Civil Engineering Reference
In-Depth Information
found experimentally that the softening coefficient was not only a function of the perpendicular
tensile strain ¯
ε
1
, but also a function of the concrete compressive strength
f
c
. The softening
coefficient was then improved to become a function of both ¯
ε
1
and
f
c
(see Section 6.1.7.2).
ε
1
and
f
c
, Chintrakarn (2001) and Wang (2006) showed that
the softening coefficient is also a function of the deviation angle
In addition to the variables ¯
(see Section 6.1.7.3)
when the fixed angle shear theory is used (including SMM). A new function
β
f
(
β
) was
established to relate the softening of concrete struts to the deviation angle (
β
) and, in
ρ
ρ
t
). By treating the softening co-
efficient (see Section 6.1.7) as a function of all three variables (
turn, to the ratio of longitudinal to transverse steel (
f
c
and
), the fixed
angle shear theory (including SMM) becomes very powerful, applicable to 2-D elements
with any
ε
1
,
β
ρ
ρ
t
ratio from unity to infinity, and any concrete strength up to 100 MPa (see
Section 6.1.7.3).
Using the strain-control feature of UPT, Zhu and Hsu (2002) quantified the Poisson effect
of RC 2-D elements (see section 6.1.3) and characterized this property by two Hsu/Zhu ratios
(see Section 6.1.4). Taking into account the Poisson effect, Hsu and Zhu (2002) developed
the softened membrane model (SMM). SMM can satisfactorily predict the entire monotonic
response of the RC 2-D elements, including both the ascending and the descending branches,
as well as both the pre-cracking and post-cracking responses.
The servo-control system of UPT also allows the researchers at UH to conduct reversed
cyclic shear tests of RC 2-D elements (Mansour and Hsu, 2005a,b). The constitutive relation-
ships of concrete and steel under unloading and reloading were established from these tests,
and a cyclic softened membrane model (CSMM) was developed. This model allows us to
predict the shear stiffness, the shape of the hysteresis loop, the shear ductility, and the energy
dissipation capacity of a 2-D element. CSMM will be studied in Section 6.3.
6.1.3 Poisson Effect in Reinforced Concrete
6.1.3.1 Biaxial Strains versus Uniaxial Strains
c
2
as shown in Figure 6.3(a). The side lengths of the 2-D element are taken as unity. When the
Hsu/Zhu ratios (
c
1
A 2-D concrete element defined in the 1-2 coordinate is subjected to two stresses
σ
and
σ
ν
12
and
ν
21
) are considered, the strains
ε
1
and
ε
2
are expressed as follows:
1
2
E
2
σ
E
1
−
ν
12
σ
ε
1
=
(6.9)
2
1
E
1
σ
E
2
−
ν
21
σ
ε
2
=
(6.10)
where
where:
ν
12
=
ratio of the resulting strain increment in the 1-direction to the source strain
increment in the 2-direction;
ν
21
=
ratio of the resulting strain increment in the 2-direction to the source strain
increment in the 1-direction;
E
1
,
E
2
=
moduli of concrete in the 1- and 2-directions, respectively, when a panel is
subjected to uniaxial loading; or subjected to biaxial loading, but assuming the
Hsu/Zhu ratios to be zero.
