# Moments of Fixed-Fixed Beam

I have a quick question related to the topic of mechanics of materials.

If I were to have a beam that is fixed at both ends and has a central point force, do moments develop at the fixed supports?

e.g. I was thinking, wouldn't there be an absence of moments at the fixed supports since the balanced vertical reaction forces at both ends would prevent the development of a moment?

When trying to figure out whether or not a given reaction will exist at a given support, it's worth remembering what a reaction actually is.

A reaction is the means by which the support resists the movement of the beam at that point. Force reactions resist the beam's attempts to deflect up-down or left-right at the support. Moment reactions resist the beam's attempts to rotate at the support.

So, let's assume there's no bending reaction at the beam's extremities. In that case, we'll be simply dealing with a simply-supported beam with pinned supports at the ends. These pinned supports will generate vertical forces which resist the beam's deflection at those points.

But how would the beam deflect in this case? Well, it'd obviously be shaped like a parabola (well, it'd actually be a cubic function, but it looks parabola-ish). But that means there was rotation at the extremities: the beam was previously horizontal at the supports, but now it's tilted.

But the definition of a fixed support is that it doesn't allow for rotations at that point.

So what does the fixed support do to stop that rotation? Well, it applies a concentrated bending moment to the beam, rotating it back to horizontal: Figures created with Ftool, a free structural analysis program.

• Thank you so much for the response! It was very clear and well explained. I think I understand where my misunderstanding came from now. Jul 9 '19 at 16:51
– Wasabi
Jul 9 '19 at 16:53

This is a statically indeterminate problem. I don't know what exactly you mean by

moments develop at the fixed supports

But I guess you mean if there exist any non-zero components of moment at the fixed points. The answer is yes. A simple and lengthy calculations yields:

$$M_{\text{Fixed points}} = \left|\frac{FL}{8}\right|$$

The sign of the moments at the endpoints are different, so the total moment is zero, otherwise the beam would rotate. But the moment itself at the endpoints is not necessarily zero.

So the moment is not zero there. The graph of the moment vs the length of the beam is a parabola.

• Hello Sam! Thank you for the response. So if I were to design a system say that were attached to two fixed supports, I would then have to account for potential failure at the support location due to a non-zero moment developed at the supports for the described load? Jun 14 '19 at 19:17
• @Justin Yes because the maximum shear force happen to be at the end points. The shear force graph is a butterfly if recall correctly. Take your time, i studied these things as an electromechanical engineer, there are real experts in this community who have an amazing insight into this subject, so take your time and wait.But my answer is also correct. Jun 14 '19 at 19:23
• Thank you! I really appreciate your help. Jun 14 '19 at 19:48

The moments are Pl/8 at support and at the middle.

Edit

This beam if it was not fixed end, supports would rotate at the ends over the supports by an angle, $$\theta= Pl^2/16EI ,$$ but we know it's a fixed end, so we know the support has created a moment which has bend the beam back from $$\theta$$ into horizontal angle of zero.