Civil Engineering Reference
In-Depth Information
Figure 9.9
Coordinate systems for reinforced concrete elements: (a) applied principal stresses in local
coordinate (b) reinforcement component in local coordinate
The stress and strain vectors in
x-y
coordinates and 1-2 coordinates are denoted as
σ
x
σ
y
τ
xy
T
,
ε
x
2
γ
xy
T
,
σ
1
σ
2
τ
12
T
and
ε
1
ε
2
2
γ
12
T
, respectively. Here
1
1
ε
y
τ
12
=
0 because the 1-2 coordinate represents the principal stress directions.
By using the transformation matrix [
T
(
α
)], the stresses and strains can be transformed
between different coordinates. [
T
(
α
)] is given by
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
.
c
2
s
2
2
sc
s
2
c
2
[
T
(
α
)]
=
−
2
sc
(9.30)
c
2
s
2
−
sc
sc
−
where
c
=
cos (
α
) and
s
=
sin (
α
), and the angle
α
is the angle between the two coordinates.
For example, if the angle
α
is from the
x-y
coordinate to the 1-2 coordinate the angle
α
will
be written as
α
1
x
. If the angle
α
is from the 1-2 coordinate to the
x-y
coordinate the angle
α
will be written as
α
x
1
=−
α
1
x
.
The stresses and strains transformed from the
x-y
coordinate to the 1-2 coordinate using
the transformation matrix are expressed as follows:
⎧
⎨
α
x
1
. Of course,
⎫
⎬
⎭
=
⎧
⎨
⎫
⎬
σ
1
σ
2
0
σ
x
σ
y
τ
xy
[
T
(
α
1
x
)]
(9.31)
⎩
⎩
⎭
⎧
⎨
⎫
⎬
⎭
=
⎧
⎨
⎫
⎬
ε
1
ε
2
ε
x
ε
y
[
T
(
α
1
x
)]
(9.32)
⎩
⎩
⎭
1
1
2
γ
12
2
γ
xy
9.4.2 Implementation
For a 2-D membrane element, a 'material constitutive matrix' is needed to relate the state
of stresses and strains. The material constitutive matrix (which may also be known as the
'material stiffness matrix') can be expressed in terms of secant or tangent formulations.
The secant material constitutive matrix relates the absolute values of strains and stresses
of the element, while the tangent material constitutive matrix defines the relationship between
the increment of the stresses and strains of the element. In this section, the material constitu-
tive matrix for the reinforced concrete membrane material using CSMM is derived in terms
