If the load be terminal and the depth constant, the form of the beam in breadth should be wedge-shaped, the breadth increasing as the length of the beam (the latter measured from the loaded end). II. If the breadth be constant, the square of the depth must vary as the length, or the vertical section will be a parabola. III. When both breadth and depth rary, the section should present a cubical parabola. IV. When the beam supports only its own weight, it should be a double parabola, that is, the upper as well as the lower surface should be of a parabolic form, the depth being as the square of the length. V. When a beam is loaded evenly along its surface, the upper surface being horizontal, the lower one should be a straight line meeting the upper surface at the outer end, and forming a triangular vertical section; the depth at the point of support being determined by the length of the beam and the load to be sustained. VI. If an additional terminal load be added to such a beam, the under surface should be of a hyperbolic curvature. VII. And in a flanged beam, the lower flange should describe a parabolic curve (as in example IV.). 16289. "Beims supported at both ends. I. A beam loaded at any one point, as scale beams and the like, should have a parabolic vertical section each way from the loaded point, A Fig. 613b. A, fig. 613b. II. In flunged beans, the lines may be nearly straight, and approach the straight lines more as the flanges are thinner. III. A beam loaded uniformly along the whole of its length, should have an elliptic outline for the upper surface the lower one being straight. This form applies to girders for bridges and other purposes where the load may be spread. IV. With thin flanges, a beam so circumstanced should be of a parabolic figure. V. If a flanged beam have its upper and lower sides level, and be loaded uniformly from end to end, the sides of the lower flange should have a parabolic curvature." (Gregory.) VI. In the case of example III., Fairbairn observes that the greatest strength will be attained, while the breadth and depth is allowed to be diminished Fig. 613c. B towards the ends. This diminution should take place in curved lines which are strictly parabolic. The most convenient way of doing this is by preserving a horizontal level in the bottom flange, diminishing its width, as well as the height of the girder, as fig. 613c. Thus the spaces bb should be square on plan for the bearings on the wall, &c., and equal to the width of the bottom flange at the centre; the intermediate length 7 tɔ be curved to the form prescribed. The width of the bottom flange is to be reduced near the ends to one half of its size in the middle, and the total depth of the girder reduced at the ends in the same proportion. At the middle of the bearing, a flange may be cast on to connect the upper and lower flanges, and this will give additional stiffness to the girder. 1628r. Gregory further remarks on this subject: when the depth of the beam is uniform, and (VII.) the whole load is collected in one point (as A, fig. 613d.) the sides of the bean. E B Fig. 613d. PLANS. Fig. 613e. should be straight lines, the breadth at the ends, B, being half that where D the load is applied. VIII. When the load is uniformly distributed (fig. 613e.) the sides should be portions of a circle, the radius of which should equal the square of the length of the beam divided by twice its breadth. When the breadth of the beam is uniform and (IX.) the load is collected in one point, the extended (under) side should be straight, the depth at the point where the load is applied twice that at the ends, and the lines connecting them straight (fig. 613b.) See example I. When the load is uniformly distributed, X. the extended (under) side should be straight, and the compressed (upper) side a portion of a circle whose radius equals the square of half the length of the beam divided by its depth. See examples III. and VI. When the transverse section of a beam is a similar figure throughout its whole length; XI. if the load be collected at one point, the depth at this point should be to the depth at its extremities as 3:2: the sides of the beam being all straight lines. XII. When the load is uniformly distributed, the depth in the centre should be to the depth at the end as 3: 1, the sides of the beam being all straight lines. VARIOUS LAWS AFFECTING BEAMS AND GIRDERS. 16288 The principles on which the rules subjoined are founded may be seen in Gregory, Mechanics, &c. and Barlow, Strength of Materials, but divested, certainly, of the refine ment of Dr. T. Young's Modulus of Elasticity, and some other matters, which we cannot help thinking unnecessary in a subject where, after exhausting all the niceties of the question, a large proportion of weight is considered too much for the constant load. 1628t. The transverse strength is that power, in the case of a beam, exerted in opposing a force acting in a direction perpendicular to its length. The following formulæ and rules apply to the various positions in which a beam or girder is placed. I. If a beam be loose (or supported) at both ends, and the W = W 11. If a beam be loose at both ends, and the weight be ap- V. If a beam be fixed at both ends, and the weight be ap- VI. If a beam be fixed at both ends, and the weight be ap- VII. If a beam be fixed at one end, and the weight be applied at the other end, it will bear only one fourth of the weight carried by beam No. 1, of the same length or