IRLF LIBRARY UNIVERSITY OF CALIFORNIA. . ,8 9 3~~~ B tcsions ^o^ r~lass No. COMPOUND RIVETED GIRDERS, AS APPLIED IN THE CONSTRUCTION OF BUILDINGS. WITH NUMEROUS PRACTICAL ILLUSTRATIONS AND TABLES. BY WILLIAM H. BIRKMIRE, tt AUTHOR OF " ARCHITECTURAL IRON AND STEEL " ANI> " SKELETON CONSTRUCTION IN BUILDINGS." NEW YORK: JOHN WILEY & SONS, 53 EAST TENTH STREET. 1893- COPYRIGHT, 1893, BY WILLIAM H. BIRKMIRE. PREFACE. IN order to facilitate the calculation attending the construc- tion of Wrought Iron and Steel Riveted Girders, the author has endeavored in this work to supply the link which separates Theory from Practice. Its object may be briefly stated. A riveted girder is to be designed ; the span, depth, and loads are known, the strains are calculated b$ the well-known bending- moment formulae, and largely by the graphic method ; lastly, the details of construction are fully illustrated. Touching the question of accuracy, it is scarcely necessary to notice the slight difference that may arise between the two methods, i.e., working out the usual formulae, or by measuring from the graphic diagrams. The time consumed in wading through a complicated series of equations to reach a few meas- urements is objectionable when at least such measurements can at once be had by the graphic method. This work does not investigate exceptional or extremely scientific riveted girders, but more especially those of a type now extensively adopted and constructed by well-known archi- tectural iron workers. The diagrams and the various examples explaining the Author's method are submitted to architects and architectural students with the hope that they will become a medium of use- fulness to them in the routine of office work. WILLIAM H. BIRKMIRE. NEW YORK, 1893. TABLE OF CONTENTS. PART I. STRAINS IN COMPOUND RIVETED GIRDERS. PAGE Compound riveted girders described I Bending moments 2 Flanges 4 Shearing forces on the webs 5 Buckling of webs 7 Stiffeners 8 Riveting 8 Frict on of plates , . 10 Proportioning rivets n Rivets connecting webs with flanges 12 Spacing rivets according to strain produced in the flanges by the bending moments , 15 Proportioning girders 16 Shearing and bearing resistance of rivets (Table) 16 Details of construction 17 Extract from the New York Building Law in relation to riveted girders 18 To calculate the approximate weight of girder before its dimensions are fixed IQ Splicing 20 PART II. QUALITY OF MA TERIAL. Wrought-iron 21 Limit of elasticity of wrought-iron 21 Ultimate strength of wrought-iron 21 Rivet iron 21 Mild steel 21 v VI TABLE OF CONTENTS. PACK Ultimate strength and elongation 22 Rivet steel 22 Painting 22 PART III. EXAMPLE /. Girder supporting a concentrated load at centre of span. 23 Construction of flanges in a girder supporting a concentrated load at centre 25 Flanges reduced in area towards the supports in a girder supporting a con- centrated load at centre 26 Webs proportioned in a girder supporting a concentrated load at centre of span 28 Stiff eners in a girder supporting a concentrated load at centre of span 2& Rivet spacing in a girder supporting a concentrated load at centre of span. 29, Graphical representation of bending moments and shearing forces in a girder with a concentrated load at centre of span 30 List of material and details of a girder supporting a concentrated load at centre 32 Areas of angles with even legs (Table) 33 " " " uneven legs (Table) 33 Sectional area in inches of rivet-holes in plates of various thicknesses (Table) 34 Gross area of plates of various thicknesses (Table) 35 Safe buckling value of web plates in wrought-iron (Table) 35 Shearing value of wrought-iron web plates (Table) 36 " " " steel web plates (Table) 37 EXAMPLE II. Girder supporting one concentrated load not at centre of span 3& Construction of flanges in a girder supporting one concentrated load not at centre 4<> Flanges reduced in area in a girder supporting one concentrated load not at centre 41 Webs proportioned in a girder supporting a concentrated load not at centre 42 Stiffeners in a girder with one concentrated load not at centre 43 Spacing of rivets in a girder with one concentrated load not at centre 43 Graphical representation of bending moments and shearing forces in a girder with one concentrated load not at centre of span 44 List of material and details of a girder supporting one concentrated load not at centre of span 4& TABLE QF CONTENTS. VU EXAMPLE III. PAGE Girder supporting a uniformly distributed load 47 Construction of flanges in a girder supporting a uniformly distributed load. 49 Flanges reduced in area towards the supports of a girder supporting a uni- formly distributed load ., 49 Webs proportioned in a girder supporting a uniformly distributed load. ... 50 Stiffeners in a girder supporting a uniformly distributed load 51 Spacing of rivets in a girder supporting a uniformly distributed load 51 Method of drawing parabolas 52 Parabola by the construction of a diagram 53 Graphical representation of bending moments and shearing forces in a girder supporting a uniformly distributed load 54 List of material and details 'of a girder supporting a uniformly distributed load 55 EXAMPLE IV. Girder supporting two concentrated loads 56 Construction of flanges in a girder supporting two concentrated loads 58 Flanges reduced in area towards the supports in a girder supporting two concentrated loads 59 Webs proportioned in a girder supporting two concentrated loads 60 Stiffeners in a girder supporting two concentrated loads 61 Spacing of rivets in a girder supporting two concentrated loads 61 Graphical representation of bending moments and shearing forces in a girder supporting two concentrated loads 62 List of material and details of a girder supporting two concentrated loads. 64 EXAMPLE V. Girder supporting two concentrated loads and a uniformly distributed load 65 Construction of flanges in a girder supporting two concentrated loads and a uniformly distributed load 67 Webs proportioned in a girder supporting two concentrated loads and a uniformly distributed load 68 Stiffeners in a girder supporting two concentrated loads and a uniformly distributed load 69 Spacing of rivets in a girder supporting two concentrated loads and a uni- formly distributed load .69 Graphical representation of bending moments and shearing forces in a girder supporting two concentrated loads and a uniformly distributed load , 70 Flanges reduced in area towards the supports in a girder supporting two concentrated loads and a uniformly distributed load by the funicular polygon 7 1 List of material and details of a girder supporting two concentrated loads and a uniformly distributed load 73 TABLE OF CONTENTS. EXAMPLE VI. PAGE Girder supporting three concentrated loads 74 Construction of flanges in a girder supporting three concentrated loads. . . 76 Webs proportioned " " " " " " 77 Stiffened " " " " " "... 78 Spacing of rivets " " " " " "... 78 Flanges reduced in area towards the supports in a girder supporting three concentrated loads 79 Graphical description of bending moments and shearing forces in a girder supporting three concentrated loads 80 List of material and details of a girder supporting three concentrated loads 82 EXAMPLE VII. Girder supporting four concentrated loads 83 Construction of flanges in a girder supporting foiir concentrated loads 85 Webs proportioned " " 85 Stiffeners " " " " " 86 Spacing of rivets " " " " " " 86 Flanges reduced in area towards the supports in a girder supporting four concentrated loads 87 Graphical representation of the bending moments and shearing forces in a girder supporting four concentrated loads 88 List of material and details of a girder supporting four concentrated loads. 90 EXAMPLE VIII. Steel girder supporting five concentrated loads 91 Determination of bending moments in a girder supporting five concentrated loads 93 Construction of flanges in a girder supporting five concentrated loads 94 Stiffeners " 95 Spacing of rivets " 95 Flanges reduced in area towards the supports in a girder supporting five concentrated loads 96 Girder fixed at one end and loaded with a concentrated load at the other, as a cantilever 97 Girder fixed at one end supporting a uniformly distributed load, as a canti- lever 98 Girder fixed at one end supporting more than one load, as a cantilever. ... 99 The relative strength of simple and cantilever girders; maximum vertical shear, bending moments, and deflection (Table) 99 Modulus of elasticity of wrought-iron and steel in riveted girder as com- pared with solid sections, as I-beams 99 Moment of Inertia for Rectangular sections loo TABLE OF CONTENTS. IX PART IV. TABLES. PAGE Average weight in pounds of a cubic foot of various substances 101 Weight of loo rivets in pounds 104 Decimal equivalents for fractions of a foot 105 Number of U. S. gallons contained in circular tanks . : 106 Decimal equivalents for fractions of an inch 106 Weight per lineal foot of cast-iron columns 107 Weight of square cast-iron columns per lineal foot 108 Weight per foot of flat iron 109 Table of squares and cubes. ... in Table of circles 115 Shearing and bearing resistance of rivets 16 Areas of angles with even legs 33 ' ' uneven legs f 33 Sectional area in inches of rivet-holes in plates of various thicknesses 34 Gross area of plates of various thicknesses 35 Safe buckling value of web plates (wrought-iron) 35 Shearing value of wrought-iron web plates 36 " " " steel web plates 37 LIST OF ILLUSTRATIONS. PART I. FIG. PAGE 1. Plate-girder section I 2 . Plate-girder section with single flange plate I 3. Plate-girder section with three flange plates I 4. Box-girder section with three flange plates I 5. Box girder-section with three webs I 6. Girder with two loads supported upon a fulcrum 3 7. A simple girder supported at each end and load in middle 3 8. A lever held up with a weight at either end 5 9. A simple girder with load out of centre 6 10. A simple girder with a specified load out of centre 6 1 1. Two plates riveted with rivets in single shear 12 12. Plate-girder section with rivets in double shear 12 i2a. Box-girder section with rivets in single shear 12 13. Girder illustrating the strains on rivets connecting flange with web. . . 12 PART III. 14. Diagram of a girder with one concentrated load at centre 23 15. Diagram determining position of flange plates in a girder of one con- centrated load at centre 26 16. Section of plate girder with stiffeners bent around chord angles 29 17. Section of plate girder with straight stiffeners and fillers 29 18. Diagram of the graphical representation of bending moments and shearing forces of a plate girder with one concentrated load at centre 31 19. Detail of girder of one concentrated load at centre 32 20. Diagram of a girder with one concentrated load not at centre 38 21. Diagram determining position of flange plates in a girder of one con- centrated load not at centre 41 22. Diagram of the graphical representation of bending moments and shearing forces in a girder with one concentrated load at centre. . 45 23. Detail of girder of one concentrated load not at centre 46 24. Diagram of a girder with a uniformly distributed load 47 25. Diagram determining position of flange plates in a girder supporting a uniformly distributed load 50 xi Xii LIST OF ILLUSTRATIONS. FIG. 26. Diagram of a parabola, by ordinates from a tangent to a parabola at its vertex * 27. Diagram of a parabola, by lines to two sides of an isosceles triangle.. 53 28. Diagram of the graphical representation of bending moments and shearing forces in a girder supporting a uniformly distributed load 54 29. Detail of girder supporting a uniformly distributed load 55 30. Diagram of a girder supporting two concentrated loads 56 31. Diagram determining position of flange plate in a girder supporting two concentrated loads 59 32. Diagram of the graphical representation of bending moments and shearing forces in a girder supporting two concentrated loads 62 33. Detail of girder supporting two concentrated loads 64 34. Diagram of a girder supporting two concentrated loads and one uni- formly distributed load 66 35. Diagram of the graphical representation of bending moments and shearing forces in a girder supporting two concentrated loads and one uniformly distributed load 70 36. Diagram determining position of flanged plates in a girder supporting two concentrated loads and a uniformly distributed load 72 37. Detail girder supporting two concentrated loads and a uniformly dis- tributee? load 73 38. Diagram of a girder supporting three concentrated loads 75 39. Diagram determining position of flange plates in a girder supporting three concentrated loads 79 40. Diagram of the graphical representation of bending moments and shearing forces in a girder supporting three concentrated loads. . . 80 41. Detail of girder supporting three concentrated loads 82 42. Diagram of a girder supporting four concentrated loads 84 43. Diagram determining position of flange plates in a girder of four con- centrated loads 88 44. Diagram of the graphical representation of bending moments and shearing forces in a girder supporting four concentrated loads. ... 89 45. Detail of girder supporting four concentrated loads 90 46. Diagram of a girder supporting five concentrated loads 92 47. Diagram determining position of flange plates in a girder supporting five concentrated loads 96 48. Diagram of a girder secured at one end (as a cantilever) and supporting a concentrated load at the other .. 98 49. Section of a plate girder, determining the notation used in the calcula- tion of the section for the moment of inertia 100 50. Section of a box girder, determining the notation used in the calcula- tion of the section for the moment of inertia 100 PART I. THE STRAIN IN COMPOUND RIVETED GIRDERS. PART I. STRAINS IN COMPOUND RIVETED GIRDERS. For buildings, as well as railway and highway bridges, there is probably no other form of girders more extensively used than those made up of plates and angles, called Compound Riveted Girders. Some of the principal reasons for this lies mainly in the simplicity of their construction ; they can be adopted for any load or number of loads, and accommodated to any span usually met with in the construction. The single web or plate girder is more economical, more accessible for painting and inspection. Formed of a single web and four angles, as Fig. I, suitable for light loads and short spans ; for heavier loads a single plate is added to the top and bottom flanges, as shown in Fig. 2 ; for still heavier loads additional plates, as in Fig. 3. nr FIG. i. FIG. 2. FIG. 3. FIG. 4. FIG. 5. Where thick walls are to be supported and lateral stiffness is required, the double web or box girder, Fig. 4, or the triple 2 COMPOUND RIVETED GIRDERS. web, Fig. 5, is employed. It also becomes necessary in many cases to place two plate or two box girders side by side. The box girders as represented in'section, Fig. 4, is consid- ered superior to the plate girders represented in Figs. I, 2, and 3 ; but the preference should be given the latter on account of its simplicity of construction, and although inferior in strength to the box girder it has nevertheless other valuable properties to recommend it. On'comparing the strengths of these separate girders, weight for weight, it will be found that the box girder is as I to .93, or nearly as 100 to 90. The difference in strength does not arise from want of proportion in the top and bottom section of either girder, but from the position of the material ; which in that of the box girder offers greatly superior powers of re- sistance to lateral flexure. The box girder, it will be observed, contains larger exterior sectional area, and is consequently stiffer and better calculated to resist lateral stress, in which direction the plate girder generally yields before its other re- sisting powers of tension and compression can be brought fully into action. Taking this girder, however, in a position similar to that in which it is used in supporting floor-beams and floor- arches of buildings, its strength is very nearly equal to that of the box shape, and, as previously mentioned, is of more simple construction, less expensive, and more durable, from the cir- cumstance that the web-plate is thicker than the web-plates of the box girder, and it admits of easy access to all its parts for purposes of painting, etc. Bending Moments. Generally, the strength of a com- pound riveted girder is founded on the equality that must always exist between the resultant of the various loads tending to cause its rupture and the strength of the material of which the girder is composed. The former may be resolved horizon- tally into strains, depending for their value upon what are known as moments of rupture, bending moments, or leverage, of THE STRAINS IN COMPOUND RIVETED GIRDERS. 