Advanced Methods of Structural Analysis

482 13 Stability of Elastic Systems

P

h

l

EI

kEI

a b

12

Primary system

c d

Elastic curve

r 11

3 EI

h l

3 kEI

Z 1 = (^1) r
21
r 21
M 1 M 2
Elastic curve
r 12
r 22
r 22
Z 2 = 1
j 1 (u)^3 EI
h^2
j 1 (u)
3 EI
h^3
(^3) hEI 2 j 1 (u) h 1 (u)
1
Primary system
e
Elastic curve
1
Z 1 = (^1) D r
11
3 kEIl EIhu tan u
M 1
fg
Fig. 13.18 (a) Design diagram of the frame. (b–d) First approach – primary system and cor-
responding bending moment diagrams in the unit conditions. (e–g) Second approach – primary
system and corresponding bending moment diagrams in the unit conditions
and bending moment diagrams caused by the unit rotation and linear displacement
of induced constrains 1 and 2.
Bending moments diagram is curvilinear for compressed vertical member of the
frame. The ordinates are found in accordance with the Table A.22. The bending
moment diagrams yield
r 11 D
3 EI
h
' 1 . /C
3kEI
l
I r 12 D
3 EI
h^2
' 1 . /
r 21 D
3 EI
h^2
' 1 . /I r 22 D
3 EI
h^3

1 . /
where parameter of stabilityDh
q
P
EI. Again, the subscript 1 at functions'and
is concerning to the pinned-clamped member subjected to angular and linear dis-
placements of the clamped support.