Torsion with Fiber Sections

Original Post - 06 Oct 2019 - Michael H. Scott

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I won’t blog about every pull request to OpenSees on GitHub, but I will blog about pull requests that could affect backward compatibility of user scripts. Pull request #142 is one such case. It affects how torsion is added to fiber sections in three-dimensional models.

The frame elements require torsional stiffness in order to prevent a local rigid body mode of spinning. The fibers in a uniaxial fiber section do not provide shear stress, so torsion is not one of the stress resultants. Torsion has to be added to the fiber section.

There are two methods to add torsion to a FiberSection3d object: 1) use the SectionAggregator to attach a UniaxialMaterial; 2) use the -GJ option in the fiber section definition.

# Method 1
section Fiber 3 {
   patch ...
   layer ...
}
uniaxialMaterial Elastic 5 $GJ ;# Doesn't have to be elastic
section Aggregator 1 5 T -section 3

# Method 2
section Fiber 1 -GJ $GJ {
   patch ...
   layer ...
}

Most people use Method 2 for which this pull request will not break backward compatibility. If you use Method 1, read on.

Years ago when I implemented Method 2, I put a bogus GJ value in to the FiberSection3d class to accommodate folks who might forget to use either Method 1 or Method 2. The force-based element would always have a full rank flexibility matrix to invert and the displacement-based element would have something to interpolate. No harm, no foul, right?

Wrong! This messed up folks who used Method 1, which apparently was a small group because it took years for anyone to notice this issue. It’s likely they struggled in silence with OpenSees convergence problems just like everyone else.

The issue with Method 1 is that the FiberSection3d object is created with an artificially high GJ value, giving a section stiffness matrix with the following topology:

\[{\bf k}_s = \left[ \begin{array}{cccc} X & X & X & 0 \\ X & X & X & 0 \\ X & X & X & 0 \\ 0 & 0 & 0 & 1e14 \end{array} \right]\]

The 3x3 block is the axial-flexural stiffness that comes from the uniaxial fibers. With Method 2, the absurdly large 1e14 is overwritten with the -GJ option and everything is fine. But, with Method 1, when the SectionAggregator adds the torsional UniaxialMaterial to the fiber section, the section stiffness matrix ends up with two torsional modes–one that’s junk and the other that the user requests:

\[{\bf k}_s = \left[ \begin{array}{ccccc} X & X & X & 0 & 0 \\ X & X & X & 0 & 0 \\ X & X & X & 0 & 0 \\ 0 & 0 & 0 & 1e14 & 0 \\ 0 & 0 & 0 & 0 & GJ \end{array} \right]\]

The frame elements in OpenSees are able to handle two torsional modes in the section–that’s not the problem. The problem is that the conditioning of the element stiffness matrix and/or stability of the element state determination could be poor due to the extremely stiff torsional response, leading to global convergence issues. What was I thinking?

This pull request fixes the issue by forcing the user to specify either a GJ value or a UniaxialMaterial for torsion when defining the fiber section.

# New Method 1
uniaxialMaterial Elastic 5 $GJ ;# Doesn't have to be elastic
section Fiber 1 -torsion 5 {
   patch ...
   layer
}

# Method 2 (same as before)
section Fiber 1 -GJ $GJ {
   patch ...
   layer ...
}

With the new Method 1, the FiberSection3d class is the aggregator, not the aggregated. Unfortunately, backwards compatibility is broken for the old Method 1. After this pull request, you will receive an error stating that there is no torsion in your fiber section. You have to use either the -GJ or the -torsion option.

The SectionAggregator did not go anywhere. You can still aggregate shear to a fiber section and create other aggregations of section response without any problems.

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Torsion with Fiber Sections