Chapter 13.3, Problem 45P

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_cr = 20.19 EI/L². 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 d²v/dx² + (P/EL)v =(R'/EI)(L − x). The solution is of the form v = C₁ sin $\left(\sqrt{P/EIx}\right)+C_2\cos \left(\sqrt{P/EIx}\right)+\left(R\text{'}/P\right)\left(L-x\right).$. After application of the boundary conditions show that tan $\left(\sqrt{P/EIL}\right)=\left(\sqrt{P/EIL}\right)$. Solve numerically for the smallest nonzero root.