Design and Optimization of Laminated Composite Materials

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Design and Optimization of Laminated Composite Materials. Zafer Gurdal Virginia Polytechnic Institute and State University

Raphael T. Haftka University of Florida

Prabhat Hajela Rensselaer Polytechnic Institute

@ A Wiley-lnterscience Publication

JOHN WILEY & SONS, INC. New York I Chichester I Weinheim I Brisbane I Singapore I Toronto

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi This book is printed on acid-free paper.

§ 1

Introduction

1

Copyright© 1999 by John Wiley & Sons. All rights reserved.

Introduction to Composite Materials I 4 1.1.1 Classification of Composite Materials I 5 1.1.2 Fiber-Reinforced Composite Materials I 8 1.2 Properties of Laminated Composites I 9 1.2.1 Material Orthotropy I 9 1.2.2 Rule of Mixtures, Complementation, and Interaction I 11 1.2.3 Laminate Definition I 16 1.3 Design of Composite Laminates I 19 1.3.1 Historical Perspective I 19 1.3.2 Material-Related Design Issues I 21 1.4 Design Optimization I 24 1.4.1 Mathematical Optimization I 24 1.4.2 Stacking Sequence Optimization I 29 Exercises I 31 References I 32 1.1

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Library of Congress Cataloging-in-Publication Data: Giirdal, Zafer. Design and optimization of laminated composite materials/Zafer Giirdal, Raphael T. Haftka, Prabhat Hajela. p. em. Includes index. ISBN 0-471-25276-X (hardcover : alk. paper) I. Laminated materials. 2. Composite materials. 3. Structural optimization. I. Haftka, Raphael T. II. Hajela, Prabhat, 1956- . III. Title. TA418.9.L3G87 1998 620.1'18--dc21 98-22855 Printed in the United States of America 10 9 8 7 6 5 4 3 2 I

2

Mechanics of Laminated Composite Materials . . . . . . 33

Governing Equations for Elastic Medium I 34 2.1.1 Strain-Displacement Relations I 34 2.1.2 Stress-Strain Relations I 34 2.1.3 Equilibrium Equations I 36 2.2 In-Plane Response of Isotropic Layer(s) I 39 2.2.1 Plane Stress I 39 2.2.2 Single Isotropic Layer I 41

2.1

v

VI

CONTENTS

2.2.3 Symmetrically Laminated Layers I 43 2.3 Bending Deformations of Isotropic Layer(s) I 49 2.3.1 Bending Response of a Single Layer I 50 2.3.2 Bending Response of Symmetrically Laminated Layers I 52 2.3.3 Bending-Extension Coupling of Unsymmetrically Laminated Layers I 54 2.4 Orthotropic Layers I 61 2.4.1 Stress-Strain Relations for Orthotropic Layers I 62 2.4.2 Orthotropic Layers Oriented at an Angle I 64 2.4.3 Laminates of Orthotropic Plies I 69 2.4.4 Elastic Properties of Composite Laminates I 73 2.5 Properties of Laminates Made of Sublaminates I 83 Exercises I 87 References I 88

3

Hygrothermal Analysis of Laminated Composites . . .

4.1.2 Mathematical Optimization Formulation I 133 4.2 Graphical Solution Procedures I 138 4.2.1 Optimization of Orientations of Layers I 138 4.2.2 Graphical Design of Coefficients of Thermal Expansion I 153 4.2.3 Optimization of Stack Thicknesses I 155 4.3 Dealing with the Discreteness of the Design Problem I 160 Exercises I 166 References I 167

5 Integer Programming

3.1

4

Laminate In-Plane Stiffness Design . . . . . . . . . . . 127 4.1

Design Optimization Problem Formulation I 128 4.1.1 Design Formulation of In-Plane Stiffness Problem I 128

. . . . . . . . . 169

5.1 Integer Linear Programming I 170 5.2 In-plane Stiffness Design as a Linear Integer Programming Problem I 172 5.3 Solution of Integer linear Programming Problems I 177 5.3.1 Enumeration I 177 5.3.2 Branch-and-Bound Algorithm I 180 5.4 Genetic Algorithms I 184 5 .4.1 Design Coding I 185 5.4.2 Initial Population I 188 5.4.3 Selection and Fitness I 191 5.4.4 Crossover I 201 5.4.5 Mutation I 206 5.4.6 Permutation, Ply Addition, and Deletion I 208 5.4.7 Computational Cost and Reliability I 209 Exercises I 218 References I 219

89

Hygrothermal Behavior of Composite Laminates I 90 3.1.1 Temperature and Moisture Diffusion in Composite Laminates I 91 3.1.2 Hygrothermal Deformations I 94 3.1.3 Residual Stresses I 99 3.1.4 Hygrothermal Laminate Analysis and Hygrothermal Loads I 102 3.1.5 Coefficients of Hygrothermal Laminate Expansion I 105 3.2 Laminate Analysis for Combined Mechanical and HygrothermaL Loads I 108 3.3 Hygrothermal Design Considerations I 118 Exercises I 124 References I 125

vii

CONTENTS

6

Failure Criteria for Laminated Composites 6.1

. . . . . . 223

Failure Criteria for Isotropic Layers I 226 6.1.1 Maximum Normal Stress Criterion I 226 6.1.2 Maximum Strain Criterion I 227 6.1.3 Maximum Shear Stress (Tresca) Criterion I 229 6.1.4 Distortional Energy (von Mises) Criterion I 230

VIII

CONTENTS

Failure of Fiber-Reinforced Orthotropic Layers I 231 6.2.1 Maximum Stress and Maximum Strain Criteria I 233 6.2.2 Tsai-Hill Criterion I 237 6.2.3 Tsai-Wu Criterion I 241 6.3 Failure of Laminated Composites I 245 6.3.1 Failure under In-Plane Loads I 249 6.3.2 Failure under Bending Loads I 257 Exercises I 258 References I 259 6.2

7 Strength Design of Laminates .

7.1

. . . . . . . . . . . 261

Graphical Strength Design I 262

7 .I .1 Design for Specified Laminate Strain Limits I 262 7.1.2 Design of Laminates with Two-Fiber Orientations I 266 7.1.3. Design of Multiple-Ply Laminates with Discrete Fiber Orientations I 271 7.2 Numerical Strength Optimization Using continuous Variables I 274 7.2.1

Strength Design with Thickness Design Variables I 274 7 .2.2 Strength Design with Orientation Angle Design Variables I 286 7.3 Numerical Strength Optimization Using Discrete Variables I 288 7.3.1

Integer Linear Programming for Strength Design I 289 7.3.2 Genetic Algorithms for Strength Design I 292 Exercises I 294 References I 295 8

Laminate Design for Flexural and Combined Response . . . . . . . . . . . . . . . . . . . . . . . . . 297

8.1

Flexural response Equations I 298

8.2 Stiffness Design by Miki's Graphical Procedure I 301

CONTENTS

IX

8.2.1 Linear Problems I 30 l 8.2.2 Changes in Stacking Sequence I 306 8.2.3 Nonlinear Problems I 308 8.3 Flexural Stiffness Design by Integer Linear Programming I 310 8.3.1 Ply-Identity and Stack-Identity Design Variables I 310 8.3.2 Stiffness Design with Fixed Thickness I 313 8.3.3 Buckling Load Maximization with Stiffness and Strength Constraints I 317 8.3.4 Stiffness Design for Minimum Thickness I 322 Exercises I 328 References I 328 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

PREFACE

With rapid growth of the use of composite materials in many commercial products ranging from sports equipment to high-performance aircraft, literature on composite materials has proliferated. At the time of the publication of this book a simple search of a popular web site for books with the words "composite materials" in their title yielded more than 250 entries. Many of these titles are well-written textbooks on mechanics of composite materials and have been adopted by educational institutions for introductory courses. As the application of composites to commercial products has increased, so has the need for literature that focuses on the design aspects of these materials. However, the number of titles that focus on the mechanics of composites far outnumbers those dealing with design. In particular, books that focus on optimal design of composite materials virtually do not exist. It is the intent of this book to introduce readers to the emerging field of optimal design of laminated composite materials. The first and the foremost reason for writing this book was the desire to acquaint students with the latest techniques in the field. For many years, designers have treated optimization problems involving composite materials with continuous optimization techniques that were ill suited for these problems. The design of a composite laminate stacking sequence generally involves selecting discrete layer thickness and orientation angles-a discrete optimization problem. Researchers in this field have more recently focused on numerical and graphical methods useful for the solution of such problems; this book mirrors that focus. In particular, the book places emphasis on graphical design techniques developed by Professor Miki from Japan. These techniques allow representation of even the most complicated stacking sequences using two parameters bounded by a parabola and provide extremely valuable insight into the multiplicity of solutions available for laminate design problems. Another important motivation for the book was the need to provide condensed coverage that would be of use to the design engineer. Dexi

'"'

PREFACE

sign of composite materials and structures requires both a thorough understanding of the mechanics of laminated composites and familiarity with optimization techniques that enable designers to find practical laminate configurations in an efficient manner. At present, a student wanting to learn about the application of optimization techniques to composite design will need to take a separate course in each subject. This is somewhat difficult within the constraints of an undergraduate curriculum. The book combines the study of the mechanics of composite laminates with optimization methods that are most useful for the design of such laminates. This book has been developed for senior-level undergraduate or early graduate courses in numerical design methods for laminated composite materials. Applications of composite materials have traditionally originated in weight-critical aerospace structures. More recently, these materials have become popular in civil engineering infrastructure applications (such as bridge and building construction) and mechanical engineering applications (such as mechanisms and lightweight robotic structures). Therefore, the book may be used in aerospace engineering, civil engineering, engineering science and mechanics, and mechanical engineering curricula. In addition, the book has technical material useful for practicing engineers in related fields. Researchers in composite materials are likely to benefit from state of the art methods introduced in the book. The first chapter reviews the types of composite materials in use and the terminology established for their description. The types of composites considered in this book are then identified and their properties are discussed within the context of mechanics. A brief review of the design issues relevant to composite materials is included. The chapter concludes with an introduction to the terminology and formulation of mathematical optimization problems, with special emphasis on laminate design problems. The second and third chapters introduce the basic equations and assumptions used in the analysis of laminated composites under mechanical and thermal loads. They emphasize the computation of elastic properties as functions of variables that can be changed during the design process and the effects of such changes on response quantities such as stresses and strains. Chapter 4 formulates the in-plane stiffness design as an optimization problem and introduces a simple graphical technique for its so-

PREFACE

XIII

lution. Also provided is a technique to handle the discrete nature of the thickness and orientation design variables. Two formal procedures, namely integer linear programming and genetic algorithms, suitable for handling discrete optimization problems specific to composite laminate design are introduced in Chapter 5. In particular, the formulation and solution of the in-plane stiffness design· problem is demonstrated. Chapters 6 and 7 address strength analysis and design, respectively. Commonly used failure criteria for laminated composite materials and their implementation for strength analyses are introduced in Chapter 6. Chapter 7 describes the implementation of strength constraints in design optimization based on graphical and mathematical optimization procedures. Finally, Chapter 8 introduces analysis and design for bending requirements. These include the transverse displacement of a simply supported laminate loaded by transverse loads, its natural vibration frequencies, and the buckling response of a simply supported laminate under in-plane loads. . We have used the material in our respective institutions for a combined senior-level undergraduate and first-year graduate course for several years. For a one-semester course with students who have no previous background in composites or optimization, we recommend that the course cover most of the material in Chapters 1, 2, 4, and 5, and parts of Chapter 6. In addition, it is probably possible to cover material from one more chapter. Depending on the emphasis of the course, Chapters 3, 7, or 8, which focus on thermomechanical, strength, or bending design characteristics, respectively, may be added. It is also possible to cover a combination of sections from the remaining chapters as the instructor sees fit. The authors wish to express their appreciation for many valuable suggestions from former students in courses that led to the development of this book. Thanks also go to the authors' respective universities for providing the opportunities to teach courses directly relevant to the book's content, and their fellow faculty members for providing valuable input and stimulating discussions. Special thanks are also due Professor Mitsunori Miki of the Doshisha University, Japan for introducing us to the graphical representation of laminate optimization problems, which is heavily used in the book. The authors also appreciate the input and suggestions provided by various individuals, in

'"" PREFACE

particular, Professor Valery V. Vasiliev of Moscow State University of Aviation Technology, Professor Ron Kander of Virginia Tech, Professor Giinay Anla~+ of Bogazic;i University, Dr. Walter Dauksher of Boeing, for reviewing some of the chapters of the book. The authors would also like to thank Dr. James H. Starnes, Jr., of NASA Langley Research Center. He has sponsored research leading to many of the results reported in the textbook, and his insight into design issues for composite materials enriched our appreciation of the subject. Zafer Gtirdal Raphael T. Haftka Prabhat Hajela

f,

I

1 INTRODUCTION

Structural designers seek the best possible design, be it a vehicle structure, a civil engineering structure, or a space structure, while using the least amount of resources. The measure of goodness of a design depends on the application, typically related to strength or stiffness, while resources are measured in terms of weight or cost. Therefore, the best design often means either the lowest weight (or cost) with limitations on the stiffness (or strength) properties or the maximum stiffness design with prescribed resources and strength limits. Traditionally, engineers have relied on experience to achieve such designs. For a given application, first a set of essential requirements are identified and designs that satisfy these requirements are obtained. Next, structural modifications that are likely to improve the performance or reduce the weight or the cost are implemented. The tasks, stated so simply in two sentences, are often extremely tedious and require a large number of iterations by the designer. For example, changes in the structural dimensions (cross-sectional areas, thicknesses, member lengths) implemented to improve the performance may yield designs that violate the strength or stiffness requirements. Sometimes these requirements are difficult to satisfy and may require many iterations by the designer. In some cases, it is so difficult to satisfy these requirements that the first design ever identified to satisfy them becomes the final design. 1

3 2

INTRODUCTION

Over the past two decades, mathematical optimization, which deals with either the maximization or minimization of an objective function subject to constraint functions, has emerged as a powerful tool for structural design. The use of mathematical optimization for design relieves the designer of the burden of repeated iterations and transforms the design process into a systematic well-organized activity. Within the context of optimization, the weight or the performance becomes the objective function and the variables controlling the structural dimensions (such as thicknesses or cross-sectional areas of members) are design variables that are sized to achieve the best configuration. Formulation of the engineering design optimization, structural optimization, and the methods of solution are topics covered in extensive detail in a number of textbooks;, see, for example, Kirsch (1993), Vanderplaats (1998), Arora (1989), and Haftka and Gtirdal (1993). However, none of these books covers in detail a subset of the field of structural optimization, namely optimization of structures made of composite materials, with the exception of the last one, which has a chapter devoted to optimization of laminated composites. Use of composite materials in structural design has gained popularity over the past three decades because of several advantages that these materials offer compared to traditional structural materials, such as steel, aluminum, and various alloys. One of the primary reasons for their popularity is their weight advantage. Composite materials such as Graphite/Epoxy and Glass/Epoxy have smaller weight densities compared to metallic materials. For example, the weight densities of high-strength Graphite/Epoxy and Glass/Epoxy are 0.056 lb/in3 and 0.065 lblin 3 , respectively, compared to the weight density of Aluminum which is 0.10 lb/in 3 • In addition to their weight advantage per unit volume, some composites provide better stiffness and strength properties compared to metals as well. That is, structural members made out of composite materials may undergo smaller deformations, and carry larger static loads than their metallic counterparts. Stiffness of high strength Graphite/Epoxy, for example, is around 22 x 106 lb/in 2 compared to Aluminum's stiffness of 10 x 106 lb/in 2 . The weight advantage may also be reflected in the performance properties described above by introducing specific stiffness and specific strength, defined by dividing the respective property with the material density. Therefore, even if the stiffness and/or strength performance of a composite material is comparable to that of a conventional alloy, the advantages

INTRODUCTION

of high specific stiffness and/or specific strength make composites more attractive than alloys. Composite materials are also known to perform better under cyclic loads than metallic materials because of their fatigue resistance. It was these advantages that initially stimulated widespread use of composites in structural design, especially for weight-critical applica~ tions such as design of spacecraft and aircraft; see, for example Niu (1992) and Zagainov and Lozino-Lozinski (1996). Beyond their weight advantage, the use of composite materials in structural design had significant implications in terms of the design problem formulation, particularly in terms of optimal design. As will be explained in · this chapter and in a subsequent chapter, with the introduction of composite materials into a design problem formulation, designers obtained a new flexibility through the use of variables that directly change the properties of the material, and therefore optimal design of structures has acquired a new meaning. In order to improve structural performance and meet the requirements of a specific design situation, it is now possible to tailor the properties of the structural material in addition to structural dimensions. Design of a material, however, is not a task that can be taken lightly. Many of the properties of composite materials are unfamiliar to many design engineers who have no formal training in composite materials. There are also various composite material response mechanisms that are not fully understood and are still topics for continuing research. Therefore, most designers limit themselves to aspects of material response that are rather well defined for engineering applications. Another area which is not fully investigated is the integration of the design and manufacturing aspects of composite structures. There are a number of different ways to manufacture a composite part or a structure, but not all configurations designed to tailor the material properties are manufacturable. Therefore, most designers limit themselves to configurations known to be manufacturable, thereby limiting the performance gains that can be achieved through the use of composites. However, the fast pace of advances in composite material manufacturing technology is forging ahead of the present state of established design practices. Therefore, there is a continued need for developing tools and methodologies for design of composite materials and structures to achieve maximal performance gains.

4

INTRODUCTION

The application of the methods of mathematical optimization to the design of structures made of composite materials attracted the attention of many researchers. However, the approach has not been fully accepted by practicing engineers, primarily because of the impracticality of many designs obtained via the optimization process. Early design optimization research studies employed mainly extensions of the approaches used for designing structures made of traditional materials and lacked the ability to handle features that are unique to composite materials. For example, composite material design calls for use of design variables that assume values only from a discrete set of prescribed values. Use of traditional optimization techniques, which treat design variables as continuous valued, produces designs with limited practical value. Only recently has research on the design of composite structures started to introduce features that are important to .attain practical designs. This book attempts to address the unique design features associated with a limited class of composite materials, called laminated composite materials, via the application of the methods of mathematical optimization. In subsequent sections of this chapter, we first provide an introduction to the types of composite materials in use, with the intent of familiarizing the reader with general terminology in the materials field. The types of composites t.hat are specifically considered in this book are then identified, and the properties of such composites are discussed within the context of mechanics. A very brief and probably incomplete review of composite material design issues is included to warn designers about the pitfalls associated with tailoring the properties of a composite material. The chapter concludes with a brief discussion of the terminology and formulation of mathematical optimization problems, with a special emphasis on laminate design problems.

1.1

INTRODUCTION TO COMPOSITE MATERIALS

The most general definition of a composite material is very closely related to the dictionary definition of the word composite, meaning made up of different parts or materials. Composite materials are constructed of two or more materials, commonly referred to as constituents, and have characteristics derived from the individual constituents. Depending on the manner in which the constituents are put together,

1.1


5

the resulting composite materials may have the combined characteristics of the constituents or have substantially different properties than the individual constituents. As explained later, sometimes the properties of the composite even exceed those of the constituents. The following classification is given by Schwartz ( 1984).

1.1.1

Classification of Composite Materials

One approach to the general classification of composite materials is based on the nature of the constituent materials. The constituents may be either organic or inorganic. The designation of organic refers to materials originating from plants or animals or materials of hydrocarbon origin (natural or synthetic) such as carbon. Inorganic materials are those that cannot be classified as organic matter, for example, metals and minerals. Historically, perhaps the most common structural material of organic nature has been wood. Wood is a composite material because it is composed of two distinct constituents: stiff and strong fibers surrounded by a supporting structure of softer cells. In fact, most modern composite materials imitate wood in that they consist of strong fibers embedded in softer supporting material. The supporting material in such an arrangement is commonly referred to as matrix material. Before getting into a more complete description of the modern composite materials, we continue with general descriptions. For most highly loaded single-material structures, the use of purely organic materials is avoided because of their generally low stiffness and strength properties. They are also sensitive to environmental effects such as temperature and moisture and have low resistance to chemicals and solvents. Despite their disadvantages, organic materials such as polymers are commonly included in composites to bring in certain characteristics that may not be achievable by using only inorganic constituents. For inorganic composites, the constituents may be all metallic, all nonmetallic, or a combination of the two. The most common inorganic materials used for highly loaded structures are metals. Despite their good stiffness and strength properties, metals have high specific weight-a disadvantage in weight-critical applications, such as aircraft and spacecraft. Therefore, metallic composites are used mostly when there is need for specific properties, such as resistance to high temperatures. As a matter of fact, in contrast to single

8

INTRODUCTION

r !

1.1

material structures, which are mostly metallic, most frequently used composite materials are made up of all nonmetallic constituents, which include polymeric constituents as well as inorganic glasslike substances. Polymer matrix materials are made of long chain of organic molecules, generally classified as thermoset or thermoplastic. The two categories have substantially different characteristics as matrix materials. Fibers preimpregnated with thermoset matrix (thermoset prepregs) are soft and malleable. They are tacky and can be draped easily, hence allowing fabrication of t:omplex shapes with ease. They require low processing temperatures but long curing times. They undergo a chemical change (cross-linking of polymer chains) during curing, and the process is irreversible. Thermoplastics, on the other hand, do not require curing and they can be reshaped and reused. However, thermoplastic prepregs are hard and boardlike and lack drape and tack; making them harder to handle. They also require very high processing temperatures to shape them, although processing time can be very short. Thermoplastics also offer high resistance to moisture and can operate at higher temperatures of up to 500°F (260°C) compared to thermoset operating temperatures of about 250°F (121 °C). Typical thermoset matrices include Epoxy, Polyimide, Polyester, and Phenolic materials. Among popular thermoplastics are Polyethylene, Polystyrene, and Polyetheretherketone (PEEK) materials. A more complete discussion of advantages and disadvantages of thermosets and thermoplastics is provided by Niu (1992). A more traditional classification than the one based on the nature of the constituents is derived from their form. Several examples of composite materials with different forms are presented in Fig. 1.1. Particulate composites are generally made up of a randomly dispersed hard particle constituent in a softer matrix. Examples of particulate composites are metal particles in metallic, plastic, or ceramic matrices. A widely used particulate composite is concrete in which gravel is mixed in cement. Flake composites, as the name suggests, are formed by adding thin flakes to the matrix material. Although flake dispersion in the matrix is generally random, the flakes may be made to align with one another, forming a more orderly structure compared to particulate composites. This alignment provides for more uniform properties in the plane of the flakes than with particulate composites. Typical examples of flake

7


Particulate composite

Fiber reinforced composite

Flake composite

Laminated composite

Figure 1.1. Composite materials with different forms of constituents.

materials are glass, mica, metals, and carbon. The size, shape, and material of the flake to be used depend on the type of application, which also determines the amount of matrix material. The matrix material (either plastics, metals, or epoxy resins) used in a flake composite may make up the bulk of the composite or be in small amounts just sufficient to provide bonding of the flakes. For example, aluminum flakes are sometimes used in molded plastic parts to provide decorative effects. In such a composite, the plastic matrix makes up the bulk of the composite. Fiber-reinforced composites (or fibrous composites) are the most commonly used form of the constituent combinations. The fibers of such composites are generally strong and stiff and therefore serve as the primary load-carrying constituent. The matrix holds the fibers together and serves as an agent to redistribute the loads from a broken fiber to the adjacent fibers in the material when fibers start failing under excessive loads. This property of the matrix constituent contributes to one of the most important characteristics of the fibrous composites, namely, improved strength compared to the individual constituents. This is elaborated upon in Section 1.2.2, which deals with the different mechanisms by which the properties of composites are obtained from those of its constituents.

l

INTRODUCTION

The last form of composite materials is thin layers of material fully bonded together to form so-called composite laminates. The layers of a laminated composite material may be different single materials, such as clad metals that are bonded together or the same material, such as wood put together with different orientations. The layers may be composites themselves, such as fibrous composite layers placed so that different layers have different characteristics. This type of composite is the most commonly encountered laminated composite material used in design of high-performance structures and is the primary subject of this book. discussed next.The various forms of laminated composite materials are

9

1.2 PROPERTIES OF LAMINATED COMPOSITES

.~~~/ Random short fibers

Oriented short fibers

Continuous fibers

.7

Plain Fibrous Layers

1.1.2 Fiber-Reinforced Composite Materials An individual layer of a laminated composite material may assume a number of different forms, depending on the arrangement of the fiber constituent (see Fig. 1.2). The layers may be composed of short fibers embedded in a matrix. The short fibers may be distributed at random orientations, or they may be aligned in some manner forming oriented short-fiber composites. A typical example of a random short-fiber composite is fiberglass. Continuous fiber-reinforced composite layers are made up of bundles of small-diameter circular fibers. Typically, the radii of these fibers are in the order of 0.0002 in (0.005 mm), such as the radius of carbon fibers. The largest diameter fibers, such as boron fibers, are of the order of 0.002 in (0.05 mm). Continuous fiber-reinforced composite materials are commercially available in the form of unidirectional tape, with fibers aligned along the length of the tape. The fibers of the tape are preimpregnated with the matrix material, and for this reason the tape is sometimes referred to as a prepreg. As discussed in Section 1.2, this arrangement of aligning the fibers in a given direction ·provides the unique feature of the material properties of fiberreinforced composites. Another form of continuous fiber-reinforced composite layers is the woven fabric type composite, where fiber tows, which are large bundles of fibers (generally 10,000 or more fibers), are woven in two or more directions (see Fig. 1.2). Various fabric types with different weave features producing different material characteristics are available (see, for example, Middleton, 1990).

1.2 PROPERTIES OF LAMINATED COMPOSITES 1.2.1

Material Orthotropy

Composite layer properties (such as strength, stiffness, thermal and moisture conductivity, wear and environmental resistance) strongly depend on the form of the reinforcement in the laminate. The directional nature of the fibers in a fiber-reinforced laminate introduces directional dependence to most of those properties. Materials whose properties are independent of direction are called isotropic materials. Conversely, those materials with different properties in different directions are called anisotropic. A special case of anisotropy is the existence of two mutually perpendicular planes of symmetry in material properties. Such materials are referred to as orthotropic. Fibrous composites with either oriented short fibers or continuous fibers are orthotropic in nature. In such composites, properties are defined in

IV

INTRODUCTION

the plane of the layer in two directions-the direction along the fibers and the direction perpendicular to the fiber orientation. A typical example of an orthotropic material property of unidirectional fiber-reinforced composites is the stiffness. The matrix portion of the composite, which holds the fibers together, is generally isotropic. For all practical purposes, the fibers, which usually have much higher stiffness than the matrix material, are also isotropic. However, when combined together, the properties of the composite are no longer isotropic. Consider, for example, the fiber-reinforced material shown in Fig. 1.3 loaded along either the fiber direction, xl' or transverse to the fiber direction, x2 . When the material is loaded along the fiber direction, the deformation is smaller compared to the deformation under a load of the same magnitude applied in a direction perpendicular to the fiber. Since the amount of deformation under specified load reflects the stiffness of a material, the unidirectional composite has different stiffness properties along these two mutually perpendicular directions. The stiffness of the composite in the fiber direction is much closer to that of the fiber stiffness, E1 , and the stiffness perpendicular to the fiber direction is governed mostly by the properties of the matrix material, Em, Gm, and vm. Directions that are along the fiber and perpendicular to the fiber directions are commonly referred to as the principal material directions or principal axes of the material. Four elastic stiffness properties, commonly referred to as the engineering constants, fully describe the mechanical properties of an orthotropic layer in its plane (more detailed discussion of the stiffness properties is given in Section 2.4). These are two Young's moduli, 1 E and E2, along the fiber and transverse to the fiber directions, respectively, and the shear modulus G 12 and Poisson's ratio v in the 12

~ ·g

Table 1.1.

Ef' Gf' v1

Em, Gm, vm

Properties of various fiber-reinforced composite layers

E1 Constituents

Material T300/5208 AS4/3501 8(4)/5505 Kevlar49/Ep Scotchply1002 Source:

Graphite/Epoxy Graphite/Epoxy Boron/Epoxy Aramid/Epoxy Glass/Epoxy

E2

r;

, '"

106 psi (GPa) 106 psi (GPa) 106 psi (GPa) 26.3 (181) 20.0 (138) 29.6 (204) 11.0 (76) 5.60 (38.6)

1.49 (10.3) 1.30 (8.96) 2.68 (18.5) 0.80 (5.50) 1.20 (8.27)

1.04 (7.17) 1.03 (7.10) 0.81 (5.59) 0.33 (2.30) 0.60 (4.14)

1/12

0.28 0.30 0.23 0.34 0.26

vJ 0.7 0.66 . 0.5 0.6 0.45

Tsai and Hahn, 1980.

plane of the layer. As discussed in the next section, these properties are related to the amount of fiber present in the composite layer. A quantity, "f. referred to as the fiber volume fraction, is commonly used to measure the amount of fiber in a composite material. Typical stiffness properties of various unidirectional composite materials systems are given in Table 1.1 along with the fiber volume fraction of the composite. Some of the properties of a fiber-reinforced composite can be estimated from the properties of its constituents by simple calculations. Others have to be measured. There are three different mechanisms Jl~Jo._o_htain_properties of the composite from those of its constituents. As shown in the next section, properties of different natures are determined through different m:ecnanisms.

1.2.2

.. XI

11


( \_..&:. ~ '-"-,p ~ Vf.-rj ML C+Urv.0 Rule of Mixtur-es,_ Complementation, and Interaction

One of the ways to estimate composite material properties is summation of the properties of the individual constituents based on their contribution to the overall material volume. This method is commonly referred to as the rule of mixtures. The rule of mixtures employs the volume fraction of the constituents to estimate the properties of the composite. For example, in the case of a continuous fiber-reinforced composite layer, we have a fiber volume fraction V1 and a matrix volume fraction Vm, which must satisfy

Figure 1.3. Elastic properties of a unidirectional fiber-reinforced composite layer.

VI+ Vm

= 1.

(1.2.1)

\

12 INTRODUCTION

Based on the rule of mixtures, a property p is estimated from the constituent properties, p and Pm' as

13


matrix

fiber

matrix

1

P =P1~+ Pm Vm

=P1~+ Pm(l -

~).

(1.2.2)

For example, the longitudinal (fiber direction) stiffness property, EP of the composite may be calculated from the Young's moduli of the constituents E1 and Em, using this rule of mixtures as El =EI~+EmVm.

(1.2.3)

Equation (1.2.3) may be derived with analogy to the calculation of the overall stiffness of two springs connected in parallel; see Fig. 1.4. The fibers are assumed to be fully bonded to the matrix, and the total end-deformation o of the composite is identical in the fiber and the matrix, <51 = om= o. The total force causing the deformation is carried partly by the fibers and partly by the matrix P=P1 +Pm.

(1.2.4)

Assuming that each constituent acts as an axial bar with a forcedisplacement relation o= PU AE, where A is the cross-sectional area, Eq. (1.2.4) becomes oAE1

-y- =

oA1Et oAmEm L + -y--.

p

i-··-:::1~)------,', ,,"

,-t.-·'

!--.!

' o= B1 +B2

unit volume

stiffness representation

Figure 1.5. Rule-of-mixtures model of a transverse stiffness of a composite layer.

For the elastic modulus in a direction perpendicular to the fibers, the calculations and assumptions may require a more sophisticated model that includes the fiber shape and the amount of fiber in the matrix. However, good estimates for the modulus are obtained by modelling the fiber and the matrix as two elastic springs connected in series; see Fig. 1.5. With this model, the total deformation at the point of load application in a direction perpendicular to the fiber is the sum of the deflections experienced by the fiber and the matrix. The resulting expression for the transverse modulus (derivation is left to the reader as an exercise) is

vm

_!___~ E-E+E.

(1.2.5)

2

Since the cross-sectional areas are proportional to fiber fractions ~A, Am= VmA), we obtain Eq. (1.2.3).

(A1=

'f

(1.2.6)

m

Equation (1.2.6) may be interpreted as the rule of mixtures, Eq. (1.2.2), applied to the flexibility liE. Using similar arguments, Poisson's ratio and the shear modulus for the composite can be obtained in the following form: V12 = v1~+ Vm Vm

and

_l_=_!i+ Gl2

G!

vm em·

(1.2.7)

p

:--.: unit volume

Example 1.2.1

0 stiffness representation

Figure 1.4. Rule-of-mixtures model of a longitudinal stiffness of a composite layer.

The elastic modulus and the weight density of a typical graphite fiber and an epoxy resin are E1 = 230 GPa, p1 = 17.2 kN/m 3 , and Em= 3.45 GPa, Pm = 12.0 kN/m 3 , respectively. Assuming the

14

INTRODUCTION

packing of the fibers to be represented by the unit volume shown in Fig. 1.4, plot the longitudinal specific modulus, E/p, and the transverse specific modulus, E 2 /p, as a function of the fiber volume fraction 0.0 ~ vf~ v__rax for a unidirectional fiber-reinforced Graphite/Epoxy composite system. Note that p is the weight density of the composite system. We first express the specific moduli in the longitudinal and transverse directions, E 1, E 2, and the composite density, p, in terms of the fiber volume fraction. From Eq. (1.2.2), the longitudinal modulus and the weight density are E 1 = Er~+ Em(l- ~)

= 3.45 + 226.55 V1 (GPa)

P = p!Vf+ Pm(l- V1) = 1~.0 ::f" 5.2 V1 (kN/m3), and from Eq. (1.2.6) the transverse modulus is E2 =

E!Em ~Em

793.5

=

= p

E2

p

(3.45 + 226.55

Vd X 10

12.0 + 5.2

6

vr

vf".. = !t0.52 =o.7854. 0.8 3 0.6

+--..J.___ 0

0.2

__!__-1...._--.J.

0.4 (a)

0.6

0.8

v, 0

0.2

0.4

0.6

0.8

v,

(b)

Figure 1.6. Normalized composite longitudinal (a) and transverse (b) specific stiffnesses as a function of the fiber volume fraction.

(GPa).

+ ( 1 - Vr) E1 230 - 226.55 Vr

The specific stiffnesses are then given as, E1

15


(m)

'

( ) 7.935 X 105 2 m. 2.76- 1.5226 v1 - 1.17806 v

1

The fiber volume fraction can vary from zero to the maximum value of vrx. where the maximum value is determined from the geometry of the fiber packing in the unit volume. Based on the packing geometry provided in Fig. 1.4, the maximum possible fiber radius is equal to half the unit edge of the volume. Therefore,

vrx = 1t 0.5 2 = 0.7854. The composite longitudinal specific stiffness (E/p) normalized by the fiber specific stiffness (E/p1 = 13.37 x 106 m), and the transverse specific stiffness (E2 / p) normalized by the matrix spe-

cific stiffness (Em!Pm = 276 km) are shown in Fig. 1.6 as a function of the fiber volume fraction. The maximum longitudinal and transverse stiffnesses are achieved for the largest allowable fiber volume fraction of vrx = 0.7854. The magnitude of the longitudinal specific stiffness at that point is about 84% of that of the fiber material, and the magnitude of the transverse specific stiffness is more than 3.3 times that of the matrix material specific stiffness. Even larger values of the specific stiffnesses may be possible for other fiber packing models (see, for example, Exercise 2, at the end of this chapter) which allow larger fiber volume fractions. Finally, we point to the fact that, since the fiber weight density is larger than that of the matrix, as the fiber volume fraction increases the density of the composite also increases linearly according to the rule of mixtures equation for the composite density. Therefore, even though the largest stiffnesses are achieved for large fiber volume fraction composites, weight considerations may favor low fiber volume fraction composites as long as stiffness requirements are met. This is demonstrated later in Example 1.4.1, in which we formulate a simple design optimization problem.

16

INTRODUCTION

Complementation is another mechanism for the composite material to inherit properties from those of its constituents. Complementation means that one of the constituents contributes a distinct property that is not present in other constituents, and the resulting composite acquires this property. For example, most matrix materials do not conduct electricity. Combining such matrix materials with fibers having good electrical conductivity results in a conductive composite. Also, many polymer matrix composite materials are sensitive to environmental conditions such as moisture. Adding a surface layer that seals the composite against such effects is an effective solution and demonstrates the complementation mechanism. The last mechanism, interaction, is one of the most important mechanisms for obtaining superior composite properties. It is the only mechanism that can cause a composite property to exceed those of the individual constituents. For example, the tensile strength of a glass fiber-reinforced plastic is larger than both the tensile strengths of the plastic matrix and the glass fiber. The explanation of this phenomenon is in the interaction of the fiber and the matrix via load transfer from one to the other. The tensile strength of a fiber is generally governed by defects in the fiber, which are likely to repeat over the length of the fiber. For an unsupported fiber, failure of a single defect is enough to limit the load-carrying capability. On the other hand, for fibers supported in a matrix, upon failure of a fiber due to a defect at one location, the load in the fiber is locally transferred through the matrix to neighboring fibers. Therefore, the load can be increased beyond the first fiber failure load to a point where multiple fiber failures degrade the load-carrying capability of a larger region of the composite and result in an unstable growth of the fiber failures.

1.2.3 Laminate Definition In the preceding paragraphs we discussed the properties of a single fiber-reinforced composite layer. As stated earlier, the main emphasis of this book is on the design of laminated composite material where the layers of unidirectional fiber-reinforced composites are stacked on top of one another. Such laminates are described according to a standard notation called stacking sequence, which is described in the following.


17

The stacking sequence lists fiber orientations measured_from a reference axis of the laminate. If the orientation is counterclockwise from the reference direction, it is considered to be positive. The standard stacking sequence lists orientations of the different layers, starting from the top of the laminate to the bottom, in a string separated by slashes. For a laminate with N layers, each made of the same composite material and of the same thickness, t, starting with the top layer with a fiber orientation e" the laminate is represented as

[9/9/ · · · ;eN].

( 1.2.8)

The thickness of each layer, t, in a consolidated form in the laminate is generally provided by the manufacturer's specifications. The total thickness, h, of the laminate is h = tN. Layers oriented at an angle from the reference axes of the laminate are called off-axis layers. When the orientation of a layer coincides with one of the reference axes of the laminate, e = oo ore= 90°, that layer is referred to as an on-axis layer. When several layers with the same orientation are adjacent to one another, it is common to group them together and represent the total number of adjacent layers with the same orientation as a subscript to that particular orientation. For example, the laminate [Oi45 4 / - 45 2] has the top two layers oriented along the reference axis of the laminate, followed by four layers oriented at 45° from the reference axis, followed by two layers of a -45° orientation. When a group of layers is repeated, then the number of repetitions is used as a subscript to the repeating group enclosed in parenthesis. For the following laminate, [Oi(02 /45 2 /90)/0 2], sandwiched between the two layers of oo on the top and two layers of oo at the bottom, the 02 /45 2 /90 group is repeated three times. A laminate is symmetric when the fiber orientations of the bottom half of the laminate are mirror images of the fiber orientations above the mid-plane of the laminate, for example, [-45/30/0/45/45/0/30/45Jr (the subscript T is used after the bracket to indicate that the designation is for the total laminate). Symmetric laminates with an even number of layers are represented by the portion of the stacking sequence above the laminate mid-plane followed by a subscript s after the closing bracket, [-45/30/0/45],. If the layers within the brackets are repeated, the number of repetitions can also be placed after the

18

INTRODUCTION

bracket before the subscript s. That is, the laminate [4510z14510zl 4510z!Oz14510z14510z145Jr is represented in a compact form as [45!02] 3s. In some situations it is desirable to place a negative 9 orientation for every occurrence of the positive 9 layer. Such laminates are referred to as balanced laminates. Pairs of positive and negative orientations do not have to be placed adjacent to each other, but if they are adjacent, the laminate designation may be condensed by putting a plus-minus sign in front of the orientation angle. For example, the [30/ -30/30/ -30h laminate is simply designated as [± 30zJr The [30/30/30/-30h stacking sequence, on the other hand, is designated by [+30]s. In addition to the classifications presented above, several unique cases of laminate stacking sequence definitions are sometimes used. Laminates that have alternating orientations of 0° and goo plies are called cross-ply laminates. In cross-ply laminates, grouping of more than one layer with the same orientation angle is allowed. For example, the symmetric laminate [g02/0J, is a cross-ply laminate although the first two layers near the surfaces of the laminate do not alternate but are of the same orientation angle of goo. Another special case is the angle-ply laminate. All the layers of an angle-ply laminate have the same fiber orientation angle, say 9 = a, with an alternating sign. For example, a 30° symmetric angle-ply laminate could be [30/-30/ 30/-30], (or [±30JJ Finally, a laminate is said to be antisymmetric if the magnitude of the ply orientation angle below the laminate mid-plane is a mirror image of the ply orientations above the mid-plane with signs reversed. For example, [-45/45/ -45/45/ -45/45/ -45/45Jr is an antisymmetric laminate. Antisymmetric laminates with a mixture of off-axis ply orientations (i.e., 9 * oo or goo) are also possible, such as [+301+45/ +451+30h. Note that this last laminate cannot be designated as [+301+45]. because of antisymmetry with respect to the mid-plane (expanded representation of such a symmetric laminate would have been [+301+45/±45/±30h). Antisymmetric laminates always have an even number of equal thickness layers. In the case of cross-ply laminates, antisymmetry implies change of the ply orientation angle from oo to goo or from goo to 0° from one side of the laminate mid-plane to the other. For example, [O/go;o;go]r and [OfgOiOigO]r laminates are antisymmetric cross-ply laminates.

1.3 DESIGN OF COMPOSITE LAMINATES

19

1.3 DESIGN OF COMPOSITE LAMINATES Laminated fibrous composite materials are finding a wide range of applications in structural design, especially for lightweight structures that have stringent stiffness and strength requirements. While they are attractive replacements for metallic materials for many structural applications, the analysis and design of these materials are considerably more complex than those of metallic structures. As mentioned earlier, finding an efficient composite structural design that meets all requirements of a specific application can be achieved not only by sizing the cross-sectional areas and member thicknesses, but also by tailoring of the material properties through selective choice of orientation, number, and stacking sequence of the layers that make up the composite laminate.

1.3.1

Historical Perspective

The field of composite design engineering has continued to evolve over the years. In spite of the distinct advantage of composites for tailoring material properties, most early applications of composites were aimed strictly at weight reduction. Metals were replaced with lighter composites with little or no emphasis placed on tailoring the properties. In some instances, designers created quasi-isotropic laminates that largely suppressed the directional properties of the unidirectional layers and made the laminate behave in a manner similar to that of an isotropic material. One such laminate has equal percentages of 0°, +45°, -45°, and goo layers placed symmetrically with respect to the laminate mid-plane, for example [±45/gO/Ol,. Like isotropic materials that have the same elastic properties in every direction, quasiisotropic laminates have elastic properties that are independent of the direction in the plane of the laminate (an in-depth description of quasiisotropic laminates is provided in Chapter 2). Quasi-isotropic laminates were, therefore, a convenient replacement for isotropic materials in weight critical applications; weight savings could be achieved by simply replacing the isotropic material with a similar stiffness laminate that was lighter and probably stronger. For example, the elastic modulus of a [±45/0/gOL T300/5208 Graphite/Epoxy laminate is about E = 70 GPa, which is less than half of the stiffness of the unidirectional material. This stiffness value is comparable to the elastic modulus of Aluminum,

20

INTRODUCTION

which is roughly Eat = 73 GPa. However, the weight density of the Graphite/Epoxy is approximately p = 15 kN/m 3 compared to the weight density of Aluminum, Pat = 26 kN!m\ yielding a specific stiffness of Elp = 4.7 x 106 m for the composite laminate versus E/p = 6 2.8 x 10 m for Aluminum. Therefore, the specific stiffness of the Graphite/Epoxy is 1. 7 times that of the aluminum. Such retrofitting requires a minimal amount of redesign effort and was suggested for aircraft structural components with minimal change in the structural configuration-so-called black aluminum structures. As the number of design engineers with formal training in composite materials increased, tailoring of material properties gained more acceptance. By varying the percentage of layers in the laminate with different orientations, material properties of the entire laminate in different directions can be adjusted to meet the requirements of a design situation. As an example, for a unidirectional material with a fiber direction modulus to transverse modulus ratio of 10, the laminate stiffness ratio E/Ex of an angle-ply laminate [±9], can be varied from 0.1 to 10 by varying the fiber orientation angle 9 from 0° to 90°. Similarly, the bending stiffness of a laminate can be altered by rearranging the relative through-the-thickness location of layers with various orientations. For example, although the in-plane stiffnesses of laminates [Oz190], and [90z10L are identical (same number of layers with same orientations) as will be shown later, the bending stiffnesses of these two laminates are markedly different. The design challenge is, therefore, to find the stacking sequence of a laminate, with different properties in different directions, so as to maximize the utility of the directional nature of the material properties. Despite the weight savings and the material property-tailoring advantages of laminated composites, one factor remains a serious obstacle for widespread use of composite laminates-cost. The cost per pound of a typical high-performance composite material is still much higher than the costs of most structural metals. Therefore, the principal supporter of the composite materials technology so far has been the aerospace industry, where weight savings are often crucial to good performance and can also provide substantial savings in operating costs. In order to make composite structures competitive with metallic counterparts, there is a need to reduce the overall product cost. One area that seems to have substantial potential for savings is the manufacturing of composite parts. Based on advances in various manufac-


21

turing techniques, such as filament winding, thermoforming, and towplacement, it is possible to make net shape components. Many manufacturing processes produce parts that need additional machining for final shape or have limitations on producing the desired shape, requiring manufacture of the component in two or more pieces and subsequent assembly. Net shape parts are largely ready to use in the final product and greatly reduce assembly and manufacturing time. The challenge for composite designers is to incorporate manufacturing considerations into the design process. This is an area in which previous design work has lagged, but is now gaining momentum.

