Question 2.9.1: A simply supported beam AB of span 20 m is crossed by travel...
A simply supported beam AB of span 20 m is crossed by travelling loads. Determine the greatest shear and bending moment experienced at a point C, distant 8 m from the left hand support due to passage across the span of:
(a) A uniformly distributed load of 5 kN/m extending over a length of 6 m.
(b) Two point loads of 3 kN and 2 kN, 6 m apart. They may cross in either direction, and either may lead.
The influence lines for shear and bending moment at C are shown in Fig. 2.9-2(a) and (b).
(a) The greatest positive shear at C will occur when the front of the load extends to the left of C, and appears as shown in Fig. 2.9-2(c). Numerically that shear will be (the load intensity) x (the area of the influence line covered) and will be
5 \times (6/2)(0.4 + (2/8)0.4) = 15 \times 0.4 \times 1.25 = 7.5 kN
The greatest negative shear will occur when the load is just to the right of C, as in (d), and the magnitude will be
5 \times (6/2)(0.6 +0.3) = 15 \times 0.9 = 13.5 kN
which is the larger of the two possibilities.
The load will cause the greatest bending moment at C when it is so arranged over C that the ordinates cd and ef, at the two ends of the load, are equal—as shown in Fig. 2.9-2(e). Thus the load is placed in such a way that C divides the span and the load in the same ratio. The statement is obvious enough from that figure—if the load is moved a small distance to the right (as shown), or left, the area of the influence line thus uncovered would be greater than the additional amount covered at the other end.
(M_{c})_{max}=(5\times 6/2)(3.36+4.80)=122.40 kNm
It will be noted that the influence line technique may be used to position the load to
