STRENGTH OF MATERIALS in a beam. But 5M/S.C is the whole shearing force Q on the section of the beam. Hence . and this is also the intensity of vertical shearing stress at the dis- tance 2/0 from the neutral axis. This expression may conveniently be written q = QA.y/z l, where A is the area of the surface AG and y the distance of its centre of gravity from the neutral axis. The intensity q is a maximum at the neutral axis and diminishes to zero at the top and bottom of the beam. In a beam of rectangular section the value of the shearing stress at the neutral axis is q max. = IQ/6A. In other words, the maximum intensity of shearing stress on any section is f of the mean intensity. Similarly, in a beam of circular section the maximum is ^ of the mean. This result is of some importance in application to the pins of pin-joints, which may be treated as very short beams liable to give way by shearin^. In the case of an I beam with wide flanges and a thin web, the above expression shows that in any vertical section q is nearly 'con- stant in the web, and insignificantly small in the flanges. Practi- cally all the shearing stress is borne by the web, and its intensity is very nearly equal to Q divided by the area of section of the web. Principal 64. The foregoing analysis of the stresses in a beam, which stresses resolves them into longitudinal pull and push, due to bending moment, along with shear in longitudinal and transverse planes, is generally sufficient in the treatment of practical cases. If, how- ever, it is desired to find the direction and greatest intensity of stress at any point in a beam, the planes of principal stress passing through the point must be found by an application o f ' the general method given in the article ELASTICITY, chapter iii. In the present case the problem is excep- tionally simple, from the fact that the stresses on two planes at right angles are known, and the stress on one of these planes is wholly tan- gential. Let AC (fig. 39) be an indefinitely small portion of the horizontal section of a beam, on which there is only shearing stress, and let AB be an indefinitely small portion of the vertical section at the same place, on which there is shearing and normal stress. Let q be the intensity of the shearing stress, which is the same on AB and AC, and let p be the in- tensity of normal stress on AB: it is required to find a third plane BC, such that the stress on it is wholly normal, and to find r, the in- tensity of that stress. Let 6 be the angle (to be determined) which BC makes with AB. Then the equilibrium of the triangular wed^e ABC requires that rBCcos0=p. AB+ q. AC, and rBCsin0=y. AB ; or (r-p) cos 6=*q sin0, and ?-sin = gcos 0. Hence, q z = r(r j>) , tan26=2q/p, The positive value of ? is the greater principal stress, and is of the same sign asp. The negative value is the lesser principal stress, which occurs on a plane at right angles to the former. The equa- tion for gives two values corresponding to the two planes of principal stress. The greatest intensity of shearing stress occurs on the pair of planes inclined at 45 to the planes of principal stress, and its value is V^ + %p 2 (by 5). 65. The above determination of r, the greatest intensity of stress due to the combined effect of simple bending and shearing, is of some practical importance in the case of the web of an fbeam. We have seen that the web takes practically the whole shearing force, distributed over it with a nearly uniform intensity q. If there were no normal stress on a vertical section of the web, the shearing stress q would give rise to two equal principal stresses, of pull and push, each equal to q, in directions inclined at 45 to the section. But the web has further to suffer normal stress due to bending, the intensity of which at points near the flanges approxi- mates to the intensity on the flanges themselves. Hence in these regions the greater principal stress is increased, often by a consider- able amount, which may easily be calculated from the foregoing formula. What makes this specially important is the fact that one of the principal stresses is a stress of compression, which tends to make the web yield by buckling, and must be guarded against by a suitable stiffening of the web. The equation for 6 allows the lines of principal stress in a beam to be drawn when the form of the beam and the distribution of loads are given. An example has been shown in the article BRIDGES ( 13, fig. 12), vol. iv. p. 290. )eflexion 66. The deflexion of beams is due partly to the distortion caused f beams, by shearing, but chiefly to the simple bending which occurs at each vertical section. As regards the second, which in most cases is the only