Discussion
A simply supported beam has a distributed load of linearly varying intensity with zero at one end to w per unit run at the other end. The value of maximum bending moment is:
Answer: B
The maximum bending moment is \(\frac{wl^{2}}{9\sqrt{3}}\).
The total load is \(wl/2\), acting at \(2l/3\) from the zero end, so the reactions are \(R_{A} = wl/6\) at the zero end and \(R_{B} = wl/3\). The shear force is zero where \(wl/6 = wx^{2}/2l\), that is at \(x = l/\sqrt{3}\). There
\[ \begin{aligned} M &= \frac{wl}{6}x - \frac{wx^{3}}{6l} \\ &= \frac{wl^{2}}{6\sqrt{3}} - \frac{wl^{2}}{18\sqrt{3}} = \frac{wl^{2}}{9\sqrt{3}} \end{aligned} \]
\(wl^{2}/8\) is the result for a uniform load w over the whole span, which carries twice this total load. \(wl^{2}/4\) and \(wl^{3}/4\) match no standard case, and the last has the wrong dimensions.
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