A column with rectangular cross-section is pin-supported at its two ends (Figure 8.9). It is restrained in the plane of the figure but free to deflect in a plane perpendicular to it. Find out the ratio h:b such that the critical load is same for buckling in two principal planes of the column.

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Accepted Answer

A rectangular section possesses two principal axes of buckling as shown in Figure 8.10. From the figure

[latex] I_{x x}=\frac{1}{12} b h^3 \quad ext { and } \quad I_{y y}=\frac{1}{12} \lambda b^3 [/latex]

Now radius of gyration,

[latex] K_{x x}=\sqrt{\frac{1}{12} b h^3 imes \frac{1}{b h}}=\frac{h}{2 \sqrt{3}} [/latex]

[latex] K_{y y}=\sqrt{\frac{1}{12} h b^3 imes \frac{1}{b h}}=\frac{b}{2 \sqrt{3}} [/latex]

From the cross-section configuration, [latex] h>b ext {, so } K_{y y}>K_{x x} ext {. Thus, } K_{y y}=K_{\min } [/latex]. As the column is free to buckle in any direction, it will buckle in a plane whose normal is the y-axis. It means, in this case, the column with buckle in the ‘in-plane’ or in the plane of the figure. Now, for in-plane buckling in the present configuration,

[latex] \left(P_{ Cr } ight)_I=\frac{\pi^2 E(I)_{\min }}{L_{ e }^2} [/latex]

where [latex] L_{ e } [/latex] is the effective length; here it is L/2. Therefore, substituting, we get

[latex] \left(P_{ Cr } ight)_I=\frac{\pi^2 E(I)_{\min }}{(L / 2)^2}=\frac{\pi^2 E\left(h b^3 / 12 ight)}{(L / 2)^2} [/latex]

[latex] \left.\left(P_{ Cr } ight)_I=\frac{h b^3}{3} \lambda \quad ext { (as } \lambda=\pi^2 E / L^2 ight) [/latex] (1)

For ‘out-of-plane’ buckling, [latex] L_{ e }=L [/latex] as there is no additional constraint at B in the plane perpendicular to the plane of figure. So

[latex] \left(P_{ Cr } ight)_0=\frac{\pi^2 E}{L^2}\left\lgroup \frac{b h^3}{12} ight group [/latex]

Thus,

[latex] \left(P_{ Cr } ight)_0=\lambda \cdot \frac{b h^3}{12} [/latex] (2)

According to the problem, [latex] \left(P_{ Cr } ight)_I=\left(P_{ Cr } ight)_0 [/latex], therefore

[latex] \frac{h b^3}{3}=\frac{b h^3}{12} [/latex]

or [latex] \frac{h}{b}=2 [/latex]

So, h : b = 2 : 1.

Question 8.3: A column with rectangular cross-section is pin-supported at ...

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