Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 653

FIGURE 11.4
Stability and instability from the standpoint of energy.

Suppose the particular variation that corresponds to the critical load is δ . Then

1
2 δ

1
2 δ

δ ˚

2 ˚ 2 ˚

δ() = δ + +··· = δ = 0 (11.5)

or

1
2 δ

2 ˚

δ() = δ = 0 (11.6)

Equation (11.6) can be used as the criterion for computing the critical load. It holds only for
conservative loading.

EXAMPLE 11.2 A Simple Example
Return to t he weightless, rigid rod with the elastic base attachment of Fig. 11.2. The potential
energy for H = 0 is given by

= (strain energy of the spring) − (potential energy of the applied loading)

1
2 k

2

= φ − PL ( 1 − cos φ) (1)

˚

For equilibrium δ = 0or

k φ − PL sin φ = 0 (2)

which corresponds to the value of P of Eq. (3) of Example 11.1.