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Proc. R. Soc. A (2009) 465, 1123–1143
doi:10.1098/rspa.2008.0380
Published online 23 December 2008
On feasible regions of lamination parameters
for lay-up optimization of
laminated composites
B Y M. W. B LOOMFIELD , C. G. D IACONU
AND
P. M. W EAVER *
Department of Aerospace Engineering, University of Bristol, Queen’s Building,
University Walk, Bristol BS8 1TR, UK
The stiffness tensors of a laminated composite may be expressed as a linear function of
material invariants and lamination parameters. Owing to the nature of orienting
unidirectional laminae ply by ply, lamination parameters, which are trigonometric functions
of the ply orientation, are interrelated. In optimization studies, lamination parameters are
often treated as independent design variables constrained by inequality relationships to
feasible regions that depend on their values. The relationships between parameters enclose a
convex feasible region of lamination parameters which is generally unknown. The convexity
properties allow the efficient optimization of laminated composite structures where
lamination parameters are used as design variables. Herein, a two-level method is presented
to determine the feasible regions of lamination parameters where potential ply orientations
are a predefined finite set. At the first level, the feasible region of the in-plane, coupling and
out-of-plane lamination parameters is determined separately using convex hulls. At the
second level, a nonlinear algebraic identity is used to relate the in-plane, coupling and outof-plane lamination parameters to each other. This general approach yields all constraints
on the feasible regions of lamination parameters for a predefined set of ply orientations.
Keywords: composite materials; lamination parameters; feasible region; optimization
1. Introduction
The stiffness tensors of a laminated composite may be expressed as a linear
function of material invariants and lamination parameters. Owing to the nature
of orienting unidirectional laminae ply by ply, lamination parameters, which are
trigonometric functions of the ply orientation, are interrelated. In optimization
studies, lamination parameters are often treated as independent design variables
constrained by inequality relationships to feasible regions that depend on their
values. The relationships between lamination parameters enclose feasible regions
of lamination parameters which were subsequently proved to be convex
(Grenestedt & Gudmundson 1993). In this paper, the feasible regions describing
the in-plane, coupling and out-of-plane stiffness are shown to be three
* Author for correspondence ([email protected]).
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2008.0380 or via
http://journals.royalsociety.org.
Received 24 September 2008
Accepted 26 November 2008
1123
This journal is q 2008 The Royal Society
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1124
M. W. Bloomfield et al.
Table 1. Nomenclature.
A
B
c2(i ), c4(i )
D
E11, E22
Gij
Hj
H L, H U
h
j
hi
h ji
ki
M
m
N
n
p
Qij
Ui
V
v
s2(i ), s4(i )
zi
ai , bi , ci
^
G, G
30
xA;B;D
i
x
qi
k
n12
f
ji
in-plane stiffness tensor
coupling stiffness tensor
cos 2qi , cos 4qi
out-of-plane stiffness tensor
longitudinal and transverse Young’s moduli
shear moduli
hyperplane constraints
lower- and upper-bound values of the hyperplane constraints
plate thickness
coefficients of hyperplane constraints
coefficients of normalized hyperplane constraints
constraint scaling factor
vector of out-of-plane moments
size of f
vector of in-plane loads
dimension of the in-plane, coupling or out-of-plane feasible region
number of plies in a stacking sequence
reduced stiffness matrix
material invariants
set of vectors of lamination parameters
vector of lamination parameters
sin 2qi , sin 4qi
normalized through thickness coordinate
real-valued scalars
vector of lamination parameters
vector of mid-plane strains
lamination parameters
random vector of lamination parameters
the ith ply orientation
vector of plate curvatures
Poisson’s ratio
set of ply orientations
scalar quantities
interdependent convex subspaces related by a nonlinear algebraic identity. Note
that this particular function has physical meaning; it makes links between the
linear, quadratic and cubic volume fractions of each ply in the lay-up. The
purpose of this paper is to provide a method that would enable the reader to
derive explicit relationships between lamination parameters for any predefined
finite set of ply orientations. It is expected that the results will prove useful in the
efficient optimization of laminated composite structures (table 1).
The excellent performance of composite materials has been well publicized in
recent years. Studies have shown that they possess excellent stiffness and
strength properties. As such, the aviation industry is rapidly employing
composite materials for primary structures such as wings and fuselages. Their
use is clearly seen with the development of many recent commercial and military
aircraft, including Airbus A400M, A380 and the Boeing 787 Dreamliner. Despite
Proc. R. Soc. A (2009)
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Feasible regions of lamination parameters
1125
their insertion in high-profile aircraft, there are potential efficiency gains to be
made by undertaking stacking sequence (lay-up) optimization and reducing
dependence on the use of homogeneous properties or ‘black metal’.
Lay-up optimization of monolithic laminated composites has evolved significantly over the past 25 years. Initially, Tsai et al. (1968) and Tsai & Hahn (1980)
characterized the stiffness properties of a monolithic laminated composite in terms
of material invariants and 12 lamination parameters. Tsai and Pagano and Tsai and
Hahn demonstrated that the in-plane A, coupling B and out-of-plane D stiffness
tensors can be expressed as the linear functions of material stiffness invariants and
the so-called lamination parameters, xiA;B;D . Note that, for each stiffness tensor,
there are four corresponding lamination parameters. Using lamination parameters
for the lay-up optimization significantly reduces the number of design variables in
comparison with using ply orientations and thicknesses. However, it is important to
note that the 12 lamination parameters are interrelated. That is, when the values of
any number of parameters are fixed, the remaining parameters take a range of
values constrained to certain feasible regions. The extent of each feasible region is
itself a function of the number and value of the fixed lamination parameters.
Explicit relationships between lamination parameters can be derived to determine
the boundaries of feasible regions. Grenestedt & Gudmundson (1993) proved that
the feasible regions of lamination parameters are convex. This fact reduces the
complexity and computational expense of structural optimization problems.
Moreover, if ply orientation and thickness are used as design variables the resulting
feasible region is often non-convex and shows a complex response surface. In this
case, resulting optimization studies may obtain suboptimal solutions by converging
to a local optimum. Recent work (Grenestedt 1991; Le Riche & Haftka 1993; Liu
et al. 2004; Todoroki & Terada 2004; Herencia et al. 2006, 2007; Matsuzaki &
Todoroki 2007; Bloomfield et al. 2008) has focused on optimizing laminated
composites using lamination parameters.