3 greater or less complexity, tending to cause the failure of the girder by tearing asunder its fibres in the bottom flange, crush- ing them together in the top flange, and vertically upon the web into what are known as shearing forces, due to the trans- mission of the vertical pressure of the loads to the points of support. The strain produced in the flanges is resisted by a leverage equal to the depth of the girder, that is, between the centre of gravity of the flanges, and the amount per square inch of sec- tion with which the metal may be safely trusted. O w O FIG. 6. The bending moment is a compound quantity resulting from the multiplication of a force by a distance, and desig- nated by the letter M. The forces are expressed in tons or pounds, and the distances in feet or inches ; then the bending moments are in ton-feet or pound-inches. FIG. 7. If b or a is the arm of leverage, Fig. 6, and a load R or R' acts for a distance from W, M at W is equal to the load R or R' multiplied by the distance b or a. Then M "= Rb or Ra. The direction of the stresses upon the girder are vertical, those at the ends being downwards, while that at the middle is 4 COMPOUND RIVETED GIRDERS. upwards. In Fig. 7 we have a girder supported at both ends,, and a load W resting upon the middle of its length. Compar- ing this with Fig. 6, we see that the stresses here are also verti- cal,* but in reversed order, the one at the middle being downwards, while those at the end are upwards. In other re- spects we have the same conditions as in Fig. 6. M does not represent strain, being independent of depth, but is converted into flange strain by dividing by the depth ; the strain then found, divided by the maximum unit strain, determines the number of square inches to be given to the flanges. The maximum unit strain herein adopted is 6 tons (12,000 pounds) per square inch for wrought-iron and 7 tons (14,000- pounds) for steel. Here A = Area of flange d depth in feet, s = unit strain in tons, and M = bending moment. Then A- M ~Ti Flanges. Compound girders are unlike rolled beams, in which every fibre is connected ; but have strains transmitted only through rivets which are distributed only at certain dis- tances apart ; consequently the flange angles are at every point more or .less subjected to strains in addition to their own. This additional strain will evidently increase with the amount of plates. It i9 good practice, therefore, to make the girder so deep that the flanges do not require a number of plates to be packed one upon another, and then to choose angles as heavy as possibfe consistent with the total flange area required. * "We have to distinguish between the outer forces which may act at various portions of the girder tending to cause motion of its parts, and the inner forces which prevent this motion. The first we may call stresses, and the second strains. We therefore speak of the ' stresses ' upon a girder and the ' strains * in a girder." THE STRAINS IN COMPOUND RIVETED GIRDERS. 5 In order^to give the single-web girders the greatest amount of resistance, it is usual to use angles with unequal legs with, the longer leg horizontal. It sometimes becomes important to have the plates of the top flange extend from end to end, even when angles may be found which alone are sufficient to make up the required sec- tion, as it gives great lateral stiffness to the flange, and also helps to distribute the stress more uniformly than with. the angles alone. In box girders the flange plate adjoining the angles are required to extend from end to end. In making up the bottom flange, rivet-holes must be de- ducted to obtain the net section, and in so doing the diameter of the rivet-hole should be taken at least -J inch larger ; this latter provides to a certain extent for the damage done to the strength of the metal in the process of punching or drilling. For the top flange the gross sectional area may be taken as making up the same, providing the riveting is well done, i.e., the rivet completely filling up its own hole. Shearing Forces on the Webs. It is by the law of the lever that we are enabled to determine precisely what portion of a given load resting upon a girder is sustained by either point of support ; the loads balancing each other at either end of a girder, or lever, on any point are to each other invers as their distances (called lever arms) from the point or ful- crum. For example : suppose we have a girder held up as in Fig. 8, with a load at either end, the point of support being to 6 COMPOUND RIVETED GIRDERS. one side of the centre, say one-fourth of the lever arm from one end. In order that the lever be balanced, the load at W must be one-fourth the sum of Wand W, and that at W three- fourths that sum, for W multiplied by JZ, must always equal W multiplied by \L, and the sustaining force P must of course equal the sum of Wand W . Again : supposing that there is one load as in Fig. 9. This condition is the same as before, only reversed ; and, according to the law of the lever, we find that for equilibrium a force must be applied to R equal to J W. This example is precisely the same as that of a girder, only R and R' are now called reactions of the supports, the sum of which must always be equal to the load or number of loads causing them. In order, then, to know just how much of the load or num- ber of loads at any point of the girder is supported by either support, all that is necessary to be done is to multiply the shorter or longer distance by the load and divide the product by the span L. Example : Suppose we have a girder R and R' y Fig. 10, of 25 feet span, and there is a load of 20 tons 5 feet from R f . Then each sustains a certain amount of this load proportionately to THE STRAINS IN COMPOUND RIVETED GIRDERS. J its distance from the load, the sum of the reactions being equal to the load, 20 X 5 /L supports = 4 tons ; 20 X 20 R r " - = 1 6 tons. 25 The most practical way of proportioning the web is then to make its section sufficient to resist this entire shearing force at either end of the girder. Under the supposition that the flanges alone .resist the entire bending moment, and the web only the shearing action, the following formula can be adopted : Let S = shearing stress ; A area at point of stress ; K = effective resistance to shearing ; / thickness of web ; d = depth of web in inches. c A td and 5 = ktd, or t . dK The safe shear on the webs per square inch herein adopted Is 6000 Ibs. for wrought-iron and 7000 Ibs. for steel. Example. Suppose we take the above wrought-iron plate girder, Fig. 10, and have 16 tons (32,000 Ibs.) shear on the web at R' support, the web being 12 inches in depth. 5 32000 / . = -, = .44 more than yV of an inch in dk 12 X 6000 thickness. Buckling of Web. The web is still in danger of buckling under this compression stress ; consequently the web with its thickness as already proportioned for shearing must now be examined for its strength as a column. The depth being the vertical distance between the upper and lower rows of rivets in 8 COMPOUND RIVETED GIRDERS, the web (but to facilitate the calculation the height will be taken the full depth of web). This condition is attained when the shear per square inch of cross-section at any point does not exceed The safe resistance to buckling per square inch = - when d and / are the depth and thickness of web in inches. Stiffeners. If the result obtained by the above formula is less than that adopted by our safe shear, vertical angles or Stiffeners are riveted each side of the web at intervals. They should always be used at the bearings and where concentrated loads occur. The spacing of Stiffeners is more a matter of experimental judgment than of mere calculation. Of several rules given by engineers, one is, " that Stiffeners in girders over three feet in depth shall be placed at distances apart (centre to centre) generally not exceeding the depth of the full web-plate, with the maximum limit of 5 feet." In girders under 3 feet in depth Stiffeners may be placed 3 feet apart, and in some special cases where there is little or no shearing, at greater distances. This refers, of course, to those parts of the girders where there are no concentrated loads. Another rule worthy of notice says " that when the least thickness of web is less than ^ the depth of the girder, the web shall be stiffened at intervals not over twice the depth." Riveting. The rivets in girder work are generally the same size as those adopted for boiler work, i.e., f", f", and i" in diameter; but as girders do not require caulking like a boiler, the pitch or distance of rivets from centre to centre is much greater, and usually varies from 2^ diameters to 16 times the thickness of the outside plate joined. By diameters is under- stood the diameter of the shank. When the edges of the plate are often roughly shorn, the margin, or distance between the THE STRAINS IN COMPOUND RIVETED GIRDERS. 9 rivet-hole and edge of the plate, is seldom less than \\ times the diameter of the rivet. As the effect of punching is to weaken the plate some distance all round the punched hole, the above proportions should be adopted. Nearly all experimenters on the subject agree that punch- ing generally reduces the tenacity of iron and steel plates to a greater degree than the area of the metal punched out, and a close examination of the border of each hole shows that it has been subject to a certain degree of rupture, which in most cases has reduced the ductility of the metal and made it wholly crystalline in fracture, and as some may suppose caused cracks round the edge of the hole ; but this latter seems doubtful, as Mr. Gerhard (see Experiments in Stoney's Riveted Joints) instituted an investigation as to whether there was any founda- tion for the very generally received opinion that the edges of a punched hole on the die side are injured by a ring of minute incipient cracks. For this purpose a large number of specimens 5 inches by 3 inches by inch of all kinds of steel were pre- pared. The edges were planed, the surfaces polished, holes were pierced in various ways, and the metal surrounding them was carefully examined with a microscope, but no trace what, ever of cracks was found, though the nature of the steel ranged from o.i to 0.6 per cent of carbon. Owing to its hard- ness and inability to stretch, this annulus of strained material round the punched holes, when the specimen is under tension, takes a higher proportion of the stress than the other more yielding parts, and hence it reaches the breaking point sooner, that is, the punched plate breaks in detail : first the annulus gives way, and then the more ductile portion between the holes. Reaming or boring out a zone of metal -J inch wide round the punched hole removes the annulus of strained ma- terial and neutralizes the effect of punching. In numerous experiments on the subject the loss of tenacity in iron plates from punching varies from 5 to 23 per cent of the original IO COMPOUND RIVETED GIRDERS. strength of the solid plate, but the percentage in any particular case will doubtless depend (ist) on the diameter of the holes; (2d) on the pitch ; (3d) on the width of the strip punched, for wide plates are apparently less injured than narrow strips ; (4th) on the condition of the punching tool i.e., the sharpness of its cutting edges and the maintenance of the proportion of size between the punch and the die ; (5th) on the quality and thickness of the metal, hard iron generally suffering more than ductile iron, and thin plates less than thick ones. Probably the most accurate method for making an allowance for the in- jurious effect of punching is to allow a certain percentage when calculating the effective net area of a punched plate, and, as heretofore mentioned under Flanges, page 5, \ of an inch more than the diameter of the rivet is adopted. Friction of Plates. Rivets contract in cooling and draw the plates together with such force that the friction produced between their surfaces is generally sufficient to prevent them from sliding over each other so long as the stress lies within limits which are not exceeded in ordinary practice. The friction of plates is an important factor in boiler work ; and as it is usual, to test them hydraulically, to double their working pressure, the joints are so designed that this water test, as well as the expansion and contraction due to changes in temperature, will not cause the joints to slip. Though the friction of riveted plates may be sufficient to convey the normal working load without subjecting the rivets to a shearing stress, it does not follow, nor do experiments indicate, that the ultimate strength of a riveted joint is increased by this friction. When several plates are riveted together with numerous rivets, as in the piled flanges of a girder, the slipping of plates does not seem to have occurred in Mr. Baker's experiments, for with two wrought-iron girders with 5 and 8 plates, respectively,, in their flanges, each 20 feet span and 2 feet in depth, which he tested to failure, there was no movement in the flanges, and THE STRAINS IN COMPOUND RIVETED GIRDERS. II the pile of plates behaved almost like a welded mass of iron, and Mr. Baker states " that he invariably found that badly punched girders, with the holes partly blind and the rivets tight but not filling the holes, deflected neither more nor less than the most accurately drilled work." Whatever value may justly be attached to the above, an inspection of the riveted joint when being tested to destruction dispels all idea of the ultimate stress being in any degree affected by it ; for when the stress is considerable, the joints open at each end of the plates, and the higher the stress the greater the amount of opening is observed. Under such condi- tions it is not customary to take into consideration the friction of the joints. Proportioning Rivets. Rivets, as used for girders, must be proportioned to resist shearing, and the area of their bearing must be such that the metal against which they bear shall not be crushed. The stresses allowed on these members are : shear- ing* 75 to 9000 pounds, and crushing, 15,000 pounds per square inch. The shearing strain is measured on the area of the cross- section of the rivet ; the crushing, on the area obtained by the product of the diameter of the rivet by the thickness of the web or plate upon which it bears. To illustrate the shearing and bearing area of a rivet, we take for example two plates of wrought-iron 8 inches wide by J inch thick, which overlap each other for a joint, with 45,000 pounds strain on the plates ; what number of rivets will be re- quired to resist the strain on the joint ? The area of a rivet of an inch in diameter is 0.4417 square inches; this multiplied by 7500 pounds, the safe shearing, = 3312.75 pounds, the safe amount of strain each rivet can sus- tain without shearing; dividing 45,000 by this, we get 13.6, say 14 rivets, as shown in single shear, Fig. n. If constructed as shown in Fig. 12, as in the flange of a plate girder, the rivets 12 COMPOUND RIVETED GIRDERS. would be in double shear and have twice the value ; then 7 rivets would be sufficient. i o o o o 10000 o o o o FIG. IT. FIG. FIG. i2A. In box girders, as Fig. I2a, the rivets connecting the angles with the webs are in single shear. The bearing area of each rivet is f inch by inch = f square inch; this multiplied by 15,000 pounds for crushing would equal 5625 pounds. Dividing 45,000 by this we obtain 8 rivets. This latter calculation should not be overlooked in riveted work. Its observance in most cases of riveted girders with single webs gives the size and number of rivets to be used, and in thin webs the bearing area may be small, necessitating a thicker web than would otherwise be required. Rivets Connecting Web with Flanges. The strain which the rivets connecting the web and flanges sustain is evidently due to the strain which is transmitted from one to the other ; this strain is horizontal, and is the maximum increment of flange strain at every section of the girder, and is found by the maximum shear at any point by the height of the For 'example: suppose a girder of 24 feet span (Fig. 13), 3 feet in depth, sustains a load of 150,000 pounds uniformly dis- tributed over its whole length. Tak- half the load over half the girder, at i 5 Jp ^e support R' the shearin r strain is ~"*F 75,000 pounds or half the w ,ole load ; FlG - J 3- at a or 3 feet it is equal to f of half the load or 56,250 pounds ; at b or 6 feet it is equal to of half the load or 37,500 pounds ; at c or 9 feet it is equal to J of half THE STRAINS IN COMPOUND RIVETED GIRDERS. 