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 ques tion, a large proportion of weight is considered too much for the constant load. 16284. 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 weight be applied in the middle II. If a beam be loose at both ends, and the weight be applied uniformly along the same length, it will bear twice the load placed in the middle III. If a beam be loose at both ends, and the weight be applied at an intermediate point; the spaces m with n=1IV. If a beam be fixed at both ends, and the weight be applied in the middle, it will bear one half more than if both ends be loose (I.) V. If a beam be fixed at both ends, and the weight be applied uniformly along the same length, it will bear three times more than the load in the middle of No. 1, than if both ends be loose 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 distributed, 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 (fig. 613u.). 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. 1628u. 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 200 tions are obtained by dividing a diameter Fig. 613f. Fig. 613g. Fig. 615h. 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 df 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. 1628x. 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 : I B 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. 613i. B Fig. 615j. breaking weight; I feet, or length between supports; d whole depth; d, depth to bottom flange; b breadth of bottom flange; and by thickness of vertical rib. The simpler rule usually emW tons, or the breaking weight which should axdxC I feet ployed, as 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 if 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. 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 5 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. ad C 6 x 22 x 75 .........> 16296. Fairbairn has formed a comparison between a wrought iron and a cast iron girder 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 andths thick; with angle iron 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 beam, taking the constant at 75, would be "d=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. Fig. 613n. bor beam, in which there are two vertical webs. Fig. 613n. is a plite girder of greater dimensions than fig. 6131.; 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 beam. The thickness of the web is seldom made less than ths 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. 6137, 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 form -and being united by properly proportioned covering plates at top and bottom, and to the joints (par. 1630y.), and all the riveting be well executed, the beam will be equal in strength 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. 1629f. The results of various testings of a new manufacture of girder patented by Messrs. Phillips have just been reported. 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 present sections of rolled irons are so limited in depth that they have been hitherto only valuable where light loads and limited spans The new system consists in riveting plates to the top and bottom flanges of rolled occur. B 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. 6137.) 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. A Fig. 6130. Fig. 613 p. VARIOUS FORMS IN USE FOR BEAMS, GIRDERS, AND IRONS. K 1629g. PROBLEM I.-To find the breaking weight of a beam, the load being in the middle and all the dimensions known. The ends loose or supported. For A, Timber beams : — bd2 S lbs. I feet bd2 S cwt. = W cwt. | bd2 S tons = - W tons. a4d S W lbs. / ins. = W lbs. Here S, in the first formula, represents the value of the breaking weight in pounds in the middle, taken from the preceding table: in the other two it would be deduced from it; thus taking Riga fir -3.21 for cwt.; and 160 for tons: b breadth in inches; d depth in inches; distance between the points of support, in feet; a area of These letters will be continued in these problems, until other values are attached to them. W will always represent the breaking weight. section. 359 lbs. 3.21 20 cwt. Here F represents the weight in pounds borne by a rod 1 inch square, when the strain is as great as the rod will bear without destroying part of its elastic force,15,300 for cast iron. From a mean of 265 experiments by Hodgkinson and Fairbairn, it appears (by Gregory) that a weight of 454'4 pounds in the middle of a bar of cast-iron, 1 inch square and 4-5 feet bearing, produced fracture. Therefore, for a bar of any other dimensions, we have: bd F 61 - × (1 − q × p3) = W lbs. Here q difference between the breadth in the middle and the extreme breadth='625 as found to answer in practice; and p, depth of the narrow part in the middle="7 as found to answer. When the middle part of the beam is omitted, except sufficient uprights to connect the top and bottom bars, then · × (1—p3) = W lbs. Here d whole depth; and p depth of part omitted. If the thickness of the web be about th orth of the depth of the beam, then 514 cwts. 20 for tons bd2 F 536 26.8, called 27 tons. Here 514 cwts. may be used for side castings, or 536 cwts, for erect castings. The other quantities are obtained thus: =25·7, called 26 tons; When I is used in feet, 2·166: a represents area of bottom flange in inches. 1629k. For D, cast iron girder (Hodgkinson's pattern): adC=514 /inches ad C=26 = W tons. ad C=2.14 =W cwts. Here a and d as before; P permanent load distributed, or about one-fourth of the breaking weight distributed; and multiplied by 2 when the ends are fixed = one-half BW. From the experiments above quoted from Gregory, we obtain— 16291. Gregory's work also states an arbitrary formula given by Mr. Dincs, which he found to be tolerably correct in all cases where the length of the girder did not exceed 25 feet; its depth in the centre not greater than 20 inches; the breadth of the bottom flange not less than one-third, or more than half, the depth; the thickness of the metal not less thanth of the depth. Then 16. 1792 [b d2 - (b-b2)d,2] = W lbs. | [bd2 (b-b1)d'] = W cwts. Here b entire breadth of bottom flange; b, thickness of the vertical part; d depth of whole girder; d, depth without the lower flange, all in inches; 7 length in feet. = 2d 1629m. Hurst, Handbook, notices that the area of the top flange should be of that of the bottom flange when the load is on the top; and when the load is on the bottom flange: Molesworth, Formule, has for the latter; he notes that if the depth of the girder be of the span, then a4-17 W tons, the weight being distributed. When the depth is to a5=W tons, the weight being distributed. The depth at the ends may equal. Approximate rules for these girders have been given in the Pocket-book for 1865, as 1 x P=dxa, = a. Here I feet; P tons distributed; d depth of girder; a area of bottom flange, both in inches. Here a area of the bottom flange; C coefficient determined for this particular form of tube. In the table given by Fairbairn (pp. 116-17), a area of the whole cross section, the constant C=178 tons, for a tube having the top flange=142 thick, twice the area of the bottom one; the tube being 9-6 inches square, and 17.5 feet long between the supports. Such a beam deflected 176 inches with a breaking weight of 7,148 lbs. 16290. Hurst states it is usual to camber a riveted girder, so that on receiving the permanent load it may become nearly horizontal. If the required rise or camber equals e in the middle in inches, d being in inches and I in feet, we have Ke. For girders uniformly loaded and of uniform section throughout the length, K='018. When the section is also made to vary so that the girder will be of equal strength throughout, K-021. Molesworth notes the area of top flange as al 18; Hurst says al 75. If the depth of the girder be of the span, then W = 13.3a tons; if then W=16a tons. The rivets to be inch and inch in diameter, placed 3 inches apart in the top, and 4 inches apart in the bottom, flange. |