important cause of deflexion, we have seen ( 59) that the radius of curvature R, at any section, due to a bending moment AI, is EI/AI, which may also be written Ey^/p^ Thus beams of uniform strength and depth (and, as a particular case, beams of 605 uniform section subjected to a uniform bending moment) bend into a circular arc. In other cases the form of the bent beam, and the resulting slope and deflexion, may be determined by integrating the curvature throughout the span, or by a graphic process (see BRIDGES, 25), which consists in drawing a curve to represent the beam with its curvature greatly exaggerated, after the radius of curvature has been determined for a sufficient number of sections. In all practical cases the curvature is so small that the arc and chord are of sensibly the same length. Calling i the angle of slope, and u the dip or deflexion from the chord, the equation to the curve into which an originally straight beam bends may be written du . d 2 u di El dx ' dx*~dx~bl ' Integrating this for a beam of uniform section, ot span L, supported at its ends and loaded with a weight W at the centre, we have for the greatest slope and greatest deflexion, respectively, i = WL 2 /16EI, " 1= , W r L , 3 / 48EIp If the load W is uniformly distributed over L! S J 24 and Ul = 5WL3/384EI - For other cases > see BRIDGES, The additional slope which shearing stress produces in any originally horizontal layer is q/C, where q is, as before, the intensity of shearing stress and C is the modulus of rigidity. In a round or rectangular bar the additional deflexion due to shearin^ is scarcely appreciable. In an I beam, with a web only thick enough to resist shear, it may be a somewhat considerable proportion of the whole. 67. Torsion occurs in a bar to which equal and opposite couples Torsion are applied, the axis of the bar being the axis of the couples, and of solid gives rise to shearing stress in planes perpendicular to the axis, and Let AB (fig. 40) be a uniform circular shaft held fast at the end A, hollow and twisted by a couple applied in the plane BB. As- suming the strain to be within the limits of elasticity, a radius CD turns round to CD', and a line AD drawn at any dis- tance r from the axis, shafts. Fig. 40. and originally straight, changes into the helix AD'. Let 6 be the angle which this helix makes with lines parallel to the axis, or in other words the angle of shear at the distance r from the axis, and let i be the angle of twist DCD'. Taking two sections at a distance dx from one another, we have the arc 6dx = rdi. Hence q, the intensity of shearing stress hi a plane of cross-section, varies as ?, sinceg'=C5 = CV ^- . The resultant moment of the whole shearing stress on each plane of cross-section is equal to the twistinf moment AI. Thus _ Calling ?-j the outside radius (where the shearing stress is greatest) and q l its intensity there, we have q = rq l /r 1 , and hence, for a solid shaft, q l = 2AI/irr 1 . For a hollow shaft with a central hole of radius ?- 2 the same reasoning applies : the limits of integration are now r-i and r. 2 , and 2 Jfr The lines of principal stress are obviously helices inclined at 45 to the axis. If the shaft has any other form of section than a solid or sym- metrical hollow circle, an originally straight radial line becomes warped when the shaft is twisted, and the shearing stress is no longer proportional to the distance from the axis. The twisting of shafts of square, triangular, and other sections has been investigated by AI. de St Venant (see ELASTICITY, 66-71, where a comparison of torsional rigidities is given). In a square shaft (side = /t) the stress is greatest at the middle of each side, and its intensity there 1 is <7,=AI/ For round sections the angle of twist per unit of length is a 2M 211
- = r^~ = ~~f "7 in solid and ^ j r . in hollow shafts.
Crj TrGVj 4 *(-! 4 - r 2 4 ) 68. In what has been said above it is assumed that the stress is within the limit of elasticity. When the twisting couple is increased so that this limit is passed, plastic yielding begins in the outermost layer, and a larger proportion of the whole stress falls to be borne by layers nearer the centre. The case is similar to that of a beam bent beyond the elastic limit, described in 57. If we suppose the process of twisting to be continued, and that after passing the limit of elasticity the material is capable of much distortion without further increase of shearing stress, the distribu- tion of stress on any cross section will finally have an approximately uniform value q 1 , and the moment of torsion will be / 2irr-q'd,-
v 9['( r i 3 ~ ?V 3 ). Iu the case of a solid shaft this gives for AI a
Rankine, Applied Mechanics, 324.
