Civil Engineering Reference
In-Depth Information
W
B
D
W
C
2 m
E
4 m
A
2 m
2 m
F
IGURE
18.16
Portal frame of
Ex. 18.8
(a)
(b)
4
u
4
u
4
u
3
u
u
u
u
u
B
D
B
D
B
D
2
u
2
u
2
u
2
u
C
C
2
u
2
u
u
u
E
E
E
A
A
A
(c)
(a)
(b)
F
IGURE
18.17
Collapse mechanisms for the frame of Ex. 18.8
the beam and sway mechanisms. However, in this case, the moments at B due to the
vertical load at C and the horizontal load at B oppose each other so that the moment
at B will be the smallest of the five peak moments and plastic hinges will form at the
other locations. We say, therefore, that there is a
hinge cancellation
at B; the angle
ABC then remains a right angle. We shall now examine each mechanism in turn to
determine the value of
W
required to cause collapse. We shall designate the plastic
moment of the frame as
M
P
.
BEAMMECHANISM
Suppose that BC is given a small rotation
θ
. Since CD
CB then CD also rotates
through the angle
θ
and the relative angle between CD and the extension of BC is 2
θ
.
Then, from the principle of virtual work
=
W
2
θ
=
M
P
θ
+
M
P
2
θ
+
M
P
θ
(i)
which gives
W
=
2
M
P