1.3.2 Material-Related Design Issues Even if one does not consider manufacturing related aspects of designing a composite part, there are a number of other issues that makes the design task complex and in some instances intractable. The mechanics of laminated composite materials is generally studied at two distinct levels, commonly referred to as macromechanics and micromechanics. At the macromechanical level the properties of the individual layers are assumed to be known a priori. Macromechanics involves investigation of the interaction of the individual layers of a laminate with one another and their effects on the overall response quantities of the laminate. For example, elastic stiffness properties and the influence of temperature and moisture on the response of laminated composites can be predicted well by macromechanics. Compared to isotropic materials, we need to deal with a more complex model that addresses the material orthotropy and anisotropy and requires more material constants for characterization of the mechanical response of the laminate. However, macromechanics of composite laminates is reasonably well understood in formulating the stiffness analysis of laminates. For a given stacking sequence, the stress-strain relations of a composite laminate (commonly referred to as the constitutive equations) can be derived, and the various coupling mechanisms between the in-plane and out-of-plane deformation modes of a composite laminate can be explained. Thus, the use of macromechanics formulation in designing composite laminates for desired stiffness characteristics is well established. In the first two chapters of this book, the development of macromechanics-based analysis for the stiffness and hygrothermal

22

INTRODUCTION

properties of laminated composites is discussed. Use of such analyses along with automated design tools for the in-plane problems are then discussed in Chapters 3-5. Composite properties that must be predicted on the basis of micromechanics, on the other hand, are not as well understood. Micromechanics deals with the interaction of the fiber constituent with the matrix material. Earlier in this chapter, we have shown that the elastic properties of an individual layer can be derived from the elastic properties of the fiber and matrix constituents based on the rule of mixtures. The level of accuracy of those equations is, unfortunately, not as good as the laminate constitutive equations. Rather than using the rule of mixtures, for most engineering applications the elastic stiffness properties of a composite layer (Ep ~' v 12, and G 12) are determined through tests. Thus, the selection of the fiber and matrix combination that will suit the needs of a design situation is largely a decision made prior to the design optimization. Moreover, the factors that go into deciding the fiber and matrix materials are not limited to stiffness properties, but involve other considerations (such as response to environmental effects, damage tolerance, cost, fatigue) which are more difficult to include in design optimization. Nevertheless, in some cases this decision is made after experimenting with different fiber and matrix combinations in automated design optimization and evaluating the effects of the different combinations on the overall design. Composite materials fail due to the failure of either the individual constituents (fiber or matrix) or their interface, and a proper modeling of those failures can only be made at the micromechanical level. In a single fibrous composite layer under a complex loading, failure may be predicted by comparing stresses in the layer to the strength of the individual constituents. However, there are only a handful of cases where the failure of a composite laminate is dictated by the failure of a single layer. Stresses in a laminate generally vary in different layers, and failure of a laminate may involve a complex sequence of failure events in different layers. Consequently, there is no well-accepted analytical expression for the determination of the strength of a laminated composite material based on the strengths of the individual constituents. Moreover, there are failure modes, such as the delaminations between the composite layers, that are unique to laminated composite materials. Even though such failures appear at the macroscopic level, initiation of these failures is mostly related to


23

microscopic events. For example, in the case of delaminations, the transfer of applied loads from one layer to another creates a complex state of stress between the fibers and the matrix near the layer interface region that is likely to cause failure at the fiber-matrix interface. For most engineering design applications, using a level of detail for analysis that addresses the fiber-matrix interface stresses is unrealistic. The common approach taken by designers is to ignore the micromechanical nature of the failures and to limit the analysis to phenomenological failure models that address the specific features of certain classes of commonly observed laminate failures. In this book we limit ourselves to only constant thickness laminates without discontinuities under in-plane and bending loads. As such, the laminate failures are related to a small number of failure modes that are well correlated with the stress and strain levels in the individual layers of the laminate. These models are discussed in Chapter 6 along with a short discussion of the nature and modes of failures in laminated composite materials. The failure models discussed in Chapter 6 are suitable for a range of applications. However, at this point we would like to caution the reader about the varied nature of failure modes that can be observed in laminated composite structures. Most engineering structures include local details such as notches, holes, and thickness variations in the form of ply drop-oft's that give rise to stress concentrations and threedimensional local stresses. For example, for a two-dimensional plate with a circular hole under uniaxial loading, the primary stress field coincides with the direction of loading in the plane of the laminate. However, there are secondary stresses (i) in the plane of the laminate in a direction transverse to the loading direction generated due to Poisson's effect and (ii) in the through-the-thickness direction of the laminate due to the free edge at hole boundary. Even though the primary loading is in the plane of the laminate, failures in such regions of the laminate may be due to these secondary stresses, such as the throughthe-thickness stresses, which are usually lower in magnitude than the primary stresses along the applied load direction. Such through-thethickness failures lead to delaminations at the free edges of the laminate that may affect the in-plane stress state. In this book, local details in the form of ply drop-offs, notches, and holes that give rise to critical secondary stresses are ignored. Only failure modes that are associated

24

INTRODUCTION

such that h,(x) =0,

with in-plane stresses in plate-type laminates without discontinuities under in-plane and bending loads are discussed.

gix)

The increased number of design variables is both a blessing and a curse for the designer. There are more controls to fine-tune the structure to meet design requirements, but the increased number of variables brings the additional burden of selecting those design variables that are important and identifying their values for the best solution of the design problem. The possibility of achieving an efficient design that is safe against multiple failure mechanisms, coupled with the difficulty in selecting the values of a large set of design variables makes mathematical optimization a natural tool for the design of laminated composite structures.

Mathematical Optimization

In general, an optimization problem has an objective function which measures the goodness or efficiency of the design. The maximization of the goodness is generally performed within some limits that constrain the choice of design. Such limits are called constraints. Finally, an optimization problem has design variables, which are the parameters that are changed during the design process. Design variables can be continuous or discrete depending on whether they can take values from a continuum or are limited to a number of discrete values. A special case of discrete variables are integer variables. The commonly used notation for design variables, objective function, and constraints is as follows. We use a vector x with n components to describe the design variables. To allow for the case that some of the design variables may be discrete or limited otherwise, we use X to denote the domain of these design variables. For the objective function, we use the notation f(x) and for constraints, the notation g(x) for inequality constraints and h(x) for equality constraints. The standard form of the optimization problem is written as

minimize f(x)

~

0,

i = 1, ... 'ne, j

XE

X

(1.4.1)

= 1, ... , n8 , (1.4.2)

xL ~x~xu,

1.4 DESIGN OPTIMIZATION

1.4.1

25


where the elements of the vectors xL and xu are the lower and upper bounds on the values of the design variables. Note that in this standard form we minimize rather than maximize the objective function. In many problems we may wish to maximize the objective function. For example, we may wish to maximize stiffness for a given laminate thickness rather than minimize thickness subject to stiffness constraints. To convert a maximization problem into a minimization problem, we just need to change the sign of the objective function. That is, instead of maximizing f(x) we can minimize -f(x). Similarly, note that the constraints are written with a zero on the right-hand side and for a particular choice of the sense of the inequality. Obviously, it is easy to arrange for a zero on the right-hand side of the equalities or inequalities, and the sense can be changed by multiplying the inequality by -1. For example, an inequality relation in material strains, (1.4.3)

£xu~ £x'

can be converted into £xu- Ex~

0

£x- £xu~

0.

(1.4.4)

and then into

The set of design points in X that satisfy all the constraints is called the feasible domain. At a given design point x, a constraint may be satisfied or violated. An inequality constraint for which the equality condition holds at a particular design point x is called an active constraint. All satisfied equality constraints are active, while inequality constraints can be active or may be satisfied with a margin, in which case they are called inactive. Formulation and graphical representation of a design problem is shown in the next example, in which the op-

26

INTRODUCTION

timal solution along with active constraints is identified based on visual inspection of the design space.

27


such that

g 1(Vr)

EI =0.2- E f

Example 1.4.1

_

For the fiber packing geometry presented in Fig. 1.4 and fiber and matrix properties of Kevlar and Epoxy provided below, formulate the optimization problem to minimize the composite material weight as a function of the fiber volume fraction. The longitudinal modulus of the composite should at least be 20% of that of the fiber modulus, and the transverse modulus should be larger than 15% of the composite longitudinal modulus. Plot the stiffness constraints to determine the optimal solution for the fiber volume fraction graphically. The elastic modulus and the weight density of a typical Kevlar fiber are E1 = 124 GPa, p1 = 14.1 kN/m 3 , and of the epoxy resin are Em = 3.45 GPa, Pm = 12.0 kN/m3 , respectively. The density of the composite to be minimized is the objective function j(~) given in terms of the fiber volume fraction using Eq. (1.2.2), f(~) = p = PFJ+ Pm(l -

v,) = 12 + 2.1

~ (kN/m 3).

The stiffnesses that are constrained are given by

g2 (V1)- 0.15

or 21.35- 120.55 V1 ~ 0,

_ E2

E or

180.83

v,- 180.83 V{- 30.16 < VI

I

_

-0,

10 .29

0:::; V1 ~ Vj,

where v_rax is calculated to be 0.7854 in Example 1.2.1. Since the composite weight density linearly increases with the fiber volume fraction, the lowest weight would be achieved by reducing the V1 to zero. However, an all-epoxy material would not satisfy the requirement on the longitudinal modulus to be greater than 20% of the fiber modulus. A plot of the longitudinal composite modulus normalized by the fiber modulus is shown in Fig. 1.7a along with a horizontal line that indicates the 20% limit on the modulus. The vertical dashed line in the figure indicates the value of the fiber volume fraction at which the longitudinal composite stiffness is equal to 20% of the fiber modulus. For values of V1 less than approximately 0.18, the longitudinal stiffness constraint is violated. Similarly, a plot of the transverse composite modulus normalized by the longitudinal composite modulus is shown in Fig. 1.7b.

E 1 = EF1 + Em(I - V1) = 3.45 + 120.55 V1 (GPa),

E2=

E1Em V1Em + (1- V1) El

427.8 124- 120.55 VI (GPa).

The constraints on the longitudinal and transverse stiffnesses, E 1 and £ 2, respectively, require that E 1 :?: 0.2 E1 and

£ 2 :?: 0.15 E 1•

Using the standard formulation of an optimization, we pose the problem in the following form: minimize

f(~)

= 12 + 2.1

V1

0.5

E11Et

0.4

0.3

____ j __,______ _ I

0.2

I

0.1

I

0

0

0.2

0.4

0.6

0.8

v,

(b)

Figure 1.7. Normalized logitudinal and transverse stiffnesses versus fiber volume fraction.

28

INTRODUCTION

The horizontal dashed line indicates the 15% limit on the transverse modulus, and the vertical dashed line shows the value of the fiber volume fraction at which the transverse composite stiffness is equal to 15% of the longitudinal composite modulus. In this case, the requirement on the composite transverse modulus is violated for values of 'j greater than approximately 0.22. The horizontal line in the figure intersects the normalized constraint function once more at a value of the V1 slightly greater than the v_rax which is not a physically acceptable solution. Based on the these constraints, the feasible range of fiber volume fraction for this problem is approximately 0.18 ::;; 'j::;; 0.22. The minimum composite weight is, therefore, achieved for "r= 0.18.

The mathematical statement of an optimization problem defined in Eq. (1.4.1) is also called a nonlinear program, and the mathematical methods for solving the problem are called nonlinear programming techniques. The special case of Eq. ( 1.4.1) where the objective function and constraints are linear functions of x is called a linear program, and the corresponding solution techniques are referred to as linear programming techniques. When some of the design variables are integer variables, it may be difficult to write expressions for the objective function and constraints using these variables without the use of multiple "if-then" clauses. As will be shown later in Chapter 5, this problem is solved by introducing 0-1 or binary design variables. There are a number of efficient classical algorithms available for the solution of these linear and nonlinear programming problems. However, the usefulness of these algorithms and their efficiency is determined to a large extent by the characteristics of the design space, which is in turn determined by the nature of the objective and constraints functions. For example, in the case of a design space corresponding to a mixed set of discrete, integer, and continuous design variables, the application of classical mathematical programming algorithms becomes quite cumbersome. For a detailed discussion of various design optimization algorithms, the reader is advised to refer to one of the textbooks on the subject matter (see, for example, Arora, 1989, or Haftka and Gtirdal, 1993).

1.4

DESIGN OPTIMIZATION

29

In recent years, there has been considerable interest in exploring new optimization methods such as genetic algorithms and simulated annealing, which are emerging as strong contenders against the traditional search methods. These methods belong to a generic category of stochastic search techniques and have been used with some success in problems where the design space may be difficult and where a mix of integer, discrete, and continuous variables is encountered.

1.4.2 Stacking Sequence Optimization In the context of optimization, tailoring of the material properties is also associated with maximization or minimization of a performance criterion. The performance criterion may be the weight of the laminate, and the desirable response quantities, such as stiffness, become the constraints. Alternatively, a response quantity can be maximized or minimized subject to a constraint on weight. In either case, the variables typically used for design optimization define the fiber orientation, number of plies, and stacking sequence of plies that make up the composite laminate. Most commercially available composite materials come as a unidirectional tape with a fixed thickness. Hence, when laminate thickness is optimized, the optimal number of layers in the laminate needs to be determined; this is an integer problem. Despite the integer nature of the problem, most of the early work in design optimization of composite laminates was based on the use of continuous-valued ply thicknesses as design variables. This was partly due to the unavailability of easy-to-use commercial integer programming software and the high computational cost of solving integer programs. When ply thicknesses are used as design variables, they usually control the total thicknesses of contiguous plies of the same orientation. Once the optimization is completed, the final ply thicknesses can be rounded off to integer multiples of the commercially available ply thickness. It is usually safe to round all thicknesses up, but this may lead to substantially heavier designs. It makes sense to round some thicknesses down. For example, if two thickness design variables have values of 3.1 and 4.7 times the commercially available thickness, it may be appealing to round off the first one to 3 and the second to 5. However, for a large number of design variables, finding a rounded-off design that does not violate any constraint is often difficult. If such

30

INTRODUCTION

a design is found, it may be a suboptimal design-other designs obtained by rounding off differently will have better objective function values. With 20 design variables, for example, there are 220 "' 1,000,000 possible rounded-off designs. Another problem with using thickness design variables occurs when orientations are limited to a finite set of angles. Then, the problem must be formulated with a given stacking sequence, rather than letting the optimization obtain the best stacking sequence. That is, relative through-the-thickness placement of the distinct fiber orientations needs to be decided before the thicknesses are determined by the optimizer. Still another problem occurs in enforcing constraints on having too many contiguous plies of the same orientations. Laminates with more than four contiguous layers of the same fiber orientation direction are generally assumed to be not practical because of thermal stresses created during the curing process, which can lead to matrix cracking. Therefore, if at the end of the optimization the total ply thickness of a certain fiber orientation comes out to be larger than the thickness of four layers, the designer would be forced to restart the optimization process by limiting that particular orientation to having an upper bound and redefining the stacking sequence so that the ply orientation with the excessive thickness in the previous run appears elsewhere in the laminate. Orientations of layers are also occasionally used as design variables, with angles taking any value between oo and 90°. Again, one of the major difficulties in a realistic design situation is the need for a practical laminate, which is generally made up of plies with only oo, 90°, and ±45° orientations (or occasionally orientations with 15° increments between 0° and 90°). For these reasons, laminate design or the "stacking sequence design" is primarily an integer problem that calls for discrete programming techniques. There is a growing interest in the application of integer programming methods to laminate design. The primary approach in this book is to employ procedures that lead to practical laminate design, and therefore mathematical programming based on continuous variables will be avoided unless a design situation specifically calls for the use of such an approach. In subsequent chapters, we introduce various approaches that address the discrete nature of the stacking sequence optimization problem.

31

EXERCISES

EXERCISES

1. Using the stiffness representation models discussed in Section 1.2.2, show that the transverse stiffness, £ 2 , of a fiber-reinforced composite material is given by Eq. (1.2.6). 2. Assuming the packing of the fibers to be represented by the unit volume shown in Fig. 1.8, plot the longitudinal and transverse specific stiffnesses, E/p and E/p. and the weight density of the composite, p, as a function of the fiber radius (0.0 $: r1 $: r[
90i±45!-45Jr. (f) The following laminate is a symmetric laminate: [Oi±45/0/

90i0 /±45/02) T'

unit volume

Figure 1.8. Unit volume

32

INTRODUCTION

(g) The laminate [0/±45/90/±451s is not a quasi-isotropic laminate. (h) Bending stiffness matrix of laminated composite materials is a strong function of the sequence of the fiber orientations in the laminate definition. 4. Write down the shortest possible representation of the following laminates: (a) [+45/ -45/0/0/+45/ -45/0/0/90/0/0/0/0/90/0/0/ -451+451010145/+451r(b) [ +45/-45/0/0/+45/-45/0/0/90/90/0/0/90/90/0/0/90/90/ +45/ -45/0/0/+45/-45Jr.

REFERENCES

2 MECHANICS OF LAMINATED COMPOSITE MATERIALS

Arora, J. S. (1989). Introduction to Optimum Design. McGraw-Hill, New York. Haftka, R. T. and Giirdal, Z. (1993). Elements of Structural Optimization. Third Revised and Expanded Edition. Kluwer Academic Publishers, Boston. Kirsch, U. (1993). Structural Optimization: Fundamentals and Applications. Springer-Verlag, New York. Middleton, D. H. (1990). Composite Materials in Aircraft Structures. Longman Scientific and Technical (co-published in the U.S. with John Wiley & Sons, Inc., New York). Niu, C. Y. M. (1992). Composite Aiiframe Structures-Practical Design Information and Data. Hong Kong Conmilit Press Ltd., Hong Kong. Schwartz, M. M. (1984). Composite Materials Handbook. McGraw-Hill, New York. Tsai, S. W. and Hahn, H. T. (1980). Introduction to Composite Materials. Technomic Publishing Co., Inc., Lancaster, PA. Vanderplaats, G. N. (1998). Numerical Optimization Techniques for Engineering Design. Vanderplaats Research & Development, Colorado Springs, CO. Zagainov, G. I. and Lozino-Lozinski, G. E. (1996). Composite Materials in Aerospace Design. Chapman & Hall, New York.

In order to design a structure that will meet requirements on certain response quantities such as displacements, stresses, or buckling loads, we need to be able to calculate that response for a specified loading. Elastic properties that form the relation between the stresses and strains are the fundamental quantities that govern the response of the material and, therefore, the structure to be designed. Response quantities such as deformations, stresses, buckling loads, and natural frequencies, all depend on elastic properties such as elastic modulus (Young's modulus), shear modulus, and Poisson's ratio. In designing structures made up of monolithic materials, these properties are normally given and are not included among the set of parameters that the designer varies in order to improve the performance of the structure. In contrast, for composite materials part of the effort is to design elastic properties of the composite medium. The objective of this chapter is to discuss how elastic properties are calculated for laminated composite materials from variables that can be changed during the design process. The subject of this chapter is covered more extensively in books devoted to the analysis rather than the design of composite materials and structures (e.g., Jones, 1998, Tsai and Hahn, 1980, Vinson and Sierakowski, 1987, Hyer, 33

34

MECHANICS OF LAMINATI!O COMPOSITE MATERIALS

2.1

1998, and Herakovich, 1998). In this chapter we present only a relatively brief discussion of the subject matter. We start with the fundamental relations that govern the linear elastic response of an isotropic medium in three dimensions and demonstrate how they are reduced to two-dimensional relations when two or more thin isotropic layers are laminated together. We then extend the treatment to the case of layers with directional properties.

O'x O'y O'z 'tyz 'tzx 'txy

2.1

2.1.1

GOVERNING EQUATIONS FOR ELASTIC MEDIUM

Strain-Displacement Relations

Under loading, an elastic medium undergoes displacements that vary over the domain. These displacements along the x, y, and z coordinate axes are denoted as u, v, and w, respectively. For small deformations the following linear form of the strain displacement relations holds:

au Ex= dx'

dW

dv dy

dW Ez=a;,-,

E =-, y

dV

dU

dW

c,, cl2 c,3 = c,4 c,s c,6

c,z C22 c23 Cz4

c,3 Cz3 c33 c34

cl4 c,s c,6 c24 Czs Cz6 c34 c3s c36 c44 c4s c46 Czs c3s c4s Css Cs6 c26 c36 c46 cs6 c66

Ex

(2.1.3)

I

Ey Ez Yyz Yzx Yxy

where, because of symmetry, there are 21 independent material constants. The matrix C is the material stiffness matrix, and 0' and E represent the vectors of stress (see Fig. 2.1), and strain components, respectively. There are three other stress components 'tzy' 'txz' and 'tyx" but as will be shown in the next section 'tzy='tyz' 'txz='tzx' and 'tyx = 'txy· The symmetry of the material stiffness matrix, Cij = Cji' can be proved from the existence of a strain energy density U; see Boresi (1987). For example, upon differentiation of the strain energy density, one can show au/dEX = O'x andiJU!dEY= O'y. Further differentiation of the stresses, ax and crY, with respect to the strains, EY and Ex, respectively, yields dO'/dEY = dO'/dEx, from which one can show that

c,z = cz,· dV

dU

~=~+~, ~=~+~, ~=~+~,

(2.1.1)

where Ex, EY, Ez denote the normal strains in the x, y, z directions, respectively, and Yy::: Yzx' Yxy denote shear strains.

2.1.2 Stress-Strain Relations

In the case of a three-dimensional isotropic material, which has the same properties in every direction, Eq. (2.1.3) reduces to the following form,

O'J

~ c,,

e,z c,z

0

0

0

cry I

I e,z ell e,z

0

0

0

0

0

0

0

0

I jYyz

0

I IYzx

In addition to the strain-displacement relations, solution of elasticity problems requires the relations between stresses and strains in the medium. The stress-strain relation for a three-dimensional anisotropic linear elastic medium, also known as Hooke's law, is expressed in the following matrix form:

O'z

I = I c,2 el2 ell

'tyz

I

O'=CE,

or

35


(2.1.2)

Io

0

0

c"- e,z

'tzxl

I o

0

0

0

'txyJ

Io

0

0

0

2

ell- e,z 2

0

ell- e,z 2

Ex

Ey Ez

J lYv (2.1.4)

....

MECHANICS OF LAMINATID OOMIIOSITE MATERIALS

az

rft.

7PJtcr,

cr~ ~~x

~ Figure 2.1.

x

xy

Stress components in a three-dimensional medium.

with only two independent material constants. The stiffness coefficients ell and cl2 are related to modulus of elasticity (elastic modulus or Young's modulus) E, and Poisson's ratio J..l by (1- v)E

ell = (1 + v)(l -

and

2J..I)

Ciz=

vE

(1 + v)(l _ v). 2

(2.1.5)

The diagonal terms that relate the shear strains to the shear stresses are commonly referred to as the shear modulus G and are obtained in terms of the elastic modulus and Poisson's ratio as c44

= c55 = c66 =

ell- c1z _ 2

E - 2(1 + v)

=c.

2.1

faces of the element are calculated from the corresponding stresses on those faces. The rectangular parallelepiped differential element of dimensions &, t..y, and & shown in Fig. 2.1 is used to represent the stress state at the center of the element. Since stresses vary within the material; the actual stresses on the faces of the differential element are different from the ones at the center. A first-order approximation to the stresses at the faces of the differential element is shown in Fig. 2.2 looking down along the z axis. For the sake of clarity, stress components along the z direction are not shown in the figure. For the element in Fig. 2.2, in the absence of body forces, summation of forces in the x direction yields

L\yJ t..x&

dax t..xJ ( dax t..xJ ( d'tyx - ax- dx 2 L\y& + ax+ ax 2 t..y&- 'tyx- ay 2 (

~

&J

(

&J

-

+ 'tyx + ay t..yJ 2 t..x& - ('tzx - d'tl.X dz 2 t..x~y + 'tz.x + a-rl.X az 2 ~xt..y- o. ( (2.1.7) The terms involving ax, 'tyx• and'tz.x cancel, and we divide the rest of the terms by t..x ~y t..z to get (2.1.8)

daX a'tyx d'tzx -+ +-=0. dx ay dz

(2.1.6)

Depending on the degree of directionality of the material properties, there are other special cases of material response between the fully anisotropic case defined by Eq. (2.1.3) and the simplest case of isotropic behavior defined by Eq. (2.1.4). However, for many of the equations that we will introduce in the following sections the degree of material anisotropy is not of consequence. Therefore, for the sake of clarity we postpone the discussion of special cases of material properties until Section 2.4.

acrx

t:..y

(y- dy 2dtyx ----1----- 'tyx - dy ...-::.t-zy \

~;--z~• - z ' t -

t:..y 2 aax t:..x

-=;::_

0

x-

2.1.3 Equilibrium Equations tzx -

For a body to be in equilibrium under the applied loads, every differential element in the material needs to be in equilibrium under the internal and external forces acting on it. The net forces acting on the

37


""'ax~ - "" 1 I 'J ~~-· ""r + -ay

ox+

ax

2

dtxy t:..x I txy + ax- 2

il<, Az dz z

Figure 2.2.

oy +

Ay

2

Variation of stresses over a differential element.

38

MECHANICS OF LAMINATED COMPOSITE MATERIALS

This is the governing partial differential equation of equilibrium in the x direction for the variation of stress components. Summation of the forces in the y and z directions produces two other equations of equilibrium, (}rxy dx

+

d
+

d'tzy dz -

d'txz d'tyz dx + dy

0 and

+

d(Jz _ dz -

(2.1.9)

O.

Similarly, equations of equilibrium for the moments of the stresses about each of the coordinate axes yield 'txy ='tyx'

'txz ='tzx

and 'tyz ='tzy•

(2.1.10)

Note that these equations apply everywhere in the material and are independent of the assumptions for the material behavior or the straindisplacement relations. As we will see later, they can be simplified for two-dimensional problems into forms commonly used for plateand shell-type structures. The stresses must also be in equilibrium with the external forces (tractions) acting on the boundaries. We denote the external forces per unit area in the x, y, and z coordinate directions by X, Y, and Z, respectively, and then the equilibrium conditions for the boundary stresses are

2.2 IN-PLANE RESPONSE OF ISOTROPIC LAYER(S)

39

From these equations, the absence of one of the external force components will imply the vanishing of the associated boundary stress component at that point.

2.2 IN-PLANE RESPONSE OF ISOTROPIC LAYER(S) Thin structures made up of thin layer(s) of material can carry loads primarily through two different types of mechanisms. One mechanism involves stretching or compression of the layer(s) in their own plane. This mechanism is sometimes referred to as the membrane action. The other mechanism involves bending of the layer(s). In this section we analyze the in-plane problem where all deformations are in the plane of the layer(s). This holds for two-dimensional plate problems where all loading is limited to the plane of the plate, and the loads do not have any bending component. Following the customary notation we use the z axis to denote the direction normal to the layers.

2.2.1

Plane Stress

If there are no forces acting on the z-normal surfaces of the layer(s),

then from Eq. (2.1.12) the
lax+ m'txy + n'txz =X,

(Jz = 0 l'txy +may+ n'tyz l'txz

= Y,

+ m'tyz + n
(2.1.11)

where l, m, and n (referred to as direction cosines) are cosines of the angles between the direction normal to the boundary and the coordinate axes x, y, and z, respectively. In special situations where the surface normal is in the same direction as one of the coordinate axes, two of the direction cosines vanish. For example, if the surface normal is in positive z direction, l = m = 0 and n = 1, and therefore, Eq. (2.1.11) yields (Jz =Z,

'txz =X

and

'tyz

= Y.

(2.1.12)

and 'tyz ='tzx = 0.

(2.2.1)

The remaining stresses, ax, crY, and 'txy' are all in the plane of the layer(s); hence, the state of stress is called plane stress. In case of plane stress, the equilibrium equations (2.1.8) and (2.1.9) are simplified. Setting all stress components in the z direction to zero, the last equilibrium equation of Eq. (2.1.9) vanishes, and the remaining two are given as d
+

d'tyx _ dy -

0

and

d'txy dx

+

d
O.

(2.2.2)

Similarly, the three-dimensional form of Hooke's law for an isotropic material, Eq. (2.1.4), can also be simplified. The fact that no normal stress exists in the z direction does not imply that the corresponding

40

MECHANICS OF LAMINATED OOMIIOSITE MATERIALS

E ~X

Q,,

~

~if;:ilr---- I I

I

= 1- vz'

vE Q,z = 1- vz

and

E Q66 = G = 2(1 + v)

(2.2.6)

"x

-.L-Ji' I

41

2.2 IN-PLANE RESPONSE OF ISOTROPIC LAYER(S)

crx

I I

In matrix notation Eq. (2.2.5) is generally represented as

I

0'

= QE,

(2.2. 7)

cry

Figure 2.3.

Two-dimensional plane stress state.

strain fz is zero. Substituting az = 0 into Eq. (2.1.4 ), and solving for in terms of the fx and fY, we obtain

fz

where the matrix Q is commonly referred to as the reduced material stiffness matrix. We will see in Section 2.4.1 that the matrix notation of Eq. (2.2.Eis, a.lso applicable for orthotropic layers where the definitions of t e Qii's in Eq. (2.2.6) are modified to take into account the directio nature of the stiffness properties. If the stresS-Jstate is known, the strain field at every point can be obtained from the inverse of the stress-strain equation of (2.2.5),

1/

fz=- - - (fx + £).

(2.2.3)

1-v

Substituting f 2 from Eq. (2.2.3) back into Eq. (2.1.4) and collecting the coefficients of the l\ and fY terms together, we get the stress strain relation for the pi~~, stress case as 1/

ax I

I

ell - I -

1/

v

c,z c,z- 1 -

c,z

0

Cu---c,z 1-l/

0

1/

a yI 1xyl

=I c,z-1-=-C,z 1/

I

v

1/

0

r~

(2.2.4)

fy

-lQll J{fx} Q Q Q 12

where

12

0

11

0

00

EY

Q66

'Yxy

,

where

0]

sll s12 S= S~z sll 0 [ 0 s66

. With sll

1

= E'

1-l/

sl2 =~

and

s66 = 2(Sll- S12).

The Sii's in the above equation are referred to as the compliance coefficients.

2.2.2 Single Isotropic Layer

Substituting the values of C11 andC12 from Eq. (2.1.5), we obtain a new material stiffness matrix

ax} aY { 'txy

(2.2.8)

=Sa,

(2.2.9)

E -2(1 + v) I I'Yxy

0

E

(2.2.5)

A simple in-plane problem may be formulated for a rectangular plate loaded by either uniform edge stresses or uniform edge displacements. For example, for a rectangular isotropic layer with unrestrained edges loaded by a uniaxial load P along the x axis, the plate equilibrium equations and boundary conditions tell us that the in-plane shear stress 'txy and the transverse stress aY are zero everywhere, and the axial stress is uniform throughout the layer (see Fig. 2.4) and is equal to

ax=PIA,

(2.2.10)

42

MECHANICS OF LAMINATI!D COMFIOSITE MATERIALS

Ox=!0_/··~

-~P/A /Ox/

l£xl

I00-- -iffi-ox - distribution

Figure 2.4.

ex - distribution

Stress and strain distribution in a single isotropic layer.

where A is the cross-sectional area of the layer normal to the loading axis. Since the stresses are constant, the strains are also constant throughout the layer as shown in Fig. 2.4 and are computed from Eq. (2.2.8) as

Ex=

p AE'

-vP

cy= AE.

(2.2.11)

To obtain displacements of the laminate, we integrate the first two of Eq. (2.1.1) and obtain longitudinal and transverse displacements as,

u~(:E

)x+c,(y)

and

v~-v({E}+c2(x),

(2.2.12)

respectively. Since the loading is uniaxial tension and the shear stress is zero, the shear strain Yxy is identically zero everywhere:

'txy

du

Yxy

= dy

dV

+ dx = c;(y) + c~(x) = 0.

(2.2.13)

In order to satisfy Eq. (2.2.12), c 1 andc2 are, in the most general case, linear functions of y and x, respectively (c 1 =Ay + B, for example). Based on uniformity of the stresses and from symmetry considerations, u(x, y = +y0 ) = u(x, y = -y0 ), c 1 and c2 are constants in this case and can be obtained from boundary conditions on u and v. Finally, it is important to emphasize that the uniformity of the stresses over the domain allowed us to reach the solution discussed above. For problems with nonuniform applied loads or domains that

2.2 IN-PLANE RESPONSE OF ISOTROPIC LAVER(S)

43

cause nonuniform internal distribution of stresses (such as plates with holes or openings), the stresses and displacements are functions of both x andy coordinates. Determination of the displacements and stresses in such cases typically requires solution of the two equilibrium equations as a coupled system, often by numerical schemes such as finite differences or finite elements. Nevertheless, at a given point in the domain, · the through-the-thickness distribution of the stresses and the strains is still uniform as long as there are no applied loads that create bending of the layer. The bending response of two-dimensional layers is discussed in Section 2.3.

2.2.3 Symmetrically Laminated Layers When several layers are stacked together, a set of additional assumptions is needed in order to derive equations that govern the constitutive behavior of the laminate. In addition to the layers being in a state of plane stress, each layer is assumed to be bonded perfectly to the adjacent layers, so that the laminated layers deform in unison without experiencing any discontinuity in displacements. Most commonly used laminates have layers with mirror image elastic properties with respect to their mid-plane. That is, the stacking sequence of the layers (the elastic properties) above the mid-plane is the mirror image of the one below it. Such laminates are referred to as symmetric laminates or mid-plane symmetric laminates. For symmetric laminates, properly distributed in-plane loads produce only inplane deformations throughout the laminate thickness. At the edges of the laminate, the in-plane loads must be applied in such a manner that their resultants are at the mid-plane of the laminate. Any offset from the mid-plane would cause bending moments with respect to it and force the laminate to bend. The in-plane deformation assumption together with the plane-stress assumption means that the strains and the stresses are constant throughout the thickness of each layer. The perfect bond assumption guarantees that strains are constant through the entire thickness of the laminate and, therefore, the laminate strains could be characterized by the strains in a single layer, say the mid-plane layer, represented by t 0 as shown in Fig. 2.5. Having a constant strain distribution through the thickness does not imply, however, a constant stress distribution. Although the stresses

......

MECHANICS OF LAMINATI!C OOMIIOSITE MATERIALS £

' • •:·>=:

material2 material!

I

=£0X

~

=distribution

{ax} N1 N: = J aY dz, {

ax

~. -1- -j-----,pEx

Figure 2.5. layers.

X

crx = distribution

Stress and strain distribution in symmetrically laminated isotropic

Nxy

where Nx and NY are in-plane normal stress resultants, and Nxy is the shearing stress resultant. Since the stress resultants are obtained by through-the-thickness integration of stresses, dividing them by the laminate thickness h yields average laminate stresses,

=

= ~k) Eo,

where k is the layer number. This variation is the main reason for the common use of symmetric stacking sequences (or through-the-thickness elastic property distribution) of the layers. For an arbitrary stacking sequence, the stress distribution through the entire laminate thickness given by Eq. (2.2.14) will have a net bending moment. Such a moment will force the laminate to bend, unless the bending of the laminate is prevented by some external means to keep the laminate in its plane. The possibility of bending is precluded if we limit ourselves to laminates that are symmetric with respect to the laminate mid-plane. With regard to overall stiffness, the laminate behaves as a single layer with properties that are some averages of the properties of the individual layers. The fact that the stresses vary from one layer to another makes the characterization of the stress-strain behavior of a laminate more difficult. In order to define a simple stress-strain relation we need a quantity that represents the overall effect of a stress component on the laminate response. This is achieved by through-thethickness integration of the stress components. That is, for a laminate of thickness h we define

t {Nx1

(2.2.16)

NY · Nxy

Since the stresses are constant within a layer, integration can be replaced by summation over individual layers, N1 {ax} N: = L J aY { N

(J'(k)

'txy

-h/2

xy

(2.2.14)

(2.2.15)

h/2

crx} {~y

are constant within each layer, they will vary from one layer to the next (see, for example, Fig. 2.5) depending on the stiffness of the individual layers (which are assumed to be isotropic in this section) following the stress-strain relation of Eq. (2.2.7) as

45

2.2 IN· PLANE RESPONSE OF ISOTROPIC LAVER(S)

Nxy

zk

k=i zk-1

't

xy

dz =

L {ax} aY N

'txy

/<;=I

(k)

(2.2.17)

(zk-zk_,), (k)

where N is the number of layers, the subscript (k) denotes quantities in the kth layer, and zk's are the through-the-thickness locations of the interfaces between the layers; see Fig. 2.5. Substituting the stressstrain law, Eq. (2.2.14), we get

Nx} NY \ Nxy

= lN L lQ" Q,2 Q,2 Q" k=i

0

0

0

J

(zk- zk_,)

J{£~} E~

Q66

'(

(2.2.18) ,

xy

(k)

where the superscript "o" is used to denote mid-plane strains (recall that strains are constant through the thickness and can be represented by the mid-plane values). Equation (2.2.18) can be rewritten as

J{~} Ey '

Nxl-lAu NY - A 12 A,2 A22 00 0 0 A66 \ Nxy

'Y~y

where the coefficients of the A matrix are given by

(2.2.19)

MI:OHANICS OF LAMINATIO OOMIIOSITE MATERIALS N

A.=~ lj .4..J Q..lj(k) (zk- zk_ 1) or

(2.2.20)

which yield

k~J

_1_ (A 11 A22- Az12

Eett- h N

N

A -A - ~ u -

22 -

E
~ ~kl E
A 12 =£...J 1 ~I

(k)

2 V(k)

-

N

tk,

A66

= L G(k) tk

N=At 0 •

!Nx)

[All

l;:, ~ * ~ ~ *ci'

12

01 {E~J

A A22

0

E~

0

A66

Y~y

.

1- v2 --h eff

and

1-

A 12

v;tf =h,

A,2 vett= A

(2.2.25)

.

22

(2.2.26)

Geff = A66/h.

Example 2.2.1 The sandwich laminate shown in Fig. 2.6 is made up of an Aluminum core of thickness ta1 = 0.2 in and brass face sheets of tbr = 0.05 in each. Determine the effective elastic properties, Eeft' Geft' and veff of this laminate. The aluminum and brass have the following properties: Ea1 = lO.Ox 106 psi, Ga1 = 3.70 X 106 psi, Ebr = 15.0 X 106 psi, and Gbr = 5.60 X 106 psi. Given the elastic and shear moduli, Poisson's ratios for the two materials can be calculated from Eq. (2.2.6) as G=

E 2(1 + v)

=>

V=

E 2G - 1;

therefore val= 0.351 and j/br = 0.339.

The reduced stiffness matrices for the two materials [see Eqs. (2.2.5) and (2.2.6)] are

_L

t-r (br) =0.05 ---__L

t veffEeff

_Au

J and

(2.2.23)

Equation (2.2.23) is identical in form to the stress-strain relation of Eq. (2.2.5); therefore, by replacing the elastic properties in Eq. (2.2.6) with unknown effective properties, Eeffand veft' we can write the following two equations in terms of the two unknowns for effective properties: Eeff

A22

The effective shear modulus is obtained simply as

(2.2.22)

Once the in-plane stiffness matrix is determined, effective engineering constants of a laminate can be obtained by relating the average stress [see Eq. (2.2.16)] in the laminate to the strains

ax)

l

~I

(2.2.21) and where tk = zk- zk-J is the thicknesses of the kth layer. For isotropic laminates considered in this section, the ~ 2 term is identical to A 11 term. However, for the sake of generality we keep the matrix representation of A as shown in Eq. (2.2.19). For more general laminates made of anisotropic layers (such as fiber-reinforced orthotropic layers; see Section 2.4) these two terms may not be the same. The matrix A is generally referred to as the extensional material stiffness matrix (or in-plane stiffness matrix) and Eq. (2.2.19) is written in matrix notation as

47

2.2 IN·PLANE RESPONSE OF ISOTROPIC LAYER(S)

(2.2.24)

Figure 2.6. tion.

Brass

---------------

---I

Aluminum Brass

I

.(.1)

~

0.2"

Symmetric sandwich laminate of aluminum and brass construc-

-

MECHANICS OF LAMINATID COMPOSITE MATERIALS

I 1.408

4.008

Ql = 4.~08

11.408

[

~

0

Qr =

ing to applied displacements exist), through-the-thickness integration may be used to define applied traction resultants,

and

3.700

16.951

5.751

5.751

16.951

0

0

[

6

] x 10 psi,

Nx} = h12!Xl BY dz. jNxy J K Z

~

A, A., Al2 A22

O

0.1

0 0

J= 0.2

O A 66

6

] x 10 psi.

5.600

[ 16.951 5.751 5.751 16.951 0

0

r

0 0

1.408 4.008 4.008 11.408

l [ X

106

=

0 0

1

0

3.700

3 977 1.377 1.377 3.977

5.600

X

0

0

0 0

106 +

l X

106 lb/in.

1.300

Using Eq. (2.2.25), we then calculate the effective elastic properties E

ff

e

=

1 3.98 0.2 + 2 X 0.05

_ 1.38 _ 3 98 -0.3467

Veff-

o

and

X

3.98 - 1.382 3.98 _ Geff-

Unfortunately, the literature does not make any distinction in notation between the internal stress resultants and applied boundary traction resultants, and Nx, NY, andNxy are often used to denote the applied traction resultants.

2.3

0

X

6

10

=11.67 X 106 psi,

1.30 X 106 {)

,....,

I

,...,

(2.2.27)

-h/2

From Eq. (2.2.21), the in-plane stiffness matrix can be calculated to be

r

49

2.3 BENDING DEFORMATIONS OF ISOTROPIC LAYER(S)

".;

£\ £\~

_

- 4.33

.

X

106 psi.

Note that these properties are the weighted average of the elastic moduli E's and G's for the two materials, and very close to the weighted average of the Poisson's ratios v's. Finally, note that the definition of the in-plane stress resultants discussed in this section is based on the internal stresses. At a boundary point where tractions X, Y, and Z are applied (or tractions correspond-

BENDING DEFORMATIONS OF ISOTROPIC LAYER(S)

In addition to in-plane loads, laminated plates are often loaded by forces that bend the laminate. This important deformation state is best studied by isolating the loads, so that only pure bending deformation of the layer(s) is present without in-plane deformations at the midplane. For bending of laminates we invoke the classical KirchhoffLove assumptions of pure bending. One assumption is that a straight line perpendicular to the mid-plane before deformation remains straight and perpendicular to the same. A second assumption is that the length of that line remains unchanged. The assumption of unchanging length of the mid-plane normals of the Kirchhoff hypothesis implies that through-the-thickness deformations in the laminate are zero. That is, the strains in the out-of-plane direction z are neglected, and the w displacement is constant throughthe-thickness. The layer(s) are also thin compared to the in-plane dimensions of the layer(s), and the stress components in the z direction are assumed to be negligible so that an approximate state of plane stress prevails. The assumptions invoked in this section, along with the assumption of perfectly bonded layers (with infinitesimally thin nondeformable bonding agents) constitute classical lamination theory or CLT. Note, however, that the plane stress assumption discussed earlier in Section 2.2.1 does not limit the strain Ez to being small. In fact, as can be observed from Eq. (2.2.3), the magnitude of Ez can be comparable to the in-plane strains. Therefore, the assumption of zero through-the-thickness deformation of the Kirchhoff hypothesis is in-

150

MECHANICS OF LAMINAT!C OOMIIOSITE MATERIALS

consistent with the plane stress assumption. In spite of this inconsistency, the assumptions of classical lamination theory provide a basis for many of the analyses that are published in the engineering literature and have proved to be adequate for most engineering applications.

2.3.1


51

The negative of the second derivative of the out-of-plane displacement is an approximation to the curvature of the layer in the x-z plane and is denoted by Kx, so that Ex= zKx. We can get similar expressions for the other two strains and write in vector representation

Bending Response of a Single Layer

The cumulative effect of the Kirchhoff-Love assumptions can be best described by studying Fig. 2. 7. In the figure, the dashed lines denote the undeformed configuration and the solid line the deformed one, so that points A', B', and C' move after deformations to positions A", B", and C". Since there are no in-plane deformations of the mid-plane, the length of the undeformed line BB' remains unchanged in the deformed positions as shown by BB". But if the point of interest is either below (positive z direction) or above the mid-plane of the layer, there will be an elongation or shortening of the line segment along the x axis. For example, the line AA' will shorten, indicating compressive strains at z = -t/2. Strain values at points z :t:- 0 can be obtained from the rotation of the line segment A'B'C'. The displacement of a point along the x axis, denoted by u, at any z location is related to the slope by

aw ax·

u=-z8x--z_

(2.3.1)

From the strain displacement relation of Eq. (2.1.1) we have

au aw ax = -z ar . 2

Ex =

x

(2.3.2)

I££1- tk3Ex - distribution

crx - distribution

c z

Figure 2.7.

a2w ax2 a2w I ayl a2w 2-----axay -

Bending deformations of a single layer.

{::} Yxy

~ 3} z{

where {

xy

3}

=-

xy

(2.3.3)

In contrast to the in-plane deformation case, where the strain within a layer is constant, it is clear that the strain distribution is a linear function of z with Ex(z = 0) = 0 [antisymmetric with respect to the midplane Ex(z) =-Ex( -z)] and, therefore, the mid-plane strains cannot be used to represent the response of the layer in pure bending. Because w is constant through the thickness, the curvatures are also constant through the thickness and are used in a role similar to the mid-plane strains to represent the deformation of a layer. Stresses corresponding to these strains are not constant even for a single layer and may be calculated from Eq. (2.2.14). Since the strains are antisymmetric with respect to the mid-plane of the layer, the stresses are also antisymmetric; that is,
1

M: = h/2 J {O'crYx} z dz,

M

1

Mxy

-h/2

(2.3.4)

'txy

where h is the thickness of the layer. Substituting the expressions for the stress components in terms of the strains from Eq. (2.2.5) and using the strain-curvature relations, Eq. (2.3.3), we get

Mxl

MY

!

lkfxy

=

f rQQ,z Q Q2z

hl2

-h/2

11

0

12

0

00

l

Q66

MECHANICS OF LAMINATIO COMPOSITE MATERIALS

{Ex} EY z dz = [Q J z dz J{k' K'Yx} • h/2 2

'Yxy

-h/2

(2.3.5)

1(

xy

After integration, we define the matrix that relates the curvatures to the moment resultants as the material flexural stiffness matrix, D,

{Mxl lkfy

=

lkfxy

rDD 12 11

0

12

l

{Kx} K), 0 D66 Kxy

D Dzz

00


hy the moment resultants introduced for a single layer. However, in the case of a laminated medium in bending, stresses not only vary linearly through the thickness but also make discrete jumps across layer interfaces; see Fig. 2.8. These jumps are the result of the change in material properties from one layer to another and are governed by Eq. (2.2.7). Through-the-thickness integration of the stresses to establish the moment resultants involves a summation

l !

(2.3.6)

lkf: M

or in matrix notation, (2.3.7)

where elements of the D for the single layer isotropic case are given by Eh 3

D" = Dz2 = 12(1- v) 2'

=

lkfxy

M=DK,

D,2 = vDil

and

D 66

= (1 -

Dll

v) T·

(2.3.8) For isotropic laminates considered in this section, Q = Q and, there11 22 fore, D22 is equal to D11 • However, for the sake of generality we keep the matrix representation of D as shown in Eq. (2.3.6).

53

J {crx} crY z dz = L J

h/2

-h/2

N

't

/r-1 zk-l

xy

(2.3.9)

zk {ax}

crY

z dz.

't

xy (k)

Substitution of the stress-strain relation of Eq. (2.2.7) and the straincurvature relations of Eq. (2.3.3) into Eq. (2.3.9) yields

l ! lMkf:

lkfxy

f jE:

N =L ~k)

k=l

zk

zk-1

E}

z dz

"{ xy (k)

N = (L

~kl J z dz J{K:K } .

k=l

zk

2

zk-1

K

xy

After integration, the moment-curvature relation for the laminate can be rearranged to express it in the matrix form of Eq. (2.3.7) where the coefficients of the flexural matrix D are now defined by N

1 D I].. =3

2.3.2 Bending Response of Symmetrically Laminated Layers For multiple layers of isotropic material stacked symmetrically with respect to the mid-plane of the laminate, the assumptions described for laminated plates under in-plane loads and the assumptions for a single layer in bending are combined. That is, the layers of the laminate are perfectly bonded together and deform in a way that straight lines normal to the mid-plane remain straight and normal without changing their length. Therefore, following Eq. (2.3.3) the throughthe-thickness distribution of the bending strains remains linear. The curvature is still independent of the through-the-thickness location in the laminate and can be used as the representative quantity for the laminate bending deformation. The stresses can also be represented

(2.3.10)

I

(2.3.11) 3 3 )· Q lj(k) .. (zk-zk-1

k=l

By substituting the strains from Eq. (2.3.3), which are linear in terms of the curvatures, into the definition of the in-plane stress resultants we have

material I

- material2 - - - material I

-\-~ Ex -

Figure 2.8.

distribution

ax - distribution

Bending strains and stresses in laminated layers.