In laminated composites design, potential ply orientations are often limited to
predefined finite sets, e.g. 0, 90, G45 degrees. This particular set is mainly driven by
manufacturing and certification limitations. However, studies have shown
(Grenestedt 1991; Bloomfield et al. 2008) that using a larger set of ply orientations
(including non-standard fibre orientations) imparts greater orthotrophic and
anisotrophic elastic tailoring capabilities which may bring associated gains in
efficiency or, indeed, functionality. Although, the feasible regions of lamination
parameters have been derived by Diaconu & Sekine (2004) for 0, 90, G45 degree
plies, such information is not available for expanded finite sets of ply orientations. It
is noted that approximate expressions are known for the general feasible region
(Setoodeh et al. 2006) of lamination parameters where no restrictions are placed on
potential ply orientations. However, the number of expressions is significantly large
and may reduce the efficiency of an optimization routine. Our motivation is to
provide a method to determine the exact feasible regions of lamination parameters
where potential ply orientations are a predefined finite set. This information may
subsequently be used for the more efficient design of composites. Specifically, it has
been shown by Herencia et al. (2006, 2007) that lay-up optimization using
lamination parameters provides a fast and robust method for multi-part structures.
In the paper, after a brief description of lamination parameters including their
characteristics and usage, a two-level method is proposed and used to derive the
explicit expressions describing the feasible region of lamination parameters where
Proc. R. Soc. A (2009)
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1126
M. W. Bloomfield et al.
ply orientations are a predefined finite set. The first level determines, separately,
the feasible region of the in-plane, coupling and out-of-plane lamination
parameters using convex hulls. It will be shown that the respective feasible
regions are enclosed by sets of hyperplane constraints. The second level, using
data from the first level, establishes relationships between the hyperplane
constraints determined at the first level. As such, the feasible regions may be
viewed as essentially three separate four-dimensional (four lamination parameters) spaces with interconnections between them provided by the second level.
This general approach yields all relationships or constraints, between lamination
parameters and thus determines the entire feasible regions of lamination
parameters for predefined finite sets of ply orientations.
2. Background
Using the classical laminate plate theory, Tsai & Hahn (1980),
(
) (
) A B
N
30
Z
;
k
M
B D
ð2:1Þ
where N is a vector of resultant running loads; M is a vector of resultant outof-plane moments; 30 is the vector of mid-plane strains; and k is the vector of
plate curvatures. The in-plane, coupling and out-of-plane stiffness tensors are
defined in terms of the lamination parameters and material invariants
0
1
2
1
A11
6
C
B
61
B A22 C
6
C
B
6
C
B
6
B A12 C
60
C
B
6
B A C Z h6
B 66 C
60
C
B
6
BA C
6
@ 16 A
60
4
A26
0
0
B11
1
2
0
6
C
B
60
B B22 C
6
C
B
6
C
B
6
2
B B12 C h 6 0
CZ 6
B
BB C
4 6
B 66 C
60
C
B
6
BB C
6
@ 16 A
60
4
B26
0
Proc. R. Soc. A (2009)
xA
1
xA
2
0
KxA
1
xA
2
0
0
KxA
2
1
0
KxA
2
0
xA
3 =2
xA
4
0
A
xA
3 =2 Kx4
0
xB1
xB2
0
KxB1
xB2
0
0
KxB2
0
0
KxB2
0
xB3 =2
xB4
0
xB3 =2 KxB4
0
3
70 U 1
1
07
7B C
7B U C
2
7
C
0 7B
B
7B U3 C
C;
7
C
1 7B
C
7B
U
A
@
4
7
07
5 U5
0
0
3
70 U 1
1
07
7B C
7B U C
2
7
C
0 7B
C
7B
;
U
B
3
7B C
0 7B C
C
7@ U4 A
7
07
5 U5
0
ð2:2Þ
0
ð2:3Þ
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1127
Feasible regions of lamination parameters
0
1
2
1
D11
6
C
61
B
6
B D22 C
C
6
B
C
6
B
B D12 C h 3 6 0
CZ 6
B
B D C 12 6
60
B 66 C
C
6
B
6
BD C
16
A
60
@
4
D26
0
xD
1
xD
2
0
KxD
1
xD
2
0
0
KxD
2
1
0
KxD
2
0
xD
3 =2
xD
4
0
D
xD
3 =2 Kx4
0
3
70 U 1
1
07
7B C
7B U C
2
7
C
0 7B
C
7B
;
U
B
7B 3 C
1 7B C
C
7@ U4 A
7
07
5 U5
0
0
where the lamination parameters are
ð
1 1
A
x½1;2;3;4 Z
½cos 2qðzÞ; cos 4qðzÞ; sin 2qðzÞ; sin 4qðzÞdz;
2 K1
Ð1
½cos 2qðzÞ; cos 4qðzÞ; sin 2qðzÞ; sin 4qðzÞz dz;
xB½1;2;3;4 Z K1
ð2:4Þ
9
>
>
>
>
>
>
>
>
>
=
>
>
>
ð1
>
>
3
>
2
D
>
½cos 2qðzÞ; cos 4qðzÞ; sin 2qðzÞ; sin 4qðzÞz dz; >
x½1;2;3;4 Z
>
;
2 K1
ð2:5Þ
and q(z) is the distribution function of the ply orientations through the
normalized thickness coordinate zZ ð2=hÞz. For further details regarding q(z),
see Diaconu et al. (2002a). Additionally, the material invariants are defined as
9
U1 Z ½3Q11 C 3Q22 C 2Q12 C 4Q66 =8; >
>
>
>
>
U2 Z ½Q11 K Q12 =2;
>
>
=
U3 Z ½Q11 C Q22 K2Q12 K4Q66 =8;
ð2:6Þ
>
>
>
U4 Z ½Q11 C Q22 C 6Q12 K4Q66 =8; >
>
>
>
;
U5 Z ½Q11 C Q22 K2Q12 C 4Q66 =8;
where Qij are reduced stiffnesses for unidirectional lamina and defined as
9
Q11 Z E 211 = E11 K E22 n212 ; >
>
>
>
>
2
Q22 Z E11 E22 = E11 K E22 n12 ; =
ð2:7Þ
>
>
Q12 Z n12 Q22 ;
>
>
>
;
Q66 Z G12 :
In equation (2.7), E11, E22, G12 are the longitudinal, transverse and shear moduli,
respectively; n12 is the Poisson’s ratio for a unidirectional laminate. Note that the
12 lamination parameters defined in equation (2.5) are integrals through the
thickness of the sines and cosines of the lay-up orientations. In practice, this
integration is replaced by a through the thickness summation at ply level.
Moreover, only these 12 lamination parameters are necessary to model the
stiffness properties of any monolithic laminated composite.
