Advanced Methods of Structural Analysis

478 13 Stability of Elastic Systems

Stability equation in general and expanded forms are
ˇ
ˇ
ˇ
ˇ

r 11 r 12

r 21 r 22

ˇ

ˇ

ˇ

ˇD^0

4



' 2 . /C

1

ˇ





' 2 . /Ck' 2 . 2 /C

1

ˇ

' 32 . /D0:

Let ̨D3; ˇD1; kD 4. In this case, 2 Dand stability equation becomes

4Œ' 2 . /C1Œ5' 2 . /C1' 32 . /D0:

The root of this equation isD4:5307. The critical load is

PcrD

^2 EI

h^2

D

4:5307^2 EI

h^2

:

Example 13.5.The frame in Fig.13.16a is loaded by two forces at the joints. De-
rive the stability equation and find the critical loadP.

h=10m

l=5m

P 1.4P

EI EI

2 EI

Primary system

12

i=0.4EI

3 4

i=0.1EI

ab

r 12

r 22

M 2

Z 2 = 1

Z 2 = 1

M 31 M 42

M (^13) R 2
R 1
M 13 = 4 ij 2 (u) = 0. 4 EIj 2 (u) M^13 =^6 ij^4 (u) = 0.^6 EIj^4 (u)
12 i
l^2
R 1 = h 2 (u) = 0. 12 EIh 2 (u)
3 i
l^2
R 2 = h 1 (1.1832u) = 0. 003 EIh 1 (1.1832u)
R = 6 ij 4 (u) = 0. 6 EIj 4 (u)
r 11 Z 1 = 1
3 i=1.2EI
r 21
M 1
M 13
M 31
R
R
M 13
M 31
R
R
cd
Fig. 13.16 (a, b) Design diagram of the frame and primary system. (c, d) Unit bending moment
diagrams