Civil Engineering Reference
In-Depth Information
assumed to be zero for the whole post-cracking range as
ν
21
=
0
(6.34)
It is interesting to note that twelve full-size RC 2-D elements (panels) have been tested to
establish the two Hsu/Zhu ratios. These panels include four variables: the percentage of steel
(
0
◦
and 45
◦
), the ratio of steel percentages
ρ
=
0.77-3.04%), the steel bar orientation (
α
1
=
in the transverse and longitudinal direction (
0.24-1.0), and the strength of concrete (50
and 90 MPa). Test results show that the Hsu/Zhu ratios are not a function of any of these four
variables within the usable ranges.
η
=
6.1.4.3 Effect of Hsu/Zhu Ratios on Post-peak Behavior
Now let us demonstrate the important effect of Hsu/Zhu ratios on the post-peak behavior of
RC 2-D elements by examining the equilibrium equation (6.1). In order to simplify Equation
(6.1) we take the following three measures. (a) Under pure shear loading, the applied stresses,
σ
, on the left-hand side of Equation (6.1) should be zero. (b) For a specimen with the same
steel ratios in both the longitudinal and the transverse directions, the smeared shear stress
of concrete
12
is zero, because the deviation angle
τ
β
=
0. (c) The smeared tensile stress of
1
can be neglected, because its magnitude is very small when compared with the
smeared compressive stress of concrete (
concrete
σ
2
) and the smeared steel stresses (
σ
ρ
f
and
ρ
t
f
t
).
12
1
σ
=
τ
=
σ
=
Setting
0,
0, and
0, Equation (6.1) is simplified to:
c
2
sin
2
0
=
σ
α
1
+
ρ
f
(6.35)
ρ
f
in Equation (6.35) is treated as a uniaxial steel stress, Equation (6.35) represents the
basic 'truss-concept' of the internal balance between the compressive stress of concrete struts
σ
If
c
2
and the tensile stresses of steel bars
ρ
f
. In the ascending branch of the load-deformation
2
and the steel stress
curves, both the concrete stress
ρ
f
increase, and the internal equi-
librium is maintained. The peak point of a load-deformation curve represents physically the
concrete compressive stress
σ
2
reaching its peak. Beyond the peak point, the concrete stress
σ
2
begins to decrease, while the steel stresses
σ
ρ
f
continue to increase. As a result, the
equilibrium equation (6.35) cannot be satisfied and the computer operation comes to a halt at
the peak point.
If
ρ
f
in Equation (6.35) is treated as a biaxial steel stress subjected to Poisson effect,
however,
ρ
f
in Equation (6.35) should be replaced by the expression in Equation (6.24)
to give:
ν
12
¯
α
1
+
ρ
ν
21
¯
α
1
E
s
E
s
E
s
c
2
sin
2
ε
2
cos
2
ε
1
sin
2
0
=
σ
α
1
+
ρ
ε
+
ρ
(6.36)
Equation (6.36) includes two additional terms with Hsu/Zhu ratios. The last term is zero
because
ν
21
=
0. The remaining term with Hsu/Zhu ratio
ν
12
is caused by compressive strain
¯
ε
2
and should be negative. This term, which is induced by Poisson effect, reduces the uniaxial
steel stress,
E
s
ρ
ε
, so that equilibrium can be restored in the post-peak range.
