Question 2.8.4: A beam of span L, of uniform flexural rigidity, carries a di...
A beam of span L, of uniform flexural rigidity, carries a distributed load which varies linearly in intensity from zero at the left-hand support to w per unit length at the right, as shown in Fig. 2.8-12. Find the distance from the left-hand support of the point of maximum bending moment, sketch the shear force and bending moment diagrams, and give numerical values if L = 10 m and w = 2 kN/m.
The total load on the beam is (wL/2), and taking moments about A will show that R_{B} = (wL/3) and R_{A} = (wL/6).
The bending moment at a point x from
A = (wL/6)x – (wx/L)(x/2)(x/3)
= (wLx/6) – (wx^{3}/6L)
Differentiating with respect to x, and equating to zero, shows that a maximum will occur at x = 0.577L where M=\frac{wL^{2}}{9\sqrt{3} } =0.064 wL^{2}.
The shapes of the shear force and bending moment diagrams are shown at (b) and (c)