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

moment, then the system will tend to return to its initial configuration. However, if the
upsetting moment is larger than the restoring moment, then the system will be unstable in
the displaced configuration and pin B will move farther to the right until either equilibrium
is attained or the model collapses. The magnitude of axial load P at which the restoring
moment equals the destabilizing moment is called the critical load P cr . To determine
the critical load for the column model, consider the moment equilibrium of bar BC in
Figure 16.1c for the load P = P cr :
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INTROduCTION
Σ M = P ( L/2)sin ∆θ
− K(2 ∆ θ) = 0
B cr (a)
Since the lateral displacement at B is assumed to be small, sin ∆θ ≈ ∆θ, and thus, Equation (a)
can be simplified and solved for P cr :
P ( L/2) ∆ θ = K(2 ∆θ)
cr
4K
∴ Pcr
=
L
(b)
If the load P applied to the column model is less than P cr , then the restoring moment is
greater than the destabilizing moment and the system is stable. However, if P > P cr , then the
system is unstable. At the point of transition, where P = P cr , the system is neither stable nor
unstable, but rather, is said to be in neutral equilibrium. The fact that ∆θ does not appear
in the second line of Equation (b) indicates that the critical load can be resisted at any value
of ∆θ. In other words, pin B could be moved laterally to any position, and there would be
no tendency for the column model either to return to the initial straight configuration or to
move farther away from it.
Equation (b) also suggests that the stability of the elementary column-buckling model
can be enhanced by increasing the stiffness K or by decreasing the length L. In the sections
that follow, we will observe that these same relationships are applicable to the critical loads
of actual columns.
The notions of stability and instability can be defined concisely in the following manner:
Stable—A small action produces a small effect.
Unstable—A small action produces a large effect.
These notions and the concept of three equilibrium states can be illustrated by the
equilibrium of a ball resting on three different surfaces, as shown in Figure 16.2. In all three
cases, the ball is in equilibrium at position 1. To investigate the stability associated with
each surface, the ball must be displaced an infinitesimally small distance dx to either side
of position 1. In Figure 16.2a, a ball displaced laterally by dx and released would roll back
to its initial position. In other words, a small action (i.e., displacing the ball by dx) produces
a small effect (i.e., the ball rolls back a distance dx). Therefore, a ball at rest at position 1
on the concave upward surface of Figure 16.2a illustrates the notion of stable equilibrium.
By contrast, the ball in Figure 16.2b, if displaced laterally by dx and released, would not
return to position 1. Rather, the ball would roll farther away from position 1. In other
words, a small action (i.e., displacing the ball by dx) produces a large effect (i.e., the ball
rolls a large distance until it eventually reaches another equilibrium position). The ball at
rest at position 1 on the concave downward surface of Figure 16.2b illustrates the notion of
unstable equilibrium. The ball in Figure 16.2c is in a neutral equilibrium position on the
horizontal plane because it will remain at any new position to which it is displaced, tending
neither to return to nor to move farther from its original position.
1
(a)
1
(b)
1
(c)
dx
dx
dx
FIGURE 16.2 Concepts of
(a) stable, (b) unstable, and
(c) neutral equilibrium.