Why don't we use principal maximum stresses when designing a structural element?

Consider a concrete column is under compression from top loading and also carries some shear stresses.

If take a plane 2d element in the column with these stresses and rotate it to the point it gives out the it maximum normal stress then rotate it to give out maximum shear stress which both values should be higher than our original calculated stresses.

Why don't we compare those value to the concrete's compression and shear strength instead?

I apologize if my question is too simplistic, I'm still a civil engineering freshman.

• Maximum principle stress is mostly only useful for failure criterion of brittle materials. On the other hand, structural elements tend to be ductile and require Distortion Energy Theory to characterize failure; in which case, Von Mises Stress is the way to go.
– Paul
Mar 18 '16 at 19:10
• So in the case of concrete, taking the principal stresses for design would be appropriate since it's a brittle material? Mar 19 '16 at 11:21

The short answer is because it is too complicated/impossible to do so.

Here is a diagram of the principal stress trajectories for an uncracked concrete beam under both flexure and compression:

As you can see the orientation and magnitude of the principal stresses will change depending on the point you are interested in and the applied loads. We know that concrete is weak in tension. So if we are looking at a location of principal tensile stress we can compare this to the tensile capacity of the concrete (which is often considered to be a function of $\sqrt{f_c'}$).

What if the principal tensile stress exceeds the tensile capacity of the concrete?

Well at that point the concrete may fail. But this doesn't mean the whole element will fail. It means that it will crack at that location. But that is OK, that is what reinforcement is for!

So now we have a concrete element with a crack (or many cracks!), and reinforcement to hold the pieces together:

If we now want to calculate our principal stresses, what is the state of stress at a particular point? We have some stress being carried by the reinforcement, some stress being carried by aggregate interlock along the cracks, some being carried by compression, and some voids where no stress can exist - how much goes into each mechanism? We can't simply use formulas like $\nu = \frac{VQ}{It}$ since this only applies to a uniform material.

We can't determine the state of stress with any reasonable certainty in a cracked concrete section$_1$.

So what can we do now? Well, we do lots, and lots of tests and then fit a design equation to the results.

You mentioned columns in your question. Columns are dominated by compressive stresses, so cracking is often not as much of an issue. However, there are still complicating factors which will make it difficult/impossible to determine the stress state. In fact, the commentary of ACI 318 says:

The actual distribution of concrete compressive stress is complex and usually not known explicitly. ... The Code permits any particular stress distribution to be assumed in design if shown to result in predictions of ultimate strength in reasonable agreement with the results of comprehensive tests.

So again, we are forced to take the easier route of assuming a simplified stress state and confirming that is safe according to tests.

The uncertainty related to using these simplifications is incorporated into the safety factors used in building codes.

It would be much more satisfying to have a design methodology which is built on the principal stresses. This has apparently been tried in the past, but was always unsuccessful due to the difficultly in determining the stress state$_2$.

1. Kong, F. K., & Evans, R. H. (2013). Reinforced and prestressed concrete. Springer.

• Another reason for taking a simplified approach in situations like this is to allow "low tech" civil engineers to actually work for a living. With adequate computer software and hardware, and knowledge of how to use them, you can get pretty close to the details of the load paths in the concrete itself and in the reinforcement bars. But it's not practical to make a small-town builder with maybe 30 years of practical experience, but zero formal education in finite element analysis methods, use that level of analysis to select the correct size of beam to repair somebody's house safely. Mar 19 '16 at 20:42
• This is a fantastic answer. May 12 '16 at 14:16