Influence Lines - Structural and Stress Analysis - page 651

also suppose that we require values of shear force and bending moment at the section
K; the S K and M K influence lines are then constructed using either of the methods
described in Sections 20.1 and 20.2.

Since the system is linear we can use the principle of superposition to determine the
combined effects of the loads. Therefore, with the loads in the positions shown, and
referring to Fig. 20.5(b)

S K = W 1 s 1 + W 2 s 2 + W 3 s 3 (20.9)

in which s 1 , s 2 and s 3 are the ordinates under the loads in the S K influence line.

Similarly, from Fig. 20.5(c)

M K = W 1 m 1 + W 2 m 2 + W 3 m 3 (20.10)

where m 1 , m 2 and m 3 are the ordinates under the loads in the M K influence line.

Maximum shear force at K

It can be seen from Fig. 20.5(b) that, as the loads W 1 , W 2 and W 3 move to the right,
the ordinates s 1 , s 2 and s 3 increase in magnitude so that the shear force at K increases
positively to a peak value with W 1 just to the left of K. When W 1 passes to the right of
K, the ordinate, s 1 , becomes negative, then

S K =− W 1 s 1 + W 2 s 2 + W 3 s 3

and the magnitude of S K suddenly drops. As the loads move further to the right the
now negative ordinate s 1 decreases in magnitude while the ordinates s 2 and s 3 increase
positively. Therefore, a second peak value of S K occurs with W 2 just to the left of K.
When W 2 passes to the right of K the ordinate s 2 becomes negative and

S K =− W 1 s 1 − W 2 s 2 + W 3 s 3

so that again there is a sudden fall in the positive value of S K . A third peak value is
reached with W 3 just to the left of K and then, as W 3 passes to the right of K, S K
becomes completely negative. The same arguments apply for negative values of S K as
the loads travel from right to left.

Thus we see that maximum positive and negative values of shear force at a section of a
beam occur when one of the loads is at that section. In some cases it is obvious which
load will give the greatest value, in other cases a trial and error method is used.

Maximum bending moment at K

A similar situation arises when determining the position of a set of loads to give the
maximum bending moment at a section of a beam although, as we shall see, a more