Advanced Methods of Structural Analysis

464 13 Stability of Elastic Systems

Unknown parametersC 1 ;C 2 ,andfmay be determined using the following bound-

ary conditions:

.1/ y .0/D 0 I .2/ y^0 .0/DI .3/ y .l /Df

1.The first boundary condition leads to equation

C 1 Cf



k 1 l

N

 1



D0:

2.At pointA.xDyD0/the reactive moment equalsMADf.k 1 lP/, thus the
angle of rotation atAis

'D

MA

k 2

D

f

k 2

.k 1 lP/;

so the second boundary condition leads to the following equation

C 2 nf



k 1

P

C

k 1 lP

k 2



D0:

3.The third boundary condition leads to the following equation

C 1 cosnlCC 2 sinnlD0:

Conditions 1–3 may be rewritten in the form of the homogeneous algebraic equa-

tions with respect to unknownsC 1 ;C 2 ,andf. Equation for critical load is presented

as determinant from coefficients at these unknowns, i.e.,

DD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

10

k 1 l

P

 1

0n



k 1

P

C

k 1 lP

k 2



cosnl sinnl 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

D0or

DD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

10

k 1 l

P

 1

0n



k 1

P

C

k 1 lP

k 2



1 tannl 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

D 0

This equation may be rewritten as the transcendental equation with respect to pa-

rametern

tannlDnl

k 1 l

n^2 EI

 1

k 1 l

n^2 EI

C



k 1 ln^2 EI



l

k 2

:

For given parametersl;EI;k 1 andk 2 of the structure, solution of this equation

leads to parameternof critical load. The critical load isPcrDn^2 EI.Table13.1