Advanced Methods of Structural Analysis

466 13 Stability of Elastic Systems

13.3.2 Initial Parameters Method....................................

This method may be effectively applied for stability analysis of the columns with
rigid and elastic supports and columns with step-variable cross section. Moreover,
this method allows us to derive the useful expressions, which will be applied for
stability analysis of the frames.
Let us consider a beam of constant cross section. The beam is subjected to axial
compressed forceP. Differential equation of the beam is

EI

d^2 y

dx^2

CPyD0;

whereyis a lateral displacement. Twice differentiation of this equation leads to
fourth-order differential equation

EI

d^4 y

dx^4

CP

d^2 y

dx^2

D0; or

d^4 y

dx^4

Cn^2

d^2 y

dx^2

D0; (13.8)

where

nD

r

P

EI

Solution of (13.8) may be presented as

y.x/DC 1 cosnxCC 2 sinnxCC 3 xCC 4 ; (13.8a)

whereCiare unknown coefficients. It is easy to check that this solution satisfies
(13.8).
The slope, bending moment and shear are

'.x/Dy^0 .x/DC 1 nsinnxCC 2 ncosnxCC 3 ;

M.x/DEIy^00 .x/DEI.C 1 n^2 cosnxCC 2 n^2 sinnx/;

Q.x/DEIy^000 .x/DEI.C 1 n^3 sinnxC 2 n^3 cosnx/:

AtxD 0 the initial kinematical and static parameters become

y.0/Dy 0 DC 1 CC 4 ;

'.0/D' 0 DC 2 nCC 3 ;

M.0/DM 0 DC 1 n^2 EI;

Q.0/DQ 0 DC 2 n^3 EI;

wherey 0 ;' 0 ;M 0 ;Q 0 are lateral displacement, angle of rotation, bending moment,
and shear at the origin (Fig.13.9,y-axis is directed downward). They arise because
the rod lost the stability.