Static rollup of an isotropic beam

In this benchmark test, we examine the static pure bending of an isotropic beam of 10 units in length. Under the application of a specific moment, the beam rolls up into a circular curve and the exact analytical tip displacement (in the direction of the beam length) is -10. Details of the benchmark problem are described in [@Simo-VuQuoc:1986]. The following table shows the tip displacement calculated with a single Legendre Spectral Finite Element (LSFE) with the number of nodes shown. N-Point Gauss-Legendre quadrature was employed, where N is the number of nodes. The results show the expected spectral convergence to the exact solution.

# of nodes	Tip Displacement
3	-15.6418448333196
6	-10.9854783002967
9	-10.058403575783
12	-10.0000013123339
15	-10.0000000000645

Note

This benchmark is included as a regression test in the Kynema code base. The test is implemented in:

https://github.com/kynema/kynema/blob/main/tests/regression_tests/regression/verification/static_isotropic_beam_rollup.cpp

Simo, J. C., and L. Vu Quoc. 1986. “A three-dimensional finite-strain rod model. Part II.” Computer Methods in Applied Mechanics and Engineering 58: 79-116.