VIII. If a beam be fixed at one end only, and the weight be applied in the middle, it will bear half as much again as at the end. IX. If a beam be fixed at one end, and the weight be applied uniformly along its length, it will bear double the load at the end. X. If a beam be fixed at one end only, it is as strong as one of equal breadth and depth, and twice the length which is fixed at both ends. XI. If a beam be supported in the middle and loaded at each end, it will bear the same weight as when loose at both ends and loaded in the middle (as I.) XII. If a beam be continued over three or four points and the load be uniformly dis tributed, it will suffice to take the part between any two points of support as a beam fixed at both ends. XIII. If some of the parts have a greater load than the others, it will be near enough in practice to take the parts so loaded as supported at the ends only. XIV. If a beam be inclined and supported at both ends, it has its breaking weight equal to that of the same beam when horizontal, multiplied by the length of the inclined beam and divided by the horizontal distance, NOTE. In calculating for the strength of a beam or of a girder, it is usual to reckon on the ends being loose, from the difficulty of fixing the ends in a sufficient manner to warrant the rule in that case being followed: and when the ends are solidly embedded, they should penetrate the wall for a distance equal to at least three times the depth of the beam or girder (par. 1630m.); but this precaution is seldom carried out in practice. 1628. For the effect of running loads over bars, we must refer to Professor Willis's experiments at Cambridge, given at the end of Barlow's Strength of Materials, &c., 1851. 1628v. Two geometrical methods of finding the best proportion of a beam to be cut out of a given cylinder have been propounded. The stiffest beam, says Tredgold, that can be cut out of a round tree, is that of which the depth is to the breadth as 3 to 1, or nearly as 1.7320508 to 1; this is in general a good proportion for beams that have to sustain a considerable load. The required propor tions are obtained by dividing a diameter as ab in fig. 613f, into two equal parts, ac and cb, then drawing with a and b as centres two arcs through e to cut the circle ine and f; the points aehf being joined, the figure is that of the stiffest beam that can be cut out of a cylinder, to resist a perpendieular strain. It is also observed by Tredgold that the strongest beam which can be cut out of a round tree is that of which the depth is to the breadth as √2 is to 1, or nearly as 1-4142136 to 1; or as 7 to 5. Its two sides must be to the diameter of the tree as the and to 1. The required proportions are obtained by dividing a diameter ab, fig. 613h., into three equal parts ac, cd, db, and drawing the lines ce and dƒ at right-angles to ab; the points aebf being joined, the figure is that of the strongest beam that can be cut out of a cylinder. The strength of a square beam, fig. 613g., cut from the same cylinder, is to the strength of the strongest beam nearly as 101 to 110, although the square beam would contain more timber nearly in the ratio of 5 to 4-714. The stiffest beam is to the strongest as 0-97877 to 1, as regards power of bearing a load; but as 1'04382 to 1 as regards amount of deflection, in equal lengths between the supports. 1628w. Buffon, during his extensive series of experiments on oak timber, from 20 to 28 feet in length, and from 4 inches to 8 inches square in section, found that the heart-wood which was densest was also strongest, and the side on which the beam was laid also affected the strength; for when the annual layers were horizontal, and the strength 7, the layers laid vertically gave a strength of 8. He also found that beams which had each supported, without breaking, a load of 9,000 lbs. during one day, broke at the end of five or six months with a weight of 6,000 lbs., that is to say, they were unable to carry for six months two-thirds of the weight they bore for one day. TRANSVERSE SECTIONS. 1628r. The transverse section of a cast iron girder previous to Hodgkinson's experiments was that of Tredgold, consisting of equal flanges at top and bottom, as A, fig. 6131; and that of Lillie and Fairbairn, in 1825, with a single flange, as B; Hodgkinson deduced a section of greatest strength having areas of flanges as 6 to 1, as C. Taking this form as unity, the ratios will stand : 1:754 1:820 For Hodgkinson and Fairbairn, as For Hodgkinson and Tredgold, as For Fairbairn and Tredgold, as (Fairbairn, Application, &c. p. 25: Tredgold, Cast Iron, 1824, p. 55, describes the advantages of his own form of section.) 1628y. Hodgkinson's complete section for a cast iron girder is shown in fig. 613j. Its chief principle is, that the bottom flange must contain six times the area of the top flange. The several dimensions are taken thus:-I. For the depth, the total dimension D. II. For the bottom flange, the width B, and for the two thicknesses, one is taken at the centre bb; the other b at the end. III. For the top flange, the width T, and for the two thicknesses, one is taken at the centre tt, the other t at the end. In this manner the dimensions of the flanges are Fig. 6136. Fig. 613j. quite independent of the thickness of the rib. IV. For the rib the two dimensions r and rr are measured as continued to the extreme top and bottom surfaces of the girder, with the same view of making these dimensions independent of those of the flanges, and promoting exactness in defining the entire section. Hodgkinson's complex rule for obtaining the weight a girder will carry, is 3 ¿ ¿ } b d3 − (b − b ̧ 3) d'=W. Here W = tons, or beaking weight; I feet, or length between supports; d whole depth; d, depth to bottom flange, b breadth of bottom flange, and b, thickness of vertical rib. The simp'er rule usually em ployed, as 2 axdxC = W tons, or the breaking weight which should not be less than four times the permanent load distributed; and it gives a result less by 7 per cent. than the complex rule above described, therefore an excess of strength is obtained. 