Consider an ideal column as in Fig. 1310d, having one end

Chapter 13, Problem 13-45

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Consider an ideal column as in Fig. 13–10d, having one end fixed and the other pinned. Show that the critical load on the column is \(P_{\mathrm{cr}}=20.19 E I / L^{2}\). Hint: Due to the vertical deflection at the top of the column, a constant moment M will be developed at the fixed support and horizontal reactive forces \(\mathbf{R}^{\prime}\) will be developed at both supports. Show that \(d^{2} v / d x^{2}+(P / E I) v=\left(R^{\prime} / E I\right)(L-x)\). The solution is of the form \(v=C_{1} \sin (\sqrt{P / E I} x)+C_{2} \cos (\sqrt{P / E I} x)+\left(R^{\prime} / P\right)(L-x)\). After application of the boundary conditions show that tan \((\sqrt{P / E I} L)=\sqrt{P / E I} L\). Solve numerically for the smallest nonzero root.