Advanced Methods of Structural Analysis

6.5 Graph Multiplication Method 183

a

EI 1 EI 3

EI 2

l

P

h

A B

CD

Design diagram Unit state for Dhor

A B

1 ⋅h 1 ⋅h

M

H = 1 P^ =1

y 1

y 2

Actual state

P P⋅h

A MP B

l

RA RB

H=P

Centroid

W (^2) •
W (^1)
EI 1 EI 3
EI 2
l
P
h
A B
CD
Design diagram Unit state for qB
A B
M M=1
1
1
1/l 1/l
yC
RB
Actual state
PP⋅h
A MP B
R l
A
H=P
Centroid
W (^2)
W (^1)
b
P
A B
CD
qB
ΔB
c
Fig. 6.23 (a) Portal frame. Actual and unit states and corresponding bending moment diagrams.
(b) Portal frame. Actual and unit states and corresponding bending moment diagrams. (c)Design
diagram and elastic curve for portal frame
Solution.As usual, the analysis starts from construction of bending moment dia-
gram in actual state.
Reactions of supports areH DP,RADRB DPh=l. The real directions
of reactions are shown in Fig.6.23a. The tensile fibers on elementsCDandACare
located below and right from the neutral lines of the elements, respectively. Bending
moment ordinates at pointCfor vertical and horizontal members arePh.
(a)Horizontal displacement at B.For required displacementhor, the unit state
presents the same frame with horizontal forcePD 1 , which is applied at pointB.
Direction of the unit force is chosen in arbitrary way. Only horizontal reaction
HD 1 is induced. The tensile fibers are located outdoor of the frame. The bending
moments at rigid jointsCandDand within cross bar equal to 1 h.
Multiplication of the bending moment diagrams should be performed for mem-
bersAC,CD,andDBseparately. For horizontal memberCD, the area of the bending
moment diagram in actual state is ̋ 1 D.1=2/P hland corresponding ordinate
from unit state isy 1 D 1 h. We assume that horizontal portionCDof both bending