Civil Engineering Reference
In-Depth Information
strain components,
ε
1
,
ε
2
and
γ
12
, and the stationary strain components,
ε
,
ε
t
and
γ
t
,isthe
strain transformation.
In Figure 4.7(c) a positive unit length on the
-axis will have projections of cos
α
1
and
−
sin
α
1
on the 1- and 2-axis, respectively. A positive unit length on the
t
-axis, however, should
give projections of sin
α
1
and cos
α
1
on the 1- and 2-axis, respectively. Hence, the rotation
matrix [R] is
cos
α
1
sin
α
1
[R]
=
(4.31)
−
sin
α
1
cos
α
1
Notice that the rotation matrix [R] for strains in Equation (4.31) is identical to matrix [R]
for stresses in Equation (4.1). The relationship between the strains in the 1-2 coordinate [
ε
12
]
−
ε
t
]is
and the strain in the
t
coordinate [
ε
t
][R]
T
[
ε
12
]
=
[R][
(4.32)
or
⎡
⎣
ε
1
⎤
⎡
⎣
ε
⎤
γ
12
2
γ
t
2
cos
cos
α
1
sin
α
1
α
1
−
sin
α
1
⎦
=
⎦
(4.33)
γ
21
2
γ
t
2
−
sin
α
1
cos
α
1
sin
α
1
cos
α
1
ε
2
ε
t
Performing the matrix multiplications and noticing that
γ
t
=
γ
t
and
γ
12
=
γ
21
results in
the following three equations:
α
1
+
γ
t
2
cos
2
α
1
+
ε
t
sin
2
ε
1
=
ε
(2 sin
α
1
cos
α
1
)
(4.34)
α
1
−
γ
t
2
ε
2
=
ε
sin
2
α
1
+
ε
t
cos
2
(2 sin
α
1
cos
α
1
)
(4.35)
cos
2
α
1
γ
12
2
=
α
1
+
γ
t
2
sin
2
(
−
ε
+
ε
t
)sin
α
1
cos
α
1
−
(4.36)
Equations (4.34)-(4.36) can be expressed in the matrix form by one equation:
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎤
ε
1
ε
2
γ
12
2
ε
ε
t
γ
t
2
sin
2
cos
2
α
1
α
1
2sin
α
1
cos
α
1
⎣
⎦
sin
2
cos
2
α
1
α
1
−
2sin
α
1
cos
α
1
(4.37)
(cos
2
sin
2
−
α
1
cos
α
1
α
1
cos
α
1
α
1
−
α
1
)
sin
sin
3 matrix in Eq. (4.37) is the
transformation matrix
[
T
] for transforming the strain
in the stationary
This 3
×
t
coordinate to the strains in the rotating 1-2 coordinate. Using the tensor
notation, Equation (4.37) becomes:
−
ε
12
]
=
[
T
][
ε
t
]
(4.38)
[
It is interesting to note that the transformation matrix [
T
] for strain is the same as that for
stress.
4.2.2 Geometric Relationships
Equations (4.34)-(4.36) can be illustrated strictly by geometry in Figure 4.8. Figure 4.8(a)
gives the geometric relationships between the three strain components
ε
1
,
ε
2
and
γ
12
/2, in
