13.1 Fundamental Concepts 451
Unstable equilibrium state means that if astructure under compressed load is dis-
turbed from an initial equilibrium state and after all disturbing factors are removed,
then the structure does not return to the initial equilibrium state. In this case, we say
that the acting compressed load is larger than the critical one.
Change of configuration of a structureunder the action of compressed load is
called a loss of stability of the initial form of equilibrium or a buckling. If the com-
pressed load is a static one, then this case is referred as the static loss of stability.
In this chapter, we will consider absolutely rigid and absolutely elastic structures
subjected to static compressed loads only. If a structure switches to other state (as
a result of loss of stability) and remains in this state in equilibrium, then this new
equilibrium state is called the adjacent form of equilibrium.
The static load may be of two types: conservative and nonconservative. The work
done by conservative forces is determined only by the initial and final position of
points of application of a force. Example of a conservative force is a force that keeps
its direction (Fig.13.2a).
P P>Pcr P
PabFig. 13.2 The column under conservative and tracking force
The work done by nonconservative forces is determined by trajectory between
the initial and final position of points of application of the force. Example of non-
conservative force is atrackingforce whose orientation depends on the slope of the
elastic curve at the point of application of the force (Fig.13.2b).
The corresponding systems are called conservative and nonconservative. In this
chapter, only conservative systems are considered.
Thecritical forcePcris the maximum force at which the structure holds its ini-
tial equilibrium form (the structure is still stable), or minimum force, at which the
structure no longer returns to the initial state(the structure is already unstable) if all
disturbing factors are removed.
The state of a structure that corresponds tocritical load is called the critical state.
The switching of a structure into new state occurs suddenly and as a rule leads to
the collapse of a structure. The theory of static stability of the structures is devoted
to methods of calculation of critical loads.
Degree of freedomis a fundamental concept of stability analysis. Degrees of
freedom present independent parameters, which define the structure’s configuration.
The structures, which contain only rigid elements, have the finite number degrees
of freedom. Each deformable element should be considered as a member with dis-
tributed parameters, so the structures that contain only deformable elements have