Civil Engineering Reference
In-Depth Information
Example 4.3 Three-span continuous beam composed of precast
elements
Three precast prestressed simple beams are prestressed and made con-
tinuous at age
t
0
by a reinforced concrete joint cast
in situ
(Fig. 4.4(a) ).
It is required to
nd the bending moment diagram at a later age
t
. The
prestress tendon pro
fi
le for each beam is as shown in Fig. 4.4(b). The
following data are given. The initial prestress at age
t
0
creates a
uniformly distributed upward load of intensity (2/3)
q
; thus,
fi
2
3
q
8
Pa
l
2
=
where
P
is the absolute value of the prestress force;
a
and
l
are de
ned in
Fig. 4.4(b);
q
is the weight per unit length of beam. Prestress loss is to be
assumed uniform and equal to 15 per cent of the initial prestress. Creep
coe
fi
cient
φ
(
t
,
t
0
)
=
2.5; aging coe
cient,
χ
(
t
,
t
0
)
=
0.8. Ignore cracking
at the joint.
Two statical systems need to be analysed. (a) Simple beams with
modulus of elasticity
E
c
(
t
0
) subjected to the self-weight
q
/unit length
downwards plus a set of self-equilibrating forces representing the initial
prestress (Fig. 4.4(c) ); the bending moment for this system is shown in
the same
gure. (b) A continuous beam subjected to a set of self-
equilibrating forces representing the prestress loss and redundant con-
necting moments caused by creep; the modulus o
f
elasticity to be used
with this loading is the age-adjusted modulus,
E
c
(
t
,
t
0
). The analysis
for the statically indeterminate bending moment due to loadings is
calculated below.
A statically determinate released structure is shown in Fig. 4.4(d).
Because of symmetry, the two coordinates representing the connecting
moments at B and C are given the same number 1.
Change in displacement in the released structure during the period
t
0
to
t
(Equation (4.3) ) are
fi
∆
D
1
=
(
∆
D
1
)
load
+
(
∆
D
1
)
prestress loss
(
∆
D
1
)
load
=
D
1
(
t
0
)
φ
(
t
,
t
0
)
where
D
1
(
t
0
) is the instantaneous displacement of the released structure
due to the loading in Fig. 4.4(c). Using Equation (C.6), Appendix C,
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