Civil Engineering Reference
In-Depth Information
w
u
A
f
B
B
A
C
L
u
+
f
R
A
R
B
F
IGURE
18.14
Collapse
mechanism for a
propped cantilever
x
L
x
(a)
(b)
The total load on AC is
wx
and its centroid (at
x
/2 from A) will be displaced a vertical
distance
δ
/2. The total load on CB is
w
(
L
−
x
) and its centroid will suffer the same
vertical displacement
δ
/2. Then, from the principle of virtual work
wx
2
+
x
)
2
=
w
(
L
−
M
P
θ
+
M
P
(
θ
+
φ
)
Note that the beam at B is free to rotate so that there is no plastic hinge at B.
Substituting for
δ
from Eq. (i) and
φ
from Eq. (ii) we obtain
M
P
θ
wL
θ
x
x
2
=
M
P
θ
+
+
θ
L
−
x
or
M
P
θ
2
wL
θ
x
x
2
=
+
L
−
x
Rearranging
2
L
2
M
P
Lx
−
x
w
=
(iii)
L
−
x
For a minimum value of
w
,(d
w
/d
x
)
=
0. Then
−
d
w
d
x
=
2
M
P
L
x
(
L
−
x
)
−
(2
L
−
x
)(
L
−
2
x
)
=
0
x
2
(
L
x
)
2
−
which reduces to
x
2
2
L
2
−
4
Lx
+
=
0
Solving gives
x
=
0
.
586
L
(the positive root is ignored)
Then substituting for
x
in Eq. (iii)
11
.
66
M
P
L
2
w
(at collapse)
=
