Civil Engineering Reference
In-Depth Information
ε
2
=
¯
uniaxial smeared (average) principal strain in the 2
−
direction
ε
ci
=
¯
initial strain in concrete
¯
ε
cx
=
extra strain in concrete after decompression
ε
c
=
maximum compression strain normal to the compression direction under considera-
tion (always negative)
ε
cr
=
concrete cracking strain
concrete cylinder strain corresponding to peak compressive stress
f
c
ε
0
=
ε
1
=
biaxial smeared (average) principal strain in the 1
−
direction
ε
2
=
biaxial smeared (average) principal strain in the 2
−
direction
γ
12
=
−
smeared (average) shear strain in the 1
2 coordinate
decompression modulus of concrete, given as
kf
c
ε
0
E
c
=
k
=
1.4-1.5
E
c
=
modulus of concrete in tension before cracking
f
cr
=
cracking tensile strength of concrete
ε
c
/ε
o
)
D
=
damage coefficient
=
1
−
0
.
4(
≤
1
.
0
f
c
=
cylinder compressive strength of concrete
ζ
=
softening coefficient
The constant, 400 in Equation (9.25), is changed to 250 when sequential loading is applied.
The unloading and reloading paths defined in the concrete constitutive model in CSMM for
reinforced concrete were simplified in ConcreteL01 and are illustrated in Figure 9.7. The
unloading and reloading paths from the ascending branch of the compressive envelope are
simplified as one straight line and the slope is taken as the initial modulus of concrete
E
c
0
, where
kf
c
/ε
0
E
c
0
=
(9.29)
The slope of the unloading and reloading paths from the descending branch of the com-
pressive envelope are simplified as one straight line with slope of 0
8
E
c
0
. The reloading paths
from tension to compression reflect the full closing of the concrete cracks before reaching the
peak point in the compressive envelope and the partial closing of the concrete cracks after
exceeding the peak point.
.
9.3 1-D Fiber Model for Frames
The fiber element approach, originally developed by Professor Filippou and his co-workers
(Taucer
et al
., 1991) has been recognized as one of the most promising methods for static
and dynamic analysis of RC frame structures since the late 1980s. By using the fiber element
approach, the hysteretic behavior of RC members can be captured directly from cyclic stress
and strain relationships of the materials; the axial force-biaxial bending interaction can be
rationally accounted for. Also, it has been found that the state-of-the-art force-based approach
generally has superior robustness and requires fewer degrees of freedom for comparable
accuracy when dealing with strength softening elements such as RC members. Hence the
force-based fiber element approach is utilized in the model.
A schematic figure from Taucer
et al
. (1991) to conceptually illustrate the configuration of
a fiber beam-column element is replicated in Figure 9.8. Each of the control cross-sections
is subdivided into many fibers or filaments; some are concrete fibers while the others may be
steel fibers. There are a number of control cross-sections along the element. These sections
are located at the control points of the numerical integration scheme. The characteristics of the
