§5.3 Slope and Deflection of Beams 103 But at B,x =a Wa2 C,=-"2+wa2+Cg Wa2 C3=C1- 2 Substituting in (5), Edy-Mx+22- Wa2 dx -W+Wax+C- B张=Mk+e-ar+c: (7) dx Also,for the same deflection at B equating (3)and(6),with x =a ccCC. Ma2 Qa3 C:a+C:-Wat ha +2+C3a+C 2 +(-"加+c Wa3 C4=C2+ 6 Substituting in (6), x2 x3 Wa2\ Ely M 2+ (9-2x+w +C2 2 x3 =M 2+06-w-a 6+C1x+C2 (8) Thus,inspecting (4),(7)and(8),we can see that the general method of obtaining slopes and deflections (i.e.integrating the equation for M)will still apply provided that the term W(x-a)is integrated with respect to (x-a)and not x.Thus,when integrated,the term becomes w(x-a)2 2 and w(x-a) 6 successively. In addition,since the term W(x-a)applies only after the discontinuity,i.e.whenx>a,it should be considered only when x a or when (x-a)is positive.For these reasons such terms are conventionally put into square or curly brackets and called Macaulay terms. Thus Macaulay terms must be (a)integrated with respect to themselves and (b)neglected when negative. For the whole beam,therefore, Ei dx2=M+Qx-W[(x-a)]$5.3 Slope and Deflection of Beams 103 But at B,x = a .. Wa2 2 c + Wa2 + C3 1- L Substituting in (9, .. dY x2 x2 Wa2 El-=Mx+Q-- W-+ Wax+C,--- dx 2 2 2 dY x2 w El- = Mx + Q- - -(x-a)’ +C, dx 22 Also, for the same deflection at B equating (3) and (6), with x = a Ma2 Qa3 Ma2 Qa3 Wa3 Wa3 -+-+C,a+C, =-+---- + ~ + C3a + C, 2 6 2 6 6 2 .. .. Substituting in (6), Wa3 Wa3 C,a+C2 = -- +- + C3a + C, 6 2 = -__ Wa3+wa3 + ( c,--- y2)a+c, 6 2 Wa c,=c2+- 6 (7) x2 x3 (x - a)3 2 6 6 = M- + Q- - W- + c,x + c, Thus, inspecting (4), (7) and (8), we can see that the general method of obtaining slopes and deflections (i.e. integrating the equation for M) will still apply provided that the term W(x -a) is integrated with respect to (x -a) and not x. Thus, when integrated, the term becomes (x - a)2 W- 2 (x - a)3 and W- 6 successively. In addition, since the term W(x - a) applies only after the discontinuity, i.e. when x > a, it should be considered only when x > a or when (x - a) is positive. For these reasons such terms are conventionally put into square or curly brackets and called Macaulay terms. Thus Macaulay terms must be (a) integrated with respect to themselves and (b) neglected when negative. For the whole beam, therefore, d2Y El, = M+Qx- W[(x-a)] dx