Civil Engineering Reference
In-Depth Information
Notice that the three compatibility equations for fixed angle theory contain the terms with
concrete shear strain,
γ
12
. In all the fixed angle models, it is necessary to have a constitutive
relationship that relates
12
in the equilibrium equations.
γ
12
in the compatibility equations to
τ
Constitutive relationships
The solution of the three stress equilibrium equations and the three strain compatibility equa-
tions requires two constitutive matrices, one for concrete and one for steel. Assuming reinforced
concrete to be a
continuous orthotropic
material, the concrete constitutive matrix is:
⎡
⎣
⎡
⎣
⎤
⎦
⎤
⎦
=
⎡
⎣
⎤
⎦
=
ε
1
ε
2
γ
12
2
1
E
1
ν
12
E
1
σ
0
2
ν
21
E
2
E
2
σ
0
(6.7)
12
G
12
τ
0
0
3 matrix in Equation (6.7) contains three diagonal elements
E
1
,
E
2
and
G
12
.
Element
E
1
is a nonlinear modulus, representing the tensile stress-strain curve of concrete
(
The 3
×
1
−
ε
1
curve). Modulus
E
2
represents the compressive stress-strain curve of concrete
σ
2
−
ε
2
curve); and
G
12
is the shear stress-strain curve of concrete (
τ
12
−
γ
12
/
(
2 curve).
The first two diagonal elements
E
1
and
E
2
in the matrix represent curves obtained from
experiments, and the third diagonal element
G
12
is the shear modulus in the 1-2 coordinate.
In the past,
G
12
was thought to be an independent material property, which must also be
established from experiments. The resulting experimental expressions for
G
12
were often very
complicated and made the analytical method very difficult to apply. One great advantage of
using the smeared crack concept is that the cracked concrete can be treated as a
continuous
ma-
terial, which requires only two independent moduli, rather than three. A theoretical derivation
of
G
12
as a function of the two independent moduli
E
1
and
E
2
will be derived in Section 6.1.10.
The two off-diagonal elements
σ
ν
21
E
2
in Equation (6.7), represent the Poisson
effect, i.e. the mutual effect of normal strains in the 1-2 coordinate. The symbol
ν
12
E
1
and
ν
12
is the ratio
of the resulting strain increment in 1-direction to the source strain increment in 2-direction,
and the symbol
ν
21
is the ratio of the resulting strain increment in 2-direction to the source
strain increment in 1-direction.
The ratios
ν
21
are the well-known Poisson ratios for continuous isotropic materials,
and vary from zero to 0.5. In a linear, isotropic unit cube subjected to triaxial compressive
stresses, the volume will expand if the Poisson ratio is larger than 0.5 (Boresi,
et al
., 1993). For
cracked reinforced concrete, however, Equation (6.7) is assumed to be a constitutive matrix
for
continuous orthotropic
materials, so that the smeared (or average) behavior of this cracked
composite can be evaluated by continuum mechanics. In this case,
ν
12
and
ν
12
and
ν
21
are the Hsu/Zhu
ratios, and
ν
12
is allowed to exceed 0.5 because of the smeared cracks.
The steel constitutive matrix is:
⎡
⎤
⎡
⎤
⎡
⎤
ρ
E
s
ρ
f
ρ
t
f
t
0
0
0
ε
ε
t
0
⎣
⎦
=
⎣
⎦
⎣
⎦
ρ
t
E
t
0
0
(6.8)
0
0
0
ρ
E
s
ρ
t
E
t
, are moduli for steel bars in the
The two diagonal elements,
and
- and
t
-
and
E
t
represent the smeared
stress-strain curves of mild steel bars embedded in concrete (
f
−
ε
curve or
f
t
−
ε
t
curve).
ρ
t
are the steel ratios.
E
s
directions, respectively.
ρ
and
