many instances will entirely account for the so-called error. Similar figures corrected in this manner are shown below, from which it will be seen that the difference is much greater in the circle than in the rolled joist, and, for obvious reasons, it will be seen that the difference is greatest in those sections in which much material is concentrated about the neutral axis. 8 I FIG. 384. FIG. 385. But before leaving this subject the author would warn readers against such reasoning as this. The actual breaking strength of a beam is very much higher than the breaking strength calculated by the beam formula, hence much greater stresses may be allowed on beams than in the same material in tension and compression. Such reasoning is utterly misleading, for the apparent error only occurs after the elastic limit has been passed. are applied at opposite ends of a bar in such a manner as to tend to rotate it in opposite directions, the bar is said to be subjected to a bending moment. Thus, in Fig. 386, the bar ab is subjected to the two equal and opposite couples R. ac and W. be, which tend to make the two parts of the bar rotate in opposite directions round the point c; or, in other words, they tend to bend the bar, hence the term "bending moment." Likewise in Fig. 6387 the couples are Rac and Rbc, which have the same effect as the couples in Fig. 386. The bar in Fig. 386 is termed a cantilever." The couple R. ac is due to the resistance of the wall into which it is built. Support Support Support Load (4) FIG. 388. The bar in Fig. 387 is termed a "beam." When a cantilever or beam is subjected to a bending moment which tends to bend it If there be more than two couples, they can always be reduced to two concave upwards, as in Fig. 388 (a), the bending moment will be termed positive (+), and when it tends to bend it the reverse way, as in Fig. 388 (6), it will be termed negative (-). W Bending-moment Diagrams.-In order to show the variation of the bending moments at various parts of a beam, we frequently make use of bending-moment diagrams. The bending moment at the point c in Fig. 389 is W. bc; set down from the ordinate c = W.bc on some given scale. The bending moment at d = W. bd; set down from d, the ordinate dd' W. bd on the same scale; and so on for any number of points: then, as the bending = d W,bd FIG. 389. moment at any point increases directly as the distance of that point from W, the points b, d', c', etc., will lie on a straight line. Join up these points as shown, then the depth of the diagram below any point in the beam represents on the given scale the bending moment at that point. This diagram is termed a "bending-moment diagram." In precisely the same manner the diagram in Fig. 390 is obtained. The ordinate dd, represents on a given scale the bending moment Rad, likewise c1 the bending moment R1ac or Rabc, also ee, the bending moment Robe. The reactions R, and R are easily found by the principles of moments thus. Taking moments about the point b, we have W.bc R2 In the cantilever in Fig. 389, let W = feet, bd = 45 feet. The bending moment at c = W. be = 800 (lbs.) × 6'75 (feet) Let 1 inch on the bending-moment diagram = 12,000 (lbs. feet), or a scale of 12,000 (lbs.-feet) per inch, or 12000 lbs.-feet I (inch) Then the ordinate cc1 = 5400 (lbs.-feet) 12000 (lbs.-feet) =0'45 (inch) I (inch) Measuring the ordinate dd,, we find it to be 0'3 inch. Then 03 (inch) × 12000 (lbs.-feet) (3600 (lbs.-feet) bending I (inch) moment at d In this instance the bending moment could have been obtained as readily by direct calculation; but in the great majority of cases, the calculation of the bending moment is long and tedious, and can be very readily found from a diagram. bc In the beam (Fig. 390), let W 1200 lbs., ac = 5 feet, = 3 feet, ad = 2 feet. R1 = W.bc 1200 (lbs.) × 3 (feet) = 450 lbs. the bending moment at c = 450 (lbs.) × 5 (feet) =2250 (lbs.-feet) Let 1 inch on the bending-moment diagram = 4000 lbs. 4000 (lbs.-feet) feet), or a scale of 4000 (lbs.-feet) per inch, or I (inch) Then the ordinate cc1 = 2250 (lbs.-feet) = 0.56 (inch) 4000 (lbs.-feet) I (inch) Measuring the ordinate dd, we find it to be o°225 (inch). Then 0'225 (inch) x 4000 (lbs.-feet) 1900 (lbs.-feet) bending General Case of Bending Moments.-The bending moment at any section of a beam is the algebraic sum of all the moments of the external forces about the section acting either to the left or to the right of the section. Thus the bending moment at the section f in Fig. 391 is, taking moments to the left of f— Raf- Wcf - W2df or, taking moments to the right of ƒ— R.bf - Wef That the same result is obtained in both cases is easily shown by taking a numerical example. Let W1 30 lbs., W2 50 lbs., W3 cd=25 feet, df = 1.8 feet, fe : = 40 lbs. ; ac = 2 feet, = 22 feet, eb = 3 feet. W3 R2 FIG. 391. We must first calculate the values of R, and R. Taking moments about b, we have R1ab W1cb Wądb + Web R1 R1 = = = = W1cb + Wodb + Web ab 30(lbs.) X 9'5(feet) +50(lbs.) × 7(feet)+40(lbs.) × 3(feet) R2 = 30 (lbs.)+50 (lbs.)+40 (lbs.) - 65'65 (lbs.)=54'35 (lbs.) The bending moment at ƒ, taking moments to the left off, = 65.65 (lbs.) x 6'3 (feet) 30 (lbs.) x 4'3 (feet) × — 50 (lbs.) The bending moment at f, taking moments to the right off, 54 35 (lbs.) X 5.2 (feet) — 40 (lbs.) × 2'2 (feet) = 1946 (lbs.-feet) Thus the bending moment at ƒ is the same whether we take moments to the right or to the left of the point f. The calculation of it by both ways gives an excellent check on the accuracy of the working, but generally we shall choose that side of the section that involves the least amount of calculation. Thus, in the case above, we should have taken moments to the right of the section, for that only involves the calculation of two moments, whereas if we had taken it to the left it would have involved three moments. |