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

668
COLuMNS
Stability of Equilibrium
The concept of stability of equilibrium with respect to columns can be investigated with the
elementary column-buckling model shown in Figure 16.1a. In this figure, a column is modeled
by two perfectly straight pin-connected rigid bars AB and BC. The column model is
supported by a pin connection at A and by a slotted support at C that prevents lateral movement,
but allows pin C to move freely in the vertical direction. In addition to the pin at B, a
rotational spring with a spring constant K connects the bars, which are assumed to be
perfectly aligned vertically before the axial load P is applied, making the column model
initially straight.
Since the load P acts vertically and the column model is initially straight, there should
be no tendency for pin B to move laterally as load P is applied. Furthermore, one might suppose
that the magnitude of load P could be increased to any intensity without creating an
effect in the rotational spring. However, common sense tells us that this supposition cannot
be true: At some load P, the pin at B will move laterally. To investigate the issue further, we
must examine the column model after pin B has been displaced laterally by a small amount.
In Figure 16.1b, the pin at B has been displaced slightly to the right so that each bar
forms a small angle ∆θ with the vertical. The rotational spring at B reacts to the angular
rotation of 2∆θ at B, tending to restore bars AB and BC to their initial vertical orientation.
The question is whether the column model will return to its initial configuration from the
displaced configuration after it is subjected to the axial load P or whether pin B will move
farther to the right. If the column model returns to its initial configuration, the system is
said to be stable. If pin B moves farther to the right, the system is said to be unstable.
To answer this question, consider the free-body diagram of bar BC shown in Figure 16.1c.
In the displaced position, the forces P acting at pins B and C create a couple that tends to
cause pin B to move farther away from its initial position. The moment created by this
couple is called the destabilizing moment. In opposition, the rotational spring creates a
restoring moment M, which tends to return the system to its initial vertical orientation. The
moment produced by the rotational spring is equal to the product of the spring constant
K and the angular rotation at B, which is 2∆θ. Therefore, the rotational spring produces a
restoring moment M = K(2∆θ) at B. If the restoring moment is greater than the destabilizing
P
P
P
L
2
L
2
C
B
Slotted
support
Rotational spring
with stiffness
(spring) constant K
C
∆θ
∆θ
B
2∆θ
∆θ
C
M = K(2∆θ)
P
L
2
B
A
A
(a) Initial configuration
FIGURE 16.1 Elementary column-buckling model.
P
(b) Displaced configuration
(c) Free-body diagram