Advanced Methods of Structural Analysis

13.2 Stability of Structures with Finite Number Degrees of Freedom 453

13.2 Stability of Structures with Finite Number

Degrees of Freedom

This section is devoted to the calculation of critical load for structures containing

only absolutely rigid bars, with elastic constraints. Three types of elastic constraints

will be considered. They are the following:

1.Elastic support, which allows anangulardisplacement. Rigidity of support is
krot. The reactive moment of such support and its angular displacementare
related asMDkrot.
2.Elastic support, which allows alineardisplacement. Rigidity of support isk.
Reaction of support and its linear displacementfare related asRDkf.
3.Elastic connections between absolutely rigid members (elastic hinged joint) of
a structure. Rigidity of connection iskrot. The moment of this joint andmutual
angulardisplacementof two adjacent absolutely rigid bars are related asMD
krot.
From methodical point of view, it is reasonable that all structures with finite number
degrees of freedom divide into two large groups. They are structures with one degree
of freedom and structures with two or more degrees of freedom.

13.2.1 Structures with One Degree of Freedom....................

A simplest structure with one degree of freedom is shown in Fig.13.3a. Absolutely

rigid vertical weightless column is placed on rigid supporting plate. Since the foun-

dation is flexible, the column in whole may rotate around the fixed pointO.The

column is subjected to axial compressed static forceP. Assume that anangular

rigidity of elastic support iskrot; such type of elastic support is presented astwo

springs of equal stiffness. Within their deformation, the forces that arise in both

springs create the elastic couple.

P

l

krot

P

q

M=krotq

O

f

O

P

q

D

2

U q^2

0 = krot

W = −PΔ

abc

Fig. 13.3 Absolutely rigid vertical cantilever bar on the elastic support: (a) Design diagram;

(b) Static method; (c) Energy method