Wheel load train: BM at 8 m and absolute max

Data: span $L = 25$ m. Wheel loads (from the rear): 80, 160, 160 and 120 kN at 2 m, 2 m and 3 m spacing; the 120 kN load leads (the train moves left to right). Total load $W = 520$ kN.

(i) Maximum BM at the section C, 8 m from the left end

ILD for $M_C$: peak ordinate $\dfrac{ab}{L} = \dfrac{8\times 17}{25} = 5.44$ m. Condition for maximum: the average load on the left of $C$, $\dfrac{W_L}{a}$, must be approximately equal to $\dfrac{W}{L}$ (it should exceed $W/L$ with the load at $C$ counted on the left, and fall below it without that load).

Try the rear 160 kN load (the second load from the rear) at $C$: the load on the left of $C$, including it, is $80 + 160 = 240$ kN.

$$\frac{240}{8} = 30 > \frac{520}{25} = 20.8, \qquad \frac{80}{8} = 10 < 20.8$$

So the condition is satisfied with the rear 160 kN load at $C$. Positions from the left support: 80 kN at 6 m, 160 kN at 8 m (at $C$), 160 kN at 10 m, 120 kN at 13 m.

$$M_{C,max} = 80(4.080) + 160(5.440) + 160(4.800) + 120(3.840) = 2425.6\ \text{kN m}$$

(Check by reaction: $R_A = 323.2$ kN, $M_C = R_A\times 8 - 80\times 2 = 2425.6$ kN m.)

(ii) Absolute maximum bending moment

Resultant $R = 520$ kN. Its distance from the 120 kN (leading) load $= \dfrac{80\times 7 + 160\times 5 + 160\times 3}{520} = \dfrac{1840}{520} = 3.538$ m behind it. The load nearest to the resultant is the 160 kN load that is 3 m behind the 120 kN load, at 0.538 m from the resultant. Place this load and the resultant symmetrically about mid-span: this 160 kN load is 0.269 m on one side of the centre and the resultant 0.269 m on the other.

• 160 kN (critical) at 12.769 m from the left support; resultant at 12.231 m; 120 kN at 15.769 m; other 160 kN at 10.769 m; 80 kN at 8.769 m (all on the girder).
• $R_A = 520\times\dfrac{25-12.231}{25} = 265.60$ kN.

$$M_{max} = R_A\times 12.769 - 80\times(12.769-8.769) - 160\times(12.769-10.769) = 2751.5\ \text{kN m}$$

Answer: (i) $M_C = 2425.6$ kN m; (ii) absolute maximum $BM = 2751.5$ kN m under the 160 kN load.