Consider an ideal column as in Fig. 13-10c, having both ends fixed. Show that the critical load on the column is \(P_{\text {cr }}=4 \pi^{2} E I / L^{2}\). Due to the vertical deflection of the top of the column, a constant moment \(\mathbf{M}^{\prime}\) will be developed at the supports. Show that \(d^{2} v / d x^{2}+(P / E I) v=M^{\prime} / E I\). The solution is of the form \(v=C_{1} \sin (\sqrt{P / E I x})+\) \(C_{2} \cos (\sqrt{P / E I x})+M^{\prime} / P\).