Civil Engineering Reference
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and the plasticity condition is violated. The statical method can now be used to
obtain a lower bound by reducing
Q
until
M
3
=
M
p
, whence
M
3
Q
=
4.5
M
p
L
M
p
=
3
M
p
L
and so
3
M
p
/
L
<
Q
ult
<
4.5
M
p
/
L
.
The true collapse load can be obtained by assuming a collapse mechanism
with plastic hinges at point 3 (the point of maximum moment for the previous
mechanism) and at points 1 and 4. For this mechanism
−
M
1
=
M
3
=−
M
4
=
M
p
,
and so, from the second of equations 5.72,
Q
ult
≤
3.6
M
p
/
L
.
If this result is substituted into the first of equations 5.72, then
M
2
=
0.6
M
p
<
M
p
,
and so the plasticity condition is also satisfied.Thus
Q
ult
=
3.6
M
p
/
L
.
5.11.2.3 Graphical solution
Acollapsemechanismcanbeanalysedgraphicallybyfirstplottingtheknownfree
moment diagram (for the applied loading on a statically determinate beam), and
then the reactant bending moment diagram (for the redundant reactions) which
corresponds to the collapse mechanism.
For the built-in beam (Figure 5.38a), the free moment diagram for a simply
supported beam is shown in Figure 5.38c. It is obtained by first calculating the
left-hand reaction as
R
L
=
(
Q
×
2
L
/
3
+
2
Q
×
L
/
3
)/
L
=
4
Q
/
3
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