Buckling of Columns

*Example: A slender beam (shown below) has length, L, and modulus of elasticity, E. It
is supported at each end by pins and by rollers at mid-span that prevent motion in the x-direction

but not in the z-direction. The x-section of the beam (shown below) is b x h. Determine the
ratio of h to b so that the buckling of the column in the z and x directions will occur
simultaneously.

Strategy: Calculate the critical load by applying P_cr = n π²EI/L² for buckling in the z-direction

and for buckling in the x-direction. Set them equal and solve for h/b.

Note: The critical buckling load depends on the sizes of the moment of inertia and the
buckling mode shape resulting from the mid-span restraint.

The moments of inertia, I, are as follows: I_zz = (1/12)hb³ and I_xx = (1/12)bh³

Next consider the buckled mode shape. The mode shape for buckling in the z-direction is

a single sine wave. Whereas the mode shape for buckling in the x-direction, due to the

restraint at mid-span, will be a double sine wave.

Click here to continue this example.

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