Civil Engineering Reference
In-Depth Information
example is at the centre of span of a simply supported beam. The
absolute value
P
of the prestressing force at time
t
0
is assumed constant
at all sections, while the dead load bending moment,
M
, is assumed to
vary as a parabola. The pro
le of the prestress tendon is assumed a
parabola, as shown. The graphs in this
fi
fi
gure show the variation over
the span of
∆
σ
ps
which are respectively the axial
strain and curvature at
t
0
and the changes during the period (
t
ε
O
(
t
0
),
ψ
(
t
0
),
∆
ε
O
,
∆
ψ
,
t
0
) in
axial strain, in curvature and in tension in prestress steel due to
the combined e
−
ff
ects of creep, shrinkage and relaxation. The values
of (
) will be used in Example 3.5 to calculate
displacement values at time
t
.
ε
O
+ ∆
ε
O
) and (
ψ
+ ∆
ψ
Example 2.3 Pre-tensioned section
Solve the same problem as in Example 2.2 assuming that pre-tensioning
is employed (the duct shown in Fig. 2.6(a) is eliminated).
(a) Stress and strain at age t
0
The prestressed steel must now be included in the calculation of the
properties of the transformed section at
t
0
. With this modi
cation and
considering that there is no prestress duct in this case, calculation of the
area properties of the transformed section in the same way as in Table
2.1 gives:
A
fi
48.77 × 10
−3
m
4
.
The forces applied on the section at
t
0
are the same as in Example 2.2.
Equation (2.32) gives the strain and the curvature at the reference point
immediately after prestress transfer:
=
0.3805 m
2
;
B
=
4.413 × 10
−3
m
3
;
I
=
ε
O
(
t
0
)
=−
120×10
−6
;
ψ
(
t
0
)
=−
153×10
−6
m
−1
.
The change in stress in the prestressed steel at transfer (Equation
(2.38) ) are
(
∆
σ
ps
)
inst
=
200[
−
120
+
0.45(
−
153)]10
3
=−
37.8 MPa.
Multiplying this value by the area of the prestressed steel gives the
instantaneous prestress loss (
43 kN).
The stresses and strain introduced at transfer and the corresponding
resultants of stresses are shown in Fig. 2.8(a).
−
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