tendency to give way by bending sideways. This includes Stone and brick pillars and blocks, of ordinary proportions; Pillars, struts, and rods of cast iron, in which the length is not more than five times the diameter, approximately; Pillars, struts, and rods of wrought iron, in which the length is not more than ten times the diameter, approximately ; Pillars, struts, and rods of dry timber, in which the length is not more than twenty times the diameter. Let W == Crushing load in Ibs. C= Ultimate resistance of material to crushing in Ibs. per square inch. A = Sectional area of pillar, &c., in square inches. Then will TP = A X C; and A = -^- C TENSION, causes the material to be torn asunder. (1) AND GENERAL PRINCIPLES. Resistance of bars, &<?., to teaiing: the ultimate strength of a bir (co :te.fcring) is : whssa T= Ultimate resistance of the material to tearing, in Ibs. per square inch. W= Tearing load in Ibs. A = Sectional area of bar, in square inches. W Then will TF= A X T; and A = SHEARING causes the fibres of the material to be shorn by each other ; when a bolt pulls out of its nut, the threads of the screw are said to be sheared. Resistance of bars, bolts, &c., when sheared at one place, is: when S = Ultimate resistance, of material to shearing, in Ibs. per square inch. W= Shearing load in Ibs. A = Sectional area of bar, &c., in square inches. W Then will W= A X 8; and A = -~ o CROSS-BREAKING a beam, &c., supported at one or both ends, with a force at right angles to its length, sufficient to cause rup ture, is said to be broken across. Resistance to cross-breaking is the resistance of the material to compression, tension, and shearing combined ; . The flanges or booms, in beams or trusses, resist the bending moment, or moment of rupture. The web or braces, in beams or trusses, resist the shearing forces. CAPACITY means the load or pressure a structure is capable of sustaining with safety. DEFLECTION is the displacement of a beam from a horizontal, caused by its own weight or the applied load, or both. CAMBER is given a beam to counter balance the deflection, so that the beam may be horizontal when loaded ; the camber should be equal to the computed deflection. To find the effect of combining several loads on one beam, whose separate actions are known: for each cross section, the shearing force is the sum of the shearing forces, and the bending moment the sum of the bending moments, which the loads would produce separately. When a load on a structure is partly static and partly moving, multiply each part of the load by its proper factor of safety, and DEFINITIONS AND GENERAL PRINCIPLES. add together the products : the sum will be the load to which the structure is to be adapted. For a Bridge with two platforms, one carrying a road and the other a railway, those two loads are to be combined. Of Iron Ties, Struts, and Beams. In designing ordinary structures of wrought iron, it saves time and expense to use iron bars of such forms of cross section as are usually to be met with in the market. An engineer should avoid introducing new sections for bars into his designs, except when, by so doing, some important purpose is to be served, or some decided advantage to be gained. The most common forms of rolled bars is shown by the following enumerated figures : No. of figure. Name of Form. Applicable for 4 Square iron Ties. 13 Round iron Ties, bolts, and rivets. 2 Flat iron Ties. 29 I or double T-iron Beams rafter* and struts 30 Channel iron . Rafters and struts. 37 T-iron Rafters and struts 47 L or angle iron Various purposes. 1 Deck Beam Beams and rafters Avoid the use of cast iron for ties, also trussed cast-iron beams. When a member of a structure acts alternately as a strut and as a tie, it must have sufficient total sectional area, and sufficient stiffness, to resist the greatest compressive strain that can act, and sufficient effective sectional area to resist the greatest tensional strain which can act along it. Let all pins, bolts, rivets, &c., exposed to a shearing strain, fit tight in its hole or socket. Roof trusses, the divisions of a rafter, and also the struts, may be considered as hinged at the ends. In members under a compound strain, as for instance a trussed beam with a distributed load, be careful to take into account the additional compression, caused by the bending moment. The best distribution of the material in a section of a cast-iron T s Q beam, for cross-breaking, is that = ; or = - s s / s T When T= Ultimate resistance of the material to tension. C= Ultimate resistance of the material to compression. s Distance from neutral axis to most extended fibres. s, = Distance from neutral axis to most compressed fibres. That is, the fibres most in tension should be nearest the neutral axis of beam. DEFINITIONS AND GENERAL PRINCIPLES. In wrought-iron beams, the section may be made alike above and below the neutral axis. THE MODULUS OF RUPTURE should be applied to beams with full section, or beams with continuous web only ; for all open web beams, or beams with very thin web, the resistance of the mate rial to crushing or tearing, respectively, must be used instead. EXPANSION AND CONTRACTION of long beams, which arise from the changes of atmospheric temperature, are usually provided for by supporting one end of each beam on rollers of steel or hard ened cast iron. The following table shows the proportions in which the length of a bar of certain materials is increased by an elevation of temperature from the melting point of ice (32 Fahr., or Centigrade) to the boiling point of water under the mean atmospheric pressure, (212 Fahr., or 100 Cent.;) that is, by an elevation of 180 Fahr., or 100 Cent,: METALS. Brass 0.00216 Bronze 0.00181 Copper 0.00184 Cast iron 0.00111 Wrought iron 0.00120 Tin 0.00225 Zino 0.00294 Lead 0.00290 EARTHY MATERIALS. Brick, common 0.00355 Brick, fire 0.00050 Cement 0.00140 Glass, average 0.00090 Granite 0.00085 Marble 0.00087 Sandstone 0.00105 Slate 0.00104 Reference. Let u Value given in above table, for a certain material. I Length of a bar at Centigrade, and Z x its length at a given number of degrees Centigrade. a Given number of degrees, at which I, is required. A = Superficial area of a plate ; and A, its area at a given number of C. B = Cubic contents of a body, and .#,= its contents at a given number of C. Then will I, = I (1 +au); A, = A(l + 2au) , B / = B (1 -j- 3 a u). Example : A bar of wrought iron 2 inches square, is 10 feet long at a temperature of Centigrade ; what is its length at an increased temperature of 30 ? Ans : I, = 10 (1 + 30 X 0.00120) = 10.036 feet. THE NEUTRAL Axis, in a cross section of a beam, is that layer of fibres which are neither in compression or tension, by the action of a load on the beam ; it always passes through the centre of gravity of the section : provided the limits of elasticity of the material is not exceeded. A beam supported at both ends, with a load acting perpendicular between the supports, will cause the fibres above the neutral axis to be compressed, and those below, extended: the farther from the fibres to the neutral axis, the greater the stress. MOMENTS OF INERTIA AND RESISTANCE. 5 Under MOMENT OF INERTIA of a cross section, may be under stood : an internal force at rest ; a static force resisting an exter nal force; it means the sum of all the area elements, multiplied by the square of their perpendicular heights from the neutral axis of the section. The moment of inertia, which we have denoted with I, depends on the form and dimensions of the cross section, and is expressed in square inches. MOMENT OF RESISTANCE of a cross section is that static force resisting an external force of compression or tension ; it is equal to the moment of Inertia divided by the distance of the most ex tended or compressed fibres, respectively, from the neutral axis. MOMENTS OF INERTIA AND RESISTANCE OF VARIOUS SECTIONS. To find the moment of inertia of any given cross section FIRST. Divide the section into as many simple figures as possi ble. (See diagram, fig. 1.) SECOND. Find the moment of inertia of each of the simple figures about its own neutral axis, and insert the value in the following formula : Reference. Letters A, A /t A //t = area of simple figure, respectively; and d, d /t d //t = its distance from its centre of gravity to that of the whole section. i Vi V/ = moment of inertia of simple figures, re spectively. For neutral axis see centre of gravity. Fig. 1. B y Formula. 1= (i + dU) + (v + d,*A,) + (i // -{- djfAji) + <fcc., = moment of inertia of whole section. MOMENTS OF INERTIA I AND MOMENTS OF RESISTANCE I Reference. m n = neutral axis of section. r = radius. s = distance from neutral axis to most compressed or extended fibres. 6, h, &c. = dimensions. A = area. MOMENTS OF INERTIA AND RESISTANCE. No. of Figure. Form of Section. 2 and 3 It Tfo LJL7I JHfr m n 771 MOMENTS OF INEETIA AND EESISTANCE. Moment of Inertia 7. Moment of Resistance- = T v Ah* bh* h* 6 12 h* - Qh 12 1/2 MOMENTS OF INERTIA AND RESISTANCE. No. of Section. VI. No. of Figure, Form of Section. VII. VIII. 10 IX. 11 12 MOMENTS OF INERTIA AND RESISTANCE. Moment of Inertia /. Moment of Resistance - A A A 1 A jly 6^3 = 10 MOMENTS OF INERTIA AND RESISTANCE. No. of Section. No. of Figure. XI. 13 Form of Section. XII. 14 XIII. XIV. 15 16 XV. 17 and 18 MOMENTS OF INERTIA AND RESISTANCE. 11 Moment of Inertia /. Moment of Resistance JL } TT r = J Ar J i* s = 0.576/fc = (1 A_ ) h 12 MOMENTS OF INERTIA AND RESISTANCE. No. of Section. No. of Figure. XVI. 19 Form of Section. XVII. 20 XVIII. 21 XIX. XX. 22 23 TTU - - MOMENTS OF INERTIA AND RESISTANCE. 13 Moment of Inertia /. Moment of Resistance - bh* = & Alt 15 - 10 MOMENTS OF INERTIA AND RESISTANCE. No. of Section. No. of Figure. Form of Section. XXI. 24 XXII. XXIII. 25 26 ? /* / XXIV. 27 XXV. 28, 29, and 30 MOMENTS OF INERTIA AND RESISTANCE. 15 Moment of Inertia /. Moment of Resistance - I A [i V cos*u + Jf A 2 sin*v] A [A 2 cos 2 ?; + V siri*v] 12 I */, 16 MOMENTS OF INERTIA AND RESISTANCE. No. of Section. No. of Figure. Form of Section. XXVI. 31 XXVII. XVIII. 32 33 fel kM XXIX. 34 XXX. 35 MOMENTS OF INERTIA AND RESISTANCE. 17 Moment of Inertia /. 12 Moment of Resistance . b (h* - A/) ~~~~ 1 ~o/r 18- MOMENTS OF INERTIA AND RESISTANCE. No. of Section. XXXI. XXXII. No. of Figure. 36 and 37 38 Form of Section. FF^li . V \ . A . XXXIII. XXXIV. XXXV. 39 41 -^~"~^j3i MOMENTS OF INERTIA AND RESISTANCE. 19 Moment of Inertia /. Moment of Resistance - A ( bh * + W) 6/1 A (*6 + &,*) hb* + 66 A l(**S 3625 A A t^/ 4 6 s &/ (A Z)) 4 s ] 0.049 Id* IT 20 MOMENTS OF INERTIA AND RESISTANCE. No. of Section. Nj. of Figure. Form of Section. XXXVI. 42 XXXVII. XXXVIII. XXXIX. XL. 44 45 46, 47, and 48 MOMENTS OF INERTIA AND RESISTANCE. 21 Moment of Inertia I. Moment of Resistance - s - V) A __ 1*7 12(M" 6, 22 MOMENTS OF INERTIA AND RESISTANCE. No. of Section No. of Figure. Form of Section. XLI. 49, 50, and 51 XL1I. XLIII. 52 53 -i XLIV. XLV. 55 MOMENTS OF INERTIA AND KESISTANCE. 23 Moment of Inertia /. (bh* - 6 A 2 ) 2 - *&/*&/ 12 ^ r* \X3~0.5413r 4 Moment of Resistance (bh*- b^/) *- 4bhb / h / (h-h / ) > = O.G381 = 0.5413 (r* r/) = 6381 (?* r/) MOMENTS OF INERTIA AND RESISTANCE. No. of Section. No. of Figure. XLVI. XLVII. XLVIII. 56 57 58 Form of Section. ^ T I i ""7 - A XLIX. 59 \ I L. 60 MOMENTS OF INERTIA AND RESISTANCE. 25 Moment of Inertia J. Moment of Resistance I . s n / = number of sides. ^j w/r 4 sin . v (2 -f- cos . v) A n / r3 s ^ ?l v (^ 4~ cos v ) n / = number of sides. b = length of a side. TV ^ (3/& 2 -}~ J O 2 ) ^1 i _^ (3^2 ^_ 1 12) h oJfi b/hf -j- b/h/fi w _ w ; w 12 6A 2b^hh // " " / / 7 // 26 STRENGTH OF MATERIALS. STRENGTH OF MATERIALS, &o. f In H>s., avoirdupois, per square inch of cross-section. Materials. rt ^j o| "5 Ultimate Resistance to Modulus of elasticity. Tearing. Crushing. Shearing. Oross-br k Modulus o Rupture. METALS. Brass, ca^t, average 505.7 533 524 537 540 18000 40000 30000 10000 30000 3;ooo 00000 105(10 13400 to 20000 10300 9170000 14230000 9900000 17000000 17000000 14000000 to 22900000 29000000 25300000 15000000 29000000 to 42000000 720000 4000000 13000000 1000000 1350000 " wire. Bronze or gun metal, (cop per 8, tin 1) Copper cast 117000 " she^t " bolt** " \viro Iron ca^t avcnvo 445 4)54 to 450 112000 80000 to 115000 27700 " various " beams, average.. 28800 17000 33000 to 43500 38000 " solid rect. bars, various quailities. Iron, wrought, average 481 05000 30000 to 40000 50000 plates joints, d ble riveted. Iron, wrought, joints, single riveted. Iron, wrought, bars and bolts. hoop, best best wire 51000 35700 28000 GdOOO to 70000 04000 70000 to 100000 00000 wire ropes.... Steel, average 490 80000 " bars 100000 to 130000 80000 3:500 4(300 7000 to 8oOO 17000 0300 11500 12000 7730 15500 15000 " plates Lead, sheet 712 402 430 47 43 Tin, cast Zinc TIMBER, (well seasoned and dry.) Ash 9000 9300 1400 12000 to 14000 9000 to 20000 Bamboo Beech STRENGTH OF MATERIALS. 27 Materials. Weight of a cubic foot. Ultimate resistance to If Tearing. Crushing. Shearing Cross-br k Modulus o Rupture || TIMBER Continued. Birch 44 80 33.4 34 74.5 37 37 33 52 47 52.5 44 62 35 49 52.5 47.4 15000 200O( 100O( to 1300< 140iK 120CX to 14000 12401 9QOC to 10* -00 25000 20000 23400 1COOO ll.XOO 8000 to 21800 10600 10000 o 19800 6400 10300 5300 10300 19-)00 5375 to 6200 5900 5570 11000 7300 9000 9900 8200 6500 10000 7700 6100 6000 5300 5400 12000 12000 11000 6500 4000 550 to 800 1100 1700 417 to 612 11701 1086 6004 to 2700 7 UK to 9541 990, to 1 JjOt 50. < to 10001 1735! 1 1 ..00 1200 10000 10000 to 13(00 87 0< 10601) 9600 12000 to 17460 6600 1645000 1140000 700000 to 1340000 146 000 to 1900000 UOO 00 to 1800000 900000 to 1 .".60000 10-tUOOO 1255000 1200000 to 1750000 21 50000 2400000 Box.... .... Chestnut Elm 1400 Ebony, West Indian.... Fir Red Pine .. 500 to 8(iO 600 970 to 1700 " Spruce " Larch Hickory Hornbeam Lance wood Locust Lignum vitse Mahogany Maple...... 2300 Oak, British " Dantzic " American white.... 42 54 346 29 37 48 62.5 18000 10250 11500 15000 13000 15000 red Pine, American, white " yellow Sycamore Teak, Indian Water gum Walnut 40 25 50 125 135 37.5 loo 8000 14000 8000 280 to 300 Willow, various Yew STOXES, (natural and arti ficial.) Brick, weak red " strong red " fire " work Cement 89 280 to 300 STRENGTH OF MATERIALS. Materials. Weight of a cubic foot. Ultimate resistance to Modulus of elasticity. Tearing Crushing. Shearing. >oss-br k. Jodukis ot Rupture. STONES Continued. Chalk 145.5 173 168 118 9400 330 8000000 13000000 to 16000000 Glas=! Granite 5500 2360 1100 5000 Limestone marble 172 to 11000 5500 4000 to 4500 About 4-10 cut stone. 5500 3300 to 4400 " granular 197 100 to 170 50 109 116 Rubble masonry Sandstone, strong ") " ordinary ( 144 " weak ) Slate 178 9600 to 12800 25000 14000 6300 4200 5200 7700 MISCELLANEOUS. Flaxen yarn Hempen ropes... Hide, ox undressed ... Leather ox Silk fibre Whalebone MODULUS OF RUPTURE R. According to Professor Rankine, the modulus of rupture is eighteen times the weight that is required to break a bar of a given material one inch square (section) and one foot between supports, the weight concentrated at the middle. MODULUS OF ELASTICITY E Is that power (in Ibs. generally) through which a prismatic body of a given material, of section = 1, is assumed to be extended double its length, or compressed to 0. Let A = Sectional area of a rod of the material. W= Weight or power in Ibs., which causes the extension or compression of the rod. I = Length in inches of rod before W is applied. Y = The extension or compression of the rod in inches, caused by W. Wl W RESISTANCE TO CROSS -BREAKING AND SHEARING. 29 FACTORS OF SAFETY k. The ultimate resistance of material should be divided by A Wrolgift t I 1 e on nd For Proof stren S th - Foi> Working Stress. Steady load 2 Moving load 4 to 6 Cast Iron. Steady load 2 to 3 3 to 4 Moving load 6 to 8 Timber. Average 3 8 to 10 RESISTANCE TO CROSS-BREAKING AND SHEARING. CAPACITY AND STRENGTH OF BEAMS. Reference. A = Area of cross-section of beam. D Deflection of beam from a horizontal. E = Modulus of elasticity. J= Moment of inertia of cross section. M Maximum moment of rupture, or bending moment. R = Modulus of rupture. S = Vertical shearing force. V = Pressure on supports. W= Capacity or weight of load, c, d, I == Dimensions in units of length. k = Factor of safety. w = Weight of load per unit of length. = Moment of resistance of cross-section. I R I For the stability of a beam : M=. K = . . k s The web of a metal beam must have sufficient area to resist the shearing force 8; that is, A = -rrr-: : : Ultimate resistance to shearing. The weight of the beam must be added to W, except in small beams, under 60 Ibs. per lineal foot, when it may be disregarded. [NOTE. Always use the same units of dimensions or weight.] 30 RESISTANCE TO CROSS BRKAKING AND SHEARING. No. of Figure. Manner of loading and fastening beams. 61 . -P " " " V x 62 63 64 . If o p xim ent ure W.I - W. W. 5.333 ity sec K I 4 K ~ 5.333- K- ------ - ------- - .65 r "V RESISTANCE TO CROSS BREAKING AND SHEARING. 31 Maximum deflec tion D. Distance from A to point of maximum/). Shearing force S. Pressure on sup ports V. W I 3 .EM 3 J At any point. W W W I s E I 8 I At any point. w .d W W p - lO ls" z 2 At any point. IF ~Y V - V - W 1/1 - ^2--^ IF Z 3 "J 0.00931 J-j. J. 0.553J 1 TT.-L ".-".-?- & W P 2 "T At any point, d<d / ; (i-0 fi-F._i 8 E.I 48 32 RESISTANCE TO CROSS-BREAKING AND SHEARING: 66 68 69 Manner of loading and fastening beams. Maximum mo ment of rup ture M. \ /MM A ^ <rl- A w|< \ \ 2- - ^ \-L r*-f * i ^ -------- _ ------ ^ - IF. 12 W .1 + W tion. Capaci any 8. 12 "T- -.K RESISTANCE TO CROSS BREAKING AND SHEARING. |CQ ^"S Maximum deflec tion D. 11 p r Shearing force 5. Pressure on sup ports F. "DQTJH C P W Z 3 j ir W E.I 4.48 O ^ W Z 3 00 r )4 Q. 572.1 n tt 1 " ,T\ v v w E.I ( 8 1 W P l *<** E.I 8.48 o -(!-) ^ (Irr) + At any point be / W l ^ 3 > i tween loads. ( y - ~ ) ~t~ ^ S= W .S l = II 1 \\ l \ n 2 /.J. ) ~m~ _i_ ~nr ^ (\ W 4- w l + W z E.I ~3") At any point and under any load. _ }y 1 2 V -?- W W /3 72/2 1 fj 6j I 1 E.I 3 I 2 I 2 Constant bet, A & IF J s w ll V 2 = 1- W Constantbet. Z?& IF 3-1 RESISTANCE TO CROSS-BREAKING AND SHEARING. i Maximum moment of Manner of loading and fastening beam?, j rupture M. A x:.:v ^ /C>v rrK I 1 m ^ l ^ T^ ! IF./, W.I, When ^>^ [(4-) - . ] When Z < Zi \/8 ; RESISTANCE TO CROSS-BREAKING AND SHEARING. 35 ^"c Capacity W of any section. Maximum deflection D. sll I s ! Shearing force 8. Pressure on sup ports V. 5 W E.I K h a 2 W w 2 h i 2 w l * F 1= = Kl ~~i~w ^( 1 -^) 1 1 2 Ti 1 i w W 1 2 t l - 8 E.I K Wl^ w F! = F 2 h D,- l W \ 2 3 2 (1 + ^i) g (4) - .- w .l or w.-JL- W The greater value to be 2 taken. 2(1+21,) K " 36 RESISTANCE TO CROSS- BREAKING AND SHEARING. Manner of loading and fastening beams. Maximum moment of rupture M. When c? >(Z c) ; w- W- 78 1^ (!+,)] RESISTANCE TO CROSS-BREAKING AND SHEARING. 37 Capacity W of any section. MAximum deflection D. S oq lag a| 1-1 |^l Shearing force 8. Pressure on sup ports F". 1 K e (l__d) A 2K (l-r-d)* 213 E W -^ 3 E.I IS(1IJ 1^(31-1^(1-1^) I 2 P K Vff-y" 38 EESISTANCE TO CROSS-BREAKING AND SHEARING. EXAMPLE. Capacity of wrought-iron l-shaped beams; top and bottom flange alike ; load equally distributed ; ends not fixed. Dimensions of Cross-section. h = Height = 10 inches. b = Width of flange 4 inches. t = Thickness of flange = 0.8 inches. t / = Thickness of web = 0.5 inches. 7t /= = h 2t; b, = b t,. Distance between supports = 20 feet = 240 inches. Factor of safety = 3. MOMENT OF RESISTANCE. _Z_ ft/* 3 M/ 3 4 x 1Q 3 3.5 X 8.4* _ V = 6* ~ 6 X 10 Capacity W. w = (4 X 0.8 X 2 + 8.4 x 0.5) x 240 X 0.28 = 712.32 Ibs. K=*. ^J*. 32.09 = 406473.33. . k s 3 W=8^--w^8.~^ -- 712.32 = 12836.72 Ibs. li ^41) EXAMPLE. Capacity of cast-iron i-shaped beams; load equally distributed; ends not fixed; flange down. Dimensions of Cross- section. Let h = Height = 18 inches. b = Width of flange = 9 inches. t = Thickness of flange = 1.25 inches. t / = Thickness of web = 1 inch. 7i / = h t; by = b t/. Area = 28 square inches. Distance between supports = 20 feet = 240 inches. Factor of safety k = 4. MOMENT OF RESISTANCE. sh, (h h,Y * L bh 2 2b / hh / + b,h/ -*[- 2b / hh / + (9 x 18 2 8 x 16.752)2 9 X 18 2 2 X 8 X 18 X 16.75 + 8 X 16.75 2 4X9X18X8X 16.75 (18 16.75) 2 9 X 18 2 2 X 8 X 18 X 16.75+8 X 16. 5) 2 n 6.75 2 J RESISTANCE TO CROSS-BREAKING AND SHEARING. 39 _ rj452256.25_ _135675.00_-] = 15- * L 336.5 336 5 J . Capacity W. w = 28 x 240 X 0.261 = 1754. lbs. JT^;* J- = -^-. 157 = 1099000. k s 4 JF= 8 A - = 8 . -- ~S" - 1757 = 34879 lbs - For light beams no attention need be paid to weight of beam w. CAPACITY If OF ROLLED I -SHAPED BEAMS. Load equally distributed. The calculations are based upon (lie patterns or section* used by the Phcenixville Iron Company. Practically this applies to all similar beams rolled in the United States, the difference in the profile of section being slight. In the following table the factor of safety k = 2.53: Reference. W= Load in tons of 2,000 lbs., equally distributed. w = Weight of beam in tons of 2,000 lbs. L = Distance between supports in feet. I = Distance between supports in inches. iu / = Weight per square foot of floor. W,= Capacity of coupled or trebled beams in tons of 2,000 lbs. D = Deflection in inches at centre, between supports. d = Distance between centres of beams, when spacing for floors, in feet. W W, r> W+w J? 7.0 tons, d -- , or d= - -, D f . . L.to, L.w / &.L 4b K 1 = Constant, computed by formulas. (See under examples.) 40 RESISTANCE TO CROSS -BREAKING AND SHEARING. The rivets for coupled or trebled beams should be about inch in diameter, and 8 inches apart. Trebled Beams. Coupled Beams. W,= WX 5.33. :JQ ; = 17.2 tons. This is also found at the intersection of Fig. 79. Examples explanatory of the following Table. EXAMPLE.- What is the capacity of a 15-inch light beam, load equally distributed, distance between supports = 20 feet? 7-1 /f 8 1 TT7 Kl K 1 A = r^ , and W ; for 15-inch light beam ----- = -Li J_j 345. 19 20 ~ 20 feet and column under capacity W. EXAMPLE. What distance apart should 9-inch medium beams be placed, the distance between supports being 20 feet, and to carry a total load of 140 Ibs. per square foot of floor surface? Ans. 4.4 feet; being found at the intersection of the horizontal line from 20 feet and the vertical column under 140 Ibs. EXAMPLE.- What is the capacity of 12- inch light beams trebled . load equally distributed, distance between supports = 25 feet? Ans. W for 12-inch light beam == 9.19 and W, = W X 5.33 = 9.19 X 5.33 = 48.98 tons. RESISTANCE TO CROSS-BREAKING AND SHEARING. 41 CAPACITY OF ROLLED BEAMS. Explanation of Tables for I Beams. The first column gives the distance between supports in feet. The second column gives the capacity in tons of 2,000 Ibs., equally distributed. The third column gives the deflection in inches at centre of beam. The fourth column gives the weight of beam in Ibs. for length between supports. The fifth to fifteenth column (inclusive) gives the distance in feet that the beams should be spaced from centre to centre, for weight in Ibs., per sq. ft. of surface for floors. Pounds in decimals of a ton. Ibs. tons. GO == 0.03 70 = 0.035 80 = 0.0-1 90 = 0.045 100 = 0.05 140 = 0.07 160 = 0.03 180 = 0.085 200 = 1 250 = 0.125 300 == 0.15 In using these beams for floors, with brick arching, the ends resting on supports should have a bearing of about 8 inches, resting on a cast-iron plate, 8 X 12 in. sq,, by 1 in. thick. Tie rods should be used where floors are subject to heavy con centrated moving loads, (as trucks with merchandise, &c.;) these rods should be about 8 times the depth of beam apart, fastened about -J from the bottom of beam. When beams are used to support walls, or as girders to carry floor beams, and put side by side (II,) they should be fastened to gether with cast-iron blocks, fitting between the flanges, so as to securely combine the two beams. The blocks may be put about the same distance apart as the tie-rods. 42 RESISTANCE TO CROSS BREAKING AND SHEARING. Fig. 81. 15" "Heavy " Beam. Weight per If. = 66.66 Ibs. Sectional area = 20.0" Moment of inertia / = 652.42 Constant # =434.95 K W . L s "S 5 a fl 05 Deflec. in inches. Weight in Ibs. 1 Distance d bet. centres of beams in feet, for weight in Ibs. per sq. foot of 1 8 i .0 i o | GO .D g <n .0 j B 1 8 i 8 i i 1 20.1 17.6 14.7 12.8 io!2 8.9 8.0 7.2 6.7 5.9 5.5 50 4.7 4.2 3.9 3.6 3.4 3.2 3.0 2.8 2.6 2.5 2.3 2.2 2.1 2.0 1.9 1.8 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 72.49 62.13 54.35 4832 43.48 39.54 36.24 33.45 31.05 28.99 27.18 2558 24.16 22.89 21.73 20.71 19.58 18.91 18.12 17.39 16.72 16.10 15.53 14.99 14.49 14.03 13.59 13.17 12.79 12.42 12.08 11.75 11.43 11.15 10.87 0.037 0.050 0.065 0.084 0.104 0.126 0.150 0.177 0.205 0.236 0.270 0305 0.342 0.383 0.426 0471 0.515 0.569 0.623 0.677 0.735 0.795 0860 0.925 0.994 1.067 1.141 1.219 1.304 1.384 1.473 1.564 1.656 1.754 1.854 400.0 466.6 5333 600.0 666.6 733.3 800.0 866.6 933.3 1000.0 1066.6 1133.3 1200.0 1266.6 1333.3 1400.0 1466.6 1533.3 1600.0 1666.6 1733.3 1800.0 1866.6 1933.3 2000.0 2066.6 2133.3 2200.0 2266.6 2333.3 2400.0 2466.6 2533.3 2600.0 2666.6 20.9 17.7 15.5 13.5 12,0 10.7 9.6 8.6 7.9 7.1 6.7 6.0 5.6 5.2 4.7 4.4 4.1 3.8 3.6 3.3 3.1 3.0 2.8 2.6 2.5 2.4 2.2 2.1 22.1 19 > 79 3 21.2 19.6 16.7 15.2 13.5 12.5 ll.l 10.5 9.4 8.6 8.0 7.4 6.9 6.8 6.0 5.6 5.3 4.9 4.7 4.4 4.1 39 3.7 3.5 3.3 18.8 17.0 14.9 13.4 12.0 11.5 9.8 9.4 8.3 7.7 7.2 6.7 6.2 5.7 5.3 5.0 4.7 4.4 4.1 3.9 37 3.5 3.3 3.1 2.9 LG.9 15.0 13.4 12.0 10.8 9.8 8.9 8.2 7.5 6.9 6.4 5.9 5.5 5.1 4.8 4.5 4.2 3.9 3.7 3.5 3.3 3.1 3.0 2.8 2.7 21.4 19.1 17.6 15.5 14.7 127 11.8 10.7 10.2 9.2 8.5 7.9 7.4 6.9 6.4 6.0 5.7 53 5.0 4.7 4.4 4.2 4.0 3.8 2-i o 19 7 21.7 19.7 17.8 17.1 15.1 14.4 12.8 11.9 11.0 10.7 9.8 9.0 8.4 7.9 7.5 7.1 6.6 e.3 6.0 5.7 5.4 21.0 18.8 17.3 16.7 14.9 13.8 12.9 12.0 11.3 10.0 9.9 9.4 8.8 8.4 7.9 7.5 7.1 6.7 18.9 16.7 15.5 15.2 13.4 12.3 11.5 10.7 10.0 9.4 8.8 8.2 7.9 7.4 7.0 6.6 6.3 6.0 21.4 19.8 18.2 17.2 16.1 15.0 14.0 13.3 12.5 11.8 11.1 10.8 10.0 9.5 9.0 21.5 19.9 18.3 17.1 15.8 14.8 13.8 12.9 12.0 11.4 10.7 10.1 9.5 9.1 8.5 8.1 7.7 RESISTANCE TO CROSS BREAKING AND SHEARING. 43 Fig. 82. 15" "Light" Beam. Weight per If. = 51.66 Ibs. Sectional area = 15.5" Moment of inertia / 517.78 Constant K =345.19 K W =-. L 3 00 <D Distance d bet. centres of beams in feet, for o a c o 00 weight in Ibs. per sq. foot of d CO CJ ^ a 3 |.s | i lip | .0 Ul .C Vl .0 ?] | & JS j5 09 X3 72 & ft 1 ^c <S 9 8 o Srj o & 1 >g 1 CM 1 6 57.52 0037 310.0 7 49.31 0.050 361.6 8 43.13 0.065 413.3 9 38.35 0.084 4650 10 34.50 0.103 51 6.6 23.0 11 31 .38 0.124 567.3 22.9 19.0 12 28.76 0.150 620.0 19 T15.9 13 26.55 0.176 671. 6 22.7 ^0 4 164 13.6 14 24. G5 0.205 793 ^ 22.0 19.5 17.6 14.0 11.7 15 23.01 0.236 775.0 21.9 19.1 17.0 15.3 12.3 10.2 16 21.57 0.269 806.6 19.2 16.8 14.9 13.4 10.71 8.9 17 20.30 304 858.3 17 9 14.9 13.2 11.9 95! 7.Q 18 19.16 OJ341 830.0 21.3 15^2 13.3 11.8 10.6 8.5 j 7.1 19 18.15 0.381 981.6 21.3 19.1 13.6 11.9 10.6 9.5 7.6 6.3 20 17.24 0.424 1033.3 21.5 19.1 17.2 12.3 10.7 9.5 8.6 6.9 5-7 21 16.43 0.469 1085.0 19.5 17.4 15.6 11.1 9.8 8.7 7.8 6.2 5.2 22 15.68 0.515 1136.6 20.3 17.8 15.8 14.2 10.1 8.9 7.9 7.1 5.7 4.7 23 15.00 0.565 1187.3 21.7 18.7 16.3 14.5 13.0 9.3 8.1 7.2 6.5 5.2 4.3 24 14.38 0.620 1240.0 19.9 17.1 14.9 13.3 11.9 8.5 7.4 6.6 59 4.8 3.9 25 13.80 0.674 1291.6 18.4 15.8 13.8 12.3 11.0 7.8 6.9 6.1 5.5 4.4 3.6 2(5 13.27 0.732 1343.3 17.0 14.5 12.8 11.3 10.2 7.2 6.3 5.6 5.1 4.0 3.4 27 12.78 0.791 1395.0 15.7 13.6 11.8 10.5 9.4 6.7 5.9 5.2 4.7 3.7 3.1 28 12.32 0.855 1446.6 14.6 12.5 11.0 9.7 8.8 6.2 5.5 4.8 4.4 3.5 2.9 29 11.93 0.921 1498.3 13.7 11.8 10.2 9.1 8.2 5.8 5.1 4.5 4.1 3.2 2.7 30 11.50 0.989 1550.0 12.7 10.9 9.5 8.5 7.6 5.4 4.7 4.2 3.8 3.0 2.5 31 11.13 1.060 1601.6 11.9 10.3 8.9 8.0 7.1 5.1 4.4 3.9 3.5 2.8 2.3 32 10.78 1.133 1653.3 11.2 9.( 8.4 7.4 6.7 4.8 4.2 3.7 3.3 26 2.2 33 10.46 1.211 1705.0 10.5 9.0 7.9 7.0 6.3 4.5 3.9 3.5 3.1 2.5 2.1 34 10.14 1.292 1750.6 9.9 8.5 7.4 6.6 5.9 4.2 3.7 3.3 2.9 2.3 1.9 35 9.86 1.375 1808.3 9.3 8.( 7.0 6.2 5.6 4.0 3.5 3.1 2.8 2.2 1.8 36 9.58 1.463 1860.0 8.8 7.6 6.6 5.9 5.3 3.8 3.3 2.9 2.6 2.1 1.7 37 9.32 1.553 1911.6 8.3 7.2 6.2 5.6 5.0 3.5 3.1 2.7 2.5 2.0 1.6 38 9.08 1.645 1963.3 7. 6.8 5.9 5.3 4.7 3.4 2.9 2.6 2.3 1.9 1.5 39 8.85 1.742 2015.0 7!5 6.5 5.6 5.0 4.5 3.2 2.8 2.5 2.2 1.8 1.4 40 8.62 1.841 2066.6 7.1 6.1 5.3 4.7 4.3 3.0 2.6 2.3 2.1 1.7 1.4 RESISTANCE TO CROSS- BREAKING AND SHEARING. Fig. 83. 12" "Heavy" Beam. Weight per If. = 56.66 Ibs. Sectional area = 17.0" Moment o l inertia / = 373.53 Constant K = 311.28 K W = . g d 00 Distance d bet. centres of beams in feet, for o ft . G x> weight in Ibs. per sq. foot of ft-g d d "-I 2 u d .2 I S o Gj 6 o I 00 32 _Q OQ & i A J | n ft C3 o ft o o o o o o o g 1 6 51.88 0.046 340.0 7 44.54 0.063 396.6 8 38.70 0.082 453.3 <) 34.58 0.105 510.0 10 31.12 0.131 566.6 20.7 28.29 0.158 623 ? " 17.1 12 25.9^ 0.188 680.0 21.6 17.2 14.4 -iq 23.94 0.222 736.6 23.0 20.4 18.4 14.7 12.2 lo 14 22/22 258 793.3 22.