Advanced Methods of Structural Analysis

462 13 Stability of Elastic Systems

y

x

P P

l

x

f y

0

Fig. 13.7 Buckling of clamped-free column

This equation may be transformed to the form

d^2 y

dx^2

Cn^2 yDn^2 f;nD

r

P

EI



1

length

: (13.6)

Equation (13.6) is nonhomogeneous linear differential equation of order two in one

variablexwith constant coefficientn^2. Therefore, the solution of this equation

should be presented in the form

yDAcosnxCBsinnxCy;

whereAandBare constants of integration. The partial solutionywe will find

in the form of the right part of (13.6). Since the right part is constant (does not

depend ony), then supposeyDC. Substitution of this expression into (13.6)

leads ton^2 C Dn^2 f !C Df. Therefore, the general solution of (13.6)and

corresponding slope are

yDAcosnxCBsinnxCf;

y^0 DAnsinnxCBncosnx:

To determine unknown parameters, let usconsider the following boundary condi-

tions:

1.AtxD 0 (fixed end) the slopey^0 D 0. Expression for slope leads to theBD 0
2.AtxD 0 the displacementyD 0. Expression foryleads to theADf

Thus the displacement of the column becomes

yDf.1cosnx/ :

AtxDl(free end) the displacementyDf. Therefore,fDf.1cosnl/,which

holds if

cosnlD0:

This equation is called the stability equation for given column; the smallest root

equalsnlD=2.ThevaluenD=2lis called the critical parameter. Thus the

smallest critical load for uniform clamped-free column becomes

PcrDn^2 crEID

2 EI

4l^2

: