How can we assume that structural supports are rigid?

In engineering classes we made the assumption that structural supports (fixed, roller, etc.) are rigid. Thus, the displacements of a beam ,for example, at a fixed support are set to zero in the horizontal and vertical directions as boundary conditions. However in real situations shouldn't there be some displacement due to deformation in the support? Wouldn't that affect stresses/strains in both the beam and the support considerably?

• It depends. Sometimes, a problem is meant to illustrate what happens to the span between supports, rather than the supports themselves. In other cases, the effect of the settlement of the supports may be included. – user16622 Jun 24 '16 at 10:55
• "First, we assum a spherical cow with a uniform distribution of milk." You always start with a simplified model so you can learn something. Then you add in more and more real-world constraints as your knowledge increases. – Carl Witthoft Jun 24 '16 at 13:15
• "You always start with a simplified model so you can learn something." Also, in real engineering work, you try not to design things you can't analyse easily. Many real-world designs contain features whose only purpose is to make these simplifying assumptions "accurate enough". en.wikipedia.org/wiki/Expansion_joint has a few examples, with pictures. – alephzero Jun 25 '16 at 2:39

Like many things in engineering this one often comes down to a judgment on if this effect is signficant. By significant we would mean does it change the results enough for us to worry about it.

A few examples may be worth considering.

1. A simple beam under uniform load. Take:

$$w = 10\ \mathrm{kN/m} \\ L = 2\ \mathrm{m} \\ E = 30\ \mathrm{GPa} \\ I = 6.75 \times 10^{-4}\ \mathrm{m^4}$$

In which the standard deflection formula gives: $$\frac{5wL^4}{384 E I} = 0.103\ \mathrm{mm}$$ Which assumes small deflections and therefore no horizontal movement of the supports.

Unfortunately there is no simple analytical way to consider the horizontal displacement. Using non-linear finite element analysis the maximum vertical deflection is $0.108\ \mathrm{mm}$ and the right support moves by $1.4 \times 10^{-5}\ \mathrm{mm}$. Clearly the horizontal movement is very, very small and considering it doesn't make enough of a difference in the deflection or stresses for us to worry about. For beams with realistic sizes, materials, and loads this will usually be the case.

2. A support settlement.

A support settlement is one situation where even small displacements of the supports need to be considered. For example a fixed-fixed beam where one end is displaced by $\Delta$:

In this case the resulting bending moment in the beam due to the support displacement is:

$$M= \frac{6EI\Delta}{L^2}$$

Using the same properties as above and a displacement of $\Delta=1\ \mathrm{mm}$ results in $M=30\ \mathrm{kNm}$. Which, if our beam was made of concrete, could be enough to cause it to crack. In this case we definitely would want to consider the displacement of the support.

Some other cases which displacements of the supports would be considered:

• In tall buildings the effect of axial displacement of the columns is considered (it is not significant in low rise buildings)
• When temperature fluctuations are considered, the fact that an element may not actually be free to move is considered, as this will result in (sometimes very significant) stresses. This is a common source of cracking in concrete.
• When second-order effects are considered, the so called $P-\delta$ effect, where the structure is in a displaced configuration (due to wind, earthquake, etc), and the loads are then applied.

I recently saw a picture on Facebook of a t-shirt quite similar to this one.

This is as good a definition of engineering as any. And that's because you are correct.

To say that any support is perfectly rigid is a lie, since there will always be some deformation of the support. But, then again, stating that a material is perfectly elasto-plastic (Hooke's Law) is also a lie, since micro-imperfections will always make a member deform in a slightly chaotic (as in, not perfectly predictable) manner. Euler-Bernoulli beam theory is a lie since it ignores shear deformations. Timoshenko beam theory (which does consider shear) is also a lie, since it considers perfectly elastic material behavior, which we saw above is a lie.

There is not a single "theory" (in engineering) which is not a lie, and that is simply because engineering doesn't care about providing a perfect description of reality. Engineers just want a theory that's good enough.

That's why we use engineering stress (as opposed to true stress). That's why we use Euler-Bernoulli or Timoshenko beam theory. And here I'm talking about the fundamental laws governing structural engineering. If we can accept simplifying assumptions here, in the bedrock of our field, we can accept them on a case-by-case basis as well.

And that's why we usually consider supports to be rigid. Because it's a simplification which makes our lives infinitely easier and which is good enough.

Could you create a complex model, where you consider the stiffness of the columns? Sure. But the columns are resting on the foundations, so you'll need to consider the stiffness of those as well, and they are resting on the soil, so you'll need to consider the effect of that as well. There comes a time when you need to decide whether it's worth it. Is the deformation of the supports really going to make a difference?

If you have reason to believe that your supports may be significantly flexible (for example, your foundations are resting on a complex soil), then yes, you absolutely must consider the effect that will have on your structure. If, however, there is no reason to believe the support's deformations will be all that relevant, then, well, it's not relevant and there's no reason to consider it.

And, as a detail, in the simple case of isostatic structures (such as simply-supported beams), small support deformations don't cause any stress increase at all. Well, actually, that's a lie since it arises from Euler-Bernoulli theory. But if you pretend that Euler-Bernoulli is true (you really should), then yeah, no internal forces emerge.

Depending on the level of engineering course you are taking, as previously mentioned this can be a simplified case, and most engineers love to keep things simple. In addition, if you start tossing in various movements, your solution for the beam can go from being statically determinant to indeterminant. As a result you may not have the math courses or engineering formulas developed yet to handle those situations. You will eventually come across differential settlement where one or more supports will drop by different amounts. In addition you will be taking on cases where both end of the beam are restrained and the beam contracts. These are real issues that as a structural engineer you will deal with. And at that time you will have your complete degree to help you, along with analysis programs, and a myriad of resources available on the net and elsewhere to assist you.

• I fail to see why this was downvoted. – Carl Witthoft Jun 24 '16 at 13:15