The maximum bending moment caused by a set of concentrated moving loads is

Concept:

Maximum moment under a load:

Let a train of concentrated loads W₁, W₂, W₃,… move on a simply supported beam AB from left to right as shown in Figure 5.18. Now, the condition for moment to be maximum under wheel load W₂ is required. Let R be the resultant of all loads. Let its distance from W₂ be d and from support A be x (Refer figure 5.18)

Now, \({R_A}\; = \;\frac{{R\left( {L - x} \right)}}{L}\)

Therefore, moment under load W₂

M = R_A (x + d) – Rd

\(= \;R\left[ {\frac{{L - x}}{L}} \right]\left( {x\; + \;d} \right) - Rd\)

\(= \;\frac{R}{L}\left[ {Lx - {x^2}\; + \;Ld - xd} \right] - Rd\)

For the moment to be maximum

\(\frac{{dM}}{{dx}}\; = \;0\; = \;\frac{R}{L}\left[ {L - 2x - d} \right]\)

Or  \(x\; = \;\frac{L}{2} - \frac{d}{2}\)       ---(1)

Distance of W₂ from A

\(= \;x\; + \;d\; = \;\frac{L}{2} - \frac{d}{2}\; + \;d\)

\( = \;\frac{L}{2}\; + \;\frac{d}{2}\)         ---(2)

From (1) and (2) we conclude, for moment to be maximum under any particular load, the load and the resultant should be equidistant from the mid-span.

To get maximum bending moment under a chosen wheel load,keep the loads such that the chosen wheel load and resultant are at equal distance from centre. Then, maximum BM occurs under choosen wheel load (But not at centre).  

So here the appropriate option will be option D.