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Deflection and Stiffness 181
Note that we could have easily incorporated the stiffness of the support at B if we
were given a spring constant.

4–11 Compression Members—General

The analysis and design of compression members can differ significantly from that of
members loaded in tension or in torsion. If you were to take a long rod or pole, such as
a meterstick, and apply gradually increasing compressive forces at each end, nothing
would happen at first, but then the stick would bend (buckle), and finally bend so much
as to fracture. Try it. The other extreme would occur if you were to saw off, say, a 5-mm
length of the meterstick and perform the same experiment on the short piece. You would
then observe that the failure exhibits itself as a mashing of the specimen, that is, a
simple compressive failure. For these reasons it is convenient to classify compression
members according to their length and according to whether the loading is central or
eccentric. The term column is applied to all such members except those in which fail-
ure would be by simple or pure compression. Columns can be categorized then as:

1 Long columns with central loading
2 Intermediate-length columns with central loading
3 Columns with eccentric loading
4 Struts or short columns with eccentric loading

Classifying columns as above makes it possible to develop methods of analysis and
design specific to each category. Furthermore, these methods will also reveal whether
or not you have selected the category appropriate to your particular problem. The four
sections that follow correspond, respectively, to the four categories of columns listed
above.
4–12 Long Columns with Central Loading

Figure 4–18 shows long columns with differing end (boundary) conditions. If the axial
force P shown acts along the centroidal axis of the column, simple compression of the
member occurs for low values of the force. However, under certain conditions, when
P reaches a specific value, the column becomes unstable and bending as shown in
Fig. 4–18 develops rapidly. This force is determined by writing the bending deflection
equation for the column, resulting in a differential equation where when the boundary
9
conditions are applied, results in the critical load for unstable bending. The critical
force for the pin-ended column of Fig. 4–18a is given by
2
π EI
P cr = 2 (4–42)
l
which is called the Euler column formula. Equation (4–42) can be extended to apply to
other end-conditions by writing
2
Cπ EI
P cr = 2 (4–43)
l
where the constant C depends on the end conditions as shown in Fig. 4–18.

9 See F. P. Beer, E. R. Johnston, Jr., and J. T. DeWolf, Mechanics of Materials, 5th ed., McGraw-Hill,
New York, 2009, pp. 610–613.