Stability

Abstract

Stability is concerned with the maximum compressive load a structure can withstand before failing in a catastrophic manner.

Appendix: Solutions

• 12.2.1 Show that Eq. 12.4 is a solution to the second-order differential Eq. 12.3.

Solution

$$v^{\prime \prime} + k^{2} v = 0$$

$$v = C_{1} \sin kx + C_{2} \cos kx$$

$$\begin{aligned} v^{\prime} (x) & = C_{1} k\,\cos kx - C_{2} k\,\sin kx \\ v^{\prime \prime} (x) & = - C_{1} k^{2} \sin kx - C_{2} k^{2} \cos kx = k^{2} \left( { - C_{1} \sin kx - C_{2} \cos kx} \right) \\ \Rightarrow v^{\prime \prime} + k^{2} v & = - k^{2} \left( {C_{1} \sin kx + C_{2} \cos kx} \right) + k^{2} \left( {C_{1} \sin kx + C_{2} \cos kx} \right) = 0 \\ \end{aligned}$$

• 12.2.2 Compare the critical buckling loads for a pinned-end yardstick and a pinned-end, foot-long ruler of the same material and cross-sectional area.

Solution

$$P_{cr} = \frac{{\pi^{2} EI}}{{L^{2} }}$$

$$\begin{aligned} P_{cr}^{YS} & = \frac{{\pi^{2} EI}}{{3^{2} }} = \frac{{\pi^{2} EI}}{9} \\ P_{cr}^{R} & = \frac{{\pi^{2} EI}}{{1^{2} }} = \pi^{2} EI \\ \end{aligned}$$

The critical buckling load of the ruler is 9 times that of the yardstick.