Civil Engineering Reference
In-Depth Information
external force. Alternatively, when a tendon is idealized as a member (Fig.
6.1(a)), two axial restraining forces are to be entered for this member:
A
r
(
t
j
)
prestress
=
A
ps
σ
p
(
t
j
)
(6.4)
where
A
ps
and
σ
p
(
t
j
) are the cross-sectional area of the tendon and its stress at
time
t
j
, respectively. The minus and the plus sign are, respectively, for the force
at the
fi
rst and second ends of the member (Fig. 6.3).
Computer run 2
: In this run the structure is idealized wi
th
the modulus of
elasticity of concrete being the age-adjusted modulus,
E
c
(
t
k
,
t
j
) given by
Equation (1.31), which is repeated here:
E
c
(
t
j
)
E
c
(
t
k
,
t
j
)
=
(6.5)
1
+
χφ
(
t
k
,
t
j
)
where
φ
(
t
k
,
t
j
) is creep coe
cient at time
t
k
for loading at time
t
j
;
χ
(
≡
χ
(
t
k
,
t
j
))
is the aging coe
cient;
E
c
(
t
j
) is the modulus of elasticity of concrete at time
t
j
.
The vector of
xed-end forces {
A
r
(
t
k
,
t
j
)} is to be entered as loading data;
where {
A
r
(
t
k
,
t
j
)} is a vector of hypothetical forces that can be introduced
gradually in the period
t
j
to
t
k
to prevent the changes in nodal displacements
at member ends. The elements of the vector {
A
r
(
t
k
,
t
j
)} for any member com-
prise a set of forces in equilibrium. Calculation of the elements of the vector
{
A
r
(
t
k
,
t
j
)} is discussed below, considering the separate e
fi
ff
ect of each of creep,
shrinkage and relaxation.
Member
xed-end forces due to creep
: The member end forces that restrain
nodal displacements due to creep are:
fi
E
c
(
t
k
,
t
j
)
E
c
(
t
j
)
{
A
r
(
t
k
,
t
j
)}
creep
=
−
φ
(
t
k
,
t
j
) {
A
D
(
t
j
)}
(6.6)
The vector {
A
D
(
t
j
)} is given by Equation (6.3), using the results and the
input data of Computer run 1. For the derivation of Equation (6.6), consider
the hypothetical displacements change [
(
t
k
,
t
j
) {
D
*(
t
k
)}] as if they were
unrestrained. Premultiplication of this vector by [
φ
A
u
] and substitution of
Equation (6.2) give the values of the restraining forces fo
r
a member whose
elasticity modulus is
E
c
(
t
j
). Multiplication of the ratio [
E
c
(
t
k
,
t
j
)/
E
c
(
t
j
)], to
account for the fact that the restraining forces are gradually introduced, gives
Equation (6.6).
−
Member end forces due to shrinkage
: The change of length of a concrete
member subjected to shrinkage
ε
cs
(
t
k
,
t
j
) can be prevented by the gradual
introduction of axial member-end forces:
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