Calculate the Euler's limiting value of slenderness ratio for which it is not valid for long columns.Take $E=2 \times 10^{5}MPa, \sigma

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written 5.6 years ago by

teamques10 ★ 69k

$\begin{array}{l}{\text { Data: } E=2 \times 10^{5} \mathrm{MPa}, \quad \sigma_{\mathrm{c}}=320 \mathrm{N} / \mathrm{mm}^{2}} \ {\text { Find: } \lambda_{\text {limiting }}} \ {\text { For long column, Euler's formula, }} \ {\quad \mathrm{P}_{\mathrm{E}}=\frac{\pi^{2} \text { EImin }}{\left(\mathrm{L}_{\mathrm{c}}\right)^{2}}}\end{array}$

$\begin{array}{l}{\text { But } \mathrm{K}=\sqrt{\frac{\mathrm{I}}{\mathrm{A}}}} \ {\mathrm{K}^{2}=\frac{\mathrm{I}}{\mathrm{A}}} \ {\mathrm{I}=\mathrm{AK}^{2}} \ {\mathrm{P}_{\mathrm{E}}=\frac{\pi^{2} \mathrm{E} \times \mathrm{A} \times \mathrm{K}^{2}}{\left(\mathrm{L}_{\mathrm{e}}\right)^{2}}} \ {\frac{\mathrm{P}_{\mathrm{E}}}{\mathrm{A}}=\frac{\pi^{2} \mathrm{E}}{\left(\frac{\mathrm{Le}}{\mathrm{K}}\right)^{2}}}\end{array}$

$\begin{array}{l}{\sigma_{c}=\frac{\pi^{2} \mathrm{E}}{(\lambda)^{2}}} \ {(\lambda)^{2}=\frac{\pi^{2} \mathrm{E}}{\sigma_{c}}} \ {\lambda=\sqrt{\frac{\pi^{2} \mathrm{E}}{\sigma_{c}}}}\end{array}$

$\begin{array}{l}{\lambda=\sqrt{\frac{\pi^{2} \times 2 \times 10^{5}}{320}}} \ {\text { for column to be safe }} \ {\sqrt{\frac{\pi^{2} \times 2 \times 10^{5}}{320}} \leq \lambda} \ {78.54 \leq \lambda}\end{array}$

This Euler's limiting value is 78.54. If it is less than 78.54 Euler's formula for long column is not valid.

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