Answer: The critical loads of thin columns depend on the end conditions of the column

Chapter 3, Problem 24

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The critical loads of thin columns depend on the end conditions of the column. The value of the Euler load \(P_{1}\) 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 \(\delta\) as shown in FIGURE 3.9.9 or does not cause such a deflection. In either case the differential equation for the deflection y(x) is

\(E I \frac{d^{2} y}{d x^{2}}+P y=P \delta\).

(a) What is the predicted deflection when \(\delta=0\)?

(b) When \(\delta \neq 0\), show that the Euler load for this column is one-fourth of the Euler load for the hinged column in Example 4.