Instability and Buckling

The structural instability, or the loss of the equilibrium stability, is listed in the Lecture

16 as one of the possible failure modes characterizing a civil structure. Contrary to the

majority of failure modes, extensively studied, which are predicted using the equations of

motion written on the initial, undeformed con?guration, the prediction of the structural

instability requires that the equations of motion to be formulated on the deformed con?g-

uration of the structure. Since the deformed con?guration is not known in advanced, ob-

taining a solution of the equations of motion is not a trivial task and frequently encounter

mathematical di? culties. The generality involved by using the equations of motion are

often reduced in the engineering practice, by accepting the quasi-static nature of the load-

ing as the consequance of its slow application, to static equilibrium equations. Altrue, the

dynamic method provides a general de?nition of the instability phenomenon, the static

method, a more simpler method, is able to correctly describe the phenomenum for the

case of conservative forces. Mane of the static loads considered in the structural investi-

gations are of this nature. The Classical Theory of Structural Stability refers maintly to

the investigation of the elastic buckling of individual columns and frames, an extremely

important subject for any structural practitioner. The inelastic behavior, in the classical

sense, extends the elastic behavior of the material over the proportionality limit, com-

plicating somehow the theoretical approach. Over the years the structural stability area

of reaserch enlarge in mane other theoretical directions covering not only columns and

frames, but all sorts of structural components as: plates, shells and three-dimensional bod-

ies. The material behavior considered addressed also more complicated, but more closely

related to reality, constitutive laws as: viscoelastic, visco-elasto-plastic and recently the

theory of damage.

The subjects treated in this lecture, intend to familiarize the novice student of Mechan-

ics of Materials with the structural buckling (de?nition, assumptions, limitations), by

2 18. Instability and Buckling

considering ?rst the elastic behavior of the individual column and extending later to its

inelastic behavior. In this lecture the static method of investigation is employed.

For someone interested to extend his/hers knowledge base to the case of the instability

of more complicated material behavior or structural geometry, a very extensive technical

literature is available in English and Roumanian languages. Some of these remarkable

textbooks and handbooks are listed in the reference section of this testbook.

18.2 Stability of Equilibrium. De?nition.

From the previous lectures it can be concluded that the study of the deformable body,

in general, and of the linear plane beam, in particular, always required some form of

equilibrium equations to be written (global equilibrium, part equilibrium, in?nitesimal

equilibrium etc.). A new concept related with the equilibrium, the stability of equilibrium,

is introduced by analyzing ?rst the equilibrium of a rigid sphere, as pictured in Figure

17.1, located on three di?erent surfaces: (a) concave, (b) ?at and (c) convex.

Figure 17.1 Stability of Equilibrium

Initially, in all three con?gurations (a), (b) and (c), the sphere is in equilibrium and

consequently, the static equilibrium equations are satis?ed:

X

Fx  0 (18.1)

X

Fy = ?W + N = 0

X

P

M  0

where: P is the original contact point between the sphere and the surface and N is the ver-

tical reaction force due to the self weight of the sphere In the con?guration (a) any change

of the sphere position in the neighborhood of the original position will be only temporary

and the sphere will move back regaining the original position. This con?guration implies

18.3 Buckling and Stability of Simple Mechanical Systems 3

a stable equilibrium. Contrary, the con?guration (c), charcterized by a movement of the

sphere away from its initial position, indicates an unstable equilibrium. In con?guration

(b), called neutral equilibrium, the sphere has not any tendency to move either to the

left or right of its initial position nor to regain its original location if moved. It can be

concluded that despite the fact that the equilibrium is satis?ed in all three situations, the

stability of equilibrium is di?erent in each one of the con?gurations considered.

The general de?nition of stability, due to the theoretical contribution of Lyapunov in

1892, is stated as:"a system is stable if a small change in its input leads to a small change

in the response of the output". Translated in terms more appropiated for the structural

engineering a structure is stable if a ?nite change in the initial conditions (input) does

not produced an in?nite change in the solution (response). Otherwise, the structure is

unstable. Mathematically, the de?nition stated above is expressed as:

De?nition 1 If for an arbitrary positive number  there exists a positive number  such

that every solution v(t);obtained from the integration of the equation of motion, with

initial value v(t = t0)   satis?es the inequalities v(t)  " for all times t > t0:

The static equilibrium being a particular case of the equation of motion, the time variable

t looses its meaning and consequently, the condition of stability refers exclusively to the

position of the equilibrium, as exemplify in the cases illustrated in Figure 17.1.

Instability and Buckling (Columns)