Proc. R. Soc. A (2009)
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1128
M. W. Bloomfield et al.
The derivation of relationships between lamination parameters to provide a
feasible region has been partially developed in an incremental fashion by several
authors. The earliest use of lamination parameters in composite design appears
to have been undertaken by Miki (1982) and Miki & Sugiyama (1993), who
pioneered their use in the optimization studies. In doing so, Miki defined the
feasible regions between some of the lamination parameters. Specifically, he
determined the feasible regions needed to describe both the in-plane or outof-plane stiffnesses of an orthotrophic laminate using two in-plane or two outof-plane lamination parameters, respectively,
2ðxj1 Þ2 K1% xj2 ;
ð2:8Þ
where jZA, D. These feasible regions were used to provide an efficient, accurate
and graphical approach to the design of laminated composites. Later, Fukunaga &
Sekine (1992) derived the feasible regions of the four in-plane and separately, four
out-of-plane lamination parameters
2ð1 C xj2 Þðxj3 Þ2 K4xj1 xj3 xj4 C ðxj4 Þ2 Kðxj2 K2ðxj1 Þ2 C 1Þð1Kxj2 Þ% 0;
ð2:9Þ
2 2
xj1 C xj3 % 1;
ð2:10Þ
where jZA, D. For a solely out-of-plane problem, such as initial buckling,
knowledge of the complete feasible region makes the optimization process
more efficient. At this time, the feasible region of the coupling lamination
parameters, where ply orientations are unrestricted, had not been derived.
Next, Grenestedt & Gudmundson (1993) used a variational approach to
implicitly determine the feasible region of orthotrophic symmetric laminates.
Furthermore, Grenestedt & Gudmundson (1993) derived explicit expressions
between certain sets of the in-plane and out-of-plane lamination parameters.
For example,
3
3
1 A
1 A
%
K1
C 1;
xi C 1 K1% xD
x
i
4
4 i
ð2:11Þ
where iZ1, ., 4. Note that, for a given value of xiiA , there exists a range of
A
A
D
values for xD
i for all values except for xi ZG1 when xi Z xi . Grenestedt &
Gudmundson (1993) additionally proved that the feasible region was necessarily
convex. Diaconu et al. (2002a) used the approach of Grenestedt & Gudmundson
(1993) to obtain an implicit mathematical formulation of the general feasible
region of all 12 lamination parameters. Also, Diaconu et al. (2002b) derived
the feasible region of lamination parameters linking the in-plane, coupling and
out-of-plane lamination parameters, where the index of the parameter was
the same
Proc. R. Soc. A (2009)
D
A
4
B 2
4ðxA
i C 1Þðxi C 1ÞR ðxi C 1Þ C 3ðxi Þ ;
ð2:12Þ
D
A
4
B 2
4ðxA
i K 1Þðxi K1ÞR ðxi K1Þ C 3ðxi Þ ;
ð2:13Þ
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Feasible regions of lamination parameters
1129
where iZ1, ., 4. It is noted that in the above studies, no restrictions were
placed on potential ply orientations. Later, Diaconu & Sekine (2004) derived
explicitly the feasible regions of lamination parameters for 0, 90, G45 degree
plies. They derived explicit expressions that related the in-plane, coupling and
out-of-plane lamination parameters to each other. It is noted that Diaconu &
Sekine (2004) did not provide a general method to determine the constraints on
the feasible region for in-plane, coupling and out-of-plane lamination parameters
for finite sets of ply orientations. One of the aims of the current work is to
provide such a method.
Recently, Setoodeh et al. (2006) developed a method to approximate the
boundary of the general feasible region for lamination parameters. They
generated lay-ups of varying thickness and ply orientations and calculated
the corresponding lamination parameters for each lay-up. The convex hull
of the set of lamination parameters was taken and noted to increase with
the growing number of different ply orientations. By monitoring convergence,
they determined vectors of lamination parameters on the boundary of the
feasible region. The convex hull was then used to determine 12-dimensional
linear approximations of the feasible region of lamination parameters.
While it is noted that this method could be used for a finite set of ply
orientations, only approximations to the feasible region would be determined.
By contrast, the method presented in this paper yields exact constraints
on the feasible region for any finite set of ply orientations. Furthermore,
the nature of the feasible region, as three interconnected discrete spaces, is
not represented in the work of Setoodeh et al. With respect to the method
presented by Setoodeh et al., it is noted that this method yields a relatively
large number of constraints, which may be computationally inefficient for
optimization routines. It is further observed that computing the convex
hull in higher dimensions is computationally expensive. Moreover, currently,
industry generally restricts itself to discrete sets of ply orientations and
as such, the work presented in the current paper may be useful for
such purposes.
At this time, the general feasible region remains unknown in analytical form.
However, the method detailed in the paper yields all the constraints on the
feasible region of lamination parameters for any finite set of ply orientations.
Before constructing new relationships between lamination parameters, it is
helpful to make some definitions.
Feasible region. A region in an abstract design space of lamination parameters
that contains all feasible vectors of lamination parameters. For each feasible
vector of lamination parameters, there exists at least one real lay-up. Note that
for any given vector of lamination parameters outside the feasible region, no real
lay-up exists. Also note that the dimension of the design space (or of the feasible
region) equals the number of lamination parameters considered—maximum of 12
for the most general case.
Hyperplane constraint. A linear inequality constraint on the in-plane, coupling
or out-of-plane feasible regions are denoted as HA, HB, HD, respectively.
Constraint on the feasible region. Any constraint that forms part of the
boundary of the feasible region of lamination parameters.
Proc. R. Soc. A (2009)
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M. W. Bloomfield et al.
Using this background information and initial definitions, the method to
derive the feasible region of the in-plane, coupling or out-of-plane lamination
parameters is presented in §3.
3. Feasible regions of in-plane, coupling and out-of-plane
lamination parameters
In this section, the feasible region of the in-plane, coupling and out-of-plane
lamination parameters is determined separately, using convex hulls. Formally,
the convex hull of a finite set of points X is defined as
CH ðXÞ Z
(
N
X
li Xi j li R 0;
i Z1
i Z 1; .; N ;
N
X
)
li Z 1 :
ð3:1Þ
i Z1
Note that the convex hull of a set of points X is the minimum convex set
containing X. For a finite set of ply orientations, it will be shown in this
section that the feasible region of the in-plane, coupling or out-of-plane
lamination parameters is a convex polyhedron (or polygon). Furthermore, the
convex polyhedron is formed by taking the convex hull of the minimum
number of vertices on the boundary of the feasible region. It is observed that a
convex polyhedron is bounded by a set of hyperplanes. As such, each of
the feasible regions (in-plane, coupling or out-of-plane) are enclosed by a set
of hyperplane constraints, each of the dimension n where n is the number
of the in-plane, coupling or out-of-plane lamination parameters noting nZ1, 2,
3 or 4.
