Chapter 17.3, Problem 45P

Consider an ideal column as in Fig. 17–10d, having one end fixed and the other pinned. Show that the critical load on the column is $P_{\text{cr}}=20.19EI/L^2$. 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 R′ will be developed at both supports. Show that $d^{2}v/d{x}^2+\left(P/EI\right)v=\left(R^{\prime }/EI\right)\left(L-x\right)$. The solution is of the form $v=C_1\sin \left(\sqrt{P/EI}x\right)+C_2\cos \left(\sqrt{P/EI}x\right)+\left(R^{\prime }/P\right)\left(L-x\right)$. After application of the boundary conditions show that $\tan \left(\sqrt{P/EI}L\right)=\sqrt{P/EI}L$. Solve numerically for the smallest nonzero root.