# Problem while solving for the moment of a cantilever beam

As in the first image, Moment at any point between C and B should be 0.

But when I try to solve it by substituting equal and opposite forces at end C so that anti-clockwise moment of $$F(l-a)$$ and force $$F$$ act at C, moment diagram comes out different. Why is there discrepancy?

• If you draw shear force and bending moment diagrams for the two sets of loads it is obvious the bending moments are different everywhere along the beam. I'm not sure what you are trying to ask here. – alephzero Jan 11 at 14:49
• I have included the picture. – BJKShah Jan 12 at 9:36

This is cheating a bit, but let's start backwards by looking at the correct diagrams you've already shown us and comparing them to your solution. This will hopefully let you notice something which hints at the problem with your solution.

Your solution is the bending moment equation $$M = F(\ell - a) - Fx$$.

Now, what is the bending moment equation for the correct moment diagram? Think about it.

Well, that's a trick question. What I should have asked is: what are the bending moment equations (plural!) for the correct moment diagram?

There is an obvious discontinuity in the slope of the bending moment diagram: it is linear from A to B, and then constant at 0 from B to C. That's my point: you could never solve this with a single equation, you'll always need two: one from A to B, another from B to C.

But what's the source of that discontinuity? Well, for that we need to remember that bending moment is the integral of shear force. Since you haven't asked about the correct shear diagram, I'm going to assume you understand why it looks as shown in the correct answer (simply put, it's constant and equal to $$F$$ between the support and the force, and zero elsewhere).

So, if the shear equation is discontinuous, that means the derivative (slope) of the bending moment equation will also be*. And since the shear equation gives us the slope of the bening moment equation at any point, we can trivially see that the slope of the line will be equal to $$F$$ from A to B, and zero from B to C.

Given this, we can derive the two bending moment equations (here using $$x=0$$ at A):

$$M = \begin{cases} Fx - Fa &\text{if }x\in[0, a]\\ 0 & \text{otherwise} \end{cases}$$

* Note that a discontinuous slope means the slope can change instantaneously, but the value of the bending moment equation on both sides of the discontinuity must be equal; the slope is discontinuous, not the value.

• Thank you for the answer. I just wanted to know what's wrong in the 2nd method? Is the substitution of forces not allowed here ? – BJKShah Jan 12 at 19:08
• @BJKShah The problem is that you've used the same equation for both parts of the beam. Whenever you have a discontinuity (concentrated force or bending moment), you'll end up with two different equations, one for each side of the discontinuity. – Wasabi Jan 12 at 21:04