1628z. The proportions of the rib are undetermined, but it is evident that they should bear some ratio to those of the flanges. It must be sufficiently rigid to prevent lateral weakness. Moreover the very theory which maintains the principle of the neutral axis (par. 1630c) also recognises the increase of the forces of compression and extension upward and downward from the neutral axis, and would therefore lead to the adoption of a rib tapered in both directions. In practice it is found desirable to taper the rib so as to meet each of the flanges with a thickness corresponding to that of the flange, for any very great disproportion exists, the operation of casting the beam cannot be so perfectly performed, from unequal shrinkage of the metal, and an imperfect casting or one having flaws in it, renders futile all calculations of strength. if 1629. Hodgkinson gradually varied the form of section of girder in his experiments until the widths and depths of the flanges were as follows:-Top flange 2.33 inches wide, 0:31 inch deep; bottom flange 6-67 inches wide, 0-66 inch deep; the areas being 720 and 4.4 inches. The rib was 266 inch thick, and the total depth 54 inches. The constant or C was found to be 514 for cwts., or 26 for tons. (Warr.) 1629a. It will scarcely be within our province to describe all the forms of sections. and the results of the experiments made by Fairbairn in obtaining his box beam or plate girder in wrought iron, but it is to be noted that all the cylindrical tubes broke by extension at the rivets before the tube could fail by compression. Fairbairn in his Application of Cast and Wrought Iron to Building Purposes, edit. 1857-8, p. 80, notices that although the plate girder be inferior in strength to the box beam, it has nevertheless other valuable properties to recommend it. On comparing the strength of these separate beams, weight for weight, it will be found that the box beam is as 100: 93. The plate beam is in some respects superior to the box beam; it is of more simple construction, less expensive, and more durable, from the circumstance that the vertical plate is thicker than the side plates of the box beam. It is also easier of access to all its parts for the purposes of cleaning, &c. 16296. Fairbairn has formed a comparison between a wrought iron and a cast iron g rder for a span of 30 feet. The plate girder, fig. 613k, would be 31 feet 6 inches in length, and would be composed of plates 22 inches deep and ths thick; with angle iron ad C ths thick, riveted on both sides at the bottom of the plate, and angle iron inch thick at the top, the width over the top being 7 inches, and the bottom 5 inches. The breaking weight of this 6 x 22 x 75 beam, taking the constant at 75, would be = W; or 360 =27.5 tons in the middle, or 55 tons distributed equally over the surface. In the edition of 1857-8, the angle irons are described as 3 inches by 3 inches, inch thick for the bottom, and 4 inches by 4 inches, inch thick at the top; it would, therefore, be 8 inches over at the top, and about 6 inches at the bottom. Now a cast iron girder of the best form and strongest section and calculated to support the same load, would weigh about 2 tons, the plate beam about 18 cwts., or less than one half. To secure uniformity of strength in a rectangular box beam, the top is required to be about twice the sectional area of the bottom; hence resulted the use of cells in that portion. Fig. 613k. -12 1629c. Fig. 6131. is a plate beam having a single plate for the vertical web, while each of the flanges consists of a flat bar and a pair of angle irons riveted to each other and to the vertical web. Fig. 613m. is a Fig. 6131. Fig. 613m. box beam, in which there are two vertical webs. Fig. 613n. is a plate girder of greater dimensions than fig. 6134; the flanges contain more than one layer of flat bars, and the web, which consists of plates with their largest dimensions vertical, is stiffened by vertical T iron ribs at the joints of those plates, as shown in the plan or horizontal section lettered A. The pieces should abut closely and truly against each other, having end surfaces made exactly perpendicular to the axis of the Fig. 613n. beam. The thickness of the web is seldom made less than 3ths inch, and except for the largest beams, this is in general more than sufficient to resist the shearing stress. Above each of the points of support, the vertical ribs must be placed either closer or made larger, so that they may be jointly capable of safely bearing as pillars the entire share of the load which rests on that point of support. A pair of vertical T iron ribs riveted back to back through the web plates (as A, fig. 613n.) may be held to act as a pillar of cross-shaped section. 