Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 654

To study the stability of the system, note that for φ = 0

2 ˚

δ = k − PL (3)

Thus,

> 0 stable

k − PL = 0 critical (4)

< 0 unstable

We conclude that P L is the cr iti cal level of the applied loading. For example,

the rod in its vertical position is stable for P

= P cr = k /

< P cr .

11.1.2 General Criterion

As already demonstrated by the examples given in the last section, stability problems
in structural mechanics can be viewed from several standpoints and placed in distinct
categories.
In addition to the energy criterion just described, the classical equilibrium method is
based on the description of the structural response with differential equations and leads,
for example, to the Euler load of columns. In these differential equations, some nonlinear
deformation terms are included that are linearized by additional assumptions. The asso-
ciated eigenvalue problem is solved, with a load parameter introduced. The bifurcation
of equilibrium is defined as the stability limit and is calculated as the lowest value of the
load factor for which the eigenvalue determinant is zero. The energy criterion and the clas-
sical equilibrium categories for stability are related (Pfluger, 1964), both being based on
descriptions of the structure without imperfections.
From a practical standpoint, it is important to consider structures with imperfections
in geometry. Of particular interest is the identification of load characteristics leading to
deformations that exceed prescribed limits. Regions are sought corresponding to relative
extrema for the load-deformation diagrams and their particular effects and conditions for a
change in strategies. These are the limit point or snap-through problems. Insight into this with
respect to the stability of a structure can be obtained from a mechanical or a mathematical
viewpoint.
To use a static formulation, the applied loading must be conservative. For non-
conservative systems, such as systems wherein the change in direction of the loading is
taken into account, a dynamic formulation must be employed. A kinetic stability criterion
needs to be introduced, utilizing the dynamic equation of motion (Ziegler, 1977). To obtain
a tractable solution with this approach normally involves a major effort.
With proper definitions, the various methods for formulating a stability analysis can
be based on a common fundamental foundation. This formulation is of general character
and has to be specialized for particular physical problems. The steps for implementing this
generalized criterion are summarized below. The terms in parentheses in the different steps
are referring to the application of the general stability criterion in structural mechanics.

Summary of the General Criterion for Stability
•

Establish the unperturbed state (in equilibrium)—the Fundamental state.

•

Disturbance (of equilibrium) yields perturbed state—the Neighboring state.

•

Establish a characteristic (change of displacement) that is a critical feature of the state.