An introduction to bending moment calculations
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Example case 4

Step 1: calculate the reactions at the supports

The reactions at the encastré support of this cantilever beam can be determined by equilibrium calculations only, so the beam is statically determinate. The support provides a vertical reaction and a moment reaction. The equilibrium equations I will use are those for vertical equilibrium and moment equilibrium. The equilibrium calculations are made easier if I convert the triangularly distributed load in to an equivalent point load.

View video tutorial for step 1 (pop-up video)

Step 2: form imaginary cut and draw free body diagram

In order to be able to calculate the shear force and bending moment at D, I introduce an imaginary cut in to the beam at D. This splits the actual beam in to two imaginary free bodies. It also splits the triangularly distributed load in to two, one for each free body. I’ll choose to analyse the left hand section A to D, and draw it’s free body diagram. As before, I turn the triangularly distributed load on this section in to an equivalent point load. I treat the trapazoidally distributed load on this free body as the sum of a uniformly distributed load and a trangularly distributed load, which is more convenient.

View video tutorial for step 2 (pop-up video)

Step 3: calculate force and moment at the imaginary cut

I will calculate the shear force and bending moment at D using my free body diagram and the equilibrium equations for vertical equilibrium and moment equilibrium. This is a similar process to what I did in step 1 to calculate the reactions at the encastré support.

View video tutorial for step 3 (pop-up video)

Step 4: check calculation by analysing the second free body

In the check calculation, I choose to analyse the right hand section D to C. This must give me the same values I have already found for the shear force and bending moment at D, if I’ve already calculated these correctly. This calculation is actually easier than the one I already did.

View video tutorial for step 4 (pop-up video)

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