Can bending moments for a combination of loads on a simple beam, be calculated by adding the individual B.M. of each load?

Question

This problem is about bending moment of a simple beam subject to a mixture of 4 or 5 UDLs and point loads. The beam is assumed to be rigid enough that deflection of the combined load is "small".

Solving this analytically for all load combinations simultaneously, in order to graph the result, is a tricky formula.

I'm wondering if, in this kind of simple case without significant distortion, one can simply use standard formulae for bending moments for single UDLs and point loads on a simple beam, and sum the results, which would be a much easier approach.

Intuitively, it should work. But is this actually a correct intuition? Can bending moments be summed in this way for multiple loads acting on a beam, to generate a graph of total bending moment acting at different positions on the beam?

Answer 1

YES

This is known as the superposition principle. For beams composed of linear-elastic materials where all loads are constant and independent from changes in beam geometry (i.e. not rain or snow loads, which increase as the beam deflects, creating an ever-deeper "pool"), the loads can all be considered independently. This means you can model each load alone, obtain the relevant results and then add them all up.

Just note that determining which results are "relevant" is not necessarily easy to do without modeling everything simultaneously. For example, a beam with complicated loading won't necessarily have maximum bending moment at midspan (unless all the loads are maximized at midspan, of course). In these cases, it may be hard to determine a priori which points to sample the bending moment, so finding the maximum bending moment may require solving all the loads simultaneously.

For example:

This beam has a maximum bending moment 4.10 m from the left support, which couldn't be determined a priori since the loading can be split into three "sub-loads":

1. UDL of 10 kN/m along the entire span, which we trivially know will cause maximum bending at midpsan;
2. UDL of 10 kN/m on the middle 2 m, which we trivially know will cause maximum bending at midpsan;
3. UDL of 30 kN/m applied 2nbsp;m to 4 m from the left support. This one is harder to determine, but has maximum bending 3.40 m from the left support.

Given this information, you couldn't possibly know that the most important point along the span is actually at 4.10 m. The only way to know this is by considering all the loads simultaneously.

But if the loading is more trivial or if you already know which points to sample, then the superposition principle is an excellent friend to have.