Advanced Methods of Structural Analysis

13.5 Stability of Arches 487

3.Using expression (13.17a), the slope is

dw

d

DBncosnC

Ccos

n^2  1

: (13.19)

The slope at the support is

dw

ds

D';

the negative sign means the reactive momentM 0 and angle'have the opposite

directions. On the other hand

dw

ds

D

dw

Rd

;

so

dw

d

DR':

According to (13.15b)weget

'DC

EIsin ̨

kR^2

;so

dw

d

DC

EIsin ̨

kR

:

IfD ̨then the expression (13.19) becomes

Bncosn ̨C

Ccos ̨

n^2  1

DC

EIsin ̨

kR

:

After rearrangements this expression may be presented in form

Bncosn ̨CC



cos ̨

n^2  1

C

EIsin ̨

kR



D0: (13.20)

Equations (13.18)and(13.20) are homogeneous linear algebraic equations with re-

spect to unknown parametersBandC. The trivial solutionBDCD 0 corresponds

to state of the arch before the loss of stability. Nontrivial solution occurs if the fol-

lowing determinant is zero:

DD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

sinn ̨

sin ̨

n^2  1

ncosn ̨I

cos ̨

n^2  1

C

EIsin ̨

kR

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

D 0 (13.21)

The stability equation (13.21) may be presented as follows:

tann ̨D

nk 0

k 0 cot ̨C.n^2 1/

;k 0 D

kR

EI

: (13.22)