84

l


jNxy

= L Q
N ) N:

N

.

k=l

J 1E: zk

zk-l

E}

Yxy (k)

dz

=

L ~k) Jz dzJ1K:1C } . N

zk

k=l

zk-1

(2.3.12)

p ~ material I Eo +- - - · - · - · - · - · - · - - - - - material 2

Kxy

Since the Qk)'s are symmetric with respect to the mid-plane, the summation inside the big parenthesis can be shown to be zero (see Exercise 2). So for this case, the in-plane stress resultants are zero.

E( ) 2

3

For the symmetric laminates considered so far, the sequence of layers on one side of the mid-plane is the mirror image of the ones on the other side. In-plane loads for such a configuration produce strains that are constant through the thickness and stresses that are constant within each layer but change from one layer to another. The moment resultants are zero under in-plane loads. In the case of pure bending deformations, on the other hand, through-the-thickness strain and stress distributions are both linear (stresses are piecewise linear) and are distributed antisymmetrically with respect to the laminate mid-plane. In this case, the integrals of stress components, defined to be the stress resultants, are zero because of the symmetry. The response of unsymmetric laminates to in-plane loads and pure bending is substantially different. As we will discuss in the following, the application of in-plane loads produces bending strains as well as in-plane strains. Conversely, application of bending loads or moments generate in-plane strains as well as bending strains. Consider first, for example, an unsymmetric laminate with two materials subjected to bending moment M that generate curvatures K; see Fig. 2.9. Also assume that the in-plane laminate mid-plane strains, 0 £ are suppressed to be zero by adjusting the load P. The reason for the latter assumption will be explained later. Based on Kirchhoff-Love assumptions, through-the-thickness distribution of the strains are linear and, because £ 0 = 0, are antisymmetric with respect to the midplane. The corresponding stresses are also linearly distributed within each layer, but their magnitudes, governed by the stiffness properties of the individual layers, are distributed unsymmetrically with respect to the mid-plane. In contrast to pure bending of symmetric laminates,

-

P

X

---+ M

~

-f±-=t- t~-r3--ex - distribution

2.3.3 Bending-Extension Coupling of Unsymmetrically Laminated Layers

55

2.3 BENDING DEFORMATIONS OF ISOTROPIC LAVER(S)

Figure 2.~.

E(l) > E( 2)

ax- distribution

Strain and stress distribution in unsymmetric laminates.

where the in-plane stress resultants are zero, through-the-thickness integrations of the stresses may not vanish and may indicate nonzero in-plane stress resultants. For a more general laminate with N layers, these in-plane stress resultants are given as

N ) = h/2 N Nx f {cr} ax dz = L

I

N:Y

-h/2

't;

k=l

2

f {ax} cry k

'txy

2k-l

dz.

(2.3.13)

(k)

Substituting the stress-strain Eq. (2.2.7) and the strain-curvature relations Eq. (2.3.3) into Eq. (2.3.13), we obtain

l l Nx) NY Nxy

=

L N

k=l

Q(k)

J z dz J{KYKx} = [BB

zk

Kxy

zk-l

12

11

12

0

B B22 0

l{

00 B66

Kx} , JS Kxy

(2.3.14)

where the components of the B matrix are given by N

1

Bij

= 2 L Qij(k) (z~- zLI).

(2.3.15)

k=l

The matrix B is called the coupling matrix, and Eq. (2.3.14) is represented in matrix notation as N=BK.

(2.3.16)

MECHANICS OF LAMINATID COMF'OSITE MATERIALS

Oct

Note that the in-plane strains are assumed to be suppressed at this point, and the implication of the above equation is that any type of curvature is also associated with in-plane stress resultants. From inplane equilibrium, the net result of these in-plane stress resultants at the edge of the laminate is equal to the applied load P. Conversely, any in-plane stress resultant induced by edge loads will induce curvature. The coupling between the in-plane and out-of-plane response of an unsymmetric laminate has one more element. In a more general case the in-plane strains at the laminate mid-plane will not vanish under pure bending moment. In order to visualize this, consider the top layer in Fig. 2.9 and assume it to be very thin compared to the overall laminate thickness and to have infinitely large in-plane stiffness, so that it cannot experience any in-plane strains. For this laminate, bending deformations would cause zero strain at the top surface of the laminate (because of the infinite in-plane stiffness) and finite strains at the laminate mid-plane induced by the curvature. For more realistic unsymmetric laminates, the actual strain distribution is a combination of pure bending strains with zero mid-plane value and the finite midplane strain due to the asymmetry of the in-plane stiffnesses of the layers. This can be incorporated into Eq. (2.3.16) easily by adding the relation between the in-plane strains and the in-plane stress resultants from Eq. (2.2.22),

N=BK+At

u-r

L.l --,.--- ----

material I

1 e

material2 '----------' e

material!

-·-------+ __. material2

X

---- -,f

0

--8- -F Ex -

Figure 2.10.

•

=tl E =£

(2.3.17)

0

distribution

aX - distribution

Unsymmetric laminate under in-plane strain state.

'I :1

57

BENDING DEFORMATIONS OF ISOTROPIC LAYER(S)

Next, consider a case where we subject the unsymmetric laminate to in-plane strains only by suppressing curvatures. This can be m:hieved by placing the laminate between two rigid walls to suppress the curvatures and applying in-plane loading at the laminate midplane. Even though through-the-thickness distribution of strains represented by mid-plane strains £ 0 is constant, and stresses within each layer are constant, the unsymmetric profile of the stresses (see Fig. 2..1 0) generates moment resultants about the laminate mid-plane. Starting from the definition of the moment resultants, Eq. (2.3.10), we use lhc fact that strains are constant to obtain

M} N ~k) J {££~ M: =L { 0

zk

Mxy

k=l

zk-1

"(

}

xy

z dz

l

0

~k) Jz dz {£Ei ·

= lN L

zk

k=l

zk-1

(k)

}

(2.3.18)

£xy

Note that the matrix terms in front of the strain vector are the same us the ones obtained earlier [see Eq. (2.3.14)] for pure bending. Therefore, the relation between the moment resultants and the in-plane strain components is represented as

Mx} _ MY Mxy \

[B11 B 12 0

B12 B22 0

0

.l {£~} £~ '

B66

(2.3.19)

Y~y

or in matrix notation,

M=BE

0 •

(2.3.20)

Equation (2.3.20) suggests that under pure in-plane deformations, internal moment resultants are generated. In effect, these moment resultants are the result of external actions that keep the laminate in flat equilibrium configurations. Another way of inducing in-plane deformations without laminate curving is to apply in-plane loads with an 0 eccentricity with respect to the laminate mid-plane. Thus a given £ 0 and zero K can be achieved by applying in-plane loading of N = AE and bending moments M= BE 0 • Note that different eccentricities may be needed for different components of the in-plane loads. Conversely, if external bending moments are applied without in-plane loads, Eq.



(2.3.20) suggests that in-plane deformations would result. Since applied bending moments also cause curvatures as given by Eq. (2.3.7), the resulting deformation state is a combination of the in-plane and bending deformations governed by

M=BE 0 +DK.

Example 2.3.1 A two-layer laminate is made up of an aluminum sheet of thickness ta1 = 0.2 in on top and a brass sheet of tbr = 0.05 in at the bottom. The elastic properties of the aluminum and brass are same as the ones given in Example 2.2.1. Under the action of bending moments with no in-plane loads, the laminate bends such that Kx = 0.06 1/in, ~ = -0.006 1/in, and Kxy = 0.0 1/in. Determine the in-plane strain that the laminate experiences, and determine the magnitude of the bending moment required to cause this bending. Using the reduced stiffness matrices Qa1 and Qbr given in Example 2.2.1, the in-plane stiffness matrix for the laminate configuration is calculated to be

An AI2

r0

12

A

Ol = 0.2 Qal + 0.05 ~r

A22

0

0

A66

4.008

4.008

11.408

0

0

16.951

5.751

5.~51

16.951

= 0.2 [

(2.3.21)

It is important to note, however, that for a given applied bending moment Eq. (2.3.21) alone is not enough to solve for the deformation state. There are two sets of unknowns, E0 and K, but only one set of equations. The second set of equations is Eq. (2.3.17) and, therefore, the solution depends also on the in-plane stress resultants. If in-plane stress resultants are zero, then Eq. (2.3.17) can be used to solve for the curvatures in terms of the in-plane strains (or vice versa). After that Eq. (2.3.21) can be used to solve for the in-plane strains (or the curvatures). The following example demonstrates a situation in which the in-plane stress resultants are zero, and the bending deformations are known.

+ 0.05

[

~

1.089 3.129 [

0

0

l ~l X

106

3.700

0

3.129 1.089

=

~

11.408

59

X

106

5.600

6 ] X 10 lb/in.

1.020

The distances to the layer boundaries are z0 = -0.125 in, z1 = 0.075 in, and z2 = 0.125 in, and the bending-extension stiffness matrix for the laminate is

l

Bll Biz Biz Bzz

0

0

~ l = k{Q., (0.075' - (-o.J25)') + 0.. (0.125

B66J

27.72

=

8.72

8.72 27.72 [

0

0

~

l X

2

-

0.07 5'))

10' lb,

9.5

and the bending stiffness matrix is

l

DII

DI2

Diz Dzz

0

0

~l

=

D66J

t{Q., 17.68

=

(0.075' - (-o.J25)') + Q,, (0.125 3 - 0.0753))

6.11

6.11 17.68 [

0

0

~

] x 103 lb-in.

5.79

80


Given the curvatures of the laminate, we can use Eq. (2.3.17) to solve for the values of the in-plane laminate mid-plane strains. Since the in-plane stress resultants are zero, A11 A, 2 A12 A22

r

0

0

l

00 A 66

ll

{E~} Biz E~ + rBll B B 12

ty

00 B66

22

0

0

0.06 } -0.006 0.0

= {0} 0 , 0

and we can solve this equation to obtain

£~} = {-540.52} Ei X10-6. {Yxy 74.18 0.0 Once the curvatures and the in-plane strains are known, the moments that generate those can be calculated from Eq. (2.3.21),

Mx} My { Mxy

=

0] X 10 {-540.5} 74.2 X 10-

[27.72 8.72 8.72 27.72 0 0 0 9.5 17.68

+

6.11

6.11 17.68 [

0

0

3

6

0.0

J 10

00 5.79

X

3

I

J=

0.06 -D.006 0.0

lb-in 1010 258 in 0

2.4 ORTHOTROPIC LAYERS

61

in the more general case when both N-:~; 0 and M and (2.3.21) have to be solved simultaneously.

2.4

-:~;

0, Eqs. (2.3.17)

ORTHOTROPIC LAYERS

Replacing isotropic layers by fiber-reinforced layers affects only the layer stress-strain relations without requiring any changes in the assumptions that form the basis for the laminate constitutive relations. However, individual layers then have directional properties with planes of symmetry that are not necessarily oriented along the general x-y axes of the laminated medium. For a unidirectional fiber-reinforced lamina, there are two perpendicular planes of symmetry that define two principal axes of material properties; see Fig. 2.11. These principal axes correspond to the direction of the fibers and a direction transverse to the fibers and are denoted by subscripts 1 and 2, respectively, in the remaining chapters of this book. Such layers are referred to be orthotropic, and the 1-2 axes are the principal axes of orthotropy (sometimes referred to as the principal material directions). When the principal axes of orthotropy of a lamina are oriented at an angle, e, with respect to the axes of the laminate (an off-axis layer), the lamina behavior will appear to have the characteristics of an anisotropic material. However, the lamina is still orthotropic and is referred to as generally orthotropic.

1 1

2

Another easy case is when the in-plane loading of Fig. 2.10 is at the mid-plane without applied moments (M = 0). In this case an unsymmetric laminate must bend. Since the moment resultants are zero, the curvature can be calculated from Eq. (2.3.21) as K

=-D-l Bt 0 •

(2.3.22)

Knowing the curvatures in terms of the in-plane strains, one can solve Eq. (2.3.17) for either in-plane stress resultants given the in-plane strains or in-plane strains for given in-plane stress resultants. Finally,

Figure 2.11.

Planes of material symmetry for an orthotropic layer.

MEOHANICS OF LAMINATIO COMPOSITE MATERIALS

2.4.1

Stress-Strain Relations for Orthotroplc Layers

For an orthotropic lamina, the stress-strain relation in the principal material directions is given by the following set of equations with nine independent constants:

cr, 0"2 0"3

c, 2

o o c,2 c2 2 o o cl3 c33 o o = 0 c23 o o C44 o 0 o o o C55 Cu

C 13 C23

0 0 0 0 0 c66

I

't23 't31

0

'tl2

0

0

0

0

£1

E 1 (1 - v23 v32)

E2

(v12

c21

c31

,

c12

+ v13 v32) c22

=

c32

=

Ll

=

E3

(v,3

=

E 1 (v21 + J.l23

+ v,2 v23) A

,

E2 (1 -

E3

v31)

·

cl3

·

c23

(v13 1,.121

~11

1/13

v23

1/32

£2

£1

£3

£1

£2

£3

=0

and

all [Q Ql2 0"2 = Ql2 Q22 {'t12 ° 0 11

(vl2

=

= 0,

(2.4.6)

{£1] £2 '

(2.4.7)

't31

v31 + v32)

E3 (1 -

1

Q66

El

Qll

Y12

E2 Q22

1 - 1/12/,./21 ' v 12E 2

v12 v21)

A

0 0

where the Qu's are the reduced stiffnesses and are given in terms of four independent engineering material constants in principal material directions as

Ll

Q

12

1-

1/121/21

v2,EI

= 1 - v12 v21 = 1 -

V12v21

and

(2.4.2)

where

Q66=

A= 1 -

't23

which reduces the stress-strain relations to

E 1 (v31 + v21 v32) = A ,

c33

(2.4.5)

For thin layers with no applied forces acting in the out-of-plane direction, we invoke the plane stress assumptions of Section 2.2.1 in the 1-2 principal material plane,

Y31

E2

,

v,2

(2.4.1)

yl2

+ v23)

A

v2,

cr3 = 0,

Y23

l-'13 v31)

.

63

- = - , - - - and - = - .

£2 £3

Expressions for the Cu's in terms of the orthotropic engineering material constants, E" E2 , E3 , G12 , G23 , G3 " and vu's are: ell=


J.l12 v21 -

V13

v31 - v,2

v23

V31 - J.113 J.l21

J.l32-

J.l23

v32•

(2.4.3)

and C44 = G23•

Css = G!3

and

C66 = G,2.

(2.4.4)

In addition to the three elastic moduli and three shear moduli, there are six Poisson's ratios in the preceding equations giving a total of twelve engineering constants. However, only nine of the constants are independent. Three reciprocity relations provided below without proof may be used to express the Cu's in terms of nine independent material constants:

Gl2'

(2.4.8)

Two of the four independent material constants are the elastic moduli in the 1 and 2 directions, £ 1 and£ 2, respectively. The other two material constants are the shear modulus in the 1-2 plane G 12 and the major Poisson's ratio v12• The major Poisson's ratio is defined as the negative of the ratio of strain in the 2 direction, £ 2 , to the strain in the 1 direction, £ 1, under loads acting in the 1 direction only. The minor Poisson's ratio is defined as v 21 = -£/£2 under loads along the 2 direction, and is related to the other elastic properties through the resciprocity relations of Eq. (2.4.5) E2

v21

=£ v12. I

(2.4.9)

84

MECHANICS OF LAMINATI!D COMPOSITE MATERIALS

As stated earlier for the isotropic case, the plane stress assumption does not preclude the existence of through-the-thickness strains. From the first equation of (2.4.6), the third row of Eq. (2.4.1) can be used to solve for the through-the-thickness strain, cl3 c23 E3=-cE1-cE2· 33 33

(2.4.10)

:1.4 ORTHOTROPIC LAYERS

T=

m2

n2

2

m2

r

E3 =

1-

E1-

v12v2I

+ V31V12

1-

E2

vl2v21

(2.4.11)

Ex} Ey

{ :~} 't12

= T {::}

and

'txy

{:~} E12

E=b_ xy 2'

E12=2·

and the transformation matrix is given by

Exy

Te=

m2

n2

n2

m2

{::1= .1

1

mn . -mn

(2.4.16)

m2-n2

QRTR

1

{

'txyf

(2.4.12)

(2.4.17)

::J. Yxy

-]

}

{d [~:: ~: ~:: k' Qll Q12 Q16

ax

Exy

-

-

Ex

(2.4.18)

=

where the transformed reduced stiffnesses (2.4.13)

(2.4.15)

Substituting the transformation relations for the stresses and engineering strains into Eq. (2.4.7), we obtain

where Exy and E12 are the tensor shear strains which correspond to one-half of the engineering shear strains

Y12

Yxy

R= 0 1 0 . 0 0 2

This transformation produces = T {::].

n =sin e.

[1 001

= R {Ex} Ey '

r-2mn 2mn

Since orthotropic layers are generally rotated with respect to a reference coordinate system, x-y, the stress-strain relations given in the principal directions of material orthotropy, Eq. (2.4. 7), must be transformed to the reference axes. The transformation of stresses and strains can be accomplished by

and

The transformation matrix for the engineering strains is, therefore, given by

where the various Poisson's ratio V;j can be computed from the reciprocity relations of Eq. (2.4.5).

2.4.2 Orthotropic Layers Oriented at an Angle

(2.4.14)

In order to convert the strain transformation relation from tensor strain notation to engineering strain notation, the transformation matrix is pre- and postmultiplied by a matrix R and the inverse of R, respectively, where

{ v32

1

2mn , m=cose -2mn

n -mn mn m2-n2

By using the definition of the Cy's from Eq. (2.4.2), Eq. (2.4.10) may be expressed in terms of the engineering properties of the material v31 + v32l-'21

65

Qll = Qll

4 COS

rlu are related to

e + 2(Q12 + 2Q66) sin2 e cos2 8 + Q22 sin4 e,

Q12 = (Qll + Q22- 4Q66)

COS

2 8 sin2 e + Ql2 (sin 4 e + cos 4 9),

the

Qij

by

au

MECHANICS OF LAMINATED COMFIOSITE MATERIALS

Q22 =Qll sin 9 + 2(Q 12 + 2Q66) Sin 2 9 COS 2 9 + Q 22 COS 4 9, 4

I Q26 =2 u2 sin 29-

Ql6 = (Qll- Ql2- 2Q66) Sin 9 COS 3 9 + (QI2- Q22 + 2Q66 ) sin 3 9 COS 9, Q66

Q26 = (Qil Q66

= (Qll

-

67

2.4 ORTHOTROFIIC LAYERS

3

= us -

u3 sin 48, (2.4.21)

U3 cos 49.

Ql2- 2Q66) sin e cos e + (Ql2- Q22 + 2Q66) sine cos 3 e,

+ Q22-'-- 2Ql2- 2Q66) sin2 e cos 2 e + Q66 (sin4 e + cos4 9).

(2.4.19) These equations can be put into a simpler form by using trigonometric identities to replace the terms with powers of sines and cosines of the fiber orientation angle 9 with the sin and cos of multiple angles 29 and 49. Tsai and Pagano (1968) defined the following material properties: ul

Example 2.4.1

1

=8 (3Qu + 3Qzz + 2Qlz + 4Q66), 1

u2 = 2 (Qil -

An off-axis unidirectional Graphite/Epoxy (£ 1 = 181 GPa, E2 = 10.3 GPa, G 12 = 7.17 GPa, v 12 = 0.28) ply with an orientation of 8 is loaded such that only a uniform state of ax = 100 MPa exists. Determine the in-plane strains resulting from this loading as a function of the fiber orientation angle. The reduced stiffness matrix for the material is obtained from Eq. (2.4.8),

Qzz),

1

u3 =8 (Qll + Qzz- 2Ql2- 4Q66), 1 u4 = 8 (Qll + Q22 + 6Q1z- 4Q66), 1

Us= 8 (QII + Q22- 2Ql2 + 4Q66),

Ql2

= U1 + U2 cos 2e + U3 cos 49, = U4 -

U3 cos 49,

Q22 = U1 - U2 cos 2e + U3 cos 49,

u2 sin 29 + u3 sin 48,

2.90

2.90

10.35

0

0

[

~

1GPa 7.17

and the corresponding U values are, U1 = 76.374 GPa,

U2 = 85.73 GPa,

U4 = 22.61 GPa,

and

U3 = 19.71 GPa,

Us= 26.88 GPa.

Substituting the U's into Eq. (2.4.21), we obtain the terms of the fiber orientation angle 8, -

Qll

Ql6 = ~

181.81

Q=

(2.4.20)

which yield a simpler form of the transformed reduced stiffnesses of Eq. (2.4.18) as Qll

Both the U's and the Q's are independent of the ply orientation, but . Lhe Q's are dependent on ply orientation for orthotropic materials. An advantage of the form of the reduced stiffnesses in Eq. (2.4.21) is the constant parts of {211, Q 1z, Q22 , and Q66 given in terms of the Up U4 , and Us which do not depend on the ply orientation. Based on this feature, the U's are sometimes called the material invariants. As we will see in Chapter 4, the form of Eq. (2.4.21) is also useful for design optimization.

Q matrix

= 76.37 + 85.73 cos 29 + 19.71 cos 48, Ql2 = Q21 = 22.61- 19.71 cos 49,

in

MECHANICS OF LAMINATED OOMI'OSITE MATERIALS Q22

=76.37-85.73 cos 28 + 19.71 cos 48, Q16 = Q61

= 42.87 sin 28 + 19.71 sin 48,

Q26 = Q62 = 42.87 sin 28- 19.71 sin 48, Q66 = 26.88- 19.71 cos 48.

Ex} Ey

{Yxy

=

{5.553- 4.578 COS 28-0.422 COS 481 -0.577 + 0.422 COS 48 -4.578 sin 28 - 0.844 sin 48

X

10-ll

O'x.

Using these equations, normalized values of the three strain components are plotted in Fig. 2.12 as a function of the fiber orientation angle e. In order to show all three strain components on the same figure, the transverse strain component is multiplied by -10 and the shear strains by -1. From the figure, we see that the axial strain Ex increases as the fiber orientation angle increases from oo to 90°. The compressive transverse strain is symmetric with respect to the e = 45° orientation with a peak value of almost 10% of the maximum axial strain obtained for e = 90° at e = 45°. The shearing strain Yxy is zero for the two on-axis

I

0.8 O'x

~~

O'y

·§

0.6

"'

0.4

ax= JOOMPa

The same assumptions used in deriving Eq. (2.2.22) in Section 2.2.3 and Eqs. (2.3.7), (2.3.16), and (2.3.20) in Section 2.3 are applied when plies with unidirectional properties are bonded together. That is, classical lamination theory (CLT) assumes that the N orthotropic layers described above are perfectly bonded together with an infinitely thin bond line and the in-plane deformations across the bond-line are continuous. The assumptions of constant through-the-thickness strain distribution E0 in case of in-plane loading and a linear throughthe-thickness variation, Eq. (2.3.3), in case of bending loads can be superimposed. The strain distribution is, therefore,

Ex} = {E~} +z {l(x} , { Y~y

Strain versus orientation angle for unidirectional lamina in

E~

Ey

Yxy

(2.4.22)

~

Kxy

where the superscript "o" represents the mid-plane strains, and the curvatures K that are constant through the thickness. Therefore, the stresses in the kth ply can be expressed in terms of the reduced stiffnesses of that particular ply by substituting Eq. (2.4.22) into the stress-strain relationship, Eq. (2.4.18),

kt

0.2 10 20 30 40 50 60 70 80 90 fiber orientation 9

Figure 2.12. tension.

Laminates of Orthotropic Plies

(jx

!:":

~

configurations, 8 = 0° and 8 = 90°. However, because of the nonzero Q 16 and Q26 terms, the layer experiences shearing strains at off-axis orientations. The maximum shearing strain is at about e = 36°.

2.4.3

For the specified stress state of O'x = 100 MPa and cry = 'txy = 0, strains are obtained from the inverse of Eq. (2.4.18) symbolically in terms of the fiber orientation angle

89


--.1 } [ ~ ~:: ~: ~:,., lkJ+zk} Qll

-

Ql2

-

Ql6

-

E~

Kx

(2.4.23)

The stress resultants and moment resultants (stress couples) per unit width of the cross-section acting at a point in the laminate (see Fig. 2.11) are obtained by through-the-thickness integration of the stresses in each ply,

-· ...... ·~· •w vvmr""\JCIIII: Ml\1 I: RIALS

N)

N:

I

=

Nxy

M) M: Mxy

i

I

{ax}

h/2

crY

't

-h/2

=

I

xy

dz = (k)

{ax}

h/2

z dz =

crY

't

-h/2

xy

(k)

L J{ax} crY dz, N

zk

k=l

zk-l

zk

k=l

zk-l

(2.4.24)

71

N=AE"+BK,

(2.4.31)

M=BE 0 +DK.

(2.4.32)

't

xy

{ax}

L J crY N

2.4 ORTHOTROPIC LAVERS

(2.4.25)

z dz.

't

For some design formulations, the use of sines and cosines of multiple angles [see Eqs. (2.4.21)] proves to be more useful. Starting with the integral form of Eqs. (2.4.28)-(2.4.30), for example,

xy h/2

Substituting the stress-strain relations of Eq. (2.4.23), we obtain the following constitutive relations for the laminate:

nr~~ NY Nxy

=

Ey + A, A"JrJ Y'/ry

A,2 A22 A26 A,6 A26 A66

r· B., n B"J

B,2 B22 B26 B,6 B26 B66

Ky Kxy

{A 1" B 1" D 11 }

=I Q11

{ 1,

(2.4.33)

z, z2 } dz

-h/2

and assuming all layers are of the same material, we have (2.4.26)

h/2

{A 11 , B 1" D 11 }

= U1 {h. O,

~;} + U I cos 28 {1, z, z

2

2

and

}

dz

-h/2

r·) My Mxy

= rBH B,2

n., B"j

B22 B26 B,6 B26 B66

r}

h/2

r~~ o., o"H KY~}

EY + Dl2 D22 D26 Y'/ry D,6 D26 D66

Kxy

I

+ u3 cos 48 {1. z. z2 } dz.

(2.4.34)

-h/2

(2.4.27) Similar expressions can be found for the other stiffness terms and are summarized in Table 2.1 where the expressions for the V's are given by

where N

Aii =

L (Qii)
(2.4.28)

VO{A, B, D}

Bii =2 ~ (Qii)(kl (z~- zL), k=l

N

tL k=l

~; }•

h/2

N

1~-

Dii =

={h, 0,

(Qii)(kJ

(z~ -

(2.4.29)

VI{A,B,DJ

h/2

= Jcos28 {1,z,z

2

}

dz,

v2{A,B,D}

-h/2

(2.4.30)

V3{A, B, D}

z2 } dz,

=JCOS 48 {1, Z, z } dz,

Z,

z2 } dz.

h/2

2

-h/2

Z,

-h/2

h/2

zL).

=I sin 28 {1,

v4{A.B,D}

=I sin 48 {1, -h/2

(2.4.35)

In matrix notation, Eqs. (2.4.26) and (2.4.27) are represented as For example, the expressions for A 12 and B 11 are given by

'~

MECHANICS OF LAMINATID OOMFIOSITE MATERIALS

Table 2.1. A, B, D matrices In terms of lamina Invariants {A 1 ~> Bl!, Dl!}

B22• D22} {A 12> B 12 , D 12 } {A66• B66• D66} 2{A 16, B 16 , D1d 2{A26• B26• D26} {A22,

Vo{A.B,DJ

Vl{A,B,D)

V2{A,B,D)

V3{A,B,D)

V4{A,B,D)

UI UI u4 Us 0 0

u2 -u2 0 0 0 0

0 0 0 0

u3 u3 -u3 -u3 0 0

0 0 0 0 2U3 -2U3

u2 Uz

h/2

I

A,2 = u4 VOA - u3 v3A = h u4- u3 cos 49 dz

and

-h/2 h/2

B 11

= u, V08 + U2 VIB + U3V38 = U2

h/2

I cos 29 {z} dz + U I cos 49 {z} dz,

-h/2

3

-h/2

(2.4.36) respectively. The set of integrals in Eq (2.4.35) can be replaced by summations since the fiber orientation angle 9 is constant through the thickness of each layer. After replacing the integrals with summations and performing the integrations across the individual layer thicknesses, the V terms take the following form: N

V, {A, B, DJ

= L cos 29(kJ { tk, t;ik, tk (z~- 2z~k-I + zL)}, k=l

N v2{A, B, D)

=L sin 29(k) { tk, t;ik, tk (z~- 2z~k-i + zL)}, k=l

N v3{A,B,D)

= L cos 49(k) {tk, t;ik, tk (z~- 2z~k-l + zL)J, k=l

N v4{A, B, D)

=L

sin 49(k) { tk, t;i"' tk (z~- 2z~k-i + zL)}'

k=l

(2.4.37)


73

where tk =zk- zk_ 1 represent the layer thicknesses and zk =(zk + zk_ 1)12 is the z coordinate of the mid-plane of the kth layer measured from the mid-plane of the laminate.

2.4.4

Elastic Properties of Composite Laminates

The in-plane, bending, and coupling stiffness matrices derived in the previous section are sufficient to describe fully the elastic response characteristics of a laminate under combined in-plane and bending loads. However, we may be tempted to follow the engineering tradition of associating the stiffness properties of a laminate with quantities that are similar to the classical elastic material properties. This is, of course, not possible for general laminates because of the complex couplings that exist between the different deformation modes. We saw that there is coupling (B :F. 0) between in-plane and bending deformations of a laminate, unless the lamination sequence is symmetric with respect to the laminate mid-plane. This is true even in the case of laminates made of isotropic layers. For example, from Eqs. (2.4.26) and (2.4.27), it can be observed that existence of a B 11 term implies coupling between in-plane stress resultant Nx and the laminate curvature along the x axis Kx, as well as between moment resultant Mx and the mid-plane strain E~. A more intricate coupling is due to the B 16 and B 26 terms. These terms may induce twisting deformations in laminates even if there is no applied twisting action. Laminates loaded by in-plane loads inducing uniform in-plane Nx and NY stress states will experience twisting curvatures Kxy· The terms B 16 and B26 are often referred to as the extension-twist coupling terms and are caused by the existence of off-axis layers that are not symmetric with respect to the laminate mid-plane. If the laminate is made of isotropic layers, then B 16 = B26 = 0 even if the laminate is not symmetric. Clearly, there are no traditional material properties that can reflect this complex material behavior. When restricted to symmetric stacking sequences, vanishing of the B matrix eliminates coupling between the in-plane and out-of-plane responses of a laminate. In this case, the mid-plane strains are only related to the in-plane stress resultants and the curvatures to the moment resultants (stress couples). Equations (2.4.31) and (2.4.32) can be solved independently from one another: one set for the in-plane response and another set for the out-of-plane response. Based on this

74


simplification, it may again become tempting to define material properties that mirror the classical material properties of monolithic materials. However, even in the uncoupled case, there is nothing simple about the composite laminate deformation characteristics. There are still terms in the in-plane and bending stiffness matrices that display nonclassical material behavior. Starting with the bending stiffness matrix, the stiffness terms D 16 and D26 couple the moment resultants Mx and MY with the twisting curvature. The resulting effect is the tendency of the laminate to curl under applied uniform bending moments. Thus, the terms D 16 and D 26 are commonly referred to as the bending-twisting coupling terms. These terms exist for all laminates that have layers with off-axis orientations. Their relative magnitude compared to the other bending stiffness terms may be made small by manipulating the stacking sequence (as mentioned in the next paragraph), but unless the off-axis layers are removed from the laminate they will have a finite value. Similarly, the existence of the A 16 andA 26 terms in the in-plane stiffness matrix yields a coupling behavior termed shear-extension coupling. The net effect of these terms on the laminate response is the induction of shearing deformation under in-plane normal stress resultants Nx and NY. Although the primary reason for the existence of these terms is the presence of layers with off-axis fiber orientations, unlike the bending-twisting terms, these terms may be eliminated q uiteeasily by enforcing a balanced condition of the off-axis layers. Balanced laminateshave a layer with a negative 9 orientation for every layer with a positive 9 orientation. The positive and the negative parts of the pair do not have to be placed adjacent to each another, but can be at separate through-the-thickness locations. However, the distance between them has a direct influence on the D 16 and D 26 terms. Balanced laminates with adjacent layers of plus and minus 9 orientations will have smaller bending-twisting terms compared to ones that have layers with the same orientations grouped together and separated from the layers with negative angles. The following example demonstrates some of the coupling characteristics of unsymmetric laminates.

Example 2.4.2 A 16-layer Glass/Epoxy laminate with [(9/(9 + 90h/9)2 / (45/ -45 2/45) 2h stacking sequence is loaded such that the in-plane

75

:J 4 ORTHOTROPIC LAYERS

and the bending stress states are characterized by {Nx, NY, Nx)T = {0.0, 0.0, O.O}T and {Mx, My, Mx)T = { 1.0, 0.0, O.O}T MN, respectively. The fiber orientation angle of the layers above the mid-plane can take values of 9 in the range 0° ~ 9 ~ 45°. Determine the mid-plane strains and curvatures experienced by the laminate for the range of 9 under consideration. The elastic properties of the Glass/Epoxy are E 1 = 181 GPa, E 2 = 10.3 GPa, G = 7.17 GPa, v 12 = 0.28, and the individual layer thickness is 12

0.001 m. The elements of the in-plane, coupling, and bending stiffness matrices for the laminate are computed in terms of the angle 9 from Eq. (2.4.35) and Table 2.1, A II= 300.6+ 61.55 cos 29 + 13.32 cos 49

+ 61.55 cos(180 + 29) + 13.32 cos(360 + 49), A

12

= 114.8- 13.32 cos 49- 13.32 cos(360 + 49),

A 16 = 30.78 sin 29 + 13.32 sin 49

+ 30.78 sin(180 + 29) + 13.32 sin(360 + 49), A

22

= 300.6 -

61.55 cos 29 + 13.32 cos 49

-61.55 cos(180 + 29) + 13.32 cos(360 + 49), A 26

= 30.78 sin 29- 13.32 sin 49 + 30.78 sin(180 + 29)- 13.32 sin(360 + 49),

A 66

B

11

= 146.1 -

13.32 cos 49 - 13.32 cos(360 + 49)

(all Au in MN /m),

= -0.1065-0.2462 cos 29-0.05327 cos 49 -0.2462 cos(180 + 29)- 0.05327 cos(360 + 49),

B

12

=0.1065 + 0.05327 cos 49 + 0.05327 cos(360 + 49),

B 16 =-0.1231 sin 29-0.05327 sin 49

- 0.1231 sin( 180 + 29) - 0.05327 sin(360 + 49),

78

MECHANICS OF LAMINATED COMIIOSITE MATERIALS

B 22 =-0.1065 + 0.2462 cos 28 - 0.05327 cos 48

+ 0.2462 cos(180 + 28)- 0.05327 cos(360 + 48), B 26

=-0.1231 sin 28 + 0.05327 sin 48 - 0.1231 sin( 180 + 28) + 0.05327 sin(360 + 48),

B 66 = 0.1065 + 0.05327 cos 48

+ 0.05327 cos(360 + 48) (all Bu in MN), D 11

=6.412 + 1.375 cos 28 + 0.2974 cos 48 + 1.252 cos(180 + 28) + 0.2708 cos(360 + 48),

D 12 =2.449- 0.2974 cos 48- 0.2708 cos(360 + 48), D 16 = 0.06155 + 0.6873 sin 28 + 0.2974 sin 48

+ 0.6258 sin( 180 + 28) + 0.2708 sin(360 + 48), D 22

=6.412- 1.375 cos 28 + 0.2974 cos 48

77


shear-extension coupling terms vanish for the entire laminate. Since there is a 90° difference between the pairs of the layers on the top half, the only angles that yield a balanced laminate are 8 = 0°, 8 = 90°, and 8 = 45o. For the first two values, the section of the laminate above the mid-plane is cross-plied (0/90/0) 2 and (90/0zf90) 2 stacking sequences, respectively. For 8 = 45°, the section of the laminate above the mid-plane is the balanced angle-ply (45/-45z145) 2 laminate. The variation of the A16 andA 26 terms with the orientation angle 8 is shown in Fig. 2.13. The maximum values for the shear-extension coupling terms reach a maximum value for 8 = 22.5° for this laminate. They vanish for 8 = 0° and 90° angles. Also, the curves for A 16 and-A 26 coincide. Although the expression for A 26 is not the same as -1 x A 16 above, it can be shown through trigonometric manipulation that A 16 =-A 26 • Next we investigate the terms of the B matrix. Again, through trigonometric manipulation it can be shown that for this laminate B n = B 22 and B 11 = -B 12 regardless of the 8 values. Also the extension-twisting coupling terms are B16 = -B 26 • Variation of the magnitudes of the various B terms is shown in Fig. 2.14. Note that for 8 = 45°, not only the section of the laminate above the mid-

- 1.252 cos(180 + 28) + 0.2708 cos(360 + 48), D 26 = 0.06155 + 0.6873 sin 28-0.2974 sin 48

+ 0.6258 sin(180 + 28)- 0.2708 sin(360 + 48), D 66 = 3.118-0.2974 cos 48

-0.2708 cos(360 + 48)

A (MN/m} 25 20

(all Du in kN- m).

Note that the three stiffness matrices are fully populated, indicating highly coupled deformation modes involving mid-plane strains and curvatures. However, the terms are trigonometric functions of the 8 and some may vanish for certain values of the orientation angle. For example, the bottom half of the laminate (45/-45z145) 2 is balanced. For orientation angles that satisfy the balanced condition for the top half of the laminate, the in-plane

A16 -A26

15 10

5

5

10

15

20

25

30

35

40

45

e

Figure 2.13. Variation of A1s and A2s as a function of 8 for a [{8/(8 + 90)2/8)2/(45/-452/45)2] r laminate.

78


79

:1.4 ORTHOTROPIC LAYERS

B (MN)

0.2r

~Bu

0.15

0.1

0.05

5

10

15

20

25

30

35

40

45

e

Figure 2.14. Variation of 811, 822, 816, and 826 as a function of 8 for a [{8/{8 + 90)2/8)2/(45/-452/45)2] r laminate.

sin 48 and sin ( 180° + 28) terms are slightly different. The trigonometric simplifications mentioned earlier are not possible. The resulting curves for the laminates as a function of the orientation angle are distinct. Although the D 26 curve crosses the zero axis, there is no orientation angle that will make both terms vanish simultaneously. To determine the mid-plane strains and curvatures, coupled equations of the form in Eqs. (2.4.26) and (2.4.27) need to be solved. For space considerations, the form of the equations will be shown for two numerical values of the orientation angle, namely, 8 = 30° and 8 = 45°. For 8 = 30°, the stacking sequence is [(30/120z130)zl(45/-45zl 45) 2 Jr, or equivalently [(30/-60z130)/(45/-45z145) 2 Jr. The constitutive equations become 287.2 128.1 23.07] 128.1 287.2 -23.07 [ 23.07 -23.07 -23.07

plane is balanced, but the entire laminate becomes symmetric, [(45/-45z145) 2 ],, causing all the B terms to vanish. Finally, the bending-twisting coupling terms D 16 andD are 26 shown in Fig. 2.15. The expressions for the D 16 andD terms 26 seem to possess the kind of similarities that exist between the A 16 andA 26 and B 16 andB26 terms. However, a closer look reveals that the coefficients of the sin 28 and sin ( 180° + 28) terms, and

l

E~)

X

106 E~

yoxy

-53.27 53.27 -92.27] { Kx) + 53.27 -53.27 92.27 X 103 Ky [ -92.27 92.27 53.27 Kxy

= {0.0~ 0.0

(2.4.38)

0.0

and D (kN-m) 0.6

-53.27 53.27 -92.27] { E~~ 53.27 -53.27 92.27 X 103 E~ [ -92.27 92.27 53.27 y~y

D16

6.189 2.733 0.6069] X 103 {Kx) + 2.733 6.066 -D.3772 Ky [ 0.6069 -0.3772 3.402 Kxy

= {1.0} 0.0 X 106. 0.0 (2.4.39)

Solution of the six equations for the six unknowns yields E~

Figure 2.15.

Variation of D16 and fk6 as a function of 8 for a [{8/{8 + 90)2/8)2/(45/-452/45)2] r laminate.

and

=0.05095 m/m,

E~

=0.05095 m/m,

Y~y

=0.1875 m/m

ou

MECHANICS OF LAMINATID OOM,.OSITE MATERIALS Kx

= 216.2/m,

KY

= -104.3/m,

Kxy

Curvatures ( /m)

= -50.32/m.

For the [(45/-45i45) 2], stacking sequence, which corresponds to e = 45°, the constitutive equations governing the in-plane and bending responses uncouple 273.9 141.4 0.0 0.0

141.4 273.9 [

0.0

l

200 150

0

X106 {£Ef = {00:00} }

172.8

0.0

81

2.4 ORTHOTROF'IC LAYERS

(2.4.40)

0.0

Yxy

100 50

and

s 5.844 3.018 [

3.018 0.1231] 5.844 0.1231 X 103

0.1231 0.1231

3.686

{K} = 110:00} K:

Figure 2.17. X 106.

(2.4.41)

= Eoy = yoxy = 0 •0

and Kx=

233.4 /m,

KY

=-120.4 /m,

Kxy

=-3.772 /m.

Strains (m/m)

0.2 Yxy

Ey

5

Figure 2.16.

10

15

20

25

30

40

45

e

Mid-plane strains as a function of 8 for a [(8/(8 + 90)2/8)2/(45/-

452/45)2] r laminate.

15

20

25

30

35

40

45

e

Curvatures as a function of 8 for a [(8/{8 + 90)2/8)2/(45/-

452/45)2] r laminate.

0.0

Kxy

Solution of the six equations for the six unknowns yields Eox

10

Solution of the mid-plane strains and the curvatures for the complete range of the e is presented in Figs. 2.16 and 2.17, respectively. For balanced and symmetric laminates where the coupling characteristics are the least pronounced, use of effective engineering material properties may be attempted. This practice is still not encouraged in composite engineering. This is mostly because the utility of such properties is limited. For example, given the elastic modulus E of a material, the in-plane and bending stiffnesses of a monolithic plate may simply be expressed by using EA and EI, respectively. In this representation, the geometric properties of the plate, which are the crosssectional area A and the second moment of the area /, are conveniently decoupled from the material property. A single material property value and a single thickness value uniquely identify both the in-plane and the bending stiffness. This is not the case in composite laminates. There is no simple material property that can be decoupled from the laminate geometry in describing the bending stiffness. The bending stiffness of a laminate is strongly influenced by the relative positioning of the layers with different orientations. For a given total laminate thickness, there can theoretically be infinitely many different bending stiffnesses, even if the properties of the individual layers along their on-axis are identical.

MECHANICS OF LAMINATID OOMFIOSITE MATERIALS

Nevertheless, as we will see in later chapters, there arc situations in which the representation of the in-plane stiffness properties of a laminate by quantities that emulate traditional elastic stiffness properties may be useful. Following the approach used in Section 2.2.3, the following effective elastic properties may be derived in terms of the values of the in-plane stiffness matrix: ptt X

=_!_h (A 11A 22 - A2J2 J A22

c;e_ff xy

1 =hA66

pn y = _!_ h

and

(A A 11

22 -

An

AJ2

xy-

J' (2.4.42)

veff- ·--.

and

A212

A22

Note the difference between the elastic moduli along the x and the y coordinates of the laminate. In a direction other than the two geometric axes, the stiffness property will be different and needs to be computed using the transformation relations discussed in Section 2.4.2. Another perspective to this problem will be to imagine that we rotate the entire laminate under consideration, which is balanced and symmetric, through an angle -<1> from the x axis and we want to express the new stiffnesses along the x-y coordinates, £!, ~. G~Y' and ~Y in terms of the stiffnesses described in Eq. (2.4.42). It can be shown that the new stiffnesses are given by

J

1 2 2 (- 1 2 1/xy -=E..._+!l:_+ FJ Ex E G - E p q, xy

Y

X

£t

~y

J

1 =4 [ -+--+1 1 pq+-(p-q)2, 1 -<1>

Gxy

~y =£! [

Ex

1/xy

Ex

Ex

Ey

Gxy

(p2 + q2) - (__!_ + __!_ - _1

Ex

EY

Gxy

)p ql '

PROPERTIES OF LAMINATES MADE OF SUBLAMINATES

83

In the preceding paragraphs, we have made a series of assumptions with respect to the laminate construction to simplify the elastic response characteristics. One final assumption is to enforce A 11 to be lhc same as A22• Such balanced symmetric laminates, for example I±45].. or [0/90]., will have equal stiffnesses E;ff =E'/f. Unfortunately, however, they will still not behave like isotropic materials, which have identical stiffnesses in every direction, because their stiffness in directions other than the x and y directions may be different. There are only a small class of laminates, such as the [0/±45/90].,, [0/±60]., and 190/±30]s laminates, which meet the test of true in-plane isotropy. These laminates will have identical extensional stiffness in every direction and can be shown to have A 66 = 112 (A 11 - A 12), which reduces lhe shear stiffness to

£";ff G~~ = 2 ( 1 + v~~~ .

(2.4.44)

Although such laminates possess complete in-plane isotropy, they are only referred to as quasi-isotropic because in directions other than the in-plane direction the elastic isotropy disappears. Also, the laminates mentioned above are not the only quasi-isotropic laminates. A more general condition for laminate quasi-isotropy will be shown in Chapter 4.

2.5 PROPERTIES OF LAMINATES MADE OF SUBLAMINATES

X

1 n2 n2 ( -1 - 2 1/xyJ -=::J.__+L_+ - pq Ex EY Gxy Ex ' 2

;us

(2.4.43)

where p = cos 2 <1> and q = sin 2 <1>, and the superscript "eff' was eliminated from the right-hand sides for clarity.

The stiffness matrices of a laminate made of two or more sublaminates may be derived by using the stiffness matrices of the individual sublaminates directly, rather than using the Q matrices of the individual plies of the sublaminates. Besides convenience, the method presented here provides understanding of laminate stiffness properties and can be used to construct laminates with certain desired properties hased on the properties of selected sublaminates. Consider, for example, a laminate made up of two sublaminates denoted by M and N with nM and nN plies, respectively; see Fig. 2.18. We represent the stiffness matrices (A, B, and D) of the two laminates using subscripts M and N and the combined laminate of nM + nN =n plies with subscript MN.

"'"

84

MECHANICS OF LAMINATI!D COMPOSITE MATERIALS

Using the definition of the in-plane stiffness matrix from Eq. (2.2.20), the in-plane stiffness of the combined laminate is given by n

(A;)MN=

n

L

QiJ(kJ (zk- zk-I)

k=l

(2.5.1)

k=l

where the tk's are the thickness of the kth layer of the combined laminate. If the laminate is made up of S sublaminates, each with nq plies where q = 1, ... , S, then the summation in Eq. (2.5.1) can be divided into S summations, n,

(Aij)MN=

L

Qij(k) tk +

L

• -·

k =nM

=n

k

, . +

Z,nk

Qij(k) tk + ...

h=n,+l

+

L

(2.5.2)

Qij(k) tk.

h=n+·+n. +I l .'i-1

The tk's and Qk's under the summations are independent of the z coordinate and, therefore, the summations can be redefined with respect to the reference axes of the individual sublaminates. Rewriting the summations for the individual sublaminates, we have nJ

n2

~ .£....

