Suppose that a uniform thin elastic column is hinged at the end x = 0 and embedded

Question:

Suppose that a uniform thin elastic column is hinged at the end x = 0 and embedded at the end x = L.

(a) Use the fourth-order differential equation given in Problem 23 to find the eigenvalues ln, the critical loads P_n, the Euler load P₁, and the deflections y_n(x).

(b) Use a graphing utility to graph the first buckling mode.

Problem 23

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 d 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

d'y El- + Py = P8. dx2