Chapter 13
Stability of Elastic Systems
Theory of structural stability is a special branch of structural analysis. This theory
explores the very important phenomenon that is observed in the behavior of the
structures subjected to compressed loads. This phenomenon lies in the abrupt
change of initial form of equilibrium. Such phenomenon is called loss of stabil-
ity. As a rule, the loss of stability of a structure leads to it collapse. Engineering
practice knows a lot of examples when ignoring this feature of a structure led to its
failure.
This chapter is an introduction to stability analysis of engineering structures sub-
jected to compressed loads. Among them are structures that contain nondeformable
members as well as beams, frames, and arches. Classical methods of analysis will
be discussed.
13.1 Fundamental Concepts.................................................
We will differentiate two types of structures, mainly, the structures consisting of
absolutely rigid bodies connected by elastic constrains and structures consisting of
deformable members; it is possible to combine in one structure both types of mem-
bers, i.e., absolutely rigid discs with deformable members.
Some examples of these structures under compressed loads are shown in
Fig.13.1. Structures that contain absolutely rigid members.EI D1/are shown
in Fig.13.1a, b; these design diagrams present structures with elastic joints. Elastic
joint means that the angle between two adjacent members is changed upon load
application. Figure13.1c, d presents structures, which contain deformable ele-
ments; structure in Fig.13.1e contains the absolutely rigid partACand deformable
partCB.
Structures, which are subjected to compressed loads, may be either in stable or in
unstable equilibrium. Stability is a property of a structure to keep its initial position
or initial deformable shape. Stable structure will regain to its original state if any
disturbed factor changes the initial state and after it is removed.
A structure subjected to compressed loads may be disturbed from initial equilib-
rium state by, for example, a small lateralload. After removing this disturbance, a
I.A. Karnovsky and O. Lebed,Advanced Methods of Structural Analysis,
DOI 10.1007/978-1-4419-1047-913,cSpringer Science+Business Media, LLC 2010
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