This rule for determining the maximum shear at any section of a beam is equally applicable to bridge-trusses under certain conditions, and has an important bearing upon the determination of the greatest stresses in some of the members of bridge-frames, although it has less importance now than it had in the earlier days of bridge-building. =85. Bending Moments and Shears for Cantilever Beams.=—The case of a loaded overhanging beam or cantilever bracket, as shown in Fig. 16, is sometimes found. In that figure a single weight _W_ is supposed to be applied at the end, while a uniform load _w_ per unit of length extends over its length _l_. The bending moment at any point _C_ distant _x_ from the end will obviously be _wx²_ _M = Wx_ + -----. (24) 2 [Illustration: FIG. 16.] The greatest value of the bending moment will be found by placing _x_ equal to _l_ in equation (24), and it will have the value _wl²_ _M₁ = Wl_ + -----. (25) 2 The shear at any point and at the end _A_ respectively will be _S = W + wx_ and _S₁ = W + wl_. (26) The shear due to _W_ is equal to itself and is constant throughout the whole length of the beam. The second term of the second member of equation (24) is the equation of a parabola with its vertex at _B_, Fig. 16. Hence if _AF_ be laid off equal to (_wl_²)/2, and if the parabola _FHB_ be drawn, any vertical intercept, as _HK_, between that curve and _AB_ will represent the bending moment at the corresponding point. On the other hand, the first term of the second member of equation (24) shows that the bending moment due to _W_ varies directly as the distance from _B_. Hence if _AG_ be laid off vertically downward from _A_ equal to _Wl_ to any convenient scale, then any intercept, as _KL_, between _AB_ and _BG_ will represent the bending moment due to _W_ at the corresponding point of the beam. =86. Greatest Bending Moment with any System of Loading.=—One of the most important positions of loading to be established either for simple beams or for bridge-trusses is that at which any given system of loading whatever is to be placed on any span so as to produce the maximum bending moment at any prescribed point in that span. In order to make the case perfectly general a system of arbitrary loads, like that shown in Fig. 17, is assumed and the system is supposed to be a moving one. [Illustration: FIG. 17.]