Biomedical Engineering Reference
In-Depth Information
s
x
s
y
s
xy
s
1
s
2
s
12
Q
26
¼ðQ
11
Q
12
2
Q
66
Þ
sin
3
F
cos
F
ðQ
22
Q
12
2
Q
66
Þ
sin
F
cos
3
F
¼½T
1
:
(3.2.12.16)
(3.2.12.26)
where [
T
-1
]is
Q
16
¼ Q
26
¼
0 for a laminated symmetric composite.
The transformation ma
tri
x[
T
1
] and the transformed
reduced stiffness matrix
½Q
ij
are very important matri-
ces in the macromechanical analysis of bothe laminae and
laminates. These matrices play a key role in detemining
the effective in-plane and bending properties and how
a laminate will perform when subjected to different
combinations of forces and moments.
cos
2
F
sin
2
F
2sin
F
cos
F
sin
2
F
cos
2
F
2sin
F
cos
F
sin
F
cos
F
sin
F
cos
F
cos
2
F
sin
2
F
½T
1
¼
(3.2.12.17)
The
x
and 1 axes form angle V. This matrix is also valid
for the transformation of strains,
Macromechanics of a laminate
The development of the
A, B,
and
D
matrices for lami-
nate analysis is important for evaluating the forces and
moments to which the laminate will be exposed and in
determining the stresses and strains of the laminae. As
given in Eq. (3.2.12.19),
3
x
3
y
3
1
3
2
¼½T
1
:
(3.2.12.18)
1
1
2
g
xy
2
g
12
Finally, it can be demonstrated that
s
x
s
y
s
xy
3
x
3
y
g
xy
ð
s
k
Þ¼½Q
ij
ð
3
k
Þ
(3.2.12.27)
¼½Q
ij
:
(3.2.12.19)
where s
¼
normal stresses, 3
¼
normal strains, and [
Q
ij
]
¼
stiffness matrix. The
A, B,
and
D
matrices are equivalent
to the following:
where
½Q
ij
is the transformed reduced stiffness. The
transformed reduced stiffness matrix is
2
4
3
5
½A
ij
¼
X
n
Q
11
Q
12
Q
16
Q
21
Q
22
Q
26
Q
16
Q
26
Q
66
ðQ
ij
Þ
k
ðh
k
h
k
1
Þ
(3.2.12.28)
½Q
ij
¼
(3.2.12.20)
k¼
1
2
X
n
½B
ij
¼
1
ðQ
ij
Þ
k
ðh
k
h
k
1
Þ
where,
(3.2.12.29)
k¼
1
Q
11
¼ Q
11
cos
4
F
þ Q
22
sin
4
F
þ
2
ðQ
12
þ
2
Q
66
Þ
sin
2
F
cos
2
F
3
X
n
½D
ij
¼
1
(3.2.12.21)
ðQ
ij
Þ
k
ðh
k
h
k
1
Þ
(3.2.12.30)
k¼
1
Q
22
¼ Q
11
sin
4
F þ Q
22
cos
4
F
þ
2
ðQ
12
þ
2
Q
66
Þ
sin
2
F
cos
2
F
The matrix [
A
] is called the
extensional stiffness
matrix
because it relates the resultant forces to the
midplane strains, while matrix [
D
] is called the
bending
stiffness matrix
because it relates the resultant moments
to the laminate curvature. The so called
coupling stiffness
matrix,
[
B
], accounts for coupling between bending and
extension, which means that normal and shear forces
acting at the laminate midplane are causing laminate
curvature or that bending and twisting moments are ac-
companied by midplane strain.
The letter
k
denotes the number of laminae in the
laminate with a maximum number (
N
). The letter
h
represents the distances from the neutral axis to the
edges of the respective laminae. A standard procedure
for numbering laminae is used where the 0 lamina is at
the bottom of a plate and the
K
th lamina is at the top.
(3.2.12.22)
Q
12
¼ðQ
11
þ Q
22
4
Q
66
Þ
sin
2
F
cos
2
F
þ Q
12
ð
sin
4
F
cos
4
FÞ
(3.2.12.23)
Q
66
¼ðQ
11
þ Q
22
2
Q
12
2
Q
66
Þ
sin
2
F
cos
2
F
þ Q
66
ð
sin
4
F
cos
4
FÞ
(3.2.12.24)
Q
16
¼ðQ
11
þ Q
12
2
Q
66
Þ
sin
F
cos
3
F
ðQ
22
Q
12
2
Q
66
Þ
sin
3
F
cos
F
(3.2.12.25)
