Timoshenko Beam Analysis using FEM and VEM

This repository contains an implementation of the Finite Element Method (FEM) and Virtual Element Method (VEM) for the static analysis of Timoshenko beam theory.

The focus of this work is on:

• Verification of numerical formulations
• h-convergence studies
• Slenderness effects
• Comparison between FEM and VEM
• Symbolic derivation of the stiffness matrix

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Description of Examples

1. Cantilever Beam

• Static analysis of a Timoshenko cantilever beam
• Includes:
  • Comparison of results for different order and method
  • h-convergence study
  • Comparison of results for different slenderness ratios

2. Simply Supported Beam

• Static analysis of a simply supported Timoshenko beam
• Includes:
  • Comparison of results for different order and method
  • h-convergence study
  • Comparison of results for different slenderness ratios

3. Symbolic Stiffness Matrix

• Symbolic derivation of the Timoshenko beam stiffness matrix

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Numerical Methods

• FEM: Standard displacement-based formulation for Timoshenko beams
• VEM: Virtual Element formulation formulation for Timoshenko beams
• Comparison between FEM and VEM is performed through numerical experiments

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Key Features

• Modular and extensible source code
• Clear separation between solver and example problems
• Numerical convergence studies
• Slenderness parameter investigation
• Symbolic stiffness matrix derivation

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Requirements

• Programming language: * Python *
• Additional libraries: NumPy, SymPy

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How to Run

1. Navigate to the examples directory
2. Run the desired example file:
   • Cantilever beam
   • Simply supported beam
   • Symbolic stiffness matrix
3. Modify parameters inside the example files to perform additional studies

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Author

Muhammad Hamza
University of Stuttgart

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Reference

1. Wriggers, P. (2023). A locking-free virtual element formulation for Timoshenko beams. Computer Methods in Applied Mechanics and Engineering, Article 116234.
2. Wriggers, P. (2022). On a virtual element formulation for trusses and beams. Archive of Applied Mechanics, 92, 1655–1678.
3. Wriggers, P., Aldakheel, F., & Hudobivnik, B. (2024). Virtual Element Methods in Engineering Sciences Springer.
4. Öchsner, A. (2021). Classical Beam Theories of Structural Mechanics Springer.
5. von Scheven, M., Bischoff, M., & Ramm, E. (2024/2025). Computational Mechanics of Structures: Lecture Notes. Winter Term 2024/2025

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License

This project is intended for academic and research use.