8.4 Sidesway Frames with Absolutely Rigid Crossbars 295

ab

h

P

EI EI EI

EI=∞

P

1

i

R 1 P

cd

Z 1 =1

× × ×

6 i

h

6 i

h

6 i

h

6 i

h

r 11

1

M 1

Z 1 =1

(^1) r
11
M
Q
Q
Q M
Q
Q
Q M=6i/h
Q=12i/h^2
Q
Q
M M M=6i/h
(^1) R
P 1 P
e EI=∞
× × ×
Ph
6

M 0 M (^0) M
0
M 0 M 0 M 0
P
R 0 R 0 R 0 =P/ 3
MP
Deflection
curve
Inflection
point
f
Fig. 8.10 Frame with infinitely rigid crossbar: (a) Design diagram; (b) Primary system; (c) Bend-
ing moment diagram in unit state; (d) Free-body diagram of the crossbar. (e) Free-body diagram
of the crossbar in loaded state; (f) Final bending moment diagram
The primary unknown becomes
Z 1 D
R1P
r 11
D
Ph^3
36 EI
:
The bending moment at each of the specified sections of the frame is calculated
by the formula
MPDMN 1 Z 1 CMP^0 :
Since applied loadPdoes not produce bending moments in the primary system,
thenMP^0 D 0 and the resulting bending moments due to applied loadPare de-
termined asMP DMN 1 Z 1. The corresponding bending moment diagram and the
reactions of supports are presented in Fig.8.10f.
The bending moment at the top andbottom of each vertical member is
M 0 D
6i
h
Ph^3
36 EI
D
Ph
6
: