Civil Engineering Reference
In-Depth Information
6.1.4.4 Strain Conversion Matrices
To solve the equilibrium and compatibility equations, (6.1)-(6.6), the constitutive relationships
of concrete and steel have to be introduced. As pointed out previously, the compatibility
equations (6.4)-(6.6) are expressed in terms of biaxial strains (
ε
1
,
ε
2
,
ε
,
ε
t
). These biaxial
2
,
f
,
f
t
) in the equilibrium equations
(6.1)-(6.3) by experimental constitutive laws, because the Poisson effect could not be ignored
in the tests. Therefore, the biaxial strains (
1
,
strains could not be related directly to the stresses (
σ
σ
ε
1
,
ε
2
,
ε
,
ε
t
) must first be converted to the uniaxial
strains (¯
ε
1
,¯
ε
2
,¯
ε
,¯
ε
t
).
Uniaxial strains (
¯
ε
2
) of smeared concrete
The uniaxial strains in the 1-2 coordinate (¯
ε
1
,
¯
ε
1
,¯
ε
2
) of smeared concrete can be calculated from
the biaxial strains in the 1-2 coordinate (
ε
2
) using Equations (6.13) and (6.14). These strain
conversion relationships can be generalized into in a 3
ε
1
,
×
3 matrix equation as follows:
⎡
⎤
1
ν
12
⎡
⎤
⎡
⎤
0
ε
1
¯
¯
ε
1
ε
2
γ
12
2
⎣
1
−
ν
12
ν
21
1
−
ν
12
ν
21
⎦
⎣
⎦
=
⎣
⎦
ε
2
γ
12
2
ν
21
1
(6.37)
0
1
−
ν
12
ν
21
1
−
ν
12
ν
21
0
0
1
The 3
×
3 matrix in Equation (6.37) is called the strain conversion matrix [
V
] for smeared
concrete.
In Equations (6.37), or (6.13) and (6.14), the two Hsu/Zhu ratios
ν
12
and
ν
21
have been
determined experimentally in Section 6.1.4.2. When
ν
21
=
0 after cracking, Equation (6.37)
can be reduced to:
¯
ε
1
=
ε
1
+
ν
12
ε
2
(6.38)
ε
2
=
ε
2
¯
(6.39)
It is interesting to observe in Equation (6.39) that the uniaxial strain ¯
ε
2
is identical to
the biaxial strain
ε
2
in principal compression. In Equation (6.38), however, the term
ν
12
ε
2
represents the Poisson effect. Since
ε
2
is always negative and
ν
12
=
1.9 after the yielding of
the steel, the uniaxial strain ¯
ε
1
is equal to the biaxial strain
ε
1
in principal tension minus 1.9
times the biaxial strain of principal compression
ε
2
.This
ν
12
ε
2
term can be quite large in the
post-peak range of the load-deformation curves.
The uniaxial strains (¯
ε
2
) obtained using the conversion matrix [
V
] in Equation (6.37),
or from Equations (6.13) and (6.14), can be used to calculate the stresses in the concrete (
ε
1
and ¯
1
σ
2
) in the equilibrium equations (6.1)-(6.3). The constitutive relationships between the
stresses (
and
σ
1
,
2
) and the uniaxial strains (¯
σ
σ
ε
1
,¯
ε
2
) are obtained directly from tests, and are given
in the following Sections 6.1.6-6.1.8.
Uniaxial strains (
¯
ε
t
) of smeared steel bars
The uniaxial strains in the
ε
and
¯
ε
t
) of the smeared steel bars can be
calculated from the uniaxial strains in the 1-2 coordinate (¯
−
t
coordinate (¯
ε
and ¯
ε
2
) using the transformation
equations (6.20) and (6.21). These strain transformation relationships can be generalized into
ε
1
,¯