( 19.8 17.6 15.8 12.6 10*.5 15 20.75 0.297 850.0 10 7 17.2 15.3 138 11.0 9.2 16 19*.5( 0.339 906.6 17.4 15.2 13.5 12.1 9.7 8.1 17 0.383 963*3 21.5 15.3 13.4 11.9 10.7 8.6 7.1 18 17/29 0.431 1020.0 21. P 19.2 13.7 12.0 10.6 9.6 7.6 6.4 19 16.38 0.481 1076.6 21.5 19.1 17.2 12.3 10.7 9.5 8.6 6.8 5.7 20 15.61 0.538 1133.3 19.5 17.: J 15. ( 11.1 9.7 8.6 7.8 6.2 5.2 21 14.82 0.592 1190.0 20.1 17. ( 15.( 14.1 10.0 8.8 7.8 7.0 5.6 4.7 22 14.1-1 0.652 1246.6 21.4 18.3 16.( 142 12.8 9.1 8.0 7.1 6.4 5.1 4.2 23 13.53 0.717 1303.3 19.0 16.8 14."~ 130 11.7 8.4 7.3 6.5 5.8 4.7 3.9 24 12.9" 0.786 1360.0 18.0 15.4 13.5 12.( 10-8 7.7 6.7 6.0 5.4 4.3 3.6 25 12.4 0.855 1416.6 16.6 14.2 12.4 11.0 9.r 7.1 6.2 5.5 4.9 3.9 3.3 26 11.9 0.927 1473.3 15.3 13.1 11.5 10.1 9.1 6.5 5.7 5.1 4.G 3.6 3.0 27 11.5 1.003 1530.0 14.2 12.1 10.0 9.4 8-5 6.C 5.3 4.7 4.2 3.4 2.8 28 11.1 1.084 1586.6 13.2 11 .f 9.9 8.8 7-i 5.C 4.9 4.4 3.9 3.1 2.6 29 10.7 1.170 1643.3 12.3 10.5 9.2 8.2 7.4 5.2 4.0 4.1 3.7 2.9 2.4 30 10.3 1.257 1700.0 11.5 9.8 8.0 7.G 6.9 4.! 4.3 as 3.4 2.7 2.3 31 10.0 L350 1756.6 10. 9.2 8.0 7.1 6.4 4.C 4.0 3.6 3.2 2.5 2.1 32 9.7 1.443 1813.3 10.1 8.6 7.5 67 6.0 4.3 3.7 3.4 3.0 2.4 2.0 33 9.4 1.546 1870.0 9.5 8.2 7.1 6.P 5.7 4.( 3.5 3.1 2.8 2.2 1.9 34 9.1 1.650 1926.6 8.9 7.6 6.7 5.9 5.3 3 .8 3.8 2.9 2.6 2.1 1.7 35 8.8 1.758 1983.3 8.4 7.2 6.; 5.( 5-0 3.f 3.1 2.8 2.5 2.0 1.6 36 8.6 1.871 2040.0 8.0 6.8 6.0 5.3 4.8 3.4 3.0 2.6 2^4 1.9 1.6 37 8.4 1.987 2096.6 7.5 6.4 5.6 5.0 4.5 3.L 2.8 2.5 2.2 I .s 1.5 38 8.1 2.104 2153.3 7.1 6.1 5.3 47 4.3 3.( 2.6 2.3 2.1 1.7 1.4 39 7.9 2.234 2210.0 6.8 5.8 5.1 4.5 4.0 2.9 2.5 2.2 2.0 l.C 1.3 40 7.7 2.336 2266.6 6.4 5.5 4.8 4.3 3.8 &u 2.*4 2.1 1.9 1.5 1.2 RESISTANCE TO CROSS BREAKING AND SHEARING. 45 Fig. 84. 12" "Light" Beam. Weight per If. =41.66 Ibs. Sectional area = 12.5" Moment of inertia 1= 275.92 Constant K = 229.94 K 1 CB d Distance d bet. centres of beams in feet, for ft^5 3 d .2 1 weight in Ibs. per sq. foot of CO O d 6 d GO GO ^ o! CD P CO CO* GO .0 ja J3 .0 JO ,0 ft 1 1 .Q i 1 i g 8 o 1 | o I 1 6 39.31 0.047 250.0 7 32.84 0.063 291.6 8 28.74 0.083 333.3 24.0 9 25.54 0.105 375 23.0 18.9 10 22.98 0.131 416.6 22.0 18.3 15.3 11 20.90 0.158 458.3 23.0 21.0 19.0 15.2 12.6 12 19.16 0.189 500.0 22.0 19.9 17.7 15.9 12.7 10.6 13 17.68 0.222 541.6 10 4 17.0 15.1 13.6 10.9 9.0 14 16.42 0.258 583.3 16.7 14.6 13.0 9.3 7.8 15 15.32 0.297 625.0 22.0 20.0 14.5 12.7 11.3 iol2 8.1 6^7 16 14.37 0.339 666.6 22.0 19.9 17.9 12.8 11.2 9.9 8.9 7.1 5.9 17 13.52 0.383 708 3 19.9 17.7 15.9 11.3 9 9 8.8 7.9 6.3 5.3 18 12.77 0.431 750.0 20.0 17.7 15.7 14.1 10.1 8.8 7.8 7.1 5.6 4 .7 19 12.10 0.481 791.6 21.0 18.3 15.9 14.2 12.7 9.1 7.9 7.0 6.3 5.1 4.2 20 11.48 0.538 833.3 19.1 16.4 14.3 12.7 11.4 82 7.1 6.3 5.7 4.5 3.8 21 10.94 0.592 875.0 17.3 15.0 13 ll.( 10.4 7.4 6 5 5 7 5 2 4 1 3.4 22 10.44 0.652 916.6 15.8 13^5 111 8 me 9^5 G!? 5.9 5.2 4.7 3.7 3.1 23 9.99 0.717 958.3 14.4 12.5 10.8 9.7 8.6 6.2 5.4 4.8 4.. 3.4 2.8 24 9.58 0.786 1000.0 13.3 11.4 9.9 8.8 7.9 5.7 4.9 4.4 3.9 3.1 2.6 25 9.19 0.855 1041.6 12.2 10.5 9.1 8.2 7.3 5.2 4.5 4.0 3 ( 2.9 2.4 26 8.84 0.927 1083.3 11.3 9.7 8.5 7.5 6.8 4.8 4.2 3.7 3.4 2.7 2.2 27 8.51 1.003 1125.0 10.6 9.0 7.8 7.0 6.3 4.5 3.9 3.5 3.1 2.5 2.1 28 8.21 1.084 1166.6 9.7 8.3 6.5 5.8 4.1 3-.C 3.2 2.9 2.3 1.9 29 7.92 1.170 1208.3 9.1 7.8 6.8 6.1 5.4 3.8 3.4 3.( 2 .7 2.1 1.8 30 7.66 1.257 1250.0 8.5 7.2 6.3 5 ( 5.1 3.6 3.1 2.8 2.f 2.0 1.7 31 7.41 1.350 1291.6 7.9 6.8 5.9 5^3 4.8 3.4 2.9 2,( 2.3 1.9 1.5 32 7.18 1.443 1333.3 7.4 6.4 5.6 4.9 4.4 3.2 2.8 2.4 2.2 1.7 1.4 33 6.96 1.542 1375.0 7.C 6.0 5.2 4.7 4.2 3.0 2.6 2.3 2.1 i.r 1.4 34 6.75 1.645 1416.6 6.6 5.6 4.9 4.4 3.9 2.8 2.4 2.2 2.0 1.5 1.3 35 6.57 1.754 1458.3 6.2 5.3 4.7 4.1 3.7 2.6 2.3 2.0 1.8 1.5 1.2 36 6.38 1.871 1500.0 5.9 5.0 4.4 3.9 3.5 2.5 2.2 1.9 1.7 1.4 1.1 37 6.21 1.987 1541.6 5.5 4.8 4.2 3.7 3.3 2.3 2.0 1.8 1.6 1.3 1.1 38 6.05 2.109 1583.3 5.3 4.5 3.9 3.5 3.1 2.2 1.9 1.7 l.f 1.2 1.0 39 5.89 2.229 1625.0 5.0 4.J 3.7 3.3 3.0 2.1 1.8 1.6 1.4 1.1 1.0 40 5.74 2.366 1666.6 4.7 4.1 3.5 3.1 2.8 2.0 1.7 1.5 1.3 1.0 0.9 46 RESISTANCE TO CROSS BREAKING AND SHEARING. 10.5 Fig. 85. 10.5" Seam. Weight per If. = 35 Ibs. Sectional area = 10.5" Moment of inertia I = 179.44 Constant K =170.903 K -2 c Distance d bet. centres of beams in feet, for g; , c d weight in Ibs. per sq. foot of E" c "7 2 c5 1 c c5 o 09 1 K A .c <R J | | | 1 | 0* & 3 Q s o Q I I o 00 I I I I 1 I 1 6 2U8 0.053 2100 2L41 0.072 2450 23.2 8 21.3(5 0.095 280.0 21.3 17.8 9 18 OS 0.120 315.0 23.4 21.1 17.0 14.0 10 17.09 0.149 350.0 21 3 18 9 17.0 13.6 11.4 11 15.53 0.181 385.0 20.1 17.6 15.6 14.1 11.3 9.3 12 14.21 0.216 420.0 1(5.9 14.8 13.1 11.8 9.4 7.9 13 13 14 0.254 455.0 20,6 202 144 12.6 11.1 10.1 8.1 6 7 14 12.2(1 0.295 490.0 21.7 19.2 17.4 12.4 10.9 9.6 8.7 6.9 5*.8 15 11.38 0.340 525.0 21.9 18.9 17.0 15.1 10.8 9.4 8.4 7.5 6.0 5.0 16 1068 0.389 560.0 22.2 19.0 16.6 14.9 13.3 9.5 8.3 7.4 6.6 5.3 4.4 17 10.05 0.439 595.0 19.7 17.0 14.7 13.2 11.8 8.4 7.3 6.5 5.9 4.7 3.9 18 9.49 0.494 630.0 17.5 15.0 131 11.7 10.5 7.6 6.5 5.8 5.2 4.2 3.5 19 8.99 0.553 6650 15.7 13.0 11.7 10.5 9.4 6.7 5.9 5.2 4.7 3.7 3.1 2) 8.54 0.614 700.0 14.2 12.2 10.6 9.4 85 6.1 5.3 4.7 4.2 3.4 2.8 21 8.13 0.681 735.0 12.9 11.1 9.6 8.6 7.7 5.5 4.8 4.3 3.8 3.1 2.5 22 7.75 0.752 770.0 11.7 10.0 9.1 7.8 7.0 5.0 4.4 3.9 3.5 2.8 2.3 2> 7.43 0.823 805.0 10.7 9.2 8.0 7.2 6.4 4.6 4.0 3.5 3.2 2.5 2.1 24 7.12 0.903 840.0 9.8 8.4 7.4 6.5 5.9 4.2 3.7 3.2 2.9 2.3 1.9 25 6.83 0.980 875.0 9.1 7.8 6.8 6.0 5.4 3.9 3.4 3.0 2.7 2.1 1.8 2 , 657 1.067 910.0 8.4 7.2 6.3 5.6 5.0 3.6 3.1 2.8 2.5 20 1.6 27 G.32 1.154 945.0 7.8 6.7 5.8 5.2 4.6 3.3 2.9 2.6 2.3 1.8 1.5 28 6.10 1.251 980.0 7.2 6.2 5.4 4.8 4.3 3.1 2.7 2.4 2.1 1.7 1.4 2!) 5.89 1.346 1015.0 6.7 5.8 5.0 4.5 4.0 2.9 2.5 2.2 2.0 1.6 1.3 30 5.69 1.450 1050.0 6.3 5.4 4.7 4.2 3.7 2.7 2.3 2.1 1.8 .5 1.2 31 5.51 1.556 1085.0 5.9 5.1 4.4 3.9 3.5 2.5 2.2 1.9 1.7 A 1.1 32 5.31 1.672 1120.0 5.5 4.7 4.1 3.7 3.3 2.3 2.0 1.8 1.6 .3 1.1 33 5.17 1.783 1 155.0 5.2 4.4 3.9 3.4 3.1 2.2 1.9 1.7 1.5 .2 1.0 34 5.02 1.906 1190.0 4.8 4.2 3.6 3.2 2.9 2.1 1.8 1.6 1.4 !T 35 4.88 2.033 1225.0 4.6 4.0 3.4 3.1 2.7 1.9 1.7 1.5 1.3 .1 36 4.69 2.143 1200.0 4.3 3.7 3.2 2.8 2.6 1.8 1.6 1.4 1.3 1.0 37 4.61 2.297 1295.0 4.1 3.5 3.1 2.7 2.4 1.7 1.5 1.3 1.2 38 4.50 2.444 1330.0 3.9 3.3 2.9 2.6 2.3 1.6 1.4 1.3 1.1 39 4.38 2589 1365.0 3.6 3.2 2.8 2.5 2.2 1.6 1.4 1.2 1.1 40 4.20 2.711 1400.0 3.5 3.0 2.6 2.3 2.1 1.5 1.3 1.1 1.0 RESISTANCE TO CROSS- BREAKING AND SHEARING. 47 Fig. 80. 9 /x " Heavy" Beam. Weight per If. = 50 Ibs. Sectional area = 15.0" Moment of inertia / = 188.55 Constant /f 209.50 K 03 03 Distance d bet. centres of beams in feet, for N o fl C 03 weight in Ibs. per sq. foot of 3 o O3 O U "3 J.S 6 6 o il 03 a 03 g 03 OB w g JC -A ,0 .2 Q ft CS o o Q I p ,Q O So i 8 O 1 1 8 1 6 36.91 0.065 300.0 7 29.92 0.084 3500 8 26.1 8 0.111 400.0 21.8 9 23.27 0.141 450.0 20. (i 17.2 10 20.95 0.174 500.0 23.2 20.9 16.7 13.9 11 19.01 0.211 550.0 21.6 19.1 17.3 13.7 11.5 12 17.45 0.253 600.0 20.7 18.1 16.1 14.5 11.6 9.6 13 16.11 0297 650.0 17.C 15.4 13.8 12.3 9.9 8.2 14 14.96 0.345 700.0 21.3 15.2 13.3 11.8 10.C 8.5 7.1 15 13.96 0.398 750.0 20.6 18.0 13.2 11.6 10.3 9.3 7.5 6.2 16 13.09 0.454 800.0 20.4 19.5 16.3 ll.(j 10.2 9.0 8.1 6.5 5.4 17 12.32 0.515 850.0 20.7 18.1 16.1 14.5 10.3 9.0 8.0 7.2 5.7 4.8 18 11.63 0.580 900.0 21.5 18.4 16.1 14.3 12.9 9.2 8.0 7.1 6.4 5.1 4.3 19 11.02 0.648 950.0 19.3 16.5 14.5 12.8 11.6 8.2 7.2 6.4 5.8 4.6 3.8 20 10.47 0.722 1000.0 17.4 14.9 13.0 11.4 10.4 7.4 6.5 5.8 5.2 4.1 3.4 21 9.97 0.799 1050.0 15.8 13.5 11.8 10.5 9.4 6.8 5.9 5.2 4.7 3.4 3.1 22 9.52 0.883 11000 14.4 12.3 10.3 9.6 8.2 6.1 5.4 4.8 4.3 3.4 2.8 23 9.10 0.968 1150.0 13.1 11.3 9.8 8.7 7.9 5.6 4.9 4.3 3.9 3.1 2.6 24 8.72 1.002 1200.0 12.1 10.3 9.0 8.0 7.2 5.1 4.5 4.0 3.6 2.9 2.4 25 8.34 1.152 1250.0 11.1 9.5 8.3 7.4 6.6 4.7 4.1 3.7 3.3 2.G 2.2 26 8.05 1.258 1300.0 10.3 8.8 7.7 6.8 6.1 4.4 3". 8 3.4 3.0 2.4 2.0 27 7.75 1.363 1:550.0 9.5 8.0 7.1 6.3 5.7 4.1 3.5 3.1 2.8 2.2 1.9 28 7.48 1.477 1400.0 8.9 7.6 6.6 5.9 5.3 3.8 3.3 2.9 2.6 2.1 1.7 29 7.22 1.593 1450.0 8.3 7.2 6.2 5.5 4.9 3.5 3.1 2.7 2.5 1.9 1.6 30 6.98 1.718 1500.0 7.7 6.6 5.8 5.1 4.6 3.3 2.9 2.5 2.3 1.8 1.5 31 6.75 1.846 1550.0 7.3 6.2 5.4 4.8 4.3 3.1 2.7 2.4 2.1 1.7 1.4 32 6.54 1.982 1GOO.O 6.7 5.8 5.1 4.4 4.0 2.9 2.5 2.2 2.0 1.6 1.3 33 6.34 2.119 1650.0 6.4 5.6 4.8 4.2 3.8 2*.7 2.4 2.1 1.9 1.5 1.2 34 6.16 2.265 1700.0 6.0 5.1 4.5 4.0 3.6 2.5 2.2 2.0 1.8 1.4 1.2 35 5.98 2.416 1750.0 5.6 4.8 4.2 3.7 3.4 2.4 2.1 1.8 1.7 1.3 1.1 36 5.81 2.577 1800.0 5.3 4.6 4.0 3.5 3.3 2^3 2^0 1.7 1.6 1.2 1.0 37 5.66 2.742 1850.0 5.0 4.3 3.8 3.3 3.0 2.1 1.9 1.6 1.5 1.2 38 5.51 2.918 1900.0 4.8 4.1 3.6 3.2 2.9 2.0 1.8 1.6 1.4 1.1 39 5.37 3.098 1950.0 4.5 3.9 3.4 3.0 2.7 1.9 1.7 15 1.3 1.1 40 5.27 3.289 2000.0 4.3 3.7 3.2 2.9 2.6 1.8 1.6 1.4 1.3 l.C 43 RESISTANCE TO CROSS BREAKING AND SHEARING. 9" " Medium" Beam. Weight per If. = 30 Ibs. Sectional aren = 9.0" Moment of inertia I = 111.32 Constant K = 123.09 K f oc p <D Distance d bet. centres of beams in feet, for o 1 weight in Ibs. per sq. foot of <n i ft .2 ~j G5 S .0 O 9 Itf) J d g " J3 | 00 JO s JO 1 n .0 s 6 O ft 9 p o s 8 8 1 1 1 6 20.00 0.062 1800 22 7 17.07 0.085 210.0 25.0 20. ( 16.0 8 15.40 0.111 240.0 21.0 21.0 in n 15.0 12^0 9 13.74 0.141 270.0 21.0 19.0 100 15.0 11.0 10.0 10 12.36 0.174 300.0 17.0 15.0 13.0 12.0 9.8 8^2 11 11.24 0.211 330.0 22.0 20.0 14.0 12.0 11.01 10.0 8.1 6.8 12 10.30 0.252 300.0 21.0 19.0 17.0 12.0 10.0 9.5 1 8.5 6.8 5.7 13 9.51 0.297 390.0 2J.O 18.0 10.0 14.0 10.0 9.1 8.1! 7.3 5.8 4.8 14 8.83 0.345 420.0 21.0 18.0 15.0 14.0 12.0 9.0 7.8 7.0i 6.3 5.0 4.2 15 8.24 0.398 450.0 18.0 15.0 13.0 12.0 10.0 7.8 6.8 6.11 5.4 4.3 3.6 16 7.73 0.455 480.0 10.0 13.0 12.0 10.0 9.6 6.9 6.0 5.3; 4.8 3.8 3.2 17 7.21 0.511 510.0 14.0 12.0 10.0 9.4 8.4 6.0 5.3 4.7i 4.2 3.2 2.8 18 6.87 0.580 540-0 12.0 10.0 9.5 8.4 7.6 5.4 4.7 4.2^ 3.8 3.0 2.5 19 6.51 0.650 570.0 11.0 9.7 8.5 7.0 6.8 4.8 4.2 3.8 3.4 2.7 2.2 20 6.18 0.722 000.0 10.0 8.8 7.7 6.8 6.1 4.4 3.8 3.4 3.0 2.4 2.0 21 5.88 0799 630.0 9.3 8.0 7.0 6.2 5.6 4.0 3.5 3.1 2.8 2.2 1.8 22 5.02 0.884 660.0 8.5 7.2 6.3 5.0 5.1 3.6 3.1 2.8 2.5 2.0 1.7 23 5.37 0.969 6DO.O 7.7 6.6 5.8 5.1 4.6 3.3 2.9 2.5 2.3 1.8 1.5 24 5.15 1.065 720.0 7.1 6.1 5.3 4.7 4.2 3.0 . 2.0 2.3 2.1 1.7 1.4 25 4.94 1.157 750.0 6.5 5.6 4.9 4 3 3.9 2.8 2.5 2.1 1.9 1.5 1.3 26 4.83 1.277 780.0 6.1 5.3 4.6 4J 3.7 2.6 2.3 2.0 1.8 1.4 1.2 27 4.58 1.365 810.0 5.6 4.8 4.2 3.7 3.3 2.4 2.1 1.8 1.6 1.3 1.1 28 4.41 1.476 840.0 5.2 4.5 3.9 3.5 3.1 2.2 1.9 1.7 1.5 1.2 1.0 29 4.20 1.593 870.0 4.8 4.1 3.6 3.2 2.9 2.0 1.8 1.6 1.4 1.1 30 4.12 1.718 900.0 4.5 3.9 3.4 3.0 2.7 1.9 1.7 1.5 1.3 1.0 31 3.99 1.846 930.0 4.2 3.0 3.2 2.8 2.5 1.8 1.6 1.4 1.2 1.0 32 3.80 1.982 900.0 4.0 3.4 3.0 2.6 2.4 1.7 1.5 1.3 1.2 33 3.74 2.119 9900 3.7 3.2 2.8 2.5 2.2 1.0 1.4 1.2 1.1 34 3.03 2.235 1020.0 3.5 3.0 2.6 2.3 2.1 1.5 1.3 1.1 1.0 35 3.53 2.416 1050.0 3.3 2.8 2.5 2.2 2.0 1.4 1.2 1.1 1.0 36 3.43 2.577 1080.0 3.1 2.7 2.3 2.1 1.9 1.3 1.1 1.0 37 3.34 2.742 1110.0 3.0 2.5 2.2 2.0 1.8 1.2 1.1 1.0 38 3.25 2.918 1140.0 28 2.4 2.1 1.9 1.7 1.2 1.0 39 3.17 3.098 1170.0 2.7 2.3 2.0 1.7 1.6 1.1 1.0 40 3.09 3.289 1200.0 2.5 2.2 1.9 1.6 1.5 1.1 RESISTANCE TO CROSS BREAKING AND SHEARING. Fig. 88. 9" "UgU" Beam. Weight per If. = 23.33 Ibs. Sectional area = 7.0" Moment of inertia /== 91.00 Constant K r =101.2 K 3 CQ e CO o Distance d bet. centres of beams in feet, for 1 3 .5 JG o c rc ,Q weight in pounds per sq. foot of CO 0) G fl J.S o d HP a! i -; 02 1 .0 | ,0 1 .a JD 01 ft 6 ft 8 i 8 5 g 1 g 1 i 1 g 16.86 0.002 140.0 22.0 18.7 7 14.45 035 103.3 i;> 5 79 () 20.0 .0.0 13.7 12.05 0.111 180.0 25.0 19.7 17.5 15.8 [2.0 10.5 9 11.24 0.141 21o o 17.8 15.6 13.8 10.1 8*.6 10 10.12 0.175 233.3 22.0 20.0 14.4 12.0 11.2 iai 8.0 6.7 11 9.20 0.212 250.0 21.0 18.7 10.7 11.9 10.4 9.2 8.3 0.7 5.5 12 8.43 0.253 280.0 23.0 20.0 17.5 15.0 14.0 10.0 8.7 7.8 7.0 5.6 4.6 13 7.78 0.297 303.3 19.9 17.9 14.9 13.4 11.9 8.5 7.4 0.0 5.9 4.8 3.0 14 7.22 0.345 320.0 17.1 14.7 12.8 11.4 10.3 7.3 0.4 5.7 5.1 4.1 3.4 15 6.74 0.398 350.0 14.9 12.9 11.2 10.0 8.9 6.4 5.0 5.0 4.5 3.0 2.$ 16 6.31 0.453 373.3 13.1 11.2 9.8 8.7 7.8 5.0 4.9 4.3 3.9 3.1 2.6 17 5.95 0.510 390.0 11.6 10.0 8.7 7.8 7.0 5.0 4.3 3.8 3.5 2.8 2.3 18 5.02 0.579 420.0 10.4 8.9 7.8 0.9 0.2 4.4 3.9 3.4 3.1 2.5 2.0 19 5.32 0.048 443.3 9.3 8.0 7.0 0.2 5.0 4.0 3.5 3.1 2.8 2.2 1.8 20 5.00 0.721 466.6 8.4 7.2 03 5.0 5.0 3.0 3.1 2.8 2.5 2.0 1.0 21 4.81 0.797 490.0 7.6 G.5 5.7 5.1 4.5 3.2 2.8 2.5 2.2 1.8 1.5 22 4.59 0.879 513.3 0.9 5.9 5.2 4.0 4.1 2.9 2.0 2.3 2.1 1.7 1.3 23 4.40 0.908 536.6 0.3 5.5 4.7 4.2 3.8 2.7 2.3 2.1 1.9 1.5 1.2 24 4.21 1.000 500.0 5.8 5.0 4.3 3.8 3.5 2.5 2.1 1.9 1.7 1.4 1.1 25 4.04 1.151 583.3 5.3 4-G 4.0 3.5 3.2 2.3 2.0 1.7 1.0 1.2 1.0 26 3.89 1.254 GOO.G 4.9 4.2 3.7 3.3 2.9 2.1 1.8 1.0 1.4 1.1 27 3.74 1.359 030.0 4.0 3.9 3.4 3.0 2.7 1.9 1.7 1.5 1.3 10 28 3.00 1.400 653.3 4.2 3.0 3.2 2.8 2.5 1.8 1.0 .4 1.2 1.0 29 3.48 1.582 676.6 4.0 3.4 3.0 2.0 2.4 1.7 1.5 .3 1.1 30 3.37 1.711 700.0 3.7 3.2 2.8 2.5 2 2 1.0 1.4 1.2 1.1 31 3.20 1.837 723.3 3.5 3.0 2.0 23 2.1 1.5 1.3 .1 1.0 32 3.10 1.968 746.6 3.2 2.8 2.4 2.1 1.9 1.4 12 l.o 33 3.00 2.104 770.0 3.0 2.0 2.3 2.0 1.8 1.3 1.1 34 2.97 2.250 793.3 2.9 2.4 2.1 1.9 1.7 1.2 l.o 35 2.89 2.399 810.G 2.7 2.3 2.C 1.8 1.0 1.1 36 2.81 2.505 840.0 2.0 2.2 1.9 1.7 1.5 1.0 37 2.73 2.723 803.3 2.4 2.1 1.8 1.0 1.4 38 2.06 2.898 880.6 2.3 2.0 1.7 1.5 1.4 39 2.59 3.070 910.0 9 9 1 i r 1 4 1 .} 40 2.52 3.250 2.1 1.8 1.5 1.4 1.2 50 RESISTANCE TO CROSS-BREAKING AND SHEARING. 8" Beam. Weight per If. -= 21.66 Ibs. Soctionnl area = 6.5" Moment of inertia I = 05.09 Constant, K = 82.49 K -2 "tf =Q C o> Distance d bet. centres of beams in feet, for p. P.*- c c 3 "3 c ri JB weight in pounds per pq. foot of "c c ffi c; c d 09 32 e 2 tC 1 1 1 Ja J | 5 B _C i 1 w X> cr s 6 & 3 i 55 i 1 g i o o i 1 G 13.74 0.070 130.0 oo 9 18 3 Ti 7 11.78 0.095 151.6 21. 18. "> 10.8 13.-; 11.2 g 10.30 0.124 173.3 >- Y 18.3 1 0.0 14. 3 12.8 10.3 8.5 9 9.16 0.158 195.0 . ().:> 1 l.fi 12.7 11.3 10.1 8.1 6.7 10 8.23 0.198 216.6 J0.5 1X.3 10.4 11.7 10.2 9.1 8.2 6.5 5.4 11 7.49 0.238 238.3 22.C 19.4 17.0 15.1 l:j.G 9.7 8.5 7.5 6.8 5.4 4.5 12 6.87 0.284 260.0 19.0 163 L4.8 12.7 11.4 8.1 7.1 6.3 5.7 4.5 3.8 13 634 0.335 281.6 16.2 14.0 12.1 10.9 7 0.9 6.0 5.4 4.8 3.9 3.2 14 5.89 0.390 303.3 14.0 12.0 10.5 9.:} 8.4 0.0 5.2 4.0 4.2 3.3 2.8 15 6.49 0447 325.0 12,2 10.5 9.1 8,1 7.3 5.2 4.5 4.0 3.6 2.9 2.4 16 5.15 0.511 346.6 10.7 9.1 8.0 7,1 6.4 4.5 4.0 3.6 3.2 2.5 2.1 17 4.85 0.580 368.3 9.5 .0 7.1 6.3 5.7 4.0 3.7 3.2 2.8 2.3 1.9 18 4.58 0.653 390.0 8.4 7.2 6.3 5.6 5.1 3.8 3.1 2.7 2.5 2.0 1.7 19 4.34 0.731 411.6 7.6 65 5.7 5.1 4.5 3.3 29 B5 22 1.8 1.5 20 4.11 0.810 433.3 6.8 5.8 5.1 4.5 4.1 29 2.5 2.2 2.0 1.6 1.3 21 3.92 0.898 455.0 0.2 5.3 4.6 4.1 3.7 2.6 2.3 2.0 1.8 1.4 1.2 22 3.73 0.989 476.6 5.6 4.8 4.2 3.7 3.4 2.4 2.1 1.8 1.6 1.3 1.1 23 358 1.090 498.3 5.1 4.4 3.8 3.4 3.1 2.2 1.9 1.7 1.5 1.2 1.0 24 3.42 1.192 520.0 4.7 4.0 3.5 3.1 2.8 2.0 1.7 1.5 .4 1.1 25 3.29 1.300 541.6 4.3 3.7 3.2 2.9 2.6 1.8 1.5 1.4 1.3 1.0 26 3.17 1.417 563.3 4.0 3.4 3.0 2.7 2.4 1.7 1.4 1.3 1.2 27 3.05 1.536 585.0 3.7 3.2 2,8 2.5 2.2 1.6 1.4 1.2 1.1 28 2.94 1.602 606.6 3.5 3.0 2.6 2.3 2.1 1.5 1.3 1.1 1.0 29 2.84 1.795 628.3 3.2 2.8 2.4 2.1 1.9 1.4 1.2 1.0 30 2.73 1.923 650.0 3.0 2.6 2.2 2.0 1.8 1.3 1.1 | 31 2.66 2.080 671.6 2.8 2.4 2.1 1.9 1.7 1.2 1.0 2.56 2.219 693.3 2.6 2.2 2.0 1.7 1.0 1.1 33 2.49 2.383 715.0 2.5 2.1 1.8 1.6 1.4 1.0 34 2.42 2.550 736.6 2.3 2.0 1.7 1.5 1.4 35 2.35 2.722 758.3 2.2 1.9 1.6 1.4 1.3 36 2.29 2.907 780.0 2.1 1.8 1.5 1.4 1.2 37 2.22 3.084 801.6 2.0 1.7 1.5 1.3 12 38 2.17 3.290 823.3 1.9 1.6 1.4 1.2 1.1 39 2.11 3.484 845.0 1.8 1.5 1.3 1.2 1.0 40 2.06 3.702 866.6 1.7 1.4 1.2 1.1 1.0 ] BESISTANCE TO CEOSS-BBEAZING AND SHEABING. 51 Fig. 90. 7" Seam. Weight per If. = 18.33 Ibs. Sectional area == 55" Moment of inertia / = 44.84 Constant # =64.06 K f e jj 02 Distance d bet. centres of beams in feet, for O p. 1 A o m .0 weight in Ibs. per sq. foot of 13 O 2 .s """ OD CD 5 c5 d JbK) 03 so , | t | .JO i i X s OS O 1 i | i O 05 3 1 1 1 6 10.67 0.080 110.0 25.4 22 2 H).7 17.7 14.2 11 8 7 9.15 0.109 18.6 16.3 14/> 13,0 10.5 8 .7 8 8.00 0.143 14&6 25. 6 22.2 20. (! 14.2 12.5 11.1 10.0 8.0 6.6 9 7.11 0.181 165.0 2*2.9 19.7 17.5 16.8 11.2 9.8 8.7 7.9 6.3 5.2 10 6.40 0.224 183.3 2L3 18.2 16.0 14.2 12.8 9.1 8.0 7.1 6.4 5.1 4.2 11 5.82 0.272 201.6 17.6 15.3 13.2 11.7 10.5 7.5 6.6 5.8 5.2 4.2 3.5 12 5.33 0.325 220.0 14.8 12.6 11.1 9.8 8.8 6.3 5.5 4.9 4.4 3.5 2.9 13 4.92 0.382 238.3 12.6 10.9 9.4 8.3 7.5 5.4 4.7 4.1 3.7 3.0 2.5 14 4.56 0.444 256.6 10.8 9.3 8.1 7.2 6.5 4.6 4.0 3.6 3.2 2.6 2.1 15 4.27 0.513 275.0 9.4 8.2 7.1 6.3 5.7 4.0 35 3.1 2.8 2.2 1.8 16 3.99 0.585 293.3 8.3 7.1 6.2 5.5 4.9 3.5 3.1 2.7 2.4 1.9 1.6 17 3.76 0.665 311.6 7.3 6.5 5.5 4.9 4.4 3.1 2.7 2.3 2.1 1.7 1.4 18 3.55 0.749 330.0 6.5 5.6 4.9 4.3 3.9 2.8 24 2.1 1.9 1.5 1.3 19 3.37 0.840 348.3 5.9 5.1 4.4 3.9 3.5 2.5 2.2 1.9 1.7 1.4 1.1 20 3.20 0.936 366.6 5.3 4.5 4.0 3.5 3.2 2.2 2.0 1.7 1.6 1.2 1.0 21 3.05 1.038 385.0 4.8 4.1 3.6 3.2 2.9 2.0 1.8 1.6 1.4 1.1 22 2.91 1.146 403.3 4.4 3.7 3.3 2.9 2.6 1.8 1.6 1.4 1.3 1.0 23 2.78 1.257 421.6 4.0 3.4 3.0 2.7 2.4 1.7 1.5 1.3 1.2 24 2.66 1.381 440.0 3.6 3.1 2.7 2.4 2.2 1.6 1.3 1.2 1.1 25 2.56 1.504 458.3 3.4 2.9 2.5 2.2 2.0 1.5 1.2 1.1 1.0 26 2.45 1.630 476.6 3.1 2.6 2.3 2.0 18 1.4 1.1 27 2.37 1.775 495.0 2.9 2.5 2.1 1.9 1.7 1.3 1.0 28 2.27 1.871 513.3 2.7 2.3 2.0 1.8 1.6 1.2 29 2.20 2.075 531.6 2.5 2.1 1.8 1.7 1.5 11 30 2.12 2.229 550.0 2.3 2.0 1.7 1.5 1.4 1.0 52 RESISTANCE TO CROSS- BREAKING AND SHEARING. 6" Seam. Weight per If. = 13.33 Ibs. I 0.28" ^pfc Sectional area =. 4.0" Moment of inertia I = 22.5 Constant K = 37.64 K W . - K or: 0) Distance d bet. centres of beams in feet, for a . 1 00* weight in Ibs. per sq. foot of * d .5 s ?c d d ".S 05 6 o a n J j 03 jQ > JO 2 J J s Bi o o Q 1 i i i o Ci 8 o o CD 1 1 1 6 6.27 094 80.0 23.2 20.9 14.9 13.0 11.6 10.4 8.3 6.9 7 5.37 o!l28 93.3 19.1 17.3 L5.3 lo .o 9.5 8.4 7.6 6.1 5.1 8 .168 106^6 19.5 16.8 14.6 13.0 11.7 8.5 6.5 5.8 4.7 3.9 9 4J8 0.213 120.0 15.4 13.4 11.6 10.4 9.2 6.6 sis 5.1 4.G 3.7 3.1 10 3.75 0.263 133.3 12.5 10.7 9.3 8.3 7.5 5.3 4.7 4.1 3.7 3.0 2.5 11 3.42 0.320 146.6 10.3 9.0 7.7 6.9 6.2 4.4 3.8 3.4 3.1 2.4 2.0 12 3.13 0.382 160.0 8.6 7.0 6.5 5.7 5.2 3.7 3.2 2.9 2.6 2.0 1.7 13 2.89 0.450 173.3 7.4 6.4 5.5 4.9 4.4 3.1 2.7 2.4 2.2 1.7 1.4 14 2.68 0.524 186.6 6.3 5.4 4.7 4.2 3.8 2.7 2.3 2.1 1.9 1.5 1.2 15 2.51 0.607 200.0 5.5 4.8 4.2 3.7 3.3 2.3 2.1 1.8 1.6 1.3 1.1 16 2.34 0.689 213.3 4.8 41 3.6 3.2 2.9 2.0 1.8 1.6 1A 1.1 17 2.21 0.786 226.6 4.3 3.7 3.2 2.8 2.5 1.8 1.6 1.4 1.8 18 2.09 0.888 240.0 3.8 3.3 2.9 2.5 2.3 1.6 1.4 1.2 1.1 19 1.98 0.995 253.3 3.4 3.0 2.6 2 3 2.1 1.4 1.3 1.1 20 1.88 1.110 266.6 3.1 2.7 2.3 2.1 1.8 1.3 1.1 21 1.79 1.231 280.0 2.8 2.4 2.1 1.9 1.7 1.2 1.0 22 1.70 1.350 293.3 2.5 2.2 1.9 1.7 1.5 1.1 23 1.63 1.493 306.6 2.3 2.0 1.7 1.5 A 1.0 24 1.56 1.641 320.0 2.1 1.8 1.6 1.4 .3 25 1.50 1.787 333.3 2.0 1.7 1.5 1.3 .2 26 1.44 1.950 346.6 1.8 1.5 1.3 1.2 .1 27 1.39 2.129 360.0 1.7 1.4 1.2 1.1 28 1.33 2.286 373.3 1.5 1.3 1.1 29 1.29 2.489 386.6 1.4 1.2 1.0 30 1.25 2.698 400.0 1.3 1.1 RESISTANCE TO CROSS-BREAKING AND SHEARING. 53 CAST-IRON BEAMS. Factor of rupture for cast-iron beams of various sections. The factor C is based on practical experiments by Hodgkinson- Its value alters with the different proportions of the cross-sections of beam. Beam supported at the ends ; load concentrated at the center. Reference. = Factor of rupture. W= Breaking weight in Ibs. A = Sectional area of beam in square inches. I = Distance between supports in inches. h = Height of beam in inches. Dimensions in inches. 6 = Thickness of web at center is the unit. Fig. 92. 0.32 = 0.726 5.125 = 11.646 0.44=6 0.47 = 1.066 10.52 = 1.186 Fig. 93. 2.27 = 5.156 . = 3.20 (7=27292 1.74 = 5.86 JJT 5.125 = 17.086 .0.26 = 0.866 0.30 = 6 . 55 = 1.736 l. 78~="5?936 = 2.73 (7=28513 BESISTANCE TO CEOSS-BEEAKING AND SHEAEING. Fig. 2^. 1.07 = 3.346 =0.946 5.125 = 166 2.10 = 6.566 = 2.88 (7=30330 . 95. 1.6 = 4.216 " 5.125 = 13.486 . 315 = 0.826 0.38 = 6 0.53 = 1.396 Fig. 96. 4.16 = 10.946 4.33 C= 35262 2.33 = 8.756 5.125=19.266 0.31 = 1.166 = 6 = 2.486 6.67 = 25.076 6.23 (7=44176 RESISTANCE TO CROSS-BREAKING AND SHEARING. 55 10 r^ CD CO a (M JO O 56 EESISTANCE TO CROSS-BREAKING AND SHEARING. -5 ^ o"* C -<3j rn ~ & S jo O 8 RESISTANCE TO CROSS-BREAKING AND SHEARING. 57 o CD r-i W) ^ T3 2 >> fl Cf- -FH -ua o O> 00 ^D S 1 f 1 IS! 3 -s a CX .2 cj a K ^ T 1 2 H H g^ O rH a g 5 .2 CN O - 1-1 -Is I S r& O CD CD CQ T-\ "* " ^ -j e8 | a 3 1 S S o CD O M 58 RESISTANCE TO CROSS- BREAKING AND SHEARING. Load concentrated at center: W= r-, or K l L W. t Beam fixed at one end; principal flange at top. Load equally distributed: TF= -^-p or JT 1 = 2 .1. W. . V K l Load concentrated at free end: W -7-7-, or K l = 4 .1. W. . 4 .1 [NOTE. The more the sectional area is contained in coefficient ITi, the more is the section economical.] EXAMPLE. Section No. 34. Load equally distributed; beam supported at both ends; thickness of web = 1 inch: thickness of flange = 1} inch; height = 10 inches; width of flange = 5.9 inches. Distance between supports = 20 feet = 240 inches. K l 658 W = -jy- = ^- = 5.48 tons capacity. T Tfi The moment of resistance of cross-section = - r- RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. ^3 2P-S |IB Sectional area in square incnes. Coefficient K.I Fig. 102. 1 6 5.0 10.0 238 2 6} 5.2 10.7 280 ^ ft ^ i 3 7 5.5 11.5 322 ^ j JT 4 7} 5.7 12.2 364 1 5 8 6.0 13 420 N**. _Z"]lpl^iP^ _.$., 6 gi 6.2 13.7 476 Fig. 103. 7 9 6.5 14.5 532 \ lj. 8 9} 6.7 15.2 602 9 10 6.9 15.9 672 4c 10 10} 7.1 16.6 742 11 11 7.4 17.4 812 > ^^^i _ N /___ 12 11} 7.6 18 1 882 K""*" -4 13 12 79 18.9 966 Fig. 104. t27 14 12} 8.1 19.6 1050 ! * 15 13 8.4 20.4 1134 ^ i 16 13} 8.6 21.1 1232 % \ . ~ 17 14 8.8 21.8 1316 % i 1 Q 1/11 5^^^4^~. 1 " lo 14} ; 9.0 22.5 1428 k-"--B -i" 19 15 9.3 23.3 1526 Fig. 105. 20 15} 9.5 24.0 1624 : * I F 21 16 9.8 24.8 1750 .: 1 I / ! \ -\ 22 16} 10.0 25.5 1848 \k I ! 23 17 10.3 26.3 1960 1 ! 24 17} 10.5 27.0 2086 _ .. J. " ;/;:,;% * [ jg ->;_ 25 18 10.8 27.8 2212 60 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. .2 Width B of lower flange in inches. Sectional area in square inches. Coefficient K.i Fig. 106. 26 6 4.5 10.4 224 i- --> 27 6} 4.6 11.1 266 t i 28 7 4.8 11.8 322 y- 29 7j 5.0 12.5 364 - \ P 30 8 5.2 13.2 420 31 gl 5.4 13.9 476 Fig. 107. 32 2 9 5.6 14.7 532 \J"/ 33 9} 5.7 15.4 588 P -- -y 34 10 5.9 16.2 658 " 35 10$ 6.1 16.9 728 36 11 6.3 17.6 798 | i 1 37 11} 6.5 18.3 882 ,- k >^- >; 38 12 6.7 19.1 952 Fig. 108. 39 12} 6.9 19.8 1036 ^ ir 40 13 7.1 20.6 1134 | i 41 13} 7.3 21.3 1218 rr , 42 14 7.5 22.1 1316 I ^ 1 ! 43 14} 7.7 22.8 1414 E]l^ !< :. -#- >! ?* 44 15 7.9 23.6 1512 Fig. 109. 45 15J 8.0 24.3 1610 >i \/gi i IT 46 47 16 16J 8.2 8.4 25.1 25.8 1722 1834 i 48 17 8.6 26.5 1946 1 * ^ 1 1 49 1VJ 8.8 27.2 2072 sk KA I Q 9.0 28.0 2198 ,_..-S- H- O\J J.O RESISTANCE TO CKOSS-BREAKING AND SHEAEING. 61 Number of section. Height H in inches. *!t Sectional area in square inches. Coefficient Ji I ig. 110. 51 6 4.2 10.5 224 \ 2 ! 1" T 52 6} 4.3 11.4 266 1 \ % i 53 7 4.5 12.3 308 " f 54 7} 4.6 12.9 364 ^ I 55 8 4.7 13.6 406 z^flltll ^ 1 A O K B > 56 8 i 4.8 14.3 462 I fy. 111. 57 9 5.0 15.0 532 V 7 58 91 5.1 15.7 588 m % 1 \ 59 10 5.3 16.5 658 1 J 60 10} 5.4 17.2 728 1 \ \ r^r, 61 11 5.6 17.9 798 62 11} 5.7 18.6 868 * ^-- - 63 12 5.9 19.4 952 f 7 ^. 112. 64 12} 6.0 20.1 1036 p"~* ^ 65 13 6.3 20.9 1120 3T 66 13} 6.4 21.6 1204 M w ! 67 14 6.6 22.4 1302 W 1 ^ \ -** i A f\r\ HHf wSm^. 68 14} 6.7 23.1 14UU <_ J J- >| 69 15 6.9 23.8 1498 J V 113. 70 15} 7.0 24.5 1610 \jj _^ i f hf Q oc q i >7ns 1 lo / .4 ZiO . O 1 <UO 1 \ 72 16} 7.3 26.0 1820 i 73 17 7.5 26.8 1932 i } 1 74 17} 7.7 27.5 2058 4^ fer-L B- ! 75 18 7.9 28.3 2184 62 RESISTANCE TO CROSS -BREAKING AND SHEARING; o . ^g c^c -so tt s ^ o 31&I SM 1 I SgJ o^ * a> W.S 1 * %- o o Fig. 114. I frr? 76 6 4.0 12.0 224 i 77 7 4.1 13.1 308 i * 78 8 4.2 14.4 406 -F?>. 115. 79 9 4.4 15.7 518 VT ~A~ 1 i 80 10 4.6 17.1 644 * | I 81 11 4.8 18.6 784 "|P|| ~~] I i *- -H 82 12 5.0 20.0 938 %, 116. 1" ^ "" i 83 13 5.2 21.4 1106 aJ, ^ 84 14 5.5 22.9 1288 1 TS2P 7 <L jB >| 85 15 5.7 24.4 1484 Fig. 117. ^ ^T- 86 16 5.9 25.8 1694 1 1 1 i 1L 87 17 6.2 27.3 1918 1 1 i i_ 88 18 6.4 28.8 2156 ^---^""^ RESISTANCE TO CROSS-BREAKING AND SHEARING. 63 Number of section. 51 II "o gj (33 ^ fl 5 -^ rf^ Sectional area in square incnes. Coefficient JZX Fig. 118. 89 6 5.6 12.9 294 W. ^ 90 6} 5.8 13,8 336 I U I I 91 7 6.0 14.7 392 B _ 92 ff 6.2 15.5 448 v 93 8 6.4 16.4 518 j0gH|BHB-i* -. 94 ii 6.6 17.3 588 s \e. .ft _>| 2 Fig.V&l 95 9 6.9 18.3 658 . // 96 9} 7.1 19.2 742 T 97 10 7.4 20.2 826 & 98 10} 7.6 21.1 910 99 11 7.9 22.1 1008 i 100 Hi 8.1 23.0 1106 . 101 12 8.4 23.9 1204 Fig. 120. 102 8.6 24.8 1302 w~ \~ 103 13 8.9 25.8 1414 ^ 7T m JL 104 13} 9.1 26.7 1526 1 t - ^-" : I ! 105 1 HA 14 9.4 9r> 27.7 1652 1 7/2/1 i&" lUb *?J . b 28.5 1754: E- -J5- r^- 107 15 9.8 29.4 1890 Fig. 121. \% " \%? 108 15} 10.0 30.3 2030 ITT 109 16 10.3 31.3 2156 % \ | i 110 16} 10.5 32.2 2296 tj *i 111 17 10.8 33.2 2436 1 ! 112 171 11.0 34.1 2590 -i 113 18 11.3 35.0 2730 <____JB V; 64 RESISTANCE TO CROSS-BREAKING AND SHEARING. I \ ig. 122. 2%r Number of section. 1 X3 O bJD _e T3~ G ^ Sectional a r e a i n square inches. Coefficient jffl. 114 115 116 6 6} 7 5.3 5.4 5.6 13.6 14.4 15.3 280 336 392 I J5 r 117 7} 5.7 16.1 448 M 118 119 120 8 3} 9 5.9 6.0 6.2 17.0 17.8 18.7 518 588 658 ^^iltl^^s v Fig. 123. \ S- -* 121 9 i 6.4 19.6 742 1 122 10 6.6 20.5 814 r |l23 10} 6.8 21.4 910 v ; 124 11 7.0 22.4 994 T&T 4 l ^H i 125 126 127 128 11} 12 12} 13 7.2 7.4 7.6 7.8 23.3 24.2 25.1 26.1 1092 1190 1288 1400 Fig. 124. \ A ^ J r 129 13} 8.0 27.0 1512 H i 1 130 131 132 14 14} 15 8.2 8.4 8.6 27.9 28.8 29.8 1624 1750 1876 i ;!%* . ~i 5 Fig. 125. \5/? \5/ o y 133 15} 8.8 30.7 2002 = $ /a t-x 134 16 9.0 31.6 2142 1 ~ | | 135 16} 9.2 32.5 2282 1 < ^ k 136 137 138 17 18 9.4 9.6 9.8 33.5 34.4 35.3 2422 2562 2716 B- EESISTANCE TO CROSS-BREAKING AND SHEARING. 65 Jf I Fig. 127. W Fig 128. . 129. o . II C o I s 139 140 141 142 143 144 145 146 147 148 149 150 151 10 11 12 13 14 15 16 17 18 5.0 5.1 5.3 5.5 5.7 6.0 6.3 6.5 6.8 7.4 7.7 8.0 15.0 16.4 18.0 19 7 21.4 23.2 25.0 26.8 28.6 30.5 32.3 34.2 36.0 66 RESISTANCE TO CROSS -BREAKING AND SHEARING. VH N ( ^> 2 d 2 2*00 ^ .5 2 cr d 2 jo^ ^"o Z 1^2 J CJ ^J jj>jj E "^ c o 5 WH.m J CC * " " Fig. 130. 152 6 6.3 16.2 336 \ 8 A 153 6} 6.5 17.2 406 1 154 7 6.7 18.3 476 155 7} 6.9 19.3 546 ^ts i 156 8 7.1 20.4 616 T / Tl^ ~^1 ^~_lMi__ ,) v/ 157 8} 7.3 21.5 700 (<.__-.._ >i -F </. 131. 158 9 7.5 22.6 784 \ 7^ / 159 91 7.7 23.6 882 : A : JB: 160 161 10 8.0 8.2 24.7 25.8 980 1078 ; 162 11 8.4 26.9 1190 ""// //^i " .- " l ~~ 7 ~ r ] w 163 11} 8.6 28.0 1302 !<- j 5 -->; 164 12 8.9 29.1 1428 fty. 132. \& / 165 12} 9.1 30.1 1554 P ~A~ 166 13 9.3 31.2 1680 IT 167 13} 9.5 32.3 1806 P 168 14 9.8 33.5 1960 %, 1S>F 169 14} 10.0 34.6 2100 \<-jB >i 170 15 10.3 35.7 2254 g. 133. 171 15} 10.5 36.8 2408 1 ? it - 172 16 10.8 38.0 2562 1 1 i ^ ! 173 16} 11.0 39.1 2730 i | J 174 17 11.3 40.2 2912 1 1 ! 175 17} 11.5 41.3 3080 1 1 i_ 176 18 11.8 42.5 3262 -a 5.; RESISTANCE TO CEOSS-BEEAKING AND SHEARING. 67 Number of section. b/Ofl w.s *! Sectional area in square inches. Coefficient AI. Fig. 134. "WJT~ -: 177 6 6.0 18.0 336 \ x 178 7 6.1 19.7 402 ^1 m v 179 8 6.3 21.6 602 ,,- Fig. 135. 180 9 6.6 23.6 770 ~ ~ /T" VOX 1 ; j A- x 181 10 6.9 25.7 966 *1 182 11 7.2 27.9 1176 Fig. 136. 183 12 7.5 30.0 1400 1 f 184 13 7.8 32.2 1652 ^t ft 185 14 8.2 34.4 1932 fc~ ^.^...^" 186 15 8.5 36.7 2212 Fig. 137. V%t/ ^ W ^ % ! 187 16 8.9 38.8 2534 1 ^ * 188 17 9.2 41.0 2370 ^ to v 189 18 9.6 43.2 3220 <-_.- "3- > 68 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. ta5 ^ o> Xi^C SJC^ Sectional area in square inches. Coefficient Kl. Fig. 138. 190 6 7.0 21.0 392 H -A- ^^ *i 191 7 7.1 23.0 532 2 i 192 8 7.4 25.2 714 Fig. 139. 193 9 7.7 27.6 896 ^ f 194 10 8.0 30.0 1120 2" i ^ 195 11 8.4 32.5 1372 -%. 140. 196 12 8.8 35.0 1638 i fil ^ I ! 197 13 9.1 37.5 1932 .,: J.^._ 198 14 9.6 40.1 2240 ^ig^jr 1 199 15 10.0 42.7 2590 Fig. 141. ^ ^"/ M/e"/ \ IT 1 -^f 200 201 16 17 10.4 10.8 45.2 47.8 2954 3346 v 202 18 11.2 50.4 3766 22* >i RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. ^ 03 ill Sectional area in square incnes. Coefficient JO. ^. 142. L5$ 203 6 8.0 24.0 448 /</ : . A ^8 i 204 7 8.1 26.2 616 3T r /.,,/ yjjl \ 205 S 8.4 28.8 812 ,u^ -B ->" " , J ^. 143. 206 9 8.8 31.5 1036 1 2", P ! 207 10 9.1 34.3 1274 pP ^ -9" ft i 208 11 9.6 37.1 1554 I i ;-- -^- H 209 12 10.0 40.0 1862 %. 144. 2" m A 210 13 10.4 42.9 2198 fc \ \ te 211 14 10.9 45.8 2562 j ^ 212 1 ^ 11.4 48.7 2954 E, \ Fig. 145. \ 2 ? \J 7_ 213 16 11.8 51.7 3374 <l 1 t | 1 i If :|jg- 214 17 12.3 54.6 3822 ^ 1 1 ? C > ^y 215 18 12.8 57.6 4298 ,-- ! 