Corotational02

This page develops 5 examples that demonstrate improvements of the Corotational02 transformation over the existing CrdTransf classes in OpenSees.

To install the required dependencies create a virtual environment and run:

python -m pip install -Ur requirements.txt

All tests can be run in a Posix shell by sourcing the script test.sh.

1) Objectivity and Self-Stressing  

This example demonstrates a severe bug in the existing corotational formulation.

Setup  

A cantilever is discretized by 8 beam elements. No loading is applied. Two methods of static analysis are considered: load and displacement control.

Expectation  

Because no loading is applied, both methods of analysis should not produce any displacement.

Outcome  

The original corotational formulation gives rise to non-zero nodal displacements, and therefore fails the experiment. The severity of the problem is particularly clear in the case of displacement control, where after 200 steps a drift over 10% is produced.

When Corotational02 is used, the displacements remain at exactly 0 at all nodes, in all DOFs.

2) Accuracy in pure flexure  

Setup  

This problem furnishes one of the few known closed form solutions of a beam undergoing finite rotations. A cantilever oriented along the $\mathbf{e}_1$ coordinate basis is subjected to a concentrated moment at its tip with magnitude $\lambda \pi 2 EI/L \mathbf{e}_3$ . A discretization of 5 finite elements is employed and the load factor $\lambda$ is scaled linearly from $0$ to $1$ in 5 increments.

Expectation  

The analytic solution of the geometrically exact boundary value problem is given by:

which furnishes a tip displacement $\mathbf{e}_1 \cdot \boldsymbol{x}(L) = -1$ .

Outcome  

3) Convergence with Bathe’s Cantilever  

This example demonstrates the convergence characteristics of the new corotational transformations. The standard problem of a curved 45-degree cantilever is implemented. The following variants are investigated:

Test02-Geom01

This is the formulation that is currently available in OpenSees as the Corotational transformation. It can be executed with both the OpenSees executable, and the xara executable.

Test02-Geom02

4) Reliability under large rotations  

5) Computational efficiency  

6) Shear deformations  

This example demonstrates the use of the Corotational02 transformation to represent a shear-deformable cantilever. The setup is that of Section 4.2.2 from Perez and Filippou (2024).

Test06-Prism-Geom01

This is the formulation that is currently available in OpenSees as the Corotational transformation. It can be executed with both the OpenSees executable, and the xara executable.

Test06-Prism-Geom02