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190 Mechanical Engineering Design
4–16 Elastic Stability

Section 4–12 presented the conditions for the unstable behavior of long, slender columns.
Elastic instability can also occur in structural members other than columns. Compressive
loads/stresses within any long, thin structure can cause structural instabilities (buckling).
The compressive stress may be elastic or inelastic and the instability may be global or local.
Global instabilities can cause catastrophic failure, whereas local instabilities may cause
permanent deformation and function failure but not a catastrophic failure. The buckling
discussed in Sec. 4–12 was global instability. However, consider a wide flange beam in
bending. One flange will be in compression, and if thin enough, can develop localized
buckling in a region where the bending moment is a maximum. Localized buckling can
also occur in the web of the beam, where transverse shear stresses are present at the beam
centroid. Recall, for the case of pure shear stress τ, a stress transformation will show that
at 45 , a compressive stress of σ =−τ exists. If the web is sufficiently thin where the shear
◦
force V is a maximum, localized buckling of the web can occur. For this reason, additional
support in the form of bracing is typically applied at locations of high shear forces. 10
Thin-walled beams in bending can buckle in a torsional mode as illustrated in
Fig. 4–24. Here a cantilever beam is loaded with a lateral force, F. As F is increases
from zero, the end of the beam will deflect in the negative y direction normally accord-
3
ing to the bending equation, y =−FL /(3EI). However, if the beam is long enough
and the ratio of b/h is sufficiently small, there is a critical value of F for which the beam
will collapse in a twisting mode as shown. This is due to the compression in the bottom
fibers of the beam which cause the fibers to buckle sideways (z direction).
There are a great many other examples of unstable structural behavior, such as thin-
walled pressure vessels in compression or with outer pressure or inner vacuum, thin-walled
open or closed members in torsion, thin arches in compression, frames in compression, and
shear panels. Because of the vast array of applications and the complexity of their analyses,
further elaboration is beyond the scope of this book. The intent of this section is to make the
reader aware of the possibilities and potential safety issues. The key issue is that the
designer should be aware that if any unbraced part of a structural member is thin, and/or
long, and in compression (directly or indirectly), the possibility of buckling should be
investigated. 11
For unique applications, the designer may need to revert to a numerical solution
such as using finite elements. Depending on the application and the finite-element code
available, an analysis can be performed to determine the critical loading (see Fig. 4–25).

Figure 4–24 y
Torsional buckling of a
thin-walled beam in bending.
z

z h
x
y
b
F
Figure 4–25 10 See C. G. Salmon, J. E. Johnson, and F. A. Malhas, Steel Structures: Design and Behavior, 5th ed.,

Finite-element representation Prentice Hall, Upper Saddle River, NJ, 2009.
of flange buckling of a channel 11 See S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, 2nd ed., McGraw-Hill, New York, 1961.
in compression. See also, Z. P. Bazant and L. Cedolin, Stability of Structures, Oxford University Press, New York, 1991.