Civil Engineering Reference
In-Depth Information
structure; {
F
} are the redundant forces. The displacements {
D
} represent
inconsistencies in the released structure (with respect to the actual
structure). The redundants {
F
} must, therefore, be applied to eliminate the
inconsistencies.
Any element of the
exibility matrix,
f
mn
is equal to the displacement at
coordinate
m
due to unit load applied at coordinate
n
. Because of creep of
concrete, the value of any element of the matrix [
f
] depends upon the time
for which the displacement is considered and the age of concrete at the time
of the introduction of the unit load. Thus, we use here the symbol [
f
(
t
i
+
2
,
t
j
)]
to represent the matrix of
fl
fl
exibility at time
t
i
+
2
when the age at loading is
t
j
.
2
,
i
and
i
2
respectively refer to the beginning, the middle
The subscripts
i
−
+
and the end of interval
i
.
The forces {
F
}
i
+
2
and the displacement {
D
}
i
+
2
at the end of any interval
i
may be expressed as the sum of incremental forces {
∆
F
}
j
and displacements,
{
∆
D
}
j
occurring at the middle of the intervals
j
=
1, 2, . . . ,
i
. Thus,
i
{
F
}
i
+
=
{
∆
F
}
j
(4.21)
1
2
j
= 1
i
{
D
}
i
+
=
{
∆
D
}
j
(4.22)
2
j
=
1
The compatibility Equation (4.20) applied at the end of the
i
th interval
may be written in the form:
i
i
{[
f
(
t
i
+
2
,
t
j
)]{
∆
F
}
j
}
=−
{
∆
D
}
j
(4.23)
1
j
= 1
j
= 1
F
}
i
can be done in steps; in each step a new increment is
calculated. In the
i
th step, the values {
The analysis for {
∆
F
}
i
− 1
are known
from the preceding steps and Equation (4.23) can be used to determine {
∆
F
}
1
, {
∆
F
}
2
, . . . , {
∆
F
}
i
.
Equation (4.23) may be rewritten by separating the last term of the summa-
tion on the left-hand side and the substitution of Equation (4.22):
∆
i
−
1
[
f
(
t
i
+
2
,
t
i
)] {
∆
F
}
i
=−
{
D
}
i
+
−
{[
f
(
t
i
+
2
,
t
j
)]{
∆
F
}
j
}
(4.24)
2
j
=
1
This recurrent equation can be solved successively with
i
=
1, 2, . . . to
determine the values of the vector {
∆
F
}
1
, {
∆
F
}
2
, . . . and so on.
The
fl
exibility matrices involved in the analysis di
ff
er only in the modulus
of elasticity and the creep coe
cient to be employed in the calculation.
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