Beam of Equal Strength.—Continuing the above consideration, suppose first that the greatest stress allowed on the material either tension or compression is for definiteness, say, 4 tons per sq. inch. If the shape of the section be such that the neutral axis is at the centre of depth, then, when the beam is so much bent that there is 4 tons tension at one edge, there is 4 tons compression at the other, and hence the resistance of the metal on both sides of the neutral axis is developed as much as in bending it possibly can be. But if the neutral axis were nearer the tension side, say, then, when the compressive stress was 4 tons, the tensile would be less, since рау, but the beam must not be bent any more, otherwise the compression will be more than 4 tons, which is not allowed. Hence the metal on the tension side of the neutral axis has not its full resistance developed; and the full power of the beam is not put forth. Similar reasoning applies if the neutral axis be nearer the compression side. It follows that the neutral axis should in this case lie in the centre of depth, and our sections should be shaped accordingly. If, however, the values of ƒ for tension and compression are as usual unequal, the best position of the neutral axis will not be at the centre of depth, but the same reasoning will show us where it should be. f=compressive stress allowed, ft = tensile Let y=distance of compressed edge from neutral axis, Then, to fully develop the resistance of the metal, the neutral axis should be so situated that, when the bending is such that the stress at the tension edge is ft, that at the compression edge should be fe, when each part will be resisting as much as it possibly can. This gives us at once the required position, for so the position of the neutral axis is determined. When the condition just found is satisfied, the section apply this to the case of an I beam (Fig. 267). A area of top flange, Let bottom flange, C= h = given depth, and we will suppose the beam bent as usual, so as to be convex below, and in consequence the top flange is in compression. In some cases the flanges are thin compared with h, so their thickness may be neglected, and the section treated as if the flanges were simple lines (Fig. 268), giving them, however, their proper sectional area in calculating. Then, NN being the neutral axis for equal strength, the moment of B is negative, since B is below NN. which is the relation which must hold between A, B, and C. For the full determination of A, B, and C to resist a given moment we must wait till the next chapter, but we have investigated here the above relation in order to show clearly that it depends entirely on the relations between the stresses at different points of the section, and not on their absolute amounts under a given bending moment. This it certainly shows, for so far we have not seen how to find what these amounts are. EXAMPLES. 1. An I beam 18 ft. span is 12 ins. deep over all. Each flange is 6 ins. by in., and the web is inch thick. Find the total stress in one flange under its own weight. Material cast-iron. Ans. 1730 lbs. 2. The beam in the preceding carries 1 ton at the centre. Find the greatest total stress in one flange. Ans. 11810 lbs. 3. In the preceding questions, find in each case the maximum intensity of the shearing stress on the web. Ans. 73, 287 lbs. 4. A beam is 12 feet long, and is loaded with 4 cwt. at one end and 5 cwt. at the other. Find the positions of two supports 8 feet apart so that there may be pure bending between them. Ans. 2 ft.; 2 ins. from 4 cwt. 5. Find the diameter of the least circle into which inch steel wire can be coiled, the stress being limited to 6 tons per sq. inch. Ans. 45 ft. 2 ins. 6. A steel ribbon, width 8 times its thickness, weighing the same per foot as the wire in the preceding, is bent into a circle the diameter of the smallest coil there found. Find the stress produced. Ans. 7.7 tons per sq. inch. 7. Find the position of the neutral axis of a T beam; flange 4 ins. by in., web 8 inches by inch. If the beam be bent into a circle 4000 ft. diameter, the flange being nearest the centre; find the greatest tensile and compressive stresses. Material wrought-iron. Ans. 1.9 ins. from centre of web; 34 and 1ğ tons per sq. inch. 8. If the ratio of tensile to compressive stress allowed in the preceding be 5 to 3; find what should be the proper thickness of the web for equal strength. Ans. .583 ins. 9. Find the neutral axis of an I beam. Top flange 6 ins. by in., bottom flange 10 ins. by 1 in., web 14 ins. by in. Ans. 5.83 ins. from bottom. 10. Find the position of the neutral axis of a channel iron. Flange 8 ins. by in., each web 7 ins. by inch. Ans. 2 ins. from outer edge of flange. II. A beam is built up of a plate 15 ins. deep, to the top of which are riveted a pair of angle irons 3 by 3 by inch, and to the bottom a pair 4 by 5 by ğ inch, the 4-inch side being riveted to the plate. Find the position of the neutral axis. Ans. 6.78 ins. from bottom. 12. Find the neutral axis of a box girder. Top plate 14 ins. by in., bottom plate inch thick; each side plate 16 by inch; angle irons 3 by 3 by 3 in. Ans. 8 ins. from top. CHAPTER XIX BENDING (continued) IN the last chapter we found a relation between the stress produced and the radius of bending. What we usually require, however, is the stress produced by a given moment, hence we now proceed to consider this. The figure (258) on page 356 is here reproduced (Fig. 269), and we proceed to inquire further into the equilibrium of the piece KCDK. In chap. xviii. we resolved the forces, so we will now take moments about some axis, i.e. consider its equi K C D Fig. 269. librium against rotation. The axis we will select is NN, its projection in the plane of (a) being the point R, so if we thought of the forces as all in the plane of (a) we should speak of taking moments about R; but since we know they are not all in the one plane we must not as in two dimensions speak of a point, but of an axis. We then have Moment of stresses on KK_moment of couple M about NN about NN. But it is proved in Statics that the moment of a couple |