1629d. The rib or web of a plate beam, as fig. 6134, having little or nothing to do with the pressure directly, has been replaced in some cases by simple upright struts or diagonal braces between the flanges, which in cast iron girders are in one casting, but experience has proved this not altogether politic, particularly in cast iron. Hodgkinson remarked that such beams were weaker than those with a solid rib. Rankine observes that transverse ribs or feathers on cast iron beams are to be avoided, as forming lodgments for air bubbles, and as tending to cause cracks in cooling. Open work in the vertical web is also to be avoided, partly for the same reasons, and partly because it too much diminishes the resistance to distortion by the shearing action of the load. 1629e. "Where the span renders it impracticable to roll a beam in one piece," Fairbairn, page 91, notices that "convenient weights might be rolled into sections of the proper forin and being united by properly proportioned covering plates at top and bottom, and t the joints (par. 1630y.), and all the riveting be well executed, the beam will be equal in trength to one" of an entire length. "This construction may be carried to a span of 40 to 50 feet. In practice it is found necessary to confine the use of cells to spans exceeding 100 or 150 feet: within these limits the same objects are most economically obtained by the use of thicker plates” (page 215). "The more nearly the bottom approximates to a solid homogeneous mass, the better it is calculated to resist a tensile strain" (see pages 248 to 256 for full instructions as to riveting plates; and Kirkaldy, Experiments, &c., page 196, for comparison of strength). As the bending moment of the load on a girder diminishes from the middle towards the ends, and the shearing force from the ends towards the middle, it follows that the transverse sections of the bottom plates may be diminished from the middle towards the ends, and that of the vertical web from the ends towards the middle, so as to make the resistance to bending and shearing respectively vary according to the same law. Consequently, towards the centre of a girder for a large span, the bottom plate is usually increased by additional plates to secure the requisite strength in the sectional area, giving the underside of the plate a bellied form. C. Graham Smith, Wrought Iron Girder Work, deserves attentive perusal by the student. It is printed in the British Architect, for June 1877, pages 582 to 385. 1629f: The results of various testings of a new manufacture of girder patented about 1866 by Messrs. Phillips are here recorded. A double weight in a cast iron girder is required to give equal strength with one of wrought iron. A riveted plate girder is not always adaptable for general purposes. The new system consists in riveting plates to the top and B bottom flanges of rolled iron beams, and so strengthening them as to obtain results apparently disparaging to ordinary plate girders. The experiments noticed here in an abridged form were on a patent girder of 22 lbs per foot run, with a web plate, as A, fig. 6130., and 20 feet bearing, as compared with a riveted plate girder of 9 in depth; it gave a breaking weight of 7 tons and a safe load of 4 tons; the formula for the breaking weight of an ordinary plate girder would give 3 tons. When two of the 8-inch rolled girders were riveted together with a plate on the top, as B, the metal being about 40 lbs. per foot run, the girder was found to resist 20 tons, even then not breaking, but becoming twisted. An ordinary riveted plate girder of 40 lbs. per foot run, with a web of 12 inches, with double angle irons of 3 inches by 3 inches and inch thick, would break with a strain of 9 tons. A simple web plate girder, with angle irons top and bottom (fig. 613k.), gives C = 60; a plate on top and bottom in addition (fig 613/.) gives C=75; and a box beam (fig. 613m.) gives C = 80. The rolled girders made by the Butterley Company give C=57 to 88. The example A gives C=210; and the example B, 300. Other experiments are required fully to prove the superiority of the new system over the beams and girders of the old sections. The details of the above testings are given in the Builder, p. 148; Mechanics' Magazine, p. 129; Engineering, p. 139; &c., all for the year 1866. Fig. 6130. Condition of Breaking Weight in the Middle. ! ! !IDIIT+ A B Ꭰ E G H Ι K Fig. 613p. VARIOUS FORMS IN USE FOR BEAMS, GIRDERS, AND IRONS, Application to the manufacturer selected must be made for any special lengths and strengths of rolled iron joists and girders, riveted and compound, &c. The former can be obtained from 3 inches deep by 12 inches up to 22 inches deep by 8 inches, being from 6 feet to 36 feet in length, with top and bottom flanges of usual proportions. The latter can be obtained of the same lengths. One manufacturer advertises the following makes.— Rolled girders up to 19 inches deep and to 38 feet long. Zore's patent girders up to 8 inches deep and to 34 feet long. Channel iron to 12 inches wide and to 32 feet long. Angle iron to 12 united inches and to 30 feet long. Tee iron to 12 united inches and to 30 feet long. Flitch and sandwich plates to 14 inches wide and to 36 feet long. Riveted girders made up from stock to all sections. Bulb tees up to 10 inches deep. Rounds to 6 inches. Squares to 5 inches. Flats to 14 inches. Chequered plates up to 8 feet by 4 feet. The opinion is gaining ground that most of the constants in use for calculating the strength of beams are too high. A comparison of Tredgold, Barlow, and Clark, will show |