With respect to the in-plane or out-of-plane lamination parameters, the
minimum number of plies that form the boundary of the feasible region is one, as
shown by Fukunaga & Sekine (1992). Therefore, for any one ply of arbitrary
orientation, the four in-plane or the four out-of-plane lamination parameters
that form the boundary of the feasible region of the in-plane or out-of-plane
lamination parameters, respectively, are defined as follows:
9
xj1 Z cosð2qÞ >
>
>
>
>
>
j
x2 Z cosð4qÞ =
;
>
xj3 Z sinð2qÞ >
>
>
>
>
;
j
x4 Z sinð4qÞ
where j Z A; D:
ð3:2Þ
With respect to the coupling lamination parameters, the minimum number of ply
orientations to define the boundary of the feasible region is two (where q1sq2)
since a single ply is necessarily symmetric and two plies is the minimal needed for
non-zero xBi . Therefore, it is asserted that two plies lie on the boundary of the
feasible region of the coupling lamination parameters, xB1 , xB2 , xB3 , xB4 defined as
Proc. R. Soc. A (2009)
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Feasible regions of lamination parameters
9
Kcosð2q1 Þ C cosð2q2 Þ >
>
>
>
2
>
>
>
>
>
>
Kcosð4q
Þ
C
cosð4q
Þ
1
2 >
B
>
x2 Z
>
>
=
2
;
Ksinð2q1 Þ C sinð2q2 Þ >
>
B
>
x3 Z
>
>
2
>
>
>
>
>
>
Ksinð4q
Þ
C
sinð4q
Þ
1
2 >
B
>
>
x4 Z
;
2
1131
xB1 Z
where q1 sq2
ð3:3Þ
and q1 is the bottom ply. Equations (3.2) and (3.3) define vertices on the
boundary of the feasible region of the in-plane, out-of-plane or coupling
lamination parameters, respectively. Each vertex corresponds to a single ply
orientation for the in-plane or out-of-plane lamination parameters and to a
non-symmetric combination of two plies of equal thickness and different angles
for coupling lamination parameters. By connecting the vertices on the
boundary of the feasible region, hyperplane constraints are determined and
the explicit feasible regions are derived. The set of vertices on the boundary of
the feasible region are used in the following algorithm to analytically determine,
separately, the feasible region of the in-plane, coupling and out-of-plane
lamination parameters.
Algorithm 3.1.
(i) For each ply orientation or unique set of two orientations in f, calculate
the corresponding vertex of lamination parameters v, using equation (3.2)
or (3.3), respectively.
(ii) The set of all v for a given f is denoted by V.
(iii) The convex hull of V is computed in MATLAB using QHULL (Barber et al.
1996; Bertsekas et al. 2003) and the ‘convhull’ function (if nZ2) or
‘convhulln function’ (if nO2).
(iv) The convex hull function outputs a set of vertices for each hyperplane.
These sets of vertices are used to determine the coefficients of the
hyperplane constraints which form the boundary of the feasible region of
the in-plane, coupling or out-of-plane lamination parameters.
Note that, if nZ1, the range of xjk , with jZA, D and kZ1, ., 4 is
calculated using equation (3.2) for all q2f. The range of the coupling
lamination parameters is similarly calculated using equation (3.3). For nZ2, 3
or 4, the hyperplane constraints (which form the boundary of the in-plane,
coupling and out-of-plane feasible regions) are found in the following form
(nZ4 is shown):
j
j
j
j
j
h1 xj1 C h2 xj2 C h3 xj3 C h4 xj4 C h5 Z 0;
Proc. R. Soc. A (2009)
ð3:4Þ
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1132
M. W. Bloomfield et al.
1.0
0.5
0
– 0.5
– 1.0
1.0
0.5
1.0
0.5
0
0
– 0.5
–0.5
– 1.0
Figure 1. Feasible region of
ðxj1 ; xj2 ; xj3 Þ
where jZA, B and D and
0
1 xj21 xj31
B
B
j
j
B 1 x22 x32
j
B
h1 Z detB
B 1 xj23 xj33
@
1 xj24 xj34
0 j
x11 xj21 1
B
B j
j
B x12 x22 1
j
B
h3 Z detB
B xj13 xj23 1
@
xj41
where
for 0, 90, G30, G45, G60 degree plies where
jZA, D.
1
C
C
xj42 C
C
j C
x43 C
A
j
x44
1
xj41
C
C
xj42 C
C
j C
x43 C
A
j
x44
xj14 xj24 1
0 j
x11 xj21 xj31
B
B j
j
j
B x12 x22 x32
j
B
h5 ZKdetB
B xj13 xj23 xj33
@
xj14 xj24 xj34
– 1.0
xj4 Z 0
xj41
0
1
C
C
xj42 C
C:
j C
x43 C
A
j
x44
xj11
B
B j
B x12
j
h2 Z detB
B j
B x13
@
xj14
0 j
x11
B
B j
B x12
j
h4 Z detB
B j
B x13
@
xj14
1
xj31
1
xj32
1
xj33
1
xj34
xj21
xj31
xj22
xj32
xj23
xj33
xj24
xj34
xj41
1
C
C
xj42 C
C
j C
x43 C
A
j
x44
1
1
C
C
1C
C
C
1C
A
1
ð3:5Þ
In equation (3.5), each entry in the determinants, xjkl , corresponds to a
lamination parameter xjk with jZA, B and D and kZ1, ., 4 for a specific
vertex lZ1, ., 4. Note that at least n vertices are necessary to define a
hyperplane in an n-dimensional space. For simplicity, each hyperplane
j
constraint is normalized with respect to the constant h5 . Thus, equation
(3.4) can be rewritten as
Proc. R. Soc. A (2009)
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1133
Feasible regions of lamination parameters
1.0
0.5
0
– 0.5
– 1.0
1.0
0.5
1.0
0.5
0
0
– 0.5
– 1.0
–0.5
–1.0
Figure 2. Feasible region of ðxB1 ; xB2 ; xB3 Þ where xB4 Z 0 for 0, 90, G30, G45, G60 degree plies.
1.0
0.5
0
– 0.5
– 1.0
– 1.0
– 0.5
0
0.5
1.0
A
Figure 3. Feasible vectors of ðxA
1 ; x2 Þ for 0, 90, G30, G45, G60 degree ply assuming uniform ply
thickness.
h j1 xj1 C h j2 xj2 C h j3 xj3 C h j4 xj4 K1% 0;
ð3:6Þ
j
j
where h ji ZKðhi =h5 Þ and iZ1, ., 4 with the inequality showing the scope of
the feasible region.
j
Note that h5 is a non-zero constant since the set of lamination parameters
j
are linearly independent. Furthermore, if h5 was zero, the resulting hyperplane
would pass through the origin in lamination parameter space. Since the origin
Proc. R. Soc. A (2009)
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1134
M. W. Bloomfield et al.
j
is not on the boundary of the feasible region, h5 must always be non-zero. In order
to validate algorithm 3.1 and the assertions in the introduction of this section, the
in-plane, coupling and out-of-plane feasible regions were derived for 0, 90 and G45
degrees, and proved to be identical with those derived by Diaconu & Sekine
(2004). Moreover, using algorithm 3.1, the explicit expressions describing the
boundary of the aforementioned feasible regions for 0, 90, G30, G45, G60 degree
plies are derived and shown in appendix B in the electronic supplementary
material. These feasible regions are shown in figures 1 and 2.
