Civil Engineering Reference
In-Depth Information
Figure 4.12
Reserved transformation
sin
α
1
in Equation (4.7) results in:
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎤
⎦
c
1
σ
σ
cos
2
α
1
sin
2
α
1
−
α
1
cos
α
1
2sin
⎣
c
2
σ
t
sin
2
α
1
cos
2
α
1
2sin
α
1
cos
α
1
σ
(4.49)
sin
α
1
cos
α
1
−
sin
α
1
cos
α
1
(cos
2
α
1
−
sin
2
α
1
)
c
t
c
12
τ
τ
3 matrix in Equation (4.49) is the
inverse transformation matrix
[
T
]
−
1
This 3
×
for trans-
forming a set of stresses in the 1-2 coordinate to a set of stresses in the
−
t
coordinate. Using
the tensor notation, Equation (4.49) becomes:
σ
12
(4.50)
Substituting Equation (4.49) into Equations (4.46)-(4.48) gives the three equilibrium equa-
tions for fixed angle theories as:
σ
=
σ
t
=
[
T
]
−
1
σ
c
1
cos
2
c
2
sin
2
c
c
12
2sin
α
1
+
σ
α
1
−
τ
α
1
cos
α
1
+
ρ
f
(4.51)
c
1
sin
2
c
2
cos
2
c
12
2sin
σ
t
=
σ
α
1
+
σ
α
1
+
τ
α
1
cos
α
1
+
ρ
t
f
t
(4.52)
c
1
c
12
(cos
2
c
sin
2
τ
t
=
(
σ
−
σ
2
)sin
α
1
cos
α
1
+
τ
α
1
−
α
1
)
(4.53)
Similarly, in Section 4.2.1, Equation (4.37), we derived the same matrix [
T
] that transforms
a set of strains in the
−
t
coordinate (
ε
,
ε
t
and
γ
t
) into a set of strains in the 1-2 coordinate
(
γ
12
). Using the same process as in stress transformation, we can reverse the process,
as shown in Figure 4.12, and use the reverse transformation matrix [
T
]
−
1
to transform a set of
strains in the 1-2 coordinate (
ε
1
,
ε
2
and
ε
1
,
ε
2
and
γ
12
) into a set of strains in the
−
t
coordinate (
ε
,
ε
t
and
γ
t
) as follows:
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎤
ε
ε
t
γ
t
2
ε
1
ε
2
γ
12
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
(4.54)
(cos
2
sin
2
α
1
cos
α
1
−
α
1
cos
α
1
α
1
−
α
1
)
sin
sin
Consequently, the three compatibility equations for fixed-angle theories are:
α
1
−
γ
12
2
ε
=
ε
1
cos
2
α
1
+
ε
2
sin
2
2sin
α
1
cos
α
1
(4.55)
α
1
+
γ
12
2
=
ε
1
sin
2
α
1
+
ε
2
cos
2
ε
t
2sin
α
1
cos
α
1
(4.56)
γ
t
2
=
α
1
+
γ
12
2
(cos
2
sin
2
(
ε
1
−
ε
2
)sin
α
1
cos
α
1
−
α
1
)
(4.57)