Qij(kn)l tkn I + .£... Qij(kn)2 tkn 2 +

~

~

~

t,.

l,=~m A - - - l l ' - - -

+1

• --!!! 2 X

Zk= Zn

'¥

zk = Znm k

Figure 2.18.

Znk

=ZNn

--- ---

=n.., + nN =n

-

'NIN

ZN

IJioxM 'M-

I

!M 2

., ., XN

l

Stacking sequence of a laminate made of two sublaminates.

For the calculation of the bending-extension coupling stiffness matrix of the combined laminate, (B;)MN' we start with the definition of the matrix from Eq. (2.3.15) and write the difference of the squares of the zk terms as the product of a difference and a sum, 1

(B;)MN=

n ~-

n

1

~

S

=

L (A;)q' ~

(2.5.3) where tknq and Qij(knql are the thickness and transformed reduced stiffness matrix, respectively, of the kth layer of the qth sublaminate, and (AiJ)q is the in-plane stiffness matrix of the qth sublaminate. In the case of the two sublaminates M and N of Fig. 2.18, the combined in-plane stiffness matrix is (AiJ)MN= (A;)M+ (AiJ)N.

h=l

S

ns

· · · + .£... ~ Qij(kns'\ fkn

-

2 .£... QU
Aij =

=ZMnm

n+··+n I S

nr+nz

h=I

1

k=l

ZM

=L QiJ
85

:115 PROPERTIES OF LAMINATES MADE OF SUBLAMINATES

(2.5.4)

That is, when two or more laminates are put together to make a new laminate, the in-plane stiffness matrix of the new laminate is simply a sum of the in-plane stiffness matrices of the sublaminates. Note that this does not mean that the individual stiffness properties such as Ex and vxy of the new laminate are the sum of the same properties of the sublaminates.

n

= .L Qij(k) t;:zk,

(2.5.5)

h=l

where zk is the z coordinate of the mid-plane of the kth layer measured from the mid-plane of the combined laminate. We denote the total thickness of sublaminates M and N as tM and tN' respectively. Then the locations zM and zN of the mid-planes of the sub laminates measured from the mid-plane of the combined laminate are (see Fig. 2.18) fN zM=-2

and

zN

tM

(2.5.6)

2'

We denote the location of the mid-plane of each layer of sublaminates zMk and zNk' respectively. Then the coordinate of the mid-plane of the kth layer measured from the mid-plane of the combined laminate can be expressed in terms of the zMk and zNk as

M and N measured from their own mid-planes by

~

uu

MECHANICS OF LAMINATED COMPOSITE MATERIALS -

-

(N

zk

= zMk - 2

zk

=zNk + 2

-

for layers of sublaminate M and

tM

for layers of sublaminate N.

87

I:XEACISES tM

(BiJ)MN=

2

tN

(2.5.11)

(AiJ)N-2 (Aij)M,

(2.5.7)

~~

~

(Di))MN= (Di))M+ (Di))N+4 (A;)M+4 (AiJ)N

(2.5.12)

Substituting Eq. (2.5.7) into Eq. (2.5.5), we obtain

(Bij)MN

~ -Qij(k) tknM (-ZMk- 2tNJ + L..~ -Qij(k) tknN (-ZNk + 2tMJ ' =L.. k~I

bl

nM

=L

nM

tN Qij(k) tknM ZMk-

-

2 L Qij(k) tknM lc=J

bl nN

+

L

nN

Qij(k) tknN ZNk

tM

+2

bl

L

Qi)(k) tknN·

bl

(2.5.8)

By definition, the first and third terms in Eq. (2.5.8) are the bendingextension stiffness matrices of the sublaminates M and N. Similarly, parts of the second and fourth terms of Eq. (2.5.8) are the in-plane stiffness matrices of the sublaminates. Therefore, tM tN (BiJ)MN= (BiJ)M + (BiJ)N + 2 (AiJ)N- 2(Ai)M·

In general, the bending-extension stiffness of the combined laminate does not vanish, except when the products of the sublaminate thickness 1M and the elements of the in-plane stiffness matrix of sublaminate N ure equal to the products of the sublaminate thickness tN and the elements of the in-plane stiffness matrix of sublaminate M; see Eq. (2.5.11). One such case is that of sublaminate N being antisymmetric to sublaminate M. It is interesting to note that besides vanishing BiJ terms from Eq. (2.5.12), the bending-twisting terms D 16 andD 26 are also identically zero because the D 16 and D 26 terms of the antisymmetric laminates are equal in magnitude but opposite in sign (the A16 andA 26 terms are already zero because of the balanced condition for the individual sublaminate).

EXERCISES

I.

(2.5.9)

Similarly, starting from the definition of the bending stiffness matrix, the bending stiffness matrix for the combined laminate can be expressed (see Exercise 4) in terms of the bending, bending-extension coupling, and in-plane stiffness matrices of the sublaminates as

tJv

Aluminum and brass have the following properties: Ea1 106 psi, Gal = 3.70 X 106 psi, Ebr = 15.0 X 106 psi, Gbr 106 psi.

~

(DiJ)MN= (Di))M + (DiJ)N + 4 (AiJ)M + 4 (A;)N- tN (B;)M + tM (Bi)N·

(2.5.10) A practical case of interest is when the individual sublaminates M and N are balanced and symmetric such that (B;)M = (B;)N = 0, in which case,

Compute the A, B, and D, matrices and the effective elastic properties of laminates made up of aluminum and brass layers with the following stacking sequences. All thicknesses are in inches. (a) [tbr = 0.05/tal = 0.2/tbr = 0.05] (b) [!br = 0.025/ta] = 0.1/tbr = 0.025/tbr = 0.025/tal = 0.1/tbr = 0.025] (c) [tbr = 0.1/tal = 0.2] (d) [tbr = 0.025/tal = 0.1/tbr = 0.025/tal = 0.05/tbr = 0.05/tal = 0.05]

2.

= 10.0

x

= 5.60 X

Show that the bending-extension coupling matrix B is zero for midplane symmetric laminates, by showing that the summation inside the parenthesis of Eq. (2.3.12) vanish. [Hint: the terms (zk + zk_ 1)12 are the locations of the midplane of the individual layers with respect to the midplane of the laminate.]

88 3.


Compute the A, B, and D matrices and the effective elastic properties of the following laminates. (a) [±45/0/902] 2s (b) [±45/±30/902) 2s (c) [04 /9021zs (d) [(±45/0/902 )/(±45/0/902 ) 2 ] (e) [±45/±60/±15]2s

3

All laminates are made of Graphite/Epoxy with the following properties: E 1 = 128 GPa, E2 = 13 GPa, G12 = 6.4 GPa, and v12 = 0.3. Individual ply thicknesses are t = 0.013 em. 4.

Complete the steps of derivation of Eq. (2.5.10).

REFERENCES Boresi, A. P. and Chong, K. P. (1987). Elasticity in Engineering Mechanics. Elsevier Publishing Company, New York.

HYGROTHERMAL ANALYSIS OF LAMINATED COMPOSITES

Herakovich, C. T. (1998). Mechanics of Fibrous Composites. John Wiley & Sons, Inc., New York. Hyer, M. W. (1998). Stress Analysis of Fiber-Reinforced Composite Materials. McGraw-Hill, New York. Jones, R. M. (1998). Mechanics of Composite Materials, 2nd Edition. Taylor & Francis, Philadelphia, PA. Tsai, S. W. and Hahn, H. T. (1980). Introduction to Composite Materials. Technomic Publishing Co., Inc., Lancaster, PA. Tsai, S. W. and Pagano N.J. (1968). "Invariant Properties of Composite Materials." In Composite Materials Workshop, pp. 233-253. Tsai, S. W., Halpin, J. C., and Pagano, N.J. (eds.) Technomic Publishing Co., Westport. Vinson, J. R. and Sierakowski, R. L. (1987). The Behavior of Structures Composed of Composite Materials. Martinus Nijhof Publishers, Boston.

In Chapter 2 we have seen that the stiffness properties of a composite laminate may be tailored to desired values by arranging the orientation of the layers and the stacking sequence. Of particular interest are stiffness coupling properties such as extension-twisting and bending-twisting couplings, as they provide additional degrees of freedom in design. For example, load-adaptive structural systems for aerospace and mechanical engineering applications may be possible through the use of such coupling properties. The coupling characteristics, while providing a favorable design response under applied mechanical loading, may introduce undesirable effects with changes in moisture content and temperature. The latter behavior, referred to as the hygrothermal response of the laminate, is brought about because moisture and temperature variations influence the fiber and matrix materials differently. Therefore, just like the stiffness properties, hygrothermal characteristics of the individual layers are directional and depend on the fiber orientation angle. In a laminate under thermal loads, the individual layers will have a tendency to deform differently. If these differences are not taken into consideration in establishing the stacking sequence, the 89

90


laminate may warp as the environmental conditions change. In addition, during the curing process of laminate fabrication, these differences introduce residual stresses in the system that must be taken into consideration during the design process. In some situations, particularly when variations in temperature or humidity are known a priori, the ability to predict hygrothermal deformations may be of significant importance in the design process. The present chapter examines the behavior of laminated symmetric and unsymmetric composites under varying temperature and moisture conditions, with special emphasis on aspects related to the design of such laminates. In addition, characteristics of laminates in the presence of combined mechanical and hygrothermal loads are also discussed.

3.1 HYGROTHERMAL BEHAVIOR OF COMPOSITE LAMINATES In designing the stacking sequence of a composite laminate for anticipated loadings, one must take into consideration the effects of temperature and moisture variations on the behavior of the constituent materials. In addition to temperature-related expansions, composite materials, particularly those with resin-based matrix systems, expand by absorbing moisture from the environment. In fiber-reinforced epoxy laminates, the coefficients of thermal and moisture-related expansions for fiber and matrix materials are significantly different. For example, for a unidirectional layer, the value of the coefficient of thermal expansion in the fiber direction a 1 is one to three orders of magnitude smaller than the value Uz in a direction perpendicular to the fibers; see Table 3.1. Under these circumstances, if one considers a laminate with two layers of the same material but different orientations, the response of the laminate under thermal or hygral loads is similar to that of a bimetallic plate fabricated from two different isotropic materials. Fiber-reinforced composite laminates are generally fabricated at high temperatures, which are necessary to allow for resin curing. The fabrication typically involves cutting the prepreg sheets to size, slightly heating them, and laying them up at the desired orientation angles for the laminate. The laminate is then placed in a vacuum bag and heated in an autoclave under pressure. The increasing temperature causes a change in the molecular structure of the polymer matrix, which begins to flow. At a temperature of about l20°C, the applied

:1.1

91

HYGROTHERMAL BEHAVIOR OF COMPOSITE LAMINATES

Table 3.1. Coefficients of thermal expansion

a,;oc

Material ( lruphite/Epoxy Boron/Epoxy Uluss/Epoxy ----

Source:

X IQ-6

0.02 6.1 X 10-6 8.6 x 10-6

a 2 /°C 22.5 X IQ-6 30.2 X w-6 22.1 x 10-6

Jones, 1998.

pressure assists in removal of gases and any trapped air. Furthermore, at this temperature, new cross-linking among the polymer molecules occurs, and the matrix begins to harden. For a standard epoxy-based composite system, the temperature is further increased to about 180°C and held at this cure temperature for up to two hours. Note that for typical thermoplastic resins, the cure temperature may be as high as 380°C. After this curing at elevated temperatures, the laminate is removed from the autoclave and allowed to cool down to room temperature. The process of cross-linking also results in shrinkage of the matrix material. A laminate may consist of several layers at different orientations. Hence, in any given direction the amount of shrinkage may he different in different layers. As discussed later in this chapter, this contributes to the buildup of residual stresses in the laminate. Howl!ver, this chemical shrinkage occurs at elevated temperatures close to the cure temperature, allowing viscous flow to occur in the matrix material with consequent relaxation of the residual stresses. Even though the cure temperature is higher than the temperature at which there are no residual stresses in the system, for most practical purposes the cure temperature can be taken to represent a stress-free condition. In subsequent discussions, the material behavior is assumed to be lincar elastic with time and temperature-moisture independent material properties. In actuality, at the elevated temperatures at which the laminate is fabricated, the behavior is closer to viscoelastic and the material properties are a function of both time and temperature. 3.1.1 Temperature and Moisture Diffusion in Composite Laminates Heat conduction through the thickness of a laminate is governed by ·the Fourier equation of heat transfer

92


i_ [K

az

arJ= az P car. at

(3.1.1)

:t 1 HYGROTHERMAL BEHAVIOR OF COMPOSITE LAMINATES Odeg C(z,t)

90deg

Here, T(z, t) is the time dependent through-the-thickness temperature distribution, K is the thermal conductivity in the thickness direction z, c is the specific heat of the material, and p is the aggregate density of the composite material. If all layers of the laminate are of similar material systems and with similar fiber-to-matrix volume ratios, K may be assumed constant with respect to the thickness direction z. The resin material in composite systems has an affinity for absorb- ' ing moisture or reducing its moisture content, depending on the relative humidity of the atmosphere. For a unidirectional composite lamina, this response under varying humidity conditions is similar to its thermal response-the lamina expands or shrinks to a much greater extent in a direction transverse to the fiber than in the fiber direction (Table 3.1). Hence, in a laminate consisting of layers with various orientations, the different levels of expansion or contraction result in the introduction of residual stresses. Similar to the process of thermal diffusivity, the rate of moisture absorption or desorption is expressed by Pick's equation as D

a e2 = ae 2

az

at.

(3.1.2)

Here D is the coefficient of moisture diffusion, which is analogous to the thermal diffusivity coefficient K/(pc) and is assumed to be constant in the z-direction; C(z, t) is the amount of moisture as a fraction of the dry mass of the composite at a through-the-thickness location z (see Fig. 3.1) and time t. The term e is also referred to as the specific moisture content. Since the moisture absorption is mostly governed by the resin system, the assumption of constant coefficient of moisture diffusivity in through-the-thickness direction holds for layers with different fiber properties. A typical set of initial and boundary conditions for which the above partial differential equation may be solved can be written as

e=e;,

h

h

-2
t~

0,

93

____ plane of symmetry 90deg Odeg

z

Figure 3.1. Moisture distribution in a laminate.

e=em,

z=(-~·~}

t> 0,

(3.1.3)

where h is the laminate thickness and C; is the initial moisture content. The em is the long-term equilibrium value of the moisture content, which is a function of the relative humidity of the environment to which the composite is exposed. Here em is also a function of material properties that reflect the maximum moisture concentration that can he absorbed in the laminate and is empirically represented as Cm = al(/100)b] where is the relative humidity of the environment, and a and b are constants that reflect material characteristics. Solution of the problem expressed by Eqs. (3.1.2) and (3.1.3) is typically provided ir~ terms of a through-the-thickness average specific moisture content, ( ·• in the composite material defined as

I C(z, t) dz.

h/2

-e=-;; 1

(3.1.4)

-h/2

For example, for exposures over a long time period the average speci fie moisture concentration is given by

e- e; = 1---zexp 8 (-rch22DtJ • em- e; rc See Tsai and Hahn (1980).

(3.1.5)

94


For short time exposures, an approximate solution presented by Crank (1956), -C-C; --=4

em- ci

Hft nh -

Hygrothermal Deformations

Consider a single unidirectional layer that is subjected to a temperature change 11T from its stress-free state. Furthermore, assume that this layer also experiences a change in moisture content denoted as 11C. The resulting free expansion (or stress-free expansion) strains in the lamina can be written as

{1} = {:~} {~} = {~} {~~} +

YI2

yl2

11T+

yl2

0

I.

95

HYGROTHERMAL BEHAVIOR OF COMPOSITE LAMINATES H~--~-~-

(3.1.6)

2'

provides good estimates of moisture content. It is important to emphasize that heat conduction in composite laminates occurs much faster than moisture diffusion. Any changes in the properties of the composite that may be introduced due to variations in moisture content (e.g., aggregate density or thermal conductivity) may be ignored in the solution of the heat conduction equation. Similarly, since equilibrium temperatures are rapidly attained in the laminate, the moisture diffusivity coefficient can be assumed constant through the laminate thickness. An exception must be made in the case of very short-period transient loads or environments that have large sustained through-the-thickness thermal gradient along with moisture diffusion, such as cryogenic liquid containers.

3.1.2

:1.1

11C,

(3.1.7)

0

where al' ~I' and a 2, ~ 2 are the thermal- and moisture-related expansion coefficients in the fiber direction and transverse to the fiber direction, respectively, and the superscript F indicates stress-free expansion. The superscripts FT and FH represent stress-free thermal and hygral effects, respectively. These free expansion strains are shown in Fig. 3.2, where the dark bands indicate the fibers in a unidirectional composite layer. Note that the expansion in the fiber directions is typically smaller than in a direction transverse to the fibers.

Figure 3.2. Stress-free expansion strains in a unidirectional layer.

Also note that no hygrothermal shear deformations are induced in the unidirectional composite layer. If the strains of Eq. (3 .I. 7) were of purely mechanical origin, the stress components corresponding to the strains would be obtained from the stress-strain relations of a single layer, Eq. (2.4.3) of Chapter 2, as follows:

11 cri} -[Q Q12 {'t12 0 O"z

12

1{E1} Ez ·

Q Q22

00

0

Q66

(3.1.8)

yl2

It is important to emphasize again that the stresses may be calculated from Eq. (3.1.8) if and only if the strains are of mechanical origin. Commonly, overall strains (or total strains) in a layer, E1, are expressed as a sum of the free expansion strain, EF =EFT+ EFH, and mechanically induced strains, EM, Er

= EF +EM.

(3.1.9)

Stresses in a layer that undergoes both thermal and mechanical strains could then be calculated from

cri} = [Qn Ql2 Q12 Q22 {'tl2 0 0 O"z

00

1{Ei- Ef}

Q66

t

F

Ez-Ez

Yiz- Yfz

.

(3.1.10)

98

HYGAOTHEAMAL ANALYSIS OF LAMINATED COMPOSITES

Stresses would develop in a layer if and only if the hygrothermally induced free deformations were somehow constrained by means of an externally applied mechanical strain field. For example, if the total strains in the layer were completely suppressed to zero (t 1 =0; no expansion of the layer in any direction is allowed) by means of externally applied mechanical strains equal to the hygrothermal strains but opposite in direction, Eq. (3.1.9), then the resulting stresses in the layer would be

a,} = [Q" { 0 O'z

Q12

~12

12

Q

0J {-a ~T- ~~L\C ~ .:\C} . -a~ L\T-

Qn

0

0

Q66

3.1

HYGAOTHEAMAL BEHAVIOR OF COMPOSITE LAMINATES

, ~ f I ~ ~~r===~ tr1'W-1---.tW---

90degply

(3.1.11)

i i i

0

Figure 3.4. Constraining effect of adjacent layers on 90° layer.

individual layers are bonded together, such stress-free deformations are not possible. Adjacent layers of the laminate provide a constraining influence against free hygrothermal expansion or shrinkage of every layer. Even in the absence of any external constraints, therefore, one can think of each layer deforming against some finite stiffness constraints (as shown in Fig. 3.4), which act like applied mechanical restraints. Such a constrained deformation would result in the development of residual stresses in the individual layers, and the total strain the laminate experiences would have to be determined from the overall equilibrium of the laminate. To get a clearer understanding of the nature of these residual stresses, consider a [0/90Jr laminate shown in Fig. 3.5. If each layer were allowed to expand freely without the constraining effect of the

free expansion strains

o deg

---.tW---1 -M¥-+

1---.tW---

In the absence of any external constraints, the total strains will be equal to the stress-free strains (zero mechanical strains) and, therefore, no stresses would be induced in the layers. In a composite laminate consisting of several unidirectional layers with the same orientation, the hygrothermal deformation in each layer would be identical (provided that there is no through-the-thickness variation of temperature or moisture and hygrothermal properties of the layers are the same) and there would be no constraints on layer deformations from adjacent layers. Again, in the absence of any external constraints, no stresses would be induced in the layers. In a general laminate, however, the situation is different and somewhat more complicated because each layer with different orientation would have a tendency to expand or contract differently along the axes of the laminate. For a [45/90/0] laminate, for example, the free expansion strains in the x direction (which corresponds to the fiber direction of the 0° layer) are shown in Fig. 3.3. Since in the actual laminate the

97

I

,,~----------------------------

.................. • 1

90deg

45deg

Figure 3.3. Free expansion strains in a (45/90/0]Iaminate.

,'

,~-----------------------

Figure 3.5. A (0/90] cross-ply laminate.

"

.1111

HYQROTHERMAL ANALYSIS OF LAMINATED COMPOSITES

y

0 deg ply

:1.1

HYQROTHERMAL BEHAVIOR OF COMPOSITE LAMINATES

99

produce stresses. Calculation of the total strains of the laminate in lhis case is not trivial and requires a systematic hygrothermal analysis or the laminate, as presented in the following section.

3.1.3 Residual Stresses For a general laminate made up of layers with different orientations, lhc free expansion strains in each layer given by Eq. (3.1.7) can be lransformed to a desired coordinate system x-y following the strain lransformation relation of Eq. (2.4.12), which yields

90 deg ply

EF

= m2EF1 + n2EF2'

EF

=n2EF1 + m2EF2'

X

y

y~Y = 2mn Ef - 2mn E~,

Figure 3.6. Stress-free expansion of the oo and goo layers.

(3.1.12)

where adjacent layer under changed hygrothermal conditions, the expansion would be as shown in Fig. 3.6. The 90° layer expands much more in the x direction than the oo layer, as the expansion coefficient in this direction for the former is dominated by the matrix constituent of the lamina. Under the constraining effect of the adjacent layer, the bonded laminate would expand to some intermediate stage, as shown schematically in Fig. 3.7. Such a laminate would also bend due to the unsymmetric stacking sequence of the laminate. It is clear from these figures that the oo and the 90° layers are in tension and compression, respectively, in a direction aligned with the 0° layer and in the opposite state in the 90° direction. Even if this laminate is not subjected to an external mechanical load, the internal reactions that force the layers to remain together can be thought of as mechanical loads that

oo goo

'-------~~· Tension

1001(

Compression

Figure 3.7. Constrained deformation of the [Q/gQ]Iaminate. ·

m = cos

e

n = sin e.

and

In the case of thermal strains only (no hygral strains) and using Eq. (3 .1. 7), the free expansion strains can be written as EFT= (a mz 1

X

+ a 2nz) t:..T'

EFT= (a 1n 2 + a 2m 2) t:..T' y

y~J = 2( a 1 - a 2 )mn t:..T.

(3.1.13)

In matrix notation, these equations can be represented as

{~1 'Yxy

= {::}

t:..T

or

rfT =a t:..T,

(3.1.14)

axy

where axy is sometimes referred to as the coefficient of thermal shearing. In the case of only hygral strains, the ~·s and t:..C are substituted for a's and t:..T, respectively, and the superscript FT is replaced by FH in Eq. (3.1.14).

IVV

HYGROTHERMAL ANALYSIS 011 LAMINATED COMPOSITES

In a general laminate, when layers at different orientations are bonded together, the laminate will assume some deformed shape that depends on the stacking sequence. In a symmetric layup, there would be in-plane deformations only. However, for unsymmetric layups, the laminate may also develop curvatures (except when the layup is hygrothermally curvature stable, a concept which will be explained later). Within the framework of the classical lamination theory discussed in Chapter 2, such deformations are fully described in terms of mid-plane strains and curvatures represented by

3.1

Note that (z) indicates through-the-thickness coordinate dependence of the strains, changing from one layer to another. Dependence of the 1:~~> to the z coordinate may arise due to nonuniform through-thethickness distribution of the temperature or moisture. Defining z(k) to be the through-the-thickness location of a point in the kth layer, as opposed to its traditional usage to indicate the location of the layer interfaces, we define the following strains: X

EoN} eoN= £~N {YoN

and

xy

KN={~}·

which are the total strains and curvatures corresponding to the final shape of the deformed laminate under purely hygrothermal loads. For the remainder of this chapter, laminate strains that are of nonmechanical origin will be represented by the superscript N and referred to as hygrothermal strains. That is, the sum of the thermal, Er, and hygral, EH, strains are denoted as EN. Strain state in a laminate that experiences both in-plane and bending strains is given in Chapter 2 by Eq. (2.4.22) based on classical lamination theory. Through-the-thickness distribution of the hygrothermal strains are, therefore, given by

EN= EoN+ zt(l.

(3.1.16)

It is important to distinguish the hygrothermal strains in the individual

layers, EN, from the free expansion strains, EF, of the individual layers. The former includes the effect of the restraining action of the adjacent layers on one another as the laminate deforms as a whole. The latter ignores this constraining effect and represents the layer strains as if the layers are free to slide over one another. Definition of strains that give rise to stresses was made in Eq. (3.1.9). Using the same concept, the difference between the hygrothermal (total) strains EN of Eq. (3.1.16) at a given through-the-thickness location z(k) and the free expansion strains in that layer is the strain that causes stresses, EM

(z)

=EN (z)-

E~) (z).

(3.1.17)

X

ER} £~ = {£oN} £~N { Y~y

(3.1.15)

<

101


y~~

+ z
<

X

Y~y

(3.1.18)

It is important to note that the strains E~
{(J~}

12 1

a;

'txy

w

= [Qll Qz1 Q Qzz Q Qz661 Q61 Q62 Q66

l

[{E~N} {<11£~} £~ + < - E; , Y < Yw z

xy

xy

(3.1.19) where Qij are the transformed reduced stiffnesses that are defined in terms of the material properties and orientation angles [see Eqs. (2.4.19) and (2.4.21)]. Even though the stiffness coefficients of most

102

HYGAOTHEAMAL ANALYSIS OF LAMINATED COMPOSITES

composite materials may be functions of the temperature and moisture, this variation is ignored in the present discussion. Finally, it is important to note that the overall laminate deformations under hygrothermal loads, ~;oN and KN, are still unknowns at this point. Determination of the mid-plane strains and curvatures requires completing the)aminate analysis under purely hygrothermal effects, which leads to the definition of equivalent hygrothermal stress and moment resultants.

:1.1

EoN} {KN} J[Q] {EF} [A] E~N + [B] K~ E~ dz = 0, {'Y 2

h/

-

oN xy

[B]

KN xy

In classical lamination theory, the stress behavior of the laminate is generally characterized by stress resultants obtained by through-thethickness integration of the layer stresses; see Eqs. (2.4.24) and (2.4.25). For a purely hygrothermal response of a laminate, there are no external forces or moments acting on the laminate; hence, there cannot be any stress or moment resultants within the laminate. However, we know that for general laminates there are residual stresses within the layers. Therefore, the stress resultants associated with the residual stresses must vanish. Using the definition of the stress and moment resultants, the condition for vanishing stress resultant associated with the residual stresses is given by

NR] =Jhl2 {a~} {01o, ~ a: dz= { N~y

-hn

~

0

(3.1.20)

xy

-h/2

'tR

'Y

0

EF} z dz = 0,

J [Q] E~ {

(3.1.23)

yFxy

-h/2

where A, B, and D are the extension, bending-extension coupling, and hending stiffness matrices, respectively, derived in Chapter 2 [see Eqs. (2.4.28), (2.4.29), and (2.4.30)]. The terms under the integral above are constant in each layer for a given constant through-the-thickness distribution of !:J.T or !:J.C. For a given laminate stacking sequence, those integrations yield· vectors of constants that are commonly referred to as the induced hygrothermal or nonmechanical loads,

{NN} =

NN] N~

{

NN xy

{MN}

J[Q] {EF} E; dz

h/2

=

-h/2

MN] {MNxy

h/2

-h/2

and

(3.1.24)

yFxy

= M~ = J [Q]

{EF} E; z dz.

(3.1.25)

yFxy

Rearranging the terms, Eqs. (3.1.22) and (3.1.23) can be written in the following matrix form, which reveals why the terms defined above are called hygrothermal loads:

BJt' = NN,

(3.1.26)

BE 0 N+ DJt' = MN.

(3.1.27)

AEoN+

MR

(3.1.22)

"{F xy

h/2

-

KN xy

and the vanishing of the moments of the residual stresses by

MR] =h/2J {aR} {0} M} a~ z dz = 0 . {

-h/2

EoN} {KN} E~N + [D] K~ {'Y oN xy

3.1.4 Hygrothermal Laminate Analysis and Hygrothermal Loads

103

HYGAOTHEAMAL BEHAVIOR OF COMPOSITE LAMINATES

(3.1.21)

Substituting the residual stresses of Eq. (3.1.19) into Eqs. (3.1.20) and (3.1.21), we obtain

The computation of the strains and curvatures resulting from purely hygrothermal loading is similar to the computation of strains and curvatures for mechanical loads that create a uniform stress and moments resultants of N, and M, except that the stress and moment resultants

104 HYGROTHERMAL ANALYSIS 011 LAMINATED COMPOSITES

are replaced with the equivalent hygrothermal (or nonmechanical) loads,

=

{ toN} KN

[ABDBJ-I {NN} MN·

It should be emphasized, though, that the use of letters N and M in hygrothermal loads gives the notion that these quantities are resultants of internal stresses in the laminate. In fact, sometimes these terms are erroneously referred to as the thermal stress and moment resultants. Such a terminology should be avoided, as through-the-thickness integration of the residual stresses actually vanish under pure hygrothermal loading. Finally, the thermal loads and moments may be expressed by using the material properties (invariant U's) defined in Chapter 2, Eq. (2.4.20). Substituting the definition of the Q matrix terms, Eq. (2.4.21), into Eqs. (3.1.24) and (3.1.25) and making use of the free strains of Eqs. (3.1.12) and (3.1.7), we have

{NT}

JVT}

hi2{K1 + K 2 cos 28} K1 cos 28 11T dz -h/2 K 3sm 28

= {NJ = ~ NI;Y

= { MJ = ~ MTJ

{MT}

Mxy

I

K~

(3.1.29)

h12{K1+ K2 cos 28}

I

K1 - K: cos 28 11T z dz -h/2 K sm 28 3

(3.1.30)

K2

= U2 (a1 + a 2 ) + (U1 + 2U3 -

K3

U4 ) (a

= U2 (a1 + a 2) + 2 (U3 + U5 )

1

a

-

(a1 - a

2

).

3.1.5

Coefficients of Hygrothermal Laminate Expansion

To a practicing engineer, the coefficients of moisture- and thermalrelated expansion for individual layers are often less meaningful than an equivalent coefficient for a general multidirectional composite laminate. For a unidirectional composite laminate, the hygrothermal expansion or shrinkage is the same in each layer for a uniform through-the-thickness temperature and moisture distribution. For such a laminate, the coefficients of hygrothermal expansion for the laminate are the same as that of an individual layer. For more general laminates, a calculation of these coefficients is somewhat more complex, as they become dependent on layer stiffnesses and the stacking sequence of the laminate. Following the discussion in Section 2.4.4, it is not too difficult to extend the argument that it is not possible to represent the deformation characteristics of general laminates under hygrothermal loads by defining laminate coefficients. Complex couplings that exist between the different deformation modes discussed earlier exist under hygrothermal loads. Nevertheless, there are special situations in which meaningful definitions may be made. We first start with the classical definitions of the coefficients of thermal and hygral expansions a= ET I 11T and ~ = EH I !1C, respectively. The superscripts T and H denote thermal and hygral effects, respectively. In the most general case, involving in-plane and bending deformations, the strains at a given point in the laminate are expressed in terms of the laminate mid-plane strains and curvatures, t = t 0 + zK; see Eq. (2.4.22). Therefore, the coefficients of expansion "at a point" in the laminate are expressed as 1

+ U2 (a 1 - a 2), ), 2

(3.1.31)

105


a x = -/1T (£x T + ZKI\ xh

where the constants Kp K2, and K3 all depend on the material properties only, regardless of the orientation angle, K 1 = (U1 + U4 ) (a 1 + a 2 )

3.1

0

1 ~x = /lC (£~H + z~),

1

1

ay

= f1T (£~T + ZK~,

1- (£oH + H) ~y =11C Y ZJS '

axy

= !lT (E~~ + ZK~y),

~xy = !l1C (£~: + z~).

(3.1.32) The mid-plane strains and curvatures in these equations are computed from Eq. (3.1.28) after differentiating between temperature- and moisture-related terms, t 0 T+ E0 H =toN and KT + KH = KN, which are in turn computed from

I

106


{E"T} KT

[A B]-l {NT} MT '

[A BJ' {NH} MH .

{E"H}

= B D

= B D

KH

(3.1.33)

Definitions obtained in Eq. (3.1.32) cannot be used as parameters that are associated with a laminate as their value is a function of the through-the-thickness coordinate z. Clearly, this dependency on z can be avoided if the hygrothermally induced laminate curvatures are zero. Conditions to accomplish zero curvatures while maintaining some of the stiffness coupling terms are discussed later in Section 3.3. In the present section, we concentrate on a simpler approach for suppressing hygrothermal curvatures. By imposing mid-plane symmetry conditions for the stacking sequence, the bending-extension stiffness of a laminate can be eliminated; B = 0. In this case, hygrothermally induced mid-plane strains and curvatures are decoupled and can be solved from

AEoN= NN and

D~

= MN.

(3.1.34)

However, in order to achieve zero curvature further assumptions are needed to suppress 1\f". From the definition of hygrothermally induced moments, Eq. (3.1.25), and definitions of the free strains, Eqs. (3.1.12) and (3.1.7), we reach the following conditions: 2

h/2

)

11Tzdz=O and

(3.1.35)

1-

2

h/2

M~

[Q]

-h/2

107


For example, in the thermal case, the equivalent thermal loads of Eq. (3 .1.24) are given by

NT} Nt {Nxy

2

2

I [Q]

hn

=

{ a m2 + a n 2

l

a:n + ~~ 11T dz. 2( a, - a 2)mn

-hn

(3.1.37)

Therefore, the laminate mid-plane strains from Eq. (3.1.28) are 2

EoN)

£~N

= [Ar'

{ yN

I

h/2

+ ~n 2 ) [Q] a:n 2 + a 2m 11T dz. 2(a 1 - a0mn { a m2

-h/2

(3.1.38)

If we further assume that the 11T is constant through the thickness of the laminate, defining the laminate thermal expansion coefficients a through the expression {E 0 N} = {a}I:!T, we can conclude that the laminate coefficients of thermal expansion (and similarly the coefficients of moisture expansion) are

ax} a = [Ar' J[Q] {a:y -hn h/2

2

{

2

~n }

a 1m + a 1n2 + ~m2

2(a 1 -

dz,

a 2 )mn

2

MT} {am +an MJ = J[Q] a:n2 +~~2 {Mxy -hn 2(a ~)mn

M;} {M; = J

3.1

2

{[3 1m2 + f3 2n 2 ) 13 1n + 132m

2(f3 1 -

11C z dz = 0.

(3.1.36)

[3 2)mn

Symmetry assumption ensures that all the laminate properties under the integral are mid-plane symmetric, and would lead to zero integral if the hygrothermal profiles, 11T and/or /1C were also mid-plane symmetric. Therefore, for symmetric laminates with symmetric hygral and/or thermal profiles (of course including the constant through-thethickness profiles), no curvatures would be introduced due to hygrothermal effects. This would lead to a definition of an equivalent coefficient of expansion for the entire laminate.

{

~xl ~y f3xy

2

2

= [Ar'

J[QJ

h/2

-h/2

{[3 1m2 + f3 n2 [3 1n +[32m 2([3 1 -[32 )mn 2

)

dz.

(3.1.39)

As stated in Eq. (3.1.39), the stiffness coefficients Qij are in general functions of both the temperature T and the moisture content C. For practical applications, therefore, these coefficients can be considered simply as the instantaneous values at particular temperature and moisture conditions. It is also interesting to note that while hygrothermal effects produce no shear strains in a unidirectional layer aligned with the laminate coordinate axes, they induce shear expansion coefficients axy or l3xy for general orthotropic laminates, implying that shear strains are also introduced due to hygrothermal effects. Finally, for laminates with uniform through-the-thickness temperature distribution, the temperature term may be taken out of the inte-

108


grals along with the constants Kl' K2 and K3 in thermal loads and moments defined in Eqs. (3.1.29) and (3.1.30). What is left inside the integrals are the cosines and sines of the orientation angles. Using definitions of those integrals made in Chapter 2, Eq. (2.4.35), it can be shown that the thermal loads and moments reduce to

{NT}

=~

~ :1

G (k)

Kl VOA - K2 VIA !lT,

K3

(3.2.1)

=ofk) + G~>"

Through-the-thickness integration of the layer stresses and their molucnts yields net mechanical stress and moment resultants, NM and MM. respectively,

K 1 VoA + K2 VIA} {

109

LAMINATE ANALYSIS FOR COMBINED MECHANICAL AND HYGROTHERMAL LOADS

h/2

h/2

J o- dz = NM

(3.1.40)

and

v2A

J z dz =MM.
(3.2.2)

-h/2

-h/2

Since the stresses in the layers have two parts, we now have {MT}

=~

VIB}

K2 -K2 VIB !lT, {

K3

h/2

(3.1.41)

v2B

h/2

J o-M dz + J o-R dz = NM -h/2

and

-h/2

h/2

h/2

-h/2

-h/2

Jo-M z dz + Jo-R z dz

= MM.

(3.2.3) where the V terms are defined by the summation terms of Eq. (2.4.37). An advantage of this representation is the decoupling of the stacking sequence from the material properties. For laminates made of layers with the same material properties, all the material property information is stored in the constants Kl' K 2, and K 3. The V terms, on the other hand, depend solely on the stacking sequence of the laminate and, therefore, may be used for designing the laminate stacking sequence. With these definitions, the coefficients of thermal expansion of a symmetric laminate under uniform thermal loading take the form

ax} {KI VOA + K2 VIA} ~y = ~ [Ar Kl voA -K2 viA. { K3 1

axy

We have already mentioned in the previous section that through-thethickness integrals of the hygrothermally induced stresses are zero. llowever, for the sake of derivation we substitute the expression for 1he induced residual stresses from Eq. (3 .1.19) and rewrite the above equations in the following form: h/2

J o-M dz +AEoN+

BKN-

NN = NM

and

-h/2 h/2

(3.1.42)

v2A

3.2 LAMINATE ANALYSIS FOR COMBINED MECHANICAL AND HYGROTHERMAL LOADS For laminates under combined external mechanical and hygrothermal loads, in addition to the hygrothermally induced stresses of Eq. (3.1.19), in each layer there are stresses, G~)' resulting from the externally applied mechanical loads. Total stresses in a given layer are the sum of these two types of stresses

J o-M z dz + BEaN+ DKN- MN = ~.

(3.2.4)

-h/2

Integration of the mechanical stresses may also be expressed in a fashion similar to Eq. (3.1.19), this time in terms of the mid-plane mechanical strains and mechanical curvatures only, yielding

AE 0 M+ B~ + AEoN+ BKN- NN = NM

BE

0

M

and

+ D~ + BE N+ DKN- MN =MM. 0

(3.2.5) (3.2.6)

Expressing the total laminate mid-plane deformations as a superposition of the mechanical and nonmechanical parts,

112


:1 :1 LAMINATE ANALYSIS FOA COMBINED MECHANICAL AND HYGROTHERMAL LOADS

Solution

E~

The transformed stiffness coefficients for the 0° and goo layers can be written as follows: l81.8 2.8g7 QijOo) = 2,8g7 10.35

~

0

7.17oj

l l

(

0 0

0

7.170

0

=-130 X 0.02 X 10-6 =-2.6 X 10-6 ,

'Y~y= 0.

Using the definitions of Eqs. (3.1.24) and (3.1.25), the thermal loads induced by the temperature difference are obtained as N~ = -78.432 kN/m,

J

10.35 2.8g7 QW0ol = 2.8g7 181.8

GPa

384.3 11.5g Au= 11.56 384.3

r

0

0

0 0

J MN/m,

28.68

Note that because of the symmetry in the laminate, the bending and coupling stiffness matrices are not required. In order to include the thermal effects, consider the change in temperature dT = 40- 170= -130°C that is applied to the composite system. The free expansion strains in each layer corresponding to this change in temperature are obtained from Eq. (3.1.13) as follows: (a) For the oo layers, £~

=-130 X 0.02 X 10-6 = -2.6 X 10-6,

N~ = -78.432 kN /m,

~

= 0,

and due to symmetry of the laminate and the uniformity of the temperature, the thermal moments are identically zero leading to zero curvatures. The nonmechanicallaminate strains corresponding to the above stress resultants may be obtained from Eq. (3.1.28) as

The thickness of the laminate is 4 X 10-3 m, and the in-plane stiffness coefficients are obtained as

(

A

0

1GPa,

£~N

= £~N = -0.1g8 X 10-3 ,

where the coupling matrix B, and the thermal moment MN were set to zero. The strains that induce the residual stresses in each layer of the laminate are then computed from Eq. (3.1.18) and the residual stresses are computed by multiplying these strains with the transformed reduced stiffness matrix of the associated layer. (a) For the 0° layers, £R =EoN- £F X

{

X

(J~)

X

= -1g5 5 X 10-6' y ER = £ N- £F = 2727 X 10-6 y y ' 0

'

[181.8 2.8g7 = 2.8g7 10.35

(J~

"CR

0

xy (O')

0

~

l X

109

7.170

£~ = -130 X 22.5 X 10-6 = -2g25 X 10-6,

'Y~y= 0. X

(b) For the goo layers, £~ = -130 X 22.5 X 10-6 = -2g25 X 10-6,

113

-1g5.5) ~-27.65) 2727 X 10-6 = 27.65 MPa. 0.0 0.0

l

(b) For the goo layers, £R X

= EoN- EF = 2727 X 10-6' y ER =EoN- EF = -1g5 5 X y y ' X

X

10-6'

....


f~L

= [

~ ]x

10.35 2.897 2.897 181.8 0

0

X

{

X

10

For the given layer stiffness properties, the laminate stiffness and compliance matrices can be computed as follows:

{

37.8 1.5 0 A= 1.5 37.8 0 kN/mm, [ 0 0 2.5

J

27.65} 10--6 = -27.65 MPa. 0.0

Example 3.2.2

I-.

Consider a symmetric cross-ply laminate [0190]s that is immersed in water for a short time so that the moisture variation through the thickness assumes the distribution shown in Fig. 3.8. Given the hygroscopic expansion coefficients for the material as ~ 1 = 0.01 and~ 2 = 0.30 and the stiffness properties Q 11 = 140.9 GPa, Q 12 = 3.0 GPa,

~

p

Q22 = 10.1 GPa,

Q66 = 5.0-GPa,

determine the change in temperature from the temperature at curing, 11T, that would produce a zero residual stress cr~ in the 0° layers. Each layer is 0.125 mm thick, and thermal expansion coefficients a, = -o.3 x 10--6;oc and ~ = 28 x 10-6/°C in the material axis directions may be assumed for this problem. \

A-

1

l

J

0.0265 -0.0011 0 0.0265 0 (kN/mmt'. 0 0 0.4

= -0.0011

The moisture concentration varies linearly through the thickness of the laminate. In this example, the value of the moisture concentration, C, at the mid-point of each layer is the average value for that layer and is used in computing the hygroscopic strains. Therefore, from Fig. 3.8, the 11C for the 0° ftnd the 90° layers is calculated to be 3.75 X 10-3 and 1.25 X 10-3, respectively. The temperature T is assumed to be constant through the laminate depth. The combined thermal- and moisture-related free expansion strains can be computed for each layer as (a) oo layer, Ef

=EfH +EfT= o.o1 x 3.75 x 10-3 -

o.3

x1?~ 11T, 6

E~ = ~H + E~T = 0.3 X 3.75 X 10-3 + 28 X 10-6 !1T.

(b) 90° layer,

T 1

c

5,10-3

Odeg

Ef

= EfH + E? = 0.01 X 1.25 X 10-3 -

0.3 X 10-6 !1T,

•

E~ = EfH + E? = 0.3 X 1.25 X 10-3 + 28 X 10-6 !1T,

0.5 mm- __ -l- _

90 deg

---------------

115

Solution 9

7.170

2727}

-195.5 0.0

:1.2 LAMINATE ANALYSIS FOR COMBINED MECHANICAL AND HYGROTHERMAL LOADS

------~----

90deg Odeg

Figure 3.8. Moisture distribution in [0/90]s laminate.

and the thermal loads of Eq. (3 .1.24) are calculated to be

N~ = 3.12 + 0.081411T, N~ = 3.59 + 0.080911T, N/y= 0 Nlm. Next, the laminate strains corresponding to these nonmechanical loads are obtained from Eq. (3.1.28):

116

HYGROTHERMAL ANALYSIS OF LAMINATED COM

Ili;E~N}

= [Ar

1

,-;olution

lN~}

10.079 + 0.002068 f..T} ~ = 0.092 + 0 00205 f..T

0

X

3

10-

.

The transformed reduced stiffness coefficients for the ±45° layers arc obtained as

In order to compute the value of residual stress, cr~ in the 0° layer, we need to compute the strains induced by the constraining action of the adjacent layers, which are sometimes referred to as residual strains. These strains are obtained by subtracting the stress-free expansion strains from the nonmechanical strains as follows [see Eq. (3.1.18)]: E~

~

w-s = o.oooo415 + o.oooo0237 t-.T,

E~ = (0.092 + 0.00205 f..T) X 10-3

- ( 1.125 + o.028 t-.T) x 1o-3 = -o.oo 1033 - o.oooo2595 t-.T. The residual stress cr~ in the oo layer is computed as

~

(JR x

=

Q(Oo) £R

II

X

For the laminate layup as described above, the in-plane, coupling, unu flexural stiffness matrices may be computed as follows: 113.2 84.6 [A]= 8~6 [

113.0 0

~

JMN/m,

93.2

+ Q(Oo) ER = 0 002748 + 0 000256 f..T 12 y • • '

f..T= -10.732°C.

Example 3.2.3 Consider an antisymmetric laminate [-45/45/-45/45 4 h made from T300/5208 Graphite/Epoxy, with the stiffness properties given in Example 3.2.1. The thickness of each layer group of the laminate is 0.5 mm. Assuming a laminate cure temperature of 180°C, and that the laminate is at a uniform operating temperature of 25°C, determine the laminate curvatures. Assume the following values of thermal expansion coefficients in the material axis system:

a 1 = 0.02 X 10-6 1°C and a 2 = 22.5 X 10-6 /°C

--:7)

56.6 42.32 --42.87l Q;:so) = 42.32 56.6 --42.87 GPa. [ --42.87 -42.87 46.59

0 0

[B] =

[

0 0

21.4] 21.4 kN,

21.4 21.4

0

37.8 28.2

00

[D] = 28.2 37.8 [

and setting cr~ = 0 yields a value of the temperature change as

~

_J

56.6 42.32 42.87l Q~So) = 42.32 56.6 42.87 GPa, [ 42.87 42.87 46.59

= (0.079 + 0.002068 f..T) X 1o--3 - (3.75- o.o3 t-.T) x

117

1111 LAMINATE ANALYSIS FOR COMIINED MECHANICAL AND HYGROTHERMAL LOADS

0

l

Nm.

0 31.1

The laminate is at a uniform temperature of 25°C, which implies that each layer is subjected to a temperature change t-.T = -155°C. As in Examples 3.2.1 and 3.2.2, the free expansion strains in each layer in the fiber direction and transverse to the fibers are obtained as Ef

= -3.1

X

10-6 ,

E~ = -3.487 X 10-3 ,

which can be transformed using Eq. (3.1.12) to the laminate reference x- y axis system as follows: E~ =(cos 45) 2 Ef +(sin 45). 2 Ef = -1.745

X

3

10-

,

3

E~ =(sin 45) 2 Ef +(cos 45) 2 Ef = -1.745 X 10- ,

118

HYGROTHERMAL ANALYSIS 01' LAMINATED COMPOSITES

'(~~

=2(sin 45 cos 45) Et'- 2(sin 45 cos 45) E~· =3.484 X I o-3.

