Finite Element Codes

Codes in Matlab and Python based on Finite Element Methods

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Buckling Analysis of a Two-Element Beam

This Python script performs a linear buckling analysis of a 2D beam composed of two Euler-Bernoulli elements with circular hollow cross-sections. It computes the critical buckling load ($\mathbf{P_{cr}}$) using the eigenvalue problem formulation:

$$\mathbf{K_f} \mathbf{v} = \lambda \mathbf{K_g} \mathbf{v}$$

where:

• ($\mathbf{K_f}$) is the global stiffness matrix,
• ($\mathbf{K_g}$) is the global geometric stiffness matrix,
• ($\lambda$) are the eigenvalues, the smallest of which corresponds to the critical buckling load.

The code allows for varying material and geometric properties along elements, and applies boundary conditions by masking fixed degrees of freedom.

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Features

• Supports multiple elements with variable diameters and hollow sections.
• Computes local and global stiffness matrices.
• Solves the generalized eigenvalue problem to find the critical load.
• Efficient use of numpy and scipy.linalg.eig.

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Problem Illustration

Figure: Buckling mode of a two-element cantilever beam with different hollow circular cross-sections. Node numbers, cross-sections, and axial load (P) are shown.

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Usage

Run the script using Python:

bash python euler_critical_load.py