Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

• The Euler buckling load equation [Equation (16.5)] and the Euler buckling stress
equation [Equation (16.8)] depend only on the column length L, the stiffness of the
material (E), and the cross-sectional properties (I). The critical buckling load is independent
of the strength of the material. For example, consider two round steel rods
having the same diameter and length but differing strengths. Since E, I, and L are the
same for both rods, the Euler buckling loads for the two rods will be identical. Consequently,
there is no advantage in using the higher strength steel (which, presumably,
is more expensive) instead of the lower strength steel in this instance.
The Euler buckling load as given by Equation (16.5) agrees well with experiment, but only
for “long” columns for which the slenderness ratio L/r is large, typically in excess of 140
for steel columns. Whereas a “short” compression member can be treated as explained in
Chapter 1, most practical columns are “intermediate” in length, and consequently, neither
solution is applicable. These intermediate-length columns are analyzed by empirical formulas
described in later sections. The slenderness ratio is the key parameter used to classify
columns as long, intermediate, or short.
675
buCkLINg OF PIN-ENdEd
COLuMNS
ExAmpLE 16.1
A 15 mm by 25 mm rectangular aluminum bar is used as a 650 mm
long compression member. The ends of the compression member
are pinned. Determine the slenderness ratio and the Euler buckling
load for the compression member. Assume that E = 70 GPa.
P
x
Plan the Solution
The aluminum bar will buckle about the weaker of the two principal
axes for the cross-sectional shape of the compression member
considered here. The smaller moment of inertia for the cross section
occurs about the y axis; therefore, buckling will produce bending of
the compression member in the x–z plane at the critical load P cr .
650 mm
z
y
25 mm
SolutioN
The cross-sectional area of the compression member is A = (15 mm)
(25 mm) = 375 mm 2 , and its moment of inertia about the y axis is
I
y
3
(25 mm)(15mm)
= = 7,031.25 mm
12
4
z
y
P
15 mm
Cross section.
x
The slenderness ratio is equal to the length of the column divided by its radius of gyration.
The radius of gyration for this cross section with respect to the y axis is
P
r
y
I
4
y 7,031.25 mm
= = = 4.330 mm
2
A 375 mm
and therefore, the slenderness ratio for buckling about the y axis is
L 650 mm
= =
r 4.330 mm 150.1
y
Ans.
Note: The slenderness ratio is not necessary for determining the Euler buckling load in
this instance; however, the slenderness ratio is an important parameter that is used in
many empirical column formulas.
z
y
P