The thermal loads and moment loads are calculated from Eqs. (3.1.24) and (3.1.25) as N~ = -46.8 kN/m, M~=O,

N~ =-46.8 kN /m

M~=O

and

W:Y =0,

and M~=6.36N.

The laminate strains and curvatures for these thermal loads can be solved from

{

B]-I {NN}

(m/m)} =[A KN (m- 1) B D

EoN

MN

-3.04 X Io-4 -3.04 x w-4 = 0 0 0 0.624

Note that for such an antisymmetric laminate, only a twisting curvature is induced when the laminate cools from its curing temperature to an operating temperature of 25°C.

3.3 HYGROTHERMAL DESIGN CONSIDERATIONS Residual stresses and deformations of laminates induced by changes in hygrothermal conditions constitute a major design consideration. Bending-extension and twisting-extension stiffness coupling terms in unsymmetric laminates, for example, may cause warping of the laminate during the cure and cool cycles. Such couplings, however, may be essential from a design standpoint, and there have been several recent studies w.here these couplings have been exploited in the design of aerospace structural components. A review of the literature provides a somewhat misleading conclusion that all general unsymmetric laminates have limited practical use because of their susceptibility to warping. A study by Winckler (1986) was the first to explore a class of thermal warp-stable composite laminates. In that study, families of unsymmetric composite laminates that provide desired elastic coupling characteristics and maintain shape stability during changes in hygrothermal variations are presented.

119

:1 :1 HYGROTHERMAL DESIGN CONSIDERATIONS

Sections 3.1 and 3.2 presented computations of the stresses and Nlrains in the composite material due to moisture and temperature vari111 ions. In order to understand how hygrothermal effects influence deNign decisions, it is worthwhile to examine specific examples of l'ouplings in composite laminates and their importance in structural dl!sign. In the following discussions, we will assume that all layers url! of the same material. In the absence of such a restriction, the rl!uder should recognize that there would be hygrothermally induced Nlresses and strains in unidirectional laminates as well. Let us reexamine the general laminate equation relating stress and moment resultants to strains and curvatures: Nx

Ex

NY Nxy Mx

Ey

=

[! ~]

MY

Exy

(3.3.1)

I

Kx Ky Kxy

Mxy

If the components of the B matrix are all zero (as would be true of a symmetric stacking sequence), there is no coupling between the inplane loads and out-of-plane deformations. Likewise, the moment resultants do not induce in-plane deformations. If some components of the B matrix are nonzero, as for unsymmetric laminates, then one can expect couplings between in-plane loads and out-of-plane deformations, or conversely, between out-of-plane loads and in-plane deformations. As a specific example of the latter, consider an anti symmetric laminate [I - Jr. The stiffness matrix for such a laminate has the following nonzero components: A 11 A 12 0 0 0 0 Az, A22 0 A66 B6, 0 0 Bl6 Du 0 0 0 B26 D21 B6, B62 0 0

0 0 B62

B16 B26

oI

D22

0 0

0

D66

D12

(3.3.2)

120

HVGROTHERMAL ANALYSIS OF LAMINATED COMPOSITES

The nonzero B16 and B26 terms represent extension-twisting coupling in the laminate. Such a laminate would develop a twisting curvature, ~Y' under in-plane stress resultants Nx or NY resulting from mechanical loading. In Section 3.1.3, we have seen how variations in temperature or moisture conditions induce residual stresses in the laminate. Although the net resultant of these residual stresses is zero, to achieve such a state the laminate undergoes deformations that can be quite complex These deformations are computed by using equations that are simila1 to those used in computing the deformations under mechanical loadings [see Eq. (3.1.28)], where the stress and moment resultants due to mechanical loads are replaced by quantities that reflect hygrother- · mal or nonmechanical loads. It is clear from this equation that the existence of nonzero Bu terms would indicate deformations involving curvatures under hygrothermal loads. Hence, an easy way of ensuring hygrothermally curvature-stable laminates is to select a stacking sequence for which Bu = 0. Provided that the hygrothermal profiles are uniform through the thickness, this can be accomplished by limiting the stacking sequence to be mid-plane symmetric. However, the coupling terms Bu offer a degree of freedom in the design process and have significant merit in the design of aerospace structures. Tension-twisting coupling, for example, can be used in the design of helicopter blades for aeromechanical efficiency. The blade must be appropriately twisted along the span in order to optimize aerodynamic performance. The optimal twist distribution changes with the rotational speed of the blade. The large axial centrifugal load in the blade provides a mechanism to induce twist in the blade to a desired spanwise twist distribution if the extension-twist coupling is retained in the composite material, that is, nonzero B16. A challenge to the design engineer, therefore, is to develop a laminate stacking sequence that preserves mechanical coupling without inducing curvature instability (a terminology used to imply stability of the deformations without twisting under hygrothermal loads). Winckler (1986) presents a class of laminates referred to as HTCC (hygrothermal curvature-stable coupling) laminates that possess the desired properties. It is a useful exercise to understand the physical mechanism that gives rise to couplings in laminated composites. Consider the antisymmetric laminate [J-
121

:13 HVGROTHERMAL DESIGN CONSIDERATIONS

l'oupling. If each half of the laminate is considered separately, that is, u 1+<1>1n and a [-]n laminate, they are symmetric with respect to their own mid-plane and have no extension-twisting coupling of their own. llowever, for all cases other than = oo or 90°, there is an in-plane Nhcar coupling A16 in each half, where the shearing deformation is l'qual in magnitude but opposite in sign. This is illustrated in Fig. \.lJa. When the two halves are bonded together as a laminate, the i\ lh term of the total laminate would be zero. However, the A16 terms of each half contribute to the development of the extension-twist coupling of the combination (Fig. 3.9b); that is, a nonzero B16 is obtained for the combined laminate. Note further that the coupling terms /1 11 , B 12 , B22 , and B66 are all zero for this antisymmetrical laminate. In addition to tension-twist coupling, this laminate would also develop a twist with changes in hygrothermal conditions. It is possible, however, to introduce mechanical coupling terms in u laminate without sacrificing hygrothermal curvature stability for temperature or moisture changes. To see how this would work, the reader is referred to Eq. (3.1.27). If the only nonzero coupling term is B16, then for the twisting curvature terms K~Y to be zero, one approach would be to develop a laminate for which the components N~ and M~ are exactly zero. This is possible if the in-plane hygrothermal shear strain in the laminate is zero [see Eqs. (3.1.23) and (3.1.24)].

p

p

(a)

p

(b)

Figure 3.9. (a) Stress-free and (b) constrained deformation of a [<1>/-]T antisymmetric laminate.

122

HVGROTHERMAL ANALYSIS 01' LAMINATED COMPOSITES

This observation leads us to define in-plane hygrothermal isotropy and discuss how it differs from in-plane mechanical isotropy. Consider, for example, the in-plane stiffness properties of [0/90]s laminate. As described in Chapter 2, this laminate has stiffnesses in both x and y directions, Ex= EY. However, rotating entire stacking sequence with respect to the coordinate axes by <1> suits in a change in its in-plane stiffness properties. For an arbitr<""'' value of the angle , the laminate [/90 + <1> ls will have nonzero plane shear-extension coupling terms, A 16 and A 26 • The same laminau:: •.l however, exhibits hygrothermal isotropy. That is, temperature moisture changes will result in uniform expansion and contraction all directions and the laminate will not experience any shear This is because, unlike mechanical loads, which are applied specific directions, moisture and temperature changes are direction independent. It can be shown that any balanced symmetric laminate for which the coefficients of expansion along two mutually perpendicular directions are equal will display in-plane hygrothermal isot... ropy. A hygrothermally curvature-stable laminate that retains tensiontwist coupling can be assembled from laminates that have · hygrothermal isotropy but are not mechanically isotropic. One laminate stacking sequence can be written as follows: [/( + 90)zll- /- ( + 90)zl- ].

This laminate is clearly antisymmetric; however, each half of the laminate is symmetric and is hygrothermally curvature stable. In-plane hygrothermal isotropy of each half ensures that there is no induced thermal bending, and the absence of hygrothermal shear strains prevents laminate twisting. However, each half does have nonzero extension-shear strain coupling A 16 when the angle <1> is not oo or 90°. This coupling term is of equal magnitude for the two halves, but of opposite sign. When the two halves of the laminate are put together, the inplane shear coupling for the resultant laminate is zero; however, there . is a finite B 16 which gives rise to the desired tension-twist coupling. Other variations of such hygrothermally curvature-stable laminates are possible and are discussed in Winckler ( 1986) and Hajela and Erisman (1993).

As mentioned earlier, there may be cases where we would benefit from having mechanical coupling, yet we would prefer hygrothermal

123

:1.3 HVGROTHERMAL DESIGN CONSIDERATIONS

narvature stability to prevent warping under changing environmental ronditions. Consider a case where the coupling between in-plane load N, and twisting curvature Kxy is to be preserved. In the context of l'lussical laminate plate theory, this would require that the coupling wefficient B 16 be nonzero. To derive the condition that causes the curvatures under hygrothermul loads to vanish, we can start from Eq. (3.1.28). Representing the inverse of the combined stiffness matrices by a partitioned matrix with three 3 x 3 submatrices a, b, and d,

b] [A BJ

a [b d -

B D

1

(3.3.3) '

the second row of matrix equations forK of Eq. (3.1.28) can be set c..·qual to zero. Using the partitioned form of the inverted stiffness matrices leads to the mathematical condition, K= bNN + cJMN =0.

(3.3.4)

The HTCC laminates discussed here are a candidate solution to this problem. However, while this approach yields the desired hygrothermal curvature stability, it does not provide for a direct control on the strength of the desired extension-twist coupling term. This design problem can, therefore, be formulated as an optimizal ion problem by requiring as an objective a certain strength of coupling us indicated by the magnitude of, say B 16 , with an equality constraint on the curvatures as follows:

maximize B 16 subject to

K

= b NN + dM N = 0.

Note that constraints on strength and stiffness can be included in this problem in a routine manner. A number of optimal solutions for such hygrothermally stable laminates obtained through the use of genetic algorithms are presented in Chapter 5. It is important to recognize that the selective control of a coupling coefficient such as B 16 can result in undesirable changes in other components of the B matrix, particularly if the condition of antisymmetry in the laminate is not explicitly imposed. Any adverse effects on ad-

124


ditional terms in the B matrix would have to be controlled by adding design constraints in the optimization problem.

125

REFERENCES

4.

A unidirectional composite layer undergoes the following steady state expansions when immersed in a cooled liquid medium: £1 =

EXERCISES 1.

The thermal (a) and moisture (~) related expansion coefficients of a high-strength Carbon/Epoxy unidirectional layer in the material axes system are given as follows: a 1 =- 0.3 x 10--
E 2 = 10.3 GPa,

G 12 = 2.897 GPa,

v12 = 0.3.

2.

Consider a Graphite/Epoxy laminate with a stacking sequence of [0/90/45 2L. All layers in this laminate are of equal thickness t0 = 0.125 mm. The material axes coefficients of thermal- and moisture-related expansion and the stiffness coefficients are given as a 1 = -0.02 X 10-6 (m/m)JOC, a 2 = 22.5 X 10--
3.

Consider an unsymmetric composite laminate [Oz1902 ] fabricated from a high-strength Carbon/Epoxy unidirectional layer of thickness 0.15 mm and with properties given in problem 1. Determine the total strains in each layer of the laminate if the operating temperature is 20°C and the laminate is subjected to the following stress resultants: Nx = 15 KN/m,

NY= 10 KN/m.

Assume that the stress-free curing temperature is l25°C.

0.012 X 10-3,

£2 =

0.6 X 10-3,

"{12 =

0.0

The material axes stiffness coefficients of this layer are known to be as follows: E 1 = 140.0 GPa,

E 2 = 10.3 GPa,

G12 = 2.897 GPa,

v12 = 0.3.

If this layer is used to fabricate an angle-ply laminate [82 /-8 4 /8 2], determine the angle 8 that minimize the steady-state laminate expansion strain £x if it were submerged in an identical cooled fluid.

REFERENCES Crank, J. (1956). Mathematics of Diffusion. Oxford University Press, London. Hajela, P. and Erisman, A. (1993). "Genetic Search Methods in the Design of Thermally Balanced Composite Laminates." Proceedings of OPT/'93, Zaragosa, Spain, July 7-9. .Iones, R. M. (1998). Mechanics of Composite Materials, 2nd Edition. Taylor & Francis, Philadelphia, PA. Tsai, S. W. and Hahn, H. T. (1980). Introduction to Composite Materials. Technomic Publishing Co., Lancaster, PA. Winckler, S. J. (1986). A theory for warp-free unsymmetric composite plates and beams with extension-bending coupling. Ph.D. dissertation, Rensselaer Polytechnic Institute, Troy, New York.

4 LAMINATE IN-PLANE STIFFNESS DESIGN

In Chapter 2 we discussed the use of classical lamination theory (CLT) for calculating the stiffness properties of composite laminates. In this chapter we will use the expressions obtained in that chapter for designing laminates with desired in-plane stiffness properties. In many applications the designer is concerned with the strength as well as the stiffness of laminates. Unlike stiffness, the strength of composite laminates is not easy to predict. This is the result of the existence of many complex modes of failure that will be discussed in Chapter 6. There are also many applications where the design of the laminate is primarily governed by in-plane stiffness requirements. Furthermore, because of the incomplete understanding of the various failure modes of composite laminates, limiting laminate strains as surrogate strength constraint is a common design practice. That is, strength requirements are often transformed into stiffness requirements. Another argument for studying in-plane stiffness design problems alone is that they are simpler than most other laminate design problems. Not only is it simpler to perform the analysis, but in-plane stiffness does not depend on the arrangement or stacking sequence of the layers with various orientations in the laminate. This is because the 127

128

LAMINATE IN·PLANE STIFFNESS DESIGN

in-plane stiffness matrix A simply sums the contributions of the inplane stiffness properties of layers with various orientations, without regard to their z location. As such, in-plane stiffness design is particularly suitable for introducing the basic concepts and simplest methods of design optimization. One prerequisite for separating the in-plane response from the bending response is a zero bending-extension coupling matrix B, which can be achieved by symmetry of the stacking sequence with respect to the laminate mid-plane. To further simplify the formulation, we consider in this chapter only the design of balanced laminates (that is, laminates which have a -0 layer for each +0 layer) where A 16 = A 26 = 0. The material in this chapter is organized into three sections. The first section seeks to define and formulate the stiffness design problems that we want to solve. In this section we also review optimization formulation and terminology. The second section describes graphical solution techniques for optimization of orientation angles (sometimes referred to as ply orientations) and stack thickness treated as continuous variables. The third section shows how the graphical solution technique can be used also for the case where the thicknesses and the angles are limited to discrete values.

4.1

DESIGN OPTIMIZATION PROBLEM FORMULATION

Finding an optimal design is often handled as a three-step process. First, we formulate the design in terms specific to the problem at hand. Second, we recast the problem as a mathematical optimization problem. Finally, we solve the problem using one or more available optimization methods.

4,1

129


ure also used as stiffness measures. These elastic constants are defined in a manner analogous to the definition of the elastic moduli H1, £ 2, G12 and Poisson's ratios v12 and l/21 • That is, a specified value of the stress resultant Nx, which is equivalent to an average stress d, = N/h, is assumed to produce the following axial and transverse strains: Ex= crJE~ff and Ey = -1/~~Ex' respectively. Similarly, under load- · i ng that induces transverse stress resultant NY, the induced strains are characterized by E~tt and v~~t, and the shearing stress resultant Ntr = haxy leads to a shear strain y:~ = ax/Gxy· Altogether, for an orthotropic laminate (zero or negligible A 16 and A 26 ), this leads to the following form of the constitutive equations: __1_ (- _ z,rff- ) __1_ (N _ ettN ) Ex- Eeff ax xy ay - hEeff Vxy 'y ' X

X

X

__1_ (- 1/eff- ) __1_ (N effN) Ey- Eeff ay- yx ax - hPff y- 1/ yx X ' y

y

(4.1.1)

_ (jxy _ Nxy Yxy- G - hG . xy xy

As discussed in Chapter 2 (Section 2.4.4), comparison of these equations to the constitutive relations expressed in terms of the A matrix, such as Eq. (2.2.23), yields the effective elastic constants shown in Eq. (2.4.42) in terms of the elements of the A matrix. For the sake of convenience, the superscript "eff' will be dropped in the remainder of this chapter, and the Ex' EY' Gxy' andvxy shown below will be used to represent the elastic properties of a composite laminate:

_ .!_ (A11A22- Ar2J

Ex- h

A

and

_.!_ (AuA22Aj2J ' A

Ey- h

22

4.1.1

11

Design Formulation of In-Plane Stiffness Problem

The simplest measures of laminate in-plane stiffness to obtain are the coefficients of the A matrix. These are frequently used as design requirements in the composite materials literature. For example, the PASCO program (Stroud and Anderson, 1981), which was developed for designing composite laminates for buckling constraints, permits limits on the Aij's. Sometimes, however, effective engineering elastic constants, E~ff, E~ff, G xy and ~~f, defined for the composite laminate,

1 Gxy= h A66

and

A12

1/ -~. xy- A22

(4.1.2)

For laminate design problems, the stress resultants are typically specified based on the stress analysis of the laminate under applied loads. We seek a laminate that will not exceed given strain limits under these resultants and that will have some desirable stiffness properties. The thickness of the laminate will either be given, or we will

LAMINATE IN·FILANE STIFFNESS DESIGN .

•~u

seek the laminate of minimum thickness that satisfies the strain limits and has acceptable stiffness properties. That is, the design problem may be formulated as

4.1

131

DESIGN OPTIMIZATION FIAOILEM FORMULATION

(i) For the first load case we expect that Ex will be larger than Ev' so that from the first equation of Eq. (4.1.1) the strain limit may be expressed as

find a laminate to

Ex=

1

~~00::; 0.004 or hEx~ 2,500,000 lb/in.

(a)

X

minimize

h

such that

E~::; Ex::; E~, El::; E y

For the second load case we have only Yxy' and from the last of Eq. (4.1.1) we get

< EuY'

Yxy =

y-

iYx)::; Y~Y' E~::; Ex

or

xy

hGxy ~ 500,000 lb/in.

(b)

For this all-zero laminate, by definition, Ex= E 1 and Gxy= G 12. So that from Eq. (a) we get

::;e,,

18.5 x 106 h ~ 2,500,000

1 Ey
or

h ~ 0.135 in.

Similarly, from Eq. (b) we obtain

t
vlxy
,;~00 ::; 0.006

3

0.93 x 106 h ~ 500,000

or h ~ 0.538 in.

(4.1.3)

where h is the laminate thickness, and the superscripts u and l denote upper and lower limits, respectively, of the associated quantities. Of course, it is not necessary to have upper and lower limits for every strain component and effective engineering constant.

Since the thickness h has to be an integer multiple of the layer thickness of 0.005, we will need to have a total of I 08 layers [0] 108 yielding h = 0.540 in. (ii) For the [±45]ns laminate, the calculation is more involved. We use Eq. (2.4.8) to obtain the Qij's, Q 11

Example 4.1.1

A Graphite/Epoxy laminate (E 1 = 18.5 x 106 psi, E 2 = 1.89 x 106 psi, G 12 = 0.93 X 106 psi, v 12 = 0.3, and t = 0.005 in) is to be designed for two load conditions: (a) Nx = 10,000 lb/in; and (b) Nxy = 3,000 lb/in. The allowable laminate strains are 0.4% normal strain and 0.6% shear strain. Consider the following three laminates as design candidates: (i) [O]n, (ii) [±45]ns' and (iii) [±45190/0]ns· For each laminate, determine the smallest value of n that will satisfy the specified strain limits. Note that n is the number of layers for case (i), but is the number of sublaminates in half of the laminates for cases (ii) and (iii). Then identify the lightest laminate.

= 18.67 x

106 psi,

Q 12 = 0.5723 x 106 psi,

Q22 = 1.908 x 106 psi, Q66 = 0.93 x 106 psi.

We next use Eqs. (2.4._!2) and G.4.28) _!_o calculate the Aij's. Since the expressions for Q 1 p Q 12, Q22 andQ 66 in Eq. (2.4.19) involve only squares of cosines and sines, there is no difference between the contribution of the 45° and -45° layers. Therefore, Au= hQij' where Q;j are the constants associated with the 45° layers: -

A 22 =Au= h Qu(45°) =

= h [0.25 Q11 + 0.5 (Q 12 + 2 Q66) + 0.25 Q22]

6.361 x 106 h lb/in,

132

LAMINATE IN-PLANE STIFFNESS DESIGN

4,1


A 12 = h (2 12 (45°) = h [0.25 (Q 11 + Q22 - 4 Q 66 ) + 0.5 Q 12 ]

E)z

=4.501 x 106 h lb/in, A 66 = h Q66 (45°)

= h [0.25 (Q 11 + Q22 -

2 Q 12 - 2 Q66) + 0.5 Q66]

Then from Eqs. (a) and (b) we get 7.552 x 106 h ~ 2,500,000 lb/in, 2.894 x 106 h ~ 500,000 lb/in.

Using Eq. (4.1.2) we get

= (A 11A22 -Ai 2 )/A22 = 3.176 x

106 h lb/in,

Gxyh =A 66 = 4.858 x 106 h lb/in. Then from Eqs. (a) and (b) we obtain 3.176 x 106 h ~ 2,500,000 lb/in, 4.858 x 106 h ~ 500,000 lb/in.

The first inequality is more stringent and yields h ~ 0.787 in. Since the laminate has repeating stacks with two layers each and is symmetric, h must be an integer multiple of 4t = 0.02 in, so we need h = 0.80 in or [45!-45] 4os· (iii) The last laminate is the quasi-isotropic one. From Eqs. (2.4.19) and (2.4.28) we get A11

= 0.25 h [Q 11 (0°) + Q 11 (90°) + Q 11 (45°) + Q 11 (-45°)] = 0.25 h [Q 11 + Q22 + 2 Q 11 (45°)] = 8.325 x 106 h lb/in = A22 ,

A 12 = 0.25 h [Q 12 (0°) + "Qn(90°) + {?12 (45°) + (2 12 (-45°)]

= 0.5 h [Q12 + "(212(45°)) = 2.537 X A66

106 h lb/in,

=0.25 h [(!66(0o) + "Q66(90o) + {2"66(45o) + Q66(-45o)] = 0.5 h [Q66 + Q66(45°)] = 2.894 x

Using Eq. (4.1.2) we obtain

= (A 11A22 -Ai 2)/A 22 =7.552 x 106 h lb/in,

Gxyh = A66 = 2.894 X 106 h lb/in.

= 4.858 x 106 h lb/in.

E)z

133

106 h lb/in.

The first inequality is more stringent and yields h ~ 0.331. Since the laminate is made by repeating a stack with four layers and is symmetric, h has to be a multiple of 8t = 0.04 in. So we get h = 0.36 or [45/-45!9010] 9s, which is the lightest laminate. Note that the all-zero laminate was very flexible in shear, so that the Nxy load forced the large thickness. The situation was reversed for the (45/-45) laminate which was too flexible in the axial direction. The quasi-isotropic laminate represents a good compromise for the two load cases.

4.1.2

Mathematical Optimization Formulation

Example 4.1.1 represents a typical optimization problem for the design of a laminate subject to limits on its in-plane stiffness. For this example we limited ourselves to three possible stacking sequences, and the solution could be found easily. Without the limitation to a small number of stacking sequences, the same problem is quite formidable, and we will return to it later in this chapter. Fortunately, mathematicians and numerical analysts have developed a host of methods that are available for solving optimization problems. Due to space limitations, we can cover only a small fraction of these methods in this text. However, general terminology and notation most commonly used in mathematical optimization papers and textbooks was introduced earlier in Section 1.4.1. Once a composite design problem has been formulated in this general format, the designer can make use of optimization software packages embodying various methods. The objective of the present section is to illustrate how the in-plane stiffness design problem can be formulated in the standard format of Section 1.4.1.

lit4

LAMINATE IN·PI.ANE STIFFNESS DESIGN

An optimization problem has an objective function that measures the goodness or efficiency of the design. In Example 4.1.1 our goal was to design the lightest laminate, and the total thickness h was our measure of lightness, since the density was constant for all candidate laminates. Thus h was the objective function. An optimization problem usually also has constraints that limit the choice of design. For example, in Example 4.1.1 we had two constraints that represented limits on the strains under two load conditions. Finally, an optimization problem has design variables, which are the parameters that can be changed in the design process. In the example, one of the design variables was n, which is the number of stacks of the basic repeating sublaminate. This is an integer design variable. The example had another design variable, which was the choice of laminate. One way to characterize this choice is by an integer variable taking the values 1, 2, or 3, corresponding to the choice of all-zero, [45/-45], or quasi-isotropic laminate, respectively. Optimization methods for problems with continuous design variables are usually much more efficient than methods that have to handle integer variables. For this reason, it is common for designers to replace integer variables with continuous ones, whenever this does not introduce large errors. In the example, the number of stacks n is an integer, but not a small one. For example, for the all-zero laminate, the solution was n = I 08. If we replace this design variable with a continuous variable, we can solve the problem and then round up the optimum value of the variable to the nearest integer value. In the example we have done essentially that; we solved for the continuous variable h, and then rounded it up to get an integer value of n. The other design variable cannot be treated this way. It is a choice design variable, and noninteger values do not have much meaning and cannot be rounded up or down in an obvious way. We have skirted the difficulty by solving the problem separately for each one of the choices. Instead of solving one optimization problem with two variables, we solved three optimization problems with one variable (n). This approach is called enumeration. When the number of choices is small, enumeration is a reasonable procedure. However, for more complex problems, enumeration is not efficient. We would like to use the choice design variable and formulate the problem in the standard format of Section 1.4.1:

4.1

135

DESIGN OPTIMIZATION PAOILI!M FORMULATION

minimize

j(x),

such that

h;(x) = 0,

i = 1, ... , ne,

#X) $0,

j

x

E

X,

= 1, ... , ng.

(4.1.4)

As noted in Section 1.4.1, we use f to denote the objective function, which is the thickness h, in the example. We use g1 to denote the inequality constraints, which in the example represent the limits on I he strains. There are no obvious equality constraints hi in the example hut, as we will soon see, the definition of the design variables can require the introduction of such equality constraints. In order to formulate the problem of Example 4.1.1 in the standard formulation we have to express the thickness and the strain limits in lcrms of the choice design variables. For example, the thickness can he expressed as f(x)

=h =

nt, 4nt, 8nt,

l

if i = 1' if i = 2, if i = 3.

(4.1.5)

This is a cumbersome definition of J, but one which is quite acceptable for some optimization software dealing with integer design variables. Such programs may accept user-defined subroutines for calculating lhe objective functions and constraints. However, many programs available for solving integer optimization problems do not accept such subroutines, and the user needs to formulate the optimization problem with the objective function and constraints defined in terms of the basic four algebraic operations of addition, subtraction, multiplication, and division. Fortunately, there is a standard way to convert a choice problem into an equivalent problem involving binary design variables, which in most cases eliminates the need for conditional statements. Binary variables are integer variables which take only the values 0 or 1. If we have a choice with r options, we can define r binary design variables Xp ..• , xr, with one design variable corresponding to one option. That is, X;= 1 if the ith option is exercised, and X;= 0 if it is not. Of course, we cannot exercise more than one option, so that we need a constraint that will prevent more than one of these design variables to be nonzero. This is accomplished easily by the equality constraint,

138


i=i

Once we have these binary design variables defined, we can easily write any property of the laminate in terms of the corresponding prop· erties of the possible options. For example, consider a situation four possible types of laminates, with Poisson's ratios v~Y' v;~ . v~Y' v;Y. Then the value of Poisson's ratio of the selected laminate is obtained by specifying the design variables x" x 2, x , x in 3 4 I

2

137

A similar procedure can be followed for the strain constraints expressed in terms of the limits on the values of Ex and Gxy obtained in Example 4.1.1,

~ x.= 1• £... I

_

4 1 DESIGN OPTIMIZATION PAOILEM FORMULATION

3

hEx~ 2,500,000 lb/in,

h Gxy ~ 500,000 lb/in.

In that example we found the values of Ex associated with the three laminates to be

Ex!

4

= 18.5 X 106 psi,

Ex2 = 3.176 X 106 psi,

vxy - x, v xy + x2 v xy + x3 v xy + x4 v xy

Ex3 = 7.552 X 106 psi.

The following example demonstrates how we use binary choice vari· abies to formulate Example 4.1.1 in standard format.

Since Ex is independent of the thickness, we can write it for any of the three laminates as

Example 4.1.2

Ex= (18.5 x 1 + 3.176 x 2 + 7.552 x 3 )

Formulate the design problem of Example 4.1.1 as a standard optimization problem. The natural choice of design variables for this problem is the number of stacks, n, and a choice variable, i, that ranges from 1 to 3 and indicates the choice of laminate. That is, i = 1 corresponds to all zero laminate, i = 2 to a (±45°)n laminate, and i = 3 to a quasi-isotropic laminate. We replace this single variable with three binary variables x" x 2, x3, such that X;= 1 if we choose the ith laminate, and X; = 0 if we do not. Of course, to make sure that we choose exactly one laminate we need to add the constraint x 1 +x2 +x3 = 1.

In terms of these binary design variables, the objective function may be written as

=h = n t(x1+ 2x2 + 4x3),

where the coefficients, 1, 2, and 4, of the binary variables represent the number of layers in each of the repeating sublaminates [O]n, [±45]ns' and [±45190/0]ns' respectively.

106 psi.

Now the limit on Ex may be written as h (18.5 x 1 + 3.176 x 2 + 7.552 x 3 ) x 106 ~ 2.5 x 106 •

Substituting the expression for the thickness h and rearranging, we can rewrite the constraint as 2.5- n t (x 1 + 0.5 x 2 + 0.25 x 3 )(18.5 x 1 + 3.176 x 2 + 7.552 x 3 ) ~ 0.

Following a similar procedure, we can write the constraint on Gxy as 0.5 - n t (x 1 + 0.5 x 2 + 0.25 x 3)(0.93 x1 + 4.858 x 2 + 2.894 x 3) ~ 0.

Finally, defining a general integer variable x4 =n we can formulate the optimization problem in the form of Eq. (1.4.1) as minimize

f(x)

X

such that

t xix 1+ 2 x 2 + 4 x 3)

x 1 + x 2 + x 3 = 1, 2.5 - t xix 1+ 0.5 x 2 + 0.25 x 3) X

(18.5 x 1 + 3.176 x 2 + 7.552 x 3 )

~

0,

138

LAMINATE IN·PLANE STIFFNESS

,·onstants as a function of 9 and picking the optimum design by inNpcdion of their graphs. This is illustrated in the following example.

0.5- t x/x 1 + 0.5 x 2 + 0.25 x 3) X

(0.93 X 1 + 4.858 X 2 + 2.894 x 3)

~

0,

(xl' x2 , x 3 ) binary, x 4 positive integer.

The formulation of laminate design problems in the standard is important for applying sophisticated optimization algorithms or izing standard optimization software. However, for some problems design problem is simple enough so that it can be solved by inspectwn as was Example 4.1.1 or by simple graphical procedures. The section is devoted to such procedures.

4.2

139

4 :1 GRAPHICAL SOLUTION PROC!OUPIES

GRAPHICAL SOLUTION PROCEDURES

The in-plane properties of a composite laminate are determined the orientation angle of the layers and the number of layers used a given orientation angle. Orientation angles are continuous varia ranging from -90° to 90°, while the number of layers is an · variable. Since graphical procedures are easier to apply to variables, we start this section with a discussion of the opuuuLauu• of orientation angles.

lixample 4.2.1 Consider a laminate made of Graphite/Epoxy with E 1 = 128 GPa, H2 = 13 GPa, G12 = 6.4 GPa, and v12 = 0.3. Design an angle-ply laminate [9/-9]s with maximum Ex, subject to the requirement that Gxy is at least 25 GPa and that vxy is not larger than 1. Using Eq. (2.4.28) for the AJs and Eq. (4.1.2) for the effective clastic constants, we obtain the moduli and Poisson's ratio as a function of 9 and plot them in Fig. 4.1. From the figure we see that the requirement on Gxy is satisfied in the interval 28° ~ 9 ~ 62° and that the limit on vxy is satisfied for e ~ 21° or e :?: 36°. The range for the design variable 9 that satisfies both constraints is called the feasible domain, and it is 36° ~ 9 ~ 62°. In this range the maximum value for the objective function Ex is 36.0 GPa at 9 = 36°. Note that at the optimum one of the constraints is active. Another way to say that, using optimization terminology, is that the optimum is on the boundary of the feasible domain.

~

:>

4.2.1

Optimization of Orientations of Layers

We will consider first laminates with a fixed total thickness. The objective function in this case could be one of the effective elastic properties, while constraints are imposed on other elastic properties. simplest layer orientation design problem involves design of a laminate with a single orientation angle, such as when we want to design an angle-ply laminate [9/-9]ns· The laminate can also be more complex, provided that additional layers have fixed orientations. For example, we may want to design a laminate with some 0° and 90° layers plus some angle-ply stacks with undetermined angle, that is [(±9/0/90)nL· Since the thickness is fixed, n is not a design variable, and the optimization can be solved by plotting the pertinent elastic

1:.:)~

..

~

'0

.~ ft1

E

0 z

10

20

30

40

50

60

70

80

(+9/-9) 8

Figure 4.1.

Elastic properties for Graphite/Epoxy.

90

'


liiiU

A more general graphical procedure was introduced by Miki ( 1982, 1983) for the design of laminates with prescribed in-plane stiffness properties. The procedure is suitable for balanced angle-ply laminates made up of multiple stacks of layers with different orientation angles. The stacking sequences of laminates amenable for treatment by Miki's procedure may be written as [(±ei)NI(±8I_1)N I··· /(±8 1)N], where the total number of layers in the laminat~ is N = •f'Lf= 1 N;, and I is the number of different ±8 groups. In addition to the balanced angle-ply sublaminates, stacks of unidirectional layers with principal material axes aligned with the axes of the laminate (i.e., oo or 90°) can be included in the stacking sequence. The key to Miki's procedure is that for a balanced composite laminate the engineering constants of the laminate can be described in terms of two lamination parameters that capture all the relevant in-··· formation associated with the stacking sequence. These two paramewhich are obtained by normalizing the in-plane ters are V~ and components of V 1A and V3A in Eq. (2.4.35) by the total laminate thickness. Indeed, the elements of the A matrix divided by the laminate thickness can be determined from the following equations, which can be obtained from Table 2.1:

The other expressions in Eq. (4.1.2) can be modified in a similar manncr to obtain the other effective engineering constants. The major effort of Miki's design procedure is the construction of a lamination parameter diagram which describes the region of allowable combinations of lamination parameters. For a laminate of total thickness h, in which the volume fraction of the layers with ±ei orientation angles is v;, the lamination parameters are given as I

I

V •1

= h = L.J vk cos 28k VIA

""'

A;2 A~2 A~6

r)

v· U -v; = 1 0 v;• u2 , u4 -v! u3 Us 0 -v3 VI

I

k=l

where A~=Aufh

and where the U; are the orientation-invariant material properties, defined in Eq. (2.4.20). The effective engineering constants can then be obtained from the A~'s using Eq. (4.1.2). For example,

= A*II A*22- (A*12)2

E X

A;2

(4.2.4)

where I is the number of different ±8 groups, and

LV;= 1.

(4.2.5)

i=l

From Eqs. (4.2.4) and (4.2.5), the values of the lamination parameters are always bounded, -1 ~ (V;, v;) ~ 1. For a laminate with only one fiber orientation angle, [±8nL (/ = 1), the lamination parameters are v~

(4.2.1)

and

v3* = hv3A = ""' L.J vk cos 4 8k• k=l

v;,

A;l

141

4.2 GRAF'HICAL SOLUTION I'IIIOOIDUIII!B

= cos 28

and

v; = cos 48.

(4.2.6)

Using the trigonometric identity, cos2 28 = 112 (cos 48 + 1), the two parameters are related as

v; =2 V? -1.

(4.2.7)

Values of all possible combinations of the lamination parameters are, therefore, located along the boundary curve ABC in Fig. 4.2 described hy Eq. (4.2.7). Note that points A, B, and C correspond to laminates with [0], [±45L, and [90] orientation angles, respectively. Two other points that we will consider in some of the examples in this book are point D (0.5,-0.5), corresponding to the [±30L laminate and point E ( 0.5,-0.5), corresponding to the [±60], laminate. If the laminate consists of two or more fiber orientations, then it can be shown that Eq. (4.2.7) becomes an inequality v;~2v?-1.

(4.2.8)

142


v; c ~"

3

,~'I,,

'"' ,,,

143

4 2 GRAPHICAL SOLUTION PFIOCEOUAEB

the volume fraction of 0° layers by v1 and the volume fraction of 90° luyers by v2, point A has v =(1, 0) and point C has v = (0, 1). The distance between points A and C is I= 2 units. Point S, which is at a distance of 0.8 from A and 1.2 from C, (V~, v;) = (0.2, 1.0), has r = IU~/2.0 = 0.4. Then, from Eq. (4.2.9) we have at point S,

"'',,,A

v = 0.6(1, 0) + 0.4(0, 1) = (0.6, 0.4).

t I>J.tJ/1 I I v; 1

-1

Figure 4.2.

B

Lamination parameter diagram of a [±8]s laminate.

The allowable region of the lamination parameters is the area bounded by the curve ABC in Fig. 4.2, irrespective of the number of different layer orientation angles. Each point of this region is called a lamination point and corresponds to one or more laminates with specific stiffness properties. Any point inside the lamination parameter diagram must correspond to laminates with two or more fiber orientations. The lamination characteristic of the points inside the lamination diagram may be related to the characteristics of the points along the edges of the diagram. From Eq. (4.2.4) we see that the lamination parameters are a linear function of the volume fractions. This means that the lamination parameters along a straight line connecting two points in the diagram represent laminates with volume fractions proportional to the distance from these points. That is, let points Ps and Pq, located at a distance I of each other, be characterized by vectors of volume fractions v, and vq, respectively. The dimensionality of the volume fraction vectors is determined by the number of distinct fiber orientation angles, v = (v" v2 , ..• , v1). Then the point P lying on the line Ps Pq at a distance rl from Ps and (1 - r)l from Pq has a vector v of volume fractions

v = (1 - r)vs + rvq.

(4.2.9)

For example, all the points along the line AC in Fig. 4.2 represent laminates made of combinations of 0° and 90° layers only. Denoting

The point, then, corresponds to [0/902 ]s or to any other cross-ply laminate with the same ratio of oo and 90° layers. The line AC in the lamination diagram constitutes a special case, as it lies entirely on the boundary of the diagram. In most cases a straight line connecting two points on the diagram will pass in the interior. The rule expressed by Eq. (4.2.9) still holds. However, any point along the line can also be obtained by connecting other points to one another, and therefore we have multiple laminates corresponding to the same point. This is illustrated in the following example.

Example 4.2.2

v;

Consider the origin of the lamination diagram, V~ = = 0, and obtain four laminates that represent this point by observing that it lies along the following four lines in Fig. 4.2: (i) line AE, (ii) line CD, (iii) the vertical line V~ = 0, and (iv) the horizontal line

v;=o.

(i) Point A represents an all oo laminate and point E an all ±60° laminate. So we use v 1 to denote the volume fraction of 0° layers and v2 the volume fraction of ±60° layers. The volume fraction vectors corresponding to points A and E are v 1 = (1, 0) and v2 = (0, 1), respectively. The origin is one-third of the way from E and two-thirds of the way from A, so r = 2/3. Therefore, from Eq. (4.2.9),

v =(113)(1, 0) + (2/3)(0, 1) =(1/3, 2/3). That is, the origin represent a laminate with one-third oo layers and two-thirds ±60° layers, for example, [0/±60],. (ii) The solution along the line CD is entirely similar to the solution along the line AE, so that the origin is also the laminate [90/±30],.

·-

LAMINATE IN·PLANI! STIFFNESS DESIGN

(iii) The point (0, 1) has equal amounts of 0° and 90° layers. Point B is an all ±45° laminate. So we use v1, v2, and v3 to denote the volume fraction of 0°, 90°, and ±45° layers, respectively. Point (0, 1) then has a volume fraction vector v 1 = (0.5, 0.5, 0), while point B has a volume fraction vector v2 = (0, 0, 1). The origin is halfway between these two points, so that r = 0.5, and from Eq. (4.2.9) we get v = (0.25, 0.25, 0.5) which correspond to a laminate where we have equal parts of 0°, 90°, 45°, and -45° layers, such as [±45/0/90L laminate. (iv) From Eq. (4.2.6), along the diagram boundary we have v; =cos48. Therefore, v; =0 corresponds to either 48 = ±90° (8 = ±22.5°), or 48 = ±270° (8 = ±67.5°). The point 8 = ±22.5° corresponds to the intersection of the horizontal line v; = 0 with the boundary of the diagram between A and B. Similarly, the point 8 = ±67 .5° corresponds to the intersection of the line v; =0 with the boundary of the diagram between C and B. The origin lies in the middle between these two laminates, so that it corresponds to a laminate with equal proportions of all four angles such as [±22.5/±67.5].,. It is easy to check that the laminate in case (iv) is the same laminate as that of case (iii), rotated by 22.5°. Similarly, the laminate in case (ii) is the laminate of case (i) rotated by 90°. These laminates are called quasi-isotropic because we can rotate them by any angle without changing their in-plane properties. This attribute of quasi-isotropic laminates is not difficult to prove (Tsai and Pagano, 1968). Tsai and Pagano also show that any laminate with 11; = v; = 0 has the same quasi-isotropic property.

A point in the interior of the lamination diagram may represent a very large number of laminates with some of them including many orientation angles. However, since a point is defined by two parameters, this means that only two orientation angles, 8 1 and 82, are sufficient for designing laminates for given lamination parameters and hence for prescribed stiffness requirements. We can find combinations of two angles by passing a line through the point, finding its intersection with the diagram boundary on the two opposite sides of the point, and going through the procedure illustrated in the previous example by using the interior point and the two boundary points.

145

4.2 GRAPHICAL SOLUTION PAOCECURES

The geometrical procedure clearly indicates that if we want to have only two orientation angles, these can vary within a range that depends on the lamination point. For example, a quasi-isotropic laminate made of two orientation angles can be defined by passing a line through the =0). However, that line canquasi-isotropic lamination point (Vi= not intersect the boundary of the lamination domain along the top boundary (AC in Fig. 4.2), because points along AC are combinations of 0° and 90°. Thus a line through the quasi-isotropic point intersecting AC will correspond to a laminate with three orientation angles, oo, 90° and the angle defined by the intersection of the line with the other boundary (ABC). Inspection of Fig. 4.2 indicates that one side of the line must intersect the segment AD of the boundary, and the other side the segment EC. Thus, a two-ply-orientation quasi-isotropic laminate cannot have an orientation 8 in the region 30° < 8 < 60°. Instead of going through the geometrical procedure described above, we can solve the problem algebraically as follows. For a laminate with two orientation angles, Eq. (4.2.4) becomes

v;

V~

= V 1 COS 28 1 + Vz COS 282

and

v; = V 1 COS 48 1 + V2 COS 482,

(4.2.10) where v2 = 1- VI. We thus have three parameters, vi' 8,, and 82 for obtaining a particular lamination point. One of the angles or the volume fraction v1 can be assigned a value arbitrarily, but only within some ranges. For example, the geometric procedure for creating a quasiisotropic laminate, which was discussed above, indicated that one side of the line must lie in the segment AD, and the other side in the range CE. The volume fraction of the first orientation is the ratio of the distance between the origin and the point in the segment CE to the total length of the line. It is apparent from the diagram that the volume fraction is minimal when the line begins at point A, which corresponds lo [0/±60),, where the volume fraction is 113. Similarly, the volume fraction of the first orientation is maximal along the line DE, which corresponds to [±30/901, and a volume fraction of 2/3. For a given volume fraction in the appropriate range, the two angles can be obtained by solving Eq. (4.2.10) as 8, = 0.5 cos-' Tl' where

82 = 0.5 COS-I T2,

(4.2.11)

146

LAMINATE IN·FILANE STIFFNESS DESIGN

TI

=

2v 1 V~±.Y2v 1 vz(l+V;-2 V?)

2vi

'

~

147

2 GRAPHICAL SOLUTION FIFlOC!OUREB

und v1 + v2 + v3 = I.

_ v~- v T I I T2-

v2

Example 4.2.3

For a quasi-isotropic laminate Eq. (4.2.12) reduces to

-+ TI -

{f.

VITI

---T2Vz

0

These simple equations let us easily find quasi-isotropic 1 with only two distinct orientation angles for a given volume vp as long as v1 stays within the limits of 113 and 2/3 (T1 and T2 remain between 0 and 1). For example, for v 1 = 0.4, it is easy to that we get 9 1 = 15° and 92 = 62.63°. That is, [±15/±62.63 3 ]. is 20-layer quasi-isotropic laminate. It is also easy to show that, for laminate to be quasi-isotropic, relative total thickness of the layers with 9 1 and 92 orientations must remain between 0.5 and 1. For balanced angle-ply laminates with more than two orientations, many combinations of orientation angles and volume fractions produce the same lamination parameters and, therefore, the same stiffness properties. For the case of three angles with 9 3 = 0°, for example, we have V~ =

As we will see later, the optimum laminate for Example 4.2.1 has lamination parameters V~ = 0.5172, v; = -0.3673. Design a balanced, symmetric, 16-layer laminate with these lamination parameters. For this lamination point, the value of the quantity I+ 2V? is equal to 0.09771. The smallness of the number indicates that it is close to the boundary of the lamination diagram. Since we have only 16 layers, working with integer number of layers limits the possible choice of lamination parameters. We first consider a two-angle laminate. With a minimum of four layers per angle (to achieve the symmetric and balance requirement) we need to consider only v 1 = 0.25 and v 1 = 0.5. We cannot have v1 = 0 because the lamination point is not on the boundary. We do not need to consider v 1 = 0.75 because this will give us the same solution as v 1 = 0.25, only with the two angles switched. For v1 = 0.25, from Eq. (4.2.12), we get

v;-

T

v 1 cos 29 1 + v2 cos 292 + v3 and

1

2 X 0.25

X

v; = V 1 COS 49 1 + V 2 COS 492 + V 3 .

= 0.9000

Miki (1982) gives the two angles for a given choice of volume fractions as 1 8 1 =2cos

-I (V~- Vzv 8- V3)

,

(0.5172) ± --./2 X 0.25 2 X 0.25

-I

1 82 =2cos 8,

or

or

where

91 Kb = 2 vz(v 3 - V~),

Kc = 2 v~ + 2 (v 1 - 2V~)v3 - (1 + v;) v 1 + 2 V; 2,

0.09771

T2 =

0.5172 - 0.25 -

X

0 ·9 = 0.3896

0.5172 - 0.25 0.75

X

T2

0.l 344 = 0.6448,

so that

a

Ka = 2 viv 1 + v2),

X

and

(4.2.15)

-Kb ± ...JK~- KaKc = --=----K....::..___::........::..,

0.75

0.1344

1

8

X

=-----~~--~--------------------

= 0.5 cos- 1(0.9) = 12.9°

and

92 = 0.5 cos- 1(0.3896) = 33.5°

or (4.2.16)

91 = 0.5 cos- 1(0.1344) = 41.1° and 92 = 0.5 cos- 1(0.6448) = 24.9°.

