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

The critical load for an ideal column is known as the Euler buckling load, after the Swiss
mathematician Leonhard Euler (1707–1783), who published the first solution of the equation
for the buckling of long, slender columns in 1757. Equation (16.5) is also known as
Euler’s formula.
Equation (e) can be substituted into Equation (16.3) to describe the deflected shape of
the buckled column:
P
B
⎛ nπ
⎞
v = C1sinkx = C1 sin⎜
⎟ = …
⎝ L x ⎠
n 1, 2,3,
(16.6)
An ideal pin-ended column subjected to a compressive axial load P is shown in Figure
16.4a. The deflected shape of the buckled column corresponding to the Euler buckling load
given in Equation (16.5) is shown in Figure 16.4b. Note that the specific values for the
constant C 1 cannot be obtained, since the exact deflected position of the buckled column is
unknown. However, the deflections have been assumed to be small. The deflected shape is
called the mode shape, and the buckled shape corresponding to n = 1 in Equation (16.6) is
called the first buckling mode. By considering higher values of n in Equations (16.4) and
(16.6), it is theoretically possible to obtain an infinite number of critical loads and corresponding
mode shapes. The critical load and mode shape for the second buckling mode are
illustrated in Figure 16.4c. The critical load for the second mode is four times greater than
that of the first mode. However, buckled shapes for the higher modes are of no practical
interest, since the column buckles upon reaching its lowest critical load value. Higher mode
shapes can be attained only by providing lateral restraint to the column at intermediate
locations to prevent the column from buckling in the first mode.
Euler Buckling Stress
The normal stress in the column at the critical load is
L
A
(a) Undefined column
x
π
P cr = 2 EI
1 L 2
B
C 1
σ
cr
2
Pcr
π EI
= = (f)
2
A AL
v
A
The radius of gyration r is a section property defined as
(b) First buckling mode (n = 1)
2
I
r =
(16.7)
A
If the moment of inertia I is replaced by Ar 2 , Equation (f) becomes
σ
cr
2 2 2 2 2
π EAr ( ) π Er π E
= = =
2
2
AL L ( Lr / )
2
(16.8)
B
x
4π 2 EI
P cr = 2 L 2
The quantity L/r is termed the slenderness ratio and is determined for the axis about which
bending tends to occur. For an ideal column with no intermediate bracing to restrain lateral
deflection, buckling occurs about the axis of minimum moment of inertia (which corresponds
to the minimum radius of gyration).
Note that Euler buckling is an elastic phenomenon. If the axial compressive load is
removed from an ideal column that has buckled as described here, the column will return
to its initial straight configuration. In Euler buckling, the critical stress σ cr remains below
the proportional limit for the material.
Graphs of Euler buckling stress [Equation (16.8)] are shown in Figure 16.5 for structural
steel and for an aluminum alloy. Since Euler buckling is an elastic phenomenon,
v
C 1
FIGURE 16.4 Two examples
of buckling modes.
A
(c) Second buckling mode (n = 2)
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