Advanced Methods of Structural Analysis

486 13 Stability of Elastic Systems

Thus, differential equation (13.13a) becomes

d^2 w

d^2

C



1 C

qR^3

EI



wD

k'

EI

R^2

sin ̨

sin: (13.15)

Denote

n^2 D 1 C

qR^3

EI

: (13.15a)

CD

k' R^2

EIsin ̨

: (13.15b)

Pay attention thatCis unknown, since the angles of rotation'of the supports are

unknown. Differential equation (13.15) may be rewritten as follows

d^2 w

d^2

Cn^2 wDCsin: (13.16)

Solution of this equation is

wDAcosnCBsinnCw: (13.17)

The partial solutionwshould be presented in the form of the right part of (13.16),

mainlywDC 0 sin,whereC 0 is a new unknown coefficient. Substituting of this

expression into (13.16) leads to equation

C 0 sinCn^2 C 0 sinDCsin;

so

C 0 D

C

n^2  1

:

Thus the solution of equation (13.15) becomes

wDAcosnCBsinnC

C

n^2  1

sin: (13.17a)

Unknown coefficientsA, B,andCmay be obtained from the following boundary

conditions:

1.For point of the arch on the axis of symmetry.D0/, the radial displacement
wD 0 (because the antisymmetrical form ofthe loss of stability); this condition
leads toAD 0 ;
2.For point of the arch at the support.D ̨/the radial displacement iswD 0 ,so

Bsinn ̨C

C

n^2  1

sin ̨D0: (13.18)