s11.14 Strain Energy 271 To obtain the total strain energy we must now integrate this along the length of the cantilever. In this case o is constant and equal to W and the integration is simple. W2B D5 U,= 8G1230 dx 0 W2B D5 W2BLDS =8G730 L 240G BD3 3W2L 5AG where A =BD. Therefore deflection due to shear aU,6WL δ，=aw=5AG (11.19) Similarly,since M=-Wx W2L3 UB= (-Wx) ds = 2EI 6EI 0 Therefore deflection due to bending aU WL3 δg=aw=3E7 (11.20) Comparison of eqns.(11.19)and(11.20)then yields the relationship between the shear and bending deflections.For very short beams,where the length equals the depth,the shear deflection is almost twice that due to bending.For longer beams,however,the bending deflection is very much greater than that due to shear and the latter can usually be neglected, e.g.for L=10D the deflection due to shear is less than 1%of that due to bending. (b)Cantilever carrying uniformly distributed load Consider now the same cantilever but carrying a uniformly distributed load over its complete length as shown in Fig.11.12. The shear force at any distance x from the free end Q=wx w per unit length 333363333333333* —dx -L- Fig.11.12.511.14 Strain Energy 27 1 To obtain the total strain energy we must now integrate this along the length of the cantilever. In this case Q is constant and equal to W and the integration is simple. L W2B D5 W2BLD5 8G12 30 L= 240G (%y =-- 3 W2L 5AG -- where A = BD. Therefore deflection due to shear Similarly, since M = - Wx (- WX)2 W2L3 ds = ~ uB=[ 0 2EI 6EI Therefore deflection due to bending au WLJ gB=-=- aw 3EI (11.19) (1 1.20) Comparison of eqns. (1 1.19) and (11.20) then yields the relationship between the shear and bending deflections. For very short beams, where the length equals the depth, the shear deflection is almost twice that due to bending. For longer beams, however, the bending deflection is very much greater than that due to shear and the latter can usually be neglected, e.g. for L = 1OD the deflection due to shear is less than 1 % of that due to bending. (b) Cantilever carrying ungormly distributed load Consider now the same cantilever but carrying a uniformly distributed load over its The shear force at any distance x from the free end complete length as shown in Fig. 11.12. Q = wx w per unit lengrh Fig. 11.12