13 the load or 18,750 pounds; at 12 feet or the centre it is zero: from which we can now obtain the flange strain by dividing by the height, and again by 12 inches to get the pounds per inch of run. 75000 Then at end = - - = 2083 Ibs. per inch of run ; 3 X 12 of. ,, 3 3 1 ft)') u *< ** *< "3 X 12" at c -^ = 520 ' 3 X 12 The result thus obtained is then for the shear on the rivets. If the girder has a single web, as in a plate girder, we will take its bearing value, using a f web and a J-diameter rivet. The value of each rivet would be 15,000 X 5 X f = 4920 pounds. Then at the end where we have the stress of 2083 pounds per inch run, we must therefore space rivets : 4920 at support = 2f inches centre to centre; _ 49 20 1562 = 3* 4920 at c = - = o T V " " " *' 520 But as we have exceeded our limit at c, we will require the spacing from c to middle of girder to be 6 inches centres. 14 COMPOUND RIVETED GIRDERS. In Order that rivets in the two legs of the angle should stagger each other, both legs must have the same spacing ; and in order that the rivets may lie in straight lines vertical and horizontal, the top and bottom rows should be spaced alike. In fact, the spacing in top and bottom flanges for practical considerations are similar. Generally the vertical in addition to the horizontal strain is taken into consideration for spacing the rivets, and their resultant is therefore the strain on the rivets. The vertical strain is that due directly to the load resting upon the flange of the girder, and thence through the rivets transmitted to the web. , In the above example, Fig. 13, the flange strain at the end = 2083 Ibs. per inch of run. 3 X 12 The vertical load on the girder per inch of run is the total load, divided by the length of span for the load on one foot, and the quotient by 12 inches for the per inch of run of load, 150000 or = 520 Ibs. per inch of run. 24 X 12 Then from the above at end of girder the resultant = 1/2083" + 520" = 2147 Ibs. per inch of run. The spacing of rivets at end of girder = fff ^ 2 T 5 F inches. At a, 3 feet from end, flange strain = 1562 Ibs. per inch of run. The vertical stress as given is 520 Ibs. per inch of run. The resultant Vi$6tf + 520' 1647 Ibs. per inch of run. Then ff J4 = 3 inches, the spacing of rivet at 3 feet from end of girder. Continue in like manner to the centre. THE STRAINS IN COMPOUND RIVETED GIRDERS. 15 Spacing Rivets according to Strain Produced by the Bending Moments. The rivets are also spaced, in the angles which connect the flanges with the web, according to the strain produced in the flanges by the bending M, and the number readily found with but little calculation. The horizontal strain in the flanges diminishes in intensity either way from the posi- tion of maximum M towards either support, where it is the least, and may be found (as mentioned before under Bending Moments) by dividing by the depth. The horizontal increments of strain in the web are greater, however, at the ends, and least under position of maximum M. If the maximum strain in the flange is divided by the value of each rivet, there results the minimum number of rivets either way from maximum J/to either support. For example : in a girder of 24 feet span and 3 feet in depth, f web and -J rivets, if the maximum M equals 225 ton- feet at centre of girder, and the flange stress is at that point -2.25- 75 tonS) or 150,000 Ibs., the number of rivets 150,000 = - = 30 required either way from centre 4920 a distance of 144 inches, or spaced Yo 4 = 4ff Belies. Owing, however, to the greater intensity of the horizontal increment of strain in the web towards these supports, the rivets should be spaced closer as the ends are approached. Then for the first 3 feet, say 3 inches centres ; the next, 4 inches ; the next, 5 inches ; and the remaining 3 feet, 6 inches. Then there results a total of 34 rivets. For any further explanation or spacing of rivets refer to the various examples which follow. For convenience in selection of rivets, the following table has been prepared : 1 6 COMPOUND RIVETED GIRDERS. SHEARING AND BEARING RESISTANCE OF RIVETS. Diameter of rivet in inches. Area of rivet Single shear at Bearing resistance in pounds for different thickness of plates at 15,000 Ibs. per square inch. in 7500 Ibs. Frac- tion. Deci- mal. square inches. per square inch. i A i & * 1% i a I I 0.625 0.3068 2300 2340 2930 3520 * 0-75 0.4418 3310 2810 35io 4220 4Q20 56^0 6330 i 0.875 0.6013 4510 3280 4100 4Q20 5740 6562 7380 8200 9020 9840 I I 0.7854 5890 375C 4690 562O 6562 7500 8440 9380 10310 11250 Proportioning Girders. The first operation, that of ob- taining the kind of girder, is not always left entirely to the dis- cretion of the designer; no rules can be laid down, for the reason that various loads on fixed spans and depths are given, so that, in the nature of the construction, very little limit is allowed ; for instance, if the girders are in the floor construc- tion, the height of girder is reduced to a minimum, to give the greatest height of ceiling. The depth at centre of straight independent girders as given by H umber may be made from -fa to T 1 ^ of the span. The greatest economy of material is perhaps obtained at y 1 ^. If the depth of girder is about ^ the span, the deflection will not be too great. For many cases it would be well to find the most economical depth by a few trials, and bearing in mind that the increase of depth decreases the flange area, while it increases the weight of web and stiffeners and vice versa. It has been previously mentioned that the depth of the girder is between the centre of gravity of the flanges, where we have one or more plates. We can without much error assume the distance between the centre of gravity of the flanges to be equal to the distance from top to bottom of flange-angles as the effective depth of the girder. As the flange section increases the effective depth increases, but we can assume them to be con- stant throughout. THE STRAINS IN COMPOUND RIVETED GIRDERS. \J The span should include the length between the centre of bearings or supports, but for all practical purposes the effective span herein taken is between the supports. The following general rules should be adopted in propor- tioning girders. 1. Plate girders should be proportioned upon the supposi- tion that the bending or chord strains are resisted entirely by the upper and lower flanges, and that the shearing or web strains are resisted entirely by the web plate. 2. In members sub'ject to tensile strains, full allowance shall be made for reduction of section by rivet-holes, etc. 3. The web plates shall not have a shearing strain greater than 6000 to 8000 pounds for wrought iron and 7000 to 9000 pounds for steel per square inch, and no web plate shall be less than inch in thickness. 4. No wrought-iron or steel shall be used less than f inch thick, except in places where both sides are always accessible for cleaning and painting. DETAILS OF CONSTRUCTION. 1. All the connections and details of the several parts shall be of such strength that, upon testing, rupture shall occur in the body of the members rather than in any of their details or connections. 2. The webs of plate girders, when they cannot be had in one length, must be spliced at all joints by a plate on each side of the web. T-iron must not be used for splices. 3. When the least thickness of the web is less than -fa of the depth, the web shall be stiffened at intervals not over twice the depth of the girder. 4. The pitch of rivets shall not exceed 6 inches, nor sixteen times the thinnest outside plate, nor be less than three diameters of the rivet in a straight line. 1 8 COMPOUND RIVETED GIRDERS. 5. The rivets used will be generally { and f inch diameter. 6. The distance between the edge of any piece and the cen- tre of a rivet-hole must never be less than ij inches. 7. In punching plates or other iron, the diameter of the die shall in no case exceed the diameter of the punch more than Y 1 ^ -of an inch. 8. All rivet-holes must be so accurately punched that, when the several parts forming one member are assembled together, a rivet T ! B inch less in diameter than the hole can be entered, hot, into any hole without reaming or straining the iron by "drifts." 9. The rivets when driven must completely fill the holes. 10. The rivet-heads must be hemispherical, and a uniform size for the same sized rivets throughout the work. They must be full and neatly made, and be concentric to the rivet- hole. 11. Whenever possible, all rivets must be machine-driven. 12. The several pieces forming one built member must fit closely together, and, when riveted, shall be free from twists, bends, or open joints. 13. All joints in riveted work, whether in tension or com- pression members, must be fully spliced, as no reliance will be placed upon abutting joints. The ends, however, must be dressed straight and true, so that there shall be no open joints. 14. All bed-plates under bearings of girders must be of such dimensions that the greatest pressure on the masonry shall not exceed 250 pounds per square inch. EXTRACT FROM THE NEW YORK BUILDING LAW PASSED APRIL 9, 1892. 486. " Rolled iron or steel beam girders, or rivet ed\rov\ or steel plate girders used as lintels or as girders, carrying a wall or floor or both, shall be so proportioned that the loads which THE STRAINS IN COMPOUND RIVETED GIRDERS. 19 may come upon them shall not produce strains in tension or compression upon the flanges of more than 12,000 pounds for iron nor more than 15,000 pounds for steel per square inch of the gross section of each of such flanges, nor a shearing strain upon the web plate of more than 6000 pounds per square inch of section of such web-plate if of iron, nor more than 7000 pounds if of steel; but no web plate shall be less than one quarter of an inch in thickness. Rivets in plate girders shall not be less than f of an inch in diameter, and shall not be spaced more than 6 inches apart in any case. They shall be so spaced that their shearing strains shall not exceed 9000 pounds per square inch of section, nor their bearing exceed 15,000 pounds per square inch, on their diameter, multiplied by the thickness of the plates through which they pass, The riveted plate girders shall be proportioned upon the supposition that the bending or chord strains are resisted entirely by the upper and lower flanges, and that the shearing strains are resisted entirely by the ' web plate. No part of the web shall be estimated as flange area, nor more than one half of that portion of the angle-iron which lies against the web. The distance between the centre of gravity of the flange areas will be considered as the effective depth of the girder. Before any girder, as before mentioned, to be used in any building shall be so used, the architect or the manufacturer or a contractor for it shall, if required so to do by the superin- tendent of buildings, submit for his examination and approval a diagram showing the loads to be carried by said girder, and the strains produced by such load, and also showing the dimen- sions of the materials of which said girder is to be constructed to* provide for the said strains; and the manufacturer or contractor shall cause to be marked upon said girder, in a con- spicuous place, the weight said girder will sustain, and no greater weight than that marked on such girder shall be placed thereon." 20 COMPOUND RIVETED GIRDERS. To Calculate the Approximate Weight of Girder before its Dimensions are Fixed. It should be remarked here that the weight of the girder becomes considerable when the flanges are built up of a number of plates. It is therefore desirable to be able to calculate approximately the weight of the girder before its dimensions have been definitely fixed. 'The weight of the girder will be in proportion to its area of cross-section and to its length; or when W is the gross load to be carried, and L the length between the supports, then the weight of girder between the bearings is WL in which C is a constant, and W the load to be supported. The value of C has been taken from examples of girders from 35 to 50 feet long; its value is found to be 700. Example : We have a span of 40 feet and 70 tons to be sup- ported ; what will be the approximate weight of girder? WL 70 X 40 w = TT- = -- = 4 tons, C 700 making a total of 74 tons, uniformly distributed. Splicing. Girders 40 feet and less will not require any splicing, as the plates and angles can be readily handled in one length. In splicing the top flange, no additional cover plate will be required over the joint, but the ends should be planed true and butt solidly. The rivets to be closer near the joint. The plate covering the joint of bottom flange requires to be the same area as the plates joined, and of sufficient length to take a number of rivets equal to strength of the cover plate. PART II. QUALITY OF MATERIAL. PART II. QUALITY OF MATERIAL. Wrought-iron. All wrought-iron must be tough, fibrous, and uniform in character. It shall have a limit of elasticity of not less than 26,000 pounds per square inch. The tensile strength, limit of elasticity, and ductility shall be determined from a standard test-piece about \ square inch. The elonga- tion shall be measured on an original length of 8 inches. When taken from plates rolled to a section of not more than 4^ square inches, the iron shall show a minimum ultimate strength of 50,000 pounds per square inch, and a minimum elongation of 18 per cent in 8 inches. The same sized speci- men, taken from plates 8 inches to 24 inches in width, shall show a minimum ultimate strength of 48,000 pounds per square inch, and a minimum elongation of 15 per cent in 8 inches; plates from 24 inches to 36 inches wide, 46,000 pounds per square inch, and elongate 10 per cent in 8 inches; plates over 36 inches wide, 8 per cent in 8 inches. The same sized specimen taken from angle-iron shall have a minimum ultimate strength of 48,000 pounds per square inch, and a minimum elongation of 15 per cent in 8 inches. Rivet- iron shall have the same physical requirements as high-test iron, and in addition shall bend cold 180 degrees to a curve 21 22 COMPOUND RIVETED GIRDERS. whose diameter is equal to the thickness of the rod tested, without signs of fracture on the convex side. All iron for tension members must bend cold through 90 degrees to a curve whose diameter is not over twice the thick- ness of the piece, without cracking ; at least one example in three must bend through 180 degrees to this curve without cracking. When nicked on one side and bent by a blow from a sledge, the fracture must be nearly fibrous. Mild Steel. Specimens from finished material for test, cut to size, as for wrought-iron, shall have an ultimate strength of from 54,000 to 62,000 pounds per square inch, with a minimum elongation of 26 per cent in 8 inches; to bend cold 180 de- grees flat on itself, without sign of fracture on the outside of bent portion. All rivets of mild steel must, under the above bending test, stand closing solidly together without sign of fracture. Painting. All iron and steel work, before leaving the shop, shall be thoroughly cleansed from all loose scale and rust, and be given one good coating of best oxide of iron and pure linseed oil, and after erection to receive one additional coat of paint. PART III, EXAMPLES. PART III. GIRDER SUPPORTING A CONCENTRATED LOAD AT CENTRE OF SPAN. In a girder supported at both ends with a load concen- trated at centre of span, the maximum bending moment is at the centre, and equals half the load multiplied by half the span, or WL 4 To find the bending moment at any point in the girder when the load is at the centre. Make FD, by any scale, Fig. 14, equal M at centre of span . join RD and DR'. Then by the same scale, rp will equal the 23 24 COMPOUND RIVETED GIRDERS. bending moment at point r, and ut will equal the moment at point u. Or the moment at any point r or u will equal half the load multiplied by R'r or R'u ; R' being the nearest sup- port. Then W Air, M= R'r. W Atu, M=R'u. Example : What metal area would be required in the flanges at the centre of a girder of 30 feet span, 2 feet in depth, to sus- tain 40 tons concentrated at the centre of span; 6 tons (12,000 pounds) being the maximum unit strain per square inch allowed in the flanges? Here W = 40 tons, L = 30 feet, d= 2 feet. A = area of flanges, s = 6 tons. Then at centre, 30O A = T2 2 5 square inches. 