Civil Engineering Reference
In-Depth Information
5.11.3.2 Equilibrium equation solution
If the plastic hinge is assumed to be at the mid-span of the propped cantilever as
shown in Figure 5.31c (the other plastic hinge must obviously be located at the
built-insupport),thenequilibriumconsiderationsoftheleftandrightsegmentsof
the beam allow the left and right reactions to be determined from
R
L
L
/
2
=
2
M
p
+
(
qL
/
2
)
L
/
4
and
R
R
L
/
2
=
M
p
+
(
qL
/
2
)
L
/
4
while for overall equilibrium
qL
=
R
L
+
R
R
.
Thus
qL
=
(
4
M
p
/
L
+
qL
/
4
)
+
(
2
M
p
/
L
+
qL
/
4
)
so that
q
ult
≤
12
M
p
/
L
2
.
The left-hand reaction is therefore given by
R
L
=
4
M
p
/
L
+
(
12
M
p
/
L
2
)
L
/
4
=
7
M
p
/
L
.
The bending momentvariation along thepropped cantileveris therefore given by
M
=−
M
p
+
(
7
M
p
/
L
)
x
−
(
12
M
p
/
L
2
)
x
2
/
2
which has a maximum value of 25
M
p
/
24
>
M
p
at
x
=
7
L
/
12. Thus the lower
bound is given by
q
ult
≥
(
12
M
p
/
L
2
)/(
25
/
24
)
≈
11.52
M
p
/
L
2
and so
11.52
M
p
L
2
<
q
ult
<
12
M
p
L
2
.
If desired, a more accurate solution can be obtained by assuming that the hinge is
located at a distance 7
L
/
12
(
=
0.583
L
)
from the built-in support (i.e. at the point
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