Civil Engineering Reference
In-Depth Information
Equations (6.22) and (6.23) show that the biaxial steel strains (
ε
,
ε
t
) are the uniaxial steel
strains (¯
ε
t
) minus two strain terms. These two additional steel strains are the components
of the products of Hsu/Zhu ratios and concrete strains (
ε
,¯
ν
12
¯
ε
2
and
ν
21
¯
ε
1
), and represent Poisson
effect on the stresses and strains of smeared steel bars.
Substituting ¯
f
/
E
s
f
t
/
E
t
into Equations (6.22) and (6.23) results in:
ε
=
and ¯
ε
t
=
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦ =
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦ +
E
s
E
s
ρ
ν
12
¯
ε
2
cos
2
α
1
)
+
ρ
ν
21
¯
ε
1
sin
2
α
1
)
E
s
(
(
ρ
f
ρ
t
f
t
0
ρ
ε
ε
t
0
0
0
E
t
E
t
(
E
t
(
0
ρ
t
0
ρ
t
ν
12
¯
ε
2
sin
2
α
1
)
+
ρ
t
ν
21
¯
ε
1
cos
2
α
1
)
0
0
0
0
(6.24)
where:
where
f
,
f
t
=
smeared steel stress of steel bars embedded in concrete in the
- and
t
-directions,
respectively. The symbols
f
and
f
t
represent the
biaxial
steel stresses, as well as
the
uniaxial
steel stresses;
E
s
,
E
t
=
secant modulus of steel bars embedded in concrete in the
and
t
directions,
respectively, calculated from the constitutive law of embedded steel bars (smeared
stress versus smeared strain) obtained under uniaxial loading.
Equation (6.24) shows that, under biaxial stress conditions, the smeared steel stresses in
the longitudinal direction
ρ
t
f
t
, are each made up of two
parts. The first part is the uniaxial steel stresses using the uniaxial steel moduli without taking
into account the Hsu/Zhu ratios. The second part is the steel stresses produced by the two
Hsu/Zhu ratios and the two uniaxial strains. If the Hsu/Zhu ratios are assumed to be zero,
then the second part will disappear and Equation (6.24) is reduced to the familiar expression
of smeared steel stress under uniaxial condition (note that
ρ
f
and in the transverse direction
ε
t
).
Comparison of Equations (6.24) and (6.8) gives the relationships between the biaxial steel
moduli (
E
s
ε
becomes ¯
ε
and
ε
t
becomes ¯
,
E
t
) and the uniaxial steel moduli (
E
s
,
E
t
)as:
1
α
1
E
s
E
s
ε
2
/ε
) cos
2
ε
1
/ε
)sin
2
=
+
ν
12
(¯
α
1
+
ν
21
(¯
(6.25)
E
t
1
α
1
E
t
ε
2
/ε
t
)sin
2
ε
1
/ε
t
) cos
2
=
+
ν
12
(¯
α
1
+
ν
21
(¯
(6.26)
Equations (6.25) and (6.26) state that the biaxial moduli of steel is the uniaxial moduli of
steel plus two additional term caused by Poisson effect. This concept is applicable in both
directions of longitudinal steel and transverse steel.
6.1.4 Hsu/Zhu Ratios
ν
12
and
ν
21
6.1.4.1 Test Methods
Reinforced concrete 2-D elements subjected to shear behave very differently before and after
cracking. The Poisson effect before cracking is characterized by the well-known Poisson ratio,
but after cracking is characterized by two Hsu/Zhu ratios.
Twelve panels as shown in Figure 6.2(a) and (b) were tested to establish the formulas for
Hsu/Zhu ratios
ν
12
and
ν
21
. In the 1-2 coordinate system, the panels were subjected to a
horizontal tensile stress
σ
1
and a vertical compressive stress
σ
2
in a proportional manner. The
