Civil Engineering Reference
In-Depth Information
2
nt E
b
ε
+ρ
f
f
fe
(7.18)
f
=
k
kf
flye
e
eyyh
2
nt E
h
ε
f
f
fe
(7.19)
f
=
k
lxf
e
2
nt E
b
ε
f
f
fe
(7.2 0)
f
=
k
lyf
e
n
(
)(
)
2
()
w
bd
1
−∑
1
−
s
b
1
−
s
d
i
cc
1
6
2
2
A
A
c
c
i
=
e
c
where
k
=
from Equation (7.11),
k
e
=
for rectan-
(
)
e
1
−ρ
cc
,
2
2
for rectangular tie transverse steel,
s
is
the c/c spacing of ties,
h
c
and
b
c
are the height and width of concrete core c/c, and ρ
cc
is the ratio of longitudinal reinforcement area to the area of core (Mander et al. 1988).
The values of the ultimate confined strength of the core and cover (
f
cce
, and
f
ccf
)
are determined based on the 3-D state of stress of concrete plasticity proposed by
Willam and Warnke (1975) and explained below by substituting
f
flxe
and
f
flye
as lateral
core pressures and (
f
lxf
and
f
lyf
) as lateral cover pressures, respectively. Accordingly,
the ultimate confined concrete axial load replacing that of Equation (7.2) is
A
hs
A
bs
gular tied transverse steel, ρ= ρ=
×
tie
tie
x
y
×
c
c
(
)
φ= φ
P
0.8
0.85
f
AA
−
+
0.85
f
(
AAAf
−
)
+
(7.21)
n
cce
c
st
ccf
g
c
st
y
In selecting whether to use Lam and Teng's ascending model or Mander's
descending model,
f
lf
of the equivalent circular column described in Section 7.2.2
and substituted into Equation (7.7) will be computed. If
ff
lf
/ is greater than 0.08,
Lam and Teng's ascending model described in Section 7.2.1 is used, with the
proper
f
cce
,
and
f
ccf
for the core and the cover determined using the 3-D state-of-
stress concrete plasticity model (Willam and Warnke 1975) explained in Section
7.2.5. In this case, the ultimate compressive strain (ε
ccu
) is assumed to be different
in the core and cover, determined by substituting
f
le
and
f
lf
for
f
l
in Equation (7.8),
respectively. Accordingly, the slope (
E
2
e
) of the core will be higher than the slope
of the cover (
E
2
f
).
On the other hand, when
ff
lf
/ is less than 0.08, Mander's model (Mander et al.
1988) is used with the proper (
f
cce
,
and
f
ccf
) for the core and the cover determined
using the 3-D state-of-stress concrete plasticity model (Willam and Warnke 1975)
explained in section 7.25. The ultimate compressive strains ε
ccue
and ε
ccuf
for
the core
and the cover are selected. ε
ccuf
is taken from Equations (7.8) and (7.9) using
f
if
for
the case of the cover and ε
ccue
is taken
from the energy approach corresponding to the
fracture of the first hoop (Mander et al. 1988) for the case of the core. The peak of the
curve also takes place at different strains ε
cce
, and ε
ccf
corresponding to
f
cce
, and
f
ccf
,
with different strength values for the core and the covers. This procedure has been
programmed into the software “KDOT Column Expert” developed by the author and
coworkers and described by Al-Rahmani and Rasheed (2014).
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