Linearized Buckling

1 min read • 127 words

Corotational frame elements are used to approximate Euler's buckling load.

Problem

P_{\mathrm{euler}} = \frac{\pi^2 EI}{L^2}

This example is adapted from https://github.com/denavit/OpenSees-Examples . The files for the problem are buckling.py for Python, and buckling.tcl for Tcl.

\operatorname{det}\bm{K}(\lambda) = 0

\bm{K}(\lambda) \approx \bm{K}(0) + \bm{K}^{\prime}(0) \lambda

where $\bm{K}^{\prime}$ is the derivative of $\bm{K}$ with respect to $\lambda$ .

\begin{gathered} \lambda=\sqrt{\frac{P L^2}{E I\left[1-P /\left(k_{\mathrm{s}} G A\right)\right]}}=\sqrt{\frac{P L^2}{\chi E I}} \\ \chi=1-P /\left(k_{\mathrm{s}} G A\right) \\ P=\chi \lambda^2 E I / L^2 . \end{gathered}

\begin{gathered} \chi=\frac{1}{1+\lambda^2 E I /\left(k_{\mathrm{s}} G A L^2\right)}=\frac{1}{1+\lambda^2 \varphi / 12} \\ P=\frac{\lambda^2 E I / L^2}{1+\lambda^2 \varphi / 12} . \end{gathered}

\tan \lambda_{\text {cr }}=\chi \lambda_{\text {cr }}=\frac{\lambda_{\text {cr }}}{1+\lambda_{\text {cr }} 2 \varphi / 12}

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