70 EESISTANCE TO CROSS- BREAKING AND SHEARING. Number of section. ^ 03 -C O ^ oqs.rt 2 1|| Sectional area in square inches. Coefficient JT1. Fig. 146. 1 6 6 1.4 11.4 294 r 2 6 7 1.9 12.9 336 1 3 6 8 2.3 14.3 392 1/7 1 ^\ 4 6 9 2.7 15.7 448 ^ 5 6 10 3.1 17.1 504 ^^y^fr^- 6 6 11 36 18.6 560 Fig. 147. 7 6 12 4.0 20.0 602 *y52i , 8 6 13 4.4 21.4 658 \ . j 9 6 14 4.9 22.9 714 r\ # 10 6 15 5.3 24.3 770 **.* ^ ,i 11 6 16 5.7 25.7 826 1 9 & 1 7 69 97 9 QGQ Fig. 148. l^j 13 o 6 1 < 18 . /a 6.6 At.Zi 28.6 ODO 924 -*! __ 14 7 6 1.2 12 2 350 1 ! 15 7 7 1.7 13.7 420 Jr 2-1 16 7 8 2.1 15.1 490 % I i ^ 17 H 9 2.6 16.6 560 ^i r _Z" 18 7 10 3.0 18.0 616 [^ JFJ. ^j ^>. Fig. 149. 19 7 11 3.4 19.4 686 k^^l i^.^^, 20 7 12 3.9 20.9 756 i^S ^wiffi 21 7 13 4.3 22.3 826 1 ff" 22 23 7 7 14 15 4.8 5.2 23.8 25.2 896 966 L-LJ 24 7 16 5.7 26.7 1022 ^/" ^ \ /./ : 25 7 17 6.1 28.1 1092 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. ^ o5 gc. it ll 5 2* CS 2 JQ Coefficient *> 1 Fig. 146. 26 7 18 6.5 29.5 1162 .rfeipi A" 27 8 6 1.0 13.0 434 28 8 7 1.5 14.5 504 r ( *" 29 8 8 1.9 15.9 588 1 30 1 8 9 i n 2.4 20 17.4 I Q Q 672 k Br- >i Fig. 147. ol 32 8 1U 11 .0 3.3 lo .0 20.3 826 .. te--5->! 33 8 12 3.7 21.7 910 j i i 34 8 13 4.2 23.2 994 35 8 14 4.6 24.6 1078 . 7v 1 , , x ,.. .! 36 8 15 5.1 26.1 1148 j[" \% , \~fy O>J i A c c 9^7 ^ 1 9Q9 f \, _____ ZJ_ .>,.[ ( lo O . O Zit.O IZtoZi jFi$r. 148. 38 8 17 6.0 29.0 1316 [*&] 39 8 18 6.4 30.4 1386 z;-j 40 9 7 1.3 15.3 588 77" -,/^ tt 41 9 8 1.7 16.7 686 42 9 9 2.2 18.2 784 l^^iliil[i$ltl F 7" 43 9 10 2.6 19.6 868 Fie;. 149. 44 9 11 3.1 21.1 966 k^ * K ^ y 45 9 12 3.5 22.5 1064 - /; j^^ A"" 46 9 13 4.1 24.1 1162 f^\m i ^j ^*. // /^/; % // 47 9 14 4.5 25.5 1246 ^| -f" |^ 48 9 15 4.9 26.9 1344 1 1 1 49 9 16 5.4 28.4 1442 :;;:/ ;: ^/^tff^ <:&->] 50 9 17 5.8 29.8 1526 RESISTANCE TO CROSS-BREAKING AND SHEARING. . tH fl ^ S ^.2 .fS-S . C <D "3 O) O *c "^ -G O ,2 kJU: -^ M),c o ct ^ J; JG& Is 5 -rH iol.s r % . * ?2 5< s 55 W.^H * .g C3 OQ--H Fig. 146. 51 9 18 6.3 31.3 1624 v ^-b->\ J A 52 10 7 1.1 16.1 672 i i 1 i 53 10 8 1.5 17.5 784 1" 1 ZT 54 10 9 2.0 19.0 896 1 i i 55 10 10 2.4 20.4 1008 "fi - . 1 i x/ 56 10 11 2.9 21.9 1106 K -X>-, H Fig. 147. 57 10 12 3.3 23.3 1218 J? "^"" 58 10 13 3 8 24.8 1330 i i 59 10 14 4.3 26.3 1428 r" - i TJ 60 10 15 4.7 27.7 1540 r ^ y //y////y ~ i 61 10 16 5.2 29.2 1652 * J Fia. 148 ; >i 63 10 10 18 5.7 6.1 30.7 32.1 1750 1862 <-&>\ wmi"*" 64 11 8 1.3 18.3 896 ^ "* ! 65 11 9 1.7 19.7 1008 " j r JT 66 11 10 2.2 21.2 1134 ? i 67 11 11 2.7 22.7 1246 ^ 5" 68 11 12 3.1 24.7 1372 k jj >i .%. 149. 69 11 13 3.6 25.6 1498 Kjh !<<>! 70 11 14 4.1 27.1 1610 i tir "~A~^ ~ JZS53TTH 2y/tdJL 71 11 15 4.5 28.5 1736 // - | < ,, 72 11 16 5.0 30.0 1862 % T | \ 73 11 17 5.5 31.5 1974 i 74 11 18 5.9 32.9 2100 ./__ . \ 75 12 8 1.1 19.1 994 * ^^^ -3> EESISTANGE TO CROSS-BREAKING AND SHEARING. 73 Number of section. o ^ oc. *|s Sectional area in square inches. Coefficient Ki. Fig. 146. 76 12 9 1.5 20.5 1120 S^ U ~b ; "A" 77 12 10 2.0 22.0 1260 1 78 12 11 2.5 23.5 1400 ^1 i 79 12 12 2.9 24.9 1526 1 i 80 12 13 3.4 26.4 1666 1 i 19 3.9 97 Q 1 ROfi JSflr. 147. 82 .LvH 12 15 4.3 at j 29.3 J.OUO 1932 ^JjJ"*1_.- 83 12 16 4.8 30.8 2072 ./.jp ! u 84 12 17 5.3 32.3 2198 f\ i 85 12 18 5.7 33.7 2338 v i ^^ t 86 13 9 1.3 21.3 1232 V 1 IL ^^^MM^M [i 07 1 3 1 A i 8 99 ft 1 QQA ~ j<e-_j5 -> ! o / 10 iv loob jPia. 148. 88 13 11 2.2 24.2 1540 -->! 89 13 12 2.7 25.7 1680 rPl 90 91 13 13 13 14 3.2 3.7 27.2 28.7 1834 1988 1 j 92 13 15 4.1 30.1 2128 r 93 13 16 4.6 31.6 2282 P . II s j%. 149. 94 13 17 5.1 33.1 2422 k3-* i/ 95 13 18 5.5 34.5 2576 i lBl ~*~1P w%r 96 14 9 1.1 22.1 1358 "J if /!t J5t ?| " :% i 97 98 14 14 10 11 1.5 2.0 23.5 25.0 1512 1680 i ! i 1 ! t. 99 14 12 2.5 26.5 1834 nnnnn^i <. _g. 5,1 100 14 13 3.0 28.0 2002 74 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. tB a ^ CD -C O B.S 11 ^ ol.S Sectional area in square inches. Coefficient J2X Fig. 146. 101 14 14 3.4 29.4 2170 -A" 102 14 15 3.9 30.9 2324 103 14 16 4.4 32.4 2492 ^| 2 r 104 14 17 4.8 33.8 2660 | 105 14 18 5.3 35.3 2814 7/ 7 ill HI v 106 15 10 1.3 24.3 1638 !<: jg- >{ Fig. 147. 107 15 11 1.8 25.8 1820 >J<--5->! 108 15 12 2.3 27.3 2002 /" |p """ ^"" m 109 15 13 2.7 28.7 2170 p ! J" ifi "// 110 15 14 3.2 30.2 2352 ^ 1 P 111 15 15 3.7 31.7 2520 ^P 112 15 16 4.2 33.2 2702 .Fie/. 148. 113 15 17 4.6 34.6 2884 |-5^j 114 15 18 5.1 36.1 3052 ^*. | 115 16 10 1.1 25.1 1764 rl k 116 16 11 1.6 26.6 1960 I 1 ^ ! --- 117 16 12 2.0 28.0 2156 7" 118 16" 13 2.5 29.5 2338 j< _^, ;>i v >- ^. 149. 119 16 14 3.0 31.0 2534 i<3>- K^^- 120 16 15 3.5 32.5 2730 i ^ """" 121 16 16 3.9 33.9 2912 /7 | | /, 122 16 17 4.4 35.4 3108 1 p 123 16 18 4.9 36.9 3290 124 17 11 1.3 27.3 2100 J" V:M$s. K- --S- H 125 17 12 1.8 28.8 2310 EESISTANCE TO CROSS-BREAKING AND SHEARING. 75 Number of section. S W.2 ^*0?B ^ s||| 5 _, ~f c5 J J_ CT p X! * K ~ < Coefficient jsn. Fig. 146. 126 17 13 2.3 30.3 2506 127 17 14 2.8 31.8 2716 r| jr I v ^ 128 17 15 3.2 33.2 2926 " !< 3%- >{ Fig. 147. 129 17 16 3.7 34.7 3122 ,jn ^ Jll f" -TT L 5p ^n 130 17 17 4.2 36.2 3332 y//, \ ?3LmmA 131 17 18 4.7 37.7 3542 ">-- -=# ->s 132 18 11 1.1 28.1 2240 ^. 148. jlilz-*- 133 18 12 1.6 29.6 2464 1 * s ^1 f 134 18 13 2.0 31.0 2688 1 \ IOC 1 R 1 A 2)T on c OQQQ ^^^^^^[^^| 7" loO lo 14: . O 6Z. O /jbyo fc 33 >j " Fig. 149. 136 18 15 3.0 34.0 3122 &--^ j s^ 137 18 16 3.5 35.5 3346 *\ ^ 138 18 17 4.0 37.0 3556 139 18 18 4.4 38.4 3780 fcr& >\ 76 RESISTANCE TO CROSS-BREAKING AND SHEARING. o ^i ^s.s . o *-> a g33 . -^ a a o c o JJH -i-i i-i y X3-r3 r-T |1 ..5 S^ 1 Svilfl *3 * ^ e ffi*l g" K.S > o.~ ^ CqS.- cS .S O JFfy 150. 140 6 5 1.3 12.5 280 . i< #^->j A f> -j >7 1 A. A QQJ J/M ;/ < :\ Wm t D 1 . / 11 . D w // I 142 6 7 2.1 16.7 406 J % i 143 6 8 2.5 18.8 462 f ~\ ^ p \ 144 6 9 2.9 20.9 518 jffc ~\ * 14 r q 9 99 o KQO ^ 151. 146 6 11 O . 2 3.6 & . o 24.9 JOO 644 ^ Y~O i 147 6 12 4.0 27.0 714 7 1 / WT J i^,_ A I 148 6 13 4.4 29.1 770 A 1 f 149 6 14 4.8 31.2 826 i 150 6 15 5.2 33.3 896 1 IF; vf 1 151 6 16 5.5 35.3 952 K_. x> : % 152. 152 6 17 5.9 37.4 1022 1 *-Z^i 153 6 18 6.3 39.5 1078 "TH: if 154 7 5 1.2 13.3 364 i | n i 155 7 6 1.6 15.4 434 J ^ JBT i 1 \ 156 7 7 2.0 17.5 518 . \l / 157 7 8 2.4 19.6 602 1 \ -> i """ 158 7 9 2.8 21.7 686 Fig. 153. 1 A I Jj 159 7 10 3.2 23.8 756 *Ti^ fir" 160 7 11 3.7 26.1 840 /r^ i /A /, I ij 161 7 12 4.1 28.2 924 // ^ v Ji j ^ 162 7 13 4.5 30.3 1008 1 S 163 7 14 4.9 32.4 1092 1 ^ .__^ 5~. si 164 7 15 5.3 34.5 1162 1 1 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. ^*.S . C b/D i "!r^ Sectional area in square inches. Coefficient! an. Fig. 150. 165 7 16 5.7 36.6 1246 J?tP i" "* 166 7 17 6.1 38.7 1330 3 167 7 18 6.5 40.8 1414 ^ i J2" 168 8 5 1.1 14.2 1022 u I j 169 8 6 1.5 16.3 546 IfeWiiim 1 Q 7 2.0 18 ^ 644 Fig. 151. 171 O 8 i 8 2.4 o . u 20.6 742 ^ j<-5-> i 172 8 9 2.8 22.7 840 jf/WflfjW NT" ^nlr 1 i 173 8 10 3.2 24.8 938 * 174 8 11 3.6 26.9 1036 j i 175 8 12 4.1 29.2 1148 176 8 13 4.5 31.3 1246 ^. 152. 177 8 14 4.9 33.4 1344 f^T^I 178 8 15 5.3 35.5 1442 -- : \ 179 8 16 5.7 37.6 1540 1 J Z7" 180 8 , 17 6.2 39.8 1638 ll I 181 8 18 6.6 41.9 1750 B ini-S 182 9 5 1.0 15.0 518 { J5- 183 9 6 1.4 17.1 644 .%. 153. 184 9 7 1.9 19.4 770 *&! Si ,/ r" 185 9 8 2.3 21.5 882 K ! ^ 186 9 9 2.7 23.6 1008 1 I ^^ *77^ -</3 *;>, -fi^ B i 187 9 10 3.1 25.7 1120 i 1 188 9 11 3.6 27.9 1246 4^ k ;B >j 189 9 12 4.0 30.0 1358 78 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. I! fe o ? -2 b/D J= 1 .> 03 CC-rH Coefficient -1. % 150. 190 9 13 4.4 32.1 1484 /ii -~>j -j 191 9 14 4.9 34.4 1610 i i i 192 9 15 5.3 36.5 1722 -/ 1 i 193 9 16 5.7 38.6 1848 1 ^^~ j 194 9 17 6.2 40.8 1960 Jfi i lOK 1 S 6/5 A O Q " i^ g ^ iyo lo . O 4^. y 2086 Fig 151. 196 10 6 1.3 18.0 756 ..M ->| 197 10 7 1.7 20.1 896 .. .J PI pTT i 198 10 8 2.2 22.3 1036 // i I\ ^* 199 10 9 2.6 24.4 1176 _ i 200 10 10 3.1 26.7 1316 J/ H 201 10 11 3.5 28.8 1456 Fig 152. 202 10 12 3.9 30.9 1596 l ( -&->!_ 203 10 13 4.4 33.1 1736 I It 204 10 14 4.8 35.2 1876 1 i i 1 ^ : 1 > 205 10 15 5.2 37.3 2016 i i i 206 10 16 5.7 39.6 2156 /:U [ i^ 207 10 17 6.1 41.7 2296 fc~- ii J ^.j 208 10 18 6.5 43.8 2436 % 153. /, H q 209 11 6 1.2 18.8 854 I BT" 210 11 7 1.6 20.9 1022 >V2|%f ^i j J i 211 11 8 2.1 23.2 1176 2 /I "77" *\ % 1 212 11 9 2.5 25.3 1344 i J 213 11 10 3.0 27.5 1498 4/ K ? j 3-> 214 11 11 3.4 29.6 1666 RESISTANCE TO CROSS-BREAKING AND SHEARING. 79 o . tj fl c "S ^ s -its t I,* y i S ^ 1 fficient K l. y K g^.2 p3.s 3> CS ^- o O J^ 9 ,] [50. 215 n 12 3.8 31.7 1820 ., ^ I ^ 216 -, ^ 13 4.3 34 1974 mt 1 n 217 11 14 4.7 36.1 2128 // x -. \ ^ 7 1 f 218 219 n 11 15 16 5.2 5.6 38.3 40.4 2296 2464 ify > I ^^ ^1 I ! 990 1 7 fi 1 4-9 7 2618 ^ ,7 S ; L51. //^.iW 221 . 11 i / 18 U . JL 6.5 jt^ . < 44.8 2786 i<- J -^ I 222 12 6 1.0 19.5 966 2 P 1 i h 223 12 7 1.5 21.8 1148 / | k r 224 12 8 1.9 23.9 1330 | 225 12 9 2.4 26.1 1512 8 ^ v i ! ^ H 226 12 10 2.8 28.2 1680 / ! JLi s 1 227 12 11 3.3 30.5 1862 F L52. i r^i >i 228 12 12 3.7 32.6 2044 i T ^ A i 2 h 229 12 13 4.2 34.8 2226 w i ! 230 12 14 4.6 36.9 2408 J | i* -^ 1 j 231 12 15 5.1 39.2 2590 v : i . I ; d T- /SL 232 12 16 5.5 41.3 2772 . _ ^ ^~ >\ 233 12 17 6.0 43.5 2954 i 53. gi Lj 234 12 18 6.4 45.6 3136 l?W e ^TA"" 235 13 7 1.4 22.6 1274 n ? i i 1 1 236 13 8 1.8 24.7 1470 4 , ^ 1 237 13 9 2.3 27.0 1680 ^ 1 \ 238 13 10 2.7 29.1 1876 ^ ./ ^ ^ J H - - t 5 \ 239 13 11 3.2 31.3 2072 80 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. J3 O ^IS pi! Coefficient /ri. Fig. 150. 240 13 12 3.6 33.4 2282 j%m V~ >| 241 13 13 4.1 35.7 2478 ^Ma t *" I 242 13 14 4.5 37.8 2674 i j 1 i 243 13 15 5.0 40.0 2884 1 "i 1 244 13 16 5.4 42.1 3080 * 1 ! J$2tH 245 13 17 5.9 44.4 3276 Fig. 151. 246 13 18 6.3 46.5 3486 -^ ^~~O 247 14 7 1.2 23.3 1400 1 t 248 14 8 1.7 25.6 1624 "I j jjr 249 14 9 2.1 27.7 184? p ! 250 14 10 2.6 29.9 2058 ^K . -i" 251 14 11 3.0 32.0 2282 i< - J3- ->i 252 14 12 3.5 34.3 2506 _Ficr . 152. *--! 253 14 13 3.9 36.4 2730 3 I IT" 254 14 14 4.4 38.6 2954 i 1 i 4 255 256 14 14 15 16 4.9 5.3 40.9 43.0 3178 3388 f"T< i x ,i l7# 257 14 17 5.8 45.2 3612 f J ?-- 258 14 18 6.2 47.3 3836 Fio 153. _ n 259 15 7 1.1 24.2 1526 /,! 6 > if 260 15 8 1.5 26.3 1764 " p ^ 261 15 9 2.0 28.5 2016 1 *\ 4 \ ^ 262 15 10 2.4 30.6 2254 i. \ . " ! \ 263 15 11 2.9 32.9 2492 p 264 15 12 3.4 35.1 2744 :. ^ 3- RESISTANCE TO CROSS-BREAKING AND SHEARING. 81 o . t~i C ?l 33 III *lo 13 s ? . 9 sf JJJ: 53 g.G ^ ^"c| 2 ^ ?~~ eS; s ^ S.S >OCC.S ~ c<n.S |S? o Fig. 150. 265 15 13 3.8 37.2 2982 7 266 15 14 4.3 39.5 3220 ""~ IIP \ 267 15 15 4.7 41.6 3472 J \ j[ 268 15 16 5.2 43.8 3710 i 269 15 17 5.7 46.1 3948 </ 270 1 ^ 18 a i 48.2 4no Fig. 151. 271 i <j 16 8 u . 1 1 4 27.1 Tt^,UV 1918 ^ j<-J-> 272 16 9 1.8 29.2 2184 JuW^yj/MA "A ~ ^ i 273 16 10 2.3 31.5 2450 ;! fr % T 274 16 11 2.8 53.7 2702 3g -, i 275 16 12 3.2 35.8 2968 itmimim 276 16 13 3.7 38.1 3234 j< _2. i>; -Pfyr. 152. 277 16 14 4.1 40.2 3500 k-^j 278 16 15 4.7 42.6 3766 1^ i 279 16 16 5.2 44.8 4018 2 4 280 16 17 5.7 47.1 4284 281 16 18 6.1 49.2 4550 ^^^SMlT^ 282 17 8 1.2 27.8 2072 i -J3-- ->j ^ 283 17 9 1.7 30.1 2352 Fig. 153. -y "7 284 17 10 2.1 32.2 2632 //.-I-! i^-5 JJM \ Pf" 285 17 11 2.6 34.4 2926 t\ 286 17 12 3.1 36.7 3206 k 287 17 13 3.5 38.8 3486 ^ ^ y I ^ %. i 288 17 14 4.0 41.0 3766 4/__ \ c _ J3-.-H 289 17 15 4.5 43.3 4060 82 RESISTANCE TO CROSS-BREAKING AND SHEARING. . ^i ^ . 5.S . ~z c o -g II -3 ~ 3! - iio o ^ ~ t/.^r "* " ^ ti CJ-H* ^ 5 c S" a.s :"c. 1 ?.s r o Fig. 150. 290 17 16 4.9 45.4 4340 J i C - si""" ^ 291 17 17 5.4 47.6 4620 jf X M IP - 292 17 18 5.9 49.9 4000 *!< =3 >! Fig. 151. !,_A_Vi 293 18 8 1.1 28.7 2226 . 5p IX m \ ~ 291 18 9 1.5 30.8 2520 1 j i 1 T 295 18 10 2.0 33.0 2828 Fi 152. -H 296 18 11 2.5 35.3 3136 k- 7r->\ 297 18 12 2.9 37.4 3430 V ~~L & 1 - ! j ,- i H 298 18 13 3.4 39.6 3738 1 \ 1 i ^ 299 18 14 3.9 41.9 4056 . _j> _>; .%. 153. 300 18 15 4.3 44.0 4354 t H v, <T 301 18 16 4.8 46.2 4648 i ^ ! 4 ^ |i| 18 17 5.3 48.5 4956 ^ f j I 1 303 18 18 5.7 50.6 5269 i -E r- -! RESISTANCE TO CROSS- BREAKING AND SHEARING. Number of section. Height 11 in inches. -^ c ^^ S gjt |B| G Fig. 154. 304 6 7 1.8 17.7 378 -._ ^ ~ -. 305 6 8 2.2 19.8 448 A" ^ _\ i 306 6 9 2.5 21.8 504 2 / 2 ^ -fr 307 6 10 2.9 23.9 574 ^ \ \ 308 6 11 3.3 26.0 630 %2 " ;- / M I i_ 309 6 12 3.7 28.1 686 Fig. 155. -310 6 13 4.1 30.2 756 *LMH 311 6 14 4.5 32.3 812 1 > r/ r ^ , , i 312 6 15 4.9 34.4 882 71 i jy ,,_:, ir 313 6 16 5.2 36.3 938 i , i i 314 6 17 5.6 38.4 1008 .^ . i JB M i 315 6 18 6.0 40.5 1064 Fig. 156. 316 7 7 1.6 18.9 490 H 5- ^\ -\ - * 317 7 8 2.0 21.0 574 I _^_ -k ~"i 318 7 9 2.4 23.1 658 I |< | JT 319 7 10 2.8 25.2 742 1 ] 320 7 11 3.3 27.5 826 " 1 TT ~ J| 321 7 12 3.7 29.6 896 *Fig. 157. 322 7 13 4.1 31.7 980 gfj I \ J\ 323 7 14 4.5 33.8 1064 "A 8 : * 324 7 15 4.9 35.9 1148 * : I ; 325 7 16 5.3 38.0 1232 A * ^ 326 7 17 5.7 40.1 1302 ; -;. . 327 7 18 6.1 42.2 1386 .. > -i ,___^.__.J 328 8 8 1.9 22.4 714 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. HI 5 ^11 g iKl Sectional area in square inches. Coefficient Ki. Fig. 154:. 329 8 9 2.3 24.5 812 J ^? ^ 330 8 10 2.7 26.6 910 ; --.v p|f^ i ! 331 8 11 3.1 28.7 1008 7/P 332 8 12 3.6 30.9 1106 s 333 8 13 4.0 33.0 1218 3f 3 I 334 8 14 4.4 35.1 1316 ! Fig. 155. 335 8 .15 4.8 37.2 1414 n ! 7 1 336 8 16 5.2 39.3 1512 ]} j"^ "^* V* ,- ~ vO& 337 8 17 5.7 41.6 1610 IP *l Jr 338 8 18 6.1 43.7 1708 rf i 339 9 8 1.7 23.6 840 -i pp D i 340 9 9 2.1 25.7 966 1 ^. 156. 341 9 10 2.6 27.9 1092 K-^->j 342 9 11 3.0 30.0 1204 ]/j,"* 1^ 343 9 12 3.4 32.1 1330 TA/ ~jr il . 344 9 13 3.9 34.4 1442 SH 345 9 14 4.3 36.5 1568 isiliil^ 346 9 15 4.7 38.6 1694 Fig. 157. 347 9 16 5.1 40.7 1806 4! 348 9 17 5.6 42.9 1932 "" 349 9 18 6.0 45.0 2044 "1 350 10 8 1.5 24.8 980 1 351 10 9 2.0 27.0 1120 ,^. | ^ 1 352 10 10 2.4 29.1 1260 i 1 v ____^B____.> 353 10 11 2.8 31.2 1400 RESISTANCE TO CROSS-BREAKING AND SHEARING. 85 Number of section. II ^11 g g .S j Sectional area in square inches. Coefficient KI. / V 154. 354 10 12 3.3 33.5 1540 j 1: *^T 355 10 13 3.7 35.6 1680 300 j F 356 10 14 4.1 37.7 1820 7, *J i 357 10 15 4.6 39.9 1960 ^ H i 358 10 16 5.0 42.0 2100 %j$ i 359 10 17 5.5 44.3 2240 "" i< _J3^ _>; j ^. 155. 360 10 18 5.9 46.4 2380 j, !<- --i 361 11 9 1.8 28.2 1288 JK Pf "T r " ^v^ I i 362 11 10 2.2 30.3 1442 8 J^r 363 11 11 2.6 32.4 1610 .... i | 364 11 12 3.1 34.7 1764 7^1 r 1 365 11 13 3.5 36.8 1932 A- i -j> * J ty. 156. 366 11 14 4.0 39.0 2086 &^ 367 11 15 4.4 41.1 2240 1 568 11 16 4.9 43.4 2408 ^ | 1 369 11 17 5.3 45.5 2562 370 11 18 5.8 47.7 2730 5 Sv5^3^ 371 12 9 1.6 29.4 1442 __._g_ >t r^ j V- 157. 372 12 10 2.0 31.5 1624 i "fa i 4! 373 12 11 2.5 33.8 1806 --XjjL j pr*" 374 12 12 2.9 35.9 1988 // ^ 375 12 13 3.4 38.1 2170 /y IB jr 376 12 14 3.8 40.2 2352 \ %. 377 12 15 4.2 42.3 2534 1 w- ;./^^ v i ..JB....^ 378 12 16 4.7 44.6 2716 86 RESISTANCE TO CROSS-BREAKING AND SHEARING. . ^, c ^ ^5.5 . ^.5 . 3 G0 a C5 O "S fcC^C ;"" c~ "3 " Z~ 5 o c^ SEfcd r 3 K ^ c ^ * _C ^c^.5 O cS 0, -5 ^ K.-, ^ ^* o Fig. 154. 379 12 17 5.1 46.7 2898 jrf^i-" ~- 380 12 18 5.6 48.9 3066 w \ 381 13 10 1.8 32.7 1806 p t Hi ; J&~ 382 13 11 2.2 34.8 2002 I W/, \ 383 13 12 2.7 37.1 2212 li 384 13 13 3.2 39.3 2408 Fig. 155. 385 13 14 3.6 41.4 2618 f-^3 386 13 15 4.1 43.7 2814 w 1 vr-^ % 387 13 16 4.5 45.8 3010 , j|! J 388 13 17 5.0 48.0 3220 | % 389 13 18 5.4 50.1 3416 J%E ] i 390 14 10 1.6 33.9 1988 "< JB >! Jfy. 156. 391 14 11 2.0 36.0 2212 K-J^j 392 14 12 2.5 38 . 3 2436 pa/**" r 1 393 14 13 2.9 40.4 2660 IS ,->, i C94 14 14 3.4 42.6 2870 \ ft i 395 14 15 3.9 44.9 3094 .,..v : l ; <\l/ 396 14 16 4.3 47.0 3318 {< _B >>P^ .%. 157. 397 14 17 4.8 49.2 3542 -7 , I T | //rl-i Mr i 398 14 18 5.2 51.3 3766 M]~A" 399 15 11 1.8 37.2 2408 H if 400 15 12 3.3 39.7 2660 /^P A~f~* ] y $ 401 15 13 2.7 41.6 2898 P B 402 15 14 3.2 43.8 3136 ! 1/2 : 403 15 15 3.7 46.1 3388 -&* RESISTANCE TO CROSS-BREAKING AND SHEARING. 87 o . ^ c ^i SS.S,. .S . 3.22,- G .2 .S S c c fcc<- ^ ^J= 3 c3 ^ i Jc P s s H^ll Jsll |lrl P Jfy. 154. 404 15 16 4.1 48.2 3626 __ J<;--5-! 405 15 17 4.6 50.4 3864 J} -^ j - , : :/ : | A """ PI i 406 15 18 5.0 52.5 4116 -Z/r -L 407 16 11 1.6 38.4 2618 *i i 408 16 12 2.1 40.7 2884 -__ Y//A ! 2/2% ! i 409 16 13 2.5 42.8 3136 *" ; K J3 ->; /%. 155. 410 16 14 3.0 45.0 3402 j<--^-5>i 411 16 15 3.4 47.1 3668 ^ :ra 412 16 16 3.9 49.4 3934 j ^1 ^ 413 16 17 4.4 51.6 4186 ^P 414 16 18 4.8 53.7 4452 BF ^siik 415 17 12 1.8 41.7 3108 It ^D > ^. 156. 416 17 13 2.3 44.0 3388 i^L^__ 417 17 14 2.8 46.2 3682 llH^lj 418 17 15 3.2 48.3 3962 i ! Z^ | Jr 419 17 16 3.7 50.6 .4242 : 3 ill . 420 17 17 4.2 52.8 4522 421 17 18 4.6 54.9 4816 i< -& >\ ^ Fig. 157. 422 18 12 1.6 42.9 3332 i o i M\ 423 18 13 2.1 45.2 3626 _ _//_! g ~~2 \ ^pj" A" 424 18 14 2.5 47.3 3934 vV^ *| )> fy 425 18 15 3.0 49.5 4242 1 426 18 16 3.5 51.8 4550 1 P 427 18 17 3.9 53.9 4858 | i^iiiti v K -s- H 423 18 18 4.4 56.1 5152 RESISTANCE TO CROSS-BREAKING AND SHEARING. Numberof section. a! ii " 5 c. " 3 "c Coefficient JTi. **j 7- 158. 429 6 6 1.5 18.0 336 4 .t B 7) ; - M 430 6 7 1.8 20.6 392 A 1 ! 431 6 8 2.2 23.4 462 74 i 432 6 9 2.5 26.0 518 i V ! K i i 433 6 10 2.8 28.6 588 s" ? 1 j 434 6 11 3.2 31.4 624 *" -F7 3 7. 159. 435 6 12 3.5 34.0 714 -t ^ -> 436 6 13 3.8 36.6 770 :" ^" 2 - 437 6 14 4.2 39.4 840 7// ,., ^ 438 6 15 4.5 42.0 896 < . 439 6 16 4.8 44.6 952 5" , - PI , r 440 6 17 5.2 47.4 1022 u ,- 3- y. ] i60. 441 6 18 5.5 50.0 1078 *! 5~ _ 442 7 7 1.8 22.1 532 | 2 ^ "] 443 7 8 2.2 24.9 616 4 1 JT 444 7 9 2.6 27.7 714 p \ 445 7 10 2.9 30.3 798 :___ ~OL jl" 446 7 11 3.3 33.1 882 c fi* ?. J 161. 447 7 12 3.7 35.9 966 ; i 5 ; 1 ii 448 7 13 4.0 38.5 1050 H 1 | r 449 7 14 4.4 41.3 1134 /I i i 450 7 15 4.7 43.9 1218 \ %| \ rjr 4 1 * 451 7 16 5.1 46.7 1302 1 *7 5pr AQ ^ 1 38ft -2 - 452 7 17 .O ly . o 1OOO !<- ___ -^_ ^ 453 7 18 5.8 52.1 1470 RESISTANCE TO CROSS-BREAKING AND SHEARING. 89 o fl . ^i *%a . r* O~ """ </ ?a . 2 Q.O & X! ** S**lc g o teja r^ |~ 3 o fj^ K W.S ^ o.~ ^o. I 05 "- 5 u Ft 7- 158. 454 8 7 1.8 23.6 686 2- 455 8 8 2.2 26.4 714 1 ? 7 h ; I*" t i jS \ 456 8 9 2.5 29.0 896 % | f 457 8 10 2.9 31.8 1008 n 458 8 11 3.3 34.6 1120 2"\ X- v I a \,, * ^ 459 8 12 3.7 37.4 1232 7 K B- ? 159. 460 8 13 4.1 40.2 1344 ^ > 461 8 14 4.5 43.0 1456 " T "^ - A" 1 r~ I i i 462 8 15 4.9 45.8 1554 i $ ^ > 463 8 16 5.2 48 4 1666 1 i 464 8 17 5.6 51.2 1778 Ff i. n M i 465 8 18 6.0 54.0 1890 !< 3- 160. 466 9 7 1.7 24.9 826 . 2 """A"" M A 467 9 8 2.1 27.7 966 gsa 1 468 9 9 2.5 30.5 1106 a V. I > 469 9 10 2.9 33.3 1232 1 1 470 9 11 3.3 36.1 1372 v- ; ;; : > ^^ / \ i m , 471 9 12 3.7 38.9 1498 <- B 5 i i- 161. 472 9 13 4.1 41.7 1638 ! i \ Ii 473 9 14 4.5 44.5 1778 :?i 1 1 w 474 9 15 4.9 47.3 1904 :. , j ,J 475 9 16 5.3 50.1 2044 3, 1 %$ \ 3T /i \ Z -MJ- l 476 9 17 5.7 52.9 2184 1 S n &. 9m Li 477 9 18 6.1 55.7 2310 - 4 478 10 7 1.6 26.2 980 90 RESISTANCE TO CROSS- BREAKING- AND SHEARING. o . !-. G ?2 ^ S ,- O - fl -co . d o |1 |1 I cll ;h!l o ce ^ 2 Jo " " ~ f F < / 158 479 10 8 2.0 29 1134 -7 - c^^ -- -r 480 10 9 2.4 31.8 1302 - - L i 481 10 10 2.8 34.6 1456 7/ ~ S 482 10 11 3.2 37.4 1624 483 10 12 3.6 40.2 1778 2" 484 10 13 4.0 43.0 1946 a K I / 159 485 10 14 4.4 45.8 2100 i* __^ . 486 10 15 4.9 48.8 2268 "a_ I r 487 10 16 5.3 51.6 2422 7/ r t k 488 489 10 10 17 18 5.7 6.1 54.4 57.2 2590 2744 s i 490 11 8 1.9 30.3 1316 i .i 5~" ~ i J* .> 7. 160 7J 491 492 11 11 9 10 2.3 2.7 33.1 35.9 1512 1694 -? "t 493 11 11 3.1 38.7 1876 j/ s <-> JT 494 11 12 35 41.5 2072 ! 495 11 13 4.0 44.5 2254 1 i ^~ /."> r. ] L61. >| 496 497 11 11 14 15 4.4 4.8 47.3 50.1 2436 2632 \4 5 ; 498 11 16 5.2 52.9 2814 31 . | ^ ^ IT" 499 11 17 5.6 55.7 2996 . 3* 500 11 18 6.1 58.7 3192 41 { | ff 501 12 8 1.7 31.4 1512 , B 1 1 i 502 12 9 2.1 34.2 1722 :3 t 503 12 10 2.6 37.2 1932 RESISTANCE TO CROSS-BREAKING AND SHEARING. 91 . !2 a. 53 *- a 5 ~ 1 c fc. S3 o a tfifl 7." 5^ "^C -"^ r ? r,^ GS "^ M o ^ ;>! "S .S -"" 1 "c is r ~ ^ c 5 ^ S.S ^ * /} H ^ Fig. 158. 504 12 11 3.0 40.0 2142 ~r; ^ If *~ 505 1^ 12 3.4 42.8 2366 j 1 506 12 13 3 9 45.8 2576 JJi 4 jar 507 12 14 4.3 48.6 2786 I 508 12 15 4.7 51.4 2996 *fd I 509 12 16 5.2 54.4 3220 &--> Fig. 159. 510 12 17 5.6 57 2 343 J *b -* 511 12 18 6.0 60.0 3640 2T -..5 512 13 8 1.6 32.7 1680 i/ w 513 13 9 2.0 35.5 1932 / ; M. 514 13 10 2.4 38.3 2170 ~ 2 "r~ \ \> 515 13 11 2.9 41 3 2408 K J3- ->j Fig. 160. 516 [3 12 3.3 44.1 2646 J4--g-- i 517 13 13 3.8 47 . 1 2884 1 1 j \ 518 13 14 4.2 49.9 3122 /; I JT 519 13 15 4.6 52.7 3360 i ! 520 13 16 5.1 55.7 3598 ~ r.oi i ^ K C~) CO K 1 ^ 161. <J 1 522 io 13 18 o . y 5.9 Oo .O 61.3 4088 14 i ; 523 14 9 1.9 36.8 2142 j|| 524 14 10 2.3 39.6 2408 , , i i ! 525 14 11 2.7 42.4 2674 1 ; II i * 526 14 12 3.2 45.4 2940 i x i | ! 527 14 13 3.6 48.2 3206 K -^ r 528 14 14 4.1 52.2 3472 92 RESISTANCE TO CROSS-BREAKING AND SHEARING. o . !- H} *Z2 -|.S -r a a> a 11 -S S SS, )* ** _ a ^ ^ * O j-i g O tf-C ~~ c ~ 3 ^ G "O - 3 s^ "5 ^ a^o ^ W.S ^o_ -^ o<e.~ 3D * co< - a Fig. 158. 529 14 15 4.5 54.0 3738 Jjj |P"t 530 531 14 14 16 17 4.9 5.4 56.8 59.8 4004 4270 <# 1 *" 532 14 18 5.8 62.6 4536 i ^ 533 15 9 1.7 37.9 935 *H1 534 15 10 2.2 40.9 264(3 ;. J5......J- 7. 159. 535 15 11 2,6 43.7 2940 h-d ->i 536 15 12 3.0 46.5 3234 "> r (l j.M| I ! 537 15 13 3.5 49.5 3528 4*1 i*| 538 15 14 3.9 52.3 3822 -TIT " ! 1 --VVM ! 539 15 15 4.4 55.3 4116 2 |^, 540 15 16 4.8 58.1 4410 K ^ 7n jed. 541 15 17 5.3 61.1 4704 & jphr 542 15 18 5.7 63.9 4998 ! 543 16 9 1.6 39 2 2562 J^P ff 544 16 10 2.0 42.0 2884 p 545 16 11 2.5 45.0 3206 2 " 546 16 12 2.9 4.7 X ^ , 161. 547 16 13 3.4 T: 1 . O 50.8 3850 j |i i|j 548 16 14 3.8 53.6 4172 if" 549 16 15 4.3 56.6 4494 ^ <^ 3 ^ ftp ^r 550 16 16 4.7 59.4 4816 P j; 551 16 17 5.2 65.4 5138 m 551 16 18 5.6 62.2 5460 ^r. & >\ 552 17 10 1.9 43.3 3150 RESISTANCE TO CROSS-BREAKING AND SHEARING. 93 Number of section. ^1 <3J S W.S J|| |||| Sectional area in square inches. Coefficient ** \ %. 158. 554 17 11 2.3 46.1 3486 : S ii ! 555 17 12 2.8 49.1 3836 * ^S ^y 556 17 13 3.2 51.9 4186 ft 3 r | 557 17 14 3.7 54.9 4536 I - ._^. j, Ft^. 159. 558 17 15 4.1 57.7 4872 2 "L 1 559 17 16 4.6 60.7 5222 6 1 "f 560 17 17 5.0 63.5 5572 j" >:, Ixi^ i 561 17 18 5.5 66.5 5922 f Fig. 160. 562 18 10 1.6 44.2 3346 t <-/ fepr 1 563 18 11 2.1 47.2 3724 /; 1 "^ 1 564 18 12 2.6 50 2 4102 -^ if *r mm^m- 565 18 13 3.0 53.0 4480 g < 33 3>i 566 18 14 3.5 56.0 4868 %. 161. h i ^ ^L... 567 18 15 3.9 58.8 5236 21 f 568 18 16 4.4 61.8 5628 3 3,j^ . 569 18 17 4.9 64.8 6006 i 570 18 18 5.3 67.6 6384 94 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. tec "9 ^^^ c gtg C 1-2 2 * C5 ~ Coefficient .121. Fig. 102. 571 6 9 2.3 26.6 504 ~2 E ] i 572 6 10 2.7 29.4 574 ! 573 6 11 3.0 32.0 630 . . 2 -2" 574 6 12 3.3 34.6 700 -1 1 575 6 13 3.7 37.4 756 1 1 ^ A C\ C\ OO/j Fig. 163. 576 577 6 15 4 .0 4.3 40. U 42.6 bZb 882 "F P: * _.. ... ~ 578 6 16 4.7 45.5 952 i 4 ir i 579 6 17 5.0 48.0 1008 2 p & 580 6 18 5.3 50.6 1064 - JW ^v" ~~~ 1 i i 581 7 9 2.3 28.6 686 Ej^ ~i^ 582 7 1 2.7 31.4 770 -%. 164. 583 / 7 11 3.0 34.0 854 U--Z -, 584 7 12 3.4 36.8 938 f \ _?-. t 585 7 13 3.8 39.6 1036 f/ 586 7 14 4.1 42.2 1120 2 s ^ T 587 7 15 4.5 45.0 1204 1 588 7 16 4.9 47.8 1288 M m i 1 2" 589 7 17 5.2 50.4 1372 -%. 165. 590 i 7 18 5.6 53.2 1456 , ^ , 5 591 8 9 2.2 30.4 868 ?L ^~^r 592 8 10 2.6 33.2 980 IIP P ! 593 8 11 2.9 35.8 1092 . .- << 7 | ! 594 8 12 3.3 38.6 1204 i i 595 8 13 3.7 41.4 1302 | i 596 8 14 4.1 44.2 1414 - s ? .. 597 8 15 4.5 47.0 1526 fr - ^.. -- -*_ RESISTANCE TO CROSS BREAKING AND SHEARING. 95 mberof action. 5j -H* 5 1|| |||| a J si.S ^ c.= $ Cq5 s-5 o jz ?. 162. 598 8 16 4.9 49.8 1638 -g->_ 2 " /7 r ~~^" 599 8 17 5.3 52.6 1750 _!_, if ! 600 8 18 5.7 55.4 1848 2" m JZ" 601 9 9 2.1 32.2 1064 9 i 3F i 602 9 10 2.5 35.0 1204 "if ~ |^- 603 9 11 2.9 37.8 1330 2^ g. 163. 604 9 12 3.3 40.6 1470 "6 >\ 605 9 13 3.7 43.4 I^Qft 1 1 w\ ! 606 9 14 4.1 46.2 1736 2 p -^ 607 9 15 4.5 49.0 1876 ~/T(^ 1 %%%%^ 608 9 16 4.9 51.8 2002 11 i 609 Q n r q KA_ f> i< j " U O . o O . O 2142 F* 1. 164. 610 9 18 5.7 57.4 2282 !<--Z ,._,,, 611 10 10 2.4 36.8 1414 1 IT 612 10 11 2.8 39.6 1582 "H i 613 10 12 3.2 42.4 1736 2 1 "F 614 10 13 3.6 45.2 1904 Hf i 615 in i d. 4 A AQ r\ _ 1 M / <MftMWf/t 2" IU 1 .0 4o . U 2058 616 10 i ^ A A 50 H o o o/~ 7 j > 1 U T: . TC ^c ^ . 165. 617 10 16 4.8 53.6 2380 J JJ i M ! 618 619 10 10 17 18 5.2 5.7 56.4 59.4 2595 2702 1 //I 620 11 10 2.2 38.4 1638 7 "i 7 | JZ" 621 11 11 2-6 41.2 1820 I i 622 11 12 3.0 44.0 2016 m 8 i 623 11 13 3.5 47.0 2198 j<- i 3 > 624 11 14 3.9 49.8 2380 96 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. ^ S *| .?5I| Sectional area in square inches. Coefficient X Fig. 162. 625 11 15 4.3 52.6 2576 g /. w 3 "T 626 11 16 4.7 55.4 2758 t 627 11 17 5.1 58.2 2954 t 2 ^ JL 628 11 18 5.6 61.2 3136 ,. 11. 629 630 12 12 11 12 2.4 2.9 42.8 45.8 2086 2296 Fig. 163. 631 12 13 3.3 48.1 2506 ~?W o ~7/ / %jjr _._ . -A 632 12 14 3.7 51.4 2716 J 2 1 633 12 15 4.1 54.2 2940 s 634 12 16 4.6 57.2 3150 1 \ % 1 635 12 17 5.0 60.0 3360 JmMM^m^ j m %%; -y- 636 12 18 5.4 62.8 3570 !< jg > Fig. 164. 637 13 11 2.2 44.4 2338 ,-N| 638 13 12 2.7 47.4 2576 < . 1 ii ~2 T 639 13 13 3.1 50.2 2814 | fa, i 640 641 13 13 14 15 3.5 4.0 53.0 56.0 3052 3290 w. y i i 642 13 16 4.4 58.8 3528 / t\ 2" m & j 643 13 17 4.9 61.8 3780 Fig. 165. 644 13 18 5.3 64.6 4018 i J-i 5 1 645 14 11 2.0 46.0 2604 646 14 12 2.5 49.0 2870 : i~r 647 14 13 2.9 51.8 3136 ,1 n i 1 i 648 14 14 3.4 54.8 3402 :: i 649 14 15 3.8 57.6 3668 i 650 14 16 4.2 60.4 3934 ^ & : -^ <~ ; v .y _ 651 14 17 4.7 63.4 4208 <- B- -> 1 RESISTANCE TO CROSS-BREAKING AND SHEARING. Number of section. j="o .Sf.S 5 ^ -Ifs ~ G T3 ^ 5="^ c rt * J Coefficient Jp. Fig. 162. 652 14 18 5.1 66.2 4152 f 653 15 12 2.3 50.6 3164 l^M^I 1 W# : 654 15 13 2.7 51.4 3444 <2 ||p -uL 655 15 14 3.2 56.4 3738 %ss 1 656 15 15 3.6 59.2 4032 yOC . Fig. 163. 657 658 15 15 16 17 4.1 4.5 62.2 65.0 4296 4606 }<- $--->! 659 15 18 4.9 67.8 4900 T 660 16 13 2.5 55.0 3742 sm & 661 16 14 3.0 58.0 4074 \WM \ j%^ \ 662 16 15 3.4 60.8 4396 663 16 Q Q CO 4.71 Q Fig. 164. 664 16 17 O . <J 4.3 DO . O 66.6 tfc / io 5026 i ^^~ - K- 665 16 18 4.8 69.6 5348 m^\ j : 666 17 13 2.3 56.6 4060 w 2 m & 667 17 14 2.8 59.6 4410 fm m \ 668 17 15 3.2 62.4 4760 mm%^^%^\ / ^~ 3 it tm^mM/2/m- 669 17 16 3.7 65.4 5110 Fig. 165. 670 17 17 4.1 68.2 5460 I \ \b \ 671 17 18 4.6 71.2 5810 EJ { ^~\ 672 18 13 2.1 58.2 43^2 tr v- % > . \>j^""^~" .H ^ ! 673 18 14 2.5 61.0 4746 //( * ^ 41 7 I TT ^ IT 674 675 18 18 15 16 3.0 3.4 64.0 66.8 5124 5502 ^ ^ ^illlil^ltllll; 676 18 17 3.9 69.8 5080 .. >--B H 677 18 18 4.4 72.8 6258 RESISTANCE TO CROSS BREAKING AND SHEARING. STRENGTH OF WOODEN BEAMS. Capacity W in Ibs. of American white and yellow pine beams, joists, &c., from 1" x 1" to 15 x 15 in. The modulus of rupture is taken at- - = 1250 Ibs.. or 8 times safety, K = tabulated coefficient, to be divided by I = distance between supports in inches, or length of beams in inches from support to free end of beam. a Coefficient 11 Height in "o c 1 2 3 4 5 6 7 1 1666 6666 15000 26666 41666 60000 81666 1_1^ 2500 10000 22500 39999 62499 90000 122499 2 3333 13333 30000 53333 83333 120000 163333 2*^ 4166 16666 37500 66666 104166 150000 204166 3 5000 19999 45000 80000 124999 180000 244999 31^ 5833 23333 52700 93333 145833 210000 285833 4 * 6666 26666 60000 106666 166666 240000 326(566 41^ 7499 29999 67500 119999 187499 270000 367499 5 8333 33333 75000 133333 208333 300000 408333 5% 9166 36666 82500 146666 229166 330000 449166 6 10000 39999 90000 159999 249999 360000 489999 6% 10833 43333 . 97500 173333 270833 390000 530833 7 11666 46666 105000 186666 291666 420000 571666 71^ 12500 49999 112600 199999 312499 450000 612499 8 13333 53333 120000 213333 333333 480000 653333 8 14 14166 566(56 127500 22(5666 354166 510000 694166 9 " 14998 59999 135000 239999 374999 540000 734999 9% 15831 63333 142500 253333 395833 570000 775833 10 16666 66(566 150000 266666 416666 600000 816666 10/4 17500 69999 157500 279999 437499 630000 857599 11 18333 73333 1(55000 293333 458333 660000 898533 11*^ 19166 76666 172500 306666 479166 690000 939366 12 * 20000 79999 180000 319999 499999 720000 979999 12^4 20833 83333 187500 333333 520833 750000 1020833 13 " 21666 86066 195000 346666 541666 780000 1061666 1314 22500 89)99 202500 359999 562499 810000 1102499 14 23333 93333 210000 373333 583333 840000 1143333 14K 24166 96(566 217500 386666 604166 870900 1184166 15 * 25000 99099 225000 399999 624999 900000 1224999 RESISTANCE TO CEOSS- BREAKING AND SHEARING. BEAMS SUPPORTED AT THE ENDS. K Load equally distributed, W = or K f = I W. I K Load concentrated at centre, W== or K 21W. 2 21 BEAMS FIXED AT ONE END. K Load equally distributed, W = or Kf UW. 3 K Load concentrated at free end, W = or K = SI W. 4 inches. 