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Homework 1- Nonlinear Finite Element Methods |
Nonlinear Finite Element Methods, WiSe 2024/25, Homework 1
Report all work, including any m-files (or Python code) you have written. Please write clearly and be sure to label for which problem each solution is.
Consider the following benchmark: a cantilever beam of length L = 5 m, made of steel (E = 200 GPa, G = 80 GPa) with an IPE 100 profile (A = 10.32 cm2, Aweb = 3.63 cm2, I = 171 cm4, shear correction factor κ = Aweb/A, fully fixed on the left and loaded by a single force F = 1 N on the right end.
- Derive an analytical reference solution according to the Euler-Bernoulli beam theory. Plot the displacement solution.
- Derive the one-field variational formulation (weak form) using the Timoshenko beam theory. Also draw a Tonti diagram for this case.
- Discretize the one-field weak form using standard two-node linear finite elements for both displacements and rotations. Conduct a convergence study for the displacement error in the
L 2 norm, using nele = 2, 4, 8, 16, 32, 64, 128, 256 elements and the Euler-Bernoulli analytical solution as a reference.
- Derive the corresponding two-field variational formulation (weak form), based on the HellingerReissner principle. Also draw a Tonti diagram for this case.
- Discretize the two-field weak form using standard two-node linear finite elements for all independent fields. Conduct a convergence study for the displacement error in the L 2 norm,
using nele = 2, 4, 8, 16, 32, 64, 128, 256 elements and the Euler-Bernoulli analytical solution as a reference.
- Compare and discuss the one-field and two-field results. Discuss the advantages and disadvantages of the two Timoshenko beam FE formulations and the Euler-Bernoulli beam FE formulation
that you know from FEM I.