2 X O Let r = 10 feet from R' support, and u = 5 feet from the same support. At r, M= X 10 = 200 ton-feet, A s. 1 6.66 square inches. EXAMPLE 2. 25 4O At u, M X 5 = ioo ton-feet, 100 A = -? = 8.33 square inches. 2 X O Then we require in the flanges for the above girder : At centre, 25.0 square inches. 5 ft. from ' " 16.66 " " 10 ft. " " 8.33 " Construction of Flanges. In the results of the example just given, it will be observed that the area of metal required in the flanges increases gradually from the points of support towards the centre of the girder. This will be accomplished by building up the plates of metal overlapping each other for the computed amount. To make up the 25 square inches in the top flange at the centre, we extend the angles from end to end, and making the girder 12 inches wide, using ordinary size plates (none less than f of an inch thick) and angles with the longer leg horizontal, we would require : Top flange = 2 angles* 5" X 4" X 4" = 8.50 square inches. I plate 12" X 4" = 6.00 " I " 12" X 4" = 6.00 " " I " 12" X V = 4-50 " Total, 25.09 " " For the bottom flange, the rivet-holes must be deducted to obtain the net section. By referring to the section of the con- structed girder, Fig. 19, it will be noticed that the greatest loss * For areas of angles in inches see Table, page 33. 26 COMPOUND RIVETED GIRDERS. of section is two rivet-holes opposite each other, connecting the angles with the plates of the bottom flange. Using J-inch-diameter rivets, and allowing -J of an inch more for any injury to the metal in the process of punching, we have the area of a rivet-hole equal to \" + J" + J" + f " X i" = i$ square inches, for two rivet-holes 2 X i" 3f square inches, to be added to the bottom flange, or 25" -j- 3f" = 28f square inches. Then Bottom flange 2 angles 5" X 4" X J" = 8.50 square inches. i plate 12" X f" = 7-50 " i " 12" Xf" = 7-50 " i i2 // X T V / = 5- 2 5 " Total, 28.75 " Flanges reduced in Area towards the Supports. To reduce the area of the flanges as the ends are approached, draw the diagram Fig. 15, making R and R' equal to the span of 30 FIG. 15. feet, and set off FD at centre of span equal to the bending moment at that point, or equal to DF, Fig. 14. Connect RD and DR'. Draw the rectangle RCER '. EXAMPLE I. 27 Then from F place any scale at any angle, as Fe, until it measures 25 square inches, the number required in the flanges at the centre. For two angles 5" X 4" X f", set off 4.25 square inches each at a and b\ one plate 12" X J" = 6 square inches at c ; one plate 12" X J" = 6 square inches at 5-75 16.50 18.00 19.50 21 .OO 22.50 ft 5-So 6.18 6 87 8.2 5 9.62 10.31 II .OO 12 -37 T 3-75 !4-43 15.12 16.50 17.86 19.25 20.62 5.00 5-62 6.25 7-50 8-75 9-37 10.00 11-25 12.50 13.12 13-75 15.00 16.25 I7-50 8-75 A 4-5 5-06 5.62 6 75 7.87 8-43 9.00 10.12 11.25 n.8i 13 -37 I3-50 14.62 15-75 16.87 A 4.00 4-5 5-oo 6.00 7.00 7-50 8.00 9-00 10.00 10.50 11.00 12.00 13.00 I4.OO 15.00 TS 3-5 3-94 4-37 5-25 6.12 6.56 7.00 7.8 7 8-75 9 i9 9.62 10.50 n-37 12.25 13-" 3 3-oo 3-37 3-75 4-5 5- 2 5 / 5-62 6 OO 6-75 7-5o 7.87 8.25 9 oo 9 75 10.50 ti .25 1% 2.50 3.81 3.12 3-75 4-37 4-69 5.00 5-62 6.25 6.56 6.87 7-50 8.12 8-75 9-37 i 2.OO 2.25 2.50 3.00 3 5 3-75 4.00 4-5 5 oo 5- 2 5 5-50 o.oo 6.50 7.00 7-50 EXPLANATION. Required the sectional area of a plate 26" X H" punched by six f ' rivets. The gross area by table = 17.86 square inches, and the area of six f" rivets by the previous table = 3.60 ; that is, 17.86 3.60 = 14.26, the area required. Required the sectional area of a 6" X 4" X 1" angle punched by two f '' rivets. The gross area by table of "Areas of Angles" = 7.99 square inches, and the area of two " rivets through a " plate by the table = 1.53. Then 7.99 1.53 = 6.46, the area required. SAFE BUCKLING VALUE OF WEB PLATES PER SQUARE INCH (WROUGHT-IRON). d = depth in inches. t thickness in inches. Calculated by formula i oooo d* 3000^ ' Depth in Inches. J g C g 20 24 28 30 32 36 40 42 48 50 52 60 I 3195 2455 A 4220 3365 2714 1 5134 4229 3498 3192 2889 2456 2087 1930 1548 1442 1350 T 7 * 5900 4992 4228 3896 3624 3069 2696 2455 1994 i86s I75i ? 6522 5652 4890 4546 4228 3666 3191 2983 2543 2308 2172 1724 7035 6223 5476 5133 4787 4229 3724 3498 2918 2749 2599 2087 7456 6704 5932 5656 5339 4748 4228 3992 3371 3191 3024 2456 H 78l8 7133 6 4 6f 6143 5834 5252 4726 4500 3835 3645 3465 2848 f 8044 7612 6828 6 5 22 6226 5656 5133 4889 4228 4030 3885 3J9 1 a 7752 7164 6868 6585 6045 5534 5290 4623 4420 4224 3549 Y 7184 6920 6392 5882 5649 4992 4780 4593 3891 ft 6700 6211 5988 5336 5025 4926 4237 1 D I 5674 5455 5263 4545 i N.B. If the buckling value is less than the shearing (6000 pounds for rrought-iron), the web will require to be stiffened. COMPOUND RIVETED GIRDERS. SHEARING VALUE OF WEB PLATES, WROUGHT-IRON, 6000 LBS. PER SQUARE INCH. Depth Thickness of Plate. in Inches 1 t i ft 1 H t I I 12 27000 31500' 36000 40500 45000 49500 54000; 63000 72OOO 14 31500 36750! 42000 47250 52500 57750 63000 73500 84000 16 36000 42000 48000 54000 60000 66000 72000 84000 96000 18 40500 47250 54000 60750 67500 74250 80500 94500 108000 20 45000 52500 60000 67500 75000 82500 90000 105000 I2OOOO 22 49500 57750 66000 74250 82500 90750 99000 115500 132000 24 54000 63000 72000 SlOOO 9OOOO 99000 108000 126000 I44OOO 26 58500 68250 78000 87750 97500 107250 117000 136500 I56OOO 28 63000 73500 84000 94500 105000 115500 126000 147000 168000 30 67500 78750 90000 IOI25O 112500 123750 135000 157500 ISOOOO 32 72000 84000 96000 108000 120000 132000 144000 168000 192000 34 76500 89250 102000 H4750 127500 140250 153000 179500 2O4OOO 36 81000 94500 IO8OOO J2I5OO 135000 148500 162000 189000 2I6OOO 38 85500 99750 II4OOO 128250 142500 156750 171000 199500 228000 40 90000 105000 120000 135000 150000 165000 180000 210000 24OOOO 42 94500 110250 I26OOO I4I750 157500 173250 189000 220500 252000 44 99000 115500 132000 148500 165000 181500 198000 231000 264000 46 103500 120750 138000 155250 172500 189750 207000 241500 278000 48 108000 126000 I44OOO I62OOO ISOOOO 198000 216000 252OOO 288000 50 112500 131250 1 50000 168750 187500 206250 225000 262500 300000 52 II7000 136500 156000 175500 195000 214500 234000 273000 312000 54 121500 141750 162000 182250 202500 222750 243000 283500 324OOO 56 126000 147000 168000 iSgOOO 2IOOOO 231000 252000 294OOO 336000 58 130500 152250 174000 195750 217500 239250 261000 304500 348000 60 135000 157500 ISOOOO 202500 225000 247500 270000 315000 36OOOO EXAMPLE I. 37 SHEARING VALUE OF WEB PLATES, WROUGHT-STEEL, 7000 LBS. PER SQUARE INCH. Depth in [nches Thickness of Plate. I TV * ,& f H f 1 I 12 31500 36750 42OOO 47250 52500 57750 63000 73500 84000 14 36750 42875 49000 55125 6I25O 67375 73500 85750 98000 16 42000 49000 56OOO 63000 7OOOO 77000 84000 98000 II2OOO 18 47250 55125 63OOO 70875 78750 86625 94500 110250 126000 20 52500 6I25O 70000 78750 87500 96250 105000 122500 140000 22 57750 67375 77000 86625 96250 105875 115500 134750 154000 24 63000 73500 84000 94500 105000 115500 126000 147000 168000 26 68250 79625 91000 102375 H3750 125125 136500 159250 I82OOO 28 73500 85750 98000 II0250 122500 134750 147000 171500 I96OOO 30 78750 9*875 105000 H8I25 I3I250 144375 157500 183750 210000 32 84000 98000 II2OOO 126000 14000 154000 168000 196000 224OOO 34 89250 IO4I25 IigOOO 133875 148750 163625 178500 208250 238000 36 94500 II0250 126000 141750 157500 173250 189000 220500 252OOO 38 99750 II6375 133000 149625 166250 182875 199500 232750 2660OO 40 105000 122500 I4OOOO 157500 175000 192500 210000 245000 280000 42 II0250 128625 147000 165375 183750 202125 22O5OO 257250 294000 44 H5500 134750 I54OOO 173250 192500 211750 23IOOO 269500 308000 46 120750 140875 I6IOOO I8II25 201250 221375 241500 281750 322OOO 48 126000 147000 I680OO I SgOOO 210000 231000 252000 294000 336000 50 I3I250 I53I25 175000 196875 218750 240625 2625OO 306250 350000 52 136500 159250 182000 204750 227500 250250 273000 318500 364000 54 141750 165375 iSgOOO 212625 236250 259875 283500 330750 378000 56 147000 T7I500 igOOOO 22O5OO 245000 269500 294000 343000 392OOO 58 152250 177625 203000 228375 253750 279125 304500 355250 4O6OOO 60 157500 183750 2IOOOO 236250 2625OO 288750 315000 367500 420000 COMPOUND RIVETED GIRDERS. EXAMPLE II. GIRDER SUPPORTING A CONCENTRATED LOAD NOT AT CENTRE OF SPAN. In a girder supported at both ends with a load concentrated not at centre of span, the maximum bending moment is at the load, and is equal to the load multiplied by the distance from load to left support, and from load to right support divided by the span ; or, To find the bending moment at any point in the girder when 3) FIG. 20. the load is not at centre, make FD by any scale, Fig. 20, equal M at the load ; join RD and DR'. Then *, y, z, by the same EXAMPLE II. 39 scale, will measure the moments at their respective points in the girder. *fcV*V&^E5 Lt L RF x R's Example : What metal area would be required in the flanges of a plate girder of 20 feet span, 2 feet 6 inches in depth, to sustain 60 tons concentrated at 5 feet from left support ? Here W=6o tons; L = 20 feet; d = 2 feet 6 inches; s = 6 tons. Then AtF,M=6oX ^^- 2 = 22 5 ton-feet, 225 5 X. J 5 S( l uare inches. 2.5 x 15 At x, M 60 X = 112.5 ton-feet, 112.5 z= "~ = 7 ' 5 square inches ' At 7, ^f= 60 X zr = 150 ton-feet, A = ^ = 10 square inches. 4O COMPOUND RIVETED GIRDERS. Atz,M=6oX = 75 ton-feet, A = 2. X 6 = 5 square inches - Then we require in the flanges for the above girder : At the load, 15.0 square inches. At x, 2 feet 6 inches from " 7.50 " " At 7, 5 "- " " 10.00 " " At z, 10 " " ' 5.00 " " Construction of Flanges. To make up the maximum section under the load we would require : Top flange = 2 angles 5" X 3" X J" = 7-5 square inches. I plate 12" X I" =7-5 " " Total, 15.0 " " For the bottom flange, rivet-holes to be deducted to obtain the net section. The loss of metal by rivet-holes is the same in this as the former example. Using J-inch-diameter rivets, and allowing % of an inch more for any injury to the metal by punching, area of rivet- hole equals |." + " X i" | square inches ; for two rivet-holes 2" X f" = 2.25 square inches, to be added to the bottom flange, or 15" -f- 2.25" == 17.25 square inches. Bottom flange = 2 angles 5" X 3" X i" = 7-5 square inches. I plate 12" X if" = 9-75 " Total, 17.25 " EXAMPLE II. Flanges reduced in Area towards the Supports. To place the plates in their required position for the calculated area : Draw the diagram, Fig. 2 1 , making R and R' equal to the FIG. 21. span of 20 feet as in the previous example, and set off lines FD at position of load equal to the maximum bending moment at that point, or equal to FD, Fig. 20. Connect RD and DR '. Draw the rectangle RCER '. Then from F place any scale at any angle, as Fc, until it measures 15, the number of square inches required in the flanges at F or at the load. For two angles set off 3.75 square inches at a and b, and one plate 12 X f| inches, or 7.50 square inches, at C. Horizontal lines drawn from a, b, and c to DR, DR', and carried down to base line RR' give the position of the plates. The angles to extend from end to end as shown. 42 COMPOUND RIVETED GIRDERS. Both top and bottom plates to extend 12 inches each way from FD, in addition to that determined by the diagram. Webs. The bearing area upon which a girder is supported reacts against the girder ^n amount equal to the pressure of the load upon them ; or, the sum of the loads on the girder is equal to the sum of the reactions. Hence, if there be but one support as in a cantilever, this condition gives at once the re- action. For a uniform load and a concentrated load at centre, on two supports, it is evident that each reaction equals one half of the load. We have in this example a single concentrated load situated at 5 feet from left support, whose span is 20 feet. 60 ^ i $ Then R supports = 45 tons, and/?' " = 15 tons. (Refer to article Shearing Forces on the Webs.) The thickness of the web may then be determined as in the previous example. The one-half-inch thickness to be adopted and used from end to end of girder ; otherwise we would require a variety of thicknesses to be made up between each end, which is altogether impracticable in small girders. EXAMPLE II. 43 Stiffeners. To determine whether we require stiffeners : Safe resistance to buckling 10000 10000 , 30 X 1 + lbs - P er *1- mch - r 3000/ a ' 3000 X i X i | But as we have adopted 6000 pounds for safe shearing, the web will have to be stiffened throughout, at the bearings, under the load, and every 3 feet, by 4" X 4" X f " angles riveted each side of web. We have then for the thickness f " + J" + f " = i$- inches, and the formula becomes Safe resistance to buckling = ' = 8389 lbs. 30 X 3 *~ 3000 x v- x Rivets. This example being a single-web girder, the rivets will be in double shear ; so we will take their bearing value, which is less than the shearing. The bearing area = -J" X f" X 15000 = 6562 pounds. The bending moment at load is 225 ton-feet ; this divided by the depth gives the horizontal flange strain each way from position of load to end of girder. 225 Then -- = 90 tons or 180,000 pounds. This again divided by the value of the rivet gives the total number of rivets required ; or, 180,000 = 27 nvets ' in a distance of 60 inches from the load to R support spaced about 2\ inches centres. 44 COMPOUND RIVETED GIRDERS. This is closer than they should be spaced in a straight line, so we will have to stagger them as much as possible. (See girder drawing, Fig. 23.) We will also require 27 rivets from the position of load to R', a distance of 180 inches, spaced 6.6 inches. This is more than they should be spaced in a straight line; as our maximum is 6 inches, we will space them accord- ingly. Had we used the shearing stress for the rivet spacing, we would have a uniform shear of 90,000 pounds from position of load to R, and a uniform shear of 30,000 pounds to R'. At R, - 3000 pounds per inch of run. 2.5 X 12 Spaced, = 2.19 inches centre to centre. At R 1 , -^ - = looo pounds per inch of run ; 2.5 X 12 Spaced, = 6.56 inches centre to centre. Graphical Representation of Bending Moments and Shearing Forces in a Girder with One Concentrated Load not at Centre. According to our example we have a girder of 20 feet span, sustaining a load of 60 tons 5 feet from left support. Set the load W of 60 tons to a reasonable scale off along the line PP , Fig. 22. The line PP' is thus the polygon of the given force, and P'P, its closing line, is the resultant. Take any point O as pole, equal to ten units of the scale adopted, and draw the radii OP and OP ' . Then describe the funicular polygon abc by drawing ab parallel to OP, termi- nating in ^produced, and be parallel to OP', terminating in the prolongation downwards of R'. The funicular polygon is EXAMPLE II. 45 now closed by the line ca, and a line OS is drawn through pole O parallel to ac. The bending moments are then found in the same manner as described under Fig. 18. Then the re- TT FIG. 22. action of R will equal the distance by same scale from S to P, and the reaction at R the distance from 5 to P'. Or as a condition of equilibrium, the reaction at R = PS, and at R' = SP f . The shearing forces, as in the previous example, are taken from the hatched figure. By referring to the diagram it will be noticed that the greatest shear on the web is at R support measured from P to 5, and only a small percentage of the load W\s sustained by R' support. 4 6 COMPOUND RIVETED GIRDERS. EXAMPLE III. 47 EXAMPLE III. GIRDER SUPPORTING A UNIFORMLY DISTRIBUTED LOAD. In a girder supported at both ends with a load distributed over its entire length, the maximum bending moment is at the centre, and is equal to one half the load multiplied by one quarter the span, or WL M ~~S~' To find the bending moment at any point in the girder, when the load is uniformly distributed : Make FD by scale, Fig. 24, equal M at centre of span, and draw the parabola RDR' (see method of drawing parabolas, FIG. Figs. 26 and 27). Then rp measured by the same scale will equal the bending moment at point r, and ut will equal the moment at point u. 48 COMPOUND RIVETED GIRDERS. Or the moment at any point r = half load on rR X rR f , or half load on rR f X rR. W -j- X rR Atr,M = X 2 W -J- X uR At w, M = X uR'. Example : What metal area would be required in the flanges at the centre of a box girder of 30 feet span, 3 feet in depth, to sustain 200 tons of a i6-inch wall distributed over its entire length. Here W '= 200 tons, L = 30 feet, d = 3 feet, A = area of flange, j 6 tons. At centre, M = ^ = 750 ton-feet, o A = ^ = 41.66 square inches. 