Civil Engineering Reference
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and the creep equivalent stress at the next time step is
i
∑
0
(
)
−
(
)
σ
d
ϕ
=
∆σ
ϕ
t
,
τ
+
t
−
t
ϕ
t
,
τ
+
t
−
t
(3.63)
i
j
i
+
1
0
j
0
i
0
j
0
j
=
Considering
∫
σ
e
, the creep equivalent nodal loads of an
element at time step
t
i
can be written as
T
B
∆
dv
=
K a
∆
i
i
i
∑
0
(
)
−
(
)
e
F
=
K
∆
a
ϕ
t
,
τ
+
t
−
t
ϕ
t
,
τ
+
t
−
t
(3.64)
i
e
j
i
+
1
0
j
0
i
0
j
0
j
=
where ∆
a
j
is the incremental displacements at time
t
j
corresponding to
the stress change of ∆σ
j
. From Equation 3.64, it is obvious that the cal-
culation of creep equivalent load is separated from the element stiffness
matrix. Given the history of displacement changes due to any loading
types, including redistribution loads of creep and shrinkage themselves,
creep equivalent nodal loads at the next time step can be simply obtained
by Equation 3.64, and the displacement changes at the next time step can
be solved from Equation 3.61. Iterating this process through the entire
observation history (from the first loading time to a future time) with a
small time step, displacements and internal forces due to creep and shrink-
age at any time can be analyzed. When applying this method to bridge
analysis, causes of stress changes at any time, as shown in Figure 3.12,
include different types of external loads such as construction loads, struc-
tural weights, stage changes, prestressing, and redistribution of creep and
shrinkage themselves.
3.3.3 Automatic-determining time step
Considering the behavior of concrete creep and shrinkage, these effects
may need to be analyzed at five years or even 50 years after the structure
is built (Bazant et al. 2011). The small time step used in the previous
iteration should be determined based on the performance and accuracy.
As all creep theories assert that the creep development will decrease
gradually and cease eventually, the time step can be increased from a
smaller one at an earlier age to a large time span at a more matured
age. This adjustment can be done automatically by detecting a small
displacement change. With today's advancement of modern computers,
when a bridge is modeled as a spatial frame, performance degraded due
to short time steps, such as a week or even shorter time, would not be a
consideration.
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