8 9 10 11 12 13 14 15 106666 135000 166666 201757 240000 281666 326666 375000 159999 202500 249999 302636 360000 422499 489999 562500 213333 270000 333333 403515 480000 563333 653333 750000 266666 337500 416666 504393 600000 704166 816666 937500 319999 405000 499999 605272 720000 844999 979999 1125000 373333 472500 583333 706151 840000 985833 1143333 1312500 426666 540000 666666 807030 960000 1126666 1306666 1500000 479999 607500 749999 907908 1080000 1267499 1469999 1687500 533333 675000 833333 1008787 1200000 1408333 1633333 1875000 586666 742500 916666 1109666 1320000 1549166 1796666 2062500 639999 810000 999999 1210545 1440000 1689999 1959999 2250000 693333 877500 1083333 1311423 1560000 1830833 2123333 2437500 746666 945000 1166666 1412302 1680000 1971666 2286666 2625000 799999 1012500 1249999 1513181 1800000 2112499 2449999 2812500 853333 1080000 1333333 1614060 1920000 2253333 2613333 3000000 906666 1147500 1416666 1714938 2040000 2394166 2776666 3187500 959999 1215000 1499999 1815817 2160000 2534999 2939999 3375000 1013333 1232500 1583333 1916696 2280000 2675833 3103333 3562500 1066666 1350000 1666666 2017575 2400000 2816666 3266666 3750000 1119999 1417500 1749999 2118453 2520000 2957499 3429999 3937500 1173333 1485000 1833333 2219332 2640000 3098333 3593333 4125000 1226666 1552500 1910666 2320211 2760000 3239166 3756660 4312500 1279999 1620000 1999999 2421090 2880000 3379999 3919999 4500000 1333333 1687500 2083333 2521968 3000000 3520833 4083333 4(587500 1386666 1755000 216(5666 2622847 3120000 3661666 4246666 4875000 1439999 1822500 2249999 2723726 3240000 3802499 4409999 5062500 1493333 1890000 2333333 2824605 3360000 3943333 4573333 5250000 1546666 1957500 2416666 2925483 3480000 4084166 4736(566 5437500 1599999 2025000 2499999 3026362 3600000 4224999 4899999 5625000 100 PEESSURE OF SUPPORTS. PRESSURE ON SUPPORTS. REACTION OF SUPPORTS. For a continuous beam, horizontal or inclined. Load W, equally distributed, and supports equal distance apart. Appli cable to trussed beams, rafters, or beams supported by three or more supports. Reference. (Fig. 166.) W/ = Weight of load per unit of length in Ibs. L = Distance between supports in units of length. P t Pi, P-2, = Pressure on supports in Ibs., counting frorn end support to center of beam. M t M lt M 2 = Moments of rupture over supports. ra, m 11 m 2 = Moments of rupture between supports. I, l l} 1 2 = The distance from a support to section where moments m, m lt m z occur. By this table the pressure upon any support, from 3 to 9 in number, can be ascertained; also the moments of rupture. The table is used in calculating the strains in roof trusses, &c. Fig. 166. Reactions or pressure. Number of Supports. 3 4 5 7 9 P I 0.375 W t L 1.25 W,L 0.4 W,L 1.1 W,L 0.3929TF, 1.1429 W,L 0.9286 W,L 0.3942 W,L 1.1346 W t L 0.9615 W,L 1.0192 W,L 0.3^43 W,L 1.13401^^ 0.9G29 W t L U)103W,L 0.9948 W t L M l M 2 M 3 Ml 0.125 W,L2 0.1 17^2 0.1071 W,L2 0.0714 W t L 2 0.1058 WL 2 0.0769 W t L 2 0.0865 W t L 2 0.1057 W t L 2 0.0773 W t L 2 0.0850 IF, L 2 0.0824^1/2 PRESS flUE ON SUPPORTS. 101 Reactions or pressure. Number of Supports. 3 4 5 7 9 m mi W*2 m s 0.0703 W t L 2 0.08 W t L i 0.025 W,Li 0.0772^1,2 0.0364^,1,2 0.0777 IF, 2 0.0340^1,2 0.0434:^1/2 0.0777 W 4 L 2 0.0339 ir,L 2 0.04381^1-2 0.0412 W t L 2 I 1 0.375 L 0.4 L 0.5 Z, 0.3928 0.535 L 0.3942 ^ 0.5288 L 0.4903 0.3943 I/ 0.5283 L 0.4922 0.5025 L Reference. (Figs. 167, 168, and 169.) W, TTj, PT 2 = Load in Ibs. ^, ^ 1} 1 2 = Dimensions in units of length. P, P 1} P 2 = Pressure on supports in Ibs. Three supports, unequal distances apart. Fig. 168. Load equally distributed: One support, and fixed at one end. 102 COMPRESSIVE STRAIN AND PRESSURE ON SUPPORTS. ig. 169. Load concentrated at free end: One snpport, and fixed at one end. COMPRESSIVE STRAIN AND PRESSURE ON SUPPORTS. SLOPING BEAMS, RAFTERS, &c. Load W equally distributed. For the cross- breaking strain, the rafter, &c., is to be treated as a horizontal beam of the length I. (See Compound Strains in Beam, &c.) Reference. C= Compression in direction of beam. jy= Horizontal strain acting on support. V= Pressure on supports. Lower end supported vertically and horizontally ; upper end resting on inclined support : Fig. 170. (7 = - sin . v W H= - sin.-y cos.-y 2 W x =-^- (cos.-y) 2 RESISTANCE TO CRUSE ING. Upper end fized; lower end supported horizontally : Fig. 171. 103 w ~2~ Upper end resting against a vertical surface ; lower end sup- >orted vertically and horizontally : port rtically and horizontally Fig. 172. 2 sin. u W H=- cotg v RESISTANCE TO CRUSHING. STRENGTH OF COLUMNS, PILLARS, AND STRUTS. Reference. A = Area of cross-section in inches. C== Coefficient, depending on the material. I = Least moment of inertia of cross-section. W = Capacity of column, pillar, or strut in Ibs. a = Coefficient, depending on the material in respect to flexure. c = Coefficient, depending on the material. h = The least dimension across the section in inches. k = Factor of safety. I = Length of column, &c., in inches. r = Least radius of gyration. 104 RESISTANCE TO CRUSHING. To find the square of the radius of gyration (r 2 ) of a plane about a given axis, divide the least moment of inertia by the sectional area of the plane ; that is, r 2 = . Values of For Malleable Iron. For Cast Iron. C= 36,000 Ibs. 80,000 Ibs. c= 36,000 " 3,200 " a== 0.000333 0.0025 For Dry Timber. 7,200 Ibs. 3,000 " 0.004 The factor of safety k should be, for wrought iron = 6; for cast iron = 8; for timber = 10. This applies to moving loads. Case 1. Rounded or hinged at both ends, as per Fig. 173. For square, rectangular, or circular cross-section : ir=i._ c ^_ For any other cross-section: = ~l 7^" i+-5r Case 2. Fixed, or having a flat base at one end, and rounded or hinged at the other, as per Fig. 174. For square, rectangular, or circular cross-section : W = A k p For any other cross-section: 1 CA 1 + 16 J 2 9.c.r 2 RESISTANCE TO CRUSHING. 105 Case 3. Fixed, or having flat bases at both ends, as per Fig. 175. For square, rectangular, or circular cross-section : 1 CA l+aJ ^- For any other cross-section: 1 OA 1 + EXAMPLES. Case 1. Rounded at both ends: What is the capacity of a u rought-iron strut of the annexed figure and dimensions? I = 10 feet = 120 inches. A = 4.68 inches. r __ 0.9 X 3.58+5.1x0.38 36000 X 4.68 1 + 120 2 36000 X 0.689 12 168480 57600 ~^~ 124804" = 3.227 "37322" The same as above, in Case 3, fixed at both ends: 36000 x 4.69 168480 1 + 168480 36000 x 0.689 1 + 14400 24804 106 RESISTANCE TO CRUSHING. For the annexed figure and dimensions ; otherwise, same as above : A = 7 inches. Case 1. \2: T- 1 X 43+3 x I 3 Rounded at both ends : Fig. 177. 1 = 12 = 5.( 36000 x 7 , 252000 -T*-1W~ = * 8 = 21 00 lbs . ^ A. ^ v O X _ ^ ^ 36000 X 0.8 Same as above, in Case 3, fixed at both ends : 36000 x 7 252000 i_i 36000 X 0.8 - = 1- - ~-r = 42 > 000 lbs - 1 . o Fixed ends : Case 3. What is the capacity of a cast-iro7i pillar of the annexed figure and dimensions? 1= 10 feet= 120 inches. A = 11 inches. i X 4 3 7 X 3 3 = 26.9 80000 X 11 880000 ~T90~ * S~25" = 1+0.0025 r-f- RESISTANCE TO CRUSHING. 107 For the annexed figure and dimensions; otherwise, same as above. Fig. 179. V TP=J- A 28 inches. 80000 X 28 . 1 + 0.0025 J- _2240000_ = 179)200U)3 . 1.5625 For the annexed figure and dimensions; otherwise, same as above. Fig. 180. A = 22 inches. 80000 X 22 120 2 1 + 0.0025- 1760000 1.5625 = 140,800 Ibs. To find the capacity of a Column, Pillar, or Strut of any cross-section by the following Table : Find how many times the least dimension h across the section is contained in the length I of column, &c. that is, then multiply the corresponding number on the same horizontal line, under K" , by the sectional area of cross-section. This gives the capacity in tons of 2,000 Ibs. Let I = Length of column, &c. h = Least dimension of cross-section. K" = Capacity in tons of one square inch of cross-section, to be multiplied by sectional area of desired cross- section. Various sections for which this table is applicable: Fig. 181. Fig. 182. 108 RESISTANCE TO CEUSHING. Fig. 183. Fig. 184. Fig. 186. Fig. 185. Fig. 187. Fig. 188. [NOTE. This table is strictly correct, only for columns, Ac., with circular or rectangular cross-section. As the error is small, it may be used for any cross-section.] Example explanatory of the following table. What is the capacity of a cast-iron column 10 feet = 120 inches long, fixed at both ends, and of the annexed cross-section and dimensions? Fig. 189. 7 19() -f- = -=- = 40 K" for 40=1.000 tons. h 3 Area= 6 inches. W=. 6 X 1 = 6 tons, 8 times safety. BESISTANCE TO CRUSHING. Column, &c., fixed at both ends. 109 Cast Iron eight times safety. Wrought Iron six times safety. I h K" I h K" I h K" I h K" I ~h K" I K" Tons. Tons. Tons. Tons. Tons. Tons. 1 4.987 25 1.951 49 0.714 1 2.999 25 2.487 49 1.674 o 4.950 26 1.858 50 0.689 2 2.996 26 2.452 50 1.644 3 4.890 27 1.771 51 0.666 3 2.991 27 2.418 51 1.615 4 4.807 28 1.689 52 0.644 4 2.984 28 2.383 52 1.585 5 4.705 29 1.611 53 0.623 5 2.975 29 2.348 53 1.557 6 | 4.587 30 1.538 54 0.603 6 2.964 30 2.313 54 1.529 7 | 4.450 31 1.469 55 0.584 7 2.953 31 2.277 55 1.501 8 I 4.310 32 1.404 56 0.565 8 2.938 32 2.242 56 1.474 9 10 4.158 4.000 33 34 1.343 1.285 57 58 0.548 0.531 9 10 2.921 2.905 33 34 2.206 2.172 57 58 1.4-18 1.422 11 3.838 35 1.230 59 0.515 11 2.885 35 2.136 59 1 396 12 3.676 36 1.179 60 0.500 12 2.863 36 2.101 60 1.371 13 ! 3.514 37 1.130 61 0.485 13 2.841 37 2.067 01 1.347 14 i 3.355 38 1.084 02 0.471 14 2.817 38 2.032 62 1.323 15 I 3.200 39 1.041 63 0.457 15 2.792 39 1.998 63 1.299 If. 3.048 40 1.000 64 0.445 16 2.766 40 1.963 64 1.276 17 2.902 41 0.961 65 0.432 17 2.738 41 1.930 65 1.253 18 2.762 42 0.924 66 0.420 18 2.711 42 1.896 66 1.228 19 2.628 43 0.889 67 0.409 19 2.680 43 1.863 67 1.209 20 2.500 44 0.856 68 0.398 20 2.650 44 1.831 68 1.187 21 2378 45 0.824 69 0.387 21 2.619 45 1.798 69 1.167 22 2.252 46 0.794 70 0.377 22 2.586 46 1.767 70 1.146 23 2.152 47 0.766 71 0.367 23 2.554 47 1.735 71 1.126 24 2.049 48 0.739 72 0.358 24 2.520 48 1.704 72 1.107 110 KESISTANCE TO CRUSHING. Strength of Columns, Pillars, or Struts, of seasoned wood, round or square section. Fixed at both ends. All dimensions in inches. Find how many times the least dimension across the section is TT contained in the length or height of column, &c.; that is, - ; then multiply the corresponding figures on the same horizontal line under K" by the sectional area of cross-section. This gives the capacity of column, &c., in tons of 2,000 Ibs., 10 times safety. Reference. H= Length of column, &c. D = Least dimension of cross-section. K" = Capacity in tons of one square inch of cross-section, to be multiplied by sectional area of desired cross-section. The coefficient C for white and yellow pine in the following table is taken at - h -f {} = 600 Ibs. for safety : For oak at S -J{J-- = 800 Ibs. per square inch for safety. EXAMPLE. What is the capacity of a pillar of oak, section 4x6 inches, length = 12 feet = 144 inches ? for 36 _. o.064 x 4 X 6 = 1.536 tons. Capacity K f/ of one square inch in tons of 2,000 Ibs. White and Yellow Pine. Oak. H ~D " j\" H ~D = K" H ~D ~ E H ~D ~ K" 1 0.299 26 0.081 1 0.399 26 0.108 2 0.2:)5 27 0.076 2 0.394 27 0.102 3 0.289 28 0.072 3 0.386 23 0.096 4 0.282 29 0.068 4 0.376 29 0.091 5 0.272 30 0.065 5 0.363 30 0.086 6 0.262 31 0.061 6 0.349 31 0.082 7 0.251 32 0.058 7 0.334 32 0.078 8 0239 33 0.056 8 0,319 33 0.074 9 0.226 34 0.053 9 0.302 34 0.071 10 0.214 35 0.050 10 0.285 35 0.067 11 0.202 36 0.048 11 0239 36 0.064 12 0.190 37 0.046 12 0.254 37 0.061 13 0.179 38 0.044 13 0.238 38 0.059 14 0.168 39 0.042 14 0.224 39 0.056 15 0.158 40 0.040 15 0.210 40 0.054 16 0.148 41 0.038 16 0.197 41 0.051 17 0.139 42 0.037 17 0.185 42 0.049 18 0.130 43 0.035 18 0.174 43 0.047 19 0.123 44 0.034 19 0.163 44 0.045 20 0.115 45 0.033 20 0.154 45 0.044 21 0.108 46 0.031 21 0144 46 0.042 22 0.102 47 0.030 22 0.136 47 0040 23 0.096 48 0.029 23 0.123 48 0.039 21 0.030 49 0.088 24 0.121 49 0.037 25 0.085 50 0.027 25 0.114 50 0.036 PARALLELOGRAM OF FORCES. Ill PARALLELOGRAM OF FORCES. COMPOSITION AND RESOLUTION OF FORCES. Reference. A, B, C = Forces, cr strains, acting on a single point, y, i/, = angles. Fig. 190. A = (7sin. v, sin. (v + v,) _ Csin.v . B = - ; -, when v = v /y A = B sin. (v + ^/) sec. v; whenv+^<90 C = when t> -f u y > 90 /?_ cos. (v + v,) = 90 = C cos. v = C sin. v (7 = 112 STRAINS IN FRAMES. STRAINS IN FRAMES. Reference. C = Compressive strain in units of weight, T= Tensile F= Vertical H= Horizontal " W= Load in units of weight. I = Dimensions in units of length. v = Angle between horizontal and inclined member. For 01 oss-breaking strain, see "Resistance to cross-breaking. Fig. 193. W 2 sin. v W y x = cotg. v = H Fig. 194. C / = H = ij W cotg.v = cross-breaking strain at H. H / = -j- H = j^. j- TP r cotg. v = tension in H/. I I H H, = \\. ( -- ) W cotg. v = compression in F= U W. \ I J STRAINS IN" FRAMES. 113 Fig. 195. T7 TT - V = H / tang, y = - tang, y = compression. W.I cos. y 1^, , cos. y nressiou. //= W.I. When I > ^ 3 the portion t // is in tension = V W = W Ltang.y-l) \ Ijj I When I < Z 3 the portion l y/ is in compression = IF F = F/ = . TF= tension, v /* /. 107. Ends of beams built into wall or fixed- F= -L W V, = V- W = (,--} W, = T, (tension) = C, (. V V pression.) Ill STRAINS IN BOOM DERRICKS. C= ( J = (compression) = T (tension.) V 21, J sin. v II f~ \ TFcotg. v = (tension) = H, (compression.) Ends of beams not built into wall or fixed: T/ /r= v W= ( / ) W = C / (compression) = T, (tension.) 0= = = T (tension.) sin. v I, sin. v H= Fcotg. v = TFcotg. v = (tension) = H / (compression.) STRAINS IN BOOM DERRICKS. Reference. C = Compression in boom. C / = Compression in mast. T= Tension in tackling. T,= Tension in guy. t = Tension in runner from mast head to weight. t / = Tension in runner from boom head to weight. W = Weight or load. H = Horizontal strain. V = Vertical strain. v, v lt v 2 = Angles. (See Figure.) Fifj. 198. STEAINS IN TRUSSES. 115 TTsin. v 1 sin. (v + Vj F"== / cosin. v 1 (7= Fcosec. v 2 T= Fcosec. v TFsin. v sin. (v + v x *,= <?,= PF 2^= Fcotg. -y 3 sec. v 4 STRAINS IN TRUSSES. Zoac? equally distributed. Reference. W= Load equally distributed in Ibs. I = Distance between abutments. v = Angle between horizontal and diagonal. 0= Compression in Ibs., (denoted by thick lines.) T= Tension in Ibs., (denoted by thin lines.) 2 Bays = 4- Fig. 199. T- -* ~ T6 - 3 Bays = . 200. 116 STEAINS IN TRUSSES. 4 Bays Fig. 201. C- W C 2 3(7 2 cotg. Fig. 202. C = T 6C 2 cotg. v (?!= T 1 = 2(7 2 cotg. t? W 2 rT 2 = ^ cosec. v m STRAINS IN TRUSSES. I 0= T = 90 2 8C, 2 5 a cotg. v cotg. v a 2 5(7 3 65 " "~T~ T 3 = j cosec. ^4 = 3 ^3 TABLE OF CONSTANTS, BASED ON FOREGOING FORMULA. Load equally distributed. Table of constants for strains in respective member of trusses, from 2 to 6 bays, with diagonals inclined from 5 to 45 : Reference. W = Load in Ibs., equally distributed over whole length of truss, to be multiplied by constant for strain in re- pective member. v = Angle between horizontal and diagonal. 0= Compression in Ibs. in respective member. T= Tension in Ibs. in respective member. EXAMPLE. Required, the strain in the various members of a truss of 4 bays. Length = 40 feet ; load W = 80,000 Ibs. ; angle v = 20. Members. Constants. W. Strains. C = T = 1.372 x 80,000 = 109,760 Ibs. c i= TI= 1.029 X 80,000 = 82,320 <7 2 =0.25 x 80,000= 20,000 <7 3 = 0.375 x 80,000 = 30,000 T 2 = 0.365 x 80,000= 29,200 T 3 = 1.095 x 80,000 = 87,600 [NOTE. When the trusses are inverted, the strains change in kind, but not in amount] 118 STRAINS IK TRUSSES. 2 Bays = - Fig. 204. 3 Bays = Fig. 205. V O C l T (7= T Ci Tl 5 3.572 0.625 3.584 3.810 0.333 3.820 6 2.972 " 2987 3.170 " 3.186 7 2.544 < 2.562 2.713 2.733 8 2.225 " 2.244 2.370 M 2.393 9 1.972 " 1.997 2.103 " 2.130 10 1.772 (C 1800 1.890 M 1.920 11 1.610 " 1.640 l.MO " 1.747 12 1.469 " 1.500 1.570 M 1.603 13 1.353 " 1.390 1.444 " 1.483 14 1.253 < 1.290 1.333 1.376 15 1.166 1 1.210 1.243 " 1.286 16 1.087 " 1.134 1.160 ( 1.210 17 1.022 " 1.070 1.090 M 1.140 18 0.959 " 1.013 l.<23 " 1.080 19 0.906 H 0.959 0.970 " 1.023 20 0.859 " 0.912 0.917 " 0.973 21 0.813 " 0.872 0.866 " 0.930 22 0.778 " 0.834 0.823 0.890 23 0.734 0.790 0.783 : 0.853 24 0.703 ( 0.765 0.750 0.810 25 0.668 " 0.738 0.713 " 0.786 26 0.641 0.712 0.685 0.760 27 0.613 " 0.687 0.653 0.730 28 0.587 ( 0.666 0.626 " 0.701 29 0.562 <C 0.644 0.600 .( 0.686 30 0.541 (( 0.625 0.643 M 0.666 31 0.519 0.606 0.555 " 0.646 32 0.500 M 0.591 0.533 M 0.630 33 0.481 it 0.575 0.513 0.613 34 0.463 0.559 0.493 0.596 35 0.447 0.544 0.476 " 0.580 36 0.431 0.531 0.460 0.566 37 0.416 " 0.519 0.444 " 0.553 38 0.400 " 0.506 0.426 M 0.540 39 0.384 < 0.497 0.410 0.530 40 0.372 0.487 0.396 " 0.520 41 0.359 0.475 0.385 " 0.506 42 0.347 H 0.466 0.370 0.496 43 0.334 0.456 0.357 " 0.486 44 0.322 < 0.450 0.343 0.480 45 0.312 " 0.444 0.333 " 0.473 STRAINS IN TRUSSES. 119 4 Bays = Fig. 20(>. V C = T CV-5T! C 2 Cs T 2 T 3 5 5.720 4290 0.250 0.375 1.434 4.032 6 4.700 3.570 " 11 1.200 3.600 7 4.008 3.051 " " 1.025 3.075 8 3.500 2.070 " 0.897 2591 9 3.1G4 2.373 * ;; 0.799 2.397 10 2.8-^2 2.124 0.720 2.160 11 2.508 1.926 ; 0.655 1.965 1 2 2.388 1.791 0.601 1.803 13 2.164 1623 0.556 1.608 14 2.000 1.500 0.516 1.548 15 1.804 1.398 0.482 1.446 1C 1.7-10 1.305 0.454 1.362 17 i.632 1.224 0.428 1.284 18 1.532 1.149 0.405 1.215 19 1.448 1.086 0.384 1.152 20 1.372 1.029 " 0.365 1.095 21 1.300 0.975 " 0.349 1.047 22 1.23G 0.927 ; 0.334 1.002 23 1.172 0.879 " 0.32) 0.960 24 1.124 0.843 ; " 0.306 0.918 25 1.008 0801 0.295 0.885 26 1.024 0.708 * 0.285 0.855 27 0.080 0.735 0.275 0.825 2-i 0.940 0.705 " 0.266 0.798 2y 0.900 0.675 1 0.258 0.774 30 0.804 0.048 0.250 0.750 31 0.823 0.621 0.243 0.729 32 0.800 O.COO 236 0.708 H,i 0.708 0.576 0.230 0.690 34 0.740 0.655 * 0.224 0.672 35 0.720 0.540 " * 0.218 0.654 36 0.088 0.516 0.212 0.636 37 O.G04 0.498 0.207 0.621 38 0.640 0.480 " 0.203 0.009 39 0.016 0.462 0.199 0.597 40 0.000 0.450 ; 0.195 0.585 41 0.576 0.432 0.190 0.570 42 0.500 0.420 0.186 0.558 43 0.536 0.402 0.183 0.549 44 0.520 390 0.180 0.540 45 0.500 0.375 ^ " 0.177 0.531 STRAINS IN TRUSSES. V e-jr C i = Ti C 2 ^3 Tz n 5 6.858 4.572 0.200 0.400 2.294 4.588 6 5.706 3.804 " 1.912 3.824 7 4.884 3.256 1.640 3.280 8 4.272 2.848 " 1.436 2.872 9 3.786 2.524 . 1.278 2.556 10 3.402 2.268 i 1.152 2.304 11 3.084 2.056 1.048 2.0 )6 12 2.820 1.880 " 0.962 1.01:4 13 2.598 1.732 " " 0.890 1.780 14 2.406 1.604 M 0.826 1.C.52 15 2.238 1.492 " 0.772 1.544 16 2.088 1392 " 0.726 1.452 17 1.962 1.308 " " 0.684 1.3(58 18 1.842 1.228 " 0.648 1.296 19 1.740 1.160 M " 0.614 1.228 23 1.650 1.100 " " 0.584 1.168 21 1.560 1.040 0.558 1.116 22 1.482 0.988 " 0.534 1.068 23 1.410 0.940 1 0.512 1.024 24 1.350 0.900 41 0.490 0.980 25 1.284 0.856 " 0.472 0.944 26 1.230 0.820 0.456 0.912 27 1.176 0.784 " 0.440 0.880 28 1.128 0.752 " 0.426 0.852 29 1.080 0.720 0.412 0.824 30 1.038 0.692 0.400 0.800 31 0.996 0.664 " 0.388 0.776 32 0.960 0.640 0378 0.756 33 0.924 0.616 " 0.368 0.736 34 0.888 0.592 " 0.358 0.716 35 0.858 0.572 " 0.348 0.696 36 0.828 0.552 " i 0.340 0.680 37 0.798 0.532 " 0.332 0.664 38 0.768 0.512 ; 0.324 0.648 39 0.738 0.492 " 0.318 0.636 40 0.714 0.476 M 0.312 O.C24 41 0.690 0.460 " 0.304 0.608 42 0.666 0.444 0.298 0.596 43 0.642 0.428 " 0.292 0.584 44 0.618 0.412 " 0.288 0.576 45 0.600 0.400 " 0.284 0.568 STRAINS IN TRUSSE.S 121 V t7- T Pi -21 Cj 2*2 % <7 4 e 5 1 T. T 4 T-, 5 8.5G8 7.G16 4.760 0.1 66 0.250 0.416 0.952 2.856 4.760 G 7 7.123 G.102 6.336 5.424 3.960 3.390 n u 0.680 2.379 2.041 3.965 3.402 8 5.337 4.744 2.965 ; M " 0.596 1.788 2.980 g 4.023 4.200 2.G25 " " 0.530 1.590 2.650 10 4.218 3.776 2.360 " " M 0.478 1.434 2.390 11 3.852 3.424 2.140 " " 0.435 1.305 2.175 12 3.519 3.128 1.955 " 0.399 1.197 1.995 13 3.240 2.880 1.800 " 0.369 1.107 1.845 11 3.006 2.672 1.670 " " 0.343 1.029 1.715 15 2.799 2.488 1.555 0.320 0.960 1.600 1C 2.610 2.320 1.450 < ; " 0.301 0.903 1.505 17 2.448 2.176 1.360 " 1 " 0.284 0.852 1.420 18 2.304 2.048 1.280 " 0.269 0.807 1.345 if) 2.1G9 1.928 1.205 " " , 0.255 0.765 1.275 ft) 2.001 1.832 1.145 " " 0.242 0.726 1.210 21 1.944 1.728 1080 " " 0.231 0.693 1.155 22 1.854 1.G48 1.030 " 0.221 0.663 1.105 2:> 1.7G4 1.568 0.980 " 0.212 0.636 1.060 21 1.G83 1.496 0.935 " " 0.203 0.609 1.015 23 1.G02 1.424 0.890 " 1 " 0.196 0.588 0.980 2; 1.539 1.368 0.855 " " 0.189 0.567 0.945 27 1.4G7 1.304 0.815 " " 0.182 0.546 0.910 28 1.404 1.248 0.780 t 0.177 0.531 0.885 29 1.350 1.200 0.750 " " 0.171 0.513 0.855 30 1.2)6 1.152 0.720 " " 0.166 0.498 C.830 :>L 1.242 1.104 0.690 M 0.161 0.483 0.805 32 1.197 1.0G4 0.665 " " 0.156 0.468 0.780 33 1.152 1.024 0.640 M 0.152 0.45G 0.760 34 1.107 0.984 0.615 " < ; 0.148 0.444 0.740 35 1.071 0.952 0.595 M 0.144 0.432 0.720 30 1.035 0.920 0.575 M " 0.141 0.423 0.705 37 0.999 0.888 0.555 M 0.138 0.414 0.690 88 0.954 0.848 0.530 M 0.134 0.402 0.670 39 0.918 0.816 0.510 H 0.132 0.396 0.660 40 0.891 0.792 0.495 i- t 0.129 0.387 0.645 41 0.8G4 0.768 0.480 0.126 0.378 O.C30 42 0.823 0.736 0.460 M a 0.123 0.369 0.615 4:5 0.801 0.712 0.445 " 0.121 0.363 0.605 44 0.774 0.688 0.4 JO " M 0.119 0.357 0.595 45 0.747 0.664 0.415 " " " 0.118 0.354 0.590 122 STRAINS IN TRUSSED BEAMS. STRAINS IN TRUSSED BEAMS. When a beam supported at the ends, is required to carry a greater load than its given capacity, and trussing is resorted to, it may become necessary to find what portion of the load is borne by the different members of the trussed beam. Reference. Let IF = Load acting on truss at a supported point. (See figure.) W l = That portion of IFacting on diagonals. W. 2 = That portion of IF" acting on beam. A i = Sectional area of diagonal. A 2 = Sectional area of beam. E l = Modulus of elasticity of material in diagonals. E 2 = Modulus of elasticity of material in beam. a = Length of diagonal. b = Distance between center of beam and point of support. c = Distance between abutment and point of support. { = Depth of beam. = Depth of truss. / = Distance between center of beam and abutment. [NOTE. Use the same unit of length and weight.] No. 1. Fig. 209 a 3 / 2 A 2 E 2 . -/I. A.. JIM Wj. a 3 / 2 <4 2 ^ 2 STRAINS IN TEUSSED BEAMS. - . TF When load is equally distributed W becomes | TF. No. 2. Fig. 210. 211. W<2 2 a 123 2 a 3 * / 2 A, >m a3 / 2 ^ 2 A, 2 Wi a 3 / 2 a 3 * / 2 TFl= "n7 "" PP, + 1 When load is equally distributed W becomes f TF. 124 STRAINS IN TRUSSED BEAMS. No, 3. Fig. 212. A l W 2 a (a* X _ h* (Vb^c E l w l w l *. w +1 When load is equally distributed IF becomes f PP". STRAINS IN TRUSSED BEAMS. 125 No. 4. Figs. 213 and 214. h* (I 2 - 6 2 ) c A l E l 2/2 a(a 2 +6c) ^ _ " A A, / 2 ~~ h* (Pb*)c 2W L / 2 a(a 2 +6c) W l . W W -+1 When load is equally distributed TF becomes f W. 126 STRAINS IN TRUSSES WITH PARALLEL BOOMS. STRAINS IN TRUSSES, WITH PARALLEL BOOMS. (Caused by Static and Moving Loads) The strain in the upper boom is always compressive. * The strain in the lower boom is always tensile. All braces inclined down from the nearest abutment are in tension. All braces inclined up from the nearest abutment are in com pression. The strains in the verticals and diagonals increase from the center of truss to abutment. The strains in the booms decrease from the center of truss to abutment. A moving load, advancing over a truss, &c., causes the maxi mum moment of rupture (which under an equally distributed load is at the center of truss) to shift to one side of the center, thereby changing the nature and amount of strain in web only. This requires either the enlargement of those members consti tuting the web or the addition of so-called counters, (braces, struts, or ties.) To find the point from center of truss to where the addition ol counters must commence, the following formula is used : Let d = Distance from center of truss to point where, maximum moment of rupture occurs, and where counter bracing must commence. d / = Distance from nearest abutment to ditto. Anad /= J d = -^- 2 W / These results will be found to agree with formulas for " Counter Strains" when V m becomes negative. Reference. N = Total number of bays in a truss. HK = Horizontal strains in booms. F n = Strains in verticals. y n = Strains in diagonals. V m = Vertical strains acting on counters Y m . Y m = Strains in counters, opposite in kind to Y n . STRAINS IN TRUSSES WITH PARALLEL BOOMS. 127 IF = Weight of static load, equally distributed over whole length of truss. W,= "Weight of moving load, equally distributed over whole length of truss. h = Height or depth of truss between the center of gravity of booms. I = Span or length of truss from abutment to abutment. n Number of member, counting from abutment A. in = Number of member, between center and abutment B. r = Half the length of a panel or bay. s = Length of a panel or bay. w = Weight of static load per unit of length I. iv / = Weight of moving load per unit of length I. v = Angle between horizontal and diagonal. For other designations, see diagrams and examples. The angle v for Howe Truss is generally 45. The angle v for Whipple Truss is generally 45. The angle v for Lattice Truss is generally 45. The angle v for Warren Truss is generally 60. The proportion of height h to span I is from 4 to -^ gener ally T v 128 STRAINS IN TRUSSES WITH PARALLEL BOOMS. STRAINS IN" TRUSSES WITH PARALLEL BOOMS. 129 HOWE TRUSS. (Figs. 215, 216, 217, and 218.) Additional Reference. ,T n Distance from abutment A to center of bay. y n Distance from abutment A to apex of bay. Static or Permanent Load, equally distributed over whole length of Truss, Strains in Booms. w w Strains in Verticals. w w F.= - --- -*. Strains in Diagonals. ^n= V n cosec. v. Moving and Static Load, each equally distributed per unit of length. ~ Strains in Booms. w+w, ~2h~ 2hl~ Strains in Verticals. Strains in Diagonals. Y n = F" n cosec. v. Strains in Counters. 130 STRAINS IN TRUSSES WITH PARALLEL BOOMS. EXAMPLE. (Figs. 215, 216, 217, and 218.) Moving Load, (as railway train passing over bridge.) We will assume W = 50,000 Ibs. Wt= 100,000 Ibs. I = 100 feet. h = 10 feet. v = 45, (cosec. = 1.414.) Horizontal Strains in Booms, (compression inupper, tcnsionin lower.) W-\- Wi JF + TFi 2 __ 50UOO+ 100000 //a = 2/i y " ~ 2hl 2/n = ~20 50000 + 100000_ 2 __ _ ^ 5 2 yn ~~ 2000 ~~ y& ~~ J/a 0-2/n //! = 7500 . 10 75 . 100 = 67,500 Ibs. II 2 = 7000.20 75.400 = 120,000 Ibs. 11. 3 = 7000.30 75.900 = 157,500 Ibs. 7/ t = 7500.40 75.1600 = 180,000 Ibs. J1 5 = 7500.50 75.2500 = 187,500 Ibs. Strains in Verticals. W W Wl ,\2__ 500 22__ 5000 n i> r ** "*" W ^ ". ^ 2 100 .(^ .rj 2 = 25000 500. .T n H Strains in Figs. 215 210 217 218 F 1 = 25000 500.5 +5. 05*= 67625 Ten. Ten. Corn. Com. F 2 = 25000 500. 15 4-5. 85 2 = 53625 " F, = 25000 500. 25+5. 7r> a = 40625 " F 4 = i ) 5000 500. 35 + 5. ()5 2 = 28625 " Vl = 25000 500. 45 +o.55 2 = 17625 " Counter Strains (V m ) for Strains in Counters. V 6 = 20000 500.55 + 5.45* = 7625. F 7 = 25000 500.65 + 5.85 = 5625. Strains in Diagonals. Y n = F n cosec v. Strains in Figs. 215 210 217 218 y\ = 67625 . 1.414 = 1)5,620 Ibs. Com. Com. Ten. Ten. Y 2 == 53625 . 1.414 = 75,826 Ibs. F 3 = 40625 .1 414 = 57,44 i Ibs. F 4 = 28625 . 1.4 14 = 40,476 Ibs. " F 5 = 17625 . 1.414 = 24,922 Ibs. Strains in Counters, (dotted lines, Fig. 215, for example.) Yn = F m cosec. v. Strains in Figs. 215 210 2L7 218 F G = 7625 . 1.414 ^r 10,762 Ibs. Com. Com. Tun. Ten. F 7 =5625 . 1.414= 7,954 Ibs. STRAINS IN TRUSSES WITH PARALLEL BOOMS. 131 Pig. 219. LATTICE TRUSS WITH VERTICAL NUMBERS. sssvs. Fig. 219. Load on either Boom. To compute the strains in this truss, the easiest method is to find the values of /f n , F n , F m , F n , and F m for a Howe Truss, (Figs. 215, ( 216, 217, and 218 ) loaded in the same man- | ner, (upper or lower boom.) These values in the following formulas for the above truss will i give the required strains: Strains in Booms. (8.) <?!=- 4*2~T" jC *8 /^ n r/ -"n l~ : , generally o n =: f Strains in Verticals. (U.) Upper boom loaded compression. Lower boom loaded tension. W l constant. Strains in End Post ( U .) Upper boom loaded. U = U -f- ^1= compression. Lower boom loaded. U = Si^= compression. Strains in Diagonals. (D.) D *=^~ I?_ 2 Y* Generally D n = i Strains in Counters. Generally Z> m = 132 STRAINS IN TRUSSES WITH PARALLEL BOOMS. Fig. 220. WARREN TRUSS. Fig. 220. Lower Boom Loaded. Additional Reference. #n = Distance from abutment A to center of diagonal. 3/n = Distance from abutment A to apex of bay of upper boom. 2n = Distance from abutment A to apex of bay of lower boom. Static or Permanent Load, equally distributed over whole length of Truss. Strains in Booms. Upper. 77 W W 2 ^ = -ir-2Ar* 2 Lower. W W Strains in Verticals. K = --- a? n ( F n acts at the end - I of ar n .) Strains in Diagonals. Y^ ==. F n cosec. v. Moving and Static Load, each equally dis tributed per unit of length. Strains in Booms. Upper. Lower. Strains in Verticals. --?-? +5 " Strains in Diagonals. Y* = F n cosec v. STRAINS IX TRUSSES WITH PARALLEL BOOMS. 133 Strains in Counters. EXAMPLE. (Fig. 220.) Moving Load (as railway train passing over bridge) on lower Bocm. We will assume W = 50,000 Ibs. Wi= 100,000 Ibs. I = 100 feet. A = 10 feet. ?; = 63 20 , (cosec. = 1.12.) _ Horizontal Strains in Upper Boom. (Compression.) W+Wi ^ 2 __ 50000 + 100000 2h 2hl 2.10 50000 + 100000 2 _ 150000 Zn ~~ 2.10 . 