Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 652

referred to as second order theory . Most of the stability solutions in this chapter are of second
order.
First order theory is studied by basing equilibrium on an undeformed geometry, which is
the case for this example when φ = 0.

11.1 Criteria for Stability

11.1.1 Energy Criterion

Consider an energy-based definition of the concept of stability. Begin by investigating the
potential energy of a body as a perturbation moves it from a fundamental or basic

configuration to an adjacent configuration.

˚

+ = (11.1)

Basic Perturbation Neighboring

Configuration or Incremental (Adjacent)

Change Configuration

The energy of the adjacent configuration can be written as a function of the energy of the
basic configuration

1
2 δ

=

˚ ˚ ˚

2 ˚

+ = + δ + +··· (11.2)

˚

2 ˚

variations of ˚ where δ , δ , ... are the first, second, ... . From the principle of station-

˚

ary potential energy [Chapter 2, Eq. (2.65)], δ = 0 for a body in equilibrium. Then the

incremental change can be expressed as

1
2 δ

= −

˚

2

˚

= +···

(11.3)

It is evident that the state of the potential due to the perturbation is described by the second
variation of the energy of the basic configuration. Bazant and Cedolin (1991) and Pfluger
(1964) provide a thorough development of the second variation of potential energy as a
stability criterion.
The various states of stability are illustrated in Fig. 11.4. For the transition from the
fundamental to the adjacent configuration to remain stable, i.e., for the body to be in a
stable state of equilibrium, any arbitrary perturbation should lead to an increase in potential
energy, i.e.,

2 ˚

0. If a particular perturbation can be found that leads to a decrease in
potential energy, kinetic energy can be released, and the body is in an unstable state of
equilibrium. Here, δ

δ >

0 , where δ

2 ˚

denotes the particular or special variation that leads
to instability. For the border case, wherein no change in potential energy takes place,

<

2 ˚

δ = 0

or δ

2 ˚

0 is called the neutral or indifferent configuration.
In the classical stability theory, the state of the neutral configuration serves as the cri-
terion for a critical load. This neutral configuration corresponds to a nonunique state of
equilibrium that occurs for at least a particular perturbation.
In the case of a neutral or indifferent configuration, =

=

˚

, so that

δ( −

˚

) = δ() =

0 (11.4)