Civil Engineering Reference
In-Depth Information
i.e.
R
A,V
=−
83
.
4 kN (downwards)
In the portal frame of Ex. 18.8 each member has the same plastic moment
M
P
.In
cases where the members have different plastic moments a slightly different approach
is necessary.
E
XAMPLE
18.9
In the portal frame of Ex. 18.8 the plastic moment of the member
BCD is 2
M
P
. Calculate the critical value of the load
W
.
Since the vertical members are the weaker members plastic hinges will form at B in
AB and at D in ED as shown, for all three possible collapse mechanisms, in Fig. 18.19.
This has implications for the virtual work equation because in Fig. 18.19(a) the plastic
u
u
4
u
4
u
3
u
2
u
B
B
D
B
D
D
2
u
C
2
u
C
2
u
2
u
E
E
E
u
F
IGURE
18.19
Collapse
mechanisms for
the frame of
Ex. 18.9
A
A
A
(a)
(b)
(c)
moment at B and D is
M
P
while that at C is 2
M
P
. The virtual work equation then
becomes
W
2
θ
=
M
P
θ
+
2
M
P
2
θ
+
M
P
θ
which gives
W
=
3
M
P
For the sway mechanism
W
4
θ
=
M
P
θ
+
M
P
θ
+
M
P
2
θ
+
M
P
2
θ
so that
3
2
M
P
W
=
and for the combined mechanism
W
4
θ
+
W
2
θ
=
M
P
θ
+
2
M
P
2
θ
+
M
P
3
θ
+
M
P
2
θ