148

LAMINATE IN-PLANE STIFFNESS DESIGN

Similarly, for v 1 = 0.5 we get 2 X 0.5 X (0.5172) ± .V2 X 0.5 2 X 0.09771 2x0.5

Tl

= 0.7382

or

0.2962

and T2

or

= 0.5172 T2 =

0.5 ~ ~

X

0. 7382

0.5172 - 0.5 0.5

X

= 0.2962

0. 2962

= 0. 7382,

so that, as expected, the symmetric case of v1 one pair of solutions el

= 0.5 cos- 1(0.7382) = 21.2°

= 0.5

yields only

and

82 = 0.5 cos- 1(0.2962) = 36.4°. Next we consider the case of three angles with the third one being 0°. One possibility is v1 = 0.5, v2 = 0.25, v3 =0.25. From Eq. (4.2.16) we have

Ka = 2 X 0.25 (0.5 + 0.25)

= 0.375,

4.2 GRAPHICAL SOLUTION PROCEDURES

149

To summarize the results, we found three laminates that have the same required V~ and v;, and hence the same in-plane stiffness properties. These are [±12.9/(±33.5) 3Js, [±41.11(±24.9) 3 ],, l (±21.2)/(±36.4)2],. Example 4.2.3 demonstrates that once we find the desired laminalion parameters it is possible to find many stacking sequences that lead to these parameters. This indicates that we can break the design of laminates for in-plane stiffness requirements into two steps. First we find the optimum set of lamination parameters that will maximize or minimize a selected property while satisfying constraints on other properties. Then, as a second step, we find a set of stacking sequences that we can use to achieve the desired lamination parameters. The great advantage of this two-step procedure is that the optimization phase has only two unknowns, V~ and v;, and so it lends itself to graphical solution. The design space is the area bounded by the curve ABC in Fig. 4.2, with constraints on the effective engineering constants limiting further the set of possible design points (called the feasible domain). A very useful aid in the graphical solution of the laminate design problem is to draw contours of constant effective engineering stiffnesses. The equations needed for such contours, obtained by combining Eqs. (2.4.42) and (4.2.1 ), are

Ex

contours:

V* _ u~v?- V 2Ex v~ + E,U1 - Uf + u~ 3 Ui2 VI +2 U4-EJ '

EY

contours:

v;

vxy

contours:

V*3

Gxy

contours:

v;=

(4.2.17)

Kb = 2 X 0.25(0.25 - 0.5172) = -0.1336,

Kc = 2 X 0.25 2 + 2(0.5 - 2 X 0.5172) - ( 1 - 0.3673)

X 0.5

X 0.25

U~V? + U2EYV~ + EYU 1 - Vi+ U~

U/2 V 1 + 2 V 4 - E)

(4.2.18)

+ 2 X 0.51722 = 0.07644.

For these values, we find that the quantity under the radical of the first Eq. (4.2.16), K~- KaKc is negative, and we cannot continue the computation. This indicates that even with v3 = 0.25 (25% zero layers), it is impossible to find a laminate that will have the desired lamination parameters. Since it is impossible to have a smaller volume fraction for the 0° layers, we cannot use them here.

vxyu2 v~ - vxyul + u4 (1

+ vxy)U3

Us- Gxy u3

(4.2.19)

(4.2.20)

150


Contours of constant laminate effective engineering properties are shown in Fig. 4.3. The figure indicates that, if no other constraints are specified, the maximum values of the Ex, EY, andGxy are all achieved for lamination points located around the boundary of the design space that require only one lamination angle. As expected, maximum Ex and EY are obtained for oo and 90° laminates, respectively, and the maximum shear stiffness for [±451, laminate. determination of the value of lamination angle [±9]s that maximize the effective Poisson's ratio is not so obvious and is a function of lamina properties via Eqs. (2.4.20) and (2.4.8). For example,

Ez,GPa

E1,GPa

151

4.2 GRAPHICAL SOLUTION PPIOC!OURES

TJ00/5208 Graphite/Epoxy and Scotchply 1002 Glass/Epoxy materials, the laminates that produce the maximum Poisson's ratio are [±25], and [±31],, respectively. The contours shown in Fig. 4.3 help in correlating laminate propl'rties with the two in-plane lamination parameters. The first laminalion parameter, v;, is seen to represent the relative stiffness of the laminate in the x and y directions. A positive v; represents a laminate stiffer in the x direction, while a negative v; represents a laminate stiffer in the y direction. The lamination parameter v;, on the other hand, represents the shear stiffness of the laminate, with the shear stiffness increasing as v; becomes more negative. For design problems where one or more of the effective engineering constants are constrained, appropriate contours can be superimposed lo identify the feasible design space and the lamination point that maximizes (or minimizes) the desired stiffness property. This is demonstrated in the next example.

Example 4.2.4 '\ ".... ' '\

\f

\

( -·) , II

~

•

VI

Redo Example 4.2.1 without limiting the laminate to an angleply, [±9], stacking sequence. Using Eq. (2.4.20), we first find the material invariants U; for the Graphite/Epoxy of Example 4.2.1 to be U 1 = 57.547 GPa,

G"',GP8

v"'

·-

•

Vj

·~\

~

•

Figure 4.3.

I

U4 = 17.540 GPa,

U3 = 13.604 GPa,

U5 = 20.004 GPa.

VJ

The line corresponding to Gxv = 25 GPa divides the lamination diagram into two regions (see Fig. 4.4). From Eq. (4.2.20) this line is

sr---------~--------~

.

U2 = 58.030 GPa,

-. !

~

•

V1

\

'

/

>~

-

Contours of constant effective engineering elastic properties.

v; =

20.004- 25 13.604

=-0.3673.

(a)

In the region above the line Gxy< 25 GPa, and in the region below Gxy > 25 GPa. Similarly, the line corresponding to vxy = 1 separates the lamination diagram into two regions with the bottom one corresponding to vxy > 1. The equation of this line, from Eq. (4.2.19), is

152


58.03v;-57.547+ 17 ·54 = 2 . 1328v;-t.4704. 2 x 13.604

v;

153

4.2 GRAPHICAL SOLUTION PROCEDURES

A~ 2 =

(b)

17.54 + 0.3673 x 13.604 = 22.536 GPa.

Then, from Eq. (4.2.3), The feasible domain, which is the region where both requirements are satisfied, is the unhatched part of the diagram. Note that the constraint boundaries are shaded on the side of the boundary where the constraint is violated. This is a standard graphical representation of the constraint. Figure 4.4 also shows some contours of Ex. These contours indicate that the maximum value of Ex is realized at the top right corner of the feasible domain where the two lines of Eqs. (a) and (b) intersect. The lamination parameters which correspond to the intersection point are v; = 0.5172, v; = -0.3672. At this point, from Eq. (4.2.1), A; 1 = 57.547 + 0.5172 x 58.03-0.3673 x 13.604 = 82.563 GPa,

A;

2

= 57.547-0.5172 x 58.03-0.3673 X

13.604 = 22.537 GPa,

V3 20

35

50

Ex

95

110

22.537

22.536

2

= 60 0 GP ·

a.

This value of Ex is 67% larger than the value obtained in Example 4.2.1 for an angle-ply laminate. This is because an angle-ply design has to stay on the boundary of the lamination diagram. As can be seen from Figure 4.4, the constraint on Poisson's ratio excludes a large portion of the boundary of the lamination diagram near the optimum. The optimum angle-ply design in Example 4.2.1, [36/-36] 5 , corresponds to point A in Figure 4.4 which is quite far from the optimum design, point 0. It has substantially larger Gxy than required, at the expense of the objective Ex. The previous example, Example 4.2.3, provided several laminates that realize the optimum lamination parameters of point 0.

4.2.2 80

= 82.563 X 22.537 -

Graphical Design of Coefficients of Thermal Expansion

When restricted to in-plane respose, deformation of laminated composites under thermal loads may be represented by coefficients of therfor the laminate derived in terms of the properties mal expansion, on the individual layers. In Chapter 3 the coefficients of thermal expansion of symmetric laminates are derived, Eq. (3.1.42), in terms of the in-plane V terms defined in Eq. (2.4.35). By imposing the balanced laminate condition, the expression can further be simplified as

a,

,

I

I

I

'

l 'i '! ~ '>J "\\'k'\1 ~~~l'\l'\ 1~':/ vi"\~''"

'

.....

I

I ~

..

v•1

axl JKt VoA + K V1A} {~J ~ ~ [AJ-' ( ' v,A ~ K, v,A , 2

Gxy = 25 GPa ~...

Figure 4.4.

vxy = 1.0

Lamination parameter diagram for Example 4.2.4.

(4.2.21)

where the A 16 and A26 terms also vanish. Using the definition of the remaining A terms in terms of the V's and their normalized form V*'s and performing the matrix inversion and the multiplication, it can be shown that the coefficients of expansion of a balanced symmetric laminate take the following form:

154

_

ax=


V~ (K2 U 1 - K 1 U 2 + K 2 U4 ) - V~ 2 K 2 U 2 + 2 K1 U3 v;

+ K1 (U 1 -

2(2V;U3 (U4 +U 1 )-V?U~+(Ui-U~))

155

4 2 GRAPHICAL SOLUTION PFIOCEOUFIES

U4 ) '

(4.2.22) a= y

+ V~ 2 K 2 U2 -2 K 1 U3 v;-K 1 (U 1 - U4) 2 (2 v; U3 (U4 + U1) - v? u~ +cui- u~))

V~ (K2 U 1 -K1 U2 +K2 U4 )

(4.2.23) Just as in the case of elastic properties, the above equation completely isolates the stacking sequence dependent terms from the material properties. The elastic properties of the material are in the U terms. The coefficients of thermal expansion of the individual layers (which are assumed to be the same in each layer) are captured in Kp K2 , and K 3 ; see Eq. (3.1.31). The equations shown above may be used to draw the contour lines corresponding to specified values of the coefficients of thermal expansion of laminates on the in-plane lamination diagram. Such contour lines can be used to determine laminate designs that satisfy the desired thermal characteristics. For example, the contour lines corresponding to zero values of ax andaY are shown in Fig. 4.5 for a Graphite/Epoxy material. For the material system used in this figure, the two lines do not cross. That is, no laminate has both zero ax and zero ay. Such a laminate would have no deformation in any in-plane direction under thermal loads. However, there are infinitely many laminates, defined by combinations of the V~ and v; on those lines, that have a zero value of either ax or ay. Finally, the concept of thermal isotropy introduced in Chapter 3 requires that the coefficient of thermal expansion be the same along two mutually perpendicular directions. By imposing the condition ax= a_v in Eqs. (4.2.22) and (4.2.23), we can derive the following equation for thermal isotropy: v~ (K2

U 1 - K 1 U2 + K 2 U4 )

= o.

(4.2.24)

All the terms inside the parentheses are material dependent, whereas the stacking sequence is determined by the V~ term. Obviously, one of the ways to satisfy the condition is to get the terms inside the parantheses to vanish. It can be shown (see Exercise 6), that the only way to make the material dependent part vanish is when a 1 = a 2, which

'\

Figure 4.5.

.......1:\

I

I

""'~

r I

1

v'1

Graphite/Epoxy laminates with zero coefficients of thermal expansion.

is one of the characteristics of an isotropic layer. A more interesting result of Eq. (4.2.24) is the condition that

v~

=I vk cos 2ek = o.

(4.2.25)

k=l

There are infinitely many laminates on Miki's in-plane lamination diagram, which are located along the v; line between the [0190]ns and [±45]ns laminates. Compared to the condition of mechanical isotropy that is limited to a single point, v~ = v; = 0, at the origin, there is more freedom in establishing thermal isotropy.

4.2.3

Optimization of Stack Thicknesses

Beside orientation angles, we can optimize the number of layers that we have in any given orientation. Early work mostly disregarded the discrete nature of this variable and replaced it with a variable called "ply thickness." The term generally referred not to the actual layer thickness, but to the total thickness of a stack of contiguous layers

lCU:I

LAMINATE IN·F'LANE STIFFNESS DESIGN

with the same orientation. Since this terminology is confusing, we will use the term "stack thickness" to refer to such total thickness. When the laminate thickness is large, it may be advantageous to treat the stack thicknesses as continuous design variables and find the optimum design of the continuous problem. It is then possible to look for the optimum design of the actual (discrete) problem in the vicinity of the continuous optimum by rounding up the stack thickness to nearest integer multiple of the layer thickness (as was done in Example 4.1.1 ). Such a procedure is not guaranteed to result in an optimal solution. However, it is easy to use, especially when the number of stack variables is only one or two, and it allows graphical solution technique. In this section we demonstrate the use of such a procedure. We consider again the design of a symmetric and balanced laminate with multiple angle-ply stacks [(±9JN/(±9n_1)Nn_/ · · 1(±9 1 )NJ~· Now the orientation angles are fixed, and the n unknowns are the numbers of layers of each specified orientation, Ni, where i = 1, . . . , n, and n is the number of layer groups with distinct orientation angle. We define the stack-thickness design variables to be the total thickness for each orientation; that is,

{4t N;,

x. = ' 2tN;,

if 9; ;t. oo and 9; ;t. goo, if9;=0o or 9;=goo,

(4.2.26)

where the factors of 2 and 4 are due to the symmetry and balance requirements. A typical design problem is to seek the lightest laminate that satisfies design requirements on in-plane stiffnesses. In this case, the objective function f(x) can be taken to be the total thickness n

f(x)

=LX;.

(4.2.27)

i=l

Next, we express the in-plane stiffnesses in terms of the design variables. Since the laminate is balanced, A 16 and A 26 are zero. The other four terms in the A matrix are independent of the sign of the angle (recall that the lamination parameters are defined in terms of cosines). Therefore, the contributions of the layers with orientation 9 and with orientation -9 are the same, and we can treat the laminate as if it were made of [(9n)2Nn1(9n_ 1) 2Nn-1 f ... f(9 1)2NI Js. Then, using Eq. (2.4.28) we can write the in-plane stiffnesses as

157

4.2 GRAPHICAL SOLUTION F'AOCEOURES n

Au=

L (Qu\k)xk.

(4.2.28)

k=l

Note that the xk represent the total stack thickness associated with a given orientation. Furthermore, the way Eq. (4.2.26) defined the thickness design variables, Eq. (4.2.28) applies to design variables associated with either angle-ply stacks or oo and goo stacks. With the in-plane stiffness expressed as a simple linear function of the design variables, it is usually easy to obtain expressions for constraints on engineering stiffnesses and write them in the standard form of Eq. (1.4.1). In most applications, only inequality constraints are used. As in the case of ply-orientation variables, graphical solutions are based on tracing the constraint boundaries, that is, the curves

gix) =0, j

= 1, ... , ng.

(4.2.2g)

It is usually easy to determine on which side of this boundary the constraint is satisfied, and then the feasible domain may be defined (as was done in the lamination diagram, Fig. 4.4). Next, objective function contours are drawn in the feasible domain, and the optimum design is identified. This optimum design usually corresponds to noninteger values of the N;s. So the final problem is to find a nearby integer solution. This procedure is demonstrated in the following example.

Example 4.2.5 Consider the design problem of Example 4.1.1, for a laminate with orientation angles limited to 0°, ±45°, and goo. Assume that we have determined that we need only two goo layers, so that the candidate laminate is [±45N/ON/gO]s. We seek the lightest laminate that satisfies the two design constraints of Example 4.1.1, namely,

h Ex ~ 2,500,000 lb/in

and

h Gxy ~ 500,000 lb/in,

where h is the total laminate thickness. We define the design variables as in Eq. (4.2.26), x 1 = 0.02 N 1 and x 2 = 0.01 N 2 , and the objective function is

10"

LAMINATE IN·IILANE STIFFNESS DESIGN

j{x)

= h =0.0 I + x 1 + x 2•

Next, we need to express the constraints in terms of the design variables. As a first step we write the Au's in terms of the design variables, using Eq. (4.2.28), Au = 0.01 Q u(90°) + x 1 Q u( 45°) + x 2 Qu.

Using the values of the Qu' s obtained in Example 4.1.1, we get A 11

= 1.908 X

104 + 6.361

X

106 x 1 + 18.67 X 106 x 2 ,

A 22

= 18.67 X

104 + 6.361

X

106 X 1 + 1.908 X 106 x 2 ,

A 12 = 0.5723 A 66

= 0.93 X

X

104 + 4.501

X

106 x 1 + 0.5723

104 + 4.858 X 106 X 1 + 0.93

X

X

106 x 2 ,

106 x 2 •

Next, we use Eq. (2.4.42) to write the constraint on hEx as AIIA22-

Aiz

~-=---= 2::

2,500,000 or

22

159

4.2 GRAPHICAL SOLUTION PIIIOCEOURES

Constraints boundaries can be plotted by using software packages such as Mathematica. In absence of such a capability, we can plot the constraint boundaries by selecting values of x 1 and solving for the corresponding values of x 2 by setting gi = 0. For g 1 this requires the solution of a quadratic equation, and for g 2 the solution of a linear equation. Figure 4.6 shows the constraint boundaries and the contours of the objective function. It is clear that the optimum is realized at the intersection of the constraints, where g 1 = g2 = 0. Solving these two equations we obtain x 1 = 0.0787 in and x 2 = 0.1168 in, with the objective function, the total thickness, off = 0.2055 in or 41.1 layers. From the definition of the design variables we see that this corresponds to N 1 = 3.935 and N2 = 11.68. The obvious integer solution is obtained by rounding up both n urn bers to N 1 = 4 and N 2 = 12, which corresponds to [(±45)i0 12 /90J_.. This 42-ply 0.21 in thick laminate is 38% thinner than the best laminate that we obtained in Example 4.1.1. Usually when we use this type of rounding up to obtain a laminate with an integer number of layers from the optimum continuous-thickness solution, we cannot be sure that the process will yield the optimum design. For example, the optimum design may involve

2

2,500,000 A 22 - A 11A 22 + A 12 ~ 0.

Substituting the expressions of the AiJ's into this constraint and dividing by 108 , we get g 1(xi' x 2 )

................

=-(1.908 + 636.1 x 1 + I 867 x 2 ) (18.67 + 636. I x 1 + 190.8 x 2 )

~~

........................... f = 0. 4 ................ ~~

+ (0.5723 + 450.1

X

1

+ 57.23 X 2 ) 2

X

....................

..........

2

+ 250( 18.67 + 636.1 X1 + 190.8 Xz) ~ 0.

f

~~

~~

~~

= 0.3

....................

The constraint on h Gxy is simpler. From Eq. (2.4.42) we see that it may be written as 500,000 - A 66 ~ 0.

=49.07-485.8 x 1 -

.................... ~~

.............................. ~~

.................. ...............

0

-~

0

0.04

Substituting for A 66 and dividing by 104 , we get gz(xi' x 2)

~~

93 x 2 ~ 0.

0.08

0.12

0.16

X

1

Figure 4.6.

Design space for Example 4.2.5.

0.2

160


rounding down one of the ply thicknesses and rounding up of another. Occasionally, the optimum design is not even close to the rounded up continuous solution. However, here we can tell immediately that the solution we obtained cannot be improved upon. The reason for this is that a lighter laminate will have 40 layers for a thickness of 0.2, which is less than the continuous optimum. The continuous optimum is a lower bound for the real optimum which has the additional restriction of integer number of layers. Therefore, it is impossible for the optimum design to be lighter than the continuous optimum.

4.3

161

DEALING WITH THE OISCFI!TENESS OF THE DESIGN PROBLEM

design problems that number can be very large because the stacking sequence of the laminate (the position in the laminate of the various layers) may be as important as the number of layers of a given orientation. However, as long as the total number of layers of each orientation remains the same, the stacking sequence does not change the in-plane stiffness properties. So, for in-plane stiffness design problems, the number of possible designs is not very large. Therefore, exhaustive enumeration of all possible designs is a reasonable solution strategy. This is demonstrated in the following example. Example 4.3.1

4.3 DEALING WITH THE DISCRETENESS OF THE DESIGN PROBLEM

In the last example we were fortunate that the optimum discrete solution (corresponding to integer number of layers) was obtained by simply rounding up the thicknesses obtained from the continuous optimum. Unfortunately, in many problems rounding up does not produce the optimum design. In fact, often the optimum discrete design is not even in the general neighborhood of the continuous optimum. In such cases we may want to use methods devised to solve the discrete optimization directly, methods that are developed by a branch of optimization theory called integer programming. Some of the general methods of integer programming will be discussed in Chapter 5 in more detail. Here we will introduce only two simple methods. The first method is referred to as exhaustive enumeration, which is applicable to general discrete optimization problems, while the other, an extension of Miki's lamination diagram method, was devised specifically for the in-plane stiffness design problem. There are two reasons for the discrete nature of the laminate design problem. The first is associated with the fact that we must have an integer number of layers. The second reason is preference by industry for laminates with a few fixed orientation angles. This preference is based on ease of manufacturing a laminate with commonly used angles and on the availability of test data for such laminates. Therefore, in most applications, we are restricted to a small set of prescribed orientation angles, usually 0°, ±45°, and 90°. One aspect of the discrete nature of the problem is that only a finite number of laminates need to be considered as candidates for the design. In general, for laminate

Solve the problem in Example 4.2.1 for a 16-layer balanced and symmetric laminate made of 0°, ±45°, and 90° layers. There are twenty-five combinations of orientation angles that satisfy the conditions of symmetry and balance. Since the requirement on Gxy is difficult to satisfy, only laminates with a substantial percentage of ±45° layers should be considered. Table 4.1 lists laminates with at least 50% ±45° layers in the order of increasing percentages of these layers, along with the Ex' Gxy' and vxy for each laminate. It is seen that there are four laminates that satisfy the constraint on Gxy being at least 25 GPa, and that the optimum is [(±45)/0 21, with Ex= 48.9 GPa. The constraint on vxy is not critical for any of the laminates. Note that the value of Ex is 19 percent lower than the continuous optimum of Ex= 60.0 GPa found in Example 4.2.4. This is the price we pay for limiting ourselves to the discrete set of

Table 4.1.

Candidate laminates for Example 4.3.1

Stacking Sequence [(±45)z/904ls [(±45)z/0/903ls [(±45)z/Ozf902], [(±45)zl03/90)s [(±45)z/04L [(±45h/90zL [(±45)/0/90Js [(±45)/02]., [(±45)4],

Ex (GPa) 24.98 38.77 52.20 64.91 75.78 27.16 39.07 48.90 21.87

Gxy

(GPa)

20.00 20.00 20.00 20.00 20.00 26.81 26.81 26.81 33.61

l/xy

0.203 0.243 0.305 0.408 0.615 0.373 0.480 0.672 0.709

162


angles. However, Ex is still 36% higher than the value that we found in Example 4.2.1 for an angle-ply laminate, which was determined without limiting ourselves to a discrete set of angles. When the laminate is thick, exhaustive enumeration may become cumbersome, and we may want to consider the use of Miki's lamination parameter diagrams. Miki and Sugiyama (1991) showed that the diagrams can be used for designing laminates with predetermined orientation angles. They showed that the feasible region for laminates with fixed orientation angles is a polygon with vertices located on the envelope of the lamination parameter diagram. If the design point is on the perimeter of the diagram, the laminate is an angle-ply laminate with one fiber orientation. Therefore, given a set of permissible integer orientation angles, vertices of the polygons are placed at those locations on the perimeter of the diagram that correspond to the selected angles. For example, all angle-ply laminates with orientations limited to integer multiples of 15° are shown in Fig. 4. 7. For points inside the diagram, a procedure similar to the one in Example 4.2.2 can be used to determine the possible laminates. Consider, for example, laminates made of only el = 0°, e2 = ±45°, and 83 = 90° layers. From the definition of the in-plane lamination parameters, Eq. (4.2.4), we have V~

=V1 -

V3

and

v; = V 1 -

V2

+ V 3,

vj

+ Vz + V3 = 1.

.1.1

[90)•. \..

.. , [0]

[±15]

~r--IT'

v~

1

-1

Figure 4.7. ments.

[±45]

In-plane lamination diagram for angle-ply laminates with 15° incre-

passes through the origin parallel to AB. For other permissible values of the v3 between 0 and 1, we obtain other lines that are parallel to the line AB. Similarly, by eliminating v2 and v3 from the Eqs. (4.3.1) and (4.3.2), we find equations of lines that are parallel to the line passing through A and C. Such lines are shown in Fig. 4.8a and 4.8b, respectively, for laminates made up of layers with 0°, ±45°, and 90°

(4.3.1)

where vi' v2 , and v3 are the fraction of the layers with 0°, ±45°, and 90° orientations, respectively, and VI

163

4.3 DEALING WITH THE OISCAETENESS OF THE DESIGN PROBLEM

(4.3.2)

C 190 4 llt

v;

f

[0/90 3 ]J0 2 /90 2 ]!0 3 /901, ...~ , !!:··

l

..

A

I

D

A

10 4 1,

By using Eq. (4.3.2) we eliminate v2 from the second relation of Eq. (4.3.1) and then use the first relation of Eq. (4.3.1) to eliminate v 1• The resulting expression

v; = 2V~ + 4v3 -

1

is the parametric equation of a straight line in the in-plane lamination parameter diagram. As expected, for v3 = 0 (laminates with no 90° layers), Eq. (4.3.3) corresponds to the line connecting points A and B in Fig. 4.8a. For v3 = 0.25, laminates with 25% 90° layers, the line

(a)

Figure 4.8. angles.

(b)

In-plane lamination diagram for laminates with specified orientation

164


orientations and 0°, ±30°, ±60°, and goo orientations. For laminates with 0, ±45, and goo layers, the design space is a triangle with vertices at A, B, and C as shown in the figure. For orientation angles of 0°, ±30°, ±60°, and goo, the design space is a trapezoid. The number of lamination points along the edges and interior points of the polygons corresponding to laminates with combinations of two or more orientation angles is determined by the total number of layers in the laminate. We denote the total number of layers as N and I N/2. Then, in addition to the vertices, we obtain I - l equally spaced design points along the edges and along the internal lines that join two vertices. From the nodes we obtained along the edges, we also draw lines parallel to the lines that join vertices. If such a line terminates at another discrete design point at the opposite end of the polygon, then it is easy to label the design that would be in the interior by looking at the designs at the two end points. For example, for an eight-ply (total) laminate with 0°, ±45°, and goo angles (triangular design space), there are five equally spaced design points with fiber orientations varying incrementally from one vertex to another as shown in Fig. 4.8a. Note that the design points inside the triangular region also follow an incremental pattern, but are combinations of the three available angles. The labels in Fig. 4.8a include only 45° layers, but each 45° layer can be replaced by a -45° layer, as both correspond to the same lamination parameters. However, when the laminate has to be balanced, the total number of ±45° layers has to be even. For example, the point [0/45 2], can be replaced by [0/±45],, while the point [0/45 3] is not a possible balanced laminate. To avoid the problem of obtaining points that do not correspond to a balanced laminate, we can divide the sides of the polygon into I/2 intervals instead of I intervals. This is illustrated in Fig. 4.8b, where some of the vertices corresponding to a 12-ply balanced symmetric laminate made of oo, ±30°, ±60°, and goo layers are marked. With N = 12 and I= 6, the sides of the trapezoid were divided into only three intervals. All the intersection points correspond to balanced symmetric laminates, and we note that the parallel lines indicate the existence of additional points along the line AD beside the two interior points that we started with. For this case, with four different lamination angles, we can have multiple definitions of some of the points in the diagram. Consider, for example, the origin. As we saw in Example 4.2.2, the origin lies along the line AC, so that it can correspond to

=

185

4.3 DEALING WITH THE OISOAETENESS OF THE DESIGN PROBLEM

1±60/02 Js and along the line BD, so that it also corresponds to I±30/g02J.. However, we can show that it also lies along the line v; =0, so that it corresponds also to [0/±30/±60/gO],. In Exercise 8, the reader is asked to provide all possible labels for the intersection points shown in Fig. 4.8b.

Example 4.3.2 Use the lamination diagram to solve the problem in Example 4.2.1 for a 16-layer balanced and symmetric laminate made of (a) 0°, ±45° and goo layers and (b) 0°, ±30°, ±60°, and goo layers. To solve this problem we draw the feasible domain on the diagram as in Example 4.2.4 and add the discrete points that correspond to the possible laminates. Consider first Fig. 4.ga, which represents the solution for part (a) of this Example. The number of layers is 16, so we can have eight divisions along the line AC that corresponds to cross-ply laminates (laminates with only 0° and goo layers). For example, point A corresponds to the laminate 0~6 , and the point immediately to its left corresponds to (0/gO)s. In the vertical direction, which corresponds to variation in the number of ±45° layers, we can have only four divisions because of the balanced laminate requirement. For example, along the line AB, the point closest to A is (±45/06 )s· Altogether there are 25 possible laminates, and as noted in the previous

c'

GPalv; ~ ~'A

Ex= 48.9

Ex A

Jlt ' , .....

1......

(a)

Figure 4.9.

·""·

·"·

·"·

I·

D

t

1

"

"

.n.

tA

(b)

In-plane lamination diagram for Example 4.3.2.

IVV

LAMINATE IN•PLANE STIFFNESS DESIGN

example, only four of them are in .the feasible domain. An examination of the contours of Ex in Example 4.2.4, tells us that the optimum point is at the top right region in the feasible domain, which corresponds to [(±45)/0 2], with an Ex= 48.9 GPa (see Table 4.1). Next we turn our attention to part (b) and Fig. 4.9b. This time we have two angles that have to be balanced, so the segments AB, BC, and CD are divided into only four subintervals. The labeling of all the points in the diagram is left as an exercise to the reader (Exercise 8). Of all possible lamination points, only four are in the feasible domain, and the optimum is [(±30)/±60]s. Note that this point has exactly the same lamination parameters as the solution of part (a). Indeed for V~ we get for both points V~

REFERENCES

for example [(±8/±(8 + 45°)1,,. Design such a laminate that has the lamination parameters v~ = 0.5172, v; = -0.3673.

5.

Use Miki's graphical design procedure to obtain the lightest laminate for Example 4.1.1. [Hint: Start with h = 0.34 in and draw the feasible domain on the lamination diagram, then repeat for decreasing values of h until the feasible domain vanishes.]

6.

One of the conditions for achieving a thermally isotropic laminate [see Eq. (4.2.24)] is the condition that (K2 U 1 -K1 U2 +K2 U4 )=0.

v;

= 0.75 cos(2 X 45°) + 0.25 cos(2 X 0°)

(al-~)

= 0.75 cos(4 x 30°)

II

22

=0

Vf2

(2)

'

which can be satisfied only when a 1 = a 2• 7.

Repeat Example 4.2.5 with a symmetric and balanced laminate made of oo and ±30° layers.

8.

Provide all the possible labels for all intersection points in (a) Figure 4.8b, (b) Figure 4.9a.

+ 0.25 cos(4 x 60°) = -0.5.

Therefore, the two laminates have the same A matrix and the same effective engineering constants.

E2 E Ell - E22

we get

v; =0.75 cos(4 X 45°) + 0.25 cos(4 X 0°)

(1)

By using the definitions of the K's and the U's, show that this condition can further be reduced to

= 0.75 cos(2 x 30°) + 0.25 cos(2 x 60°) = 0.25, and for

167

REFERENCES EXERCISES 1.

For the problem of Example 4.1.1, find the lightest symmetric and balanced laminate that is made only of 0°, ±45°, and 90° layers.

2.

Repeat Example 4.2.1 for a [8/-8/0°), laminate.

3.

Find a quasi-isotropic laminate that includes ±20° layers, which is not simply a rotated version of one of the quasi-isotropic laminates described in the text.

4.

Because of the availability of test data, it is desirable to work with laminates having stacking angles that are 45° or 90° apart,

Miki, M. (1982). "Material Design of Composite Laminates with Required InPlane Elastic Properties." Progress in Science and Engineering of Composites. Vol. 2, 1725-1731. Hayashi, T., Kawata, K., and Umekawa, S. (eds.) ICCM-IV, Tokyo. Miki, M. (1983). "A Graphical Method for Designing Fibrous Laminated Composites with Required In-plane Stiffness." Trans. JSCM 9, 2, 51-55. Miki, M. and Sugiyama, Y. (1991). "Optimum Design of Laminated Composite Plates Using Lamination Parameters." Proceedings of the AIAAIASME/ASCE/ AHS/ ASC 32th Structures, Structural Dynamics, and Materials Conference, Part I, pp. 275-283. Baltimore, MD. Stroud, W. J. and Anderson, M. S. (1981 ). "PASCO: Structural Panel Analysis and Sizing Code, Capability and Analytical Foundations." NASA TM 80181.

:~I

168

LAMINATE IN·F'LANE STIFFNESS DESIGN

Tsai, S. W. and Pagano N.J. ( 1968). "Invariant Properties of Composite Materials." In Composite Materials Workshop. 233-253. Tsai, S. W., Halpin, J. C., Pagano, N.J. (eds.) Technomic Publishing Co., Westport.

5 INTEGER PROGRAMMING

In the previous chapter we illustrated the use of two simple integer programming techniques for the design of laminates subject to inplane stiffness requirements. One method, enumeration, has general applicability beyond in-plane stiffness design, but for most problems it is too computationally expensive. The second method, Miki's lamination parameter diagram, can be generalized to a few other problems, but cannot handle many problems of interest. The objective of the present chapter is to describe methods that are more generally applicable to the design of laminated composites. Section 5.1 describes the formulation of linear integer programming problems. This class of problems has been studied extensively by developers of optimization methods, and efficient algorithms that are guaranteed to find the global optimum are available. Section 5.2 shows how in-plane stiffness design problems, of the type discussed in Chapter 4, can be formulated as linear integer programming problems. Section 5.3 describes the branch-and-bound algorithm, which is the most popular general algorithm for linear integer programming. For problems that cannot be formulated as linear integer programming problems, algorithms that deal with nonlinear integer programming problems must be used. Such algorithms have not reached the level of efficiency and reliability achieved for linear problems. Furthermore, the choice of algorithm is highly problem dependent. We 169

170

INTEGER PROGRAMMINQ

chose to concentrate on a single class of algorithms, genetic algorithms, with which we have had extensive and good experience in the design of composite laminates. Additionally, genetic algorithms are fairly easy to implement and to tailor to the particular laminate design problem at hand. Section 5.4 provides an introduction to genetic algorithms and their application to composite laminate design.

5.1

INTEGER LINEAR PROGRAMMING

An optimization problem is called a linear programming (LP) problem if the objective function and the constraint functions are linear functions of the design variables. The standard form of an LP problem is minimize f(x)

= cTx

ti.1

171

INTEGER LINEAR PROGRAMMING

where Cis greater than the absolute value of the lowest possible value of the unrestricted variable. More general LP problems allow design variables of unrestricted signs and general inequalities. However, frequently the values of the design variables are restricted not to exceed or drop below values that ure dictated by either availability of the resources or physical limitations. Such upper and lower bounds on the design variables are often treated differently than constraints that involve more than one design variable. The bounds, therefore, are often written separately. The general LP problem is minimize f(x)

= cTx

< such that Ax ~ b,

such that Ax= b, x::;xu ,

x;:::o,

(5.1.1)

where c is an n x I vector of constant coefficients, A is an m x n matrix of constraint coefficients, and b is an m x I vector of constants. Note that the signs of the design variables are restricted to being positive in the standard form. Some commercially available LP codes are also restricted to handle positive-valued design variables. If a variable is to have an unrestricted sign, then the standard procedure is to replace it with the difference of two positive variables. For example, an unrestricted design variable X; is replaced by two new variables as x.1 =x+-x-:I I

where both x7 and xj are limited to being positive, but depending on relative magnitudes of the two, the outcome may yield a positive or a negative value for the variable X;. A disadvantage of the approach just described is the doubling of the number of design variables. If the designer has a good idea of the lowest possible value of the variable that can assume negative values, then a more practical approach is to define the new variable by adding a positive constant C to the variable with an unrestricted sign,

X;:::: XL,

where xu and xL are the vectors of upper and lower bounds for the design variables. This more general form of LP can be converted to the standard form by a change of variables. This standard form is then solved by a variety of solution techniques, most notably the Simplex method. The reader is referred to, for example, Arora ( 1989) for more information on solution of LP problems. In the case of LP problems encountered in laminate design, some or all of the variables of a LP problem are restricted to take only integer values. If all the variables are so restricted, the problem is called an integer linear programming (ILP) problem or pure ILP problem. If some of the design variables are allowed to be continuous, the problem is referred to as mixed integer linear programming (MILP) problem, and its standard form is minimize f(x)

=cJx + ciy

such that A,x + A 2y = b, X; ;::::

x;os =x; + C,

(5.1.2)

0 integer,

YJ;:::: 0.

(5.1.3)

172

INTEGER PROGRAMMING

It is also common to have problems where design variables are used to indicate a yes-or-no type decision-making situation. Such problems are referred to as zero-one or binary ILP problems. For example, as we will demonstrate in Chapter 8, in a laminate stacking sequence design, the presence of a layer with a specified fiber orientation its absence) can be represented by a binary variable. Many pwe,uuu" for the solution of ILP problems permit the user to specify that of the variables are binary. Most of the remaining discussion in this chapter assumes to be pure ILP.

5.2 IN-PLANE STIFFNESS DESIGN AS A LINEAR INTEGER PROGRAMMING PROBLEM Linear programming (LP) problems have been important in management, economics, and engineering for many years. As a result, efficient software for solving LP problems is readily available and, therefore, there is a substantial incentive to formulate laminate design problems as LP problems. A quick review of the in-plane stiffness problem reveals that certain in-plane laminate properties are indeed linear functions of the design variables. For example, from Table 2.1 and Eq: (2.4.35) we see that the elements of the in-plane stiffness matrix A, A 11

= h(U1 + U2 v; + U 3v;),

A 22 = h( U 1 - U2 v;

A 12 = h(U4 - U3v;),

+ U3 v;), A 66 = h( U5 - U3 v;),

are all linear functions of the in-plane lamination parameters V~ and v;. Furthermore, it is demonstrated in the following that for laminates with prescribed fiber orientations, it is possible to express the in-plane lamination parameters as linear functions of the number of layers of a given orientation. Restricting ourselves to balanced symmetric laminates, we rewrite Eq. (4.2.4) by expanding the summation over each layer in one-half of the laminate: V~ = 2(vl COS 29 1 + V2 COS 292 +

V 3 COS

29 3 +

· · · + VN/2 COS

29N12 ),

173

5.2 IN· PLANE STIFFNESS DESIGN AS A LINEAR INTEGER PROGRAMMING PROBLEM

v; =2(Vl COS 491 +

V 2 COS

48 2 +

V3 COS

49 3 +

· · · + VN/ 2 COS

49N12 ). (5.2.2)

In this representation, the volume fractions of individual layers I';• i = 1, ... , N12, are equal to the ratio of the thickness of a layer, t, to total laminate thickness, h. Therefore, in the common case when all layers have the same thickness t,

v; =~(COS 29 1 +COS 292 +COS 29 3 +

· · ·+COS

29N12 ),

v; =~(COS 49 1 +COS 492 +COS 49 3 +

· · ·+COS

49N12 ).

(5.2.3)

If several of the individual layers are made up of the same orientation angles, say nP layers with SP and nq layers with Sq, then Eq. (5.2.3) takes the following form:

v~ = ~ (np cos 291' + nq cos 28q),

v; = ~ (np cos 491' + nq cos 48q).

(5.2.4)

For a constant thickness laminate with specified values of the fiber orientations 91' and 9q, the cosine terms are constants and, therefore, the in-plane stiffness parameters are linear functions of the number of layers nP and nq. Hence, the elements of the in-plane stiffness matrix A are also linear functions of the number of layers. When the laminate thickness is not constant, the lamination parameters are not linear functions of the number of layers, but the in-plane stiffness coefficients are still linear, because the h in the denominators in Eq. (5.2.4) cancels the h in Eq. (5.2.1). For example, consider a balanced symmetric laminate which contains only oo, ±45°, and 90° layers. The in-plane lamination parameters are given by

v~ =~(no -nn)

and

v; =h2t (no- 2nf+ nn),

(5.2.5)

:Ill

174

INTEGER PROGRAMMING

where no is the number of oo layers, nn is the number of 90° layers, and n1 is the number of ±45° layer groups in half of the laminate. The thickness of the laminate may be written as h = 2t(no + 2nf+ nn). Substituting this expression into Eq. (5.2.5) we see that the are no longer linear in the numbers stiffness parameters V~ and layers of distinct orientation angles. However, the coefficients of stiffness matrix are still linear functions of the number of layers. stituting the in-plane stiffness parameters into Eq. (5.2.1) yields following expressions for stiffness coefficients:

v;

A 11 = 2t[(no + 2nr+ nn) U1 + (U2 + U3) no- 2U3nr+ (U3 - U2 )nn],

A 22 = 2t[(no + 2n1 + nn) U1 + (U3 - U2) no- 2U3nr+ (U3 + U2 )nn], A 12 = 2t[(no + 2n1 + nn) U4 - Uino- 2n1 + nn)],

A66 = 2t[(no + 2n1 + nn) U5 - U3 (no- 2n1 + nn)].

The weight of the laminate is proportional to the total laminate thickness h which we have seen is a linear function of the number layers. Thus, we can formulate as LP any in-plane design problem that involves the weight and linear expressions in the elements of the A matrix. The effective engineering elastic constants are not linear functions of the numbers of layers, but two of them, the laminate shear stiffness Gxv and Poisson's ratio vxy' can be used to formulate linear constraints. For example, if we require that the laminate has a Poisson's ratio larger than a certain value vxy ~ v~Y' using Eqs. (2.4.42) and (5.2.7) we can impose the following linear constraint: (n 0 + 2n1 + nn)U4 - Uina- 2n1 + nn) ~ v~v [(n 0 + 2n1 + nn) U1

+ (U3 - U2)n

0

-

2U3nr+ (U3 + U2)nn].

This equation may be rearranged in the standard form of an inequality constraint as

5.2 IN·PLANE STIFFNESS DESIGN AS A LINEAR INTEGER PROGRAMMING PROBLEM

-1 U4 -

175

U3 - v~v (U 1 + U3 - U2 )]no- [2U4 + 2U3 - v~v (2U1 - 2U3 )]n1 -[U4

-

U 3 - v~v (U1 + U3 + U2)] ~ o.

(5.2.9)

In-plane laminate design problems can usually be classified as weight minimization or stiffness maximization. In the weight minimit.ation problem, we minimize the weight of the laminate, which is equivalent to minimizing the thickness when the laminate is made from a single material. Constraints are placed on the stiffnesses and occasionally on the maximum or minimum percentages of layers of given orientation. In the stiffness maximization problem, the objective function is one of the stiffness properties, while the weight, or the thickness, or the total number of layers is specified. Additional stiffness and layer percentage constraints may also be present. The following example demonstrates a stiffness maximization problem. The weight minimization problem is left as an exercise (see Exercise l ).

Example 5.2.1 For a 16-layer balanced symmetric laminate, formulate the problem of determining the number of 0°, ±45°, and 90° layers to maximize the stiffness coefficient A 11 while keeping the shear stiffness Gxy of the laminate larger than 12 GPa and A 22 larger than 20 MN/m. The properties of individual layers are E 1 =76 GPa, £ 2 =5.5 GPa, G 12 =2.3 GPa, v 12 =0.34, and t=0.125x 10-3 m. Using the material properties specified, we calculate the U;'s to be U 1 = 32.44 GPa,

U2 = 35.55 GPa,

U4 = 10.54 GPa,

and

U3 = 8.652 GPa,

U5 = 10.95 GPa.

For this problem, the laminate has constant thickness of 2t(n0 + 2n1 + nn) = h = 2 X 10-3 m, and so from Eq. (5.2.7) the coefficients of the stiffness matrix A are

176

INTEGER PROGRAMMINQ .

A 11

= 64.88 + 11.05 no- 4.326 n1 -

6.724 nn,

A 22 = 64.88- 6.724 no- 4.326 n1 + 11.05 nn, A 66 = 21.90- 2.163 n0 + 4.326 n1 - 2.163 nn,

all in MN/m. Therefore, the shear stiffness is Gxy = 10.95- 1.082 n0

+ 2.163 n1 - 1.082 nn GPa.

The optimization problem is then formulated as maximize f(no, n1, nn) subject to

= 11.05 n

0

-

4.326 n1 - 6. 724 nn

6.724 no+ 4.326n,- 11.05 nn $44.88, -1.082 no + 2.163 n,- 1.082 nn 2:: 1.050,

!13 SOLUTION OF INTEQER LINEAR PROGRAMMING PROBLEMS

177

5.3 SOLUTION OF INTEGER LINEAR PROGRAMMING PROBLEMS A common approach to solving MILP problems is to round off the optimum values of the variables, obtained by assuming them to be continuous, to the nearest acceptable integer value. For problems with 11 integer design variables there are 2n possible rounded-off designs, and the problem of choosing the best one is formidable for large n. For example, for a problem with 10 design variables there are 1024 possibilities. Increasing the number of design variables to 20 would increase the number of possibilities to more than 1,000,000. Furthermore, for some problems the optimum design may not even be one of these rounded-off designs, while for others, none of the rounded-off designs may be feasible. Thus, instead of using continuous optimization methods followed by rounding, it may be more reasonable to use methods specifically developed for MILP problems.

no + 2 nf + nn = 8,

5.3.1

n0 , nn E 0, ] , 2, ... , 8,

For problems with a small number of design variables, enumeration of all possible solutions is a reasonable alternative to rounding. For many laminate design problems, the cost of an analysis is low enough so that enumeration is a reasonable procedure, even if the number of possible designs is in the thousands.

n1 E 0, 1,2,3,4. Note that the constant term in the A 11 is ignored for the objective function because most linear programming codes do not allow constants in their input. Adding a constant to the objective function or multiplying it with a positive constant does not change the optimal values of the design variables. However, these operations change the value of the objective function at the optimum, and that change must be reversed in order to obtain the optimum of the original problem. Also note that the constraint equation (a) is added to the problem to insure that the total number of layers in the laminate adds up to 16. Based on this equation, the design variable n0 and nn can take any integer value between 1 and 8, but the largest integer number n1 can assume is 4, which corresponds to all ±45° laminates. The solution of the problem is no= 2, n1= 3, nn = 0 which corresponds to a [Oz1±453 ], laminate.

Enumeration

Example 5.3.1 Solve Example 5.2.1 by enumerating all possible 16-ply laminates. The possible laminates are [0no 1±45nr190nn ]5 , with the condition of having exactly 16 layers given as no+ 2nf+ nn = 8. It is convenient to enumerate on the basis of the number nf of

±45° pairs, as we can have only five possible values (0,1,2,3,4). Then for each value of n1 , we can vary no from zero to 8 - 2n1 , and use what is left for nn. The results are summarized in Table 5.1.

178 Table 5.1.

t\.3 SOLUTION OF INTEGER LINEAR PROGRAMMING PROBLEMS

Enumeration of all possible 16 balanced symmetric laminates

Laminate [90sls [0/907L [Oz/906 ], [0/9051, [Oi904], [Osf903L [OJ902], [07/901, [Osls [±45/906], [±45/0/9051 [±45/02/9041, [±45/0/903], [±45/0i902L [±45/0sf90L [±45/06], [±45zi904L [±45z101903L [±45z102/902L [±45 2/0/901 [±45 2/04], [±453/90zl, [±45/0/90L [±45/021 [±454),.

