# What is the stress distibution in a cantilever non-slender beam?

I recently had a discussion with another engineer about shear walls and the check that are necessary to establish their capacity to resist horizontal loads.

He was arguing that the walls should be checked as cantilevering beams, which I agreed with. The point we did not agree on was the use of the Euler–Bernoulli bending theory to model the beam.

In my understanding, an important assumption of the Euler-Bernoulli theory is that the beam is slender, which allows to consider that plane sections remain plane.

A shear wall often has an aspect ratio so that its depth is greater than its height, and therefore the Euler-Bernoulli assumption is invalid (although probably conservative).

Hence my questions:

• Is there any way to determine the stress distribution in a cantilevering non-slender beam?
• Would Timoshenko bending theory give a different stress distribution than Euler-Bernoulli?

Euler-Bernoulli beam theory assumes that plane sections remain plane, and also that the section remains perpendicular to the beam neutral axis.

The second part of the assumption is equivalent to ignoring the shear in the beam.

Euler-Timoshenko beam theory keeps the assumption that plane sections remain plane, but does not assume they remain perpendicular to the neutral axis.

This gives an additional term in the bending stiffness of the beam, making it more flexible than the Euler-Bernoulli assumption. Whether that is a more or less conservative assumption depends on the purpose of the beam!

It is possible to add even more correction terms including the fact that plane sections do not remain plane, the cross-sections remain plane but deform in shape (so-called anticlastic curvature), etc., but given modern computing hardware and software, at some point it is simpler just to make a three-dimensional model of the structure without making any assumptions that it behaves like a beam.

Note, making a 3-D model doesn't necessarily give you a "free lunch" solution to the problem. For example, the details of the way you model how the ends of the "beam" are connected to the rest of the structure can change the results by the same order of magnitude as assuming that beam cross sections do or don't warp.

You can valid if the assumption is correct. If the calculated stress / strain is small, very likely, the assumption is valid and the calculated result error is small.