Equivalent Stress - 4 Models of Square Hole in Plate

**What are stress singularities?**

**What are the common causes of stress singularities?**

- Corners: e.g. sharp corners has theoretically infinite stresses because the radius is zero
- Stiff Boundary Conditions: e.g. a tension bar with one end surface fixed in all direction.
- Point loads
- Constraint Equations, Coupled Nodes etc

**How does one overcome stress singularities?**

1. Submodeling

Creating a submodel at the high stress locations with enough detail (e.g. fillet radius) along with mesh convergence study is the way to go. There is still engineering judgement on deciding what radius to use to best reflect the actual part.

2. Stress Concentration Calculation

Roark's formulas for stress and strain is a classic that is often used to determine the appropriate factors relative to the nominal stress. This works great if the load pattern and geometry is close to the textbook cases. It serves as a good way to ball park a similar type geometry.

3. Hot Spot Method

This is similar to what is sometimes done to estimate stresses for weld fatigue. One first creates a path of stress to the singularity. At a certain distance away from the stress singularity (~1mm in the example plot below), it is assumed the stresses are adequately far away and are thus reported. Often times a fudge-factor on top of the extracted stress value is applied to build in some conservative margin. What distance away? and what fudge-factor to use? Well, that has to be developed on prior, more detailed, modeling of similar parts.

This method takes some upfront homework, is quick and dirty and may require good hand waving skills.

Stress Path to Square Hole

**Other points of discussion**

I'm not aware of a way to deal with stress singularities caused by boundary conditions or nodal constraint equations / coupling or point loads. These modeling short-cuts are done to get a useful stress answer elsewhere and are justifiably ignored.

PRERR in APDL lists the structural energy norm. Here's a good write up illustrating its many limitations. It doesn't overcome singularity problems but with some experience, can help reduce the mesh convergence study needed.

Everything else being equal, mesh convergence study is one of the best, full proof method of gaining confidence of the computed stress. Unfortunately it does not work for stress singularities because the computed stresses will chase infinity as finer mesh is used. It works very well in gaining confidence in stresses elsewhere.

(Admittedly mesh convergence studies are not commonly done because it is very time consuming. Many analyst claim they make up for it with experience. Unfortunately it's a painful mistake I've made in the past. Healthy skepticism should be exercised.)

How do you handle stress singularities in your models?

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