100 ** ~ ~~20 2n ~~ HI = 7500 . 10 75 . 100 = 67,500 Ibs. #2 = 7500.20 75.400 = 120,000 Ibs. H 3 = 7500.30 75.900 = 157,500 Ibs. H 4 = 7500.40 75. 1600 = 180,000 Ibs. H 6 = 7500.50 75.2500 = 187,500 Ibs. Horizontal Strains in Lower Boom. (Tension.) i JF-f Jh 50000 + 100000 " 2A 2hl 2.10 50000 -f- 100000 f 2 _ 150000 150000 2.10 .~100~ -2/n ~ 20 ^ n 2000~ 75.25 = 37500 1875= 35,625 Ibs. #2 = 7500.15 75.225 =112500 16875= 95,6251bs. H 3 = 7500 . 25 75 . 625 = 1 87500 46875 = 140,625 Ibs. Jff 4 = 7500.35 75. 1225 = 262500 91875 = 170,625 Ibs. H 6 = 7500.45 75.2025 = 337500 151875 = 185,623 Ibs. 134 STRAINS IN TRUSSES WITH PARALLEL BOOMS. Strains in Verticals. YU = F n cosec. v. W W W, , 50000 Fn = .rr n + -nT-(J *) = o 2 2 2 50000 "Too" r n + ^^T-( 100 *u) 2 = 25 000 500,r n + 5. (100 x u )* v, ss 25000 500 . 2.5 + 5 . , 9506.25 = 71281.25. r. 5= 25000 500 . 7.5 + 5 , , 8556.25 = 64031.25. F~ S=9 25000 500 . 12.5 + 5 . 7656.25 = 57031.25. P as 25000 500 . 17.5 + 5 , , 6806.25 = 50281.25. F 8 25000 500 . 22.5 + 5 , . 6006.25 = 43781.25. ^ = 25000 500 . 27.5 + 5 , 5256.25 37531.25. [ J _ 25000 500 . 32.5 + 5 . . 4556.25 = 31531.25. P- _-r^= 25000 500 . 37.5 + 5 . 3906.25 25781.25. p* 25000 500 . 42.5 + 5 , . 3306.25 SBB 20281.25. l n ( == 25000 500 . 47.5 + 5 , . 2756.25 = 14031.25. F! j= 25000 - Fj 2 = 25000 - F! 3 = 25000- = 71281.25 , = 64031.25 = 57031.25 = 50281.25 , = 43781.25 = 37531.25 , = 31531.25 = 25781.25 = 20281.25 n = 14031.25 , Counter Strains. ( F m .) 500 . 52.5 4- 5 . 2256.25 = 500 . 57.5 + 5 . 1806.25 = 500 . 62.5 + 5 . 1406.25 = Strains in Diagonals. Y" n = F" n cosec. v. 10031.25. 528125. 781.25. 1.12 = 79,835 Ibs. 1. 12 = 71, 715 Ibs. 1.12 = 63,875 Ibs. 1.12 = 56,315 Ibs. 1.12 = 49,035 Ibs. 1.12 = 42,035 Ibs. 1.12 = 35,315 Ibs. 1.12 = 28,875 Ibs. 1.12 = 22.715 Ibs. 1.12 = 15,715 Ibs. Compression in Y l and F 2 , Tension in Y 2 and Y 19 . Compression in Y 3 arid Y li Tension in Y and ]T 17 . Compression in Y 5 and Y 1 Tension in Y 6 and Y 15 . Compression in F" 7 and y r Tension in Y s and ]T 13 . Compression in Y g and Y l Tension in Y lo and Y ll . Counter Strains. = T/ m COS6C - FH= 10031.25 . 1.12 = 11,235 Ibs. Compression in F 12 = 5281.25.1.12= 5,915 Ibs. Tension in Y 9 i F 13 = 781.25 . 1.12= 875 Ibs. Compression in i and . lif j STRAINS IN TRUSSES WITH PARALLEL BOOMS. 135 Fiy. 221. WARREN TRUSS. Fig. 221. Upper Boom Loaded. Additional Reference. = Distance from abutment A to center of bay of upper boom. 2/n = Distance from abutment A to apex of bay of upper boom. 2 n = Distance from abutment A to apex of bay of lower boom. Static or Permanent Load, equally distributed over whole length of Truss. Strains in Booms. Upper. / W Wr* Lower. w 2h *"" Strains in Verticals. _ W W T a J #n i I/ Strains in Diagonals. Y n = V a cosec. v. p Moving and Static Load, each equally dis tributed per unit of length. Strains in Booms. Upper. w+Wl ( w + w, ^ 2n, a " ( 2hl * H 2hl Lower. TT+TF, 136 STRAINS IN TRUSSES WITH PARALLEL BOOMS. Strains in Verticals. - Strains in Diagonals. T n = F n cosec. v. Strains in Counters. w w w --- *-^"" (l ~* = 7 m cosec. EXAMPLE. (Fig. 221.) Moving Load (as railway train passing over bridge) on Upper Boom. We will assume W = 50,000 Ibs. W l = 100,000 Ibs. I = 100 feet. h = 10 feet. v = 63 20 , r = 5 feet. Horizontal Strains in Upper Boom. (Compression.) jT+JFi rjF+TPi 2 ,(W+W } )r*-}_ 2/i n L 2M <2a H 2AZ J "" lupOOO_ ^ 2 150000. 5* 1 _ ~200(r~ Zn ^ ^000 J ~ J50000_ rluOOO 20 Za 7500. 2 n [75.2 n 2+ 1875] jff 1= =7500.5 H 2 = 7500.15 ,= 7500.25 75.25 -f 1875" 75.225 + 1875 : 75.625 -f 1875 H= 7500.35 [75.1225 -j- 1875 H 5 = 7500.45 [75.2025 + 1875; = 33,750 Ibs. = 93,750 Ibs. = 138,750 Ibs. = 168,750 Ibs. = 183,750 Ibs. Horizontal Strains in Lower Boom (Tension.) - . ^==7500.10 75.100 = 67,500 Ibs. H,= 7500.20 75.400 = 120,000 Ibs. Hl= 7500.30 75.900 = 157,500 Ibs. J/ 4 == 7500.40 75.1600 = 180,000 Ibs. JI 5 = 7500.50 75.2500 = 187,500 Ibs. STRAINS ITS TRUSSES WITH PARALLEL BOOMS. 137 Strains in Verticals. F n = -J---^.* + _|L(Z_a; n ) 2 = 25000-500.* + 5.(Z z n ) 2 F 1= = 25000 500.5 + 5.95 2 = 67,625 Ibs. F 2 = 25000 500.15 + 5.85* = 53,625 Ibs. F 3 == 25000 500.25 + 5.75 2 = 40,625 Ibs. F 4 = 25000 500.35 + 5.65 2 = 28,625 Ibs. F 5 = 25000 500.45 + 5.55 2 == 17,625 Ibs. Counter Strains. F G = 25000 500.55 + 5.45* = 7,625 Ibs. Strains in Diagonals. F n = Fn cosec. Y 1 = 67625 . 1.12 = 75,740 Ibs. Tension in Y l and F 10 ; compression in F a and Y A . Y 2 = 53625 . 1.12 = 60,060 Ibs. Tension in Y 2 and F 9 ; compression in F b and F b . F 3 = 40625 . 1.12 = 45,500 Ibs. Tension in F 3 and Y 6 - compression in Y c and Y . F 4 = 28625 . 1.12 = 32,060 Ibs. Tension in F 4 and 7 7 ; compression in F d and F d . F 5 = 17625 . 1.12 = 19,740 Ibs. Tension in F 5 and F 6 ; compression in F e and F e . Counter Strains. F m = F m cosec. v. F 6 = 7625 . 1.12 = 8,540 Ibs. Compression in F 5 and F 6 ; tension in F fl and F. 133 STRAINS IN TUUSSES \VIT11 PARALLEL BJOMS. 8TEAINS IN TRUSSES WITH PARALLEL BOOMS. 139 LATTICE TRUSS. (Figs. 222, 223, and 224.) Lower Boom Loaded. Additional Reference. r= Half the length of a bay of simple truss. (Figs. 222 and 223.) x n = Distance from abutment A to center of bay of lower boom. 2/ n Distance from abutment A to apex of bay of upper boom. z n = Distance from abutment A to apex of bay of lower boom, The formulas are for the strains in the simple trusses, (Figs. 222 and 223.) Fig. 224 shows the simple trusses combined, con stituting the Lattice Truss. . When the upper boom is loaded, treat the strains as acting up ward and the truss inverted: the strains will be of the same amount in each member, but different in kind. Static or Permanent Load, equally distributed over whole length of Truss. Strains in Booms. Upper. H W ( 4- T \ - W (-I- T V-U Wr2 u ~ "2A r n+ TV ~~ 2/iI V B ~*" ~2~) + W Lower. _ W f r \ W f r \2 STFr 2 n ~~^" V^ u 2 / 2/iI \^ m 2 / ~ 8/ti Strains in Verticals. Fn = "4 2T * n Strains in Diagonals. Moving and Static Load, each equally distributed per unit of length. Strains in Booms. Upper. n _ W +W(.i r \ W+Wi ( , r \2 (TF+TFi)r 2 Lower. // _TF-f-IF L / __L\___^ I ^i / _ r V- 3(ir-h)rTF* 140 STRAINS IN TRUSSES WITH PARALLEL BOOMS, Strains in Verticals. W W Wi Strains in Diagonals. Y n = Vr, cosec v. Strains in Counters. W W W l = Fm cosec< v - [NOTE. The strains in Fa.b.c, .... are equal in amount, but different in kind to the strains in H,2, 3, .... EXAMPLE. (Figs. 222, 223, and 224.) Moving Load (as railway train passing over bridge) on Lower Boom. We will assume W = 50,000 Ibs. Wi= 100,000 Ibs. I = 100 feet. h = 10 feet. v = 63 20 , (cosec. = 1. 12.) r = 5 feet.. Horizontal Straws in Upper Boom. (Compression. Fig. 224.) = 7500(z n + 2.5) -75(2 n + 2.5) 2 + 468.75 H Q = 7500 . ( + 2.5) 75 . ( + 2.5) 2 + 468.75 = 18,750 Ibs. l= 7500 . ( 5 + 2.5) 75 . ( 5 + 2.5) 2 + 468.75 = 52,500 Ibs. //= 7500 . (10+ 2.6) 75 . (10 + 2.5) 2 + 468.75 = 82,500 Ibs. /= 7500 . (15 + 2.5) 75 (15 + 2.5) 2 + 468.75 = 108,750 Ibs. JI 4 = 7500 . (20 + 2.5) 75 . (20 + 2.5) 2 + 468.75 = 131,250 Ibs. //.= 7500 . (25 + 2.5) 75 . (25 + 2.5) 2 + 468.75 = 150,000 Ibs. l/ ( .= 7500 . (30 + 2.5) 75 . (30 + 2.5) 2 + 468.75 = 165,000 Ibs. H 7 = 7500 . (35 + 2.5) 75 . (35 + 2.5) 2 + 468.75 = 176,250 Ibs. /7 8 ==7500 . (40+2.5) 75 . (40+ 2.5) 2 + 468.75^ 183,750 Ibs. Hf= 7500 . (45 + 2.5) 75 . (45 + 2.5) 2 + 458.75 == 187,500 Ibs. STRAINS IN TRUSSES WITH PARALLEL BOOMS. 141 Horizontal Strains in Lower Boom. (Tension. Fig. 224.) _ -n a = - 2h V n I 1 1 2hl V yu " 3(17+ i 7\t V 75QO (y , 2.5) 75 . (y n 2.1 Shi fli = 7500 .( 5_25) 75. ( 5 2.5)21406.25 = H 2 = 7500 , ,(10 2.5) 75. (10 2.5)21406.25 = #, = 7500 H\ = 7500 . (15 2.5) 75. .(20 2.5) 75. (15 2.5) 2 1406.25 = (202.5)21406.25 = H 5 = 7500 , .(25 2.5) 75. (252.5)21406.25 = H 6 = 7500 , , (30 2.5) 75. (302.5)21406.25 = HI = 7500 .(35 2.5) 75. (352.5)21406.25 = HS = 7500 .(40 2.5) 75. (402.5)21406.25 = T 9 = 7500 # 10 = 7500 .(45 2.5) 75. .(50 2.5) 75. (45_ 2.5)2 1406.25 = (502.5)21406.25 = 2.5) 2 1406.25 16,875 Ibs. 50,625 Ibs. 80,625 Ibs. 106,875 Ibs. 129,375 Ibs. 148.125 Ibs. 163,1 25 Ibs. 174,375 Ibs. 181,875 Ibs. 185,625 Ibs. SIMPLE TRUSS. (Fig. 222.) Strains in Verticals. (F n .) "T IT - ( l ~ **) 2 = 125 - 2.5 .(Z-* n )2 II II II II II tTuV^V 12500 12500 12500 12500 12500 250 , 250 250 , 250 . 250 . + .10 + , 20 + 30 + 40 + 2.5 . 2.5 . 2.5 . 2.5 . 2.8 . 100 2 90 2 80 2 70 2 60 2 = 37,250 Ibs. 30,250 Ibs. 22,500 Ibs. 17,250 Ibs. 11, 500 Ibs. Com. in U. Counter Strains. (F m .) F 6 = 12500 250 . 50 + 2.5 . 50 2 = 6,250 Ibs. F 7 = 12500 250 . 60 + 2.5 . 40 2 = 1,500 Ibs. Y 1 Strains in Diagonals. Y n= V n C0sec - Tension in Y 1 and F 10 ; 37250 . 1.12 = 41,720 Ibs. compression in F a and Y Y 2 = 30250 . 1.12 = 33,880 Ibs. compression in Y* and Y^. Tension in Y 2 and F 9 142 STRAINS IN TRUSSES WITH PARALLEL BOOMS. compression in Y e and Y . F 4 = 17250 . 1.12 = 19,320 Ibs. Tension in F 4 and F 7 ; compression in F d and F d . F 5 = 11500 . 1.12 = 12,880 Ibs. Tension in Y 5 and F 6 compression in F e and F e . Counter Strains. Y m= V m cosec. v. F 6 = 6250 . 1.12= 7,000 Ibs. Compression in F 5 and F 6 ; tension in F e and Y e . = 1,680 Ibs. Compression in F 4 and F 7 ; SIMPLE TRUSS. (Fig. 223.) Strains in Verticals. ( F n .) l\ 12500 250 . 5 + 2.5 . 95 2 = 33812 .5. Vf= 12500 250 . 15 + 2.5 . 85 2 = 26812.5. T 7 12500 250 . 25 + 2,5 . 75 2 = 20312.5. F 4 = 12500 250 . 35 + 2.5 . 65 2 == 14312.5. F 5 = 12500 250 . 45 + 2.5 . 55 2 = 8812.5. Counter Strains. (V m .) T 7 G = 12500 250 . 55 + 2.5 . 45 2 = 3812. Strains in Diagonals. F n = V n cosec. v. F 1= 33812.5 . 1.12 = 37,870 Ibs. Compression in Y } and F IO: tension in F a and F a . Y,= 26812.5 . 1.12 = 30 ; 030 Ibs. Compression in F, and F {) ; tension in F b and F b . Y, 20312 5 . 1.12 = 22,750 Ibs. Compression in F, and F s ; tension in F c and F c . F 4 r= 14312.5 . 1.12 = 16,030 Ibs. Compression in F 4 and F 7 : tension in F d and F d . F 5 = 8812 5 . 1.12 = 9,870 Ibs. Compression in F 5 and F fi ; tension in F e and F e . Counter Strains. Y m = V m cosec - v - Y^-= 3812.5 . 1.12 = 4,270 Ibs. Tension in F 5 and F 6 ; com pression in F e and F e . STRAINS IN TRUSSES WITH PARALLEL BOOMS. 143 Fly. 225. Lower boom loaded. t 144 STRAINS IN TRUSSES WITH PARALLEL BOOMS. WHIPPLE TRUSS. (Figs. 225, 226, 227, and 228.) Additional Reference. # n , y n = Distance from abutment A to end of bay. Static or Permanent Load, equally distributed over whole length oj Truss. Strains in Booms. W W sW sW a = "2JT yB ~ W y + ~2hT **- ~4T Strains in Verticals. T7- W W Kn ~~l 2T Strains in Diagonals. Y n = V n cosec. v. Moving and Static Load, each equally distributed per unit of length. Strains in Booms. 2hl~ * 2hl s(W+W 1 ) Strains in Verticals. TF TF Strains in Diagonals. F n = F n cosec. v. Strains in Counters. STRAINS IN TRUSSES WITH PARALLEL BOOMS. 145 EXAMPLE. (Figs. 225, 226, 227, and 228.) (With 20 Bays.) Moving Load, (as railway train passing over bridge.) Let W = 50,000 Ibs. W l = 100,000 Ibs. I = 100 feet. h = 10 feet, s = 5 feet. v 45. (End diagonals v = 26 30 .) Horizontal Strains in Booms. (Compression in upper, tension in lower.) 27. S (W + "l) h-crnn .. nr o OfTK a, 1 i ft T 11 = 7500 . 75. O 2 375 yn ~ o + y\i \ 18750 = - JLo 18 7C ,7. RY= 7500 . 5 75, . 5 2 375 . 5 + 18750 = 52 , r > JL= 7500 . 10 75, . 10 2 375 , , 10 + 18750 = 82 ,51 | 3 = 7500 . 15 75 . 15 2 375 . 15 + 18750 = 108 ,7. 7500 . 20 75 . 20 2 375 . 20 + 18750 = 131 ,& // 5 = 7500 . 35 75 . 25 2 375 , , 25 + 18750 = 150 ,CM H { ~ 7500 . 30 75 , . 30 2 375 , , 30 + 18750 = 165 , !)( H 1 = 7500 . 35 75 . 35 2 375 , , 35 + 18750 = 176 ), HX 7500 . 40 75 , , 40 2 375 , , 40 + 18750 = 183 i < J/ 9 = 7500 . 45 75 . 45 2 375 , . 45 + 18750 = 187 ft _ / jfefe^A-ii 8* r" Strains in Verticals. ~ = 75,000 Ibs. Fi= F 2 = F 3 = V= F 5 = F 6 = F 7 = Strains in Figs. 2: 12500 250 . + 2.5 . 100 2 = 37,500 Ibs. C 12500 250. 5+25. 95 2 = 338121bs 12500250.10+2.5. 90 2 = 30,250 Ibs. 12500 250.15+2.5. 85 2 =r 26,812 Ibs. 12500 250 . 20 + 2.5 . 80 2 = 23,500 Ibs. 19^nn o^n of\ i o o ^7 ~9 OA 01 o i u 5 223 227 22 . C. T. T zouu zou . zo + A.A . 7o 2 = 20,312 IDS. 1 9 : >r>n 9^0 QH 1 9 Pi *7A2 1 n Of^A 11 ( izouu z,o(j . 6(J + Z.o . 70 2 = 17,250 Ibs. 10 146 STRAINS IN TRUSSES WITH PARALLEL BOOMS. Strains in Figs. 225 225 227 228 V s = 12500 250 . S5 + 2.5 . 65 2 = 14,312 Ibs. C. C. T. T. V g = 12500 250 . 40+ 2.5 . 60 2 = 11,500 Ibs. " F 10 = 12500 250 . 45+ 2.5 . 55 2 = 8, 812 Ibs. " " <l " F n = 12500 250 F r ,= 12500 250 , F w = 12500 250 . V m Acting on Counters. , 50+2.5 . 50 2 = 6,250 Ibs. 55+2.5 . 45 2 = 3, 812 Ibs. 60+25 . 40 2 = 1,500 Ibs. Strains in Diagonals. J n K n cosec. v. Strains in Figs. 225 Y l = 37500 . 1.117 = 41,887 Ibs. Ten. F 2 = 33812 . 1.414 = 47,810 Ibs. F 8 = 30250 . 1.414 = 42,773 Ibs. F 4 = 26812 . 1.414 = 37,913 Ibs. F 5 = 23500 . 1.414 = 33,229 Ibs. F 6 = 20312 . 1.414 = 28,722 Ibs. 7, = 17250 . 1.414 = 24,391 Ibs. F 8 = 14312 . 1.414 = 20,238 Ibs. F 9 = 11500 . 1.414 = 16,261 Ibs. * 10 = 8812 . 1.414 = 12,461 Ibs. Strains in Counters. F u = 6250 . 1.414 = 8,837 Ibs. F 19 = 3812 . 1.414 = 5,391 Ibs. F,;= 1500 . 1.414 = 2,121 Ibs. 223 Ten. 227 Com. 223 Com. [NOTE. If counter braces are not inserted, Vn, F"i2,and T r i3,andl r 8> Yg, and Y\Q will have an additional strain, opposite in kind and equal to V\ i, V\ 2, and V\ 3, and Y\\, ^12? and Y\ 3 ; but if counters are used, the strain V\\4 F"i2and Vis will n t occur in the structure, but will be necessary to determine the strain in FH, Fi 2 ,and F\ 3 only. FH, FI 2 , and FI 3 will then be inclined in the same direction as the diago nals from abutment A to center of truss, the character of strain being the same. (See also "Howe Truss") Keep in mind that each half truss, as to the character and amount of strain in the respective members, is alike.] STRAINS IN PARABOLIC CURVED TRUSSES. 147 STRAINS IN PARABOLIC CURVED TRUSSES " BOW STRING GIRDERS." (Figs. 229, 230, 231, 232, 233, and 234.) The strains in the lower boom (when horizontal) are the greatest, and equal in every bay, when the load is equally distributed over the whole length. The strains in the arch or upper boom are also greatest when the load is equally distributed over the whole length; the strains gradually increasing from the middle to the supports. The strains in the diagonals, whether single or double, in a bay are, when the load is equally distributed, everywhere null. When the load is unequally distributed, and one diagonal to each bay is used, they will be either in compression or tension. The character of the maximum of strains will be as follows: Assume the left half of truss to be loaded. All diagonals inclined up from left to right abutment are in tension; if inclined down, in com pression. The character of strains will be vice versa when the right half only is loaded. The strains in verticals are either compression, tension, or null. The maximum of compressive strain occurs when the diagonals in connection are under the greatest strain; that is, under an unequally distributed load. For other explanation, see diagram under variously-disposed loads. In the following formulas and examples the diagonals (for a moving load) resist a tensional strain only, and the verticals a compressive. This would not be the case if one diagonal to each bay were used. In the latter case the diagonals and verticals would have to resist an alternate compressive and tensional strain. When the trusses are inverted, the strains are different in kind, but not in amount. Reference. A, B = Reaction of support. (7= Compression in arch or upper boom. T Tension in lower boom. D and H = Rise of arch. F and / = Vertical forces. W= Weight of moving and static load per unit of span or length. V= Strain in verticals. N = Total number of bays. a = Length of a bay. c = Length of a diagonal. d and h = Ordi nates to parabola. I = Distance between supports or span. k = Total number of verticals = N 1. m = Number of bays between support and F n . 148 STRAINS IN PAEABOLIC CURVED TRUSSES. n = Number of a member, counting from support to middle of truss. t = Tension in diagonal. v and z = Angle between horizontal and member of polygon. w = Weight of static load per unit of span or length. w/= Weight of moving load, equally distributed per unit of span or length. u, x, y = Abscissas. In the following diagrams, one-half of truss only is shown, the strains being alike in the respective members of each half: Fig. 229. Lower Boom Horizontal. To find the ordinates h when H is given : The value of T given, to find h: Fig. 230. Lower Boom Curved. To find the ordinates h or d when H or D is given: 12 d = ~ % STRAINS IN PAEABOLIC CURVED TRUSSES. 14ST The value of T given, to find h: W(la)x n w ~m~ Load equally distributed Static Load. (Figs. 231 and 232.) W = The weight of construction and applied load. Fig, 231. IF/ 2 Wl * Lower Boom Loaded. , Wl wl H = (7 V= - = tension Loaded. F=null. Fig. 232. Upper Boom Loaded. (C=T.) Wl 2 Wl 2 V = = tension. 150 STRAINS ITS PARABOLIC CURVED TRUSSES. Load unequally distributed Moving Load. (Figs. 233 and 234.) (Strains in Booms, same as for Static Load.) Fig. 233. w/l - Lower Boom Loaded. Fn= ^n A= compression. Boom Loaded. TFZ T7 V n = -- = compression. . 234. n = r = compression D) STRAINS Iff PARABOLIC CURVED TRUSSES. 151 EXAMPLE. (Fig. 233.) Moving Load on Lower Boom. Reference. = 64 feet. c 1= 8.7 i eet. w = 125 Ibs. H= 8 feet, c 2 = c 3 = 10.0 feet. 10,= 625 Ibs. a = 8 feet, c 4 = c-= 10.9 feet. W=w + w,= 750 Ibs. 2V = 8, & 7. c 6 = 11.3 feet. 4 v 8 v 3(64 8^ ^i 8.0 8 = Ofeet. A 1= = X X --4^~ -== 3.5 feet, M 2 = 19.2 16= 3.2 feet. W3= 40.0 24 = 16.0 feet, ^= 128.0 -32 = 96.0 feet, 4X8X16(64-16) 2== " 2 - = . A 4 = IT =8.0 feet. Tang. t 1= ^ = = 23 Tang. v,= h ~^ = ^-^ = 3 34 30". y 1= 3,5 x 2.28 = 8.0 feet. y s = 7.5 X 5.37 = 40.0 feet. 2/2= 6.0 X 3.20 = 19.2 feet. y 4 = 8.0 X 16.00 = 128.0 feet. . = 48,000 Ibs. (7 n = C sec. v n . Q= 48000 Xl090=52,3201bs. C 3 = 48000 x 1.017 =48,816 Ibs. (72=48000x1-047=50,256^8. C 4 = 48000 X 1.0019 = 48,091 Ibs. *=*%$%-* 8.7 = 5437.5 lb, ,,- tf= t s = ^-^- X 10.0 = 6250.0 Ibs. 152 STRAINS IN PARABOLIC CURVED TRUSSES. 625 x 64 **= ** = o^o X 10.9 == 6802.5 Ibs. X o 625 x 64 ^ --- X 11-3 = 7062.5 Ibs. = 2625 2x 8 I = 1875 = 1250 j= 8(125+ 625)-- = 750 2 = 8 (125 + 625) [-^yj-] = 2250 ,= 8(125 -J 625) [J^t^L] = 45 00 4 = 8 (125 + 625) I" JL-t^^_"j = 7500 PAKABOLIC ARCHED BEAMS OR BIBS. 153 / 4 = 750 ( -- )= 562.5 V 96 -f 4 X 8 / F 1= 6000 =6,000 Ibs. F 3 = 9000 500 =8,500 Ibs. F 2 = 7812.5 312.5= 7,500 Ibs. F 4 = 9375 562.5 = 8,812.5 Ibs. CAPACITY AND STRENGTH OF PARABOLIC ARCHED BEAMS OR RIBS ORIGINALLY CURVED. Reference. (All dimensions in inches.) A = Sectional area of beam. C = Compressive strain in direction of arch. E = Modulus of elasticity. H = Horizontal thrust at abutment, or tension on tie rod. /= Moment of inertia of cross-section of beam. R = Resistance of material to crushing, (to be divided by factor of safety.) W = Concentrated load at crown of arch. a = Vertical deflection at crown. b = Horizontal deflection at abutments. h = Rise of arch. 21 = Distance between abutments = span. s = Distance between neutral axis and farthest edge of section. w = Load per unit of length, equally distributed horizontally. x = Vertical distance from crown to point of arch, intersected by y, say at on diagram. y = Horizontal distance .from middle of arch to section where the amount of strain is desired. v = Angle between horizontal and tangent to curve. Horizontal Thrust, (resisted either by abutments or tie rod.) Fig. 235. (All dimensions to line of pressure.) 154 STKAINS IN A POLYGONAL FRAME. To determine the curve or line of pressure: x 7/ 2 y =|r -f 2x 2<//^T~ Tang, v at any point = - = ^ y i 2h Tang, v at abutment = ^ t . Load concentrated at crown or middle of arch: ,_25Z_ A fty 25%2 " V 64/i 5(5Z n ^ 32Z 3 / __ 25^ IF 81 PF/s ** "cTT A T 1600J 25Z X 1600J 64A( 16007 Load equally distributed: _ + _ / . 27* p ;^ l- STRAINS IN A POLYGONAL FRAME IN EQUILIBRIUM. Load equally distributed over members of Frame. Reference. H = Horizontal strain in units of weight at foot. F n = Vertical strain in units of weight at foot. (7 n = Compressive strain in units of weight in direction of member. TF n = Load in units of weight, equally distributed over a mem ber of the polygon. fl n = Angle between horizontal and member. PARABOLIC ARCHED BEAMS OR RIBS. 155 Fig. 236. H J TFcotg. v n (7 n = F n cosec. v a 2 2 IT.+ , 3 _ , . . 1 s ^ + 2 z 2 2 _TFi , " 2 W. 2 For the equilibrium, v l being given : Tang. v 4 = -^r = tang. ^ .a 2 + TF 3 ) + H The above can be used to compute the strains in ribs for dome construction. 156 STRAINS IN ROOF TRUSSES. STRAINS IN ROOF TRUSSES. Reference. (Figs. 237 to 255.) C Weight of construction. } TF = < Pressure of wind. > Load in units of ( Pressure of snow. J weight, equally distributed over one rafter. (See Fig. 238.) C= Compression of member in units of weight. T Tension of member in units of weight. L = Total span, or distance between abutments in units of length. d, h, I, and S = Dimensions in units of length. (See Figures.) v, y Angles. (See Figures.) The diagrams show only one-half of truss, (except Fig. 238.) the thick lines indicating compression, and the thin ones tension. (See "Reaction of Supports " for pressure on joints ; also " Compound Strains in Trussed Beams") Compression in Rafters. (Trusses Nos. 1, 3, and 4.) The compressive strain in the rafter gradually increases from ridge to abutments. Let x = Horizontal distance from abutment to point where the strain is desired, and I half the span = - . tg. v C for Truss No. 1 = IF sin. v (l ) + -HL Cfor Truss No. 3 = IF sin. v(l } + -^ 1/2 tg. (v -|- i\) C for Truss No. 4 = JFsin. v(l } + - V I n 2 ig.(v~ Vl ) In the following examples the maximum of C is given : Truss No. 1. Fig. 237. W cos. v C=W&m. 2 tg.t; W T= cotg.v STEAINS Itf ROOF TRUSSES. 157 EXAMPLE. Let W 8,000 Ibs. v = 26 30 . C= 8000 X 0.44619 + -^ S5T = 10 666 lbs Com 7 8000 When a; = ~ then will C = - 7^7777^ = 8,968 Ibs. Com. A 2. X v^.TCTrOiy j_ 80Q 2.00 = 8,000 Ibs. Tension. Truss No. 2. Fig. 238. W W T= sin. v cos. v = sin. 2v 2 4 EXAMPLE. Let TT= 8,000 Ibs. v = 26 30 . 8000 C= - X 0.4462 = 1,785 Ibs. Compression. a C 1 = 8000 X 0.895 2 = 6,568 Ibs. Compression. 8000 T= -^ X 0.7986 = 1,597 Ibs. Tension. [NOTE. When the rafters are fastened together at the ridge, they are under a cross-breaking strain only. Consequently there is no horizontal thrust at the abutments ; that is, T=0, and the compression in <?i == W.] 158 Fig. 239. STRAINS IN ROOF TRUSSES. Truss No. 3. n TT7 . W cos. C = IF sin. v -\ 2 t.^- W cos. 2 sin. (v + Vj) fy. 240. Truss No. 4. 0= Ws m.v W cos. v 2 cos. v 2 sin.(v Vj) T,= TF-^-~ ^~ sm.(v vj Let TF= 8,000 Ibs. v = 26 30 X . v 1= = 5 O x . C = 8000 X 0.44619 + EXAMPLE. = 12,653 Ibs. Coin. ?1 = 9,920 Ibs. Tension. 0.087 O.ooo Bl|7201bB> Tension . STRAINS IN HOOF TRUSSES. 159 Truss No. 5. Fig. 241. = ifTFcosec. <y Ci= J TP cotg. v T= jl + -^-- TPcotg. v When there is no tie T, (7 3 is under a tensile strain == n~, 4/& 7i being the height from C l to ridge. EXAMPLE. Let TT= 8,000 Ibs. ? = 22.36 feet. / L== 11.18 feet. v = 26 3CK. C= -}-|8000 X 2.241 14,566 Ibs. Compression. Q % 8000 X 2. = 8,OQO Ibs. Compression. T= } (l + ) 8000 X 2. 9 12,000 Ibs. Tension. \ ^ii.oo / Truss No. 6. Fig. 242. 160 STEAINS IN ROOF TRUSSES. ,== 2(Wi\ W) -- EXAMPLE. Let TF= 8,000 Ibs. Z = 20 feet. / 1= = 20.6 feet. h = 10 feet. A!= 5 feet. == 22.36. 8000 x 500 1 500 (500 10 X 5) 5 X 2236 - -^^ = 29,264 Ibs. Com. Q= 0.625 x SOOOfg = 10,000 Ibs. Compression. 1500) JL = 26,780 Ibs. Tension. _ K Tj= 2(8000 1500) = 13,000 Ibs. Tension. Truss No. 7. Fig. 243. = W ~9J- - M sin. v (?!= if- IF cosec. v Let W= 8,000 Ibs. Z = 20 feet. sin. 2v EXAMPLE. h = 10 feet, v = 26 30 . Z x = 11.18 feet. vi== 26 30 7 . STRAINS IN EOOF TRUSSES. 161 n gQOO 9Q - 2 X 20 X 0.44619 = 8,964 Ibs. Compression. 08125 X 8000 X 2.2411 = 14,567 Ibs. Compression. 2== 0.625 X 8000 X 1-12 = 5,600 Ibs. Compression. = 0.625 X 8000 = 5,000 Ibs. Tension. (7 G = : T= . , . 2i= 0.8125 X 8000 X 2.0 = 13,000 Ibs. Truss No. 8. Fig. 244. c -= . = w 2sin.v 2^sin s m.(v = | T7 2 h sin - ^ i cos, vain. fa i "1 _ J ~ Let TF^8,000 Ibs. v = 26 30 / . t;!= 9 20 . up= 19 . 11 EXAMPLE. 162 STRAINS IN ROOF TRUSSES. 9000 + 0.375 x 8000 C= - - = 13,452 Ibs. Compression. 0.892 Ci 0.812 X 8000 ~r = 21,710 Ibs. Compression. u.zyo 0. 2 = 0.625 X 8000 A 8 ^ = 7,535 Ibs. Compression. T = 812 X 8000 -~- = 19,702 Ibs. Tension. (j.2t Jo Truss No. 9, 5i<7. 245. C= if fF_J: f IF sin. v sin. v 1 (;, j| FF--^ = -^fTFcosec. v r j\= |f 17 cotg. v T V IF cotg. v = -}- IF cotg. v T,= i IWcotg.v EXAMPLE. Let TF^ 8,000 Ibs. v = 26 30 7 . C = 0.812x 8000 x 2.241 0.625 X 8000 X 0.4 16 = 12,336 Ibs. Compression. Ci= 0.812 X 8000 X 2.241 = 14,56G Ibs. Compression. C 2 0.625 X 8000 X 0.895 4,475 Ibs. Compression. T 0.312 X 8000 X 2. = 4,992 Ibs. Tension. 2\= 0.812 X 8000 X 2 0.312X 8000X 2. = 8,0001bs. Tension. T 2 = 0.812 X 8000 X 2. = 12,992 Ibs. Tension. STRAINS IN ROOF TRUSSES. 163 Truss No. 1O. Fig. 246. ~~N I IF sin. v <7 2 =f JFcos.v. T _ COS. V COS. Vi 2\= if W^^--~ Tcos. (2v vi) -I Wain. cos. v W I EXAMPLE. Let TF= 8,000 Ibs. Vl = 9 20^ A = 10 feet v = 2630 / . 1 = 20 feet. A,= 2 feet. (7=0.8125 X 8000 -- 0.625 x 8000 X 0.446 = 19,517 Ibs. Compression. 987 (7i= 0.8125 X S 000 ^ 1 ^- == 21.747 Ibs. Compression. OF= 0.625 x 8000 x 0.895 = 4,475 Ibs. Compression o, 2 5 X 3000 x 0.895 2 ] = 7,163 Ibs. Tension. 164 STRAINS IN ROOF TRUSSES. Tl = _ X -j- = 10,000 Ibs. Tension. T= 0.8125 X 8000 = 19,720 Ibs. Tension. Truss No. 11. Fig. 247. (7=13 IF . C S - Vl -- - f TF sin. * cos. y T=^W (- 1 / - [-5 cos. v\ ~W I cos. v 2 h hi sin.^ Vi) EXAMPLE. Let W= 8,000 Ibs. y == 50. h = 10 feet. I = 20 feet. v = 2630 / . w 1 =920 / . Ai=2feet. S = 22.36 feet. (7= 0.8325X8000 ^ ^ 0.625 X 8000 X 0.446 = 19,517 Ibs. Compression. = 0.8125 X 8000 ~- = 21,747 Ibs. Compression, u.^yo = 0.366 x 8000 %~. = 4,070 Ibs. Compression. STRAINS IN EOOF TRUSSES. 165 2- = 0.125 X 8000 (6.5 . + 5 . 0.894) = 11,050 Ibs. Tension. 7i= 19486 X 0.986 7421 x 0.723 4930 X 0.446 = 10,000 Iba. Tension. S94 Tf= 0.812 X 8000 -^-^ = 19,486 Ibs. Tension. 0.29o Truss No. 12. Fig. 248. Let P7= 8000 Ibs. Z = 20 feet. EXAMPLE. A = 10 feet. /S = 22.36 feet. 20 C7 1= 0.366 X 8000 = 5,856 Ibs. Compression. 8000 22 sr (7 2 = 0.366 X -- 5 ~ = 3,280 Ibs. Compression. 10 = 0.866 X 10 20,992 Ibs. Tension. 3;= 1.23 X 8000 = 9,840 Ibs. Tension. 166 STRAINS IN EOOF TRUSSES. Truss No. 13. Fig. 249. ?i=*T- cos. v t C,^ f| W : -. - - 1 J 01 r> It, e 4 = ft x sin.(D L A. A. k T- Wk * W 2 ~T i? w EXAMPLE. Let TF= 20,000 Ibs. h = 20 feet, v = 21 40 X . 2 = 50 feet. 1 2 = 53.8 feet. v = 0. CO O C = 0.5 X 20000 - = 26,900 Ibs. Compression. (7 1= 0.683 X 20000 ~ = 37,018 Ibs. Compression. C 2 = 0.866 X 20000 * = 46,937 Ibs. Compression. (7 3 =0.55 x 20000 -^-i- = 11,770 Ibs. Compression. Q=0.55 x 20000 4^ =9,900 Ibs. Compression. STRAINS IN EOOF TRUSSES. 167 T= 0.683 X 20000 X 2.517 = 34,382 Ibs. Tension. T^= 0.866 X 20000 X 2.517 = 43,594 Ibs. Tension. 20000 X 20 = 14)666 lbg> Tensiont 20 T 3 = 0.183 X 20000 = 3,660 Ibs. Tension. Truss No. 14. Fig. 250. h h rp n ^2 J-b O 6 T EXAMPLE. Let TF= 24,000 Ibs. Span = 100 feet 1= Z 1= Z 2 = ? 3 = 1.25 feet. A = 20 feet. JJ^O. ^^53.85 feet. 168 STRAINS IN ROOF TRUSSES. 53 85 Q= 12000 x ~ = 32,310 Ibs. Compression. <7 2 = 49088 - 0.228 X 24000 ^L = 41,728 Ibs. Com. C 8 = 58320 0.286 x 24000 ^L = 49^88 Ibs. Com. ^ X 20 4 = 21600 -^0-- = 58,320 Ibs. Compression. C 5 = (5801 + 0.286 X 24000) ~^-- = 12,493 Ibs. Com. Q= 3432 + 5484 ~~- = 9,282 Ibs. Compression. 13 47 0,= 0.286 X 24000 ~^~ = 9,245 Ibs. Compression. T,= (24000 0.1 x 24000) -^- = 54,000 Ibs. Tension. 7!,= 5 1000 0.286 x 21000 -~- = 45,420 Ibs. Tension. 7!,= 45120 9282 --f\ 3 8> 170 Ibs. Tension. In 7^=24000 -121000= 19,200 Ibs. Tension. r 5 = 9282 J- = 5,801 Ibs. Tension, lo T G = 0.286 x 24000 ~ = 3,432 Ibs. Tension. Truss No. 15. Fig. 251. y i F STRAINS IN ROOF TRUSSES. 169 C= if W --- L^_^ -- | TFsin. v JJ TFcos. v cotg. (v vj __ sin. (v ^) 2 (A --.- tang. yi T 5 = \$W . , (h -A x ) sm.( y -yj) EXAMPLE. Let W= 20,000 lbs. h = 20 feet. v 1= = 0. I = 50 feet, v = 21 40 . -v 2 = 46 30 . tf = 0.866 X 20000 * 0.733 X 20000 X 0.369 0.183 X 0.369 20000 X 0.929 x 2.517 = 32,959 lbs. Compression. C 1= = 0.866 X 20000 X - A ~T^ 0.366 X 20000 x 0.369 0.369 = 44,236 lbs. Compression. C. 2 = 0.866 X 20000 X 7^7- = 46,937 lbs. Compression. Q= 0.55 x 20000 X 0.929 = 10,219 Ibs. Compression. C 4 = 0.366 X 20000 X 0.929 = 6,800 Ibs. Compression. 40 T= 20000 X -T^- X tang, v = Null. = 1 0,9201bs. Ten SIO n. Tf= 0.183 X 20000 X 2.5 = 9,150 Ibs. Tension. TZ= 10000 X -2Q^ = 25 -00 lbs - Tension. T,= 0.683 X 20000 X 2.5 = 34,150 lbs. Tension. 