Civil Engineering Reference
In-Depth Information
(
t
,
t
0
) calculated by the above procedure depends
mainly upon
t
0
and
t
; other factors a
The aging coe
cient
χ
ff
ecting
χ
are the time functions
φ
(
t
,
τ
)
and
E
c
(
τ
). The graphs and table presented for
χ
in Appendix A are based on
time functions for
and
E
c
in accordance with MC-90 and the ACI Com-
mittee 209 report,
7
respectively. Choice of other functions results in small
change in the value of
φ
χ
; but this change may be ignored in practice. Since
χ
is
always used as a multiplier to
, which is rarely accurately determined,
high accuracy in the derivation of
φ
ed. Appendix G
includes information about computer programs that perform the step-by-step
calculations discussed in this section. The programs can be executed on
micro-computers using the software provided on the Internet as optional
companion of this topic (See web address in Appendix G.).
χ
is hardly justi
fi
1.11 Age-adjusted elasticity modulus
The three terms in Equation (1.10) represent the strain in concrete at age
t
due to: a stress
σ
c
(
t
0
) introduced at age
t
0
and sustained during the period
(
t
−
t
0
), a stress increment of magnitude zero at
t
0
increasing gradually to a
fi
nal value
∆
σ
c
(
t
) at age
t
, and the free shrinkage occurring during the period
(
t
−
t
0
). This equation may be rewritten as follows:
=
σ
c
(
t
0
)
1
+
φ
(
t
,
t
0
)
E
c
(
t
0
)
∆
σ
c
(
t
)
E
c
(
t
,
t
0
)
+
ε
cs
(
t
,
t
0
)
ε
c
(
t
)
+
(1.30)
where
E
c
(
t
0
)
E
c
(
t
,
t
0
)
=
(1.31)
1
+
χφ
(
t
,
t
0
)
E
c
(
t
,
t
0
) is the
age-adjusted elasticity modulus
to be used in the calculation of
the total strain increment, instantaneous plus creep, due to a stress incre-
ment of magnitude developing gradually from zero to a value
∆
σ
c
(
t
). Thus,
the strain increment in the period (
t
−
t
0
) caused by the stress
∆
σ
c
(
t
) is given
by:
∆
σ
c
(
t
)
E
c
(
t
,
t
0
)
.
∆
ε
c
(
t
)
=
(1.32)
1.11.1 Transformed section
In various chapters of this topic the term
transformed section
is employed to
mean a cross-section of a reinforced concrete member for which the actual
area is replaced by a transformed area equal to the area of concrete plus
α
times the area of steel; where
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