Release the Plastic Hinge

How close to a true moment release can you make the plastic hinge at one end of an OpenSees beamWithHinges element? It’s a question I’ve thought about before, and it came up again recently.

A simple fixed-fixed beam can start to answer the question. Imposing a unit rotation at one end will produce moment reactions (fixed-end moments) of 4EI/L and 2EI/L. If there is a moment release at one end, the moment reactions will be 3EI/L and zero.

The OpenSees model is a single beamWithHinges element (modified Gauss-Radau integration) with l_{pI}=0 at the fixed end and hinge length l_{pJ}=\beta L at the end we want to release. The flexural stiffness along the beam interior is EI while the flexural stiffness of the hinge is \alpha EI.

For the imposed unit rotation, we should obtain 3EI/L when \alpha is small and $\latex beta$ is non-zero. When \beta is zero, we should recover a fixed-fixed beam with no hinge or release. Also note that we cannot make \alpha=0 because that would throw a wrench into the force-based element state determination.

Performing the imposed rotation analysis over many values of \alpha and \beta, we obtain the following relationships for the fixed-end moments at each end of the beam.

As expected, we recover 4EI/L and 2EI/L when \beta goes to zero and \alpha goes to one and we see that the moment release works fairly well when \beta \geq 0.02 and \alpha \leq 0.001.

We can generate similar plots for the other plastic hinge integration method outlined in this paper. Some of the plots get interesting due to the numerical integration characteristics of the other methods.

Even though this approach appears to work, I don’t recommend implementing it in your models any time soon, especially if you anticipate nonlinear material response. You’re better off to define two nodes and use the equalDOF command to get a moment release. But this was an interesting exercise.

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