Civil Engineering Reference
In-Depth Information
M
B
/(
ql
2
)]
load intensity
q
in this example, the dimensionless coe
cient [
−
tends to approach again the value 0.125.
Figure 13.8(b) is a plot of (
D
/
D
r
) versus (
q
/
q
r
); where
D
is the de
ec-
tion at mid-span;
q
is the load intensity and the subscript
r
refers to the
state of the
fl
rst cracking. The value of (
q
/
q
r
) is varied from zero to 2.2,
in steps of 0.3; before cracking (0
fi
1.0) the graph is linear. After
cracking, the values plotted in Fig. 13.8(b) are based on a nonlinear
analysis of
M
B
, giving the following values for the coe
≤
(
q
/
q
r
)
≤
cient: [
−
M
B
/
(
ql
2
)]
10
−3
{125, 98, 105, 122, 118} for {
q
/
q
r
}
{1.0, 1.3, 1.6, 1.9, 2.2},
respectively. In the analyses reported here the sections considered in
each span are spaced at
l
/10.
=
=
Example 13.5 Prestressed continuous beam analysed by the
incremental method
Plot the de
ection,
D
at mid-span versus the intensity,
q
of a uniformly
distributed live load covering two equal spans of the post-tensioned
beam shown in Figs. 13.9(a) and (b). The parameters
fl
σ
Oin.
and
γ
in.
de
ning the distribution of initial stresses, at various sections, existing
before application of the live load are given in Table 13.2. The vari-
ations of the parameters between sections 1 and 16 and between sec-
tions 17 and 22 are parabolic. Given data:
f
ct
fi
=
2.5 MPa (0.36 ksi);
E
c
=
30.0 GPa (4350 ksi);
E
ns
=
E
ps
=
200 GPa (29000 ksi);
β
1
=
1.0 and
β
2
=
0.5;
de
β
2
are given in Section 8.3. The prestressing force
and the self weight of the beam, producing
fi
nitions of
β
1
and
σ
O in.
and
γ
in.
, given in Table
13.2 are, respectively,
P
2200 kN (494.6 kip) and 18.0 kN/m (1.23 kip/
ft). The given value of
P
accounts for the time-dependent losses; thus
=
Table 13.2
Initial stress parameters at selected sections of the post-tensioned beam
of Example 13.5
Section
Distance from
Initial stress at
Initial slope of
number
support A
(m)
reference point O,
stress diagram,
Oin
(
MPa
)
γ
in
(
MPa/m
)
1
0.00
−
2.939
0.150
9
10.00
−
2.939
0.843
16
18.75
−
2.939
0.918
17
18.75
−
2.855
−
0.152
20
22.50
−
2.857
−
0.832
22
25.00
−
2.856
−
1.414
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