§6.4 Built-in Beams 145 Substituting in (1), Wab Wa?b MA=- L+ L2 Wab(a+b)Wa?b 宁一 L2 +2 Wab2 三一 D (6.8) 6.4.Built-in beam carrying a non-uniform distributed load Let w'be the distributed load varying in intensity along the beam as shown in Fig.6.4.On a short length dx at a distance x from A there is a load of w'dx.Contribution of this load to M Wab2 一 L2 (where W=w'dx) w'dx×x(L-x)2 w'x(L-x)2dx total M=- L2 (6.9) w'/metre A dx L-X- Fig.6.4.Built-in (encastre)beam carrying non-uniform distributed load. Similarly, MB=一 w(L-x)x2 dx (6.10) If the distributed load is across only part of the span the limits of integration must be changed to take account of this:i.e.for a distributed load w applied between x=x and x =x2 and varying in intensity, M4=- 背wxL-x好d (6.11) w(L-x)x2 MB=- L dx (6.12)56.4 Built-in Beams 145 Substituting in (l), Wab Wa2b L L2 M ---+- A￾Wab(a + b) Wa2b =- +- Wab’ LZ L2 L2 = -__ 6.4. Built-in beam carrying a non-uniform distributed load Let w’ be the distributed load varying in intensity along the beam as shown in Fig. 6.4. On a short length dx at a distance x from A there is a load of w’dx. Contribution of this load to MA Wab2 = -~ L2 (where W = w’dx) w’dx x x(L - x)’ L2 1 w’x( L;x)’dx -- total MA = - 0 w’/me tre \ Fig. 6.4. Built-in (encostre) beam carrying non-uniform distributed load Similarly, (6.9) (6.10) 0 If the distributed load is across only part of the span the limits of integration must be changed to take account of this: i.e. for a distributed load w’ applied between x = xl and x = x2 and varying in intensity, (6.1 1) (6.12)