Civil Engineering Reference
In-Depth Information
from which
5
3
M
P
Here we see that the minimum value of
W
which would cause collapse is 3
M
P
/2 and
that the sway mechanism is the critical mechanism.
W
=
We shall now examine a portal frame having a pitched roof in which the determination
of displacements is more complicated.
E
XAMPLE
18.10
The portal frame shown in Fig. 18.20(a) has members which have
the same plastic moment
M
P
. Determine the minimum value of the load
W
required
to cause collapse if the collapse mechanism is that shown in Fig. 18.20(b).
W
C
u
C
2m
C
B
D
B
D
D
5m
a
A
E
A
E
F
IGURE
18.20
Collapse
mechanism for
the frame of
Ex. 18.10
5m
5m
(a)
(b)
In Exs 18.8 and 18.9 the displacements of the joints of the frame were relatively simple
to determine since all the members were perpendicular to each other. For a pitched
roof frame the calculation is more difficult; one method is to use the concept of
instantaneous centres
.
In Fig. 18.21 the member BC is given a
small
rotation
θ
. Since
θ
is small C can be
assumed to move at right angles to BC to C
. Similarly the member DE rotates about
E so that D moves horizontally to D
. Further, since C moves at right angles to BC
and D moves at right angles to DE it follows that CD rotates about the instantaneous
centre, I, which is the point of intersection of BC and ED produced; the lines IC and
ID then rotate through the same angle
φ
.
From the triangles BCC
and ICC
CC
=
BC
θ
=
IC
φ
so that
BC
IC
=
φ
=
θ
(i)