In figures 1 and 2, the light grey lines represent the boundary of planes on the
hidden side of the surface. It is observed that the number of hyperplane constraints
that enclose the feasible region of coupling lamination parameters is significantly
greater than the number of in-plane or out-of-plane hyperplane constraints. Next,
several tests are undertaken to validate that the obtained hyperplane constraints
define the boundary of the feasible region of in-plane, coupling or out-of-plane
lamination parameters. These tests show that the boundary of the feasible regions,
obtained using algorithm 3.1, is indeed the convex hull of the minimum number of
vertices on the boundary of the feasible region.
The first test is undertaken to show that each feasible region is sufficiently
large to include all possible laminate lay-ups from 0, 90, G30, G45, G60 degree
plies. To demonstrate this feature, each vector of feasible lamination parameters
x, when substituted into each hyperplane constraint, denoted Hj, where jZA, B
and D, must be less than or equal to 0, i.e. Hj(x)%0. To achieve this, all
combinations of angles in f of one to six plies were generated using enumeration.
A
To show this feature graphically, the feasible region of xA
1 , x2 was selected and is
shown in figure 3.
A second test is undertaken to determine whether each feasible region is
sufficiently small to exclude potentially infeasible sets of lamination parameters.
For every vector of lamination parameters x that satisfies the set of hyperplane
constraints, a real lay-up is sought: if the Euclidean distance between x and the
vector of lamination parameters corresponding to the determined lay-up is small,
then the set of lamination parameters has passed the test. A discrete optimizer
(Bloomfield et al. 2008) was used to determine a lay-up from a given x. These two
tests confirm the extent of the derived feasible region and, moreover, highlight
the fact that the feasible region is formed by the convex hull of the minimum set
of vertices on its boundary.
4. Determining the feasible regions of lamination parameters
In this section, the second level necessary to determine the feasible regions of
lamination parameters is presented. Specifically, a method for establishing
relationships between the constraints on the in-plane, coupling and out-of-plane
feasible regions (as found in §3) is derived. Note that the method builds upon the
work of Diaconu & Sekine (2004) and generalizes their approach such that it is
valid for any finite set of ply orientations.
Firstly, the in-plane lamination parameters defined in equation (2.5), for a
lay-up of p plies are rewritten as a finite sum, thus
Proc. R. Soc. A (2009)
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Feasible regions of lamination parameters
9
1
>
>
xA
Z
ð1Þ
C
z
ðc
ð1ÞK
c
ð2ÞÞ
/Cz
ðc
ðp
K
1ÞK
c
ðpÞÞ
C
c
ðpÞÞ;
ðc
>
1 2
2
pK1 2
2
2
1
>
2 2
>
>
>
>
>
>
1
>
A
x2 Z ðc4 ð1Þ C z 1 ðc4 ð1ÞK c4 ð2ÞÞ /CzpK1 ðc4 ðp K 1ÞK c4 ðpÞÞ C c4 ðpÞÞ; >
>
>
=
2
>
>
>
>
>
>
>
>
>
>
>
1
A
>
x4 Z ðs4 ð1Þ C z 1 ðs4 ð1ÞK s4 ð2ÞÞ /CzpK1 ðs4 ðp K1ÞK s4 ðpÞÞ C s4 ðpÞÞ; >
>
;
2
xA
3
1
Z ðs2 ð1Þ C z 1 ðs2 ð1ÞK s2 ð2ÞÞ /CzpK1 ðs2 ðp K1ÞK s2 ðpÞÞ C s2 ðpÞÞ;
2
1135
ð4:1Þ
9
c2 ðiÞ Z cos 2qi ; >
>
>
>
c4 ðiÞ Z cos 4qi ; =
where
ð4:2Þ
s2 ðiÞ Z sin 2qi ; >
>
>
>
;
s4 ðiÞ Z sin 4qi ;
and zi (iZ1, ., pK1) is the normalized through thickness coordinate of each ply.
Note that z0ZK1 and zpZ1. Coupling and out-of-plane lamination parameters
are similarly defined by making the appropriate square and cubic substitutions
for zi in equation (4.1). Next, explicit expressions linking the lamination
parameters from each design subspaces are found. Motivated by the algebraic
identity, first used by Diaconu et al. (2002b)
2
4ðzx K zy Þ zx3 Kzy3 Z ðzx K zy Þ4 C 3 zx2 Kzy2 ;
ð4:3Þ
it is proven in appendix A that on the boundary of the feasible9region
1 A
>
A
A
>
h 1 x1 C h 2 xA
2 C h 3 x3 C h 4 x4 C h 5 Z zx K zy ; >
>
k
>
>
>
>
>
=
1 B
B
B
B
2
2
h 1 x1 C h 2 x2 C h 3 x3 C h 4 x4 Z zx Kzy ;
ð4:4Þ
k
>
>
>
>
>
1 D
D
D
D
3
3 >
>
h 1 x1 C h 2 x2 C h 3 x3 C h 4 x4 C h 5 Z zx Kzy ; >
>
;
k
!
where
5
1X
A
ð4:5Þ
hx ;
k Z max
2 i Z1 i i
and x, y in equations (4.3) and (4.4) are integers, where x, y2[1,pK1], xA
5 Z 1 and
xsy, equation (4.3) has fundamental importance for establishing relationships
between the lamination parameters since it makes links between the linear,
quadratic and cubic volume fractions of each ply orientation in the lay-up and
thus the A, B and D stiffness tensors. To make these connections, equations (4.4)
are substituted into equation (4.3) and the resulting expression is multiplied
through by k4 to obtain
!
!
!4
!2
5
5
5
4
X
X
X
X
2
A
D
A
2
B
hi xi
hi x i R
hi xi
C 3k
hi xi
;
ð4:6Þ
4k
i Z1
Proc. R. Soc. A (2009)
i Z1
i Z1
i Z1
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1136
M. W. Bloomfield et al.
where xj5 Z 1 and jZA, B and D, noting that the inequality represents the scope
of the feasible region. Furthermore, equation (4.6) is the fundamental explicit
expression that establishes relationships between the in-plane, out-of-plane and
coupling hyperplane constraints and thus lamination parameters to each other.
Note that, in equation (4.6), the inequality has been introduced to show that the
constraint forms a section on the boundary of the feasible region of lamination
parameters. For lamination parameters on the boundary of the feasible region,
the inequality in equation (4.6) becomes a strict equality. It is noted from
Diaconu & Sekine (2004) that for each constraint in the form of equation (4.6),
there were two values for h5. The two values of h5 arise from the existence of an
upper and lower bound value for each hyperplane constraint (which was
determined at the first level). This assertion can be proved as follows. Since each
lamination parameter is a simple trigonometric function, its value is bounded by
G1. It follows that any linear combination of lamination parameters has a lower
and upper bound. Therefore, each hyperplane constraint has a lower and an
L
upper bound denoted by H L and H U, respectively. Specifically, the values
P4 of HA
U
and H are defined as the minimum and maximum values of
iZ1 hi xi .
Substituting H L, H U for h5 in equation (4.6) gives
!
