Civil Engineering Reference
In-Depth Information
Figure 6.3
Biaxial strains versus uniaxial strains
2
/
E
2
in Equations (6.9) and (6.10) as the uniaxial strains;
these two equations can now be written as
1
/
E
1
and ¯
c
c
Let's define ¯
ε
1
=
σ
ε
2
=
σ
ε
1
=
ε
1
−
ν
12
¯
ε
2
¯
(6.11)
ε
2
=
ε
2
−
ν
21
¯
ε
1
¯
(6.12)
where:
where
ε
1
,
ε
2
=
smeared (average) strains in the 1- and 2-directions, respectively, when a panel is
subjected to
biaxial
loading and taking into account the Hsu/Zhu ratios;
¯
ε
1
,¯
ε
2
=
smeared (average) strains 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.
When the Hsu/Zhu ratios are assumed to be zero in Equations (6.11) and (6.12), the biaxial
strains are the same as the unixial strains (
ε
1
=
ε
1
and
¯
ε
2
=
¯
ε
2
). This condition is illustrated
c
1
c
2
in Figure 6.3(b), where the stresses
σ
and
σ
are simply related to the uniaxial strains ¯
ε
1
and
ε
2
by the uniaxial moduli
E
1
and
E
2
, respectively.
¯
6.1.3.2 Constitutive Matrices in Terms of Hsu/Zhu Ratios
Constitutive matrix of smeared concrete
Solving Equations (6.11) and (6.12) gives
1
ν
12
ε
1
=
¯
−
ν
12
ν
21
ε
1
+
−
ν
12
ν
21
ε
2
(6.13)
1
1
ν
21
1
ε
2
=
¯
−
ν
12
ν
21
ε
1
+
−
ν
12
ν
21
ε
2
(6.14)
1
1
