Question

Consider the beam and the loading shown in (Figure 1). Suppose that L = 15 m.

Determine the absolute maximum bending moment in the beam due to the loading shown. Follow the sign convention.

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Answer 1

Absolute maximum moment in a simply supported beam occurs under one of the concentrated forces,such that this force is positioned on the beam so that it and the resultant force of the system are equidistant from the beam’s centerline.

Let us first determine position of resultant load from 20 kN load

35X'= 10x2 + 5x3

X' = 1 m

i.e Resultant load 35 kN is 1 m from 20 kN load

Case 1 : Let us assume it occurs under 10 kN load

Place 10 kN and 35 kN equidistant from center and calculate reactions

Take moment about B

Clockwise moments are +ve and anticlockwise as -ve

M_B = 0 - (hinged)

R_a x 15 - 35 x7 = 0

R_a = 16.33 kN

R_b = 18.66 kN

Now calculate moment at position of 10 kN load

M = 16.33x7 - 5x1

= 109.31 kN-m

Case 2 - Let us assume it occurs under 20 kN load

Place 20 kN load 35 kN load equidistant from center and calculate reactions

Mb = 0

R_ax15 - 35x8 = 0

R_a = 18.66 kN

R_b = 16.33 kN

Take moment about 20 kN load

M = 18.66x8 - 5x3 - 10x2

= 114.28 kN-m -(larger)

So absolute maximum bending moment is 114.28 kN-m