2^= 0.866 X 20000 X 2.5 = 43,300 lbs. Tension. 170 STRAINS IN ROOF TRUSSES. Truss No. 16. Fig. 252. C 4 =f JFcos.v. Q= j|> JFcos. v + f- IF cos. v = JfTFcos. v - sm.(2v W_ __l_ ~T h F 9 TJ7" ^4= TO W ~Z sin.^ Vi) cos. v sin. (v EXAMPLE. Let TT= 20,000 Ibs. A = 20 feet. AI= 0. I = 50 feet. T; = 21 40 7 . vi.= 0. 0= 41885 0.286 X 20000 x 0.369 = 39,774 Ibs. Compression. Cj= 43567 0.228 X 20000 X 0.369 = 41, 885 Ibs. Compression. STEAINS IN ROOF TRUSSES. 171 <7 2 = 48780 5213 = 43 ; 567 Ibs. Compression. (7 3 = 0.9 X 20000 = 48,780 Ibs. Compression. <?== 0.286 X 20000 X 0.929 = 5.213 Ibs. Compression. Q= 0.514 X 20000 X 0.929 = 9,550 Ibs. Compression. T== ( 0.9 X 20000 -~ - 0.8 X 20000 X 369* 0.1 X \ 0.369 20000) * = 20,000 Ibs. Tension. / 0.686 2i= T T 5 = 20000 7188 = 12,812 Ibs. Tension. _ 2 20 99 T 3 = T T 5 = 0.757 X 20000 = 38,1 18 Ibs. Tension. U.oo 3 D99 T= 0.9 X 20000 - - = 45,306 Ibs. Tension. f= T fi = T T l= 7,188 Ibs. Tension. 6 = T^= 7,188 Ibs. Tension. Truss No. 17. Fig. 253. When the^ rafter is resting on joint A: n _ W _ TFcos. v cos.^ -y) -- - r - Go -% -- ; - - - 4 sin. v sm. v l W C? 1 =- r -T - r 4 sin.-v , TT cos. l= 0, cos. sm. ^ Bending moment at point B C! 2 sin. v, . J. 172 STRAINS IN ROOF TRUSSES. When rafter is fixed at joint A: _ C - W - r - 4 sin. v _ W cos. v cos. (i\ v) Go - % - : -- sm. v 1 T= JPFcotg. VI+TI T } = ^- cotg. v IF Bending moment at B = - . Truss No. 18. Fig. 254. TFcos. ri i T=Q Ti= C 3 cos. v 4- C 2 cos. STRAINS IN ROOF TRUSSES. 173 Truss No. 19. Fig. 255. C = \W cosec. v . v C 2 = Jf W cosec. v a= w cotg. v ^i= f TFcotg. v + JTF tang, T 2 = flF cotg. v EXAMPLE. Let W= 20,000 Ibs. v = 21 40 . t> 1= = 56 C 27,000 Ibs. C } = 36,900 Ibs. <7 2 = 46,800 Ibs. C 3 = 33,466 Ibs. Q= 6,867 Ibs. Compression. Compression. Compression. Compressian. Compression. (7 5 = 3,533 Ibs. (7 6 = 6,666 Ibs. T= 6,666 Ibs. 7\= 37,000 Ibs. r 2 = 41,831 Ibs. Compression. Compression. Tension. Tension. Tension. 174 STRAINS IN ROOF TRUSSES. M <S 2 II 8? >S 0> o 5.S bJD j: s w "p g c r 5 H | O r I CO O 1^ CD to CD CM 10 OS CD Ci Ci TO O Ci O O r- ( to 1^ ^ 1 1 1 1 cq (M II O O O II II II OC5BH ^ CM co ii ii r O^fH C j a I o co 1>- O rH IO Tin CM OO 04 10 O T 1 Ci r 1 ^ O to iO 1C 1^ r^ CM co ^ -< CM | CM CM 11 O II II II OC5^ CO CM CO II II II O<o^ a ^ D I -< v 1C O ^ 1 1 1 38 T 1 O CM CM 11 CO O Ci CM ^( T I T i Ci T i odd II II II O<o^ iO O O r-H O O CO O O CO CM CO it H II O^^H ^ I 1 o Ci O $8 CD tO r-l co CM re T ! 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S ci |e a il|S i? 25 s - d ^ci fe^S S^S) Ssa as STRAINS IN EOOF TRUSSES. 175 O 1O OC tO 1O CM 00 CM rH r-l rH CD CM CM tC CM CO r-H 01 CO rH CO CO O CM CO <M O rH CO r-H CD O rH rH CO 1C 1C rH CO CM rH 1C CO CM iC ^H 1^- CO r^ oo co co oq oo ro oo T i II II II II 0^5^^ r}H T-H O rH II IL" Ii ^5^^H ^ HH O rH CM rH II II I 1 Ii II ^Sjgjiiii CO rH O CO rH II II II II II o^cS^s^ rh CM O 1C (M ^H 01 01 1C OC CO CO TTI ^rr LO CO 1C xH Ol LQ 1.^ *5f CO CO rf O CD O rH 1C rH O 1C CD i CO rH CM CO o T i m o oo CO 1C rH 1C CO o cc GO i^ oq 1- <M 1> rH II 1! II II ^CJ^H^ CO CO II II II II ^^^^r CO O r-i (M CO II II II I! II OO^K ^i CO rH O LC rH II II II II II OCJO^iS7 CO O O 1C rH o co oq LC ic rH co CD Oq CO rH II II II 1! o^J 1 ^^ co UC UC JL^ 71 IO CO 01 CO CM co -i O co ii 11 il IL . -^j^e-i^ CO CO O O !> r-i O 5 Q 1 co co CM o oq TO O CM CO II II II II II >^Ji2li CO CO I- 1C CO CO rH rH 04 CO CO rH 1- rH oq 1C rH O 1C rH II II II II II ojg^liS rH CO OO CM O r-< F- "^ ^f iC co i- ^ CM rH O r-l CO OC T 1 O rH O O 1:- O CO 1C CO O CO O 1^ OO O O O O CO oo co r- o co X^ Oq CD 1C CM 1C CM l-C rH il i! ii ii t>cS^^ <n -- i o CM II II II II 55 O&f6>i CM O rH rH 01 II 1! 1 II II OO^^^T rH rH o rH rH II II II II II O C?0^ s iC iC O CO - i - Ol r-i CO 1 CO C r-H rH r- 1 II II II II 06^ ic i- uc r- 1^- CO CM CO 1C Ot> CO rJH 01 o o oi II 11 II Ii <oQj-i E>T 1C rH 1 O 1C 1- CO CO O rH <C iC CO 1C rH 01 o o T-H oq II II II II II Q^Mj^E? O O O iC CO i- O C^ J>- CO rH rH 1C X) Oq rH rH O CO rH II II II II II o^cS ^^T "-0 CO O 1C CO CO 00 0] CO 1C i i CO ^ -^f to r- 1 CO rH Ol CO r- OC COO rri rH O T i CO CO GO 1C CO r- LC i oq o CO 1C rH O CO 1C rH CO 1C CO ic o rH oq oq rH 1 rH T 1 II II Ii II ^CJ^^H CM O O CM i! II II II C C^^ CM o o r-i oq II II II II II ercSNesw CO O CO rH II II II II II ooo^^ 4 ?6 o ft cl o 01 oq co CO r- 1 OO T 1 il II II II C0~^^ O O iC iC 01 o CM o~i 00 L- CO CO T O T-H II II II II ^^?^^ O CO iC O 1C CM iC CM O CM CO 1C CO O CO r-- O O rH rH II II II II II c5XS^^Ts CO 01 O O CO CO CO rH 01 CO CO 1>- rH CO Oq oq o o oq T-H II II II II II ^^cJ^^ . I CO 1C 1C CO Cl 01 0> Ci rH CO CM O CM r-4 II II II II oo^^r 01 o ic co CO CO CM ( rf o CO CM T-H O CD T-H 11 II, II il t5>v>5-i ^ Cl O CO O CO co oq CD ic I-H rH 1C rH t^- CM T O O rH II II II II \l CJ^NK w co c; co o co co rH oq o co oo ic co o CM 01 O O CM rH II II II II II OOCD^^T rt< ic ic ic co CM oq oq CM CO CO CO iC C * C 0^ rr rH 1 r- 1 i. ^ CO CO iC Oj Ol O CM rH rH rH O rH r rH CO iC OO CM CO CO 1C CO OO CO 1C i^ CO co co oq co CM <M O r- 1 r-i II !LJI 11 O ^5^ E>T 1111 cc ^K" r- O O II II II II II ^j^j ^^r^ T-H O O rH rH II II II II II ?iS25iS co eT 1- ill II $ "- 381 H& ^ 2^ 5 51 II fe ^ 1 S^S: > M Oi c3 S A Hi C5 5 l.lt m ^ ?n II* SStf fi-*3 II s^l 4, r ^ 0, -, V 176 STRAINS IN ROOF TRUSSES. J _V (.1.) U.J +JJ TiJ L H GO CO TtH CO 1^ r O CO T+i O O ^H O * r-H T^H O CO CO 3 OO rH O 1C Tt< ) CO CM CO CM r-H "fOOO^OOCOO O O O O -N~t n O O CO CO O O II II 11-11 ILILIL * Cxi O O O I I I CO O O CM 00 O O O 00 C.I CM O O H, li, IL 1 1 IL i IL IL 1 O 11 r-1 TTI CO -Hi CO 1-^ OO CO O -^ 00 CO rH 1^ CO CO O 1- r-1 co o o cq co o o COOOOCOCMCMOOO I iL i UUL UULJl O O i c_; co -^r oo ^ O CO O O CO CO r- o o c: co i^ T-H 00 CO CO rH O O CO O CO CO CM O O r-i 01 O O II \\\\ II CMOOOCMCMr-HOOO IL IL I ULJLUl UL CO "ti O i-^ O -m CO CO O5 Ci O CO CO -00 CO IO ^ !> i > i^ T > N Q Q rH o4 O O 01 TO *... r O O o tx t O T ( CO CO O CO CO O CO CO CO CM CO CO O CO CO O lO ^t 1 r^i jo co co CM co o co CM -H CMOOOCMr-Hr-HOOO CMOOOr-H -HT-HOOO lULUL UUL LUl * " " " CO O O rf CO TI 00 LO CO OO CM CO CO >0 lO -^ CO O CM t T-H O O i T-^ O O O CO O J^- O CO i O CO CO ci j>- o o o co CM o ic ^f CO rt" CC CM OO T iCOCOCMr-H 11111111 CM 1 r-1 ^ r CO CO ^T CO r-H O CO CO CC CO X) CM -^ OJ CO CO 1- r-1 OO O O CO O 1 < i O CO CO CO CM CO O O lO 00 O O rti CM T*H CM 04 OO I - CO CO OJ T i r-^oo oooo ooo ll, IL ILJLJL IL ILJLJL REFE FI iH * Si to a; c p ^, CO H ^ STRAINS IN ROOF TRUSSES. CD CM iO O <M tO iO iO 1>- 1 - J>- O^> LO iO O O CO CO lO CO Ol 1C OO O LO CO CO CO CM O Ci CO^fCDociOOiOO CO CO iO CO CM O O CO lit 1 till O GO (M i i-^fO-^HCC !> CM lO T 4 -rf lO O -rH T IIOCOI>-CD1^^O co o o o o T-H oi co ^H (M ^H Cvj LO O iO O Oq O O O O rH Cq 01 (MOOOOr-l iCI 1! II II 11 II II II 1! " " T ( O O O O i i i ( II II II II II II II II -r^w^ I 1 T-HCMlOOOrHCM^HOCDCC ^OO^ IT ((MCO^OO II 1! II II II II II II H II OOicoOOOOO^OC. iO I>- CO O 1^ O O O CO TO COOO^OT-ICVICMOO II II II II II II II II II II CO CD I>- O -^H IO >O IO IO i ." <M o o r-J o T-^ r-i oi o o - ILOCDOr I O O O CO CO C^OOOOr-HT-i^OO II II II II II II II II II II. qOOO I II II II OiCMcOOIr-OOOcO CDOCDOiOOOO^- d C^J CO TjH Ol lO t Ot) T I T r-iOOOOOOOOc IIILUJMI .11,11 " - 12 177 178 PRESSURE OF WIND ON ROOFS. EXAMPLE TO TABLE OF CONSTANTS. (Tniss No. 13.) What is the amount of strain in the various members of a truss, according to Fig. 249, of the following dimensions, viz: Span 60 feet, distance between trusses 10 feet, height at center 10 feet, weight to be carried, including weight of construction, 66| Ibs. per square foot horizontally; hence total weight on one rafter = 30 X 10 X 66} = 20,000 Ibs.? L = 60 feet. r 6Q v = 18 20 . h = 10 feet. = = 6. W = 20,000 Ibs. ^ o" Member. Constant. W Strains. C 2 = 2.745 x 20,000 = 54,900 Ibs. C, = 0.660 X 20,000 = 13,200 Ibs. J. Compression. C = 0.567 X 20,000 = 11,340 Ibs. } T = 1.956 x 20,000 = 39,120 Ibs. T. = 2.606 x 20,000 == 52,120 1 K = 0.734 x 20,000 = 14^680 Ibs. , T a 0.183 X 20,000 = 3,660 Ibs. J Tension. [NOTE. la the foregoing table the proportion of 7i to L is approximate. The constants are based on the angles.] PRESSURE OF WIND ON ROOFS. In the following table the maximum pressure of wind is taken at 50 Ibs. per square foot: The angle between horizontal and direction of wind is generally 10 00 r . (See diagram.) Fig. 256. Reference. F = Force of wind in Ibs. = 50. w, = Pressure at right angles to surface per square foot in Ibs. w // = Pressure, vertical, per square foot in Ibs. w / = F sin. 2 (v + 10) w, w^= COS. V PRESSURE OF WIND ON ROOFS. Proportion of height h to span I. Angle v. Pressure w, in Ibs. Pressure w tl in Ibs. 90 00 50.00 0.00 *4 45 00 33.53 47.40 &=4- 33 41 50" 23.80 28.60 z 26 33 50" 17.64 19.70 *=-g- 21 48 13.83 14.80 ;t =4 18 26 11.23 11.80 z "7 15 54 40" 9.46 9.80 A=-L 14 02 10" 8.56 8.80 * = -r 12 31 40" 7.29 7.40 i-i 11 18 40" 6.51 6.60 180 PEESSUEE OF SNOW ON EOOFS. PRESSURE OF SNOW ON ROOFS. The average pressure of snow on a level surface, in the United States, is about 15 Ibs. per square foot. The following table gives the pressure per square foot on various inclined surfaces : Reference. P = Pressure per square foot in Ibs. = 15. p 1 = Vertical pressure in Ibs. p z = Pressure at right angles to surface in Ibs. v = Angle between surface and horizontal. p 1 = P cos. v. Proportion of height h to span I. Angle v. Pressure PI in Ibs. Pressure P 2 in Ibs. I ~2~ 45 00 10.60 7.49 --r 33 41 50" 12.48 10.38 * T 26 33 50" 13.42 12.00 *=4- 21 48 13.93 12.94 *=4- 18 26 14.23 13,50 *=4 15 54 40" 14.41 13.86 *=i 14 02 10" 14.52 . 14.05 *=4- 12 31 40" 14.64 14.29 h== Jo 11 18 40" 14.71 14.43 I 00 00" 15.00 15.00 TIE RODS AND BARS. TIE BODS AND BARS. Capacity and Proportional Dimensions of Wrought-iron Tie Rods Tie Bars, and Pins or Bolts. Ultimate resistance to tearing = 60,000 Ibs. = 30 tons pei square inch. Ultimate resistance to shearing = 50,000 Ibs. = 25 tons pe: square inch. (See Fig. 258.) Capacity of tie or bar. t d inches, d. Dimension of flat bars in in., uniform thick ness. Diamete D of pii or bolt. 2 1 d C en 4 o> O SJO CO b 3 times safety. 5 times safety. "ol C g.rH O ** 2 Q.,-1 Jj! rO O p ci Lbs. Tons. Lbs. Tons. 1 CO 1 3 I- I" 8 OJ a cc PH c C X -^ 5,000 2.50 3,000 1.50 0.25 0.56 1 0.75 0.62 0.4 6.200 3.10 3.720 1.86 0.31 0.62 \\/ 0-93 0.69 0.4 7,400 3.70 4,440 2.22 0.37 0.70 << 11 A 1.12 0.75 0.5 8,000 4.30 5,160 2.58 0.43 0.74 " 1^4 1.31 0.80 0.5 10,000 5.00 6,000 3.00 0.50 0.79 " 2 1.50 0.88 0.0 11.200 5.00 6,720 3.36 0.56 0.84 " 2i/ 1.68 0.92 0.0 12,400 6.2D 7,440 3.72 0.62 0.89 2\s 1.87 0.97 o.o 13,000 6.80 8.160 3.88 0.68 0.93 " 2 ^4 2.06 1.01 0.7 15,000 7.50 9,000 4.50 0.75 0.97 " 3 2.25 1.08 0.7 7,400 3.70 4,440 2.22 0.37 0.68 % 1 0.75 0.75 0.6 9,200 4.60 5,520 2.76 0.40 0.76 \\s 0.93 0.83 O.o 11,200 5.00 6,720 3.36 0.56 0.84 " l 1 ^ 1.12 0.92 0.0 13,000 0.50 7,800 3.90 0.65 0.91 " ]^ 1.31 0.99 0.7 15,000 7.50 9,000 4.50 0.75 0.97 " 2 1.50 1.08 0.7 10,800 8.40 10,080 5.04 0.84 1.04 " 2/4 1.68 1.13 0.8 18,000 9.30 11,100 5.58 0.93 1.09 " 2/^ 1.87 1.19 0.8 20,000 10.30 12,300 6.18 1.03 1.15 " 2^4 2.06 1.24 0.8 22,400 11.20 13,440 6.72 1.12 1.19 " 3 2.25 1.29 0.9 10,000 5.00 6,000 3.00 0.50 0.79 1 A 1 0.75 0.88 0.0 12,400 6.20 7,440 3.72 0.02 0.88 tf 0.93 0.97 0.0 15,000 7.50 9,000 4.50 0.75 0.97 |i/ 1.12 1.08 0.7 17,400 8.70 10,440 5.02 0.87 1.05 M l^i 1.31 1.16 0.8 20,000 10.00 12,000 6.00 1.00 1.13 " 2 1.50 1.24 O.S 22,400 11.20 13,440 6.72 1.12 1.20 " 214 1.68 1.32 0.9 25,000 12.50 15,000 7.50 1.25 1.26 " 2M 1.87 1.39 0.9 27.400 13.70 16,440 8.22 1.37 1.32 " 2% 2.00 1.45 1.0, 30^000 15.00 18,000 9.00 1.50 1.39 " 3 225 1.52 1.0 12,400 6.20 7,440 3.72 0.62 0.90 % 1 0.75 0.98 0.0 15.600 7.80 9,300 4.68 0.78 1.00 " \\s 0.93 1.09 0.7 18,600 9.30 11,160 5.58 0.93 1.09 * \\ 1.12 1.20 0.8, 21,800 10.90 13,080 6.54 1.09 1.18 M ]%/ 1.31 1.29 0.91 25,000 12.50 15,000 7.50 1.25 1.26 M 2 1.50 1.39 0.9! 28,000 14.00 16,800 8.40 1.40 1.34 " 2/4 1.08 1.47 1.0 30,533 15.27 18,720 9.36 1.56 1.41 " 2V 1.87 1.54 l.OJ TIE RODS AND BARS. Capacity of tie or bar. g ft a c . rt C Dimension of flat bars in in., uniform thick ness. Diameter D of pin or bolt. 7. jn .5 o GO * bb 3 times safety. 5 times safety. ol C " fl V H * .: * c || If Lbs. Tons. Lbs. Tons. I s li H t* ^ c ol l| 1 34,200 17.10 20,520 10.26 1.71 1.48 ~% 2% 2.06 1.62 1.14 37,500 18.75 22,440 11.22 1.87 1.54 " 8 2.25 1.69 1.20 15,000 7.50 9,000 4.50 0.75 0.98 % 1 0.75 1.08 0.76 18,600 9.30 11,160 5.58 0.93 1.09 M 0.93 1.20 0.85 22,400 11.20 13,440 6.72 1.12 1.19 " \\/ 1.12 1.31 0.93 26,200 13.10 15,720 7.86 1.31 1.30 " 1& 1.31 1.41 1.00 30,000 15.00 18,000 9.00 1.50 1.39 " 2 1.50 1.52 1.08 33,600 16.80 20,160 10.08 1.68 1.46 " 1.68 1.62 1.14 37,400 18.70 22,440 11.22 1.87 1.54 " 2V" 1.87 1.69 1.20 41,200 20.60 24,720 12.36 2.06 1.62 " 2% 2.06 1.77 1.26 45,000 22.50 27,000 13.50 2.25 1.69 " 3 2.25 1.86 1.32 17,400 8.70 10,440 5.22 0.87 1.05 % 1 0.75 1.16 0.82 21,800 10.90 13,080 6.54 1.09 1.18 0.93 1.29 0.91 26,200 13.10 15,720 7.86 1.31 1.29 1|| 1.12 1.41 1.00 30,600 15.30 18,360 9.18 1.53 1.40 1.31 1.53 1.08 34,800 17.40 20,880 10.44 1.74 1.49 " 2 4 1.50 1.63 1.16 39,200 19.60 23,520 11.76 1.96 1.58 " 2*4 1.68 1.73 1.23 43,600 21.80 26,160 13.08 2.18 1.66 11 2/lz 1.87 1.82 1.29 48,000 24.00 28,800 14.40 2.40 1.75 " 2% 2.06 1.89 1.34 52,400 26.20 31.440 15.72 2.62 1.83 " 3 2.25 2.00 1.42 20,000 10.00 12,000 6.00 1.00 1.13 1 1 0.75 1.39 0.80 25,000 12.50 15,000 7.50 1.25 1.26 " i/4 0.93 1.45 0.98 30,000 15.00 18,000 9.00 1.50 1.39 " \\z 1.12 1.52 1.08 35,000 17.50 21,000 10.50 1.75 1.49 " \3S 1.31 1.64 1.16 40,000 20.00 24,000 12.00 2.00 1.60 " 2 1.50 1.75 1.24 45,000 22.50 27,000 13 50 2.25 1.70 " 2/4 1.68 1.86 1.32 50,000 25.00 30,000 15.00 2.50 1.79 M 2V^ 1.87 1.96 1.39 55,000 27.50 33,000 16.50 2.75 1.87 " 2% 2.06 2.05 1.45 60,000 30.00 36,000 18.00 3.00 1.96 * 3 2.25 2.15 1.52 28,000 14.00 16,800 8.40 1.40 1.34 iy & VA 0.93 1.47 1.04 33,600 16.80 20,160 10.08 1.68 1.47 " i/^ 1.12 1.60 1.13 39,600 19.80 23,520 11.76 1.98 1.58 " i?^ 1.31 1.73 1.23 45,000 22.50 27,000 13.50 2.25 1.69 " 2 1.50 1.86 1.32 50,600 25.30 30,360 15.18 2.53 1.80 " 2/4 1.68 1.97 1.39 56,200 28.10 33,720 16.86 2.81 1.89 " 2V^ 1.87 2.09 1.48 61,800 30.90 37,080 18.54 3.09 1.98 " 2% 2.06 2.18 1.54 67,400 33.70 40,440 20.22 3.37 2.08 " 3 2.25 2.26 1.60 73,000 36.50 43,800 21.90 3.65 2.16 * 3/4 2.43 2.36 1.67 78,600 39.30 47,160 23.58 3.93 2.24 " 3/^j 2.62 2.45 1.74 84,200 42.10 50,520 25.26 4.21 2.32 * 3^4 281 2.53 1.80 90,000 45.00 54,000 27.00 4.50 2.40 4 3.00 2.63 1.86 31,200 15.60 18,720 9.36 1.56 1.41 li,/ |i/ 0.93 1.54 1.09 37,400 18.70 22,440 11.22 1.87 1.55 w li^ 1.12 1.69 1.20 43,600 21.80 26,160 13.08 2.18 1.67 1% 1.31 1.82 1.29 50,000 25.00 30,000 15.00 2.50 1.79 2 1.50 1.96 1.39 56,200 28.10 33,720 16.86 2.81 1.89 (t 2/4 1.68 2.09 1.48 62,400 31.20 , 37,440 18.72 3J2 1.99 " 2 /^ 1.87 2.19 1.55 TIE HODS AND BARS. Capacity of tie or bar. cr CO (J 03 ^ o a - Dimension of flat bars in in., uniform thick ness. Diameter D of pin or bolt. 3 times safety. 5 times safety. S o 1! 00 tn . 0> ~ *>: *! t if O <>-. 1 g | "2 si iis O J3 Lbs. Tons. Lbs. Tons. CB Q ~ g 2 03 G co 0=3 _,*- ^ 68,600 3430 41,160 20.58 3.43 2.10 i l /4 2% 2.06 2.29 1.62 75,000 37.50 45,000 22.50 3.75 2.19 " 3 2.25 2.40 1.70 81,200 40.60 48,720 24.36 4.06 2.27 3/4 2.43 2.49 1.7G 87,400 43.70 52,440 26.22 4,37 2.36 " 31^ 2.62 2.60 1.84 93,600 46.80 56,160 28.08 4.68 2.44 " 3% 2.81 2.68 1.89 100,000 50.00 60,000 30.00 5.00 2.53 " 4 3.00 2.77 1.96 41,200 20.60 24,720 12.36 2.06 1.62 Jg ^A 1.12 1.77 1.26 48,000 24.00 28,800 14.40 2.40 1.75 " 1/4 1.31 1.89 1.34 55.000 27.50 33,000 16.50 2.75 1.87 " 2 1.50 2.05 1.45 61,800 30.90 37,080 18.54 3.09 1.98 2// 1.68 2.18 1.54 68,600 31.30 41,160 20.58 3.43 2.09 < 2/<2 1.87 2.29 1.62 75,600 37.80 45.360 22.68 3.78 2.19 -% 2.06 2.41 1.71 82,400 41.20 49,440 24.72 4.12 2.29 3 2.25 2.51 1.78 89,200 44.60 53,520 26.76 4.46 2.38 3/4 243 261 1.85 96,200 4810 57,720 28.86 4.81 2.47 3^2 2.62 2.71 1.92 103,000 51.50 61,800 30.90 5.15 2.56 3% 2.81 2.81 1.99 110,000 55.00 66,000 33.00 5.50 2.65 4 3.00 2.90 2.05 45,000 22.5 27,000 13.50 2.25 1.70 1 A \y 1.12 1.86 1.32 52,400 26.20 31.440 15.72 2.62 1.83 1^4 1.31 2.00 1.42 60,000 30.00 36,000 18.00 3.00 1.96 2 1.50 2.15 1.52 67,400 33.70 40,440 20.22 3.37 2.07 2/4 1.68 2.27 1.61 75,000 37.50 45,000 22.50 3.75 2.19 2Vo 1.87 2.40 1.70 82,400 41.20 49.440 24.72 4.12 2.29 2% 2.06 2.51 1.78 90,000 45.00 54,000 27.00 4.50 2.40 3 2.25 2.63 1.86 97,400 48.70 58.440 29.22 4.87 2.49 3/4 2.43 2.73 1.93 105,000 52.50 63,000 31.50 5.25 2.59 3/^2 2.62 2.84 2.01 113,400 56.20 67,440 33.72 5.62 2.67 3% 2.81 2.93 2.08 120,000 60.00 72,000 36.00 6.00 2.77 4 3.00 3.03 2.15 127,400 63.70 76.440 38.22 6.37 2.85 4/4 3.18 3.12 2.21 135 000 67.50 81,000 40.50 6.75 2.93 41^ 3.37 3.22 2.28 142,400 71.20 85,440 42.72 7.12 3.01 4/4 3.55 3.30 2.34 150,000 7500 90,000 45.00 7.50 3.10 5 3.75 3.39 2.40 181 JOINTS OR CONNECTIONS IN IRON CONSTRUCTION. JOINTS OR CONNECTIONS IN IRON CONSTRUCTION. PROPORTIONS OF BOLTS, NUTS, RIVETS, &c. Reft re nee. A = Sectional area of bolt, rivet, or pin. AI= Sectional area of all rivets in a joint. A% Sectional area of one plate. D = Diameter of bolt, rivet, or pin. S= Ultimate resistance to shearing of material. T = Ultimate resistance to tearing of material. TI= Tensional strain on joint, &c. a = Number of times that a bolt, &c., will have to be sheared- (See 2 on Fig. 258.) d = Distance between centres of rivets. k = Factor of safety. I = Overlap, approximately If d to If d. m = Number of rivets in a joint. n = Number of lines of rivets in a joint at right angles to strain. t = Thickness of a plate. RIVETS. Fig. 257. For tension in direction of rivet: -J- T 0.7854 For shearing at right angles : If at one place D= I - Tl *__ N S 0.7854 If at two places D = I _ ?L >J S 1.5 __ .5708 Approximately : I = 3t D = 3t JOINTS OR CONNECTIONS IN IRON CONSTRUCTION. PIN, &o., IN TIE BARS. Fig. 258. PLATE JOINTS. No. I. Plate Joint Overlapped, four lines of Rivets. Fig. 259. - *. ; d =-- D + -L (0.7854 JDn) o . d .# i- Approximately c? = 1.5i to 2t i^-ct- cJ--ci->i ft A A (t)-^- 4V ^0.7854 __ 2mtS 2. Ptafe Jbm^ Overlapped, single line of Rivet Fig. 260. (Same as No. 1.) 186 JOINTS OR CONNECTIONS IN IRON CONSTRUCTION. No. 3. Plate Joint Overlapped, two lines of Rivets. Fig. 261. (Same as No. 1.) o o No. 4. Fish Joints, two lines of Rivets. Fig. 262. One fish plate. (Same as No. 1.) Two fish plates. Thickness of each fish plate = J t. />_-!_ /__**. m ** 1.570. L.5708 (Otherwise same as No. 1.) DIMENSIONS OF BOLTS AND NUTS. 187 DIMENSIONS OF BOLTS AND NUTS. (Whitworth s proportions.) Figs. 263, 264, 265, 266, 267, 268, 269, 270, and 271. Inch. 3 21 2} 21 2 If H i I A I A Dimension of Nuts and Heads. . -^ Inch. p ? Inch. Inch. Inch. 4J 5.18 5 7.07 4J 4.76 4J 6.37 3| 4.33 4J 5.83 3| 3.89 3| 5.30 3 3.46 3| 4.76 2f 3.17 3 4.24 2f 3.03 21- 3.89 ^1 2.88 2| 3.71 2J 2.59 21 3.53 2 2.30 21 3.18 H 2.16 2 2.82 11 1.87 IF 2.64 1} 1.73 If 2.29 !& 1.51 H 2.12 1* 1.38 1A 1.86 1 1.15 1A 1.67 | 1.01 i 1.41 i 0.86 1 1.23 t 0.86 f 1.06 A 0.64 I 1.06 TV 0.50 A 0.79 f 0.43 A 0.79 Dia. of No. Threads Core. per inch* Inch. Inch, 3 2 .57 3. 5 1 .50 2f 2 .35 3 .5 1 .75 2J 2 .13 4, ,0 2 .00 21 1 .91 4, ,0 2 .12 2 1 .69 4, 5 2 .25 If 1 .58 4 .5 2 .37 If 1 .47 5, ,0 2 .50 If 1 .36 5. ;0 2 .75 If 1 .25 6 3 .00 If 1 .14 6, ,0 5 .25 It 1 .08 7, ,0 3 .50 H .92 7, ,0 4 .00 l .81 8. 5 .00 i .70 9. 6 .00 i .59 10 ,0 6 .00 i .48 11, ,0 7 .00 9 T6 .42 11. 7 .00 J .37 12. 8 .00 TV .31 14, ,0 8 .00 1 .26 16. 9 .00 * .20 18 .0 9 .00 i .15 20 .0 10 .00 188 STRAINS IN HORIZONTAL AND SLOPING BEAMS. Fig. 272. Approximate proportions of bolts, nuts, and beads in incbes: d = 1.4 D -\- 0.25 = Inscribed circle. h = D = Height of nut. /*!= 0.7 D = Height of head. COMPOUND STRAINS IN HORIZONTAL AND SLOPING BEAMS. (Load equally distributed or between supports.) Area of Cross-section necessary to resist a Cross-breaking and Compressive Strain in Beams acting as a Boom in Trusses, &c,, or Beams acting as Rafters, &c. Reference. rii = Bending moment (See Page 100.) C= Compressive strain. (See Roof and Simple Trusses.) g = A factor depending on form of cross-section. /= Moment of inertia of cross-section. 8 = Distance from neutral axis to most compressed fibres. A = Sectional area of beam, &c. h = Depth of beam, &c. p = Resistance to compression with safety per square inch of section. W= Total load. I = Length of beam, &c. _ STRAINS IN HORIZONTAL AND SLOPING BEAMS. 189 For horizontal beams, &c. : For sloping beams, &c., v = angle between horizontal and beam: W r 1 / 1 \\ I cos. v~\ A = -- LT (-sTnTV + sm - v ) + T2?T J RAFTER OF A ROOF TRUSS. Fig. 273. EXAMPLE. Reference. W = 2.5 tons. = 2.8 tons. I = 10 feet. v = 26 30 p =. 5 tons per square inch. We will assume a Phoenix Go s six-inch beam of the following dimensions: h = 6 inches; A = 4 inches; J= 22.5 showing that the six-inch beam has a greater sectional area than required. If the load is concentrated at the apex of roof, the compressive strain C= 2.8 tons, and the area necessary to resist this strain 2 8 would be (taking p at five tons per square inch) - 0.56 sq. D inches, provided this is able to resist buckling. By comparing this with the above result, it will be seen how much greater the sectional area will have to be to resist a cross - breaking strain, caused by the load being distributed. These remarks also apply to simple trusses. 190 STRAINS IN HORIZONTAL AND SLOPING BEAMS. SIMPLE TRUSS, (BEAM CONTINUOUS OVER STRUT.) Fig. 274. EXAMPLE. Reference. W = 20 tons. I = 20 feet, v = 15 p = 5 tons per sq. inch. We will assume a Phoenix Go s twelve-inch beam of the fol lowing dimensions: h = 12 inches. J= 275.92 A = 12.5 inches. s = 6 inches. 275.92 = 0.5 X 12 X 12.5 m=0.0703xlXl20 2 = 84.36 (See Reaction of Supports.) C= 23.32 tons. 84.36 \ , 46.26 0.306x12 ) +23.32=- =9.25 inches. Consequently the sectional area of the twelve-inch beam is amply sufficient. [NOTE. The formulas for horizontal beams are also applicable to rafters of roof trusses, m and C being given. For the bending moments (m) the various distances are the horizontal projections of those on the rafter from abutment to ridge. The loregoing formulas also apply to beams under a cross-creaking and tensional strain. If the truss (Fig. 274) is inverted, the horizontal member will be in tension. Hence, insert the resistance of the material to tension instead of compression, and put tensional for compressive strain; other wise, the formulas remain the same.] WEIGHT OF MOVING LOADS. 191 WEIGHT OF MOVING LOADS. Variable and Accidental Loads. (Weight of construction not included.) Character of structure. How loaded. Weight in Ibs. per square foot of surface. Street bridges for horse cars and heavy traffic. Crow d with per sons. Minimum 40 Ibs. 120 " 80 " Maximum Average Street bridges for general traffic, foot passengers, &c. Persons, animals, and wagons. Public travel.... Private travel... Heavy business wagons 80 Ibs. 40 " 80 " 40 " Light business wagons Floors, &c Crowded public places. Dwellings Minimum 40 Ibs. 120 " 80 " 40 " 80 " 100 " 200 " 250 " f200 \ to " (400 80 " Maximum Average Churches, court rooms, theatres, and ball-rooms. Storage of grain... General merchan dise Warehouses Factories. Hay- lofts 192 STATIC AND MOVING LOADS ON BRIDGES. STATIC AND MOVING LOADS ON BRIDGES OF WROUGHT IRON. The following table gives an approximate weight per lineal foot in pounds of the static load or weight of construction complete for Single-Line Railway Bridges, supported at the ends, from ten to four hundred feet span; also the weight of the moving load per lineal foot of span, based on the assumption that the heaviest locomotives exert a pressure of three thousand pounds per lineal foot between their extreme bearings. The table is applicable in computing the strains in all trusses with parallel booms mentioned in this work. Weight of Construction and Moving Load of Wrought- Iron Single- Line Railway Bridges for the heaviest traffic. (From 20 to 400 feet span.) Weight of construction complete, including cross-ties and rails. Weight of moving load equal to 3,000 Ibs. per lineal foot of load. Weight in d Weight in d Weight in jj Weight in Ibs. .2 Ibs. per .2 Ibs. per .2 Ibs. per .2 a lineal foot a lineal foot a lineal foot 23 per lin. 1 of span. a r Sl of span. c3 A V) of span. a CO span. 10 427 210 1,891 10 6,300 210 2,535 20 500 220 1,964 20 5,370 220 2,495 30 573 230 2,037 30 4,250 230 2,455 40 646 240 2,110 40 3,780 240 2,375 50 719 250 2,183 50 3,550 250 L ,335 60 792 260 2,256 60 3,400 260 2,290 70 865 270 2,329 70 3,300 270 2,245 80 938 280 2,402 80 3,250 280 2,200 90 1,011 290 2,475 90 3,180 290 2,160 100 1,084 300 2,548 100 3,120 300 2,120 110 1,157 310 2.621 110 3,050 310 2,080 120 1,230 320 2,694 120 3,000 320 2,045 130 1,303 330 2,767 130 2,930 330 2,010 140 1,380 340 2,840 140 2,880 340 1,975 150 1,453 350 2,913 150 2,820 350 1,940 160 1,526 360 2,986 160 2,760 360 1,910 170 1,599 370 3,059 170 2,700 370 1,880 180 1,672 380 3,132 180 2,655 380 1,850 190 1,745 390 3,205 190 2,615 390 1,820 200 1,818 400 3,278 200 2,575 400 1,890 STATIC AND MOVING LOADS ON BRIDGES. 193 The following gives the actual weight of some well-known Bridges (single line) in America, Germany, and England: Name of Bridge. System. a Weight of con struction per lineal foot. 6 <jj <4_T3~0 ^ .5 *" c O ^ CO a CO Lbs. Lbs. Lbs. "Brenz," near Konigsbronn... "Colomak" "Iser," near Mu nich | Open Web, ^ { parallel booms, f 63.0 111.0 164.7 760 1,090 1,770 3,131 3,067 3656 7,530 9,516 8,532 "Donau," near Ingolstadt "Elb,"nearMei- ssen < 178.0 179 1,954 1 324 3,312 2 783 8,532 10390 "Rhine," near Mainz {"Pauli s," par- ") abolic arched > 345 2 170 1 970 11 660 "Royal Albert," near Saltash... "Boyne" booms. J Lattice..**... 455.0 264.0 4,418 3225 2,240 9,954 " Leven " 36 566 "Kent" M 36 580 "Harper s Ferry" Truss 124.0 770 13 MISCELLANEOUS. (195) QEOMETRY. LONGIMETEY AND PLANIMETRY. (Lines and Areas.) Reference. A = Area. - = Periphery of circle = 3.14159 when diameter = 1. r = Radius of circle. G = Length of cord of segment. p = Circumference of circle for given diameter. I = Length of circle arc, &c. h = Height of segment. v = Angles, expressed in decimals, as 15 30 / = 15.5. For other designations, see Figuies. [NOTE. Always use the same unit for dimensions.] t Values cj TT. - = \14159 x 2~ = 6.28319 ~ = 1.04720 = 0.31831 TT * = 0.78540 = 0.15915 ;r --= 0.52360 1 __ = 0.10132 - 2 = 9.86960 -3 _ 31.00628 =0.63662 ^1= L77245 &-= 1.46459 (197) 198 LONGIMETRY. Fig. 275. p = Fig. 276. 360 1 " " 360 I 180 v = - 180 TTr 180 ^ 7T r. 277. = 2(180 Fig. 278. 8h " 2h . 279. LONGIMETRY. 199 Fig. 280. Fig. 281. Ellipse. Fig. 282. [7)2 1+ ^ - + 1 256 n "J When n = , ; a -\- b b = \/a z c* a = \/6 2 -f c 2 Fig. 283. . 284. c 2 a 2 Z> 2 ~26~ 200 PLANIMETRY. 85. (Circle plane.) 286. (Circle ring.) . 287. (Sector.) li = 0.008727 w 2 . / 360 . r== N/~T~~~ 288. (Segment.) A = ?; sin.v) (0.017453 v sin. ) - 2 Fig. 28$. (Circle ring sector) 360 V1 : 0.008727 ^(ry 2 r 2 2 ) PLANIMETRY. Fig. 290. (Ellipse.) A == nab Fig. 291. (Square.) Fig. 292. (Rectangle.) A = a 2 Fig. 293. (Parallelogram.) = a sn. v 294. (Triangle.) A = = be sin. v 2 2 c 2 sin. v sin. Vj 2 sin. v 2 When the three sides are given: Let a + b + c = s CENTER OP GRAVITY OF PLANES. CENTER OF GRAVITY OF PLANES. Reference. x = Distance from a fixed base to center of gravity. r = Radius. c = Chord. b,p, h = Dimensions. A = Area. v = Angle. Fig. 295. (Quadrangle.) Fig. 296. (Triangle.) Fig. 297. (Half circle, or elliptic plane.) a and b parallel. " h h ( b a \ X 2 6" Vfi+o J - = radius = r l x = 0.4244r Fig. 298. (Concentric ring.) 4 sin. Ji; r 3 r^ "" ~3 v r 2 r, 2 CENTER OF GRAVITY OF PLANES. 203 Fig. 299. (Circle, or elliptic arc.) re 2 sin. V JF%,300. (Half circumfer ence of circle or ellipse.) x = r = 0.6366r 7T Fig. 301. (Circle sector. 4 sin. -Jv * Fig. 302. (Circle segment.) . = Area. Fig. 303. (Parabola.) 204 CENTER OF GRAVITY OF PLANES. Fig. 305. (Half parabola.) Fig. 305. Of any section, composed of any number of simple figures: Additional Reference. A, -4,, -4//= Sectional areaof simple figures. X = Distance from center of gravity of whole sec tion to axis ran. x t x /t // = Distance from center of gravity of a simple figure to a fixed axis mn. Y ^ X ~^~ ^ /x/ ~^~ A//x/ + &c- TRIGONOMETRICAL FORMULAS. 205 TRIGONOMETRICAL FORMULAS. Reference. a, 6, c = Length of sides. A, B t C= Angles opposite to a, &, c respectively. Right Angle Triangle. Fig. 306. cos. C b = a cos. C b = c cot. (7 b = a sin. B b = c tang. B c = b tang. c = a sin. Tang. (7=4-== n b cos. "cot. (7 Cotang. C= Secant (7 = Cosec. C = cos. (7 sin. C 1 cos. (7 1 sin. ~ tang. C Cos.