Civil Engineering Reference
In-Depth Information
point2(Figure5.30c), andtheotherwithahingeatpoint3(Figure5.30d).Three
different methods of analysing these mechanisms are presented in the following
sub-sections.
5.11.2.2 Equilibrium equation solution
The equilibrium conditions for the beam express the relationships between the
appliedloadsandthemomentsatthesepoints.Theserelationshipscanbeobtained
bydividingthebeamintothesegmentsshowninFigure5.30bandexpressingthe
end shears of each segment in terms of its end moments, so that
V
12
=
(
M
2
−
M
1
)/(
L
/
3
)
V
23
=
(
M
3
−
M
2
)/(
L
/
3
)
V
34
=
(
M
4
−
M
3
)/(
L
/
3
)
.
Forequilibrium,eachappliedloadmustbeequaltothealgebraicsumoftheshears
at the load point in question, and so
Q
=
V
12
−
V
23
=
1
L
(
−
3
M
1
+
6
M
2
−
3
M
3
)
2
Q
=
V
23
−
V
34
=
1
L
(
−
3
M
2
+
6
M
3
−
3
M
4
)
which can be rearranged as
4
QL
=−
6
M
1
+
9
M
2
−
3
M
4
5
QL
=−
3
M
1
+
9
M
3
−
6
M
4
.
(5.72)
If the first mechanism chosen (incorrectly, as will be shown later) is that with
plastic hinges at points 1, 2, and 4 (see Figure 5.30c), so that
−
M
1
=
M
2
=−
M
4
=
M
p
then substitution of this into the first of equations 5.72 leads to 4
QL
=
18
M
p
and so
Q
ult
≤
4.5
M
p
/
L
.
If this result is substituted into the second of equations 5.72, then
M
3
=
1.5
M
p
>
M
p
,
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