Advanced Methods of Structural Analysis

452 13 Stability of Elastic Systems

infinite number degrees of freedom. Structures presented in Fig.13.1a, b have one
degree of freedom, while the structures presented in Fig.13.1c, d have infinitely
many degrees of freedom.
The difference of this concept used in different parts of the structural theory,
such as the kinematical analysis, matrix stiffness method, and stability of structures,
is obvious.
Generalized coordinatesare independent parameters, which uniquely defines
configuration of a system in new arbitrary state. A structure withndegrees of free-
dom hasngeneralized coordinates. A structure withndegrees of freedom hasn
critical loads. Each criticalload corresponds to one specified form of equilibrium.
For structure with one degree of freedom, there exists only the unique form of the
loss of stability (Fig.13.1a, b) and its corresponding unique critical load. For struc-
ture with infinitely many degrees of freedom, there exist infinitely many critical
loads and its corresponding forms of loss of stability. Figure13.1d shows only the
first buckling form of a frame. In Fig.13.1c, the numbers 1 and 2 indicate the first
and second forms of the loss of stability of a beam. It is very important to define the
smallest critical load, because it leads to the loss of stability accordingly first form,
i.e., to the failure of the structure. The second and following forms may be realized
upon the special additional conditions.
There exist precise and approximate methods for calculating critical loads. Pre-
cise methods are static, energy method and dynamical ones. These methods reflect
the fact that the concepts “stable or unstable state of equilibrium” may be considered
from different points of view.
Static method(or equilibrium method) is based on the consideration of equilib-
rium of a structure in a new configuration. The critical force is such a minimum
force, which can hold the structure in equilibrium in the adjacent condition or max-
imum force, for which initial straight form of equilibrium is yet possible.
Energy methodrequires consideration of total energy of a structure in a new
configuration. This energyUequals to stress energyU 0 and potentialWof external
forces
UDU 0 CW: (13.1)

The potentialW of external forces equals to work, which is produced by external
forces on the displacement fromfinalstate intoinitialone. The stable equilibrium
of the structure corresponds to a minimum of the total energy.
For system withndegrees of freedom, the critical load may be calculated from
the set of equations

@U

@q 1

D 0 I

@U

@q 2

D 0 I 

@U

@qn

D0; (13.2)

whereqiare the generalized coordinates of the structure. This method is equivalent
to the virtual displacement method, according to which the sum of the work done
by all forces on any virtual displacementsis zero. The static and energy methods are
considered later.