200 X 10 At r, M = -^2 - X 20 = 666.66 ton-feet, 666.66 A = g- = 37.04 square inches. 200 At u, M = X 20 = 416.66 ton-feet, 416.66 ~ : 2x6 ~ 2 3* X 5 square inches. EXAMPLE III. 49 Then for the above girder we require in the flanges : At centre, 41.66 square inches. 5 feet from " 37.04 " " 10 " " " 23.15 Construction of Flanges. To make up the maximum section at the maximum bending moment we would require: Top flange = 2 angles 4" X 4" X If" =11.68 square inches. 3 plates,each 16" X |" = 30.00 " " Total, 41.69 " For the bottom flange the loss by rivet-holes will be the thickness of plates and one angle, and then using -J-inch-diam- eter rivets and allowing inch more, we have " -f- " + " -J- t-3." x I = 2\\ square inches, for two rivets 2 X 2-fj-" 5.375 square inches, to be added to the bottom flange at centre, or 41.66" + 5-375" = 47 square inches. Bottom flange 2 angles 4" X 4" X \\" = 1 1.68 square inches. 3 plates each 16" X f " = 36.00 " Total, 47.68 " Flanges Reduced in Area towards the Supports. To place the plates of the flanges in their required position for the calculated area, draw the diagram Fig. 25. From Fin centre of span, make FD, by the same scale as in Fig. 24, equal to the maximum bending moment at that point. Draw the rectangle RCER'. From F place the scale at any angle, as at Fe, until it measures 41.66 or 41.68 square inches. For two angles set off 5.84 square inches, each, at a and b y and three plates 16" X i", or 12 square inches, each, at c, d, and e. Horizontal lines drawn from a, b, c, d^ and e to the pa- rabola RDR\ and carried down to base line RR r , will give the position of the plates in the flanges. 50 COMPOUND RIVETED GIRDERS. The angles to extend from end to end of girder, and the adjoining plates are required to extend in like manner for prac- tical reasons, which will be readily seen in all box girders. FIG. 25. The plates of the bottom flange are, for the reasons ex- plained in the previous example, practically the same length as those of the top flange, and should extend 12 inches beyond the calculated length. Webs. The reactions on the supports of a girder sustain- ing a uniformly distributed load are each equal to one half the total load, and the shearing force on the webs at each end of 400,000 the girder is equal to = 200,000 pounds. Then * = f an inch ; EXAMPLE III. 51 but as we have two webs, each will be equal to one half of -J-J or {, say inch, for the thickness of each web. Stiffeners. To determine whether we require stiffeners, only one web need be taken into consideration ; and if stiffeners are needed, an angle can be riveted on the outside. Frequently there is considerable shear on the webs ; angles are then riveted inside and outside of each web. Safe resistance to buckling 10000 = 3 3 pounds per square m 3000 x i x The webs will have to be stiffened at the bearings. At 5 feet from the bearing or supports the shear on the webs is equal to f the shear at the bearings, or 133,333 pounds. The safe shear against buckling of a 36-inch web -J inch thick is 3663 pounds per square inch, as found above. The shearing area of the web at 5 feet from the bearing is 36 X 4 = 18 square inches. Then 3663 X 18 = 65,934 pounds safe against buckling at that point, and the shear on one web at the same point is = 66,666 pounds ; a stiffener is therefore required, and one 2' 6" towards the bearings, but none towards the centre, as the shear is theoretically nothing at the middle of a girder uni- formly loaded, but from thence increases by equal increments towards each support (refer to shearing force diagram, Fig. 28). Rivets. The rivets connecting the webs to the flanges in a box girder are in single shear; therefore the shearing value will be 7500 pounds per square inch, and is measured on the area of the cross-section of the rivet. The area of a -J-inch-diameter rivet = $" X " X .7854 = ,6013 (see Table of Shearing and Bearing Resistance of Rivets); 52 COMPOUND RIVETED GIRDERS. this multiplied by 7500 4510, the safe amount of strain each rivet can sustain without shearing. If the maximum horizontal strain in the flanges is divided by 4510, there results the minimum number of rivets required either way from the centre. Then from the example : Maximum M 750 ton-feet ; Horizontal strain = -- = 250 tons or 500,000 pounds ; and divided by 4510 = no rivets, to be placed a distance of 1 80 inches for one side, 360 inches for both, spaced about 3 inches centres. On account of the horizontal increments of strain in the web increasing towards the ends, the rivets should be spaced closer as the ends are approached, say 2f inches for the first 5 feet, 3 inches for the next, and 4 inches for the remaining dis- tance. Method of Drawing Parabolas. Draw a horizontal RF+ Fig ' 26 ' equal t0 half span f 2 irder - Set off DF perpendicular to RF, mak- ing the former equal by scale to the bending moment at that point. Through D draw CD parallel and equal to RF. The ordinates from any points in CD to the parabola will be propor- tional to the square of the distances of FIG. 26. those points from D. Thus if the ordinate at a be I, then the ordinate at b, twice the distance of a from D, must be 4 ; and so on. To proceed practically, divide CD into a number of equal parts (ri) as at a, b, c, etc. Then if RC be divided into (tf) parts, each of these parts will be the required unit, one of which is the EXAMPLE III. 53 FIG. 27. offset at a ; four at b ; nine at c ; and so on. Through the points- a', b', c', etc., thus determined, the required curve can be drawn. Parabola by the Construc- tion of a Diagram. On the span RR', Fig. 27, describe an isosceles triangle whose height is double that of the bending moment. Divide the two sides AR and AR' of the triangle into any number of equal parts, and draw lines as in the figure. These lines will be tangents to the parabola, which may then be drawn. Graphical Representation of Bending Moments and Shearing Forces for a Uniformly Distributed Load. A uniform load may be considered as a system of equal and equi- distant loads close together. Thus in Fig. 28 the load area may be divided into any num- ber of equal parts. The area of each part we may consider as the load which acts at its centre of gravity, and lay it off to any convenient scale in the force polygon, as at I, 2, 3, 4, ... 19, 20. Since the reactions at R and R r are equal, we take the pole O in a horizontal through the centre of force line PP', and draw the radii <9i, 6>2, #3, #4, . . . O2O. Then describe the funicular polygon ab, be, cd, de, . . . vw, by drawing ab parallel to PO> be to <9i, cd to Oz, de to 6>3, . . . vw to 6>2O. The funicular polygon is now closed by the line aw, and a line OS is drawn through pole O (on OH in this example), which is parallel to aw and to the girder. Any ordinate taken to this parabola from the base line aw multiplied by the distance OH will give the bending moment at any point in the girder. To avoid any error in direction carried on by the lines being 54 COMPOUND RIVETED GIRDERS. too short, one half the number of divisions will be quite suffi- cient to get the moments. The shearing forces on the webs are shown similar to the QOOOOOOOOnnnnnnnnnrv^ FIG. 28. previous examples, and are equal to the distances of the points of the polygon of forces from S. Therefore the shearing forces are taken from the force poly- gon used as ordinates as shown in the diagram, and the hatched figure is the result. EXAMPLE III. 55 bo c S CO . ^^ "0 z o * .5 o en C _0 - ^o X "o "o "o 2 ebfeo V V> "en > -lir-i K-l C4 en rS Cu O X X X X Q V X X X X $ Q O . ^ ^ ^ rt "Vfr ^ tfl fH M w ocfoo Q W 'S) ^ X H co "E. "co D M M *. X h i ti P4 bo "co H 00 1 i if 3 "co S u c .. o S II 1 ^ U c/i o CQ X O a U CO c H bb c 1 fe ^o C/3 O ^ ^ bb & t^ c - V w E a o Q a f : 3 , X > > ^.te V% QO H M M 00 >r *- c j " .bp X X X x r h-H ' " *- ^_ ^ o < 6 H o - w -ft Q aJ , J - %fr ^ re( XXX o o o * -K x x to ; en M M M H O m ^ V jQ o w i 3 1 i c/T X) a ? (U i H p H EXAMPLE V. 65 EXAMPLE V. GIRDER SUPPORTING TWO CONCENTRATED LOADS AND A UNIFORMLY DISTRIBUTED LOAD. In a girder supported at both ends, the bending moments at any point produced by all the loads is the sum of the moments produced at that point by each of the loads sepa- rately. This is a combination of the two previous examples, and each is to be taken separately. The polygon for the concentrated loads to be drawn under, and the parabola for the uniform load over, the girder. Example : What metal area would be required in the flanges of a box girder 16 inches wide, 32 feet span, 3 feet in depth, to sustain 60 tons concentrated 10 feet from right support, 40 tons 10 feet from left support, and a uniformly distributed load of 80 tons, with 6 tons unit strain per square inch in the flanges? Draw the triangles Fig. 34, having vertices at /and k, rep- resenting bending moments by M = W X STT^ for loads w * and w * (Example II). At e for W 19 M = 40 X IO X 22 = 125 ton-feet. - / At for W 3 , M = 6oX IOX 22 = 412.5 ton-feet. Draw the vertices*?/ and gh by scale equal to 125 and 412.5 ton-feet respectively, and connect each to ad\ extend ef (equal to ek) to b, and gh (equal to gi) to c, and connect #, b, c, d. 66 COMPOUND RIVETED GIRDERS. Then eb and ge measured by same scale will give the bend- ing moments produced at e and g by W l and W t . Nib / ' > FIG. 34. Draw the parabola for the uniform load, making FD, by the same scale as for the concentrated loads, equal to the moment at the centre of the girder by formula : WL 80 X 32 8 = 320 ton-feet. The bending moment due to the two concentrated loads and the uniform load at any point in the girder is equal at that EXAMPLE V. 67 point to the sum of the ordinates of the parabola RDR' and the polygon abed. Then at W^ , the ordinate of the parabola mn and the poly- gon gc, M = 812.5 ton-feet. Or at g, for W 9 , M = 60 X - =412.5 ton-feet ; f for uniform load = X 22 = 275.00 Maximum M = 812.5 " O Y n H Flange area = =45.12 square inches. Construction of Flanges. Then to make up the 45.12 square inches in the flanges at g, we would require : Top flange = 2 angles 6" X 4" X f" = 11.72 square inches. 1 plate 1 6" X i" =12 2 plates 1 6" X H" = 22 - " Total, 45.72 " For the bottom flange we deduct rivet-holes, then using f-inch-diameter rivets and allowing -| inch more, we have for one hole -Hr + if + H + rt X i = 2ff inches, for two holes X 2 = 5J square inches, to be added to the bottom flange r , or 45.12 + 5.525 = 50.62 square inches. * Refer to formula under girder of one concentrated load not at centre, f Refer to formula under girder of a uniformly distributed load. 68 COMPOUND RIVETED GIRDERS. Bottom flange = 2 angles 6" X 4" X " = ll >7 2 s q- in - i plate 1 6" X I" = 14.00 " " i " 1 6" X i-f" = 13.00 " " i " 16" X i" = 12.00 " " Total, 50.72 " " Webs. The reactions due to both uniform and concen- trated loads may be obtained by adding together the reactions due to the uniform load and each concentrated load, or they may be computed in one operation. To find the right reaction, R', the centre of moments is taken at the left support and the uniform load regarded as concentrated at its middle ; then the equation of moments is R X 32 = 60 X 22 + 40 X 10 + 80 X 1 6, from which R' = 93.75 tons. In like manner, to find R the centre of moments is taken at the right support and R X 32 = 60 X 10 + 40 X 22 + 80 X 16, from which R = 86.25. As a check the sum of R' and R is seen to be 180 tons, which is the same as the sum of the two concentrated loads and the uniform load. The thickness of web may then be determined by the for- mulas : At *',*= = - = .868 = finch. dk 36 X 6000 - 36 X M 6000 Adopting the greater thickness and having two webs, each will be one half of f or T \. EXAMPLE V. 69 Stiffeners. To determine whether we require stiffeners : Safe resistance to buckling = - ^ - ^ -- = 3069 Ibs. per square inch. *~ 3000 x A x A But having adopted 6000 Ibs. per square inch as the safe shear, we will require stiffeners to be placed throughout the length, from W^ to R and W^ to R', spaced about 3 ft. centres. No stiffeners are required between W 6 and W^ , there being little shear on the webs in that length. Refer to the diagram Fig. 35- Then to stiffen the web against buckling, place a 4" X 4" X J" angle outside of web. We then have for the thickness T \ -|- -J- = ^|- inch, and the formula becomes : Safe resistance to buckling = 6700 Ibs. per square inch. if Xtf Rivets. The shearing area is again used in this example and f-inch-diameter rivets. The safe single shear for each rivet by table = 4510 pounds per square inch. The maximum bending moment is at g, and M 812.5 ton-feet. Then divided by the depth, the horizontal strain = = 270.2 tons or 540,400 pounds. This divided by 4510 = 119 rivets, to be placed a distance of 1 20 inches for one web or 240 inches for both, spaced about 2 inches centres (staggered) from position of maximum M to R 1 support, from g to e. It will be noticed in the diagram Fig. 35 that there is but little sheer on the web ; therefore the max- imum (6 inches) spacing should be adopted. 70 COMPOUND RIVETED GIRDERS. The rivet spacing from e to R will be regulated by the horizontal strain at that point. At e, M = 737.625 ton-feet. Horizontal strain = ; __ 737 1 _5 _ 245.875 tons or 491,750 pounds, and then divided by 45;o = 109 rivets, to be placed a distance of 1 20 inches for one or 240 inches for both webs, spaced about 2^ inches (staggered) from e to R support. Graphical Representation for Two Concentrated Loads and a Uniform Load. In the following diagram, Fig. 35, we n FIG. 35- have a combination of the previous examples. The uniform load to be considered is a system of equal and equidistant loads EXAMPLE V. 71 close together, as in Fig. 28. In determining the reaction on the support, we set the given loads W^ , W^ , W^ , . . . W 19 off in succession along the line PP. The uniform load of 80 tons being divided into 16 parts, each equal to 5 tons, set W^ of 5 tons at i ; W^ of 5 tons at 2 ; W^ of 5 tons at 3 ; W t of 5 tons at 4 ; W b of 5 tons at 5 ; W t , the concentrated load, at 6 ; and so on to the end. The line PP' is then the polygon of the forces, and P'P, the closing line, is the resultant. Take any point O as a pole equal to ten units of the scale adopted, and draw the, radii OP, Oi, (92, #3, (94, (95, . . . (9i8. Then describe the funicular polygon abed . . . stuv by drawing ab parallel to OB, be parallel to Oi, terminating in the prolong- ation of W^ ; cd parallel to (92, terminating in W^ produced ; . . . finally, uv parallel to (9i8, terminating in the prolongation downwards of R r . The funicular polygon is now closed by the line va, and a line OS is drawn through the pole O parallel to va. Then as a condition of equilibrium the reaction at R = PS, and at R' = SP r . Any ordinate from av in the funicular polygon and meas- ured to the inclined lines ab, be, cd, de, . . . uv, multiplied by the pole distance OH, will give the bending moment at any point in the girder. The shearing forces on the web can be measured on the hatched figure as explained in our previous examples. The greatest shear is at the bearings, and extends from the con- centrated loads W 6 and W 13 to R and R' '. There is little or no shear on the web in centre of the girder. The Area of Flanges reduced by the Funicular Poly- gon. Construct the diagram Fig. 36, by drawing the polygon RDR' similar to the funicular polygon Fig. 35. From a point on the closing line RR f directly under the 72 COMPOUND RIVETED GIRDERS. maximum bending moment at F t draw FD at the same point ; place the scale at any angle until it meets DD' perpendicular to FD at e, measuring 45.12 square inches of the top flange. For the two angles set off 5.86 square inches each at a and b, then one plate 16 X f or 12 square inches at c, two plates ** J FIG. 36. 