Aerospace Structural Composites I
Homework 7
Anirban Chaudhuri
1.
Find 16-ply laminates that have exactly or close to the following bending lamination
parameters (A) W
1
*
=0, W
3
*
=0.5; (b) W
1
*
=-0.5, W
3
*
=0.5. For both cases use first laminates with
any angles and then limit yourself to laminates made from 0, 90 and ±45-degree plies. The
easiest way to solve this problem is to use Excel solver. Assume that you have four 2-ply groups,
and have the angles as design variables, and the objective function the distance from the target
value of the lamination parameters. When the angles are limited to zero, 90 and ±45-degree plies,
you can have integer design variables, with the angle being 45 times the integers 0, 1, 2.
Solution:
The assumed laminate configuration for this problem is
1
2
3
4
[
/
/
/
]
S
.The
equations for W
1
*
, W
3
*
are shown in equation (1). They are taken from equation 8.2.1 from the
book.
4
4
*
*
1
3
1
1
cos2
cos4
k
k
k
k
k
k
W
s
W
s
(1)
where,
3
3
1
2
2
k
k
k
z
z
s
h
h
with the total laminate thickness as h.
In this case the thickness of the laminate is not an issue. So it is assumed to be unity.
The optimization problem for this problem is shown in equation (2).
*
*
2
*
*
2
1
1 arg
3
3 arg
min
(
)
(
)
,0
90
1,2,3,4
k
t
et
t
et
o
o
k
W
W
W
W
suchthat
for k
(2)
The W
*
1target
and W
*
3target
are the target values that have to be reached as mentioned in the
question in part (a) and (b). The optimization problem is basically the minimization of the
distance to the target point in order to reach as close to the target point as possible.
The results are obtained by using the Excel solver which is very sensitive to the initial guesses.
The answer reports of the solver which have the initial guesses for the angles are given in
Appendix I for part (a) and Appendix II for part (b). The optimized stacking sequences are:
(a)
[±69.34/0/0/0]
S
with distance from the target point (objective function value) as 0.013.
(b)
[90
2
/±27.27/±67.93/90
2
]
S
with distance from the target point (objective function value) as
2.6e-7.
For the fixed angles of 0, 90, ±45-degree integer design variables have been used, with angle
being 45 times the integers 0, 1, 2. The optimization problem is shown in equation (3).

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