!
!4
!2
4
4
4
4
X
X
X
X
2
A
L
D
L
A
L
2
B
4k
hi xi KH
hi xi KH R
hi xi KH
C 3k
hi xi
i Z1
i Z1
i Z1
i Z1
ð4:7Þ
and
4k
2
4
X
!
U
hi xA
i KH
i Z1
4
X
i Z1
!
U
hi xD
i KH
R
4
X
i Z1
!4
U
hi xA
i KH
C 3k
2
4
X
!2
hi xBi
:
i Z1
ð4:8Þ
Equations (4.7) and (4.8) are the expressions that connect the in-plane, coupling
and out-of-plane hyperplane constraints, determined using algorithm 3.1, to each
other. It was observed in §3 that the number of constraints of the feasible region
of coupling lamination parameters was significantly greater than the number of
constraints on the in-plane or out-of-plane feasible regions. From equations (4.7)
and (4.8), it follows that for each constraint on the coupling lamination
parameters, there must be a corresponding constraint on the in-plane and outof-plane lamination parameters when all lamination parameters are explicitly
related to one another. Moreover, when the constraints become equalities, then
B
D
hi Z h A
i Z h i Z h i where iZ1, ., 4. Therefore, the feasible region of the coupling
lamination parameters is used to derive the constraints on the entire
12-dimensional feasible region. The process of deriving the explicit expressions
that make links between the lamination parameters is summarized in algorithm 4.1.
Algorithm 4.1.
(i) Using algorithm 3.1 determine, separately, the hyperplane constraints on
the in-plane, coupling and out-of-plane feasible regions.
(ii) For each hyperplane constraint on the coupling lamination parameters
(hi) found in step (i), determine H L, H U using the minimum and
P
maximum of 4iZ1 hi xA
i and determine k using equation (4.5).
Proc. R. Soc. A (2009)
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Feasible regions of lamination parameters
1137
(iii) Substitute H L, H U and k into equations (4.7) and (4.8) to establish
linking expressions between the in-plane, coupling and out-of-plane
feasible regions.
Using algorithm 4.1, the explicit expressions that connect the in-plane, outof-plane and coupling lamination parameters for the predefined finite set of 0, 90,
G30, G45, G60 degree plies are shown in appendix C in the electronic
supplementary material. Next, the obtained constraints are tested and validated.
5. Validation and confirmation of results
In order to confirm that the set of derived constraints formed using equations
(4.7) and (4.8) fully define the feasible region of lamination parameters for 0, 90,
G30, G45, G60 degree plies, two tests are carried out. With respect to the first
test, all feasible lay-ups should strictly obey the inequality constraint, e.g. all
feasible lay-ups must lie on the boundary of the feasible region or inside of it.
This test was detailed in §3. Concerning the second test, each vertex on the
boundary of the feasible region should correspond to at least one real lay-up. The
second test can be summarized in three levels.
(i) A random vector, x, of 12 lamination parameters is generated.
(ii) If x satisfies the set of constraints then continue to step (iii), otherwise
return to step (i).
(iii) For each x that satisfies the constraints, a corresponding lay-up is
determined using a discrete optimizer (Bloomfield et al. 2008).
(iv) The Euclidean distance between x and the corresponding vector of
lamination parameters (calculated from the obtained lay-up in step (iii))
is calculated. If the distance between the two vectors is small then it is
accepted that the test has been passed.
All lay-ups passed both tests which confirm that the derived feasible region for
0, 90, G30, G45, G60 degree plies is indeed appropriate.
6. Conclusions
A method to derive the feasible region of lamination parameters for any
predefined finite set of ply orientations has been presented. The new method was
used to rederive and confirm the feasible region for 0, 90, G45 degree plies.
Additionally, the feasible region for 0, 90, G30, G45, G60 degree plies was
derived, validated and confirmed. The information contained within on the
nature of the feasible region should prove useful for elastic tailoring purposes. For
example, it should be noted that the constraints on the feasible region of
lamination parameters can be used in conjunction with additional structural
constraints and used in efficient optimization routines for the optimization of
laminated composite structures.
M.W.B. would wish to thank both Great Western Research (GWR) and Airbus UK for his
studentship, and also his colleagues, J. E. Herencia, Michael May and Alberto Pirrera. P.M.W.
would wish to thank the EPSRC for his Advanced Research Fellowship.
Proc. R. Soc. A (2009)
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1138
M. W. Bloomfield et al.
Appendix A. Proof of equation (4.4)
Proposition A.1. It is proposed that the hyperplane constraints, hi xji , where
jZA, B and D, are related to each other by the following algebraic identity:
2
4ðzx K zy Þ zx3 Kzy3 Z ðzx K zy Þ4 C 3 zx2 Kzy2 ;
ðA 1Þ
where
9
1 A
>
A
A
>
C
h
x
C
h
x
C
h
K
z
;
h 1 x1 C h 2 xA
Z
z
3 3
4 4
5
x
y >
2
>
k
>
>
>
>
>
=
1
B
B
B
B
2
2
h 1 x1 C h 2 x2 C h 3 x3 C h 4 x4 Z zx Kzy ;
k
>
>
>
>
>
1 D
D
D
D
3
3 >
>
h 1 x1 C h 2 x2 C h 3 x3 C h 4 x4 C h 5 Z zx Kzy : >
>
;
k
ðA 2Þ
Note that k is a scaling factor and x, y are integers and 2[1,pK1].
Proof. Firstly, we seek to establish the existence of an identity in the form
of equation (A 1). There is existing evidence to suggest that the form of
equation (A 1) is appropriate because it was demonstrated by Diaconu et al.
(2002b) that
4
A
2
D
4 xA
C 3 xB1 ;
ðA 3Þ
1 C 1 x1 C 1 Z x1 C 1
which simplifies to equation (2.12) when h1Z1, h 2, h3, h4Z0, h5Z1 and kZ1 are
substituted into equation (A 2). Therefore, one solution is known to exist in the
form of equation (A 3).