16 X yi or ll square inches each at i "H. 2? X 20 At/ for W a , M= 70 X = 600 ton-feet. J At/, for W lt M=40X = 1 14.28 At/ for ^ 3 , J/= 60 X $ = 257.14 " Max. M = 971.42 " Area of flange = ^-~ = 6 4-?6 square inches. Construction of Flanges. To make up the 64.76 square inches in the flanges at / we would require : Top flange 2 angles 6" X 4" X f" = 11.72 square inches, i plate 20" X i" = 15.0 " " 3 plates 20" X f " =37-5 Total, 64.22 .54 of an inch less than required. For the bottom flange we deduct rivet-holes. Then using J-inch-diameter rivets and allowing J inch more, we have for one hole f " + |" + f " + |" + f " .X i'.' = % 6 -, for two holes iyt X 2 = y = 6| square inches. On account of the closeness of the rivets it will be noticed, by referring to the girder drawing Fig. 41, another hole in the vertical leg of the angle will require to be deducted ; we have then for one hole f X i = f, for two holes | X 2 = ^ = i square inches, to be added EXAMPLE VI. 77 to the bottom flange at/, or 64.76 + 6.375 + 1.25 = 72.385 square inches. Then Bottom flange: 2 angles 6" X 4" X I" = 11.72 sq. inches. 4 plates 20" X f " = 60 Total, 71.72 " " Webs. To find the right reaction at R f , the centre of moments is taken at the left support. Then the equation of moments is ^' X 35 =40 X 5 + 70 X 15 + 60 X 25, from which R' = - = 78.57 tons. In like manner to find J^y the centre of moments is taken at the right support, and R X 35 = 6 X 10 + 70 X 20 + 40 X 30, from which R = = 91.43 tons. As a check, the sum of R' and R is seen to be 170 tons. Then the thickness of the web becomes At /?', / = = ' 8 . nearl of an inch - 182860 ='-'5, say i inch. Adopting the greater thickness and having two webs, each will be inch thick. 78 COMPOUND RIVETED GIRDERS. Stiffeners. To determine whether we require stiffeners : Safe resistance to buckling IOOOO 3Q x 3Q = 4545 pounds per square inch. " 3000 X i X i This being less than 6000 pounds, the safe shear per square inch, the webs will require stiffeners at bearings, under each concentrated load, and the intervening distances from R to W^ , W, to W z , W 3 to R', every three feet. No stiffeners are required between W^ and W y Refer to Diagram Fig. 40. Then to stiffen web at the above-stated points, place a 4" X 4" X " angle on outside of each web, and the formula becomes : Safe resistance to buckling IOOOO 3 = 7184 pounds per square inch. 3000 X f X f Rivets. The safe shearing area of a ^-inch-diameter rivet from table = 4510 pounds. The maximum bending moment at/": M = 971.42 ton-feet. Horizontal strain = - = 388.66 tons or 777,320 pounds. 2.5 This divided by 4510 =170 rivets, to be placed a distance of 1 80 or 360 inches in both webs, spaced about 2j inches (staggered) from /to R support. EXAMPLE VI. The maximum bending moment 79 M 785.71 ton-feet. Horizontal strain = = 314.4 tons or 629,000 pounds. Then divided by 4510 = 139 rivets, to be placed a distance of 1 20 inches for one or 240 inches for both webs, spaced about if inches (staggered) from g to R'. The rivets to be spaced between W^ and W 3 , 6 inches centres. Flange Plates reduced in Area towards the Sup- ports. Draw the diagram Fig. 39, as in our previous examples, upon the span R R ', making the polygon RBDGR' similar to the bending moment polygon, Fig. 38. At F, 15 feet from R support, draw FD equal to the maximum. :,,,; FIG. 39- Bending moment due to the sum of the load W l , W,. , and W at that point, or 971.42 ton-feet. Draw the rectangle RCER' 8o COMPOUND RIVETED GIRDRRS. then from F place the scale at any angle, as at Ff t until it measures 64.76 square inches of the top flange. For two angles set off 5.86 square inches each at a and b ; one plate 20" X f ", or 15 square inches, at c ; 3 plates 20" X f ", or 1.25 square inches, each at e, d, and f. Horizontal lines drawn from a, b, c, d, e, and /to the polygon and carried down to base line RR' will give the position of plates in each flange. The angles and adjoining plates to extend the full length of girder. The plates to extend over the calculated lengths, equal to two cross-lines of rivets. Graphical Representation of the Bending Moments and Shearing Forces for Three Concentrated Loads. We have in this example a girder of 35 feet span, to sustain a FIG. concentrated load of 40 tons 5 feet from R support, 70 tons 20 feet, and 60 tons 10 feet, from R f support. We shall first EXAMPLE VI. 8 1 determine, as in our previous examples, the pressure on the supports. Set the given loads W l , W^, and W 3 , Fig. 40, off in succes- sion along a line PP' : W l of 40 tons at i, W^ of 70 tons at 2, and IV 3 of 60 tons at 3. The line PP' is thus the polygon of the forces W v , W^ , and W % , and P'P, its closing line, is their resultant. Take any point O as pole, equal to ten units of the scale adopted, and draw the radii OP, Oi, O2, O$. Then describe the funicular polygon abcde by drawing ab parallel to OP, terminating in W l produced, be parallel to Oi, cd parallel to (92, terminating in the prolongation of W^ and W 9 respectively, and de parallel to #3, terminating in the prolonga- tion downwards of R f . The funicular polygon is now closed by the line ea, and a line OS is drawn through the pole O parallel to ea. Then as a condition of equilibrium, the reaction at R = PS, and at R' = SP. The bending moments at any point in the girder can be measured by the ordinates of the funicular polygon multiplied by the pole distance OH. The shearing forces at any point on the webs to be measured from the hatched figures. The maxi- mum shear on the webs is at R support, and is equal by scale to 91.43 tons, the same as previously calculated. 82 COMPOUND RIVETED GIRDERS. EXAMPLE VIL 83 EXAMPLE VII. GIRDER SUPPORTING FOUR CONCENTRATED LOADS. The bending moment at any point produced by all the loads is the sum of the moments produced at that point by each of the loads separately, ^eing the same as two and three con- centrated loads. Example : What metal area would be required in the flanges of a box girder of 40 feet span, 3 feet in depth, to sustain 20 tons concentrated 5 feet and 60 tons 15 feet from left sup- port, 50 tons 10 feet and 30 tons 5 feet from right support, the girder to be 20 inches in breadth. Draw the triangles, Fig. 42, having vertices at n, t, *, and q, representing bending moments by = W X *? b for loads W l9 W^W % , and W, (as in Ex. II). At g, for W, , M - 20 X 12U1_ = 87.5 ton-feet 40 At i, for W 3 , M = 50 X IOX3 = 375 40 At , for W, , M = 30 X - 5 X 35 = 131.25 " 40 Draw the vertices gn, ht, ix, and kq by scale equal to 87.5, 562.5, 375, and 131.25 ton-feet respectively, and connect each 8 4 COMPOUND RIVETED GIRDERS. with R and R f . Extend gn to b, equal to gl -|- gm -f- go ; ex- tend ht to r, equal to hp -\- hr + hs ; extend ix to and k in girder by loads W l , W 9 , W^ , and W^ 4 . Then at g, M = 356.25 ton-feet ; at i, M 781.25 ton-feet, at h, M 868.75 " at&,M= 737.50 The maximum bending moment due to the several loads it can be seen at a glance on the diagram is under the load at h+ EXAMPLE VII. 85 the point in the girder requiring the greatest amount of metal in the flanges. At ,Jlf= 868.75. = 48.27 square inches in the top flange. 3X6 Construction of Flanges. To make up the required sec- tion we will construct the section as follows : Top flange = 2 angles 5" X 3i" X H" = IO -74 sq. inches. 3 plates 20" X f" = 37-5 Total, 48.24 For the bottom flange we deduct the rivet-holes, then ^"-diameter rivets and allowing -J" more, we have for one H" + y + f" + f" x i" = fi", for two holes H x 2 II = 5 j. square inches, to be added to the bottom flange at h, or 48.24 + 5.125 = 53.365 square inches. Bottom flange = 2 angles 5" X 3i" X H" = IO -74 S( l- inches. 2 plates 20" X f " = 30.00 " i " 20" X f r/ = 12.5 53.24 " Webs. To find the shear on the web at R', the centre of moments is taken at R. Then R' X 40 = 20 X 5 + 60 X 15 + 5o X 30 + 30 X 35, from which R' 88.75 tons = 177,500 pounds. In like manner to find R, the centre of moments is taken Then R X 40 = 30 X 5 + 5o X 10 + 60 X 25 + 20 X 35, 86 COMPOUND RIVETED GIRDERS. from which R 71.25 tons or 142,500 pounds. As a check, the sum of R' and R is seen to be 160 tons, which is the sum of the four loads. The thickness of web is then determined by the greatest shear. Having two webs, each will be T V thick. Stiffeners. To determine whether we require stiffeners : Safe resistance to buckling 10000 x- , . . = 3069 pounds per square inch. , 36 X 36 "*" 3000 x T V X T V Having adopted 6000 Ibs. per square inch for safe shearing, the webs will require to be stiffened at the bearings, also under each concentrated load and between W z and W^ , W^ and R! support, two stiffeners between W^ and W l , and one between W l and R support. Between loads W t and W 3 there is very little shear (see diagram Fig. ). By riveting 4" X 4" X &" angeles on the outside of webs, we have for the thickness of each web ^V -|- -$" = -j-f-". Then the formula becomes IOOOO zr 11 - i = 0392 Ibs. per square inch. 36 X 36 Rivets. Using f-inch-diameter rivets, the safe shear per square inch from table equals 45 10 Ibs. EXAMPLE VII. 87 The maximum bending moments at h\ M = 868.75 ton-feet. Horizontal strain = -^^ = 289.58 tons or 579,160 pounds, o This divided by 4510= 128 rivets, to be placed a distance of 1 80 in one or 360 inches in both webs, spaced about 2-J-f inches from h to R support. The maximum bending moment at i : M= 781.25 ton-feet. Horizontal strain = = 260.41 tons or 520,820 pounds. This divided by 4510 = 115 rivets, to be placed a distace of 1 80 in one or 360 inches in both webs, spaced about 3^ inches from i to R r support. The rivets between h and i to be spaced 6 inches centres. Flange Plates Reduced in Area towards the Sup- ports. Draw the Diagram Fig. 43 equal and similar to the polygon Fig. 42, a. &, c, d, e^ and f representing the bending moments due to the sum of all the loads at the points^-, h, i, and k. Draw the rectangle RCER '. Then from F place the scale at any angle, as at Fe, until it measures 48.2 square inches of the top flange. For two angles set off 5.37 square inches each at a and b\ three plates 20" X f" or 12.5 square inches each at c, dj and e. Horizontal lines drawn from a, b, c, d, and e to the polygon and carried down to base line RR will give the position of plates in each flange. The angles and adjoining plates to extend the full length of girder. 88 COMPOUND RIVETED GIRDERS. The plates to extend over the calculated lengths equal to four cross-lines of rivets. FIG. 43. Graphical Representation of the Bending Moments and the Shearing Forces for Four Concentrated Loads. We have in this example a girder of 40 feet span to sustain concentrated loads of 20 tons 5 feet and 60 tons 15 feet from R support, 50 tons 10 feet and 30 tons 5 feet from R' support. Set the given loads W, , W,, W 3 , and W^, Fig. 44, off in succession along a line PP ': W^ of 20 tons at I, W^ of 60 tons at 2, W^ of 50 tons at 3, and W 4 of 30 tons at 4. The line PP f is thus the polygon of the forces W^ , W^ , Wi, and W t , and PP y its closing line, is their resultant. Take any point O as pole, equal to ten units of the scale adopted, and draw the radii OP, Oi y (92, (93, 04. Then describe the funicular polygon EXAMPLE VII. 8 9 abcdef, by drawing ab parallel to OP, terminating in W, pro- duced ; be parallel to Oi, cd parallel to 02, de parallel to 3, ef parallel to 04, terminating in the prolongation of R f down- FIG. 44. wards. The funicular polygon is now closed by the line fa, and a line OS is drawn through the pole O parallel to fa. Then as a condition of equilibrium the reaction at R = PS, and at R' SP. The bending moments at any point in the girder can be meas- ured by the ordinates of the funicular polygon multiplied by the pole distance OH. The shearing forces on the webs to be measured from the hatched figure. COMPOUND RIVETED GIRDERS. o. . N o* irico O W N CO oop^oopoop^oo^o co 1-1 co mco OOOOOCO OOOOO - N mi^c>M cor>r^OM co xr>oo O N "1-vO oo O oo ON rtr^o eomoo o N TJ-I^O*-. tou-)oo TtOOOOOQOp^OO O M d co ^* u~ CO O N TfO CO OOOQOO MC7^oovOvn'^- O O r^-O O^ O M N N CO Tt tr>\O O N -I-CQ co M comt->.O>t-' com *"" Hi-iH?i-iMNNN ) NNCOCOCOCOCo5-"!l- coco SSR* NO Mvno\co NNNNNNCOCOCOCOCOTr CO O OO t^Tt OOOOOO Tj- O* Tj-CO COCO COt^N t^NO M inooo O>M N Tj-ior^co O M co T*-O^-Mxr>MTt-MrnMTl-M-l-G 1 <3-Ow>OOO'3- co ^o t^*oo O *~ to ^J~ to r^ co rf O O N COO co O co in * O N T}-\O O^ >~> co\O CO O co^mr^co O^O N corj-mOco c^o M co M Q>-'MMi-tO>-"ONOcoco^-'^- mco O O O N co Tf mO i~^co O O *-> N co Tj- mo l>cp O Q N co O r^co O O M N co Tf- mo r^co o O N r^co O O N co ^r mo i^co o O M co -3- M N co Tf m MOl^O'^-coi-tO o t^co CT>O M N co coco^t 100 r>co OOcoOO'^->-">nOO>-">-'eot^kHO O ^t M OO COM mcooco mN OO r^ *^ w co co ^* mo r^co o O O *^ M c< N coNi-iOOoor^OOmrtcONMOOOOOOmT}- M M C 1 ! N CO CO ^ *^" vo mO O I s ** t^CO CO O O O O M M corfmo t>.co OO M N corfmo t^< 8.13 8.75 9-38 IO.OO 10.63 11-25 11.88 12.50 I3-I3 13-75 3* 8.13 8.80 9.48 10. 16 10.83 11.51 12.19 12.86 *3 54 14.22 14.90 3* 8-75 9.48 IO.2I 10.94 II .67 12.40 13-13 13-85 14.58 15-31 16.04 3 9-38 10. 16 10.94 11.72 12.50 13.28 14.06 14.84 15-63 16.41 17.19 4 IO.OO 10.83 11.67 12.50 13-33 14.17 15.00 15-83 16.67 I7-50 18.33 4i 10.63 11-51 12.40 13-28 14.17 15-05 15-94 16.82 17.71 18.59 19.48 4* 11.25 12.19 13- *3 14.06 15.00 15-94 16.88 17.81 18.75 19.69 20.63 4t 11.88 12.86 13-85 14.84 15.83 16.82 17.81 18.80 19.79 20.78 21-77 5 12.50 13-54 14-58 15-63 16.67 17.71 18.75 TO fio 19.79 2O 78 20.83 21.88 21.88 22 .92 5t 3 J 3- J 3 13-75 14-38 14.22 14.90 15-57 16.04 16.77 17.19 17.97 18.33 19.17 18.59 19.48 20.36 i y . cy 20.63 21.56 21-77 22.76 22.92 23-96 22.97 24.06 25-16 25-21 26.35 6 15.00 16.25 !7-5 18.75 20.00 21.25 22.50 23-75 25.00 26.25 27.50 6} 15-63 16.93 18.23 19-53 20.83 22.14 23-44 24.74 ! 26.04 27-34 28 6 5 6* 16.25 17.60 18.96 20.31 21.67 23.02 24.38 25-73 27.08 28.44 29.79 6* 16.88 18.28 19.69 21 .09 22.50 23.91 25.31 26.72 28.13 29-53] 30.94 7 !7-5 18.96 20.42 21.88 23-33 24.79 26.25 27.71 29.17 30.62 32.08 7i 18.13 19.64 21.15 22.66 24.17 25.68 27.19 28.70 30-21 31-72 33-23 ?! 18.75 19.38 20.31 20.99 21.88 22.60 23.44 24.22 25.00 25-83 26.56 27-45 28.13 29.06 29.69 30.68 31-25 32-29 32.81 33-91 34.38 3S-5 2 8 20.00 21.67 23-33 25.00 26.67 28.33 30.00 3 I - 6 7 33-33 35-0 36-67 8i 20.63 22.34 24.06 25.78 27.50 29.22 30-94 32.66 34.38 36.09 37-8i 8i 21.25 23.02 24.79 26.56 28.33 30.10 31-88 33.65 35-42 37-19 38.96 8* 21.88 23.70 25-52 27-34 29.17 30-99 32.81 34-64 36-46 38.28 40. 