A 11 (MN/m) 11.09 28.86 46.64 64.41 82.18 99.96 117.73 135.51 153.28 20.21 37.98 55.76 73.53 91.31 109.08 126.85 29.33 47.11 64.88 82.65 I 00.43 38.45 56.23 74.00 47.58

A 22 (MN/m) 153.28 135.51 117.73 99.96 82.18 64.41 46.64 28.86 11.09 126.85 109.08 91.31 73.53 55.76 37.98 20.21 100.43 82.65 64.88 47.11 29.33 74.00 56.23 38.45 47.58

Gxy

(GPa)

2.29 2.29 2.29 2.29 2.29 2.29 2.29 2.29 2.29 6.62 6.62 6.62 6.62 6.62 6.62 6.62 10.95 10.95 10.95 10.95 10.95 15.27 F 15.27 F 15.28 F 19.60F

179

pic from Garfinkel and Nemhauser ( 1972) demonstrates the use of a constraint in an enumeration tree. The example also introduces some of the terminology that will be useful for the most important algorithm for ILP discussed later in this chapter, the branch-and-bound technique. Example 5.3.2 Consider the binary ILP problem of choosing a combination of five variables such that the following summation is satisfied: h=

I

ixi=5.

i=l

A decision tree representing the progression of the solution of this problem is composed of nodes and branches that represent the solutions and the combinations of variables that lead to those solutions, respectively (Fig. 5.1). The top node of the tree (leftmost in Fig. 5.1) corresponds to a solution for which all the variables are turned off (xi = 0, i = 1, ... , 5) with a function value of h = 0. Branching from this solution are two paths corresponding to the two alternatives for the first variable. The branch with

Only four of the laminates satisfy the constraints that A 22 be at least 20 MN/m and that Gxy be at least 12 GPa, and these are marked by the letter "F" (for feasible design). Among these the [±45/02], design has the highest A 11 • As can be seen from this example, we have used the condition that the total number of layers is 16 to reduce the number of combinations of no, n1 , andnn that we needed to check. A systematic way of taking advantage of constraints to reduce the number of cases that need to be inspected by enumeration is called an enumeration tree. An exam-

Figure 5.1. Enumeration tree for binary ILP problem with constraint h= r.t1 iXi= 5.

180

INTEGER PROGRAMMINCJ

x 1 = 1 has h = I and tolerates turning additional variables on without running the risk of exceeding the required function value of 5. The other branch is the same as the initial solution and can be branched further. Next, these two nodes are branched by considering the on and off alternatives for the second variable. The node arrived at by taking x 1 = x 2 = 1 has h = 3 and is terminated because further branching would mean adding a number that would cause h to exceed its required value of 5. Such a vertex is said to be fathomed, and is indicated by a vertical line. The other three vertices are said to be live and can be branched further by considering the alternatives for the remaining variables in a sequential manner until either the created nodes are fathomed or the branches arrive at feasible solutions to the problem. For the present example, after considering 19 possible combinations of variables, we identified three feasible solutions, which are marked by an asterisk. This is a 40% reduction in the total number of possible trials, namely 25 = 32, needed to identify all possible solutions. For a laminate design problem in which trials with different combinations of variables could require lengthy analyses, an enumeration tree can yield substantial savings.

5.3.2

Branch-and-Bound Algorithm

The concept behind the enumeration technique forms the basis for this powerful algorithm suitable for MILP problems (Lawler and Wood, 1966; Tomlin, 1970). The original algorithm, developed by Land and Doig (1960), relies on calculating upper and lower bounds on the objective function, so that nodes that result in designs with objective functions outside the bounds can be fathomed, and thereby, the number of analyses required can be reduced. Consider a hypothetical example of a MILP problem of the type of Eq. (5.1.3). The first step of the algorithm is to solve the LP problem obtained from the MILP problem by assuming the variables to be continuous valued. If all the x variables for the resulting solution have integer values, there is no need to continue; the problem is solved. Suppose several of the variables assume noninteger values and the objective function value is f 1• The f 1 value will form a lower bound JL =f 1for the MILP, since imposing conditions that require noninteger-valued variables to take integer values can only cause the objective function to increase. This initial prob-

181

!1.3 SOLUTION OF INTEGER LINEAR PROGRAMMING PROBLEMS xk+l S:

3

, ,,

, "' "'

' ' '' '

Figure 5.2.

'

Illustration of branch-and-bound decision tree for MILP problems.

Iem is labeled as LP-1 and is placed in the top node of the enumeration tree as shown in Fig. 5.2. For the purpose of illustration, it is assumed that only two variables, xk and xk+l' violate the integer requirement with xk

=x! = 4.3 and xk+I =Xk+I = 2.8.

The second step of the algorithm is to branch from the node into two new LP problems by adding a new constraint to the LP-1 that involves only one of the noninteger variables, say xk. One of the problems, LP-2, will require the value of the branched variable xk to be less than or equal to the largest integer smaller than x!, and the other, LP-3, will have a constraint that xk is larger than the smallest integer larger than x!. These two problems actually do branch the feasible design space of the LP-1 into two segments. There are several possibilities for the solutions of these two new problems. One of these possibilities is to have no feasible solution for the new problem. In that case the new node will be fathomed. Another possibility is to reach an all-integer feasible solution (see LP-3 of Fig. 5.2). In this case, the node will again be fathomed, but the value of the objective function will provide an upper bound for the optimum solution because it is feasible. We denote that upper bound as fu and update it when we find other integer solutions with lower values than the current fu· Once an upper bound has been established, any node that has an LP solution with a larger value of the objective function will be fathomed, and only those solutions that have the potential of producing an objective function between JL andfu will be pursued. If there are no solutions with an

182

objective function smaller thanfu, then the node with}~ is an optimum solution. If there are other solutions with an objective function than fu, they may still include noninteger-valued variables (LP-2 Fig. 5.2), and are labeled as live nodes. Live nodes are then again by considering one of the remaining noninteger values, and suiting solutions are analyzed until all the nodes are fathomed. The performance of the branch-and-bound algorithm depends ily on the choice of the noninteger variable to be used for and the selection of the node to be branched. If a selected node branching variable leads to an upper bound close to the objective tion of the LP-1 early in the enumeration scheme, then computational savings can be obtained because of the elimination branches that cannot generate solutions lower than the upper bound A rule of thumb for choosing the noninteger variable to be is to take the variable with the largest fraction. For the selection the node to be branched, we choose, among all the live nodes, the problem that has the smallest value of the objective function; node is most likely to generate a feasible design with a tighter bound.

Example 5.3.3 Solve Example 5.2.1 by the branch-and-bound method, using software for solving continuous linear programming problems. The decision tree for this problem is shown in Fig. 5.3. The first node shows the solution with a noninteger number of layers [0 3 _5 /±45 2 _24 ],, with All= 94 GPa. With no having the larger fraction, we branch on n0 • The branch with the additional constraint of no~ 4 does not have a solution because the constraint on Gxy cannot be satisfied. The branch with an additional constraint of noS: 3 yields the solution [0/±45 2 _5 ]. with A11 = 87.25 GPa. Now that no and nn are integers, we can branch only on nf' As shown in Fig. 5.3, the branch with n1 S: 2 does not have a solution because again we cannot satisfy the constraint on Gxy' and the branch with n1 ~ 3 yields [0/±45 3], with A11 = 74.0 GPa. This is an integer solution, and since we do not have any more live nodes, it is the optimum solution.

b.3 SOLUTION OF INTEGER LINEAR F'ROGRAMMING PROBLEMS

183

An= 94.00

no solution

Figure 5.3.

Branch-and-bound decision tree for laminate design problem.

Branch-and-bound is only one of the algorithms for the solution of ILP or MILP problems. However, because of its simplicity, it is incorporated into many commercially available computer programs (see, for example, Johnson and Powell, 1978 and Schrage, 1989). There are a number of other techniques that are capable of handling general discrete-valued problems (see, for example, Kovacs, 1980). Some of these algorithms are good not only for ILP problems but also for nonlinear programming problems with integer variables. Linear problems have a big advantage over nonlinear ones in that we can count on getting the global optimum for the problem. Algorithms for nonlinear problems, on the other hand, typically provide only a local optimum. A methodical way of dealing with multiple minima for discrete optimization problems is to use random search techniques that sample the design space for a global minimum without the use of derivatives. However, the efficiency of traditional random search techniques is poor for problems that have a large number of variables. Two algorithms, simulated annealing (Laarhoven, 1987) and genetic algorithms (Goldberg, 1989), have emerged more recently as tools ideally suited for optimization problems where a global minimum is sought. In addition to being able to locate near-global solutions, these two algorithms are also powerful tools for problems with discrete-valued design variables. Both algorithms mimic naturally observed phenomena, and their implementation calls for the use of a random selection process guided by probabilistic decisions. The next

184

INTEGER PROGRAMMING

section describes genetic algorithms, in particular some that have re· cently been applied to the design of composite laminates.

5.4 GENETIC ALGORITHMS Genetic algorithms use techniques derived from biology and rely on the application of Darwin's principle of survival of the fittest. a population of biological creatures is allowed to evolve over gene~" tions, individual characteristics that are useful for survival tend to passed on to future generations, because individuals carrying them more chances to breed. In biological populations these are stored in chromosomal strings. The mechanics of natural is based on operations that result in structured yet randomized change of genetic information (hence biological traits) between chromosomal strings of the reproducing parents and consists of duction, crossover, occasional mutation, and inversion of the somal strings. Genetic algorithms, developed by Holland (1975), mimic the mechanics of natural genetics for artificial systems based on operations that are the counterparts of the natural ones (even called by the same names). As will be described in the following paragraphs, these operations involve simple, easy-to-program, random exchanges of location of numbers in strings that represent the design variables. These operations may appear like a completely random search of extremum in parameter space based on function values only. However, genetic algorithms have been experimentally proved to be robust, and the reader is referred to Goldberg (1989) for discussion of their theoretical properties. Here we discuss the genetic representation of a minimization problem and focus on the mechanics of three commonly used genetic operations (namely, reproduction, crossover, and mutation) and a fourth operator, permutation (sometimes called gene swap), recently proposed for buckling optimization of composite plates. Unlike many search algorithms that move from one point to another in the design variable space, genetic algorithms work with a population of strings. This aspect of the genetic algorithms is responsible for increased chances of obtaining global or near-global optima. By keeping many solution points that may have the potential of being close to minima (local or global) in the pool during the search process, rather than converging on a single point early in the process, we reduce

185

!1.4 GENETIC ALGORITHMS

the risk of converging to a local minimum. Working on a population of designs also suggests the possibility of implementation on parallel computers. However, the concept of parallelism is even more basic to genetic algorithms in that evolutionary selection can improve in parallel many different characteristics of the design. Also, since the outcome of the search is random, repeated genetic optimization will often yield different good designs. This aspect can be very useful to the designer when the design space includes many local optima of comparable performance. Genetic algorithms have been applied to optimal structural design only recently. The first application of the algorithm to a structural design was presented by Goldberg and Samtani (1986), who solved a well known 10-bar truss weight minimization problem. More recently, Hajela (1990) used genetic search for several structural design problems for which the design space is known to be either nonconvex or disjoint. Rao, Pan, and Venkayya (1991) addressed the optimal selection of discrete actuator locations in actively controlled structures via genetic algorithms. Applications of the algorithm to design with composite materials are recent (see Callahan and Weeks 1992; Le Riche and Haftka, 1993; Hajela and Erisman, 1993; Nagendra et al., 1993; and Kogiso et al., 1994 ).

5.4.1

Design Coding

Application of the operators of the genetic algorithm to a search problem first requires the representation of the possible combinations of the variables in terms of bit strings that are counterparts of the chromosomes in biological genetics. For historical reasons, binary numbers were predominantly used for representing design information as bit strings. For integer design variables, this is quite simple. For example, consider the minimization problem minimize f(x),

x = {xp x 2 , x 3 , x4 },

(5.4.1)

where the variables are integer and can assume values in the range {15?x1,x4 ?0}, {7?x2 ?0}, and {3?x3 ?0}. The length of the bit string used to represent each variable is dictated in this case by the range of the variables. For example, the design variable vector can be a binary string represented as

186

INTEGER PROGRAMMING

0 I I 0 X1

--- - ---I0I

I I

X2

X3

I 0 1I , X4

(5.4.2)

where binary representations of the individual variables are strung head-to-tail, and, in this example, the variables are x 1 = 6, -X2 = 5, x3 = 3, x4 = 11. For problems where the design variables are continuous within a range :i: ::;; x. ::;; xu, the common practice is to divide the intervals into a finite number of subintervals so as to discretize the design variables. The size of the subintervals, which determines the of the solution obtained by the genetic algorithm, defines the number of bits needed. If a variable is defined in a range {.xf::;; xi::;; xf} the precision needed for the final value is _xincr, then the number m of, binary digits needed for an appropriate representation can be calculated from '

I

I

2m~

((xf- .xf)/xincr + 1).

For example, if {0.01 $xi::;; 1.81} and the required precision is 0.001, the smallest number of digits is m = 11, which actually produces increments of 0.00087 in the value of the variable, instead of the required value of 0.00 1. The conversion of variables to binary numbers introduces a degree of complexity that is not always necessary. Binary variables correspond to the smallest possible alphabet, which has some theoretical advantages. For example, with a very large alphabet it is more difficult to ensure that every possible gene is present in the initial population. However, studies comparing binary to other representations tend to find superior performance by nonbinary representations (e.g., Michalewicz 1992, Furuya and Haftka 1993). In many problems we can use the actual values of the design variables in the string, as long as we take precautions with the crossover and mutation operators, as will be described later. The use of real variables in strings has enjoyed more popularity in Europe than in the United States (see, for example, Back and Schwefel, 1993). The European school often uses the term "evolutionary strategies" to describe methods, such as genetic algorithms, based on mimicking evolution. Variants of genetic algorithms that employ real numbers and are applied to function optimization have been dubbed evolutionary algorithms by the European school. For designing the stacking sequence of a laminate, we typically employ a code assigning an integer digit to each possible orientation

5.4 GENETIC ALGORITHMS

187

angle. For example, the stacking sequence of a laminate that can only admit oo, 45°, -45°, and 90° layers with fixed thickness t can be represented by a string containing the digits 1, 2, 3, and 4, respectively. The 16-ply laminate [±45/0i902L is then represented by the genetic code [2 3 1 1 1 1 4 4 4 4 1 1 1 1 3 2]. If we make use of the symmetry condition (that is, limit ourselves to only symmetric laminates) we can simply represent the same laminate by [2 3 1 1 1 1 4 4]. This reduction in the length of the string can significantly improve the speed of the genetic search. There are 4 16 = 4.3 x 109 16-digit strings that correspond to distinct designs, but only 4 8 = 65,536 strings of 8 digits. For most applications, laminates are further assumed to be balanced, requiring a -45° layer for every 45° layer. The plus or minus pair of off-axis angles are normally placed adjacent to one another in order to keep the values of the D 16 and D 26 terms small. This allows us to reduce further the length of the string for balanced symmetric laminates. We define stack variables, which represent two adjacent layers with the same orientation. For a laminate made of only o;, ±45;, and 9~ layers we use the 1, 2, and 3 representation. Therefore, the [±45/0i902L laminate is represented by [2 1 1 3]. This reduction in the length of the string, however, does not permit us to represent laminates containing stacks with an odd number of 0° or 90o layers. It is also possible to use natural coding, where the gene looks like the stacking sequence. For example, the stacking sequence [±45/0i90 2L may be represented as [45 0 0 90]. This natural representation reduces the effort of coding and decoding, but it requires more specialized programming tailored to the specific set of orientations used. For in-plane laminate design problems, the stacking sequence is often unimportant, and we need only the number of layers in each given orientation. Then we can represent the symmetric and balanced laminate [(±8L)NI(±8L_1)N I ... /(±8 1)NL as a string with the L integers Ni, i = 1, . L, or' with a longer string containing the binary representation of these integers. For example, the in-plane design of a 48-ply balanced and symmetric laminate with ±30°, ±45°, and ±60° stacks can be represented by a string [N" N 2 , N 3 ]. The integers N" N 2, and N 3 , represent the number of stacks of ±30°, ±45°, and ±60° layers, respectively, and can vary between 0 and 12. Of course, we will need a constraint requiring that N 1 + N 2 + N 3 = 12. Instead we can I..

•

,

i1

~1

188

INTEGER PROGRAMMING

have a string composed only of [NpN2], with N3 = 12-N 1 -N2, with a constraint that N 3 ~ 0. This second representation is much more ef"' ficient. The design space associated with [Nl' N2 , N 3] has 13 3 = 2197 possible designs, while the design space associated with [Nl' N2 ] only 169 possible designs. It is easy to check that, because of constraint on N 3 , there are only 91 possible designs. So with the representation we have a much larger design space, and the great majority of the designs generated by the genetic on the way to the optimum will violate the constrai N1 + N 2 + N 3 = 12, resulting in sl'ower progress towards the optimum• With the second representation, we have a smaller design space, more than half of the designs generated by the genetic algorithm be possible designs (that is, satisfy the constraint N 3 ~ 0). Genetic algorithms generally work with a population of strings are all of the same length. This can present a problem when the ber of layers in the laminate is unknown, as in the case of thic11..m:::s:s minimization, and we use a coding with one number for each or each two-layer stack. We can solve this problem by making length of our strings correspond to an upper limit on the thickness the laminate and adding another digit to represent an absent layer stack. For example, assume that we found that all the constraints be satisfied with a 12-ply laminate, but we seek thinner laminates satisfy these constraints. Limiting ourselves to symmetric lamina this means that we can use 6-digit strings to code the laminate. will then represent the laminate [0 5 ), by [0 1 1 1 1 1], with the zero in the leftmost position corresponding to the absent layer. Similarly, we will represent [0/90/45/-45], by [0 0 1 4 2 3]. As will be seen later, with this approach we need to take the precaution of not having O's appear anywhere except for the leftmost positions in the string.

5.4.2


189

smaller population size than the design of a thicker laminate that must he represented by a longer string. Similarly, the design of a structure composed of several laminates will require an even larger population size. For example, for the design of 48-ply balanced symmetric laminates ( 12-digit string) subject to buckling and strength constraints, Le Riche and Haftka (1993) found that the optimum population size was H. For the design of a simple wing made up of 18 different laminates, the best population size was about 50 (Le Riche, 1994). The initial population is typically generated at random. Subroutines or functions that generate random numbers distributed uniformly be(ween 0 and 1 are widely available, and we use them extensively in genetic algorithms. Such random number generators typically require an input "seed" the first time they are called. The seed determines the sequence of random numbers that will be generated by repeatedly calling the function or subroutine. Thus the numbers are not truly random, and are often called pseudorandom numbers. However, for use in genetic algorithms the distinction is not important. In the following we will assume that random numbers are generated by a function RAND(ISEED) and we will denote by r the output of the function. For example, consider the case of the stacking sequence design of a laminate made of 0°, 45°, -45°, and 90° layers, so that each digit in the string can take integer values in the range (0,4), with zero corresponding to an absent layer. Using RAND, we first generate a random number r between 0 and 1.0, and then transform it to a choice of an integer between zero and four by taking the value LsrJ. where LxJ indicate the integer part of a real number x (floor function). We can then repeat this process for each layer to generate the entire laminate. Then we repeat it again to generate the requisite n5 laminates (see Fig. 5.4). Subroutine init of Fig. 5.4 will produce zeroes or "voids" inside the laminate, and these voids will have to be pushed out, for example,

Initial Population 1 0 4 2 0 3 => 0 0 1 4 2 3 => [0/90/45/-45],.

Most genetic algorithms work with a fixed-size population, so that the size of the initial population of designs determines the size of the population in all future generations as well. We will denote that size by ns, for the number of strings. Choosing the population size is at the present a matter of trial and error. In general, the optimal size of the population increases with problem size. That is, the design of a thin laminate that can be represented by a short string will require a

(5.4.4)

We can use information on the design to bias the initial population makeup. One case where we need such bias is when we encounter a singular optimum associated with 0° layers. A singular optimum is an isolated optimum that represents a discontinuity in design space. We can get a singular optimum in the design of composite laminates when the objective function favors designs with ±45° or 90° layers, but the

1"'11

190

INTEGER PROGRAMMING

subroutine init(lamin,L,ns,nmax) integer lamin(L,ns) do 100 i= 1,ns do 100 j=1,L lamin(j,i)=(nmax+ 1)*rand(iseed) 100 continue return end Figure 5.4. Fortran subroutine for generating initial population of ns strings of length L, with each digit being an integer between 0 and nmax.

loading is primarily in the 0° direction. This phenomenon was reported by Schmit and Farshi (1973) for continuous layer thickness design variables. However, a similar problem with a singular optimum occurs with discrete thickness designs, as explained in the following. Consider, for example, a 48-ply laminate under uniaxial tension, where we want to maximize Gxy subject to constraints on strains in the layers. The optimum design will usually be an all ±45° laminate. If we start with a design with a small number of 0° layers, such as (±45 8/0 4 )s, the strains in the zero fibers will be higher than the strains in the 45° or the -45° fibers (see Exercise 2 at the end of this chapter). If the load is high enough, the laminate will first fail due to breakage of the 0° fibers. If we reduce the number of 0° layers by two to (±45 9 /0 2 )s, the situation will worsen as the strains in the oo layers will increase. However, if we eliminate the last oo layer and get an all ±45° laminate, the strains in the zero direction will not matter because we will not have any fibers in this direction. The all ±45° optimum is called singular because it cannot be obtained by gradual reduction of the oo layers. When continuous optimization techniques are used, this difficulty has to be countered by considering designs that have oo layers and also designs that do not have 0° layers. With genetic algorithms we have to use a similar strategy, seeding the initial population with some designs without any oo layers. Kogiso et al. (1994) have shown that such seeding helps in getting optimal no-zero-layer designs and does not appear to have adverse effects even when the optimum is not singular.

191


Once an initial population is generated, we apply a series of genetic operators that produce a new generation and replace the initial population with the new one. The first of these operations is selection.

5.4.3

Selection and Fitness

The selection process in genetic algorithms mimics biology in giving more fit designs a higher chance to breed and pass their genes to future generations. Genetic algorithms therefore need to define fitness and use it in a procedure that selects pairs of parent designs that will be used to create child designs for future generations. For unconstrained problems, the fitness of a design can be defined as the objective function (for maximization problems) or a constant minus the objective (for minimization problems). For constrained problems the fitness also must consider constraint violations or constraint margins. Several methods exist for handling constraints, and this is a hotly debated topic (e.g., Le Riche, Knopf-Lenoir, and Haftka, 1995). The following is one simple way for handling constraints. Consider the standard formulation of an optimization problem, Eq. ( 1.4.1 ), where for simplicity we omit the equality constraints: minimize f(x),

such that g/x)

x ~

E

X

0, j

= 1, ... , ng.

We assume that the constraints are normalized so that a constraint value of -0.1 corresponds to a 10% margin, while a constraint value of 0.1 corresponds to a 10% deficiency. The design margin of safety is then defined by the most critical constraint gmax= max/g). If gmax is positive the design is infeasible, and we want to add a penalty to the objective function to help the search move into the feasible design. If gmax is negative we have a feasible design with a positive margin of safety, and we want to reduce the objective function as a bonus for that margin. We can define an augmented objective function j* as

J = fJ + p gmax + D, v-Eigma),

if gmax> 0, otherwise

(5.4.5)

r 1111

1 ' 11!1

192

INTEGER PROGRAMMING

In Eq. (5.4.5) there are two parameters, 3 and p that penalize infeasible design. The first, 3, should be chosen as a small infeasibility penalty that applies even for very small constraint violations. It should typically be a very small percentage of the average value of the objective function. The second parameter, p, penalizes substantial constraint violations. The penalty parameter p needs to be chosen large enough so that all infeasible designs have a higher value of f* than the optimum design. However, we do not want to use a very large penalty parameter because the design process often benefits from shortcuts taken via infeasible design. For example, consider a situation ' where we require the laminate to be balanced, but we have a coding that permits unbalanced designs. Now suppose we have a design that needs to replace a pair of [0/90] layers with a stack of [±45] layers. We can have intermediate designs where only one layer is replaced. For example, a 0° layer is replaced by a 45° layer, resulting in an unbalanced laminate. If we penalize constraint violation too heavily, the unbalanced laminate will have such low fitness value that it will have very little chance of being selected for reproduction. So to achieve a transition from the pair of [0/90] layers to the [±45] stack, we must have two changes in the genetic string occur simultaneously. Such double change has very little probability of happening with the standard genetic operators described later, and having to rely on a double change will slow down convergence. This example illustrates a case where the easiest transition between two feasible designs proceeds through an intermediate infeasible design. This situation is not .· unique to balance constraints, but is quite common. Therefore, large values for penalty parameters are often found to slow down the convergence of the genetic algorithm. Occasionally, we have designs that are only slightly infeasible, but which have very good objective functions. Without the first penalty parameter, 3, we would have to use very large values of p to ensure that these designs did not have the best augmented objective, because the value of 8max for these designs is very small. This situation is the reason for using two penalty parameters instead of one. The bonus parameter £ is needed when the optimum design is not unique, so that we want to find the optimum design with the highest safety margin. When we design laminates for minimum thickness we often encounter such nonuniqueness because the thickness can be only a multiple of the basic layer thickness. Thus, we often can obtain

193

5.4 GENETIC ALGOFIITHMS

several laminates, all having the same thickness, but with different constraint margins. The bonus parameter ensures that the selection process favors the design with the highest margin of safety. However, the bonus parameter £ needs to be small enough so that no design has a lower j* than the optimum design, no matter how large the constraint margin is. Genetic algorithms traditionally use the concept of fitness, which is higher for the better individuals, rather than the concept of an objective function, which could be maximized or minimized. To accommodate the minimization problem, we can define the fitness, , from j* as (5.4.6)

=c-J*,

where c is a constant chosen large enough so that <1> is always positive, but not so large as to almost obliterate the relative differences between different designs. When large changes in the magnitude of f occur during the optimization, it may be desirable to periodically update c as well, so that these conditions will be met.

Example 5.4.1 Consider the following optimization problem. maximize f(x) such that g 1(x)

~

0

and

gz(x)

~

0.

A population of four designs is calculated to have the following objective function and constraint function values:

gi = -0.11, g1 = -0.50,

Design 1:

l

Design 2:

f2 = 7.30, gi = 0.28,

Design 3:

f 3 = 6.49,

gi = -0.71,

Design 4:

r=7.o2.

gi = 0.16, gi = -0.79.

=6.69,

g~ = 0.15, g~

= -0.51,

(a) Determine the largest value of the bonus parameter £ that may be used for this set of designs without causing the GA (ge-

194

INTEGER PROGRAMMING

netic algorithm) to pick an inferior design as the best design of the population. (b) If the bonus parameter is E = 0.2 and the penalty parameter o = 0.01, determine the smallest value of the penalty parameter p that can be used without causing the GA to pick an infeasible design as the best design of the population. For part (a) of this problem, we consider only designs that satisfy both constraints. Therefore, we will add the bonus only to the objective functions of Design 1 and Design 3. Note that this is a maximization problem so we add the bonus to the objective function [as opposed to the function of Eq. (5.4.5) from which we subtract the bonus]. The bonus is calculated on the basis of the most critical constraint. For Design 1, gl = 0. 11 is the most critical and for Design 3, g~ = 0.51 is the most critical. Adding a bonus to these constraints for the two designs will change the fitness function of the respective designs, <)>

1

= 6.69 + E 0.11

and

<)>

3

= 6.49 + E 0.51.

Clearly, Design 1 has a better objective function than Design 3. However, Design 3 has a constraint margin that is larger than that of Design 1, and adding too large a bonus may make the fitness <1> of the Design 3 better than that of Design 1. This is not an acceptable situation; the largest value of the bonus parameter should be limited such that the fitness <)> 3 is smaller than the fitness <)> 1• Therefore, E(0.51 - 0.11) < (6.69- 6.49)

:::::} E::; 0.5.

For part (b), we penalize those designs that have constraint violation, namely, Design 2 and Design 4. Typically, designs that violate one or more of the constraints have better objective function values than designs that satisfy all constraints, as is the case in this example. If the penalty parameter is not large enough, the fitness function of the design with a constraint violation may be better than the fitness of the best design which satisfies all the constraints. In this example, the fitness of the best feasible design with the specified bonus parameter of E = 0.2 is <)>

1

= 6.69+ 0.2. 0.11 = 6.71.

195


For the second design, the value of the penalty parameter that will cause <)> 3 to become larger than <)> 1 is 7.30-0.01- p 0.28 > 6.71 :::::} p < 2.07. For the fourth design, the value of the penalty parameter that will cause its fitness to become larger than <)> 1 is 7.02-0.01- p 0.16 > 6.71

:::::}

p < 1.88.

Therefore, the smallest penalty parameter that can be used without causing the GA to pick an infeasible design as the best design of the population is the larger of the two values, p = 2.07. If we use a penalty parameter of p = 1.88, the fitness of Design 2 becomes <)> 2 = 6.76, which is larger than the fitness of the best feasible design and, therefore, is unacceptable.

In the preceding example we picked the permissible values of the penalty and bonus parameters by inspecting four possible designs. Of course, other designs may exist that will require more stringent values for these parameters. In general, we need to try to estimate reasonable values for these parameters based on our knowledge of the problem. It pays to begin with relatively small penalty parameters. If these parameters are inappropriate we will find that the genetic algorithm will yield infeasible designs, and then we can increase the penalty parameters. Similarly, we may want to start with fairly substantial bonus parameters. If we find that some of the designs with high constraint margins produce fitnesses larger than fitnesses for designs that have a better value of actual objective function but smaller constraint margins, we can reduce the bonus parameter. Once we define the fitness of all the designs in the population, a common procedure for selecting parent designs is to simulate a biased roulette wheel, with each design being assigned to a sector of the roulette wheel with an area proportional to its fitness. That is, with ns designs having fitness values of i, i = 1, ... , ns, the ith designs gets a fraction ri of the wheel, where

196

INTEGER PROGRAMMING

r. l

Example 5.4.2

; n

=

197


'

I

j j=l

On the computer the roulette wheel can be implemented by generating a random number r between zero and one, and then selecting the ith design if R1_ 1 :::; r
(5.4.8)

where R 0 = 0.0, and

For a generation of six designs, the following fitnesses were calculated: {<1>1' <1> 2 , 3 , <1> 4 , 5 , <1> 6 } = {0.35, 0.60, 0.38, 0.65, 0.45, 0.15}. In order to select parents for crossover, six random numbers between 0.0 and 1.0, were generated: r = {0.2825, 0.7123, 0.1560, 0.9217, 0.6120, 0.3471}. Construct the roulette wheel and determine the individuals selected for the crossover operation. For the given list of fitnesses, the sum of all fitnesses yields n,

R; =

L rj'

i

= 1' ... ' ns.

(5.4.9)

:L

<1>j

= o.35 + o.6o + o.38 + o.65 + o.45 + o.15 = 2.58.

j=l

j=l

The procedure is coded as subroutine select in Fig. 5.5 and demonstrated in the following example.

Therefore, the fractions of the wheel occupied by the different designs are given by { rl' r 2 , r 3, r 4 , r 5, r 6 }

= {0.1357, 0.2326, 0.1473, 0.2519, 0.1744,0.0581 }. subroutine select(R,ns,parent) integer parent(2) dimension R(ns) do 400 i=1,2 place=rand(iseed) do 300 j=2,ns if (place .gt. RU-1)) go to 300 parent(i)=j-1 go to 400 300 continue parent(i)=ns 400 continue return end Figure 5.5. wheel.

In order to implement the selection process using the roulette wheel, we create another list of numbers following Eq. (5.4.9), {R0 , Rl' R2, R3, R4 , R5, R6 }

= {0.0, 0.1357, 0.3683, 0.5156, 0.7675, 0.9419, 1.0}, and the wheel for this generation of designs is shown in Fig. 5.6 with the integers on the slices indicating the location of the design in the initial fitness list (also note that the angles associated with the R;'s are obtained by multiplying them by 360°). The locations of the R 1, i = 1, ... , 6 are marked along the circumference of the wheel. For each value of the random number r, we chose a parent design using Eq. (5.4.8). That is, we place the random number around the circumference going counter clockwise from R0 = 0 and find the interval in which the random number falls. For the first random number r = 0.2825 we have

Fortran subroutine for selecting two parents by simulating a roulette

R1 < 0.2825 < R 2 ;

198

INTEGER PROGRAMMING

199


r = 0.2825 ;=

r = 0.1560

n~-

(5.4.10)

i + l.

In this case the portion of the roulette wheel assigned to the design,

Eq. (5.4.7), becomes r-

(5.4.11)

2(ns- i + 1)

;- (ns

+ 1)ns

RJ

r

= 0.9217

r=0.7123 Figure 5.6.

Roulette wheel for example 5.4.1

therefore, Design 2 is selected. For the remaining random numbers, we have R 3 < 0.7123 < R4 ,

R1 < 0.1560 < R2 ,

R3 < 0.6120 < R4

and

R4 < 0.9217 < R5 ,

R 1 < 0.3471 < R2 •

Example 5.4.3 Repeat Example 5.4.2 for a fitness list of {P $2, $3 , $4, $5 , $6 } = {0.62, 0.60, 0.65, 0.61, 0.57, 0.64}, which represent a typical population towards the end of a genetic optimization when the difference between the designs becomes small. Use the same six random numbers used in the example, (a) for the standard fitness-based selection, (b) for the rank-based selection process. (a) In this case of fitness-based selection, the fractions of the wheel occupied by the different designs are given by

=

{rp r2, r3 ,r4 , r5 , r 6 }

{0.1680, 0.1626, 0.1762, 0.1653, 0.1545, 0.1734}, and the wheel for this generation of designs is shown in Fig. 5.7a. The list of summation of fractions of the area of the wheel is

The parent designs selected for the next generations are {R 1, R 2 , R3 , R4 , R 5 , R6 }

{2, 4, 2, 5, 4, 2}.

=

{0.1680, 0.3306, 0.5068, 0.6721, 0.8266, 1.0}. For the same set of random numbers used in Example 5.4.2, we have

Defining fitness on the basis of the numerical value of the objective function or the augmented function carries the disadvantage that towards the end of the optimization, when the differences between competing designs become small, the selection pressure in favor of the better design becomes small, and progress slows down. For this reason, another popular strategy is to define fitness based on the relative rank of the designs. In that case, the ith-ranked design is reassigned a fitness value of

R 1 < 0.2825 < R2 ,

R5 <0.9217
R4 < 0.7123 < R 5 ,

R 3 <0.6120
R0 < 0.1560 < R 1,

and

R2 <0.3471
yielding the following parents: {2, 5, 1, 6, 4, 3}.

Note that none of the designs is selected twice as parents.

200

INTEGER PROGRAMMING

(b) Reordering the designs according to ranking we have

{R 1, R2, R3 , R4 , R 5 , R6 } = {0.2857, 0.5238, 0.7143, 0.8572, 0.9524, 1.0}

{0.65, 0.64, 0.62, 0.61, 0.60, 0.57}, where the corresponding designs from the original list are

201


and, for the same set of random numbers used in the previous example, we have

{3, 6, 1, 4, 2, 5}.

R0 < 0.2825 < Rp

The new rank-based fitnesses for these designs are

R4 <0.9217
{6, 5, 4, 3, 2, 1}.

The fractions of the wheel occupied by the rank-based designs are given by Eq. (5.4.11) as {rp r2 , r3 , r4 , r5 , r 6 } =

and the wheel for this generation of designs is shown in Fig. 5.7b. Note that the numbers in the list above are ordered from best to poorest rather than by the original order of the designs. So r 1 = 0.2857 corresponds to the best design, or Design 3 of the original list. The list of summation of fractions of the areas of the wheel is

= 0.2825 r

= 0.1560

R2 <0.6120
R 0 < 0.1560 < Ri'

and R1 <0.3471
These random numbers select the {1,3, 1,5,3,2} designs of the ranked population as parents. The original designs corresponding to these parents are

{0.2857, 0.2381, 0.1905, 0.1429, 0.0952, 0.0476},

r

R2 < 0.7123 < R3 ,

r

= 0.1560

{3, 1, 3, 2, 1, 6}.

Designs 1 and 3 in the original population, which are among the designs with the highest fitnesses, are each selected twice as parents. On the other hand, Designs 4 and 5 of the original population with low fitnesses are not selected. Another advantage of the rank-based selection procedure is its capability to handle negative values of the fitness function, q>, as well as the combination of negative and positive ones. This feature becomes especially handy in cases where, for designs that violate the constraint, a large penalty multiplier causes the fitness function to become negative.

RJ

5.4.4

r

= 0.7123 l:z) based onfitnesses

Figure 5.7.

3

r

= 0.7123 'b) based on ranking

Roulette wheel for Example 5.4.2.

Crossover

Once pairs of parents are selected, the mating of the pair also involves a random process called crossover. The simplest crossover, the singlepoint crossover, begins by generating a random integer k between 1 and L - 1, where L is the string length. This number defines a cutoff point in each of the two strings and separates each into two substrings. Denoting, for convenience, the two parts as left (initial part) and right (final part) substrings, we splice together the left part of the string of

202

INTEGER PROGRAMMING

one parent with the right part of the string of the other parent. The random integer needed for defining the cutoff point can be generated as [(L - 1)r + 1], where r is our (0, 1) uniformly distributed random variable. Consider, for example, two symmetric laminates coded in a string of length L = 9, and a crossover point k = 5: parent 1:

0 0 1 2 1113 1 1 1

[0/45/0/-45/03),

parent 2:

0 0 0 0 1112 3 4 1

[0/45/-45/90/0Jv.

(5.4.12)

The two possible child designs are

~rossed,

choosing the endpoint randomly improves the performance of the algorithm. When integer or real variables, such as number of layers or panel dimensions, are represented by binary strings, crossover may generate ~hild designs that do not bear any resemblance to the parent designs. Consider, for example, a rectangular panel with a length variable l. Assume the length variable to have an expected range of (0.05, 3.2) in, represented to an accuracy of 0.05 in by a 6-digit binary number b. The binary number has a range from 0 to 26 - 1 = 63, so that we calculate l as l

child 1: 0 0 1 2 1 2 3 4 1 [0/45/0/45/-45/90/0],, child2: 0 0 0 0 1 3 1 1 1 [0/-45/0J,.

203


(5.4.13)

One or both of the child designs are then selected for the next generation. Note that even though both parent laminates are balanced, neither child is. This indicates the disadvantage of coding that distinguishes between +8 and -8 in balanced laminates. One solution to this problem is to use the previously discussed two-layer stacks. Another solution is discussed in Section 5.4.7. Crossover is typically implemented with some probability p c· If crossover is not indicated, then one of the parents is cloned into the next generation. Implementation of crossover begins with the generation of a single random number r uniformly distributed between 0 and 1. If the number is less than p c crossover will be performed by generating another random number for the cutoff point. The decisions on which parent to clone or which child design to select are not important, as the parents were selected at random. The processes of selection and crossover are repeated until there are n5 child designs. Multiple point crossovers in which information between the two parents is swapped among more string segments is also possible, but because of the mixing of the strings the crossover becomes a more random process and the performance of the algorithm might degrade (De Jong, 1975). An exception to this is the two-point crossover. In fact, the one-point crossover can be viewed as a special case of the two-point crossover in which one end of the string is the second crossover point. Booker ( 1987) showed that, instead of letting the end of the string be the second point that defines the string segment to be

=0.05 (b + 1).

(5.4.14)

The following crossover scenario, between two panels of lengths 0.8 and 2.45 can take place: parent 1:

0 0111 1 1 1

l = 0.8,

parent 2:

11110000

1=2.45.

(5.4.15)

The two possible child designs are

= 0.05,

child 1:

000000

l

child 2:

111111

l = 3.2.

(5.4.16)

That is, instead of getting child designs that have some intermediate value between the two parent designs, we obtain designs that are more extreme than either parent. One can check that most of the designs obtained by crossover from these two parents will have this property, which reduces the effectiveness of the crossover operator for combining existing traits from parent designs into child designs. However, it contributes to the exploration of new design alternatives, a process that is usually handled by the mutation operator described in the next section. It is probably preferable to let the crossover operator control recombination of existing traits and the mutation operator handle exploration of new traits. However, one can expect that in some situations the additional exploratory power of the binary crossover operator is beneficial. If, on the other hand, we use real numbers to represent the variables, we run into the opposite problem. In this kind of representation, the

204

INTEGER PROGRAMMING

real value of each of the variables is assigned to a single location in. the genetic string. With binary representation of these real numbe1 crossover can create child designs that have intermediate values of tho. parent genes. However, with each real number represented by a gene, crossover can only reproduce values present in the initial lation. This situation may be acceptable, since the mutation opcwLul,~ described in the next section, can create new values. Another solution to this problem is to replace the ordinary crossover with an crossover. For example, if we have a string of length L with or real variables, we generate the crossover cutoff point as a real dom number w = rL between 0 and L. Then we take the first variables from one parent, the last L - Lw J- 1 from the second and we average the LwJ + 1 variable between the two parents ac'-ulum)i to the fraction w- LwJ. That is, if in location LwJ + 1 the first p has the value x 1 and the second parent x 2, then the child design have

r = (w- LwJ) x 1 + (1- w + LwJ) r. Consider, for example, a problem where we design a balanced, metric laminated plate with the design variables as the length a width b of the plate, and the numbers of stacks of o;, ±45°, and in half of the thickness of the plate, denoted as N 0 , Nw andN90, respectively. The genetic string will be [a, b, N 0 , N 45 , N 9 oJ. The crossover scenario will correspond to a 10.1 x 7.3 [O]z0 plate crossed, with a 12.5 x 6.3 [Oi±45/904 ], plate with a cutoff point of 2.35:

parent 1:

10.1 7.3 50 0,

parent 2:

12.5 6.3 2 1 2.

The (only) child design is

child

10.1 7.3 3 1 2.

(5.4.19)

That is, the child design is a 10.1 x 7.3 [Oi±451904 ]s plate. Note that with the cutoff point being 2.35, the first two numbers in the string were taken from parent 1, the last two from parent 2, while the third number was calculated as N 0 = (2.35 - 2)5 + (1 - 2.35 + 2)2 = 3.05

(5.4.20)


205

and then rounded down to an integer. If the cutoff involved one of the real numbers in the string, the last step of rounding would have not been required. The process is coded in subroutine cross for integer variables, as shown in Fig. 5.8.

subroutine cross(parent,chldlm,prntlm,ns, pc,L) integer parent(2),prntlm(L,ns ),chldlm(L) id 1=parent(l) id2=parent(2) clone=rand(iseed) if (clone .gt. pc) then do 100 i=1,L chldlm(i)=prntlm(i,id 1) 100 continue return else endif rcut=L *rand(iseed) icut=int(rcut) if (icut .eq. 0) go to 300 do 200 i= 1,icut chldlm(i)=prntlm(i,id1) 200 continue 300 continue chldlm(icut+ 1)=(rcut-icut)*prntlm(icut+ 1,id 1) $+(1 +icut-rcut)*prntlm(icut+ 1,id2) if (icut .eq. L-1) return do 400 i=icut+2,L chldlm(i)=prntlm(i,id2) 400 continue return end Figure 5.8. Fortran subroutine for averaging crossover with integer variables in a string of length L. Crossover is performed with probability pc, otherwise the first parent is cloned.


206

5.4.5

Mutation

Mutation performs the valuable task of preventing premature loss important genetic information by introduction of occasional random alteration of a string. Inferior designs may have some good traits can get lost in the gene pool when these designs are not selected parents. Additionally, mutation is needed when integer or real c is used because in most cases there is low probability that all possible genes are represented in the initial population. Consider, example an initial population of 20 designs, each with 6 genes, each gene can take a value from 1 to 10. Let us calculate the ability that no member of the initial population will have a value 10 for one of the 6 genes. The probability that any individual does not have the value 10 in the first gene is 0.9. The that none of the designs have that value is 0.9 20 = 0.121. That is, probability that at least one design will have a gene with a value 10 is 0.879. The probability that for each one of the 6 genes we find one design (not necessarily the same for different genes) with value of 10 for that gene is 0.879 6 = 0.46. That is, there is more a 50% chance that the value 10 is missing from at least one gene at least one location in the string) for every member of the population. Because this is the highest value for the gene, crossover cannot create this gene, only mutation can. Mutation is implemented by changing, at random, the value of digit in the string with small probability. Based on small rates of currence in biological systems and on numerical experiments, the of the mutation operation on the performance of a genetic is considered to be a secondary effect. Goldberg (1989) suggested rate of mutation Pm of one in one-thousand bit operations. H when integer coding is used, higher mutation rates are required. experience with stacking sequence design suggests a rate of about in a hundred, with smaller populations requiring higher mutation to preserve diversity in the population. Mutation is implemented by generating a random number between 0 and 1 and applying a mutation if its value is smaller than Pm· Another random number can then be used to select a replacement for the digit being mutated. When we use 0 to denote absent layers, a mutation may create a void inside the laminate, so that we have to pack the laminate, as described in Section 5.4.2, after the mutation operation.

207

When real design variables are coded as binary numbers, there are situations where small changes in the design cannot be achieved by mutations. For example, consider the case when we have an integer design variable varying from 0 to 31, coded as a 5-digit binary number. The string [01111] corresponds to 15 and the string [10000] corresponds to 16. Even though the two designs are close, it is extremely unlikely that a mutation will change one string into another. This can occasionally result in slower progress for the algorithm. For example, if the optimum design corresponds to 16 and 15 is a close second best, we can run into situations where most of the members of the population have a value of 15 at some stage of the optimization. Making the final improvement from 15 to 16 can then take a very long time. To counteract this problem, one can use another binary coding, called the gray code, which does not have the abrupt change in digits caused by small changes in the value of the number. Alternatively, we can add another mutation operator, called local mutation, applied to the original variables rather than the coded ones. This local mutation operator is also useful when we do not use binary numbers but use the actual values of the variables in the string. Subroutine locmut, shown in Fig. 5.9, performs such a mutation. The size of the mutation is dictated by the parameter range. subroutine locmut(lamin,L,pm,range) dimension lamin(L) do 100 i=1,L dodont=rand( iseed) if (dodont .gt. pm) go to 100 mute=range*rand(iseed) if (dodont .lt. 0.5) then lamin(i)=lamin(i)-mute if (lamin(i) .lt. 0) lamin(i)=O else lamin(i)=lamin(i)+mute endif I 00 continue return end Figure 5.9. Fortran subroutine for local mutation that perturbs integer variables by the ±range with probability pm.

208

INTEGER PROGRAMMING

The mutation coded in subroutine locmut in Fig. 5.9 is based on a uniform distribution. When real variables are used in the string, the mutation is often a normally distributed random number with a zero mean and a specified standard deviation (see Back and Schwefel, 1993).

