## Abstract

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 out-of-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.

## 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 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 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 , coupling and out-of-plane stiffness tensors can be expressed as the linear functions of material stiffness invariants and the so-called lamination parameters, . 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, ±45 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, ±45 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 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),(2.1)where ** N** is a vector of resultant running loads;

**is a vector of resultant out-of-plane moments;**

*M*

*ϵ*^{0}is the vector of mid-plane strains; and

**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(2.2)(2.3)(2.4)where the lamination parameters are(2.5)and**

*κ**θ*(

*z*) is the distribution function of the ply orientations through the normalized thickness coordinate . For further details regarding

*θ*(

*z*), see Diaconu

*et al.*(2002

*a*). Additionally, the material invariants are defined as(2.6)where

*Q*

_{ij}are reduced stiffnesses for unidirectional lamina and defined as(2.7)In equation (2.7),

*E*

_{11},

*E*

_{22},

*G*

_{12}are the longitudinal, transverse and shear moduli, respectively;

*ν*

_{12}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.

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 out-of-plane stiffnesses of an orthotrophic laminate using two in-plane or two out-of-plane lamination parameters, respectively,(2.8)where *j*=*A*, *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(2.9)(2.10)where *j*=*A*, *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,(2.11)where *i*=1, …, 4. Note that, for a given value of , there exists a range of values for for all values except for when . Grenestedt & Gudmundson (1993) additionally proved that the feasible region was necessarily convex. Diaconu *et al*. (2002*a*) 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*. (2002*b*) 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(2.12)(2.13)where *i*=1, …, 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, ±45 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 *H*_{A}, *H*_{B}, *H*_{D}, respectively.

*Constraint on the feasible region*. Any constraint that forms part of the boundary of the feasible region of lamination parameters.

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(3.1)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 *n*=1, 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:(3.2)With respect to the coupling lamination parameters, the minimum number of ply orientations to define the boundary of the feasible region is two (where *θ*_{1}≠*θ*_{2}) since a single ply is necessarily symmetric and two plies is the minimal needed for non-zero . Therefore, it is asserted that two plies lie on the boundary of the feasible region of the coupling lamination parameters, , , , defined as(3.3)and *θ*_{1} 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.

For each ply orientation or unique set of two orientations in

*ϕ*, calculate the corresponding vertex of lamination parameters, using equation (3.2) or (3.3), respectively.*v*The set of all

for a given*v**ϕ*is denoted by.*V*The convex hull of

is computed in Matlab using QHULL (Barber*V**et al*. 1996; Bertsekas*et al*. 2003) and the ‘convhull’ function (if*n*=2) or ‘convhulln function’ (if*n*>2).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 *n*=1, the range of , with *j*=*A*, *D* and *k*=1, …, 4 is calculated using equation (3.2) for all *θ*∈*ϕ*. The range of the coupling lamination parameters is similarly calculated using equation (3.3). For *n*=2, 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 (*n*=4 is shown):(3.4)where *j*=*A*, *B* and *D* and(3.5)In equation (3.5), each entry in the determinants, , corresponds to a lamination parameter with *j*=*A*, *B* and *D* and *k*=1, …, 4 for a specific vertex *l*=1, …, 4. Note that at least *n* vertices are necessary to define a hyperplane in an *n*-dimensional space. For simplicity, each hyperplane constraint is normalized with respect to the constant . Thus, equation (3.4) can be rewritten as(3.6)where and *i*=1, …, 4 with the inequality showing the scope of the feasible region.

Note that is a non-zero constant since the set of lamination parameters are linearly independent. Furthermore, if was zero, the resulting hyperplane would pass through the origin in lamination parameter space. Since the origin is not on the boundary of the feasible region, 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 ±45 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, ±30, ±45, ±60 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, ±30, ±45, ±60 degree plies. To demonstrate this feature, each vector of feasible lamination parameters *ξ*, when substituted into each hyperplane constraint, denoted *H*_{j}, where *j*=*A*, *B* and *D*, must be less than or equal to 0, i.e. *H*_{j}(*ξ*)≤0. To achieve this, all combinations of angles in *ϕ* of one to six plies were generated using enumeration. To show this feature graphically, the feasible region of , 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 *ξ* that satisfies the set of hyperplane constraints, a real lay-up is sought: if the Euclidean distance between *ξ* 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 *ξ*. 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(4.1)where(4.2)and *z*_{i} (*i*=1, …, *p*−1) is the normalized through thickness coordinate of each ply. Note that *z*_{0}=−1 and *z*_{p}=1. Coupling and out-of-plane lamination parameters are similarly defined by making the appropriate square and cubic substitutions for *z*_{i} 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.* (2002*b*)(4.3)it is proven in appendix A that on the boundary of the feasible region(4.4)where(4.5)and *x*, *y* in equations (4.3) and (4.4) are integers, where *x*, *y*∈[1,*p*−1], and *x*≠*y*, 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 *k*^{4} to obtain(4.6)where and *j*=*A*, *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 *h*_{5}. The two values of *h*_{5} 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 ±1. It follows that any linear combination of lamination parameters has a lower and upper bound. Therefore, each hyperplane constraint has a lower and an upper bound denoted by *H*^{L} and *H*^{U}, respectively. Specifically, the values of *H*^{L} and *H*^{U} are defined as the minimum and maximum values of . Substituting *H*^{L}, *H*^{U} for *h*_{5} in equation (4.6) gives(4.7)and(4.8)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 out-of-plane lamination parameters when all lamination parameters are explicitly related to one another. Moreover, when the constraints become equalities, then where *i*=1, …, 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.

Using algorithm 3.1 determine, separately, the hyperplane constraints on the in-plane, coupling and out-of-plane feasible regions.

For each hyperplane constraint on the coupling lamination parameters (

*h*_{i}) found in step (i), determine*H*^{L},*H*^{U}using the minimum and maximum of and determine*k*using equation (4.5).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, out-of-plane and coupling lamination parameters for the predefined finite set of 0, 90, ±30, ±45, ±60 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, ±30, ±45, ±60 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.

A random vector,

*ξ*, of 12 lamination parameters is generated.If

*ξ*satisfies the set of constraints then continue to step (iii), otherwise return to step (i).For each

*ξ*that satisfies the constraints, a corresponding lay-up is determined using a discrete optimizer (Bloomfield*et al.*2008).The Euclidean distance between

*ξ*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, ±30, ±45, ±60 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, ±45 degree plies. Additionally, the feasible region for 0, 90, ±30, ±45, ±60 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.

## Acknowledgments

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.

## Footnotes

- Received September 24, 2008.
- Accepted November 26, 2008.

- © 2008 The Royal Society