458 13 Stability of Elastic Systems
and for critical force we get the same expression
PcrD
2krot
l
:
13.2.2 Structures with Two or More Degrees of Freedom
For stability analysis of such structures, first, it is necessary to define the number
of degrees of freedom, sketch the structure in any arbitrary position, and to chose
the generalized coordinates. As for structures with one degree of freedom, the static
method requires consideration of equilibrium conditions. Energy method requires
the calculation of potential of external forces and energy accumulated in elastic
constraints. The static method requires calculation of reactions ofallsupports, while
energy method requires calculation of reactions only forelasticsupports.
Static equations (for static method) and conditions (13.2) for energy method lead
tonalgebraic homogeneous equations withrespect to unknown generalized coor-
dinates. Trivial solution corresponds to initial (or unperturbed) state of equilibrium.
To obtain nontrivial solution, it is necessaryto equate the determinant of coefficients
before generalized coordinates to zero. This equation serves for calculation of criti-
cal loads. Their number equals to the number degrees of freedom. Each critical load
corresponds to specified shape of loss of stability.
Example 13.2.The structure contains three absolutely rigid bars.EID1/,which
are connected by hingesC 1 andC 2 , and supported by elastic supports at these
points; the rigidity of elastic supports isk. The structure is subjected to axial com-
pressed forceNas shown in Fig.13.6a. Calculate the critical load.
Solution.Let us consider this problem by the static and energy methods.
Static method The structure has two degrees of freedom. Displacements of the
hingesC 1 andC 2 area 1 anda 2 ; they are considered as generalized coordinates.
The reactions of elastic supports areR 1 Dka 1 andR 2 Dka 2 ; reactions of the left
and right supports are shown in Fig.13.6b.
Bending moments at hingesC 1 andC 2 are equal to zero, therefore
M 1 leftDNa 1
k.2a 1 Ca 2 /
3
lD0;
M
right
2 DNa^2
k.a 1 C2a 2 /
3
lD0:
This system may be rewritten as homogeneous algebraic equations with respect to
unknown generalized coordinatesa 1 anda 2
a 1 .3N2kl/a 2 klD0;
a 1 klCa 2 .3N2kl/D0:
(a)