Chapter 13.3, Problem 13.44P

Consider an ideal column as in Fig.13-10 c, having both ends fixed. Show that the critical load on the column is P_cr = 4π²EI/L². Hint: Due to the vertical deflection of the top of the column, a constant moment M will be developed at the supports, Show that d²v/dx² + (P/EI)v = M'/EI. The solution is of the form v = C₁ sin $\left(\sqrt{P/EIx}\right)+C_2\cos \left(\sqrt{P/EIx}\right)+M\text{'}/P$.