Chapter 5.2, Problem 24E

The critical loads of thin columns depend on the end conditions of the column. The value of the Euler load P₁ in Example 4 was derived under the assumption that the column was hinged at both ends. Suppose that a thin vertical homogeneous column is embedded at its base (x = 0) and free at its top (x = L) and that a constant axial load P is applied to its free end. This load either causes a small deflection δ as shown in Figure 5.2.9 or does not cause such a deflection. In either case the differential equation for the deflection y(x) is

$EI\frac{d^{2}y}{dx}+Py=P\delta$

Figure 5.2.9 Deflection of vertical column in Problem 24

1. (a) What is the predicted deflection when δ = 0?
2. (b) When δ ≠ 0, show that the Euler load for this column is one-fourth of the Euler load for the hinged column in Example 4.