To proceed further, the identity of equation (A 3) is generalized to fit the
form of equations (A 1) and (A 2). A solution is assumed to exist in the
following form:
9
A
A
A
a1 x A
1 C a2 x2 C a3 x3 C a4 x4 C a5 Z zx K zy ; >
>
=
B
B
B
B
2
2
ðA 4Þ
b1 x1 C b2 x2 C b3 x3 C b4 x4 C b5 Z zx Kzy ;
>
>
;
D
D
D
3
3
c1 x D
1 C c2 x2 C c3 x3 C c4 x4 C c5 Z zx Kzy ;
where ai , bi , ci are real valued scalars. Using equation (2.5), the in-plane
lamination parameters can be rewritten as
9
1
>
Z ðc2 ð1Þ C z1 ðc2 ð1ÞKc2 ð2ÞÞ /CzpK1 ðc2 ðpK1ÞKc2 ðpÞÞ C c2 ðpÞÞ; >
>
>
>
2
>
>
>
1
>
A
>
x2 Z ðc4 ð1Þ C z1 ðc4 ð1ÞKc4 ð2ÞÞ /CzpK1 ðc4 ðpK1ÞKc4 ðpÞÞ C c4 ðpÞÞ; =
2
1
>
>
xA
3 Z ðs2 ð1Þ C z1 ðs2 ð1ÞKs2 ð2ÞÞ /CzpK1 ðs2 ðpK1ÞKs2 ðpÞÞ C s2 ðpÞÞ; >
>
>
2
>
>
>
>
1
A
;
x4 Z ðs4 ð1Þ C z1 ðs4 ð1ÞKs4 ð2ÞÞ /CzpK1 ðs4 ðpK1ÞKs4 ðpÞÞ C s4 ðpÞÞ; >
2
xA
1
Proc. R. Soc. A (2009)
ðA 5Þ
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Feasible regions of lamination parameters
1139
where p is the number of plies in the lay-up. Note that the coupling and outB
of-plane lamination parameters can be similarly defined. Forming ai xA
i , bi xi ,
D
ci xi then
9
a1
>
ðc
Z
ð1Þ
Cz
ðc
ð1ÞK
c
ð2ÞÞ/Cz
ðc
ðp
K
1ÞKc
ðpÞÞCc
ðpÞÞ;
a1 xA
>
1
1 2
2
pK1 2
2
2
>
2 2
>
>
>
a
>
2
A
=
a2 x2 Z ðc4 ð1Þ Cz 1 ðc4 ð1ÞK c4 ð2ÞÞ/CzpK1 ðc4 ðp K 1ÞKc4 ðpÞÞCc4 ðpÞÞ; >
2
ðA6Þ
a3
>
>
a3 xA
Z
ð1Þ
Cz
ðs
ð1ÞK
s
ð2ÞÞ/Cz
ðs
ðp
K1ÞKs
ðpÞÞCs
ðpÞÞ;
ðs
>
1 2
2
pK1 2
2
2
3
>
2 2
>
>
>
a
>
4
A
a4 x4 Z ðs4 ð1Þ Cz 1 ðs4 ð1ÞK s4 ð2ÞÞ/CzpK1 ðs4 ðp K1ÞKs4 ðpÞÞCs4 ðpÞÞ; ;
2
9
b1
B
2
2
>
b1 x1 Z ðKc2 ð1Þ C z1 ðc2 ð1ÞK c2 ð2ÞÞ /Cz pK1 ðc2 ðp K 1ÞK c2 ðpÞÞ C c2 ðpÞÞ; >
>
>
2
>
>
>
>
b
>
2
B
2
2
=
b2 x2 Z ðKc4 ð1Þ C z1 ðc4 ð1ÞK c4 ð2ÞÞ /Cz pK1 ðc4 ðp K 1ÞK c4 ðpÞÞ C c4 ðpÞÞ; >
2
ðA 7Þ
b3
>
B
2
2
>
>
b3 x3 Z ðKs2 ð1Þ C z1 ðs2 ð1ÞK s2 ð2ÞÞ /Cz pK1 ðs2 ðp K 1ÞK s2 ðpÞÞ C s2 ðpÞÞ; >
>
2
>
>
>
>
b
4
B
2
2
;
b4 x4 Z ðKs4 ð1Þ C z1 ðs4 ð1ÞK s4 ð2ÞÞ /Cz pK1 ðs4 ðp K 1ÞK s4 ðpÞÞ C s4 ðpÞÞ; >
2
9
c1
ðc2 ð1ÞCz13 ðc2 ð1ÞKc2 ð2ÞÞ/Cz 3pK1 ðc2 ðp K 1ÞK c2 ðpÞÞ Cc2 ðpÞÞ; >
>
>
2
>
>
>
c
>
2
D
3
3
=
c2 x2 Z ðc4 ð1ÞCz1 ðc4 ð1ÞKc4 ð2ÞÞ/Cz pK1 ðc4 ðp K 1ÞK c4 ðpÞÞ Cc4 ðpÞÞ; >
2
ðA8Þ
c3
>
c3 xD
ðs2 ð1Þ Cz13 ðs2 ð1ÞKs2 ð2ÞÞ/Cz 3pK1 ðs2 ðp K 1ÞKs2 ðpÞÞCs2 ðpÞÞ; >
>
3 Z
>
2
>
>
>
c
>
4
D
3
3
c4 x4 Z ðs4 ð1Þ Cz1 ðs4 ð1ÞKs4 ð2ÞÞ/Cz pK1 ðs4 ðp K 1ÞKs4 ðpÞÞCs4 ðpÞÞ: ;
2
c1 xD
1 Z
&
Summing components of equation (A 6) gives
A
A
A
a1 xA
1 C a2 x2 C a3 x3 C a4 x4 Z
a1
a
ðc ð1Þ C c2 ðpÞ C/Þ C 2 ðc4 ð1Þ
2 2
2
a3
C c4 ðpÞ C/Þ C ðs2 ð1Þ C s2 ðpÞ C/Þ
2
a4
C ðs4 ð1Þ C s4 ðpÞ C/Þ:
2
ðA 9Þ
By collecting the constant terms from equation (A 9) and substituting them into
equation (A 4)1, it follows that:
0a
1
a
1
ðc2 ð1Þ C c2 ðpÞÞ C 2 ðc4 ð1Þ C c4 ðpÞÞ
B 2
C
2
B
C
ðA 10Þ
a5 ZKB a
C
@ C 3 ðs ð1Þ C s ðpÞÞ C a4 ðs ð1Þ C s ðpÞÞ A
2
4
2 2
2 4
Proc. R. Soc. A (2009)
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1140
M. W. Bloomfield et al.
Now define a vector G, where
GZ
9
1
>
ðcosð2q1 Þ C cosð2qp ÞÞ; >
>
>
2
>
>
>
>
>
>
1
>
ðcosð4q1 Þ C cosð4qp ÞÞ; >
>
>
=
2
ðA 11Þ
>
>
>
>
>
>
>
>
>
>
>
1
>
ðsinð4q1 Þ C sinð4qp ÞÞ: >
>
;
2
1
ðsinð2q1 Þ C sinð2qp ÞÞ;
2
It follows from equation (A 10) that:
a5 ZKð a1
a2
a3
a4 ÞG:
ðA 12Þ
It can be shown from equation (3.2) that for a given hyperplane
h 5 ZKð h 1
h2
h3
h 4 ÞG;
ðA 13Þ
when q1Zqp. Therefore, aZ f a1 a2 a3 a4 gT must be parallel to hZ
f h 1 h 2 h 3 h 4 gT . Furthermore, since q1Zqp it is evident that b5Z0.
