Advanced Methods of Structural Analysis

454 13 Stability of Elastic Systems

To determine the critical load by static or energy method, first of all, we need

to accept a generalized coordinate. Let an angular displacementof the supporting

plate being the generalized coordinate. This parameter describes completely a per-

turbed configuration of the structure. The structure in the strained state is shown by

dotted line.

Static method(Fig.13.3b). The moment that is produced in the support isMD

krot. The moment due to external loadNwith respect to support pointOisPf.

We assume that angular displacementis small, and thereforefDlsinl.

Equilibrium equation

P

M 0 D 0 leads to the stability equation

PfkrotDPlkrotD0: (13.3)

This equation is obtained on the basis of linearization procedure sin;there-

fore, the equilibrium equation (13.3) is called linearizedstability equation.

Equation (13.3) is satisfied at two special cases:

1.The angleD 0. It means that initial vertical form of the column remains vertical
for any forceP. This is a trivial solution, which corresponds to the initial form
of equilibrium.
2.The angle¤ 0. It means that the strained form of equilibrium is possible for

loadPcrD

krot

l

. This load is critical.

If the given structure is subjected to loadP<Pcr, then initial vertical position of

the column is the only equilibrium position; therefore, if the structure would be

disturbed, then it returns to its initial position. If the structure is subjected to load

PPcr, then additional equilibrium state is possible. Pay attention that the value

of the anglecannot be determined on the basis of linearized stability equation.

Energy method(Fig.13.3c). Since the vertical displacement of the point of ap-

plication of the forcePis

Dl.1cos/Š

l^2

2

;

then the potential of external loadNis

W DPDPl

^2

2

:

This expression has negative sign since the work of the forcePshould be calculated

on the displacementfromfinal to initialstate. Since the rigidity of support iskrot,

then the strain energy of elastic support is

U 0 Dkrot

^2

2

: