Advanced Methods of Structural Analysis

13.2 Stability of Structures with Finite Number Degrees of Freedom 459

The trivial solutiona 1 D0; a 2 D 0 corresponds to initial unstrained configuration
of the structure.
Nontrivial solution of this system occurs if determinant of the system equals zero:

3N2kl kl
kl 3N2kl

!.3N2kl/^2 .kl /^2 D0:

The roots of this equation present the critical loads; they are

N1crD

kl

3

;N 2 crDkl: (b)

R 2 =ka 2

a 1

( 2 a 1 + a 2 )

3

k

(a 1 + 2a 2 )

3

k

N N

C (^1) C 2
a 2
R 1 =ka 1
N N
k
l l
C 1
l
C 2
k
EI=•
f 1 C (^1) C 2 f 2
R 2 =kf 2
N N
R 1 =kf 1
b 1 b 2
Δ
A B
a
b c
N
a 1 =1
a 2 =− 1
N= N 1 cr a 1 =1 N=N^2 cr
N
a 2 =1
de
Fig. 13.6 (a–c) Structure with two degrees of freedom: (a) Design diagram; (b) Static method;
(c) Energy method. (d, e) First and second form of the loss of stability
Energy method Let the generalized coordinates be anglesˇ 1 andˇ 2 (Fig.13.6c).
The angle between portionC 1 C 2 and horizontal line isˇ 1 ˇ 2 , so horizontal dis-
placement of the point of application forceNis
Dl.1cosˇ 1 /Cl.1cosˇ 2 /Cl.1cos.ˇ 1 ˇ 2 //
Š
l
2
h
ˇ^21 Cˇ^22 C.ˇ 1 ˇ 2 /^2
i
Dl

ˇ^21 Cˇ 22 ˇ 1 ˇ 2

:
The potential of the forceNisWDN.
The vertical displacements of pointsC 1 andC 2 equal tof 1 Dltanˇ 1 Šlˇ 1
andf 2 Dltanˇ 2 Šlˇ 2 , respectively. Therefore, reactions of elastic supports are