Moments of Fixed-Fixed Beam

Question

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?

Answer 1

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:

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_{Figures created with Ftool, a free structural analysis program.}

Answer 2

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.

Answer 3

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

Edit

After comments by @Wasabi, and the fact that my answer is correct, I add this explanation.

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.

section on fixed end support from NDS, 2005 edition on fixed end supported beams with a concentrated load on the center.