10 9 22.50 24.38 26.25 28.13 30.00 31-88 33-75 35.63 37-50 39.38 41.25 9i 2 3-i3 25-05 26.98 28.91 30.83 32.76 34.69 36.61 38.54 40.47 42.40 9 23-75 25-73 27.71 29.69 31.67 33-65 35.63 37.60 39.58 41.56 43-54 9f 24.38 26.41 28.44 3 -47 32.50 34-53 36.56 38.59 40.63 42.66 44.69 10 25.00 27.08 29.17 31-25 33 '33 35-42 37-50 39.58 41.67 43-75 45-83 10* 25.62 27.76 29.90 32-03 34->7 36-30 38.44 40.57 42.71 44.84 46.98 ioi 26.25 28.44 30-63 32.81 35-0 37-*9 39-38 41.56 43-75 45-94 48.13 I0j 26.88 29.11 31-35 33-59 35.83 38.07 40.31 42-55 44-79 47-03 49-27 II 27.50 29.79 32.08 34.38 36.67 38-96 4 T - 2 5 43-54 45.83 48.13 50.42 It* 28.13 30.47 32-81 35 - l6 37-50 39-84 42.19 44-53 46.88 49.22 5I-56 "1 28.75 29.38 3i-i5 31.82 33-54 34-27 35-94 36.72 38.33 39 i7 40-73 41.61 43-13 44.06 45-52 46-51 47.92 48.96 50-31 5i-4i 52-71 53.85 12 30.00 32.50 35.00 37-5 40.00 42.50 45-00 47-50 50.00 52-5 55-00 TABLE OF SQUARES AND CUBES. Ill SQUARES AND CUBES, OF NUMBERS FROM 1 TO 440. No. Squares. Cubes. No. Squares. Cubes. I I I 56 31 36 175 616 2 4 8 57 3249 185 193 3 9 27 58 3364 195 112 4 16 64 59 3481 205 379 5 25 125 60 36 oo 216 ooo 6 36 216 61 37 21 226 981 7 49 343 62 3844 238 328 8 64 512 63 3969 250 047 9 81 729 64 40 96 262 144 10 I 00 I 000 65 42 25 274 626 ii I 21 i 33i 66 43 56 287 496 12 144 i 728 67 44 89 300 763 13 I6 9 2 197 68 4624 314432 14 I 9 6 2-744 69 47 61 328 509 15 2 25 3 375 70 4900 343 ooo 16 2 5 6 4096 7i 5041 357 9 1 * 17 2 89 49*3 72 5i 84 373 248 18 3 24 5832 73 53 29 389 017 1 9 361 6859 74 5476 405 224 20 4 oo 8000 75 56 25 421 875 21 441 9 261 76 57 76 438 976 22 484 10 648 77 5929 456 533 23 5 29 12 167 78 60 84 474 552 24 576 13 824 79 62 41 493 039 25 625 15 625 80 64 oo 512 ooo 26 6 76 17 576 81 6581 531 441 27 7 29 19 683 82 67 24 551 368 28 7 84 21 952 83 68 89 571 787 29 8 41 24389 84 7056 592 704 30 9 oo 27 ooo 85 72 25 614 125 31 9 61 29 79 * 86 73 9 6 636 056 32 10 24 32768 87 75 69 658 503 33 10 89 35 937 88 7744 681 472 34 ii 56 39 304 89 79 21 704 969 35 12 25 42 875 90 81 oo 729 ooo 36 12 96 46 656 9i 8281 753 57i 37 1369 50653 92 8464 778 688 38 1444 54 872 93 8649 804 357 39 15 21 59319 94 88 36 830 584 40 16 oo 64 ooo 95 9025 857 375 41 16 81 68 921 96 92 16 884 736 42 17 64 74088 97 9409 912 673 43 1849 79 507 98 9604 941 192 44 1936 85 184 99 98 oi 970 299 45 20 25 91 125 100 oo oo 000000 46 21 l6 97 336 IOI O2 OI 030 301 47 22 09 103 823 102 04 04 06 1 208 48 2304 no 592 103 06 09 092 727 49 24 oi 117 649 IO4 08 16 124 864 50 25 oo 125 ooo 105 10 25 157 625 51 26 oi 132 651 106 12 36 191 016 52 27 04 140 608 107 1449 225 043 53 28 09 148 877 108 16 64 259712 54 29 16 157 464 109 18 81 295 029 55 30 25 i 66 375 i no 21 00 331 ooo 112 TABLE OF SQUARES AND CUBES. SQUARES AND CUBES Continued. No. Squares. Cubes. No. Squares. Cubes. Ill 23 21 I 367631 166 275 56 4 574 296 112 25 44 404 928 167 2 78 89 4 657463 H3 2769 442 897 168 2 82 24 4 741 632 114 2996 481 544 169 2 85 61 4 826 809 H5 32 25 520 875 170 2 89 00 4 913 ooo 116 34 56 560 896 171 2 9241 5 ooo 211 117 3689 601 613 172 295 84 5 088 448 118 3924 643 032 173 2 9929 5 177 717 119 41 61 685 159 174 302 76 5 268 024 1 20 44 oo 728 ooo 175 3 06 25 5 359 375 121 4641 771 561 176 3 09 76 5 45i 776 122 4884 815 848 177 3 1329 5 545 233 123 5i 29 860 867 178 3 1684 5 639 752 124 53 76 906 624 179 3 2041 5 735 339 125 56 25 953 125 180 3 24 oo 5 832 ooo 126 5876 2 OOO 376 181 3 2761 5 929 74i 127 61 29 2 048 383 182 3 3i 24 6 028 568 128 6384 2 097 152 183 33489 6 128 487 I2Q 6641 2 146 689 184 3 38 56 6 229 504 130 69 oo 2 197 000 185 342 25 6 331 625 *3I 71 61 2 248 Ogi 186 345 96 6 434 856 132 7424 2 299 968 187 34969 6 539 203 133 7689 2 352 637 188 3 53 44 6 644 672 134 79 56 2 406 IO4 189 3 57 21 6 751 269 135 82 25 2 460 375 190 3 61 oo 6 859 ooo 136 8496 2 515 456 191 3 64 81 6 967 871 137 8769 2 571 353 192 36864 7 077 888 138 9 44 2 628 072 193 3 7249 7 189057 139 93 21 2 685 619 194 3 76 36 7 301 384 I4O 96 oo 2 744 ooo 195 38025 74M 875 141 98 81 2 803 221 196 3 84 16 7 529 536 142 2 01 64 2 863 288 197 3 88 09 7 645 373 143 2 04 49 2 924 2O7 198 3 9204 7 762 392 144 2 07 36 2 985 984 199 3 96 01 7 880 599 M5 2 IO 25 3 048 625 200 4 oo oo 8 ooo ooo 146 2 13 16 3 112 136 20 1 4 04 01 8 120 601 1 17 2 16 09 3 176 523 202 4 08 04 8 242 408 148 2 1904 3 241 79 2 203 4 1209 8 365 427 149 2 22 01 3 307 949 204 4 16 16 8 489 664 150 2 25 00 3 375 ooo 205 42035 8 615 125 151 2 28 OI 3 442 95i 206 4 24 36 8 741 816 152 2 31 04 3 511 808 207 4 28 49 8 869 743 153 23409 3 58i 577 208 4 3264 8 998 912 154 2 37 16 3 652 264 20 9 4 36 81 9 129 329 155 2 40 25 3 723 875 2IO 441 oo 9 261 ooo 156 243 3 6 3 79 6 4i6 211 445 21 9 393 93i 157 2 4649 3 869 893 212 44944 9 528 128 158 2 4964 3 944 3'2 213 4 5369 9 663 597 159 2 52 81 4 019 679 214 4 57 9 6 9 800 344 1 60 2 56 00 4 096 ooo 215 4 62 25 9 938 375 161 2 5921 4 173 281 216 4 66 56 10 077 646 162 2 62 44 4 251 528 217 4 7089 10 218 313 163 2 65 69 4 330 747 218 4 75 24 10 360 232 164 26896 4 410 944 219 4 7961 10 503 459 165 2 72 25 4 492 125 22O 4 84 oo 10 648 ooo TABLE OF SQUARES AND CUBES. SQUARES AND CUBES Continued. No. Squares. Cubes. No. Squares. Cubes. 221 4 88 41 10 793 861 276 7 61 76 21 024 576 222 492 84 10 941 048 277 7 67 29 21 253 933 223 497 29 ii 089 567 278 7 72 84 21 484 952 224 5 oi 76 ii 239 424 279 7 7841 21 717 639 225 506 25 ii 390 625 280 7 8400 21 952 000 226 5 10 76 ii 543 176 281 7 89 61 22 188 041 227 5 15 29 ii 697 083 282 7 95 24 22 425 768 228 5 19 8 4 ii 852 352 283 8 oo 89 22 665 187 22 9 5 2441 12 008 989 28 4 806 56 22 906 304 230 5 2900 12 167 OOO 285 8 12 25 23 149 125 231 5 33 61 12 326 391 286 8 17 96 23 393 656 232 5 38 24 12487 168 287 8 23 69 23 639 903 233 5 42 89 12 649 337 288 8 29 44 23 887 872 234 547 56 12 812 904 289 8 35 21 24 137 569 235 5 52 25 12 977 875 290 8 41 oo 24 389 ooo 236 5 5696 13 144 256 291 8 46 81 24 642 171 237 5 61 69 13 312053 292 8 5264 24 897 088 238 5 6644 13 481 272 293 8 58 49 25 153 757 239 5 7i 21 13 651 919 294 8 64 36 25 412 184 240 5 7600 13 824 ooo 295 8 70 25 25 672 375 241 5 80 Si 13 997 521 296 8 76 16 25 934 336 242 58564 14 172 488 297 8 82 09 26 198 073 243 5 9 49 14 348 907 298 8 88 04 26 463 592 244 5 95 36 14 526 784 299 8 94 oi 26 730 899 245 6 oo 25 14 706 125 300 9 oo oo 27 ooo ooo 246 6 05 16 14 886 936 301 9 06 oi 27 270 901 247 6 10 09 15 069 223 302 9 12 04 27 543 608 248 6 15 04 15 252 992 303 9 18 09 27 818 127 249 6 20 oi 15 438 249 304 9 24 16 28 094 464 250 6 25 oo 15 625 ooo 305 9 30 25 28 372 625 251 6 30 oi 15 813 251 306 936 36 28 652 616 252 6 35 04 16 003 008 307 94249 28 934 443 253 6 40 09 16 194 277 308 948 64 29 218 112 254 6 45 16 16 387 064 309 9 548i 29 503 629 255 6 50 25 16 581 375 310 9 61 oo 29 791 ooo 256 6 55 36 16 777 216 3ii 967 21 30 080 231 257 6 60 49 16 974 593 312 9 7344 30 371 328 258 665 64 17 173 512 313 97969 30 664 297 259 6 70 81 17 373 979 314 98596 30 959 144 260 6 76 oo 17 576 ooo 315 992 25 31 255 875 261 6 81 21 17 779 581 316 998 56 31 554 496 262 6 8644 17 984 728 317 10 04 89 3i 855 013 263 6 91 69 18 191 447 3i8 10 ii 24 32 157432 264 69696 i 8 399 744 319 10 17 61 32 461 759 265 702 25 18 609 625 320 10 24 oo 32 768 ooo 266 706 56 18 821 096 321 10 30 41 33 076 161 267 7 12 89 19 034 163 322 10 36 84 33 386 248 268 7 18 24 19 248 832 323 10 43 29 33 698 267 269 7 23 61 19 465 109 324 10 49 76 34 012 224 270 72900 19 683 ooo 325 10 56 25 34 328 125 271 7 3441 19 902 511 326 10 62 76 34 645 976 272 7 3984 20 123 648 327 10 69 29 34 965 783 273 745 29 20346417 328 10 75 84 35 287 552 274 7 5076 2O ;>7O 824 329 10 82 41 35 611 289 275 7 5625 20 796 875 330 10 89 oo 35 937 ooo TABLE OF SQUARES AND CUBES. SQUARES AND CUBES Continued. No. Squares. Cubes. No. Squares. Cubes. 331 10 95 6l 36 264 691 386 14 89 96 57 512456 332 II O2 24 36 594 368 387 14 97 69 57 960 603 333 II 08 89 36 926 037 388 15 05 44 58 411 072 334 II 15 56 37 250 704 389 15 13 21 58 863 869 335 II 22 25 37 595 375 390 15 21 OO 59 319 ooo 336 II 28 96 37 933 056 391 15 28 Si 59 776471 337 ii 35 69 38 272 753 392 15 36 64 60 236 288 338 ii 42 44 38 614472 393 15 44 49 60 698 457 339 ii 49 21 38 958 219 394 15 52 36 61 162 984 340 ii 56 oo 39 304 ooo 395 15 60 25 61 629 875 341 ii 62 81 39 651 821 396 15 68 16 62 099 136 342 i i 69 64 40 ooi 688 397 15 76 09 62 570 773 343 ii 7649 40 353 607 398 15 84 04 63 044 792 344 ii 83 36 40 707 584 399 15 92 01 63 521 199 345 ii 9025 41 063 625 400 16 oooo 64 ooo ooo 346 ii 97 16 41 421 736 401 16 08 01 64 481 201 347 12 04 09 41 781 923 402 16 16 04 64 964 808 348 12 II 04 42 144 192 403 16 24 09 65 450 827 349 12 18 01 42 508 549 404 16 32 16 65 939 264 350 12 25 OO 42 875 ooo 405 16 40 25 66 430 125 35i 12 32 01 43 243 55 ! 406 16 48 36 66 923 416 352 12 39 04 43 614 208 407 16 56 49 67 419 143 353 12 46 09 43 986 977 408 16 64 64 67 917 312 354 12 53 16 44 361 864 409 16 72 81 68 417 929 355 12 60 25 44 738 875 410 16 81 oo 68 921 ooo 356 12 67 36 45 118 016 411 16 89 21 69 426 531 357 12 74 49 45 499 293 412 16 97 44 69 934 528 358 12 81 64 45 882 712 413 17 05 69 70 444 997 359 12 88 81 46 268 279 414 17 13 9 6 70 957 944 360 12 96 OO 46 656 ooo 415 17 22 25 7i 473 375 361, 13 03 21 47 045 881 416 17 30 56 71 991 296 362 13 10 44 47 437 928 417 17 38 89 72 511 713 363 13 17 69 47 832 147 418 17 47 24 73 034 632 364 13 24 9 6 48 228 544 419 17 55 61 73 560 059 365 13 32 25 48 627 125 420 17 64 oo 74 088 ooo 366 13 39 56 49 027 896 421 17 72 41 74 618 461 367 13 46 89 49 430 863 422 17 80 84 75 151 448 368 13 54 24 49 836 032 423 17 89 29 75 686 967 369 13 61 61 50 243 409 424 17 97 76 76 225 024 370 13 69 oo 50 653 ooo 425 18 06 25 76 765 625 37i 13 76 41 51 064 811 426 18 14 76 77 308 776 372 13 83 84 51 478 848 427 18 23 29 77 854 483 373 13 91 29 5i 895 117 428 18 31 84 78 402 752 374 13 98 76 52 313 624 429 18 40 41 78 953 589 375 14 06 25 52 734 375 430 1 8 49 oo 79 507 ooo 376 14 13 76 53 157 376 431 18 57 61 80 062 991 377 14 21 29 53 582 633 432 18 66 24 80 621 568 378 14 28 84 54 oio 152 433 18 74 89 81 182 737 379 14 36 41 54 439 939 434 18 83 56 8 1 746 504 380 14 44 oo 54 872 ooo 435 18 92 25 82 312 875 38i 14 51 61 55 306 341 436 19 oo 96 82 881 856 382 14 59 24 55 742 968 437 19 09 69 83453453 383 14 66 89 56 181 887 438 19 18 44 84 027 672 384 14 74 56 56 623 104 439 19 27 21 84 604 519 385 14 82 25 56 066 625 440 19 36 oo 85 184000 TABLE OF CIRCLES. 11$ THE CIRCUMFERENCE AND AREAS OF CIRCLES FROM 1 TO 50. Diam. Circumf. Area. Diam. Circumf. Area. Diam. Circumf. Area. I-6 4 .049087 .00019 2. 1-16 6.47953 3-34TO 5. 3-16 16.2970 21.135 1-32 .098175 .00077 1-8 6.67588 3.5466 1-4 16.4934 21.648 3-64 . 147262 .00173 3-16 6.87223 3-7583 5-i6 16.6897 22. 1 66 1-16 .196350 .00307 1-4 7.06858 3.9761 3-8 16.8861 22.691 3-32 .294524 .00690 5-i6 7.26493 4.2OOO 7-16 17.0824 23.221 1-8 .392699 .01227 3-8 7.46128 4-4301 1-2 17-2788 23-758 5-32 .490874 .01917 7-16 7.65763 4.6664 9-16 17.4751 24.301 3-16 .589049 .02761 1-2 7.85398 4.9087 5-8 I7-67I5 24.850 7-32 .687223 03758 9-l6 8.05033 5.I572 11-16 17.8678 25.406 1-4 .785398 .04909 5-8 8.24668 5.4II9 3-4 18.0642 25.967 9-32 .883573 .06213 11-16 8.44303 5.6727 13-16 18.2605 26.535 5-i6 .981748 .07670 3-4 8.63938 5.9396 7-8 18.4569 27.109 11-32 1.07992 .09281 13-16 8.83573 6.2T26 15-16 18 6532 27.688 3-8 1.17810 .11045 7-8 9.03208 6.4918 6. 18.8496 28.274 13-32 1.27627 .12962 15-16 9.22843 6.7771 1-8 19-2423 29.465 7-16 1-37445 15033 3. 9.42478 7.0686 1-4 19-6350 30.680 15-32 1 .47262 .17257 1-16 9.62113 7.3662 3-8 20.0277 3i-9 T 9 1-2 1.57080 19635 1-8 9.81748 7.6699 1-2 2O.42Oz 33-183 17-32 1.66897 .22166 3-i6 0.0138 7.9798 5-8 20.8I3I 34.472 9-16 1.76715 .24850 1-4 0.2102 8.2958 3-4 21.2058 35.785 19-32 1.86532 .27688 5-16 0.4065 8.6179 7-8 21.5984 37.122 5-8 1.96350 . 30680 3-8 0.6029 8.9462 7. 21.9911 38.485 21-32 2.06167 33824 7-16 0.7992 9 . 2806 1-8 22.3838 39.871 11-16 2.15984 .37122 1-2 0.9956 9.6211 1-4 22.7765 41.282 23-32 2.25802 .40574 9-16 II.I9I9 9.9678 3-8 23.1692 42.718 3-4 2.35619 .44179 5-8 H.3883 10.321 1-2 23.56lg 44.179 25-32 2-45437 47937 11-16 11.5846 10.680 5-8 23.9546 45-664 13-16 2.55254 .51849 3-4 II.78IO 11.045 3-4 24.3473 47-173 2732 2.65072 559*4 13-16 n-9773 11.416 7-8 24.7400 48.707 7-8 2.74889 .60132 7-8 12.1737 11-793 8. 25.1327 50.265 29-32 2.84707 . 64504 15-16 12.3700 12.177 1-8 25.5254 51.849 15-16 2.94524 .69029 4. 12.5664 12.566 1-4 25.918 53.456 31-32 3.04342 .73708 1-16 12.7627 12.962 3-8 26.310 55.088 1. 3.I4I59 .78540 1-8 12.9591 13-364 1-2 26.703 56.745 1-16 3-33794 .88664 3-i6 I3.I554 13.772 5-8 27.096 58.426 1-8 3.53429 .99402 1-4 13.3518 14.186 3-4 27.4880 60. 132 3-i6 3 73064 1075 5-i6 13-5481 14.607 7-8 27.8816 61.862 1-4 3.92699 .2272 3-8 13-7445 15.033 9. 28.274 63.617 5-16 4-12334 .3530 7-16 13.9408 15.466 1-8 28.6670 65-397 3-8 4.31969 4849 1-2 14.1372 15.904 1-4 29.059 67.201 7-16 4.51604 .6230 9-ie 14-3335 16.349 3-8 29-452-1 69.029 1-2 4.71239 .7671 5-8 14.5299 16.800 1-2 29.845 70.882 g-ie 4.90874 9*75 n-i 14.7262 17.257 5-8 30.237 72 . 760 5-8 5 10509 2.0739 3-4 14.9226 17.721 3-4 30.630 74.662 n-ie 5.30144 2.2365 13-1 15.1189 18.190 7-8 31.023 76.589 3-4 5-49779 2.4053 7-8 15.3153 18.665 10. 31.415? 78 . 540 13-rf 5.69414 2.5802 I5-I 15.5116 19.147 1-8 3i.8o8e 80.516 7-8 5.89049 2.7612 5. 15.7080 19-635 1-4 32.201 82.516 i5-rt 6.0868 2.9483 i-i 15-9043 20. 129 3-8 32-594C 84.541 2. 6.2831 3.1416 1-8 16.1007 20.629 1-2 32.986 86.590 Il6 TABLE OF CIRCLES. CIRCUMFERENCE AND AREAS OF CIRCLES Continued. Diam. Circumf. Area. Diam. Circumf. Area. Diam. Circumf. Area. 10. 5-8 33-3794 88.664 17. 1-4 54.19 2 5 233.71 23. 7-8 75-0055 447.69. 3-4 33-7721 90.763 3-8 54.5852 237.10 24. 75.3982 452.39 7-8 34.1648 Q2.886, 1-2 54-9779 240.53 1-8 75 7909 457-11 11. 34-5575 95-033; 5-8 55.3706 243.98 1-4 76.1836 461.86 1-8 34-9502 97.205 3-4 55.7633 247-45 3-8 76.5763 466 . 64 1-4 35.3429 99.402 7-8 56. 1560 250.95 1-2 76.9690 47L44 3-8 35-7356 101.62 18. 56.5487 254.47 5-8 77.3617 476.26 1-2 36.1283 103.87 1-8 56.9414 258.02 3-4 77-7544 481.11 5-8 36.5210 106. 14 1-4 57-3341 261.59 7-8 78.1471 485-98 3-4 36-9 T 37 108.43 3-8 57-7268 265.18 25. 78.5398 490.87 7-8 37-3064 110.75 1-2 58.H95 268.80 i-8 78.9325 495-79 12. 37.6991 113. 10 5-8 58.5122 272.45 1-4 79-3252 500.74 1-8 38.0918 115-47 3-4 58.9049 276.12 3-8 79.7179 505.71 1-4 38.4845 117.86 7-8 59.2976 279.81 1-2 80.1106 510.71 3-8 38.8772 120.28 19. 59-6903 283.53 5-8 80.5033 5I5.72- 1-2 39.2699 122.72 1-8 60.0830 287.27 3-4 80.8960 520.77 5-8 39.6626 125.19 1-4 60.4757 291.04 7-8 81.2887 525.84 3-4 40-0553 127.68 3-8 60.8684 294.83 26. 81.6814 530-93 7-8 40.4480 130.19 1-2 61.2611 298.65 1-8 82.0741 536.05 13. 40.8407 132.73 5-8 61.6538 302.49 1-4 82.4668 54LI9 1-8 4L2334 135-30 3-4 62.0465 306.35 3-8 82.8595 546.35 1-4 41.6261 137.89 7-8 62.4392 310.24 1-2 83.2522 55L55 3-8 42.0188 140.50 20. 62.8319 314.16 5-8 83.6449 556.76 1-2 42.4115 143.14 1-8 63.2246 318.10 3-4 84.0376 562.00 5-8 42 . 8042 145.80 1-4 63.6173 322.06 7-8 84.4303 567.27 3-4 43.1969 148.49 3-8 64.0100 326.05 27. 84.8230 572- 5& 7-8 43-5896 151.20 1-2 64 . 4026 330.06 1-8 85-2157 577.87 14. 43-9823 153-94 5-8 64.7953 334.10 1-4 85.608.4 583.21 1-8 44-3750 156.70 3-4 65.1880 338.16 3-8 86.0011 588.57 1-4 44.7677 159.48 7-8 65.5807 342.25 1-2 86.3938 593-96 3-8 45 1604 162.30 21. 65.9734 346.36 5-8 86.7865 599-37 1-2 45-553 1 165.13 1-8 66.3661 350.50 3-4 87.1792 604.81 5-8 45.9458 167.99 1-4 66.7588 354-66 7-8 87-5719 610.27 3-4 46-3385 170.87 3-8 67-1515 358.84 28. 87.9646 615.75 7-8 46.7312 173-78 1-2 67.5442 363-05 1-8 88.3573 621.26 15. 47.1239 176.71 5-8 67.9369 367.28 1-4 88.7500 626.80 1-8 47.5166 179.67 3-4 68.3296 37L54 3-8 89.1427 632.36 1-4 47 993 182.65 7-8 68.7223 375.83 1-2 89.5354 637.94 3-8 48.3020 185.66 22. 69.1150 380.13 5-8 89.9281 6-13-55 1-2 48-6947 188.69 1-8 69.5077 384.46 3-4 90.3208 649.18 5-8 49.0874 I9J-75 1-4 69 . 9004 388.82 7-8 90.7135 654.84 3-4 49.4801 194.83 3-8 70.2931 393-20 29. 91.1062 660.52 7-8 49.8728 197-93 1-2 70.6858 397.61 1-8 91.4989 666.23 16. 50.2655 201.06 5-8 71.0785 402 . 04 1-4 91 .8916 671.96 1-8 50.6582 204.22 3-4 71.4712 406 . 49 3-8 92.2843 677.71 1-4 51-0509 207.39 7-8 71.8639 410.97 1-2 92.6770 683.49 3-8 51.4436 210.60 23. 72.2566 415.48 5-8 93.0697 689.30 1-2 51-8363 213.82 1-8 72.6493 420.00 3-4 93.4624 695-13 5-8 52.2290 217.08 1-4 73-0420 424-56 7-8 93-855I 700.98 3-4 52.6217 220.35 3-8 73-4347 429.13 30. 94.2478 706.86 7-8 53-0144 223.65 1-2 73.8274 433-74 1-8 94.6405 712.76 17. 53-407I 226.98 5-8 74.2201 438.36 1-4 95.0332 718.69 1-8 53-7998 230.33 3-4 74.6128 443-01 3-8 95.4259 724.64 TABLE OF CIRCLES. 117 CIRCUMFERENCE AND AREAS OF CIRCLES Continued. Diam. Circumf. Area. Diam. Circumf. Area. Diam. Circumf. Area. 30. 1-2 95.8186 730.62 37. 116.239 1075.2 43. 1-2 136.659 1486.2 5-8 96.2113 736.62 1-8 116.632 1082.5 5-8 137.052 1494.7 3-4 96 . 6040 742.64 1-4 117.024 1089.8 3-4 137.445 1503.3 7-8 96.9967 748.69 3-8 117.417 1097.1 7 ' 8 137.837 1511.9 31. 97.3894 754-77 1-2 117.810 1104.5 44. 138.230 1520.5 1-8 97.7821 760.87 5-8 118.202 mi. 8 1-8 138.623 1529-2 1-4 98.1748 766.99 3-4 118.596 1119.2 1-4 139.015 1537.9 3-8 98.5675 773-14 7-8 118.988 1126.7 3-8 139.408 1546.6 1-2 98.9602 779-31 38. 119.381 1134.1 1-2 139.801 1555-3 5-8 99.3529 785.51 1-8 119.773 1141.6 5-8 140.194 1564.0 3-4 99.7456 791-73 1-4 120.166 1149.1 3-4 140.586 1572.8 7-8 100.138 797.98 3-8 120.559 1156.6 7-8 140.979 1581.6 32. 100.531 804.25 1-2 120.951 1164.2 45. 141.372 1590.4 1-8 100.924 810.54 5-8 121.344 1171.7 1-8 141.764 1599-3 1-4 101.316 816.86 3-4 121.737 II79-3 1-4 142.157 I608. 2 3-8 101.709 823.21 7-8 122.129 1186.9 3-8 142.550 I6I7.0 1-2 102.102 829.58 39. 122.522 1194.6 1-2 142.942 1626.0 5-8 102-494 835.97 1-8 122.915 1202.3 5-8 143.335 1634.9 3-4 102.887 842 . 39 1-4 123.308 I2IO.O 3-4 143.728 1643-9 7-8 103.280 848.83 3-8 123.700 I2I7.7 7-8 144.121 1652.9 33. 103.673 855.30 1-2 124-093 1225.4 46. 144.513 1661.9 1-8 104.065 861.79! 5-8 124.486 1233.2 1-8 144.906 1670.9 1-4 104.458 868.31 3-4 124.878 I24I.O 1-4 145.299 1680. o 3-8 104.851 874.85 7-8 125.271 1248.8 3-8 145-691 1689.1 1-2 105.243 881.41 40. 125.664 1256.6 1-2 146.084 1698.2 5-8 105.636 888.00 1-8 126.056 1264.5 5-8 146.477 1707.4 3-4 IO6.O29 894.62 1-4 126.449 1272.4 3-4 146.8691 1716.5 7-8 106.421 901 . 26 3-8 126.842 1280.3 7-8 147.262) 1725.7 34. 106.814 907.92 1-2 127.235 1288.2 47. 147.655 1734.9 1-8 IO7.2O7 914.61 5-8 127.627 1296.2 1-8 148.048 1744.2 1-4 107.600 921.32 3-4 128.020 1304.2 1-4 148.440 1753.5 3-8 107.992 928.06 7-8 128.413 1312.2 3-8 148.833 1762.7 1-2 108.385 934-82 41. 128.805 1320.3 1-2 149.226 I772.I 5-8 108.778 941.61 1-8 129.198 1328.3 5-8 149.618 I78I.4 3-4 109.170 948.42 1-4 129.591 1336.4 3-4 150.011 1790.8 7-8 109.563 955-25 3-8 129.993 1344-5 7-8 150.404 ISOO.I 35. 109.956 962 . i i 1-2 130.376 1352.7 48. 150.796 1809.6 1-8 110.348 969.00 5-8 130.769 1360.8 1-8 151-189 iSig.O 1-4 IIO.74I 975.91 3-4 131.161 1369.0 1-4 I5L582 1828.5 3-8 IH.I34 982.84 7-8 I3L554 1377.2 3-8 I5L975 I837.9 1-2 III.527 989.80 42. I3I-947 1385-4 1-2 152.367 1847.5 5-8 III .919 996.78 1-8 132.340 1393.7 5-8 152.760 1857.0 3-4 II2.3I2 1003.8 1-4 132.732 1402.0 3-4 I53.I53 1866.5 7-8 112.705 1010.8 3-8 133.125 1410.3 7-8 153-545 1876.1 36. 