102 Mechanics of Materials §5.3 wx4 wx5 wL3x 23wL4 E1y=-24602+4-120 Then,for example,the deflection at the tip of the cantilever,where x =0,is 23wL4 y=- 120EI 5.3.Macaulay's method The simple integration method used in the previous examples can only be used when a single expression for B.M.applies along the complete length of the beam.In general this is not the case,and the method has to be adapted to cover all loading conditions. Consider,therefore,a small portion of a beam in which,at a particular section A,the shearing force is Q and the B.M.is M,as shown in Fig.5.10.At another section B,distance a along the beam,a concentrated load W is applied which will change the B.M.for points beyond B. Fig.5.10. Between A and B, M=3 =M+Qx () E袋=M+e吃+C (2) and * Ely -M 6+C1x+C2 (3) Beyond B d2y M=EI M+Qx-W(x-a) (4) =Mx+ 22 W2+Wax+C3 (5) ,x2 3 x2 and Ely M 乞+e-+ W6+wa2+Cx+C。 (6) Now for the same slope at B,equating(2)and(5), M+0受+G,=Mr+0写-w x2 x2 2+Wax+C3102 Mechanics of Materials 55.3 wx4 wx5 wL3x 23wL4 24 6OL 4 120 .. Ely= Then, for example, the deflection at the tip of the cantilever, where x = 0, is 23wL4 y= -___ 120EI 5.3. Macaulay’s method The simple integration method used in the previous examples can only be used when a single expression for B.M. applies along the complete length of the beam. In general this is not the case, and the method has to be adapted to cover all loading conditions. Consider, therefore, a small portion of a beam in which, at a particular section A, the shearing force is Q and the B.M. is M, as shown in Fig. 5.10. At another section B, distance a along the beam, a concentrated load W is applied which will change the B.M. for points beyond B. W 0 IX A B Fig. 5.10. Between A and B, d2Y M = El- dx2 = M +Qx x2 x3 and Ely = M- 2 + Q- 6 + C~X +C2 Beyond B d2Y M = ElT = M+Qx- W(x-a) dx and dY x2 x2 El-=Mx+Q-- W-+ Wax+C3 dx 2 2 x2 x3 x3 X2 2 6 6 Ely= M-+Q-- W-+ Wa-+C3x+C, 2 Now for the same slope at B, equating (2) and (5), X2 x2 x2 2 2 2 Mx+Q-+CC, = Mx+Q-- W-+ Wax+C3