# In FEA, why is the stress at an interior angle (<180 deg) is a singularity and an exterior angle (>180 deg) it is not?

If you observe the attached figure, I heard some FE analysts say that the stress at the interior corner is always a singularity (but its intensity depends on the value of interior angle) but the stress at an exterior corner is not. I cannot understand why. Some even say that the theoretical value at an exterior angle is supposed to be zero. My question is that is it always the case? Does the exterior angles of a geometry can never result in a singularity, for any type of loading and boundary condition applied to the geometry? Reason being? And if sufficient mesh refinement is provided, so will the stresses at the exterior angles result in values close to zero?

Fair warning, I'll be anthropomorphizing and using metaphors throughout this answer.

It helps to think in terms of how the applied force "flows" through the element. It "wants" to get from the top where it is applied over to the fixed support. To do so it has to go down and to the right (though doing so actually means going left at first).

The force likes to take the shortest possible path to the support, so it tries to make a sharp turn at the corner. This basically creates a "traffic jam" near the inner corner while leaving the outer corner much freer.

The following image is a sketch trying to show these "force lines" and now they flow from the top to the support. Notice how they bunch up near the internal angle.

You could draw as many lines as you want and you'd get arbitrarily close to the external corner, but the "density" of lines near it will be much lower than near the internal corner.

• +1 for the flow lines Feb 1, 2021 at 20:02
• So the exterior corners are never a singularity, rather it be for any type of loading and boundary condition (ofcourse assuming they are not applied at the exterior corners themselves). How would a solver know about this that the load paths are supposed to get 'traffic jammed' at the interior corner and not at the exterior corner? Or the the traffic jam analogy is developed afterwards, i mean after seeing the behavior of the stresses in FEA? Feb 1, 2021 at 20:30
• @RameezUlHaq: The concept of "flow lines" (don't remember the real engineering term for them) is very old, certainly much older than computers. As to how to calculate the stress distributions, I honestly have no idea how it was done before computers. FEA solvers simply apply analytical methods to find the deformed configuration which presents the lowest work generated (or something, I don't remember my energy methods theory very well anymore). And since work is a function of distance travelled, the deformed configuration is such that the flow lines try to stick to the inner corner.
– Wasabi
Feb 1, 2021 at 22:06