Similarly, it can be shown that bZ f b1 b2 b3 b4 gT is parallel to h and
cZ f c1 c2 c3 c4 gT is parallel to h. Therefore, by definition of parallelism
of vectors
a Z j1 h
b Z j2 h
ðA 14Þ
c Z j3 h;
where j1, j2 and j3, are scalars. Next, substituting equations (A 14) into
equations (A 6)–(A 8), respectively and making the appropriate summations
9
A
A
A
j1 h 1 xA
1 C j1 h 2 x2 C j1 h 3 x3 C j1 h 4 x4 C j1 h 5 Z zx K zy ; >
>
=
B
B
B
B
2
2
j2 h 1 x1 C j2 h 2 x2 C j2 h 3 x3 C j2 h 4 x4 Z zx Kzy ;
>
>
D
D
D
D
3
3 ;
j3 h 1 x1 C j3 h 2 x2 C j3 h 3 x3 C j3 h 4 x4 C j3 h 5 Z zx Kzy ;
ðA 15Þ
and rearranging gives
A
A
A
h 1 xA
1 C h 2 x2 C h 3 x3 C h 4 x4 C h 5 Z
h 1 xB1 C h 2 xB2 C h 3 xB3 C h 4 xB4 Z
1
ðz K zy Þ;
j1 x
1 2
zx Kzy2 ;
j2
9
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
1
3
3 >
D
D
D
D
>
zx Kzy : >
h 1 x1 C h 2 x2 C h 3 x3 C h 4 x4 C h 5 Z
>
;
j3
Proc. R. Soc. A (2009)
ðA 16Þ
Downloaded from http://rspa.royalsocietypublishing.org/ on July 31, 2017
Feasible regions of lamination parameters
1141
By fully expanding equation (A 9) in terms of zi and then combining with
equation (A 16) the coefficient of zi for iZ1, ., pK1 is
h1
h
h
ðc2 ðiÞK c2 ði C 1ÞÞ C 2 ðc4 ðiÞK c4 ði C 1ÞÞ C 3 ðs2 ðiÞK s2 ði C 1ÞÞ
2
2
2
h4
C ðs4 ðiÞK s4 ði C 1ÞÞ:
ðA 17Þ
2
It follows from equations (A 16) and (A 17) that:
1
h1
h
h
ðc2 ðiÞK c2 ði C 1ÞÞ C 2 ðc4 ðiÞK c4 ði C 1ÞÞ C 3 ðs2 ðiÞK s2 ði C 1ÞÞ
Z
j1
2
2
2
h
ðA 18Þ
C 4 ðs4 ðiÞK s4 ði C 1ÞÞ :
2
Similarly
1
h1
h
h
Z
ðc2 ðiÞK c2 ði C 1ÞÞ C 2 ðc4 ðiÞK c4 ði C 1ÞÞ C 3 ðs2 ðiÞK s2 ði C 1ÞÞ
j2
2
2
2
h
ðA 19Þ
C 4 ðs4 ðiÞK s4 ði C 1ÞÞ ;
2
1
h1
h
h
Z
ðc2 ðiÞK c2 ði C 1ÞÞ C 2 ðc4 ðiÞK c4 ði C 1ÞÞ C 3 ðs2 ðiÞK s2 ði C 1ÞÞ
j3
2
2
2
h4
ðA 20Þ
C ðs4 ðiÞK s4 ði C 1ÞÞ :
2
Comparing coefficients in equations (A 18)–(A 20), it immediately follows j1Z
j2Zj3, which is denoted by 1/k herein and hence
9
1 A
A
A
A
>
h x C h 2 x2 C h 3 x3 C h 4 x4 C h 5 Z zx K zy ; >
>
>
k 1 1
>
>
>
>
>
=
1 B
B
B
B
2
2
h 1 x1 C h 2 x2 C h 3 x3 C h 4 x4 Z zx Kzy ;
ðA 21Þ
k
>
>
>
>
>
1 D
D
D
3
3 >
>
C
h
x
C
h
x
C
c
Kz
:
Z
z
h 1 x1 C h 2 xD
>
3 3
4 4
5
2
x
y >
;
k
Therefore, the general form of equation (A 1) is now proven. To simplify the
expressions, we substitute equations (A 21) into equation (A 1) and multiplying
through by k4 yields
2
A
A
A
A
D
D
D
4k h 1 x1 C h 2 x2 C h 3 x3 C h 4 x4 C h 5 h 1 xD
1 C h 2 x2 C h 3 x3 C h 4 x4 C h 5
4
B
A
A
A
2
B
B
B 2
:
Z h 1 xA
1 C h 2 x2 C h 3 x3 C h 4 x4 C h 5 C3k h 1 x1 C h 2 x2 C h 3 x3 C h 4 x4
ðA 22Þ
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on July 31, 2017
1142
M. W. Bloomfield et al.
Solving equation (A 22) for k2 gives
5
P
hi xA
i
4
i Z1
k2 Z 5
4
2 ;
5
P
P
P
A
D
B
4
hi xi
hi xi K3
hi xi
i Z1
i Z1
ðA 23Þ
i Z1
xA;D
5 Z 1.
where
Clearly, k depends upon the values of the in-plane, coupling and
out-of-plane lamination parameters. Moreover, for each feasible lay-up, there is a
corresponding value of k. The objective is then to find maximum k denoted k max
such that equation (A 22) is convex and all feasible lay-ups are either on or
within the boundary of the feasible region. The appropriate scaling factor k max is
determined as follows:
(
4 )
5
P
A
max
hi xi
k 2max Z
i Z1
( 5
4
2 ) :
5
P
P
P
A
D
B
max 4
hi xi
hi xi K3
hi x i
i Z1
i Z1
ðA 24Þ
i Z1
P
It is observed that, maxfð 5iZ0 hi xA
i Þg is reached at a vertex which corresponds to
one ply angle. This observation holds because in a convex polyhedron the
maximum distance (positions at extreme points) from any bounding hyperplane
is obtained at a vertex on the polyhedron. Moreover, this occurs at a vertex on
the bounding hyperplane. Therefore, the maximum distance occurs between two
vertices. For details see Bertsekas et al. (2003).P
Since one ply is necessarily symmetric, then 4iZ1 hi xBi Z 0. Noting that
(
!)
(
!)
5
5
X
X
A
D
Z max
;
max
hi x i
hi xi
i Z1
i Z1
and using the following identity:
(
(
!4 )
)!4
5
5
X
X
A
A
Z max hi x i
hi xi ;
max
i Z1
ðA 25Þ
i Z1
it follows that:
max
k 2max
5
P
i Z1
Z
5
P
max 4
hi xA
i
0
0
B
B
!
Z
max
B
2
@
5
P
hi x A
i
i Z1
4
2 1
C
C
C
A
i Z1
0P
5
B
B i Z1
B
ZB
@max@
Proc. R. Soc. A (2009)
hi xA
i
4 !
112
A
hi xi
CC
2
CC :
AA
ðA 26Þ
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Feasible regions of lamination parameters
1143
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