(7= - a Oblique Angle Triangle. Fig. 307. B) a = \/6 2 + c 2 26c cos. c sin. 5 Sin. (7= Sin. 4 = c sin. 6 a sin. C b c sin. A a TRIGONOMETRICAL FUNCTIONS. NATURAL SINE Minutes. Deg. 5 10 15 20 25 30 .00000 .00145 .00291 .00436 .00582 .00727 .00873 1 .01745 .01891 .02036 .02181 .02327 .02172 .02618 2 .03490 .03635 .03781 .03926 .04071 .04217 .04302 3 .05234 .05379 .05524 .05669 .05814 .05960 .06103 4 .00976 .07121 .07266 .07411 .07556 .07701 .07846 5 .08716 .08860 .09005 .09150 .09295 .09440 .09585 6 .10453 .10597 .10742 .10887 .11031 .11176 .11320 7 .12187 .12331 .12476 .12620 .12764 .12908 .13053 8 .13917 .14061 .14205 .14349 .14493 .14637 .14781 9 .15643 .15787 .15931 .16074 .16218 .16361 .16505 10 .17365 .17508 .17651 .17794 .17937 .18081 .18224 11 .19081 .19224 .19366 .19509 .19652 .19794 .19937 12 .20791 .20933 .21076 .21218 .21360 .21502 .21644 13 .22495 .22(537 .22778 .22 )20 .23062 .23203 .23345 14 .24192 .24333 .24474 .24615 .21756 .24897 .25038 15 .25882 .20022 .25163 .26303 .20443 .20584 .26724 16 .27564 .27704 .27843 .27983 .28123 .28202 .28402 17 59237 .29376 .29515 .29654 .29793 .29932 .30071 18 .30902 .31040 .31178 .31316 .31454 .31593 .31730 19 .32567 .32694 .32832 .32969 .33106 .33244 .33381 20 ,342i)2 .34339 .34475 .34612 .34748 .34884 .35021 21 .35837 .35973 .36108 .36244 .36379 .36515 .36650 22 .37461 .37595 .37730 .37865 .37999 .38134 .38268 23 .39073 .39207 .39341 .39474 .39608 .39741 .39875 24 .40674 .40806 .40939 .41072 .412 4 .41337 .41469 25 .42232 .42394 .42525 .42657 .42788 .42920 .43051 26 .43837 .43968 .44098 .44229 .44359 .44494 .44620 27 .45399 .45529 .45658 .45787 .45917 .46046 .46175 28 .46947 .47076 .47204 .47332 .47460 .47588 .47716 29 .48481 .48608 .48735 .48862 .48989 .49116 .49242 30 .50000 .50126 .50252 .50377 .50503 .50628 .50754 31 .51504 .51628 .51753 .51877 .52002 .52120 .52250 32 .52992 .53115 .53238 .53361 .53484 .53607 .53730 33 .54464 .54586 .54708 .54829 .54951 .55072 .55194 34 .55919 .56040 .56160 .56280 .56401 .56521 .56641 35 .57358 .57477 .57596 .57715 .57833 .57952 .58070 36 .58779 .58869 .59014 .59131 .59248 .59365 .59482 37 .00182 .60298 .60414 .60529 .60(545 .60761 .60876 38 .61566 .61681 .61795 .61909 .62024 .62138 .62251 39 .62932 .63045 .63158 .63271 .63383 .63496 .63608 40 .64279 .64390 .64501 .64612 .64723 .64834 .64945 41 .65606 .65716 .65825 .65935 .66044 .66153 .66202 42 .66913 .67221 .67129 .67237 .67344 .67452 . 67559 43 .68200 .68306 .68412 .68518 .68624 .68730 .68835 44 .69466 .69570 .69675 .69779 .69883 .69987 .70091 Deg. 60 55 50 45 40 35 30 Minutes. NATURAL COSINE. TRIGONOMETRICAL FUNCTIONS. NATURAL SINE. Minutes. Deg. 35 40 45 50 55 GO .01018 .01164 .01309 .01454 .01000 .01745 89 .02703 .02908 .03054 .03199 .03345 .03490 88 .04507 .04053 .04798 .04913 .05088 .05234 87 .06250 .00395 .06540 .06685 .06831 .00970 86 .07991 .08130 .08281 .08426 .08571 .08710 85 .09729 .09874 .10019 .10104 .10308 .10453 84 .11405 .11009 .11754 .11898 .12043 .12187 83 .13197 .13341 .13485 .13029 .13802 .13917 82 .14025 .15009 .15212 .15356 .15500 .15043 81 .10048 .10792 .16935 .17078 .17222 .17305 80 .18307 .18509 .18652 .18795 .18938 .19081 79 .20079 .20222 .20364 .20507 .20649 .20791 78 .21780 .21928 .22070 .22212 .22353 .22495 77 .23480 .23627 .23769 .23910 .24051 .24192 76 .25179 .25320 .2.5460 .25601 .25741 .25882 75 .23804 .27004 .27144 .27284 .27421 .27504 74 .28541 .28080 .28820 .28959 .29098 .29237 73 .30209 .30348 .30486 .30625 .30703 .30902 72 .31808 .32006 .32144 .32282 .32419 .32057 71 .33518 .33655 .33792 .33929 .34005 .34202 70 .35157 .35293 .35429 .35565 .35701 .35837 69 .30785 .36921 .37056 .37191 .37320 .37401 68 .38403 .38537 .38671 .38805 .38939 .39073 67 .40008 .40141 .40275 .40408 .40541 .40074 66 .41002 .41734 .41866 .41998 .42130 .422i2 65 .43182 .43313 .43445 .43575 .43700 .43837 64 .44750 .44880 .45010 .45140 .45209 .45399 03 .40304 .46433 .46561 .46690 .40819 .40947 62 .47844 .47971 .48099 .48226 .48354 .48481 61 .49309 .49495 .49622 .49748 .49874 .50000 60 .50879 .51004 .51129 .51254 .51379 .51504 59 .52374 .52498 .52821 .52745 .52809 .52992 58 .53853 .53975 .54097 .54220 .54342 .54404 57 .55315 .55436 .55557 .55678 .55799 .55919 56 .50700 .56880 .57000 .57119 .572:38 .57358 55 ,58189 .58307 .58425 .58543 .58001 .58779 54 .59599 .59716 .59832 .59949 .60065 .00182 53 .60991 .61107 .61222 .61337 .61451 .61560 52 .62305 .62479 .62.395 .62706 .62819 .62932 51 .63720 .63832 .63944 .64056 .64107 .64279 50 .65055 .65166 .65276 .65386 .65496 .65000 49 .66371 .66480 .66588 .66097 .66805 .00913 48 .67600 .67773 .67880 .07987 .68093 .08200 47 .68941 .69046 .69151 .09256 .69361 .69466 46 .70195 .70238 .70401 .70505 .70008 .70711 45 23 21 15 10 5 MintitPs. NATURAL COMNE. TRIGONOMETRICAL FUNCTIONS. NATURAL SINE. Minutes. Deg. 5 10 15 20 25 30 1 45 .70711 .70813 .70916 .71019 .71121 .71223 .71325 46 .71934 .72035 .72136 .72230 .72337 .72437 .72,337 47 .73135 .73234 .73333 .73432 .73531 .73629 .73728 48 .74314 .74412 .74509 .74000 .74703 .74799 .74890 49 .75471 .75506 .75661 .75756 .75851 .75940 .70041 50 .76604 .76698 .76791 .76884 .76977 .77070 .77102 51 .77715 .77806 .77897 .77988 .78079 .78170 .78201 52 .78801 .78891 .78980 .79069 .79158 .79247 .79335 53 .79804 .79951 .80038 .80125 .80212 .80299 .80380 54 .80902 .80987 .81072 .81157 .81212 .81327 .81412 55 .81915 .81999 .82082 .82105 .822i8 .82330 .82413 56 .82904 .82985 .83006 .83147 .83228 .83308 .83389 57 .83807 .83946 .84023 .84104 .84182 .84201 .84339 58 .84805 .84882 .84959 .85035 .85112 .85188 .85204 59 .85717 .85792 .85866 .85941 .86015 .80089 .80103 60 .8(5003 .80075 .80748 .80820 .86892 .80904 .87036 61 .87402 .87532 .87003 .87073 .87743 .87812 .87882 62 .88295 .88363 .88431 .88499 .88566 .88634 .88701 63 .89101 .89167 .89232 .89238 .89303 .89428 .89493 64 .89879 .89943 .90007 .90070 .90133 .90190 .90259 65 .90631 .90692 .90753 .90814 .90875 .90936 .90996 66 .91355 .91414 .91472 .91531 .91590 .91648 .91706 67 .92.)50 .92107 .92164 .92220 .92276 .92332 .92388 68 .92718 .92773 .92827 .92381 .92935 .92088 .93042 69 .93358 .93410 .93462 .93514 .93565 .93016 .93667 70 .93969 .94019 .94068 .94118 .94167 .94215 .94264 71 .94552 .94599 .94046 .94093 .94740 .94786 .94832 72 .95106 .95150 .95191 .95240 .95284 .95328 .95372 73 .95630 .95673 .95715 .95757 .95799 .95841 .95882 74 .96196 .96166 .90206 .90246 .90235 .90324 .96363 75 .96593 .96630 .90667 .96705 .90742 .90778 .96815 76 .97030 .97065 .97100 .97134 .97109 .97203 .97237 77 .97437 .97470 .97502 .97534 .97566 .97598 .97030 78 .97815 .97845 .97875 .97905 .97934 .97963 .97992 79 .98163 .98190 .98218 .98245 .98272 .98299 .98325 80 .98481 .98506 .98531 .98506 .98580 .98004 .98629 81 .98769 .98791 .98814 .98836 .98858 .98880 .98902 82 .99027 .99047 .99067 .99087 .99106 .99125 .99144 83 .99235 .99272 .99290 .99307 .99324 .99341 .99357 84 .99452 .99467 .99482 .99497 .99511 .99526 .99540 85 .99619 .99632 .99644 .99657 .99668 .99080 .99092 86 .99756 .99766 .99776 .99786 .99795 .99804 .99813 87 .99863 .99870 .99878 .99885 .99892 .99898 .99905 88 .99939 .99944 .99949 .99953 .99958 .99962 .99906 89 .99985 .99987 .99989 .99991 .99993 .99995 .99996 60 55 50 45 40 35 30 Deg Minutes. NATURAL COSINE. TRIGONOMETRICAL FUNCTIONS. NATURAL SINE. Minutes. 35 40 45 50 55 60 Deg. .71427 .71529 .71630 .71732 .71833 .71934 44 .72637 .72737 .72837 .72937 .73036 .73135 43 .73826 .73924 .74022 .74123 .74217 .74314 42 .74992 .75088 .75184 .75280 .75375 .75471 41 .76135 .76229 .76323 .76417 .76511 .76004 40 .77255 .77347 .77439 .77531 .77023 .77715 39 .78351 .78442 .78532 .78622 .78711 .78801 38 .79424 .79512 .79300 .79088 .79776 .79804 37 .80472 .80558 .89644 .80730 .80816 .80902 36 .81496 .81580 .81664 .81748 .81832 .81915 35 .82495 .82577 .82659 .82741 .82822 .82904 34 .83469 .83549 .83629 .83708 .83788 .83807 33 .84417 .84495 .84573 .84050 .84728 84805 32 .85340 .85416 .85491 .85507 .85642 .85717 31 .86237 .86317 .80384 .80457 .86530 .80003 30 .87107 ,87178 .87250 .87321 ,87391 .87462 29 .87959 ,88020 .88089 .88158 .88226 .88295 28 .88768 .88835 88902 .88968 .89035 .89101 27 .89558 .8962:3 .89687 .89752 .89816 .89879 26 .90321 .90383 .90446 .90507 .90569 .90631 25 .91056 .91116 .91176 .91236 .91295 .91355 24 .91764 .91822 .91879 .91936 .91994 .92050 23 .92444 .92499 .92554 .92609 .92064 .92718 22 .93095 .93148 .93201 .93253 .93306 .93358 21 .93718 .93709 .93819 .93809 .93919 .93969 20 .94313 .94301 .94409 .94457 .94504 .94552 19 .94878 .94924 .94970 .95015 .95001 .95106 .95415 .95459 .95502 .95545 .95588 .95030 17 .95923 .95904 .96005 .9GC46 .90086 .90120 16 .96402 .96440 .96479 .96517 .90555 .90593 15 .96851 .90887 .96923 .96959 .90994 .97030 14 .97271 .97304 .97338 .97371 .97404 .97437 13, .97661 .97092 .97723 .97754 .97784 .97815 12 .98021 .98050 .98079 .98107 .98135 .98103 11 .98352 .98378 .98404 .98430 .98455 .98481 10 .98652 .98676 .98700 .98723 .98746 .98709 9 .98923 .98944 .98965 .98986 .99006 .99027 8 .99163 .99182 .99200 .99219 .99237 .99255 7 .99374 .90390 .99406 .99421 .99437 .99452 6 .99553 .99567 .99580 .99594 .99007 .99019 5 .99703 .99714 .99725 .99736 .99746 .99756 4 .99822 .99831 .99839 .99847 .99855 .99863 3 .99911 .99917 .99923 .99929 .99934 .99939 2 .99969 .99973 .99976 .99979 .99982 .99985 1 .99997 .99998 .99999 1.00000 1.00000 1.00000 25* 20 15 10 5 Minutes. NATURAL COSINE. 210 TRIGONOMETRICAL FUNCTIONS. NATURAL TANGENT. Minutes. Deg. 5 10 15 20 25 30 0.0000 0.0014 0.0029 0.0044 0.0058 0.0073 0.0087 1 0.0175 0.0189 0.0204 0.0218 0.0233 0.0247 0.0262 2 0.0349 0.0364 0.0378 0.0393 0.0407 0.0422 0.0437 3 0.0524 0.0539 0.0553 0.0508 0.0582 0.0597 0.0612 4 O.OG99 00714 0.0728 0.0743 0.0758 0.0772 0.0787 5 0.0875 0.0889 0.0904 0.0919 0.0933 0.0948 0.0963 6 0.1051 0.1066 0.1080 0.1095 0.1110 0.1125 0.1139 7 0.1228 0.1243 0.1257 0.1272 0.1287 0.1302 0.1316 8 0.1405 0.1420 0.1435 0.1450 0.1465 0.1480 0.1495 9 0.1584 0.1599 0.1014 0.1029 0.1644 0.1058 0.1073 10 0.1763 0.1778 0.1793 0.1808 0.1823 0.1838 0.1853 11 0.1944 0.1959 0.1974 0.1989 0.2004 0.2019 0.2034 12 0.2120 0.2141 0.2150 0.2171 0.2186 0.2202 0.2217 13 ((.2309 0.2324 0.2339 0.2355 0.2370 0.2385 0.2401 14 0.2493 0.2509 0.2524 0.2540 0.2555 0.2571 0.2586 15 0.2679 0.2695 0.2711 0.2726 0.2742 0.2758 0.2773 16 0.2867 0.2883 0.2899 0.9915 0.2930 0.2946 0.2962 17 0.3057 0.3073 0.3089 0.3105 0.3121 0.3137 0.3153 18 0.3249 0.3265 0.3281 0.3297 0.3314 0.3330 0.3346 10 0.3443 0.34(50 0.3470 0.3492 0.3508 0.3525 0.3541 20 0.3640 0.3656 0.3073 0.3089 0.3706 0.3722 0.3739 21 0.3839 3855 0.3872 0.3889 0.3905 0.3922 0.3939 22 0.4040 0.4057 0.4074 0.4091 0.4108 0.4125 0.4142 23 0.4245 0.4262 0.4279 0.4296 0.4314 0.4331 0.4348 24 0.4452 0.4470 0.4487 0.4505 0.4522 0.4540 0.4557 25 0.4663 0.4681 0.4698 0.4716 0.4734 0.4752 0.4770 26 0.4877 0.4895 0.4913 0.4931 0.4950 0.4968 0.4986 27 0.5095 0.5114 0.5132 0.5150 0.5169 0.5187 0.5206 28 0.5317 0.5336 0.5354 0.5373 5392 0.5411 0.5430 29 0.5543 0.5502 0.5581 0.5600 05619 0.5638 0.5658 30 0.5774 0.5793 0.5812 0.5832 0.5851 0.5871 0.5891 31 0.6008 0.6028 0.0048 0.6068 0.6088 0.0108 0.6128 32 0.6249 0.6269 0.0289 0.6309 0.0330 0.0350 0.6371 33 0.64!)4 0.6515 0.0535 0.0550 0.6577 0.6598 0.6619 34 0.0745 O.G70G 0.6787 0.6809 0.6830 * 0.0851 0.0873 35 0.7002 0.7024 0.7045 0.7007 0.7089 0.7111 0.7133 36 0.7205 0.7288 0.7310 0.7332 0.7355 0.7377 0.7400 37 0.7530 0.7558 0.7581 0.7604 0.7027 0.7050 0.7073 38 0.7813 0.7836 0.78GO 0.7883 0.7907 0.7931 0.7954 39 0.8098 0.8122 0.8146 0.8170 0.8195 0.8219 0.8243 40 0.8391 0.8410 0.8441 0.8466 0.8491 0.8510 0.8541 41 0.8693 0.8718 0.8744 0.8770 0.8795 0.8821 0.8847 42 0.9004 0.9030 0.9057 0.9083 0.9110 0.9137 0.9103 43 0.9325 0.9352 0.9380 0.9407 0.9434 0.9402 0.9490 44 0.9057 0.9085 0.9713 0.9742 0.9770 0.9798 0.9827 60 55 50 45 40 35 30 Deg. Minutes. NATCBAL COTANGENT. TRIGONOMETRICAL FUNCTIONS. NATURAL TANGENT. Minutes. Deg. 35 40 45 50 55 60 0.0102 0.0116 0.0131 0.0145 0.0160 0.0175 89 0.0276 0.0291 0.0305 0.0320 0.0335 0.0349 88 0.0451 0.0466 0.0480 0.0495 0.0509 0.0524 87 0.0626 . 0.0641 0.0655 0.0670 0.0685 0.0699 86 0.0802 0.0816 0.0831 0.0846 0.0860 0-0875 85 0.0978 0.0992 0.1007 0.1022 0.1036 0.1051 84 0.1154 0.1169 0.1184 0.1198 0.1213 0.1228 83 0.1331 0.1346 0.1361 * 0.1376 0.1391 0.1405 82 0.1509 0.1524 0.1539 0.1554 0.1569 0.1584 81 0.1688 0.1703 0.1718 0.1733 0.1748 0.1763 80 0.1868 0.1883 0.1899 0.1914 0.1929 0.1944 79 0.2050 0.2065 0.2080 0.2095 0.2110 0.2126 78 0.2232 02247 0.2263 0.2278 0.2293 0.2309 77 0.2416 0.2432 0.2447 0.2462 0.2478 0.2493 76 0.2602 0.2617 0.2633 0.2648 0.2664 0.2679 75 0.2789 0.2805 0.2820 0.2836 0.2852 0.2867 74 0.2978 0.2994 0.3010 0.3026 0.3041 0.3057 73 0.3169 0.3185 0.3201 0.3217 0.3233 0.3249 72 0.3362 0.3378 " 0.3394 0.3411 0.3427 0.3443 71 0.3558 0.3574 0.3590 0.3607 0.3623 0.3640 70 0.3755 0.3772 0.3789 0.8805 0.3822 0.3839 69 0.3956 0.3973 0.3990 0.4006 0.4023 0.4040 68 0.4159 0.4176 0.4193 0.4210 0.4228 0.4245 67 0.4365 0.4383 0.4400 0.4417 0.4435 0.4452 66 0.4575 0.4592 0.4010 0.4628 0.4645 0.4(363 65 0.4788 0.4805 0.4823 0.4841 0.4859 0.4877 64 0.5004 0.5022 0.5040 0.5059 0.5077 0.5095 63 0.5224 0.5243 0.5261 0.5280 0.5298 0.5317 62 0.5448 0.5467 0.5486 0.5505 0.5524 0.5543 61 0.5677 0.5696 0.5715 0.5735 0.5754 0.5774 60 0.5910 0.5930 0.5949 0.5969 0.5989 0.6008 59 0.6148 0.6168 0.6188 0.6208 0.6228 0.6249 58 0.6391 0.6412 0.6432 0.6453 0.6473 0.6494 57 0.6640 0.6661 0.6682 0.6703 0.6724 0.6745 56 0.6894 0.6916 0.6937 0.6959 0.6980 0.7002 55 0.7155 0.7177 0.7199 0.7221 0.7243 0.7265 54 0.7422 0.7445 0.7467 0.7490 0.7513 0.7536 53 " 0.7696 0.7720 0.7743 0.7766 0.7789 0.7813 52 0.7978 0.8002 0.8026 0.8050 0.8074 0.8098 51 0.8268 0.8292 0.8317 0.8341 0.8366 0.8391 50 I* 0.8566 0.8591 0.8617 0.8642 0.8667 0.8693 49 I > 0.8873 0.8899 0.8925 0.8951 0.8978 0.9004 48 0.9190 0.9217 0.9244 0.9271 0.9298 0.9325 47 0.9517 0.9545 0.9573 0.9601 0.9629 0.9657 46 0.9856 0.9884 0.9913 0.9942 0.9971 1.0000 45 25 20 15 10 5 Deg. Minutes. NATURAL COTANGENT. 212 TRIGONOMETRICAL FUNCTIONS. NATURAL TANGENT. Minutes. Deg. 5 10 15 20 25 30 45 1.0000 1.0029 1.0058 1.0088 1.0117 1.0146 1.0176 46 1.0355 1.0385 1.0416 1.0446 1.0477 1.0507 1.0538 47 1.0724 1.0755 1.0786 10818 1.0850 1.0881 1.0913 48 11106 1.1139 1.1171 1.1204 1.1237 1.1270 1.1303 49 1.1504 1.1537 1.1571 1.1606 1.1640 1.1674 1.1708 50 1.1917 1.1953 1.1988 1.2024 1.2059 1.2095 1.2131 51 1.2349 1.2386 1.2423 1.2460 1.2497 1.2534 1.2572 52 1.2799 1.2838 1.2876 1.2913 1.2954 1.2993 1.3032 53 1.3270 1.3311 1.3351 1.3302 1.3432 1.3472 1.3514 54 1.3764 1.3806 1.3848 1.3891 1.39^4 1.3976 1.4019 55 1.4281 1.4326 1.4370 1.4415 1.4460 1.4505 1.4550 56 1.4826 1.4872 1.4919 1.4966 1.5013 1.5061 1.5108 57 1.5399 1.5448 1.5497 1.5547 1.5597 1.5647 1.5697 58 1.6003 1.6055 1.6107 1.6160 1.6212 1.6265 1.6318 59 1.6643 1.6698 1.6753 1.6808 1.6864 1.6920 1.6976 60 1.7320 1.7379 1.7437 1.7496 1.7556 1.7615 1.7675 61 1.8040 1.8102 1.8165 1.8228 1.8291 1.8354 1.8418 62 1.8807 1.8873 1.8940 1.9007 1.9074 1.9142 1.9210 63 1.9626 1.9697 1.9768 1.9840 1.9912 1.9984 2.0057 64 2.0503 2.0579 2.0655 2.0732 2.0809 2.0887 2.0965 65 2.1445 2.1527 2.1609 2.1692 2.1775 2.1859 2.1943 66 2.2460 2.2549 2.2637 2.2727 2.2817 2.2907 2.2998 67 2.3558 2.3654 2.3750 2.3847 2.3945 2.4043 2.4142 68 2.4751 2.4855 2.4960 2.5065 2.5171 2.5279 2.5386 69 2.6051 2.6165 2.6279 2.6394 2.6511 2.6628 2.6746 70 2.7475 2.7600 2.7725 2.7852 2.7980 2.8109 2.8239 71 2.9042 2.9180 2.9319 2.9456 2.9600 2.9743 2.9886 72 3.0777 3.0930 3.1084 3.1240 3.1397 3.1556 3.1716 73 3.2708 3.2879 3.3052 3.3226 3.3402 3.3580 3.3759 74 3.4874 3.5067 3.5201 3.5457 3.5656 3.5856 3.6059 75 3.7320 3.7539 3.7760 3.7983 3.8208 3.8436 3.8667 76 4.0108 4.C358 4.0611 4.0867 4.1126 4.1388 4.1653 77 4.3315 4.3604 4.3897 4.4194 4.4494 4.4799 4.5107 78 4.7046 4.7385 4.7729 4.8077 4.8430 4.8788 4.9152 79 5.1445 5.1848 5.2257 5.2671 5.3093 5.3521 5.3955 80 5.6713 5.7199 5.7694 5.8197 5.8708 5.9228 5.9758 81 6.3137 6.3737 6.4348 6.4971 6.5605 6.6252 6.6912 82 7.1154 7.1912 7.2687 7.3479 7.4287 7.5113 7.5957 83 8.1443 8.2434 8.3450 8.4490 8.5555 8.6648 8.7769 84 9.5144 9.6493 9.7S82 9.9310 10.0780 10.2290 10.3850 85 11.4300 11.6250 11.8260 12.0350 12.2510 12.4740 12.7060 86 14.5010 14.6060 14.9240 15.2570 15.6050 15.9690 16.3500 87 19.0810 19.6270 20.2060 20.8190 21.4700 22.1640 22.9040 88 28.6360 29.8820 31.2420 32.7300 34.3680 36.1780 38.1880 89 57.2900 62.4990 68.7500 76.3900 85.9480 98.2180 114.5900 60 55 50 45 40 35 30 Deg. Minutes. NATURAL COTANGENT. TBIGONOMETKICAL FUNCTIONS. NATURAL TANGENT. Minutes. Deg. 35 40 45 50 55 60 1.0206 1.0235 1.0265 1.0295 1.0325 1.0355 44 1.05G8 1.0590 1.0630 1.0661 1.0692 1.0724 43 1.0945 1.0977 1.1009 1.1041 1.1074 1.1106 42 1.1336 1.1369 1.1403 1.1436 1.1470 1.1504 41 1.1743 1.1778 1.1812 1.1847 1.1882 1.1917 40 1.2167 1.2203 1.2239 1.2276 1.2312 1.2349 39 1.2609 1.2647 1.2685 1.2723 1.2761 1.2799 38 1.3071 1.3111 1.3151 1.3190 1.3230 1.3270 37 1.3555 1.3597 1.3638 1.3680 1.3722 1-3764 36 1.4063 1.4106 1.4150 1.4193 1.4237 1.4281 35 1.4595 1.4641 1.4687 1.4733 1.4779 1.4826 34 1.5156 1.5204 1.5252 1.5301 1.5350 1.5399 33 1.5747 1.5798 1.5849 1.5900 1.5952 1.6003 32 1.6372 1.6426 1.6479 1.6534 1.6588 1.6643 31 1.7033 1.7090 1.7147 1.7205 1.7263 1.7320 30 1.7735 1.7795 1.7856 1.7917 1.7979 1.8040 29 1.8482 1.8546 1.8611 1.8676 1.8741 1.8807 28 1.9278 1.9347 1.9416 1.9486 1.9556 1.9626 27 2.0130 2.0204 2.0278 2.0353 2.0428 2.0503 26 2.1044 2.1123 2.1203 2.1283 2.1364 2.1445 25 2.2028 2.2113 2.2199 2.2286 2.2373 2.2460 24 2.3090 2.3183 2.3276 2.3369 2.3464 2.3558 23 2.4242 2.4342 2.4443 2.4545 2.4648 2.4751 22 2.5495 2.5605 2.5715 2.5826 2.5938 2.6051 21 2.6865 2.6985 2.7106 2.7228 2.7351 2.7475 20 2.8370 2.8502 2.8636 2.8770 2.8905 2.9042 19 3.C032 3.0178 3.0326 3.0475 3.0625 3.0777 18 3.1877 3.2041 3.2205 3.2371 3.2539 3.2708 17 3.3941 3.4124 3.4308 3.4495 3.4684 3.4874 16 3.6264 3.6471 3.6680 3.6891 3,7105 3.7320 15 3.8900 3.9136 3.9375 3.9616 3.9861 4.0108 14 4.1921 4.2193 4.2468 4.2747 4.3029 4.3315 13 4.5420 4.5736 4.6057 4.6382 4.6712 4.7046 12 4.9520 4.9894 5.0273 5.0658 5.1049 5,1445 11 5.4397 5.4845 5.5301 5.5764 5.6234 5.6713 10 6,0296 6.0844 6.1402 6.1970 6.2549 6.3137 9 6.7584 6.8269 6.8969 6.9682 7.0410 7.1154 8 7.6821 7.7703 7.8606 7.9530 8.0476 8.1443 7 8.8918 9.0098 9.1309 9.2553 9.3831 9.5144 6 10.5460 10.7120 10.8830 11.0590 11.2420 11.4300 5 12.9470 13.1970 13.4570 13.7270 14.0080 14.3010 4 16.7500 17.1690 17.6110 18.0750 18.5640 19.0810 3 23.6940 24.5420 25.4520 26.4320 27.4900 28.6360 2 40.4360 42.9640 45.8290 49.1040 52.8820 57.2900 1 137.5100 171.8800 229.1800 343.7700 687.5500 25 20 15 10 5 Deg. Minutes. NATURAL COTANGENT. 214 TRIGONOMETRICAL FUNCTIONS. NATURAL SECANT. Minutes. Deg. 5 10 15 20 25 30 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 1.0001 1.0002 1.0002 1.00C2 1.0003 1.0003 1.0003 2 1.0006 1.0007 1.0007 1.0008 1.0008 1.0009 1.0009 3 1.0014 1.0014 1.0015 1.0016 1.0017 1.0018 1.0019 4 1.0021 1.0025 1.0021 1.0027 1.0023 1.0030 1.0031 5 1.0038 1.0039 1.0041 1.0042 1.0043 1.0045 1.0046 6 1.0055 1.0057 1.0058 l.OOGO 1.0061 1.0063 1.0065 7 1.0075 1.0077 1.0079 1.0080 1.0082 1.0084 1.0086 8 1.0098 1.0 LOO 1.0102 1.0104 1.0107 1.0109 1.0111 9 1.0125 1.0127 1.0129 1.0132 1.0134 1.0136 1.0139 10 1.0154 1.0157 10159 1.0162 1.0165 1.0167 1.0170 11 1.0187 1.0190 1.0193 1.0196 1.0199 1.0202 1.0205 12 1 0223 1.022ii 1.0229 1.0233 1.02:56 1.0239 1.0243 13 1.0263 1.0266 1.0270 1.0274 1.0277 1.0280 1.0284 14 1.0306 1.0310 1.0314 1.0317 1.0321 1.0325 1.0329 15 1.0353 1.0357 1.0361 1.0365 1.0369 1.0373 1.0377 16 1.0403 1.0407 1.0412 1.0416 1.0420 1.0425 1.0429 17 1.0457 1.0461 1.0466 1.0471 1.0476 1.0480 1.0485 18 1.0515 1.0520 1.0525 1.0530 1.0535 1.0540 1.0545 19 1.0577 1.0581 1.0587 1.0592 1.0598 1.0G03 1.0608 20 1.0642 1.0647 1.0653 1.0659 1.0664 1.0ti70 1.0676 21 1.0711 1.0717 1.0723 1.0729 1.0736 1.0742 1.0748 22 1.0785 1.0792 1.0798 1.0804 1.0811 1.0817 1.0824 23 1.0864 1.0870 1.0877 1.0884 1.0891 1.0897 1.0904 24 1.0946 1.0953 1.0961 1.0968 1.0975 1.09S2 1.0989 25 1.1034 1.1041 1.1049 1.1056 1.1064 1.1072 1.1079 20 1.1126 1.1134 1.1142 1.1150 1.1158 1.11 (56 1.1174 2T 1.1223 1.1231 1.1240 1.1248 1.1257 1.1265 1.1274 28 1.1326 1.1334 1.1343 1.1352 1.1361 1.1370 1.1379 29 1.1433 1.1443 1.1452 1.1461 1.1471 1,1480 1.1489 30 1.1547 1.1557 1-1566 1.1576 1.1586 1.1596 1.1606 31 1.1666 1.1676 1.1687 1.1697 1.1707 1.1718 1.1728 32 1.1792 1.1802 1.1830 1.1824 1.1835 1.1846 1.1857 33 1.1923 1.1935 1.1946 1.1958 1.1969 1.1980 1.1992 34 1.2002 1.2074 1.2068 1.2098 1.2110 1.2122 1.2134 35 1.2208 1.2220 1.2233 1.2245 1.2258 1.2270 1.2283 36 1.2361 1.2374 1.2387 1.2400 1.2413 1,2427 1.2440 37 1.2521 1.2535 1.2549 1.2563 1.2577 1.2591 1.2605 38 1.2690 1.2705 1.2719 1.2734 1.2748 1.2763 1.2778 39 1.2867 1.2883 1.2898 12913 1,2929 1.2944 1.29GO 40 1.3054 1.3070 4.3086 1.3102 1.3118 1.3134 1.3151 41 1.3250 1.3267 1.3284 1.3301 1.3318 1.3335 1.3352 42 1.3456 1.3474 1.3492 1.3509 1.3507 1.3540 1.3563 43 1.3673 1.3692 1.3710 3,3729 1,3748 1.3767 1.3786 44 1.3902 1.3921 1.3941 1.3960 1.3980 1.4000 1.4020 60 55 50 45 40 35 30 Deg. Minutes. NATURAL COSECANT. TRIGONOMETRICAL FUNCTIONS. NATURAL SECANT. Minutes. L>eg. 35 40 45 50 55 60 1.0000 1.0001 1.0001 1.0001 1.0001 1.0001 89 1.0004 1.0004 1.0005 1.0005 1.0005 1.0(OG 88 1.0010 1.0011 1.0011 1.0012 1.0013 1.0014 87 1.0019 1.0020 1.0021 1.0022 1.0023 1.0024 86 1.0032 1.0033 1.0034 1.0036 1.0037 1.0038 85 1.0048 1.0049 1.0050 1.0052 1.0053 1 .0055 84 1.00G6 1.0008 1.0070 1.0071 1.0073 1.0075 83 1.0088 1.0090 1.0092 1.0094 1.0096 1.0098 82 1.0113 1.0115 1.0118 1.0120 1.0122 1.0125 81 1.0141 1.0145 1.0146 1.0149 1.0152 1.0154 80 1.0173 10176 1.0179 1.0181 1.0184 1.0187 79 1.0208 1.0211 1.0214 1.0217 1.0220 1.0223 78 1.0246 1.0249 1.0253 1.0256 1.0260 1.0263 77 1.0288 1.0291 1.0295 1.0298 1.0302 1.0306 76 1.0333 1.0337 1.0341 l.< 345 1.0349 1.0353 75 1.03S2 1.0386 1.0390 1.0394 1.0399 1.0403 74 1.0434 1.0438 1.0443 1.0448 1.0452 1.0457 73 1.0490 1.0495 1.0500 1.0505 1.0510 1.0515 72 1.0550 1.0555 1.0560 1.0565 1.0571 1.0577 71 1.0644 1.0619 1.0625 1.0630 1.0636 1.0642 70 1.0082 1.0688 1.0694 1.0699 1.0705 1.0711 69 1.0754 1.0760 1.0766 1.0773 1.0779 1.0785 68 1.0830 1.0837 1.0844 1.0850 1.0857 1.0864 67 1.0911 1.0918 1.0925 1.0932 1.0939 1.0946 66 1.0997 1.1004 1.1011 1.1019 1.1026 1.1034 65 1.1087 1.1095 1.1102 1.1110 1.1118 1.1126 64 1.1182 1.1190 1.1198 1.1207 1.1215 1.1223 63 1.1282 1.1291 1.1299 1.1308 2.1317 1.1326 62 1.1388 1.1397 1.1406 1.1415 1.1424 1.1433 61 1.1499 1.1508 1.1518 1.1528 1.1537 1.1547 60 1.1616 1.1626 1.1636 1.1646 1.1656 1.1666 59 1.1739 1.1749 1.1760 1.1770 1.1781 1.1792 58 1.1808 1.1879 1.1819 1.1901 1.1912 1.1923 57 1.2004 1.2015 1.2027 1.2039 1.2050 1.2062 56 1.2146 1.2158 1.2171 1.2183 1.2195 1.2208 55 1.2296 1.2309 1.2322 1.2335 1.2348 1.2361 54 1.2453 1.2467 1.2480 1.2494 1.2508 1.2521 53 1.2619 1.2633 1.2647 1.2661 1.2676 1.2690 52 1.2793 1.2807 1.2822 1.2837 1.2852 1.2867 51 1.2975 1.2991 1.3006 1.3022 1.3038 1.3054 50 1.3167 1.3184 1.3200 1.3217 1.3233 1.3250 49 1.3369 1.3386 1.3404 1.3421 1.3439 1.3450 48 1.3581 1.3GOO 1.3618 1.3636 1.3655 1.3073 47 1.3805 1.3824 1.3843 1.3863 1.3882 1.3902 46 1.4040 1.4056 1.4081 1.4101 1.4122 1.4142 45 25 20 15 10 5 Dee Minutes. NATURAL COSECANT. 216 TRIGONOMETRICAL FUNCTIONS. NATURAL SECANT. Minutes. Deg. 5 10 15 20 25 30 45 1.4142 1.4163 1.4183 1.4204 1.4225 1.424G 1.4267 46 1.4395 1.4417 1.4439 1.44G1 1.4483 1.4505 1.4527 47 1.46G3 1.4G86 1.4709 1.4732 1.4755 1.4778 1.4802 48 14945 1.4969 1.4993 1.5018 1.5042 1.50G7 1.5092 49 1.5242 1.5268 1.5294 1.5319 1.5345 15371 1.5398 50 1.5557 1.5584 1.5011 1.5639 1.5GGG 1.5G94 1.5721 51 1.5890 1.5919 1.5947 1.5976 1.6005 l.GO:54 1.6064 52 1.6243 1.6273 1.6303 1.6334 1.6365 1.6396 1.6427 53 1.6616 1.6648 1.GG81 1.6713 1.6746 1.6779 1.6812 54 1.7013 1.7047 1.7081 1.7116 1.7151 1.7185 1.7220 55 1.7434 1.7471 1.7507 1.7544 1.7581 1.7018 1.7655 56 1.7883 1.7921 1.7960 1.7999 1.8039 1.8078 1.8118 57 1.8361 1.8402 1.8443 1.8485 1.8527 1.8569 1.8G11 58 1.8871 1.8915 1.8959 1.9004 1.9048 1.9093 1.9139 59 1.9416 1.9463 1.9510 1.9558 1.9606 1.9C54 1.9703 60 2.0000 2.0050 2.0102 2.0152 2.0204 2.0256 2.0308 61 2.0(527 2.0681 2.0735 2.0790 2.0846 2.0901 2.0957 62 2.1300 2.1359 2.1418 2.1477 2.1536 2.1596 2.1657 63 2.2027 2.2090 2.2153 22217 2.2282 2.2346 2.2411 64 2.2812 2.2880 2.2949 2.3018 2.3087 2.3158 2.3228 65 2.3662 2.3736 2.3811 2.3886 2.3961 2.4037 2.4114 66 2.4586 2.4G6G 2.4748 2.4829 2.4912 2.4995 2.5078 67 2.5593 2.5G81 2.5770 2.5859 2.5949 2.G040 2.6131 68 2.G695 2.G791 2.6888 2.6986 2.7085 2.7185 2.7285 69 2.7904 2.8010 2.8117 2.8225 2.8334 2.8444 2.8554 70 2.9238 2.9355 2.9474 2.9593 2.9713 2.9835 2.9957 71 3.0715 3.0S4G 3.0977 3.1110 3.1244 3.1379 3.1515 72 3.23G1 3.250G 3.2G53 3.2801 3.2951 3.3102 3.3255 73 3.4203 3.4366 3.4532 3.4G97 3.4867 3.5037 3.5209 74 3.6276 3.6464 3.GG51 3.68-10 3.7031 3.7224 3.7420 * 75 3.8G37 3.8848 3.9061 3.9277 3.9495 3.9716 3.9939 76 4.1330 4.1578 4.1824 4.2072 4.2324 4,2579 4.2836 77 4.4454 4.4736 4.5021 4.5331 4.5604 4.5901 4.6202 78 4.8097 4.8429 4.8765 4.9106 4.9452 4.9802 5.0158 79 5.2408 5.2803 5.3205 5.3G12 5.4020 5.4447 5.4874 80 5.7588 5.80G7 5.8554 5.9049 5.9554 5.99G3 6.0588 81 6.3924 6.4517 6.5121 6.5736 6.6363 6.7003 6.7655 82 7.1853 7.2604 7.3372 7.4156 7.4957 7.5776 7.6613 83 8.2055 8.3C39 8.4046 8.5079 8.6138 8.7223 8.8337 84 9.5608 9.7010 9.^391 9.9812 10.1270 10.2780 10.4330 85 11.4740 11.6680 11.8680 12.07GO 12.2910 12.5140 12.7450 86 14.3350 14.6400 14.9580 15.2900 15.6370 16.0000 16.3800 87 19.1070 19.G530 20.23(10 20.8430 21.4940 22.18GO 22.9250 88 28.6540 29.8990 31.2570 32.7450 34.3820 36.1910 38.2010 89 57.2990 62.5070 68.7570 76.3960 85.9460 98.2230 114.5900 GO 55 50 45 40 35 30 Deg Minutes. NATURAL COSECANT. TRIGONOMETRICAL FUNCTIONS. NATURAL SECANT. Minutes. 35 40 45 50 55 60 1.4288 1.4310 1.4331 1.4352 1.4374 1.4395 1.4550 1.4572 1.4595 1.4617 1.4640 1.4663 1.4825 1.4849 1.4873 1.4897 1.4921 1.4945 1.511G 1.5141 1.5166 1.5192 1.5217 1.5242 1.5424 1.5450 1.5477 1.5503 1.5530 1.5557 1.5749 1.5777 1.5805 1.5833 1.5862 1.5890 1.G093 1.6123 1.6153 1.6182 1.6212 1.6243 1.6458 1.6489 1.6521 1.6552 1.6584 1.6616 1.6845 1.6878 1.6912 1.6945 1.6979 1.7013 1.7256 1.7291 1.7327 1.7362 1.7398 1-7434 1.7693 1.7730 1.7768 1.7806 1.7844 1.7883 1.8158 1.8198 1.8238 1.8279 1.8320 1.8361 1.8654 1.8G97 1.8740 1.8783 1.8827 1.8871 1.9184 1.9230 1.9276 1.9322 1.9369 1.9416 1.9752 1.9801 1.9850 1.9900 1.9950 2.0000 2.0360 2.0413 2.0466 2.0519 2.0573 2.0627 2.1014 2.1070 2.1127 2.1185 2.1242 2.1300 2.1717 2.1778 2.1840 2.1902 2.1964 2.2027 2.2477 2.2543 2.2610 2.2676 2.2744 2.2812 2.3299 2.3371 2.3443 2.3515 2.3588 2.3662 2.4191 2.4269 2.4347 2.4426 2.4506 2.4586 2.5163 2.5247 2.5333 2.5419 2.5506 2.5593 2.G223 2.6316 2.6410 2.6504 2.6599 2.6695 2.7386 2.7488 2.7591 2.7694 2.7799 2.7904 2.8666 2.8778 2.8892 2.9006 2.9122 2.9338 3.0081 3.0206 3.0331 3.0458 3.0586 3.0715 3.1653 3.1792 3.1932 3.2074 3.2216 3.2361 3.3409 3.3565 3.3722 3.3881 3.4041 3.4203 3.53S3 3.5559 3.5736 3.5915 3.6096 3.6279 3.7617 3.7816 3.8018 3.8222 3.8428 3.8637 4.0165 4.0394 4.0625 4.0859 4.1090 4.1336 4.3098 4.3362 4.3630 4.3901 4.4176 4.4454 4.6507 4.6817 4.7130 4.7448 4.7770 4.8097 5.0520 5.0886 5.1258 5,1636 5.2019 5.2408 5.5308 5.5749 5.6197 5.6653 5.7117 5.7588 6.1120 6.166 1 6,2211 6.2772 6.3343 6.3924 6.8320 6,8998 6.9690 7,0390 7.1117 7.1853 7.7469 7.8344 7.9240 7.9971 8.1094 8.2055 8.9479 9.0651 9.1855 9.3092 9.4362 9.5668 10.5930 10.7580 10.9290 11.1040 11.2080 11,4740 12.9850 13.2350 13.4940 13.7630 14.0430 14.3350 16.7790 17.1980 17.6390 18.1030 18.5910 19.1070 23.7160 24,5620 25.4710 26.1500 27.5080 28.6540 39.9780 42,9760 45.8400 49.1140 52.8910 57.2990 137.5100 171.8900 229.1800 343.7700 687.5500 00 25 20 15 10 5 Minutes. 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T-< rH r-T^? ^ ^H ^-r^?CM CM^CM CM^CM^fc^CM CIRCUMFERENCE, AREA, AND CUBIC CONTENTS OF CIRCLES. 221 cqoocoocoiOT-HcocirHOi^oocooocooi 1 1 " W>3 O rt O G ^ O O^ " (Mr-Hi (r I O O O O O T li-HC^CNCOCO^lOCOi^C^Or-* ^ -rft Tf -rH ^-H iQ >O iQ >O >O i-O >O i-O iQ >O CQ CD CO CD CD CO CO ^ seqouiui .S 1 D^H ^^^ LOtOCDJr^oo .. ..,_ .. _, ,__-. ? u .rt lOOOOOOCpT^CO^^C^J^CpOp ^<5>riHOCO !!> Q ~ l cS ^J 1 1; sli 9 CO CO CO CO CO CO CO CO C soqouiui g o^o a o fl S "- 1 o ^* LCC-llOCO-t CDi IT (COl^COlOC^iLOCOCOlOOOiOlCCl O^i^rHOqj^COO^rtlOOLO ^lOasCDlOCDOCD^tHCDCi .^^ T (i lOO 1 5O^COl>-COCDlO ^COCOQT IrHOOiCOpOI^CO ^^ 2 ao a>oa 52 d S^ o? 05 crt xH^H ^ 10 10 >o >o *o co co co co co co co co co co co co co co co co c (NC^C^tMCMC^C^C^CS) 222 CIRCUMFERENCE, AREA, AND CUBIC CONTENTS OF CIRCLES T. >"~ l o on il- -p *H ob co *^ oo i> to o ^ co cp ^S o ** co cp o *H CM i O i I O i-~ T-H CO CO i I 1 O t-t O !> O-l O <M CO lO C JCOCOCOCOC OO O <M CO lO CO OO O i CO IO CD OO O CM CO O I- O O <M CO O I- O O <M OiOLOuO<X)CD tO O CO - eo co co co co co co co co co co co T > co ix^ o^ oa - <^ *" til s " (J^TtlT-IOOlOCvJOt^OCOOQOCOlOCOrHOCil CD CO QO i^ CO <NI CO Oi O O5 CO t-H ^ CD C " _ o C Jj cooo^cococooocococococococococococococo^-^ CO * O O <^ CO < ^*< O Ui ^ OO - CO CO rH J? r- CO CO C*l 00 rHr-HOCDGi^H IQOCO^DI^lOCMCDCOOOLOr-i-^llO^tiO CO OD 00 CO CO 1^- O i- O JC* CO CO 00 CD CO CO 1>- O H 1> C^lOCOr^TtlCDOir^C-O^CDl^OOOOr-lr-tC-lCMCMrH CD^fCMOoOCDCOi <MCDO^t | oO(MTMO co" ii^cMoocooo ^ai ^O OOi^LO^rlCMT-HaSOOi fOOCD^CMOOOiOCOT < saqouiui c^^cococococococooocococococococooococococo CIRCUMFERENCE, AREA, AND CUBIC CONTENTS OF CIRCLES. 