Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 649

11

Stability An alysis

Stability is not a clearly defined concept for all disciplines, but some simple analogies can
be used to illustrate an intuitive notion of stability. Often we think in terms of the dynamics
of a rigid body as shown in Fig. 11.1. There, slight disturbances to the equilibrium positions
can be demonstrated in terms of stability. If the ball tends to return to the bottom of the
trough, as the result of a slight disturbance, the equilibrium position is said to be stable .
However, as in the case of Fig. 11.1b, if a small disturbance leads to a finite motion of the
ball, the critical condition is called unstable . If the ball remains at the same vertical level, the
equilibrium configuration is referred to as being neutral.
Many phenomena exhibit instabilities. Although most instability studies in structural
mechanics deal with the elastic buckling of structural members and systems, there are
several other areas of considerable interest such as plastic stability, creep instability, which
is time dependent, and thermal stability, which is temperature dependent. Methods for the
study of stability include the equilibrium method and the energy method.
Traditionally, the problem of buckling is to ascertain the conditions for which a structure
in equilibrium is no longer stable. There is usually a parameter P , such as an applied load, for
which the structure remains stable if P is small enough and becomes unstable for sufficiently
large values of P . In the stable state, there is a unique configuration for each value of P .Ata
particular value of P , denoted P cr for a critical value or buckling load , the structure ceases to
be stable. In stability analyses of structures, we wish to find the equilibrium configurations
under specified levels of applied loadings and to determine which of these are stable.
In linear structural mechanics, displacements are proportional to applied loads. Buckling,
however, is characterized by an instability in which an inordinate increase in displacement
can result from a small increase in applied load. As a consequence, buckling is a topic that
belongs to nonlinear, rather than linear, mechanics.

EXAMPLE 11.1 Introduction to Some Stability Concepts
Apply the conditions of equilibrium to the rigid rod system of Fig. 11.2. Sum moments
about point A of the displaced configuration to obtain
M A

= 0: PL sin φ + HL cos φ = k φ (1)

or

k φ − HL cos φ

=

P (2)

L sin φ

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