Advanced Methods of Structural Analysis

450 13 Stability of Elastic Systems

k

P P

a

bde

c

EI=• EI=•

P P

k k

EI=• EI=• EI EI

PP

1

P

EI

P

2

P

EI=•

P

EI

A

C

B

Fig. 13.1 Type of structures and form of buckling: (a, b) structures with absolutely rigid members;
(c, d) structures with deformable members only; (e) structure with absolutely rigid and deformable
members

structure can return into the initial state, ortendsto return to the initial state, or re-
mains in the new state, or even tends to switch into new state. Behavior of a structure
after removing a disturbance depends on the value of compressed loads. For small
compressed load, a structure will return to the initial state, i.e., this equilibrium state
is a stable one, while for larger compressed load, a structure will not return to the
initial state, i.e., this equilibrium state is unstable. However, what does “small and
large” load mean? It is obvious, that the behavior of a structure after removing a
disturbance load depends not only on the value of the compressed load, but also on
the types of supports, the length of compressed members, and their cross sections.
Any compressed load for tall column withsmall cross section may be treated as a
large load, while for short column with large cross section the same load may be
treated as a small one. The theory of elastic stability gives the precise quantitative
characteristics of compressed loads for different types of structures, which leads to
well-defined state of a structure, and allows us to understand the influence of param-
eters of a structure (boundary conditions, cross-section, and properties of material)
on the value of this load and corresponding behavior of a structure.
Let us introduce the following definitions:
Stable equilibriumstate means that if the structure under compressed load is dis-
turbed from an initial equilibrium state and after all disturbing factors are removed,
then the structure returns to the initial equilibrium state. This is concerning to the
elastic structures. If a structure consists ofplastic or elasto-plastic elements, then a
complete returning to the initial state is impossible. However, equilibrium state is as-
sumed to be stable, if a structure even tends to return to the initial equilibrium state.
In case of absolutely rigid bodies, we are talking about stablepositionof a structure,
while in case of deformable elements, weare talking about stable equilibriumform
of a deformable state. In all these cases, we say that the acting compressed load is
less than the critical one. Definition of the critical load will be given later.