Civil Engineering Reference
In-Depth Information
E
c
(
t
0
)]
1
+
χφ
(
t
,
t
0
)
E
c
(
t
0
)
ε
c
=
ε
c
[1
+
φ
(
t
,
t
0
)]
+
ε
c
[
r
(
t
,
t
0
)
−
(1.22)
We recall that the symbol
χφ
(
t
,
t
0
) indicates the product of two functions
χ
ε
c
in
Equation (1.22) cancels out and by algebraic manipulation of the remaining
terms we can express the aging coe
and
φ
of the time variables
t
and
t
0
. The constant strain value
cient
χ
in terms of
E
c
(
t
0
),
r
(
t
,
t
0
) and
φ
(
t
,
t
0
):
1
1
χ
(
t
,
t
0
)
=
r
(
t
,
t
0
)/
E
c
(
t
0
)
−
(1.23)
1
−
φ
(
t
,
t
0
)
A step-by-step numerical procedure will be discussed in the following sec-
tion for the derivation of the relaxation curve in Fig. 1.6. The relaxation
function
r
(
t
,
t
0
) obtained in this way can be used to calculate the aging
coe
cient
χ
(
t
,
t
0
) by Equation (1.23).
1.10 Step-by-step calculation of the relaxation
function for concrete
The step-by-step numerical procedure introduced in this section can be used
for the calculation of the time-dependent stresses and deformations in con-
crete structures. It is intended for computer use and is particularly suitable for
structures built or loaded in several stages, as for example in the segmental
construction method of prestressed structures. In this section, a step-by-step
method will be used to derive the relaxation function
r
(
τ
,
t
0
). Further
development of the method is deferred to Sections 4.6 and 5.8.
The value of the relaxation function
r
(
t
,
t
0
) is de
ned as the stress at time
t
due to a unit strain introduced at time
t
0
and sustained without change during
the period (
t
fi
t
0
) (see Equation (1.16) ).
Consider a concrete member subjected to uniaxial stress and assume that
the magnitude of stress varies with time as shown in Fig. 1.7(b). At age
t
0
an
initial stress value
−
σ
c
(
t
0
) is introduced and subsequently increased gradually
or step-wise during the period
t
0
to
t
. When the variation of stress with time is
known, the step-by-step analysis to be described can be used to
fi
nd the strain
at any time
between
t
0
and
t
. Alternatively, if the strain is known, the
method can be used to determine the time variation of stress.
Divide the period (
t
τ
t
0
) into intervals (Fig. 1.7(a) ) and assume that the
stress is introduced in increments at the middle of the intervals. Thus, (
−
∆
σ
c
)
i
is
introduced at the middle of the
i
th interval. For a sudden increase in stress,
consider an increment introduced at an interval of zero length (for example
(
2
are used to refer
to the instant (or the age of concrete) at the start, the middle and the end of
∆
σ
c
)
l
and (
∆
σ
c
)
k
) in Fig. 1.7(b) ). The symbols
t
j
−
2
,
t
j
and
t
j
+
1
1
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