223 I ij saqout ui a ff o C > 0- is! IP saqouiui (M CO O ^f OO CM iO OOOr 1 !>1 <M C5 LO OO O i lC3t!LOOC iO Ci OQ 10 i O. C\l O 1^ oo O C-1 C cOOO50OQOO5T-ilCCir r-r rH ^ co j o <NI 10 *vo5 rH ^i cp oo - eo Q oo o eg ^t 1 rjH Tf ^t 1 lO 1C IQ lO O CO CD CO CO .t^ I~ L^- !> GO CO fN C<I (M <M 01 C^l 01 (M (M (M (M (M CM !M O1 <M C\l CM Oq CO OO CO GO CO GO CO GO CO OO CO rtir^oDt^^cioa xricor-icoOcMCMOco O rjH^co^cscoTicoO *it^i--iOi t^ rHoooooor- iiofMOOCsjiocDi^cot^Oi COJ^C^t-tMl.^. CNoOCOOiiOOCD CNOt^O^C-J^t^OiCMTHl^OiCM^i-^C^CvIiOl^ ^TH^JHrt<lOLOlOLOCOCOCCCOI^-t^l^J:^OOCX5OO CM CM CM CM M CM <M CNJ CM (M CM Ol CM CM CM CM CM CM C>q CN-rtlCOGOOC sJ rtlCOOOOCM-rtHcO OOOOr- 1 i i I i iT-HfM(MC<l<M CM CM CM CM CM cq C" "^ CMl^ HCMCMCMC^JCOCOCOCOTH^f"^H < ^HiOO 10 co i 110 r- Oi O O T i CM ci 111 <3>.i (N 1O l~ -- 1 ^ O Ttl CO I- CO O-l t^ rH CM r-H O5 1C CD rH O T i CM CO ^ O O UD LO O O TtH T}H !|O CM O CD l- CO G& O C>4 tQ o> rH O ^ *C> ^ Tt* *Q J>> O ^< G> ^$* i( O3 CCC co co co i L- i co co co co co as as as CD as i CO CO IO -^H Oq r-H C^> OO CO LO CO CM O Gi J-^ C setpui ui 221 SPECIFIC GRAVITIES OF MATERIALS. SPECIFIC GRAVITIES OF MATERIALS. Weight of a cubic foot in Ibs. GASES at 32 Fahr., and under the pressure of on atmosphere of 2116.4 Ibs. on the square foot: Air e 0.080728 Carbonic acid 0.12344 Hydrogen 0.005592 Oxygen 0.089256 Nitrogen 078596 Steam (ideal) 0.05022 ^Ether vapor (ideal) . 02093 Bisulphuret-of-carbon vapour (i ieal) 2137 079R LIQUIDS at 32 Fahr. (except water, which is taken at 39.4 Fahr.): Water, pure, at 39.4 Weight of a cubic foot in Ibs. avoirdupois. Specific gravity, pure water = I. 62.425 64.05 49.38 57.18 44.70 848.75 52.94 58.68 57.12 57.62 54.31 54.81 187.3 125 to 135 112 117 to 174 120 100 77. 4 to 89. 9 62.43 to 103. 6 162.3 164.2 1.000 1.026 0.791 0.916 0.716 13.596 0.848 0.940 0.915 0.923 0.870 0.878 3.00 2 to 2. 167 1.8 1.87 to 2. 78 1.92 1.602 1.24 to 1.44 1.00 to 1.66 2.6 2.63 " sea ordinary Alcohol pure proof spirit ^Ether . Mercury Naphtha Oil, linseed " olive " whale " of turpentine SOLID MINERAL SUBSTANCES, non- metallic: Basalt Brick Brickwork Chalk . .... Clay Coal anthracite " bituminous . Coke , Felspar Flint.. SPECIFIC GRAVITIES OF MATERIALS. 225 SOLID MINERAL SUBSTANCES con Weight of a cubic foot in Ibs. avoirdupois. Specific gravity, pure water = 1. tinued: Glass crown average 156 2 5 11 flint 187 3.0 " green 169 2.7 " plate 169 2 7 Granite 164 to 172 2.63 to 2.76 Gypsum 143.6 2.3 Limestone, (including marble)... magnesian 169 to 175 178 2. 7 to 2.8 2.86 Marl ... . 100 to 119 1 6 to 1 9 Masonry 116 to 144 1.85 to 2 3 Mortar 109 1.75 Mud 102 1 63 Quartz 165 2 65 Sand (damp) .... 118 1 9 " (dry). . 88.6 1 42 Sandstone average 144 2 3 " various kinds . 130 to 157 2 08 to 2 52 Shale 162 2 6 Slate 175 to 181 2 8 to 2 9 Trap... 170 2 72 METALS, solid: Brass, cast 487 to 524.4 7 8 to 8 4 " wire. 533 8 54 Bronze 524 8 4 Copper, cast 537 8 6 " sheet 549 8 8 " hammered 556 8 9 Gold 1186 to 1224 19 to 19 6 Iron, cast various 434 to 456 6 95 to 7 3 average 444 7 11 Iron, wrought various 474 to 487 7 6 to 7 8 average 480 7 69 Lead 712 114- Platinum 1311 to 1373 21 to 22 Silver 655 10 ^ Steel 487 to 493 7 8 to 7 9 Tin 456 to 468 7 3 to 7 5 Zinc 424 to 449 6 8 to 7 2 TIMBER:* Ash 47 7^3 Bamboo 25 4 Beech 43 69 15 226 SPECIFIC GRAVITIES OF MATERIALS. TIMBER :* continued. Birch . .. Weight of a cubic foot in Ibs. avoirdupois. Specific gravity, pure water = 1. 44.4 52.5 60 65.3 56.2 30.4 33.4 36.2 74.5 34 30 to 44 30 to 44 29 31 to 35 62.5 57 54 47 47 57 42 to 63 41 to 83 44 35 53 49 57 43 to 62 54 36 60 37 41 to 55 61 62 to 66 62.5 25 50 0.711 0.843 0.96 1.046 0.9 0.486 0.535 0.579 1.193 0.544 0.48 to 0.7 0.48 to 0.7 0.46 0.5 to 0.56 1.001 0.91 0.86 0.76 0.76 0.92 0.675 to 1.01 0.65 to 1.33 0.71 0.56 0.85 0.79 0.92 0.69 to 0.99 0.87 0.58 0.96 0.59 0.66 to 0.88 0.98 0.99 to 1.06 1.001 0.4 0.8 Blue-gum Box Bullet- tree Cabacalli Cedar of Lebanon Chestnut Cowrie Ebony, West Indian Elm Fir, red pine " spruce " American yellow pine Greenhart Haw thorn Ha*zel Holly Hornbeam Laburnum Lancewood Larch. (See "fir".) Lignum-vitae Locust Mahogany, Honduras Spanish . Maple Mora Oak European " American red Poon Saul Sy cam ore Teak Indian 11 African Tonka Water-gum Willow Yew *The timber in every case is supposed to be dry. WEIGHT OF A SUPERFICIAL INCH. ETC. 227 WEIGHT OF A SUPERFICIAL INCH OF WROUGHT AND CAST IRON. (From one-sixteenth to one-inch thickness.) Thickness in inches. WROUGHT IRON. Cubic foot = 480 Ibs. CAST IRON. Cubic foot = 450 Ibs. Weight in Ibs. Weight in Ibs. A 0.017356 0.0163 i 0.0347 0.0326 A 0.0520 0.0489 i 0.0694 0.0652 T 5 6 0.0867 0.0815 1 0.1041 0.0978 ft 0.1214 0.1141 i 0.1388 0.1304 T 9 * 0.1562 0.1467 i 0.1735 0.1630 0.1909 0.1793 1 0.2082 0.1956 n 0.2256 0.2119 i 0.2429 0.2282 it 0.2603 0.2445 i 0.2777 0.2608 228 WEIGHT PEE SQUARE FOOT IH POUNDS AVOIRDUPOIS. WEIGHT PER SQUARE FOOT IN POUNDS AVOIRDUPOIS. Thickness in inches. Wrought Iron. Cast Iron. Copper, sheet. Lead. Zinc. 480 Ibs. per cubic foot. 450 Ibs. per cubic foot. 549 Ibs. per cubic foot. 712 Ibs. per cubic foot. 436 Ibs. per cubic foot. TV 2.50 2.34 2.86 3.71 2.27 * 5.00 4.69 5.72 7.42 4.54 T 3 6 7.50 7.03 8.58 11.12 6.81 i 10.00 9.37 11.44 14.83 9.08 ft 12.50 11.72 14.30 18.54 11.35 1 15.00 14.06 17.16 22.25 13.62 ft 17.50 16.41 20.02 25.96 15.89 20.00 18.75 22.88 29.66 18.16 ft 22.50 21.09 25.74 33.37 20.43 1 25.00 23.44 28.60 37.10 22.70 H 27.50 25.78 31.46 40.79 24.97 f 30.00 28.12 34.32 44.50 27.24 it 32.50 30.47 37.18 48.20 29.51 * 35.00 32.81 40.04 51.91 31.78 it 37.50 35.16 42.90 55.62 34 05 i 40.00 37.50 45.75 59.33 36.33 WEIGHT OF A LINEAL FOOT, ETC. 229 WEIGHT OF A LINEAL FOOT OF FLAT AND SQUAKE BAR IRON IN POUNDS AVOIRDUPOIS. (480 pounds per cubic foot.) Breadth in inches. Thickness in inches. Weight in Ibs. d 5 2 ll Thickness in inches. Weight in Ibs. Breadth in inches. Thickness in inches. Weight in Ibs. i i 0.104 ij 1 5.000 2J 1 1.875 0.208 H 5.625 | 2.813 1 1 0.208 " if 6.250 " 3.750 0.416 " if 6.874 " 4.687 " 1 0.832 " if 7.500 M 5.624 1 | 0.312 it 4 0.739 6.562 I 0.624 i 1.459 11 1 7.500 (( I 0.937 " I 2.187 " li 8.437 (( 1.249 * 2.916 " H 9.374 II .| 1.562 <( 1 3.646 rt If 10.310 (( j 1.874 i 4.375 " If 11.250 1 i 0.416 " - 5.103 " If 12.190 " 0.833 5.833 11 If 13.120 <( -1 1.249 6.562 l| 14.060 M j 1.667 H u 7.291 11 o 15.000 -1 2.089 " if 8.020 H 2J 15.940 t 2.500 " if 8.750 11 2i 17.810 11 7 F 2.916 " if 9.478 2} | 1.041 (t 1 3.333 u it 10.930 2.089 11 i 0.521 2 1 833 " | 3.125 | 1.041 " I 1.667 11 1 4.166 (i f 1.562 3 8" 2.500 " I 5.208 2.Q89 11 | 3.333 " I 6.250 II 2.603 11 4.166 11 i 7.291 <( 3.124 11 5.000 " 8.333 3.646 c I 5.833 " li 9.398 <( 1 4.166 " 6.666 11 10.410 ii 1* 4.687 11 If 7.500 " If 11.460 <( li 5.728 " l| 8.333 " IJ 12.500 i| J 0.624 11 If 9.156 (1 If 13.540 (i \ 1.250 u if 10.000 " If 14.580 11 i 1.875 11 If 10.830 If 15.620 ii 2.500 11 If 11.660 11 2 16.660 3.125 M li 12.500 " a 17.710 ii 3.750 M 2 13.330 (( 2i 18.750 ii 4.375 2i 4 0.937 " 2| 20.820 230 WEIGHT OF A LINEAL FOOT, ETC. Breadth in inches. Thickness in inches. Weight in Ibs. Breadth in inches. Thickness in inches. Weight in Ibs. Breadth in inches. Thickness in inches. Weight in Ibs. 2J 2f 19.800 3J 2 21.660 4 2 26.660 2J ^ 1.146 i 24.370 " 21 30.000 i 2.292 " 2 27.080 " i 33.330 1 ^ 3.437 11 2| 29.790 M 2f 36.660 4 | 4.583 " 3 32.500 u 3 40.000 1 -| 5.729 41 3i 24.200 M 31 43.330 1 J 6.874 3i i 2.916 " 3* 46.660 1 I 8.020 i 5.833 11 3| 50.000 * 1 9.154 u i 8.750 " 4 53.330 * H 10.310 " 11.660 $ i 3.541 1 u 11.460 11 H 14.580 7.082 if 12.600 11 U 17 500 " | 10.620 i| 13.750 M if 20.430 M 1 14.160 1 i| 14.900 11 2 23.330 M It 16.800 1 ij 16.030 " 2 26.250 H 21.330 1 1J 17.190 2J 29.160 If 24.780 2 18.330 M 2f 32.080 i 2 28.330 2J 19.480 " 3 35.000 i 2| 31.870 i 20.620 M 3| 37.910 1 1 35.410 c 1 21.770 3 ! 40.830 1 i 38.950 1 2^ 22.910 3f 3.125 3 42.500 1 2| 24.060 j 6.250 3J 46.030 1 2| 25.200 11 j 9.375 1 3 49.570 3 i 2.500 it i 12.500 ( 3| 53.120 5.000 " 14 15.620 1 4 56.660 1 i 7.500 " 1} 18.750 60.200 1 i 10.000 M if 21.870 IF i 3.750 1 ia 12.500 " 2 25.000 7.500 1 i? 15.000 " 2* 28.120 y | 11.250 if 17.500 2J 31.250 M 1 15.000 ".": :, 2 20.000 2f 34.370 " it 18.750 1 2} 22.500 " 3 37.500 ! 9 22.500 1 2J 25.000 M 31 40.620 1 if 26.250 2| 27.500 " 3^ 43.750 2 30.000 3 30.000 " 3f 46.860 1 ^ 33.750 3| | 2.708 4 i 3.330 1 ^ 37.500 5 416 " i 6.660 < 2f 41.250 C "**; |. 8.124 " f 10.000 f 3 45.000 !t .**.] 1 10.830 * 13.330 5i 48.750 ..;_ ;,. H 13.500 " 11 16.660 1 3J 52.500 4 1| 16.250 M 1 20.000 M si 56.250 t If 18.950 a If 23.330 N 4 60.000 WEIGHT OF A LINEAL FOOT, ETC. 231 5 * 1 a/ G t, 1 - 1 PQ Thickness in inches. Weight in Ibs. a 5s ~Q-Z 06 P PQ Thickness in inches. J5 3f Breadth in inches. co CJ <P g-g !s-S ^ Weight in Ibs. 4J 4} 63 . 750 5i i 8.753 5| i 4.788 4 67.500 -1 1-3.130 i 9.587 4f 1 3.953 I 17.500 ( 1 14.370 i 7.910 1 H 21.870 i 1 19.160 " i 11.860 1 H 26.250 i 1} 23.950 15.830 1 if 30.620 i li 28.750 ti I* 19.760 f 2 35.000 1 I 33.540 li 23.750 IJ m 39.370 2 38.330 < If 27.700 u 2i 43.750 " 2t 43.120 2 31.670 1 2| 48.110 c< 2J 47.910 i 2i 35.620 3 52.500 M 2| 52.700 1 2i 39.580 1 3J 56.680 ( 3 57.500 1 2| 43.540 3i 61.250 1 3i 62.300 1 3 47.500 1 3| 65.620 4 3i 67.080 1 P 51.460 4 4 70.000 ( 3f 71.860 M 31 55.410 | 4} 74.370 4 4 76.650 (i 3| 59.370 4| 78.750 4 H 81.450 4 63.330 4 4f 83.110 1 4J 86.240 4} 67.290 5 87.500 4f 91.030 M 4 71.250 ! 5J 91.860 4 5 95.820 4| 75.200 5J J 4.587 4 5t 100.600 5 1 4.166 i 9.164 ( 5} 105.400 M i 8.330 44 1 13.750 4 5| 119.700 i 1 12.500 " 1 18.330 6 J 10.000 1 1 16.660 " H 22.900 4 - 1 20.000 1 li 20.830 u 27.500 M li 30.000 4 li 25.000 II If 32.080 44 2 40.000 li 29.160 M 2 36.660 44 2J 50.000 4 2 33.330 11 2J 41.250 44 3 60.000 21 37.500 u 2i 45.830 3J 70.000 1 2i 41.660 " 2} 50.310 44 4 80.000 4 2| 45.830 11 3 55.000 44 4 90.000 1 3 50.000 3i 59.570 44 5 100.000 < 3} 54.160 11 3J 64.160 tc 5J 110.000 1 31 58.330 11 3| 68.740 6 120.000 i 3f 62.500 (( 4 73.330 6J i 10.830 4 66.660 11 4J 77.910 21.660 t 4J 70.830 11 4J 82.500 44 li 32.500 1 4^ 75.000 " 4| 87.080 44 2 43.330 4 4 79.160 " 5 91.560 44 2i 54.160 4 5 83.330 u 5| 96.240 44 3 65.000 H i 4.376 M 5J 100.600 ii 3i 75.830 232 WEIGHT OF A LINEAL FOOT, ETC. Breadth in inches. Q ? A C 03 C Jifl Weight in Ibs. Breadth in inches. Thickness in inches. Weight in Ibs. c j| c Jl c- .2 H Weight in Ib.s. 6 ; > 4 86.66 8 4 106.60 9 8} 255.00 4J 97.50 " 4.1 120.00 " 9 270.00 it 5 108.30 " 5" 133.30 9} } 15.83 M 5 119.10 " 5} 146.60 1 31.66 6 130.00 (i 6 160.00 " 1} 47.50 " 6} 140.80 " 8} 173.30 M 2 63.33 7 i 11.66 u 7 186.60 " 2} 79.16 " 1 23.33 7} 200.00 " S 95.00 " u 35.00 " 8 213.30 " 3^ 110.80 " 2 46.66 8| } 14.16 " 4 126.60 2J 58.33 28.33 " 4} 142.50 11 3 70.00 u 1J 42.48 M 5" 158.30 11 3J 81 66 11 2 56.66 " 5-^r 174.10 " 4 93.33 u 2J 70.83 H 6 190.00 4} 105.00 11 3 85.00 " 6J 205.80 " 5 116.60 " 3} 99.16 " 7 221.60 u 5J 128.30 " 4 113.30 " 7i 237.60 6 140.00 " 4J 127.50 " 8 253 . 30 " 6J 151.60 " 5 141.60 " 8J 269.10 " 7 163.30 u 5} 155.80 " 9 285.00 n i 12.50 (i 6 170.00 11 9J 300.80 1 25.00 u 6| 184.10 10 1 16.66 " U 37.50 u 7 198.30 " 1 33.33 M 2 50.00 " 7} 212.50 " 1} 50.00 ii 2 62.50 11 8 226.60 11 2 66.66 tl 3 75.00 M 8} 240.70 " 2^- 83.33 3} 87.50 9 i 15.00 " 3" 100.00 11 4 100.00 < i 30.00 " 3^ 116.60 " 4i- 112.50 11 u 45.00 11 4 133.30 " 5 125.00 11 2 60.00 11 4J 150.00 M 5} 137.50 " 2J 75.00 M 5 166.60 11 6 150.00 11 3 90.00 " 5} 183.30 ii 6J 162.50 " 3i 105.00 (f 6 200.00 " 7 175.00 u 4 120.00 6} 216.60 M 7^ 187.50 11 4 135.00 1 7 233.30 8 | 13.33 11 5 150.00 7* 250.00 11 1 26.66 11 5} 165.00 8 266.60 11 H 40.00 " 6 180.00 1 8 283.30 11 2 53.33 11 61 195.00 1 9 300.00 " 2J 66.66 " y 210.00 ( 9J 316.60 M 3 80.00 " 7* 225.00 1 10 333.30 " 31 93.33 " 8 240.00 10} i 17.50 WEIGET OF A LINEAL FOOT, ETC. 233 Breadth in inches. Thickness in inches. Weight in Ibs. Breadth in inches. Thickness in inches. Weight in Ibs. Breadth in inches. Thickness in inches. Weight in Ibs. 10} 1 35.00 11 n 55.00 11} H 57.50 1 1} 52.50 " 2 2 73.33 2 76.66 4 $ 70.00 " 2} 91.56 44 2J 95.83 1 2} 87.50 11 3 110.00 (< 3 115.00 ! 3 105.00 11 3J 128.30 44 3} 134.10 f 3} 122.50 " 4 146.60 " 4 153.30 1 4 140.00 M 4J 165.00 " 4J 172.50 ! 4J 157.50 11 5 183.30 5 191.60 ( 5 175.00 M 51 201.60 " 51 210.80 1 5} 192.50 44 6 220.00 11 6 230.00 1 6 210.00 " 6J 238.30 < ^} 249.10 ! 6J 227.50 " 7 256.60 " 7 268.30 1 7 245.00 M 7} 275.00 71 287.50 1 7} 262.50 M 8 293.30 cJ :; 8 306.60 i 8 280.00 it 8J 311.60 " 8i 325.80 ! 8} 297.50 11 9 330.00 M 9 345.00 ! 9 315.00 44 9} 348.30 M 91 364.10 1 9J 332.50 " 10 366.60 " 10" 383.30 1 10 350.00 " 10} 385.00 M 10} 402.50 10} 367.50 11 11 403.30 M 11 421.60 11 i 18.33 11* J 19.16 " 11} 440.70 1 36.66 1 38.33 12 12 480.00 234 WEIGHT OF A LINEAL FOOT, ETC. WEIGHT OF A LINEAL FOOT OF ROLLED ROUND IRON IN POUNDS AVOIRDUPOIS. (480 pounds per cubic foot.) Diameter in inches. 1 Weight in Ibs. Diameter in inches. Weight in Ibs- Diameter in ineher. Weight in Ibs. Diameter in inches. Weight in Ibs. A 0.010 03 ^8 14.77 Bf 82.79 8J 206.2 4 0.041 2} 16.36 6| 86.52 9 212.2 A 0.091 2| 18.04 5{ 90.34 01 y f 218.0 I 0.163 2J 19.80 6 94.26 9V 223.9 iV 0.255 91 "8 21.64 6J 98.18 ^8 L 230.1 I 0.3 68 3 23.56 B| 102.20 9} 236.2 A 0.501 3J 25.56 8| 106.40 9| 242.5 i 0.655 31- 27.64 6.V 110 60 248.9 A 0.828 3| 29.82 61 114.90 ^1 255.2 I 1.022 3J 32.07 81 119.30 10 261.7 1.237 3| 34.39 6} 123 . 70 10} 268.4 I 1.473 3J 36.81 7 128.30 iQi 275.0 1 . 728 3| 39.30 n 132.90 10| 281.8 i 2.004 4 41.88 7} 137.60 10.} 288.6 2.301 45- 44.57 71 142.30 10| 295.6 i 2.618 4} 47.28 7-V 147.30 10| 302.5 if 3.310 4| 50.10 71 152.20 10^- 309.5 i} 4.094 4f 53.02 7-1 157 20 11 316.8 is 4.950 4l 56.03 n 162.40 114 323.9 ij 5.885 4J 59.05 8 167.50 lit 331.3 if 6.911 H 62.17 8i 172.80 HI 338.7 ij 8.018 5 65.49 S| 178.20 11 J 346.2 i| 9.205 5J 68.71 8J 183.60 n! 353.7 2 10.470 5} 72.13 a} 189.10 iif 3G1.5 2J 11.820 5| 75.65 81 194.80 ill 369.1 2} 13.250 5* 79.17 8f 200.40 12 376.9 BOLTS, NUTS, AND HEADS. 235 BOLTS, NUTS, AND HEADS. (Whitworth s Proportions.) Weight in Ibs. of Heads and Nuts. Diameter of bolt in in. Hexagonal. Square. Hexagonal. Square. Head. Nut. Head. Nut. Two Heads. Head &Nut. Two Heads. Head & Nut. i 0.008 0.005 0.022 0.019 0.017 0.013 0.044 0.041 A 0.014 0.007 0.027 0.021 0.029 0.022 0.055 0.048 I 0.029 0.017 0.061 0.049 0.057 0.046 0.122 0.110 A 0.059 0.040 0.069 0.050 0.119 0.101 0.138 0.119 I 0.068 0.041 0.104 0.076 0.136 0.109 0.208 0.181 0.104 0.065 0.157 0.118 0.208 0.169 0.315 0.276 0.151 0.097 0.246 0.193 0.302 0.248 0.493 0.440 0.254 0.161 0.362 0.269 0.508 0.415 0.724 0.631 0.367 0.219 0.551 0.408 0.734 0.586 1.102 0.959 i 0.546 0.326 0.683 0.463 1.092 0.872 1.366 1.146 i 0.724 411 1.109 0.797 1.448 1.135 2.217 1.906 i! 1.060 0.630 1.400 0.971 2.120 1.690 2.800 2.371 it 1.330 0.759 1.949 1.379 2.660 2.088 3.898 3.328 i 1.840 1.098 2.625 1.883 3.680 2.938 5.250 4.508 if 2.460 1.517 3.135 2.192 4.920 3.977 6.270 5.327 if 2.920 1.742 3.704 2.532 5.840 4.662 7.409 6.236 u 3.440 1.991 4.725 3.276 6.880 5.431 9.450 8.001 2 4.370 2.611 6.384 4.625 8.740 6.981 12.77 11.00 2J 6.150 3.645 8.858 6.353 12.30 9.795 17.71 15.21 2} 8.480 5.045 11.91 8.476 16.96 13.52 23.82 20.39 2J 11.32 6.747 15.59 9.019 22.64 18.06 31.18 24.61 3 14.72 8.783 21.00 15.06 29.44 23.50 42.00 36.06 1 236 WEIGHT IN POUNDS OF HOUND IRON, ETC. WEIGHT IN POUNDS OF ROUND IRON FOR Diameter in inches. Length in inches. N K % K % H y* i 2 3 i 0.002 0.003 0.005 0.007 0.008 0.010 0.012 0.014 0.027 0.041 A 0.003 0.005 0.008 0.011 0.013 0.016 0.019 0.021 0.043 0.064 1 0.004 0.007 0.011 0.015 0.019 0.023 0.027 0.031 0.062 0.093 A 0.005 0.010 0.016 0.021 0.026 0.031 0.036 0.042 0.084 0.126 i 0.007 0.014 0.021 0.027 0.034 0.041 0.048 0.055 0.110 0.166 0.009 0.017 0.026 0.035 0.043 0.052 0.061 0.069 0.139 0.208 0.011 0022 0.032 0.043 0.054 0.065 0.076 0.087 0.174 0.261 0.015 0.031 0.046 0.062 0.077 0.093 0.108 0.124 0.249 0.373 1 0.021 0.042 0.063 0.084 0.105 0.126 0.148 0.170 0.338 0.508 0.027 0.055 0083 0.110 0.138 0.165 0.193 0.221 0.442 0.663 H 0.035 0.070 0.105 0.140 0.185 0.210 0.245 0.280 0.560 0.840 4 0.043 0.087 0.131 0.173 0.217 0.262 0.304 0.347 0.695 1.043 it 0.053 0.104 0.157 0.209 0.261 0.314 0.366 0.418 0.836 1.255 if 0.062 0.124 0.186 0.249 0.311 0.373 0.435 0.497 0.995 1.493 it 0.072 0.143 0.215 0.287 0.358 0.430 0.502 0.584 1.168 1.752 if 0.084 0.168 0.253 0.337 0.421 0.506 0.590 0.677 1.354 2.032 if 0097 0.194 0.291 0.389 0.486 0.583 0.680 0.778 1.555 2.333 2 0.111 0.221 0.332 0.442 0.553 0.663 0.774 0.884 1.770 2.654 21 0.140 0.280 0.420 0.560 0.700 0.840 0.980 1.120 2.240 3.360 2} 0.174 0.347 0.521 ; 695 0.869 1.042 1.216 1.390 2.781 4.172 2| 0.209 0.418 0.627 0.836 1.045 1.254 1.463 1.673 3.346 5.019 3 0.250 0.500 0.750 1.000 1.250 1.500 1.750 1.990 3.981 5.972 EXAMPLE. Required, the weight of a bolt 1J inches diameter, 4 inches between inside of head and nut. Weight of bolt = 1.39 Weight of square head = 1.40 Weight of hexagonal nut = 1.06 taken as a hexagonal head Ans. 3.85 Ibs. WEIGHT IN POUNDS OF ROUND IKON, ETO. 237 BOLTS, ETC., BETWEEN HEAD AND NUT. Diameter 1 in inches. Length in inches. 4 5 6 - 7 8 9 10 11 12 i 0.055 0.069 0.082 0.096 0.110 0.124 0.137 0.151 0.165 A 0.086 0.107 0.128 0.150 0.171 0.192 0.214 0.235 0.257 1 0.124 0.155 0.186 0.217 0.248 0.279 0.311 0.342 0.373 A 0.167 0.209 0.251 0.293 0.335 0.377 0.419 0.461 o.5oa v 1 0.221 0.276 0.331 0.386 0.442 0.497 0.552 0.607 0.663 : 0277 0.347 0.416 0.486 0.555 0.624 0.694 0.763 0.833 0.347 0434 0.521 0.608 0.695 0.782 0.869 0.956 1.043 0.497 0.622 0.746 0.871 0.995 1.119 1.244 1.368 1.493 0.677 0.846 1.016 1.185 1.354 1.524 1.693 1.862 2.032 i 0.884 1.105 1.326 1.548 1.769 1.990 2.211 2.432 2.654 U 1.120 1.400 1.680 1.960 2.240 2.520 2.800 3.080 3.360 li 1.390 1.738 2.085 2.433 2.781 3.128 3.476 3.823 4.172 If 1.673 2.091 2.510 2.928 3.346 3.765 4.182 4.601 5.019 li 1.990 2.488 2.985 3.483 3.981 4.478 4.976 4.973 5.972 2.336 2.920 3.504 4.088 4.673 5.257 5.841 6.425 7.010 If 2.709 3.386 4.064 4.741 5.418 6.096 6.773 7.450 8.128 ll 3.111 3.888 4.666 5.334 6.221 6.999 7.777 8.547 9.333 2 3.538 4.423 5.307 6.192 7.077 7.961 8.846 9.730 10.610 2} 4.480 5.600 6.720 7.840 8.960 10.080 11.200 12.320 13.440 2J 5.562 6.953 8.343 9.734 11.120 12.510 13.910 15.290 16.690 2f 6.692 8.365 10.040 11.710 13.380 15.060 16.730 18.400 20.070 3 7.962 9.953 11.940 13.930 15.920 17.910 19.910 21.890 23.890 WEIGHT OF MATERIALS USED IN BUILDING. WEIGHT OF MATERIALS USED IN BUILDING. (Per square foot from one inch thickness to a cubic foot.) Stones, Earths, &c. 9 tt. Brick. 0) M pj <o fl i.3 9 cS g" averaj SH fi Jj of Parii 1 a 1 | 1 B.C C 73 * i 9 2 el 8s 1 o S o cT 1 1 d jj> e" <J S a 5 a 1 1 1 i 6.58 14.58 8.50 11.41 9.33 6.12 9.08 16.5 14.08 8.16 8.5 10.83 2 13.16 29.1C 17.00 22.83 18.66 12.25 18.16 33.0 28.16 16.33 17.0 21.66 3 19.74 43.74 25.50 34.24 28.00 18.36 27.24 49.5 42.25 24.50 25.5 32.49 4 26.32 58.32 34.00 45.66 37.33 24.50 36.33 66.0 56.32 32.66 34.0 43.33 6 32.90 72.90 42.50 57.08 46.66 50.61 45.41 82.5 70.40 40.83 42.5 54.16 6 39.48 87.48 51.00 68.50 56.00 J6.74 54.50 99.Q 84.48 49.00 51.0 65.00 7 46.06 102.00 59.50 80.00 65.33 42.86 63.60 115.5 98.56 57.16 59.5 75.83 8 52.64 116.64 68.00 91.32 74.66 49.00 72.66 132.0 112.64 65.32 68.0 86.66 9 59.22 131.22 76.50 102.75 84.00 55.10 81.75 148.5 126.72 72.50 76.5 97.50 10 65.80 145.80 85.00 114.16 93.33 61.23 90.83 165.0 140.80 81.66 85.0 108.33 11 72.38 160.38 93.50 125.60 102.6(5 67.35 99.13 181.5 154.90 89.82 93.5 119.16 12 79.00 175.00 102.00 137.00 112.00 73.50 109.00 198.0 169.00 98.00 102.0 130.00 Stones, Earths, &c. -^ i Granite. fl a o 7j S > a g S: o C3 <n . 11 a> o tJ a bO 5 1 I? 1 a o d 1 1 73 si OS 73 & 0? I 5* S 3 1 s | H 1 53 5 O 1 1 1 s 02 3 1 6.75 11.16 10.0 12.91 10.41 11.41 13.75 8.66 12.25 13.75 14.08 5.21 2 13.50 22.33 20.0 25.82 20.83 22.83 2750 17.33 24.50 27.50 28.16 10.42 3 20.25 33.50 30.0 38.73 31.25 34.25 41.25 26.00 36.75 41.25 42.24 15.62 4 27.00 44.66 40.0 51.64 41.66 45.66 55.00 34.66 49.00 55.00 56.32 20.83 5 33.75 55.83 50.0 6455 52.08 5708 6875 4333 61.25 68.75 70.40 26.04 6 40.50 67.00 60.0 77.46 64.50 68.50 82.50 52.00 73.50 82.50 84.48 31.24 7 47.25 78.16 7Q.O 90.37 73.00 80.00 96.25 60.66 85.75 96.25 98.56 36.45 8 54.00 89.33 800 103.28 83.32 91.32 110.00 69.22 98.00 110.00 112.64 41.66 9 60.75 100.50 90.0 116.19 93.75 102.75 123.75 80.00 110.25 123.75 126.72 4687 10 67.50 111.66 100.0 129.10 104.16 114.16 137.50 86.66 12250 13750 140.80 52.08 11 74.25 122.83 110.0 142.01 114.57 125 57 150.25 95.32 134.75 150.25 154.88 57.28 12 81.00 134.00 120.0 155.00 125.00 137.00 165.00 104.00 147.00 165.00 169.00 62.50 DIVISIONS OF A FOOT, ETC. 239 w w Q 52; CDl^-r^OOOOr-ifMCOCOLOiOCDt^oOO^ co co o :N 10 *- as i co LO r- o^ r-i co o i T-H ^ rr< Jt. <N;t>; <N !> CO OO CO CO <Q OS rt* ai -^ r OOGOOOGOCOOOGOGOGOOOOOCOOOO^aiOi CO " CO i i C cococDCDcocoi t r- GO OC Ot) Oi O O i-H T i C^l CO CO r}H TH tO IO CD lOiOlOOCDCDCDCDCDCDCDCDCDCDCDCD (M^CDoOOCVJ^CDoOOCSIiOt-OirH lOOlOOCDr-- fCDr ICD(MI^CVJ1>.C\IOO - COOiOOr-tCvlCOTjHKDCD CO IO 1> O5 i iCOlOtOCvjTtiCDOOOC^lTtH COOOCOOOrt<a5TtiailOOlOOiOi iCDrH COCO *"T^lOiOCDCDt^OOOOaiO^OOr-l COCOCOCOCOCOCOCOCOCOCOCOCO-^TjHrtl gOto Ocor- CDr-icDC^r^eqt-c CDCDJ^J>-OOOOO^CiOOT (r (C <M<N(MC^l(M(M(MCq(MCMcOCOCOCOC CDi li^<Mr-<Mt COCOCOOOCOOi ^Ci rJH CDi>-t OOOOOiOiOOr- IT ((MCMCOCO^i T _, T _ HrHr _, T _( T _i^_ i<M(M(M(M(M<M<MC^c<j r D^CS^Ci iOOT .i-HC Ot IT I <M CO ^}H lO CD CO 1>- COO^OrHClCO < 240 TABLE FOR COMPARING MEASURES AND WEIGHTS. TABLE FOR COMPARING MEASURES AND WEIGHTS OF DIFFERENT COUNTRIES. Weights. UNITED STATES AND ENGLAND. PRUSSIA. AUSTRIA. BADEN AND SWITZERLAND. FRANCE. Pound. Pound, Z. V. Pound. Pound. Kilogra e. 1 0.9072 0.8100 0.4536 1 . 1023 1 0.8928 Same as 0.5000 1.2346 1 . 1200 1 Prussia. 0.5600 1.2346 1.1200 0.9999 0.5600 2.2046 2.0000 1.7857 1 Measures of Length. Foot. Foot. Foot, Foot, Meter. = 12 inches. = 12 inches. = 12 inches. ==10 inches. = 100 Centi. 1 0.9711 0.9642 1..0160 0.3048 1.0297 1 0.9929 1.0462 0.3138 1.0371 1.0072 1 1.0537 0.3161 0.9843 0.9559 0.9490 1 0.3000 3.2809 3.1862 3.1635 3.3333 1 Measures of Surface Square Measure. Square foot. Square foot. Square foot. Square foot. Sq. Meter. 1 0.9431 0.9297 1.0322 0.0929 1.0603 1 0.9858 1.0945 0.0985 1.0756 1.0144 1 1.1103 0.0999 0.9688 0.9137 0.9007 1 0.0900 10.7643 10.1519 10.0074 11.1111 1 TABLE FOB COMPARING .MEASURES AND WEIGHTS. Cubic Measure. UNITED STATES AND ENGLAND. PEUSSIA. AUSTRIA. BADEN AND SWITZERLAND. FRANCE. Cubic foot. Cubic foot. Cubic foot. Cubic foot. Cubic meter 1 0.9159 0.8964 1.0487 0.0283 1.0918 1 0.9787 1 . 1450 0.0309 1.1156 1.0217 1 1.1699 0.0316 0.9535 0.8733 0.8548 1 0.0270 35.3166 32.3459 31.6578 37.0370 1 Weight per Unit of Length. Lbs. per lineal foot. Lbs. per lineal foot. Lbs. per lineal foot. Lbs. per lineal foot. Kil. per lineal meter 1 0.9342 0.8400 0.8929 1.4882 1.0705 1 0.8993 0.9559 1.5931 1.1904 1.1120 1 1.0629 1.7716 1.1199 1.0462 1.9408 1 1.6667 0.6720 0.6277 0.5645 0.6000 1 Weight per Unit of Surface. Lbs. per square inch. Lba. per square inch. Lbs. per square inch. Lbs. per square inch. Kil. per square cent. 1 0.9619 0.8712 1 . 2656 0.0703 1.0396 1 0.9057 1.3157 0.0731 1.1478 1.1041 1 1.4526 0.0807 0.7902 0.7601 0.6884 1 0.0556 14.2223 13.6811 12.3910 18.0000 1 16 242 RESISTANCE TO CROSS-BKEAKING. RESISTANCE TO CROSS-BREAKING. To Cut the Strongest and Stiffest Rectangular Beam from a Log, Fig. 308. (Strongest.) The diameter aa = d, divided into three equal parts, with per* pendiculars J d from a erected thereon, intersecting the circle at b, will give section for greatest capacity. Fig. 309. (Stiffeet.) The diameter aa = d, divided into four equal parts, with per- rndiculars J d from a erected thereon, intersecting the circle at , will give section with least deflection, but less capacity than Fig. 308. INDEX. Area, circumference, and cubic contents of circles 218 Axis, neutral 4 Bars, tie rods, &c 181 resistance of, to tearing 2 Beams, capacity and strength of 29 of rolled 39 of cast-iron 57 TFof rolled l-shaped 39 and strength of parabolic arched 153 cast-iron 53 iron ties, struts, and 3 sloping rafters and 102 strains in trussed 122 horizontal andsloping 188 strength of wooden 88 Bolts and nuts, dimensions of. 187 nuts, and heads 235 Boom derricks, strains in 114 Booms, strains in trusses with parallel 126 Bow-string girders 147 Bridges, static and moving loads, of wrought iron 192 Camber 2 Capacity 2 and strength of beams 29 W of rolled l-shaped beams 39 of rolled beams 41 of cast-iron beams 57 and strength of parabolic arched beams..... 153 Cast-iron beams 3, 53 Center of gravity of planes 202 Circumference, area, and cubic contents of circles 218 Columns, pillars, and struts, strength of 110 Composition and resolution of forces Ill Compound strains in horizontal and sloping beams 188 Compression 1 Compressive strain and pressure on supports 102 Contraction and expansion 4 (243) 244 INDEX. MOB, Constants for strain in trusses 117 roof trusses 174 Connections in iron construction, joints or 184 Cross-breaking 2 and shearing, resistance to 29 Crushing, resistance to 103 direct 1 Deflection 2 Derricks, strains in boom 114 Dimensions of bolts 187 Divisions of a foot, expressed in equivalent decimals 239 Expansion and contraction 4 External forces .T^J/i Factors of safety 29 Forces external ., ..r 1 % internal 1 composition and resolution of. Ill parallelogram of Ill Frame, strains in polygonal 154 Functions, trigonometrical 207 Geometry 197 Girders, strains in parabolic and bow-string 147 Gravities of materials, specific 224 Heads, nuts, and bolts 235 Horizontal and sloping beams, compound strains in 188 Howe truss 129 Inertia and resistance o various sections, moments of 5 Internal forces 1 Iron beams, capacity of cast 57 cast 53 bridges, static and moving loads, of wrought 192 construction, joints or connections in .-. 184 ties, struts, or beams 3 Joints or connections in iron construction 184 Lattice truss 139 with vertical members 131 Longimetry and planimetry 197 Materials, &c., strength of 26 Miscellaneous , 195 IffDEX. 245 Modulus of rupture 4 Moment of inertia and resistance of various sections 5 Moving loads, weight of. 191 Natural sine, cosine, &c 306 Neutral axis 4 Nuts, heads, and bolts 235 dimensions of... 187 Parallelogram of forces..... Ill Parallel booms, strains in trusses with 126 Parabolic arched beams, capacity and strength of. 153 curved trusses, strains in 147 Planimetry, longimetry, &c 197 Pillars, columns, and struts, strength of 110 Pins, &c., in tie bars 185 Polygonal frame, strains in ,. 154 Pressure on supports 100 compressive strain and 102 of snow on roofs 178 of wind on roofs 180 Rafters, &c., sloping beams _. 102 Reactions of supports 100 Resistance to direct crushing 1 of bars, &c., to tearing 2 to cross-breaking and shearing 29 crushing 103 Resolution of forces, composition, &c Ill Rolled beams, capacity of. 41 l-shaped beams, capacity of. 39 Rods and bars, tie 181 Roof trusses..., 3 strains in 156 constants for strains in 174 Roofs, pressure of wind on 178 of snow on 180 Rupture, modulus of. 4 Shearing 2 and cross-breaking, resistance to 29 Sloping beams, rafters, &c 102 and horizontal beams, compound strains in 188 Specific gravities of materials 224 Static and moving loads of wrought-iron bridges 192 Strength of materials 26 wooden beams 98 columns, pillars, and struts 110 246 INDEX. PAGE. Strength of beams, capacity, &c 29 Strains in frames 112 boom derricks 114 trusses 115 trussed beams 122 trusses with parallel booms 126 parabolic curved trusses, or bow-string girders.... 147 polygonal frame 154 roof trusses 156 constants for 174 trusses, constants for 117 Strongest and stiffest rectangular beam from a log, to cut the.. 242 Struts and beams, iron ties 3 Supports, reaction of. ... 100 compressive strain and pressure on 1 02 Table for comparing measures and weights 240 Tearing, resistance of bars, &c., to 2 Tension 1 Tie rods and bars 181 Trigonometrical functions 207 formulas 205 Truss, Howe 129 Warren 132 Whipple 144 lattice 139 with vertical members 131 Trusses parallel booms, strains in 126 parabolic curved, or bow-string 147 constants for strains in roof 1 74 constants for strains in 117 strains in 115 roof 156 Trussed beams, strains in 122 Warren truss , 132 Weight of moving loads 191 static and moving loads of wrought-iron bridges... 192 a lineal foot of flat or square bar iron 229 rolled round iron 234 materials used in building 238 superficial inch of wrought and cast iron 227 rolled round iron for bolts 236 heads and nuts 235 per square foot of metals 228 Whipple truss 144 Wooden beams, strength of. . 98 1 388 CO O O oc O co CM