Buckling
of Columns
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*Example: (continued) The figure below shows the
mode shapes for buckling about the z-axis and about the x-axis. The critical load for buckling
about the z-axis (n = 2) is Pcr |z = 4π2EIzz/L2 The critical load for
buckling about the x-axis (n = 1) is Pcr |x =
π2EIxx/L2 Recall Ixx =
(1/12)bh3 and Izz =
(1/12)hb3 Pcr |z = 4π2E(1/12)hb3
/L2 and Pcr
|x = π2E(1/12)bh3
/L2 For buckling in the z and
x-directions to occur simultaneously: 4π2E(1/12)hb3
/L2 = π2E(1/12)bh3 /L2 4b2 = h2 So
h / b
= 2 (result) Note: If there was no mid-span
restraint, the critical buckling load would occur about the axis with the smaller
moment of inertia since the buckling mode is a single sine wave in both directions. i.e. (1/12)hb3 < (1/12)bh3 assuming
h > b.
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