113.097 1017.9 1-2 133.518 1418.6 49. I53.938 1885.7 1-8 113.490 1025.0 5-8 133.910 1427.0 1-8 I54.33I 1895.4 1-4 113.883 1032.1 3-4 134.303 1435.4 1-4 154.723 1905.0 3-8 114.275 1039.2 78 134.696 1443.8 3-8 I55.H6 I9I4.7 1-2 114.668 1046.3 43. 135.088 1452.2 1-2 I55.509 1924.4 5-8 II5.06I 1053.5 1-8 I35.48T 1460.7 5-8 155.902 1934.2 3-4 "5.454 1060.7 1-4 135.874 1469.1 3-4 156.294 1943.9 7-8 115.846 1068.0 3-8 136.267 1477.6 7-8 156.687 1953-7 THE GOETZ Box ANCHOR AND POST CAP. Certainly the most admira- ble system now known for SlOW-Burning ' Posts are fastened to ca P forming continuous post from cellar to roof. Falling parts are Construction. self-releasing. In the New Schedule for rating risks, adopted by the National Board of Underwriters, November, 1892, a reduction in rate is made on all buildings that make use of the Goetz methods. A dovetail box of cast-iron built into the wall ; a notch in the girder fits over lug in box, forming a self-releasing anchor. TOP VIEW. Showing dovetail form of sides by which the box is locked in the wall. HOME OKKICE, No. 7O STATE STREET, NEW ALBANY, IND., for details and catalogue in which this self-releasing method of anchoring is applied to various conditions. Manufacturing Agents in Every Large City. SKELETON CONSTRUCTION AS APPLIED IN BUILDINGS. By \VIVE. H. Fully Illustrated with Engravings from Practical Examples of High Buildings. 8vo, Clothi, This work includes the description and practical working details of Cast Iron, Wrought Iron, and Steel Columns in the construction of the skeleton frame, and their connections with the Floor and Curtain Wall Girders; Stability of the Structure; Wind Bracing, i.e., Knee and Lateral Bracing; Construction of Joints; Experiments on the Strength of Cast Iron, Wrought Iron, and Steel Columns, such as Z Bar Columns, Phoenix Columns, Plate and Angle Columns, and various commercial rolled shape columns ; Floor Framing in the Skeleton Construction. New York Building Law of 1892 in relation to the Skeleton Frame and Curtain Walls. The same law in relation to the strength of Cast Iron, Wrought Iron, and Steel Columns. Illustration and Calculation of the Columns, Floor Plans, Tables of Material, Specification, Stairways, Elevators, and Roofs in buildings using Cast Iron, Wrought Iron, and Steel Columns as a skeleton frame, such as "The New Netherlands," a seventeen-story building with nineteen tiers of beams ; the Home Life Insurance Building, and others. To be published in February or March. FOR SALE BY JOHN WILEY & SONS, 53 East Tenth Street, New York. ARCHITECTURAL IRON AND STEEL, AND ITS APPLICATION IN THE CONSTRUCTION OF BUILDINGS. INCLUDING BEAMS AND GIRDERS IN FLOOR CONSTRUCTION, ROLLED IRON STRUTS, WROUGHT AND CAST-IRON COLUMNS, FIRE-PROOF COLUMNS, COLUMN CONNECTIONS, CAST-IRON LIN- TELS, ROOF TRUSSES, STAIRWAYS, ELEVATOR ENCLOSURES, ORNAMENTAL IRON, FLOOR LIGHTS AND SKYLIGHTS, VAULT LIGHTS, DOORS AND SHUTTERS, WINDOW GUARDS AND GRILLES, ETC., ETC., WITH SPECIFICATION OF IRONWORK. AND SELECTED PAPERS IN RELA TION TO IRONWORK, FROM A REVISION OF THE PRESENT LAW BEFORE THE LEGISLATURE AFFECTING PUBLIC INTERESTS IN THE CITY OF NEW YORK, IN SO FAR AS THE SAME REGULATES THE CONSTRUCTION OF BUILDINGS IN SAID CITY. TABLES, SELECTED EXPRESSLY FOR THIS WORK, OF THE PROPERTIES OF BEAMS, CHANNELS, TEES AND ANGLES, USED AS BEAMS, STRUTS AKD COLUMNS, WEIGHTS OF IRON AND STEEL BARS, CAPACITY OF TANKS, AREAS OF CIRCLES, WEIGHTS OF CIRCULAR AND SQUARE CAST-IRON COLUMNS, WEIGHTS OF SUBSTANCES, TABLES OF SQUARES, CUBES, ETC., WEIGHTS OF SHEET COPPER, BRASS AND IRON, ETC. BY WM. H. BIRKMIRE, OF J. B. ft J. M. CORNELL IRON WORKS, 141 CENTRE STREET. ffullB ITUustrateD. 8VO, CLOTH, S3.5O. NEW YORK : JOHN WILEY & SONS, 53 EAST TENTH STREET. 1891. CONTENTS. CHAPTER I. THE 'MANUFACTURE OF IRON. ARTICLE PAGE 1. Iron I 2. Smelting, I 3. Pig Iron, I WROUGHT IRON. 4. Puddling, 2 5. Piling, ^ 2 6. Rolling, . ..3 7. Channels, 3 8. Quality of Wrought Iron, .3 9. Testing Wrought Iron, 4 10. Cold Bend Test, 4 11. Modulus of Elasticity, 5 12. Wrought Iron in Compression, ........5 13. Weight of Wrought Iron, ......... 5 STEEL. 14. Steel, 5 15. Mild or Soft Steel 6 16. Rolling Steel 6 17. Billets, 6 18. Weight of Steel, . ........ 7 CAST IRON. 19. Cast Iron, 7 20. Castings, . 8 21. Cores, . . . 8 22. Crushing Strength of Cast Iron, 8 23. Tenacity of Cast Iron, 8 24. Weight of Cast Iron, 8 v VI CONTENTS, CHAPTER II. FLOORS. ARTICLB PAGE 25. Dead Load, 9 26. Live Load 9 27. Method of determining Rolled- iron Beams by Diagram, ... 9 28. Beams as Girders, n 29. To determine Coefficient for Beams, . . . . . . .n 30. Properties of Wrought-iron I Beams 12 31. Deflection, 12 32. Coefficients for Steel Beams, 13 33. Properties of Steel I Beams, ... .... 13 34. Channels, ...... .... 14 35. Properties of Wrought-iron Channels, . 14 56. Properties of Steel Channels, 15 37. Zee Bars, 15 38. Floors should be Rigid 15 39. Elastic Limit, . 15 40. Maximum Deflection, , . . . 16 41. Framed Beams, ..... 16 42. Tie Rods, . 16 43. Beam Connections, 18 44. Bearing for Beams, 18 45. Pressure on Brick and Stone Work, . . . . . . , 18 46. Knees for Beam Connections, . . . . . . . .18 47. Bolts and Rivets for Beam Connections 19 48. Tee Irons as Beams, 19 49. Angle Irons as Beams (Even Legs), ....... 20 50. Angle Irons as Beams (Uneven Legs), 20 51. Beams not uniformly loaded, and Beams not supported at both Ends, . 20 CHAPTER III. GIRDERS. 52. Compound Girders, 22 53. Webs 22 54. Buckling, 22 55. Flanges 22 56. Deflection, 22 57. Rivets in Girders 22 58. Strain on Flanges of Girders 23 59. Shearing, 24 60. Flanges reduced in Thickness near Ends, 24 61. Weight of Brickwork, 26 62. Separators, 27 CONTENTS. Vll ARTICLE PAGB 63. Cast- iron Plates on Girders for Walls 2$ 64. Bolts and Rivets, 28 65. Shearing and Bearing of Rivets, .28 66. Pins, 29 CHAPTER IV. CAST-IRON LINTELS. 67. Cast-iron Lintels, 30 68. Skew-back Lintels, 31 69. Lintels for Iron Fronts, 31 70. Sidewalk Lintels, ** 32 71. Window-head Lintels .32 72. Double-web Lintels, 32 73. Window Sills, 32 74. Rule for Breaking Weight at Middle 32 75. Webs, ** 33 76. Tests of Cast-iron Lintels, 33 CHAPTER V. TRUSSES. 77. Roof Trusses, . .35 78. Loads on Trusses, 35 79. Snow and Wind Pressure, . . 36 80. Ceiling Weight 36 81. The Graphic Method, . .36 82. King-post Truss, 36 83. Truss No. 2, 38 84. Truss No. 3, . . . 38 85. Truss No. 4, . . . .39 86. Truss No. 5, 40 87. Details of Iron Trusses, ,42 88. Ties and Struts 42 89. Wooden Purlins, 42 90. Connections, . . 42 CHAPTER VI. STRUTS. 91. Rolled-iron Struts, 44 92. End Connections, 44 93. Factors of Safety, 46 viii CONTENTS. ARTICLE PAGE 94. Greatest Safe Load on Struts, 47 95. Channel Struts, 48 96. Angles as Struts, ... 48 97. Tees as Struts, 49 98. Properties of Beams for Struts, , . 50 99. Properties of Channels for Struts, 51 100. Properties of Angles for Struts, 52 101. Properties of Tees for Struts, 54 CHAPTER VII. CAST-IRON COLUMNS. 102. Columns, Shafts Ornamented, 55 103. Capitals, 55 104. Cast-iron Column Connection, 56 105. Holes Drilled, 58 106. Column Flanges, . . . . 58 107. Fire-proof Column, . . . . .58 108. Dowel Columns, 59 109. Strength of Cast-iron Columns, .60 no. Weight to be Estimated for any Use, ....... 62 in. Strength of Hollow Cast-iron Columns, 62 112. Factors of Safety for Cast-iron Columns, ...... 63 113. Ribbed Base Plates, . 64 114. Flat Base Plates, 65 115. Grouting, 65 116. Bedding, 66 117. Cast-iron Dowels for Wooden Columns, 66 1 1 8. Wrought-iron Pins and Cast-iron Star-shaped Dowels, . . .66 CHAPTER VIII. WROUGHT-IRON COLUMNS. 119. Wrought-iron Column Sections, ........ 67 1 20. Zee-bar Columns, 68 121. Strength of Wrought-iron Columns, 68 122. Safe Load on Wrought-iron Columns, 69 123. Radii of Gyration for Round Column, ....... 7 124. Radii of Gyration for Square Column, ....... 70 CHAPTER IX. STAIRWAYS. 125. Close-string Stairs, 7* 126. Height and Breadth of Steps, 71 127. Cast-iron Stairs 71 CONTENTS. ix ARTICLE PACK 128. To Measure Height of Railing, ........ 72 1 29. Number of Strings regulated by Width, ...... 74 130. Wrought-iron Stairs, .......... 74 131. Circular Stairs . 76 1 32. Deck- beam Strings, 77 133. Channel Strings, . . . . . . . . . .77 134. Plate and Angle Strings, 77 135. Trea-isand Risers, .......... 78 136. Fascias, . . . . . . ... . . . -79 137. Posts or Newels, . . . . . . . . . . -79 138. Brackets for Stair Handrail, 79 1 39. Stairs to be carefully constructed, 79 CHAPTER X. ORNAMENTAL IRONWORK. 140. Ornamental Design, . . . . . . . . . .80 141. Hammered Wrought Iron, ......... 80 142. Method of Hammering Leaves, etc., 83 143. Hammered Wrought-iron Grilles, . . . . . . .83 144. Cast-iron Ornamented, . . . . . . ... . .83 145. Modelling for Ornamental Castings, 84 146. Finish of Ornamental Iron, 86 CHAPTER XI. ELEVA TOR ENCLOSURES. 147. Passenger-elevator Enclosure, 88 148. Freight-elevator Enclosure, . 90 149. Double Sliding Doors for Passenger-elevator Fronts, . . . .92 150. Elevator Guide Supports, 94 CHAPTER XII. DOORS AND SHUTTERS. 151. Circular-head Door and Frame, 95 152. Sidewalk Door, 95 153. Outside Folding Shutters, 98 154. Shutter Hinges, 99 155. Storm Hooks, . . . . . . . . . . -99 156. Shutter Eyes, ........... 99 157. Shutter Rings, 99 158. Cast-iron Brick, 99 159. Rolling Steel Shutters, 100 160. The Noiseless Shutter, .100 X CONTENTS. CHAPTER XIII. FLOOR LIGHTS AND SKYLIGHTS. ARTICLE PAGE 161. Cast-iron Floor Lights for Iron Beams 102 162. Cast-iron Floor Lights for Wooden Beams, ...... 102 163. Wrought-iron Floor Lights for Wooden Beams, 102 164. Skylights, ............ 106 165. Hip Skylight, 109 CHAPTER XIV. HOLLOW BURNT CLAY. 166. Hollow Blocks for Arches, ......,. no 167. Porp$s Terra Cotta, .-,.'/. . . . . . . . .no V r .*\ CHAPTER XV. .' " '. ; V V ANCHORS. 168. Ashler Anchors,*' .'" 113 169. Side Ajjc-tytfrfcr*' ........... 113 170. Wall Anchors, . . . . . . . . . . .114 171. Hook Anchors, . . ... . . . . . . .114 172. Drive Anchors, . . . . . . . . . . .114 173. Wedge Anchors, 115 174. Coping Anchors, ........... 116 175. Government or V Anchors, ......... 116 176. Girder Straps, 116 177. Beam Straps, 116 CHAPTER XVI. BOL TS. 178. Square-head Bolts, . . . . . . . . . .117 179. Hexagon-head Bolts, . . . . . . . . . .117 1 80. Button-head or Carriage Bolts, . . . . . . . .117 181. Countersunk-head Bolts, 117 182. Screw-head Bolts, 117 183. Tap Bolts, 117 184. Counter-sunk Tap Bolts, . . . . . . . . .118 185. Double and Single Expansion Bolts, . . . . r .118 1 86. Lag Screws, . . . . . .119 187. Upset Ends, . . . .119 1 88. Open-drop Forged Turn-buckles, Pipe Swivel and Arm Swivel, . .119 CHAPTER XVII. MISCELLANEOUS DETAILS. 189. Mail Chutes, 120 190. Folding Gates, ........... 122 CONTENTS. XI 191. Box Slides, 123 192. Hanging Ceilings, . . . . . . . . . .124 193. Flitch-plate Girders, 125 194. Sidewalk Elevator, 127 195. Wrought-iron Gratings, . . . . . . . . .129 196. Cast-iron Perforated Plates, 130 197. Knee Gratings, . . . . . . . . . . .130 198. Galvanized Iron Cornices, 199. Scuttle, ...... 200. Scuttle Ladder, .... 201. Iron Fronts, ..... 202. Plain Fire Escapes, 203. Brackets on New Buildings, . 204. Top Rails, , 205. Bottom Rails, .... 206. Filling-in Bars, .... 207. Stairs, ...... 208. Floors, 209. Drop Ladders, . . . . 136 210. Height of Railing, 137 211. Ornamental Fire Escapes, . . .137 212. Fire Escapes for Schools, Factories, etc., 137 213. Vault Cover and Frame, 137 214. Cast-iron Grating, . . .138 215. Strainers, 138 216. Plain Wrought-iron Bar Window Guards 138 217. Bridle or Stirrup Irons, 139 218. Chimney Cap, 142 219. Cast-iron Flue Door and Frame, ........ 142 220. Wrought-iron Flue Door, . . . . . . . .142 221. Flue Ring and Cover, .......... 142 222. Chimney Ladder, 143 223. Corrugated Iron, 143 224. Galvanized and Black Iron, 144 225. Finial and Crestings, 145 226. Vault Lights, 146 227. Wrought-iron Guard, 147 228. Dwarf Doors, . 148 229. Cast-iron Wheel Guards, . 148 230. Fire Pipes, ............. 148 231. Mansard Roof, 148 232. Railings for Roof Protection, ........ 148 233. Pipe Railing, 149 234. Corrugated Flooring, 149 235. Tanks, 150 236. Chains and Cables, . . . . 152 Xll CONTENTS. ARTICLE PAGE 237. Wire Work, . . . . . 154 238. Cast-iron Boiler Flues, . . . . 155 239. Wrought-iron Boiler Flues, ......... 155 CHAPTER XVIII. FINISHING IRON AND STEEL. 240. Bronzing, . . . . . . . . . . 156 241. Enamelling Cast and Wrought iron, . . . . . . .156 242. Electro-plating, 156 243. Galvanizing Sheet Iron, .-.'-. . . . . . . .157 244. Painting of Iron, . . . . . . . . . . .157 245. Malleable Castings, . . . . . . . . . .158 246. Lacquer for Iron, . . . . . . . . . . .158 CHAPTER XIX. SPECIFIC A TION. General Conditions, . . . . . . . . . . .159 Time of Completion, . . . . . . . . . .160 Payments, ............. 160 Constructive Work, 161 Wrought Iron, ............ 161 Wrought Steel, 161 Cast Iron, 162 Painting, ............. 162 Anchors, Clamps, Dowels, etc 162 Columns, ............. 162 Girders, 163 Cast-iron Lintels, 164 Stairs, 164 Bulkheads on roof, ........... 165 Miscellaneous, ............ 165 Setting, 165 CHAPTER XX. TABLES. Average Weight in Pounds of a Cubic Foot of Various Substances, 166, 167 168 Squares and Cubes, of Numbers from i to 440, . . . 169, 170, 171, 172 The Circumference and Areas of Circles from i to 50, . . . 173, 174, 175 Weight per Foot of Flat Iron, 176, 1 77 Number of U. S. Gallons (231 Cubic Inches) Contained in Circular Tanks, . 178 CONTENTS. Xlii PAGE Decimal Equivalents for Fractions of an Inch 178 Decimal Equivalents for Fractions of a Foot, . . . . . .179 Weight of 100 Bolts with Square Heads and Nuts, . . . . .180 Weights of Nuts and Bolt-heads in Pounds, . . . . . . .180 Weight of Sheets of Wrought Iron, Steel, Copper, and Brass, . . .181 Weight of Square Cast-iron Columns in Pounds per Lineal Foot, . .182 Weight per Lineal Foot of Circular Cast-iron Columns, . t . .183 CHAPTER XXI. NEW YORK BUILDING LAW. Vault Lights and Areas Protected, . . . . ^ . . .184 Buildings Increased in Size by Use of Columns and Girders, . . .185 Anchors, 185 Floors, Stairs and Ceilings of Iron, 185 Weight on Floors, 186 Framing of Beams, 187 Cast-iron Templates, ... . 188 Iron Lintels, 188 Fire-proof Columns, . . . . . . . . . . .189 Iron Fronts Backed with Brick, ......... 190 Thickness of Cast-iron Posts, ......... 190 Curtain- wall Girders, . . . . . . . . . . .190 Rolled Iron and Steel Beams and Factors of Safety, 191 Girders to be Tested, ........... 192 Beams, Lintels and Girders to be Inspected, . 192 Stirrup Irons, ............ 193 Beam Anchors, ............ 193 Flitch-plates for Girders, .......... 193 Smoke Flues Lined with Cast Iron, . 194 Iron Shutters, ............ 194 Railings around Well-holes, . 195 Elevator Wells Inclosed with Brick or Iron, ,. ...... 196 Dumb-waiters, Skylights over Elevators, . 196 Screen of Iron under Elevator Machinery, ....... 196 Mansard Roof, ............ 196 Bulkheads, ............. 197 Cornices and Gutters, . . . . . . . . . . . 197 Dormer Windows, Scuttles and Skylights, . . . . . . . 197 Iron Ladders to Scuttles, ........... 198 Fire Escapes, 198 Roof Gardens on Theatres, .......... 198 Proscenium Wall Girder, etc., ......... 199 Skylights and Doorways to Theatres, . . . . . . . .199 XIV CONTENTS. PAGE Doorways through Proscenium Wall, Doors of Iron and Wood, . . . 199 Roof of Auditorium, Main Floor of Auditorium and Vestibule, Floor of Second Story over Entrance, Lobby and Corridors of Iron and Fire- proof Materials, ........... 200 The Fronts of Galleries, Ceiling of Auditorium, Partitions in Auditorium, Entrance Vestibule, Partitions of Dressing-rooms and Doors in same to be Fire-proof, ... 200 Actors' Dressing-rooms and Fly Galleries, ... ... 201 Stage and Fly Galleries Fire- proof, .... ... 201 Proscenium Opening and Curtain, . . ... 201 RETURN TO the circulation desk of any University of California Library or to the NORTHERN REGIONAL LIBRARY FACILITY Bldg. 400, Richmond Field Station University of California Richmond, CA 94804-4698 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 2-month loans may be renewed by calling (415)642-6233 1-year loans may be recharged by bringing books to NRLF Renewals and recharges may be made 4 days prior to due date DUE AS STAMPED BELOW 1990 . 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