Advanced Methods of Structural Analysis

480 13 Stability of Elastic Systems

PP

h EI

h

3 i

h

3 i

h

j^3 i

1 (u)

3 i

h j^1 (u)

3 i

h

Z 1 =1

r 11

r 11

Q 1 Q 2 Q 3 Q 4 Q 5

3 i;

h^2

Q 1 = Q 4 = Q 5 =

3 ih 1 (u)

h^2

Q 2 = Q 3 =

a b

c

PP

h EI

P PP

221

PP

h

ll

EI

P PP

EI=•

1

de

Fig. 13.17(a–c) Multispan frame. Design diagram, primary system and free-body diagram. (d, e)
Multispan regular frames; a number of columns isk

r 11 D 3 

3i

h^2

C 2 

3i

h^2

1 . / : (a)

The stability equation becomes 3 C 2
1 . /D 0. The minimum positive root of this
equation, i.e., the parameter of lowest critical load isD2:4521. Critical load

PcrD

^2 EI

h^2

D

6:0128EI

h^2

:

Discussion. Analysis of this frame allows easy considering of two important typ-
ical cases. Both cases are related to the regular multispan frame withkcolumns
loaded by equal forcesPat the each joint. Figure13.17d presents regular frame
with hinged joints, while the Fig.13.17e shows the frame with absolutely rigid
cross-bar and fixed joints. In both cases, the frame has one unknown of the displace-
ment method. Introduced constraint prevents linear displacement; this constraint is
not shown.
For both cases, the loss of stability is possible according two different forms. The
first form occurs with horizontal displacement of the cross bar (dotted line 1) and
the second form without of such displacement (dotted line 2). For both cases, the
lowest critical load corresponds to the first form.
For case of hinged joints (Fig.13.17d), the horizontal reaction of introduced
constraint is

r 11 Dk

3i

h^2

1 . / :