5.4.6

Permutation, Ply Addition, and Deletion

For stacking sequence design, several specialized mutation operators have been suggested in the literature. One class of operators is permutation operators, which change the stacking sequence without changing the composition of the laminate. These operators change the bending stiffness of the laminate without changing its in-plane properties. They are particularly useful for laminate problems with constraints on both in-plane and bending properties, as explained in what follows. In problems of stacking sequence design, there are many laminates with the same in-plane properties and different bending properties. For example, the laminates [0/902 ]s, [90/0/90],, and [90/0], have the same in-plane stiffnesses but different bending stiffnesses. In fact, any permutation of the stacking sequence preserves the in-plane properties. In particular, if we consider the optimum stacking sequence for a combined in-plane and bending problem, there may be many laminates with identical layer composition, but with different (nonoptimal) stacking sequences. Since there are many possible laminates, the genetic algorithm typically finds laminates with optimal composition of orientation angles early in the search, and the rest of the search is devoted to shuffling the stacking sequence to produce the desired bending properties. Achieving a permutation in stacking sequence by a standard mutation operator is difficult because it calls for two simultaneous mutations, which have a low probability of occurring. For this reason, various permutation operators have been proposed for stacking sequence design (e.g., Le Riche and Haftka 1993, 1995; Marcelin, Trompette, and Dornberger, 1995; Nagendra et al., 1996). The simplest and possibly the most effective permutation operator is the ply-swap, where two layers are interchanged. For example, for a symmetric balanced laminate [45/-45/0/90/0/45/0/-45], with positions 3 and 7 chosen for a ply-swap, we have (0°, 45°, -45°, 90° coded as 1, 2, 3, 4, respectively):


209

before ply-swap:

2 3 1 4 4 1 2 1 1 3,

after ply -swap:

2 3 2 4 4 11 1 1 3,

(5.4.21)

leading to [45/-45/45/90/0/-45], laminate. Occasionally, the number of plies of each orientation is fixed, and the laminate design requires only obtaining the best stacking sequence with the given orientations. This is equivalent to finding the best permutation of a baseline laminate. Specialized codings and crossover have been developed for permutation problems arising in scheduling (e.g., Michalewicz, 1992). These can be adapted to composite optimization, and have proved to be highly efficient (Liu, Haftka, and Akgi.in, 1998). When the thickness of a laminate is not specified, we use empty layers or stacks with special codes (e.g., zero) to allow for variable thickness. The standard mutation operator in this case controls the processes of adding layers, deleting layers, and changing their orientation in a collective manner. These operators are referred to as the ply-addition, ply-deletion, and ply-orientation mutations, respectively. Assigning different probabilities to the three different forms of the mutation operator may provide performance improvements. For example, if a single mutation probability is used, we may change the laminate thickness more frequently than we need. The optimum laminate thickness is usually achieved early in the genetic search, so that the probability of adding or subtracting layers should be low compared to the probability of changing the orientation. Le Riche and Haftka (1995) showed the benefit of using separate ply-addition, ply-deletion, and ply-orientation mutations. Nagendra et al. (1995) showed similar gains and also suggested separate ply-swap operators for interchanging layers in the same laminate and between different laminates in the components of a stiffened panel (e.g., between the skin and the stiffener laminates).

5.4.7 Computational Cost and Reliability Genetic algorithms tend to be costly, requiring thousands of analyses. The cost depends on the size of the design space and the way constraints hinder movement in the design space. Therefore, to keep the computational cost manageable, it is useful to try to keep down the

210

INTEGER PROGRAMMING

size of the design space and avoid formulating constraints in such a way that they will greatly constrain the movement in the design space. The constraint reflecting the balanced laminate requirement is a good example of this principle. Consider for example, the design of a 48-ply symmetric and balanced laminate to be made of 0°, 90°, and ±45° orientations. The symmetry allows modeling of only half of the laminate, so that the string length is 24. With four possible digits to rep·\ resent the four orientations, the size of the design space is 4 24 2.8 x 1014 • As noted in the Section on crossover (Section 5.4.4), if two-parent laminates are balanced, the child laminate is often balanced. That is, we have a very large design space, and the bal constraint is violated for many laminates generated by crossover mutation. If we use stacks of two layers each 0 2 , ±45, and 902 , we automatically satisfy the balance constraint and, for the 48-ply laminate, duce the string length to 12. With only three possible digits the of the design space shrinks to 3 12 = 531,441. Of course, we may a bit on performance, because we exclude laminates with an odd ber of layers of the same orientation in a stack. If we have a where this consideration is important enough, we can still salvage, some of the performance gains. We can use, for example, the digits 1, 2, and 3 to represent the possible layers, with 1 representing 0° and 3 representing 90° layers. The digit 2 represents either a 45° or a -45° layer, with the understanding that the outermost 2 in the string: corresponds to 45°, the next 2 corresponds to -45°, and so on. The size of the design space is now 324 = 2.8 x lOll, which is still 1000 times smaller than the original design space. Also, now we can never have more than a single-layer violation of the balance condition, and violations are less common as a result of crossover. The computational cost of genetic algorithms also depends on our requirements in terms of reliability. It is important to realize that because of the random aspects of the algorithm, each execution of the algorithm can result in a different answer. High reliability means that almost always the answer is close enough to the global optimum. Le Riche and Haftka (1993) defined practical reliability as the probability that the optimum is "practically equal" to the global optimum. For the buckling and strength problems that they solved, they defined the term "practically equal" to mean within 0.1% of the optimum. Reliability can be increased by increasing the number of generations that

211


the algorithm is run and by increasing the number of strings n.1.• Alternatively, we can simply run the algorithm several times and take the best answer. All of these strategies cause an increase in computational cost. However, multiple runs may also produce multiple solutions that could be useful to the designer. This point is illustrated in the following example. The reliability of the genetic algorithm is usually established by testing with problems with known solutions. If the algorithm is run m times and is successful in finding the global optimum ms of these runs, then the apparent reliability ra is ra= ms/m.

(5.4.22)

However, if m is small, the apparent reliability may be quite different from the true reliability r. In fact, with a bit of statistics, it can be shown that the standard deviation 0'0 of the apparent reliability is 0'0

=---/ r(l

- r)/m.

(5.4.23)

For example, if we have 7 successful runs out of 10, the apparent reliability is 0.7, with a standard deviation of 0.145. Example 5.4.4

Use genetic optimization to find the thinnest balanced symmetric Graphite/Epoxy laminate made of 0°, ±30°, ±45°, ±60°, and 90° layers that satisfies the strain constraints of Example 4.1.1. Since this is an in-plane problem, we do not need to design the stacking sequence, but only the number of layers in each orientation. We therefore select the design variables as n 0 , n 30, n45 , n 60 , and n90 . With the symmetry and the balance condition, the total number of layers in the laminate, which is the objective function, is

f = 2(no + n9o) + 4(n3o + n4s + n6o)· The two constraints are a limit of 0.004 on the axial strain Ex under a load of Nx = 10,000 lb/in and a limit of 0.006 on the shear strain Yxy under a load of Nxy = 3,000 lb/in. We write the constraints in a normalized form as

212

INTEGER PROGRAMMING

e

g 1 -- 0.004 X - 1 $ 0,

~-1$0.

g2 = 0.006

For the genetic optimization we use a string of (n 0 , n30 , n45 , n60 , n90 ), and an augmented objective function of

! + p max (gp gz), if max (gp g2 ) > 0, f* ={f- e lmax (gp g2)1, otherwise. To determine the magnitude of p we note that in Example 4.1.1 the total number of layers varied between 68 and 158, depending on the choice of orientation angles. For any of these choices, one constraint is critical, that is, close to zero. Reducing the number of layers by, say 10%, will increase the strains by 10% and so will result in gmax becoming about 0.1. We need the penalty associated with this violation to result in an increment to the augmented objective of at least 10% of that objective. This means a p of between 68 and 158. To ensure feasibility we chose p = 300. To provide a bonus for feasible designs with constraint margin, we need to choose e to be small enough so that even a large constraint margin will not overpower a difference in thickness. On the other hand, we want it to be large enough to reward constraint margins for designs with the same thickness. For this we chose e = 0.1 (for a discussion of the selection of the bonus and penalty parameters, refer to Section 5.4.3 and Example 5.4.1). For the initial population we used subroutine init of Fig. 5.4, with nmax = 20. As can be seen from the equation for the objective function, this choice permits the initial population to have up to 320 layers, with an average of 160. We used the averaging crossover of Fig. 5.8 and the local mutation of Fig. 5.9. After some experimentation, we settled on a mutation rate, Pm = 0.15, a mutation range of ±5, and a crossover probability, Pc = 0.9. To determine the population size and number of generations, we ran the optimization 20 times and monitored the reliability

213


of the process. For a population size of 80 and 50 generations, 19 of the 20 optimizations gave the same final objective. Using Eqs. (5.4.22) and (5.4.23), this corresponds to a reliability of 0.95 with standard deviation of 0.049. Of course, for more expensive problems we may not be able to engage in such extensive experimentation. However, it is important to build up intuition for these parameters from simple problems. The optimizations yielded four designs with practically identical performance. These were [0 12/±45 4 ]s, [0 10/±30/±45/±60]., [09/±30/±45/90]., and (0 10/±30/±45)s. All four designs have 40 layers, and the first three have g 1 = -0.0001, g 2 = -0.0006. The last design has g 1 = -0.0365, g 2 = -0.0006. This last design is slightly superior in that it has a minimum margin of 0.06% instead of the 0.01% obtained for the other three. However, the difference is so minute that it is much smaller than the error associated with predicting the strains using classical lamination theory (or any other theory). It is worthwhile to note again that composite stiffness design problems often have multiple optima and that genetic algorithms help the designer find all or many of these optima, thus presenting a choice that could be decided on the basis of other considerations. Example 5.4.4 demonstrates that multiple genetic runs can produce near-equivalent designs. Additionally, it is often computationally advantageous to make multiple runs rather than a single long genetic optimization. For example, if we find that optimization runs that last 100 generations give us 40% reliability, then the probability that we will not find the optimum in one run is 60%, the probability that we will not find the optimum in two runs is 0.62 = 36%, and the probability that we will not find the optimum in three runs is 0.63 = 21.6%. That is, two runs give us a reliability of 64% and three runs a reliability of 78.4%. We need to compare these numbers with the reliability achieved in a single run of 200 generations and 300 generations, respectively. In general, if we denote the reliability achieved by q runs each with n generations as r/n), then rq(n)

= 1- [1- r 1(n)V

(5.4.24)

I

214

INTEGER PROGRAMMING

This equation can be used to compare the relative efficiency of multiple runs for simple cases for which we can afford to make a large number of runs. Then the results could serve as guidelines for choosing the number of runs for similar but more computationally expensive cases. Le Riche and Haftka (1995) found that for designing the stacking sequence of unstiffened panels for strength and buckling constraints, single runs are most efficient when the desired reliability is below about 80%. For higher reliabilities, multiple runs are more efficient. Most of the discussion in the section on genetic algorithm focused on integer codings. However, binary codings are also popular, and we close the chapter with an example of an application of a genetic algorithm based on binary coding.

The cantilever composite beam shown in Figure 5.10 is subjected to a transverse load P and a twisting moment Mxy· Use a genetic algorithm to design a hygrothermally curvature-stable laminate for the beam. The beam is to be designed to maximize extensiontwisting and bending-extension coupling, with constraints on stiffness in flexure and twist under the applied loads. The constraints require the tip displacement to be less than 0.1 in and the tip twist to be less than 0.01 rad. The couplings as well as the tip displacement and twist are given in terms of the inverse of the (A, B, D) matrix, with the corresponding locations of submatrices in the inverse denoted as au, h;p and d;p respectively [see Eq. (3.3.3)]. The tip displacement wtip is given as - dl1PL3 di6Mxf} 3 + 2 .

wtip-

The tip twist

tip

p ~A

'1r~

~A L=lOin - - - - - -

...,.

_l_

T

is given as tip=

di6PL2 ~6MxyL --4- + 2

Assuming that we want to maximize a linear combination of the two coupling coefficients b11 and b 16 , the objective function is (c 11b11 1+ c2 1b161). The coefficients c 1 and c2 are chosen depending

~

ce=p4 Mxy

~LOin

0.12 in

Section A-A

Figure 5.10.

Example 5.4.5

215


Cantilevered beam with a tip load and a moment.

on how we want to emphasize the two coupling terms. In addition to constraints on maximum deflection and twist at the tip, an additional constraint was imposed in this problem that limited the number of layers in the laminate to a maximum of 24. Material properties are assumed to be those corresponding to HT-S/4617 Graphite/Epoxy, and are defined as follows:

E 1 = 20 X 106 psi, G 12 = 0.65 x 106 psi,

£ 2 = 1.3 X 106 psi,

v12 = 0.304,

t = 0.005 in.

A hygrothermally curvature-stable, antisymmetric laminate [8/(8 + 90) 6 /8/-8/-(8 + 90)6 /-8 3] provides one solution that would meet the desired objective (see Chapter 3). Finding the optimum value of 8 is a one-dimensional optimization problem and can be solved by plotting the objective function and constraints as a function of 8. The hygrothermal curvature stability can also be realized in more general unsymmetric composite laminates that are obtained by stacking two symmetric layer groups [8/8 + 90], s (Type I) and [TJITJ + 60/TJ + 120]n s (Type II). Any additional symmetric layer groups with equally' spaced angles can also be added to the laminate without disturbing the hygrothermal curvature stability. The problem formulation allows

218

INTEGER PROGRAMMING

lem (ten digits for e and 11, two digits each for n 1 and n2, and one digit each for p 1 andp 2). Two variations of the above problem were considered. In each case, a total of 10,000 function evaluations were permitted. A two-point crossover was used with a probability of 0.8. Mutation with a probability of 0.02 per string was used. The first test case required the maximization of an equally weighted sum of the b11 and b16 coefficients, with a constraint only on the total allowable thickness of the beam. The best design obtained for this case was that in each of the two available slots, layer group I was selected, the basic orientation angles for these groups were 8 1 = 108° and 82 = 150°, and the coupling coefficients b 11 and b 16 were 122.44 X 10-6 in/lb-in and 436.98 X 10-6 in/lb-in, respectively. The variables n 1 and n2 were both obtained as unity; this is to be expected, as the thinner laminate would maximize the compliance coefficients. This experiment was repeated for population sizes of 100, 150, and 200, and similar solutions were identified in each case. In the second test case, the bending and twist displacement constraints were activated in addition to the maximum thickness requirement. The population size was selected as 200. For three distinct choices of weighting coefficients c 1 and c2 , five simulations of genetic search were performed. These results are summarized in Tables 5.2-5.4. For the case where c 1 = 0.75, c2 = 0.25 and c 1 = c2 = 0.5, similar results were obtained. As seen in Tables 5.2 and 5.3, the solutions corresponded to a use of both Type I and Type II groups in establishing an optimal laminate for the given design constraints. When the tension-twist coupling

for the two-layer groups to be assigned to the two available positions on the laminate. It is possible to have both layer groups be of the same type, with similar or different basic orientation angles. The optimization problem for finding the optimum laminate with the angles e and 11 was formulated as minimize

6- (c 11b11 1+ c 21b161)

such that

n 1p 1 + n2p 2 :::::; 24,

[ (I 10

[ 100

3

d PL + d 16M2xy L _11_3_

W":L'

+

2

1) 1]: : :; 0,

~,~,,LI

-

)-

1] s 0

The design variables for this problem are the basic orientation angles, e and 11, the number of layer groups, n 1 and n2, and the number of layers in each layer group, p 1 and p 2 • That is, p 1 = 2 if we select the first layer group to go with n 1, and p 1 = 3 if we select the second layer group to go with n 1• The constant 6, which was set to 10 x 10-4 , is used to convert the maximization problem into a minimization. The orientation angles e and 11 were varied discretely between 0° and 174° in increments of 6°. The 30 possible values of e were each represented by one of the unique combinations available from a five-digit binary string; the two excess binary string combinations were assigned to 0° and 90°, respectively. The variables n 1 and n2 varied between 1 and 4 and required a two-digit binary string to uniquely represent the four possibilities. It is important to indicate that there is an inherent danger in representing certain variables by more strings than others, in that the search is biased in favor of the overrepresented variables. In this problem, however, because of a large number of possible choices of e and 11 to select from, this is not a serious concern. The choice of layer group is assigned to two variables p 1 and p2 , which could assume the values of 2 or 3. Hence, a single-digit string is sufficient to represent each of these variables. A 16-digit binary string is required to represent all design variables for this prob-

217


Table 5.2. Summary of optimization results for weighting coefficients c1 = 0.5, c2 = 0.5 Run

Group Type

ID

e

11

p,

Pz

nl

nz

2 3 4 5

126 174 126 126 126

30 168 168 168 168

2 2 3 2 2

2 3 2 3 3

4 3 2 3 3

2 2 3 2 2

b,,

b,6

10-6

10-6

-16.210 -26.839 -26.031 18.544 25.743 -17.924 -26.031 18.544 -26.031 18.544

218

Table 5.3. Summary of optimization results for weighting coefficients c1 0.75, c2 0.25

=

Group Type

Run ID

9

1l

PI

Pz

ni

nz

2 3 4 5

36 0 126 0 168

168 48 168 48 42

2 3 2 3 3

3 2 3 2 2

3 2 3 2 2

2 3 2 3 3

=

bll

bi6

I0-6

10-6

-26.031 18.544 28.696 7.1758 -26.031 18.544 28.696 7.1758 28.705 -6.7722

shear modulus is less than 16 GPa, and the stiffness term A 22 is greater than 30 MN/m. Formulate the problem as a linear integer programming problem and use optimization software (e.g., Microsoft Excel) to solve it. Note that the total laminate thickness is not expected to be larger than 20 layers. Use the following material properties: E 1 = 76 GPa, E 2 = 5.5 GPa, G 12 = 2.3 GPa, v 12 = 0.34, and t = 0.125 mm. ,,

2.

For a 40-ply Graphite/Epoxy laminate under unit axial load, show that as you decrease the number of zeros in the laminate, from (±45 8/0 4 ) 5 to (±45/0 2 ) 5 and finally to (±45 10 ),, the maximum strain in any fiber direction increases and then decreases. This makes the all-±45° design singular in that it cannot be reached by gradually reducing the number of oo layers.

3.

Solve problem 1 by genetic optimization.

4.

Repeat Example 5.4.4 using binary coding of the design variables.

5.

Using genetic optimization, design 16-, 24-, and 32-ply balanced symmetric laminates to maximize the Poisson's ratio of the laminate subjected to the constraint that EY > 40 GPa. Use the following material constants:

Table 5.4. Summary of optimization results for weighting coefficients c1 = 0.25, e:z = 0.75 Run

Group Type

ID

9

1l

PI

Pz

ni

2 3 4 5

144 36 36 36 144

126 54 54 54 126

2 2 2 2 2

2 2 2 2 2

3 3 3 3 3

bii

bi6

nz

IQ-6

10-6

3 3 3 3 3

0.9194 0.9194 0.9194 0.9194 0.9194

-36.545 -36.545 -36.545 -36.545 -36.545

is emphasized to a greater degree than bending extension (c 1 = 0.25, c 2 = 0.75), the Type I group was used exclusively as shown in Table 5.4. This is easily explained by the observation that the Type II group is mechanically isotropic and would not contribute to the extension-twist coupling coefficient. In contrast, the Type I group is mechanically orthotropic and is the predominant source of the b 16 coupling coefficient. It is also worthwhile to note that the presence of the bending and twisting stiffness constraints increases the laminate thickness by introducing n 1 and n 2 values different from unity.

EXERCISES 1.

Design a minimum thickness laminate with 0°, ±45°, and 90° layers such that the laminate's Poisson's ratio is greater than 0.6, the

219

REFERENCES

INTEGER PROGRAMMING

U 1 = 57.5 GPa,

U 2 = 58.0 GPa,

U4 = 17.5 GPa,

and

U3 = 13.6 GPa,

U5 = 20.0 GPa.

Study the effects of the probability of mutation and the magnitude of the penalty multiplier on the progress and final results of the design optimization.

REFERENCES Arora, J. S., (1989). Introduction to Optimum Design. McGraw-Hill, New York. Back, T., and Schwefel, H.-P. (1993). "An Overview of Evolutionary Algorithms for Parameter Optimization." Evolutionary Computation 1(1), 1-23. Booker, L. (1987). "Improving Search in Genetic Algorithms." In Genetic Algorithms and Simulated Annealing, pp. 61-73. Davis, L. (ed.). Morgan Kaufmann Publishers, Inc., Los Altos, CA.

220

INTEGER PROGRAMMING

Callahan, J. K., and Weeks, G. E. (1992). "Optimum Design of Composite Laminates Using Genetic Algorithms." Composites Engineering 2(3), 149-160. De Jong, K. A. (1975). Analysis of the behavior of a class of genetic adaptive systems. Doctoral Dissertation, University of Michigan. University Microfilms No. 76-9381. Dissertation Abstracts International36(10), 5140B. Furuya, H., and Haftka, R. T. (1993). "Genetic Algorithms for Placing Actuators on Space Structures." Proceedings, Fifth International Conference on Genetic Algorithms, pp. 536-542. July 17-22, Urbana, IL. Garfinkel, R. S., and Nemhauser, G. L. (1972). Integer Programming. John Wiley & Sons, Inc., New York. Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Publishing Co. Inc., Reading, MA. Goldberg, D. E., and Samtani, M.P. (1986). "Engineering Optimization via Genetic Algorithm." Proceedings of the Ninth Conference on Electronic Computation, pp. 471-482. ASCE, February. Hajela, P. (1990). "Genetic Search-An Approach to the Nonconvex Optimization Problem." AIAA J. 28(7), 1205-1210. Hajela, P., and Erisman, (1993). "Genetic Search Methods in the Design of Thermally Balanced Composite Laminates." Proceedings of OPT/'93, Zaragosa, Spain, July 7-9. Holland, J. H. (1975). Adaptation of Natural and Artificial Systems. University of Michigan Press, Ann Arbor, MI. Johnson, E. L., and Powell, S. (1978). "Integer Programming Codes." In Design and Implementation of Optimization Software, pp. 225-240. Greenberg, H. J. (ed.). Nato Advanced Study Institutes Series, Urbina, Sifthoff & Noordhoff. Kogiso, N., Watson, L. T., Gtirdal Z., Haftka, R. T., and Nagendra, S. (1994). "Design of Composite Laminates by a Genetic Algorithm with Memory." Mechanics of Composite Materials and Structures 1(1), 95-117. Kovacs, L. B. (1980). Combinatorial Methods of Discrete Programming. Mathematical Methods of Operations Research Series, Vol. 2. Akademiai Kiad6, Budapest. Land, A. H., and Doig, A. G. (1960). "An Automatic Method for Solving Discrete Programming Problems." Econometrica 28,497-520. Laarhoven, P. J. M. van, and Aarts, E. (1987). Simulated Annealing: Theory and Applications. D. Reidel Puslishing, Dordrecht, The Netherlands. Lawler, E. L., and Wood, D. E. (1966). "Branch-and-Bound Methods-A Survey." Operations Research 14, 699-719. Le Riche, R. (1994). Optimization of composite structures by genetic algorithms. Ph.D. Dissertation, Virginia Polytechnic Institute and State University.

REFERENCES

221

Le Riche, R., and Haftka, R. T. (1993). "Optimization of Laminate Stacking Sequence for Buckling Load Maximization by Genetic Algorithm." AIAA/ASMEIAHS/ASCE/ASC 33rd Structures, Structural Dynamics and Materials Conference, Dallas, TX. AlsoAIAA Journal Vol. 31(5), 951-956. Le Riche, R., and Haftka, R. T. (1995). "Improved Genetic Algorithm for Minimum Thickness Composite Laminate Design." Composite Engineering, 5(2), 143- . 161. Le Riche, R., Knopf-Lenoir, C., and Haftka, R. T. (1995). "A Segregated Genetic Algorithm for Constrained Structural Optimization." In Genetic Algorithms: Proceedings of 6th International Conference on Genetic Algorithms (ICGA95) Eshelman, L. (ed.), July. Liu, B., Haftka R. T., and Akgtin, M.A. (1998). "Permutation Genetic Algorithm for Stacking Sequence Optimization." Proceedings, AIAA/ASME!ASCE! AHS/ASC 39th Structures, Structural Dynamics and Materials Conference, Part 2, pp. 1141-1152, Long Beach, CA, April20-23. Marcelin, J.-L., Trompette, P., and Domberger, R. (1995). "Optimization of Composite Beam Structures using a Genetic Algorithm." Structural Optimization 9(3/4), 236-244. Michalewicz, Z. (1992). Genetic Algorithms+ Data Structures= Evolution Programs. Springer Verlag, Berlin. Nagendra, S., Haftka, R. T., and Gtirdal, Z. (1993). "Design of a Blade Stiffened Composite Panel by a Genetic Algorithm." Proceedings, AIAAIASME/ASCE/ AHS!ASC 34th Structures, Structural Dynamics and Materials Conference, Part 4, pp. 2418-2436. San Diego, CA, April19-21. Nagendra, S., Jestin, D., Gtirdal, Z., Haftka, R. T., and Watson, L. T. (1996). "Improved Genetic Algorithm for the Design of Stiffened Composite Panels." Computers and Structures 58(3), 543-555. Rao, S. S., Pan, T.-S., and Venkayya, V. B. (1991). "Optimal Placement of Actuators in Actively Controlled Structures Using Genetic Algorithms." AIAA J. 29(6), 942-943. Schmit, L.A., and Farshi, B. (1973). "Optimum Laminate Design for Strength and Stiffness." Int. J. Num. Meth. Engrg. 7, 519-536. Schrage, L. (1989). Linear, Integer, and Quadratic Programming with LINDO, Fourth Edition. The Scientific Press, Redwood City, CA. Tomlin, J. A. (1970). "Branch-and-Bound Methods for Integer and Non-convex Programming." In Integer and Nonlinear Programming, pp. 437-450. Abadie, J. (ed.). Elsevier Publishing Co., New York.

6 FAILURE CRITERIA FOR LAMINATED COMPOSITES

Prediction of failure of structural components is ysually (!ccomplisl:u~g l?~lJ_Il~tiQIJ£ oLthe stresses or strain_§__tQ __f!!at~rial strength . . ' ~.:.The fundamental stress-strain relations for laminated composite -----~

materials are developed in Chapter 2. These relations, along with equilibrium equations and appropriate boundary conditions for a specific configuration, can be used to determine the stresses or strains under prescribed loads. Determination of the strength of a laminated composite material, on the other hand, presents several difficulties. First, unlike stresses and strains, there is no sound analytical foundation for the determination of strength. This is true even for monolithic isotropic materials, where the strength properties are determined empirically based on tension and compression coupon test data. The next difficulty is caused by through-the-thickness variation of stresses within the laminate even if the laminate is loaded by uniform in-plane loads. Variation of stresses within a structure is nothing new for stress analysts. In the case of monolithic isotropic materials, stress concentrations (such as areas around notches and holes) at which one or more stress components assume large values compared to nominal stresses in the material are likely to cause localized failures. For brittle materials, local failures generally lead to complete fracture and total 223

224

FAILURE CRITERIA FOR LAMINATED COMPOSITES

loss of load-carrying capability. For ductile materials, on the other hand, local failure is in the form of yielding that remains localized and, therefore, tolerated better than brittle failure. Most high-p~ffQ.n:nal!~~ unidirectional composite materials behave in a brittle manner .. In the case of laminated materials, however, there is an additional level of complication that is the result of stacking together several layers of material with different orientations and properties. Since the stresses and the strengths may be different in different layers, it is possible that as the loading is increased stresses in one or more of the layers of a laminate would reach their limiting strength earlier than other layers. It is, therefore, likely that the laminate would suffer damage in the form of local (and most probably brittle) failures in those layers before it fails completely and ceases to carry any loads": That is, the definition tl of failure for the laminate becomes fuzzy. For some applications first failure of any layer is not acceptable because it degrades the strength and stiffness of the laminate. Failure prediction_.based __on first faL __ _ is commonly referred to as the first-ply failure cxiterion. Yet, in some other applications the laminate is considered to be unfailed as long as it carries at least a portion of the first-ply failure load stably (that is, without unstable failure of the remaining layers). In such a case, failures progress from one layer to others as the load is increased, leading to what is commonly referred to as progressive failule. Using unidirectional fibrous composite laminates with orthotropic properties brings in additional complexities to failure prediction that we do not encounter with isotropic layers. Just as the elastic properties change as a function of layer orientation, at the macroscopic level the strength of unidirectional layers is direction dependent. The strength of a layer of unidirectional material in the fiber direction is generally high, but the strength in a direction perpendicular to the fiber is controlled by the matrix material that holds the fibers together and is usually an order of magnitude lower than the fiber direction strength. therefore, a load applied to a laminate along the fiber direction may . be carried safely, whereas the same load may cause failure if it is ,: ~pplied perpendicularly to the fibers. It is also possible that, for a lamina restrained in a direction transverse to the fibers, application of loads along the fiber direction may induce failure transverse to the fibers because of the stresses developed by Poisson's effect. ··------.-

.

.

.. · · - - - - - . · - ·

-----::;1


225

"strength evaluation becomes even more complicated if the fibers are neither perpendicular to or along a given uniaxial in-plane stress state, but are at an angle to it. We know that in-plane elastic responses of such layers demonstrate coupling between the shearing and normal deformations. Stresses calculated in the principal material axes would have components both along the fiber direction and transverse to the · fiber direction, as well as a shear stress component. However, determination of whether the failure of such a configuration is controlled by the fiber strength, matrix normal strength, matrix shearing strength, or a combination of all of them is difficult. Another difficulty that arises from the lamination of layers together is the occurrence of damage in the form of delaminations. Generally, the matrix material that holds the individual layers of the laminate together has substantially smaller strength than the in-plane strength of the layers. Under certain loading, large stresses in a direction perpendicular to the interface between the layers can be generated. These stresses are mostly localized and cause breaking of the bond between the layers within a small region. Prediction of delamination failures requires evaluation of interlaminar normal 0'2 and shear Txz and 'tyz stresses. Calculation of these stresses is complicated and usually requires expensive analyses. For the sake of simplicity, in this book we will concentrate on the in-plane stresses and failures resulting from them. However, even if the size of delaminations is small, they may affect the integrity of the laminate and can degrade the in-plane loadcarrying capability of a laminate. Therefore, in practical engineering applications it is important to check interlaminar failures and take appropriate measure against such failures. In this chapter, we provide a brief introduction to the topic of strength, with an emphasis on the strength of laminated composite materials. We first discuss some of the basic failure theories that are commonly used for isotropic materials. We then extend the discussion to fiber-reinforced orthotropic layers. Finally, we discuss the failure of laminates with emphasis on fiber-reinforced laminates. The majority of failure discussion in this chapter is limited to plane stress problems. At the end of the chapter, we include a brief discussion of the application of failure theories to bending of laminated composites. A more comprehensive coverage of the subject can be obtained in Tsai (1968), Wu (1974), Hashin (1980), and Soni (1983).

226

6.1


6.1

227

FAILURE CRITERIA FOFIISOTFIOPIC LAVERS

cr2

FAILURE CRITERIA FOR ISOTROPIC LAYERS

For monolithic isotropic materials, failure is usually associated yielding if the material is ductile or fracture if it is brittle. Limiting ourselves to thin layers under plane stress, in the most general case of combined loading we deal with only in-plane stresses ax, crY, anq 'txy- The following are the most frequently found failure criteria for · isotropic materials.

SY' Sutl

...-

-.....

"' -"':' ,

Maximum shear stress criterion

/

"' ,' ,

\ M . . 1....___ a:un:um stram crzterron

,,''

I

, I

,'

~,,,' 1,'

,, ,,'I

Sy, Sue

6.1.1

This criterion assumes that failure occurs when the maximum normal stress in the plane of the thin layer exceeds the material strength. For a given stress state ax, crY, and 'txY' the maximum normal stress corresponds to one of the principal stresses and are given by

cr,,2 =

ax+a

2

Y

±

2

, "' ,~

' -

-

)

S S Y'

uc

+ 1 xy-

For the case of isotropic ductile layers, the failure strengths in tension and compression are generally assumed to be the same and are measured by the yield strength of the material. This assumption leads to the following set of inequalities used for failure description: and

,'~

Figure 6.1. Failure envelopes in the principal stress space.

-sue< cr, < sut

-SY < cr1 < SY

Maximum normal stress criterion

2

X

(

/>

Distortion energy criterion

,'/

,,' ,

\ I

2

a -ay

,''/

I

Maximum Normal Stress Criterion

cr,

-SY < cr2 < SY,

(6.1.2)

where SY is the yield strength of the material. A graphical representation of the maximum normal stress criterion is shown in the stress space in Fig. 6.1 by solid lines, together with the other failure criteria described in the following subsections. A stress state that yields a combination of the principal stresses cr1 and
and

-sue< (J2
(6.1.3)

where sue and sut are the ultimate compressive and the ultimate tensile strengths, respectively. For most materials the ultimate compressive strength is many times greater than the tensile strength. Therefore, the square failure envelope is generally offset with respect to the principal stress axes (this offset is not shown in Fig. 6.1). 6.1.2

Maximum Strain Criterion

The maximum strain failure criterion is similar to the maximum stress criterion in that the material fails when the maximum principal strain at a point exceeds the failure strain of the same material in a simple tension or compression test. The failure envelope in the strain space is defined by the following simple inequalities: -cue < c, < Eut

and

- Euc < £2 < cut'

(6.1.4)

where cue and Eut are the ultimate compressive and the ultimate tensile strains, respectively. The failure envelope in strain space is shown by solid lines in Fig. 6.2.

1

1 11111

228


02

/

6.1.3

Figure 6.2. Failure envelopes in the principal strain space.

It is also possible to represent the maximum strain failure envelope in the stress space. Using the first two equations of the strain-stress relation, Eq. (2.2.8), for an isotropic material, the principal strains are expressed in terms of the principal stresses as

cr 1

vcr2

E

E

E =---1

and

cr E2=

vcri

l-£·

(6.1.5)

Failure stresses corresponding to the failure strains can also be obtained from the same equations, based on an additional stipulation that the failure strains are obtained from a test configuration where the stress state was a uniform uniaxial normal stress only. Therefore, we have

s

£=..._.!!!.

ut

E

and

sue

Eue =E'

(6.1.6)

-sue< 0"2- vcrl < sut•

(6.1.7)

and the failure envelope is defined by -sue< crl-

l/0"2

and

Maximum Shear Stress (Tresca) Criterion

This criterion is mostly used to predict yielding in ductile isotropic materials. Yielding is predicted when the maximum shear stress at a point exceeds the shear stress in a tension or compression test specimen at yield. The maximum shear stress in a tension or compression test specimen develops at an orientation 45 degrees from the axis of the specimen and its value is half the value of the stress in the axial direction. Therefore, for a material that has a yield strength of SY, the maximum shear stress at yield is tmax = S/2, or the yield strength in shear, Ssy' is given by

01

Eut

229

FAILURE CRITEFIIA FOR ISOTROPIC LAVERS

stress criterion, the lines for the maximum strain criterion are skewed due to Poisson's ratio in the above equations.

Maxim um strain criteri'on

Eut

6.1

The maximum strain failure envelope is shown in the stress space in Fig. 6.1 by long dashed lines. Compared to the maximum normal

ssy =s.2.

(6.1.8)

For a general state of plane stress, the maximum shear stress in the plane is

cr 1 - cr2 =± tl2 = 2

2

ax- cry (

2

2 ]

(6.1.9)

+t..,.

Yielding is prevented if the following conditions are met:

lt 12 1<

s.2

or

lcr 1 - cr21< SY.

(6.1.10)

Similar expressions for t 23 and t 31 yield equations that involve the differences cr2 - cr3 and cr3 - crl' respectively. However, for plane stress cr3 is zero, hence to prevent yield we must also satisfy the maximum normal stress criterion

lcrll < sy

and

lcr21 < sy.

(6.1.11)

For the first and third quadrants of the principal stress plane (see Fig. 6.1) the minimum and maximum principal stresses have the same sign; the shear stress criterion is, therefore, identical to the maximum stress !1111

,II 1

1111

I

230


criterion as predicted by Eq. (6.1.11 ). However, in the second and fourth quadrants, the maximum shear stress criterion predicted by Eq. (6.1.1 0) is more restrictive than the maximum normal stress criterion as shown by the short dashed line in Fig. 6.1.

6.1.4

Distortional Energy (von Mises) Criterion

The distortion energy theory is different from the other failure criteria in that instead of working directly with stresses or strains, it deals with the distortion strain energy density. The distortion strain energy is the part of the strain energy associated with shape change and it is given as 1 + j.l

ud = ~

ua,-

crz) 2+ (crz- cr3) 2+ (cr3- cr,?J

(6.1.12)

(see Section 10.6 of Beer and Johnston, 1992). The distortion energy criterion is based on the assumption that yielding occurs when the distortion energy density at a point in the material exceeds the distortion energy density of a tension or compression test specimen at yield. As with the maximum shear stress failure criterion, the distortion energy criterion is generally used for yielding of ductile materials. For a simple tension or compression test, cr2= cr3= 0 and cr, = SY at failure. Therefore,

1+v

--~--sz. Ud3E Y

(6.1.13)

For plane stress problems the safe region of the stress space can be described by setting the Ud in Eq. (6.1.12) to be smaller than its failure value in Eq. (6.1.13). This inequality yields the following quadratic expression:

crf- cr 1cr2+a~< s;;

(6.1.14)

its boundary is shown in Fig. 6.1 by the long and short dashed lines.

6.2 FAILURE OF FIBER·REINFORCED ORTHOTROPIC LAVERS

231

6.2 FAILURE OF FIBER-REINFORCED ORTHOTROPIC LAYERS Given a two-dimensional stress state, the calculation of the magnitudes of principal stresses is sufficient to predict failure of isotropic materials based on the failure criteria discussed in the previous section. That is, independent of the direction of principal stresses, the magnitudes are compared with the limiting material strength values to avoid failure. The major difference between the isotropic materials and unidirectional fibrous composite materials is the directional dependence of the strength on a macroscopic scale. As mentioned in the introduction to this chapter, composite layers are much stronger in the fiber direction than in the direction perpendicular to the fibers. For loads that are primarily parallel to the fibers, either in tension or compression, the material strength is generally governed by the failure of the fibers. For loads transverse to the fibers, failure is controlled by the failure of the much weaker matrix material. Strengths in the fiber and transverse to the fiber directions are typically determined using unidirectional test specimens (sometimes referred to as test coupons) that have fibers running along and transverse to the specimen axis, respectively, loaded by axial loads. Under general loading conditions, however, the principal stresses in layers of a laminate are commonly at an angle to the principal axes of the layers. In a test environment, principal stresses that are oriented at an angle from the principal material axes may be generated by using off-axis specimens which have fibers oriented at an angle from the specimen axis. For off-axis specimens loaded along the axis of the specimen, principal stress directions coincide with the longitudinal axis of the specimen. Strength evaluation for such a specimen as a function of the off-axis angle is not simple because it is not obvious whether the failure is controlled by the fibers or the matrix. The off-axis strength is influenced to a large extent by such factors as fiber volume fraction, ratio of fiber to matrix strength, fiber matrix interface strength, microscopic defects such as voids, and fiber matrix interfacial debonds. Use of micromechanical models is required to determine a reliable estimate of the strength (see, for example, Chamis, 1974a). However, off-axis strength prediction is difficult even if micromechanical models are used because of a number of manufacturing-related factors that are difficult to incorporate into analytical models; for example, surface treatment

232


of fibers, which influences the adhesion between the fibers and the matrix, is known to have a strong effect on off-axis strength. Of course the effects on strength of micromechanical features that bring into consideration the interaction of the matrix and the fiber constituents are not only apparent for the off-axis strength. Strength along the principal material directions is influenced by micromechanical events as well. For example, in addition to failure loads that are quite different in the fiber and transverse to the fiber directions, the failure mechanism depends on the sense of the stresses. Failure stresses of a composite specimen in tension and compression are quite different from each other and both are strongly influenced by the properties of the fiber and matrix materials, as well as the degree of adhesion achieved between the fiber and the matrix, Chamis (1974b) Typical failure mechanisms under compressive loads along the fiber direction are shown in Fig. 6.3. Depending on a variety of factors any one of those modes might dominate the failure. An in-depth review of micromechanical theories of compressive failure is provided by Schultheisz and Waas ( 1996). Discussion of the factors that affect the failure mode and differences between the failure characteristics is beyond the scope of this book.

+

+

6.2 FAILURE OF FIBER·REINFORCEC ORTHOTROPIC LAYERS

Despite the fact that the fibrous layers are made of distinct phases that control their strengths, for strength determination the layer is usually assumed to be homogeneous. Recall that the same assumption is used in determination of the elastic properties of the material. However, in contrast to the elastic properties, which can uniquely be calculated in any direction (see Section 2.4.2), because of the difficulties described in the previous paragraphs the strength of the homogeneous fiber matrix system is defined only in the fiber and transverse to the fiber directions. Strength of a composite layer in any other direction is evaluated based on various failure criteria presented in the following subsections. Prediction of the failure of laminates made of composite layers with different orientations is discussed in Section 6.3. Note that since the strengths are specified only along the fiber and transverse to the fiber directions, which are the principal axes of material orthotropy, the stresses that are used for comparison with the strengths are the stresses along the principal material axes. In the remainder of this chapter, the stresses in the principal material axes are represented as 0'1 and 0'2 . These are the stresses along the fiber and transverse to the fiber directions, not the principal stresses that are used for the case of isotropic materials. It is also important to note that, in contrast to isotropic materials in which we have generally three distinct strength quantities (tension, compression, and shear), fibrous composites typically have five distinct strength properties: tension and compression, two of each (one in the fiber direction and another one in a direction perpendicular to the fiber) and shear strength.

6.2.1

+ + + +

(a)

(b)

(c)

(d)

Figure 6.3. Compressive failure modes in unidirectional layers. (a) Fiber microbuckling-extensional mode, {b) fiber microbuckling-shear mode, (c) kink band formation, (d) shear failure on a 45 degree plane.

233

Maximum Stress and Maximum Strain Criteria

The basic premise in predicting the failure of fiber-reinforced layers using maximum stress and maximum strain criteria is the same as the one used for isotropic materials. Failure is predicted when the maximum stresses in the medium along the fiber or transverse to the fiber directions exceed the respective strengths of the tension or compression test specimens with fibers placed along the axis of the specimen or transverse to it. The failure envelope under normal stresses is defined by

234

FAILURE CRITERIA FOFI LAMINATED COMPOSITES

a, < X1,

a2 < Y1

for a 1, a 2 > 0,

(6.2.1)

a1 > -Xc,

a2 > -Yc

forcr 1,0'2 <0,

(6.2.2)

where X and Y represent the ultimate strengths along and transverse to the fiber directions, respectively. Since the strength of fibrous layers can be quite different in tension and compression, the subscripts t and c are used to distinguish the two strengths (Xc and Yc represent the. magnitude of the compressive stress allowables and are positive numbers). As noted earlier, a, and a 2 are the stresses along the fiber and·· transverse to the fiber directions, not the principal stresses that are used for the case of isotropic materials. In addition to failure initiated by normal stresses, fibrous composite layers are also vulnerable to failures caused by shearing stresses. The maximum stress criterion for shearing stresses takes the form l't 12 1 < S,

(6.2.3)

where S is the ultimate in-plane shear strength of a specimen under pure shear loading, and 't 12 is the shear stress in the principal material direction (not the maximum shear stress). The maximum strain criterion similarly states that failure when one of the following inequalities is violated: E1 <

xt c; = E,'

E2

yt < E~ = E

I

XC > -£1=--E'I

E2

>-

E~

=-

EYc

for E1,

2

s

ly12 1< Y~2 = G

E2 >

E2

235

cipal material axis oriented at an angle e from the x-axis (called the axial direction for the rest of this example) loaded by (i) pure axial tension, (ii) pure axial compression, and (iii) combined biaxial tensile loading with r=O"/ax=0.05,0.15. Typical strength properties of the Boron/Epoxy are X1 = 1260 MPa, Xc = 2500 MPa, Y1 = 61 MPa, Yc = 202 MPa, and S = 67 MPa. Determine the strength of this layer as a function of the fiber orientation angle in the range 0° :::; e : :; 90°. For a given stress state at a point (ax, O'Y, and 't_x), using Eqs. (2.4.12) and (2.4.14), the stresses in the principal material direction are expressed as 2

0'1} {'ta 2

2

= T {O'x} O'Y = [ m n

12

'txy

l

n2 -2mn 2mn {O'x} m crY . 2 2 -mn mn m - n 'txy 2

(a)

For the three cases considered in this example, we have 'txy = 0. With 0'/0'x denoted as r, the matrix equation (a) can be rewritten as cr 1 = (m 2 + rn 2 )ax, cr 2 = (n 2 + rm 2 )ax• 't 12

fort"

2

E

6.2 FAILURE OF FIBEFI·FIEINFORCED ORTHOTROPIC LAYERS

= (r-1) mnO'x.

(b)

0,

< 0,

•

12

Example 6.2.1 A unidirectional Boron/Epoxy layer (£1 = 204 GPa, E = 18.5 GPa, 2 G12 = 5.59 GPa, v 12 = 0.23, and t = 0.125 x 10-3 m) with the prin-

(i) For pure axial tension we have av = 0 and r = 0, and the failure occurs when 0' 1 or 0'2 reaches X1 or Y1, respectively. It is also possible that, for some off-axis orientations, the layer can fail because of shearing stress, in which case 't 12 = S. From Eq. (b), by substituting m = cos e and n = sin e, threeeq uations that control the maximum applied stress ax that can be allowed are

ax= X/cos 2 e,

ax= ~/sin 2 8

or

ax= S/sin e cos e.

(c)

Tensile strengths obtained from Eq. (c) are plotted in Fig. 6.4a as a function of the fiber orientation e. For very small values of the fiber orientation (up to about 3°) the tensile strength along the fiber direction controls the failure. From 3° up to approximately 42°, the strength is controlled by

lillil 1111',

IIIII I

236

FAILURE CRITERIA FOR LAMINATED COMPOSITES 3000

237

6.2 FAILURE OF FIBER·REINFOACED ORTHOTROPIC LAYERS

3000

fiber failure

Strength values as a function of the orientation are plotted in Fig. 6.5a and 6.5b for the two values of the stress ratio r. Compared to the pure tension case, adding a small transverse tension precludes fiber failure for small fiber orientations. Even for 0 = oo, the strength is controlled by transverse tensile failure. For r = 0.05, strength is controlled by shear failure for a fiber orientation range of approximately 5° ~ 0 ~ 40°. However, in the case of r = 0.15, transverse matrix tensile failure is the dominant mode for the entire range of the fiber orientation angle.

2500 fiber failure

~ 2000

~ 2000

~

~ '0~

1500

transverse matrix failure

15

90

15

30

45

60

75

90

9, (deg) (a) tensile loading

(b) compressive loading

Figure 6.4. Tensile and compressive strengths of unidirectional Boron/Epoxy.

shear failure. For orientation angles larger than 42°, the matrix strength controls the failure. (ii) In the case of pure compression, the strengths X1 and Y1 in Eq. (c) are replaced by Xc and Yc, respectively. The shear strength is independent of the sign. The compressive strength is shown in Fig. 6.4b. Compared to the tensile case, the increased strengths in compression (both in the fiber and transverse to the fiber directions) cause the material to be more vulnerable to shear strength failure, which controls the failure in the range of about 1.5 to 71°. (iii) In the biaxial tension case with a specified stress ratio cr/crx = r, the three equations that govern the applied stress at failure are

(j X

xt = --::---'---2 2 (

COS

0+r

Sin

While the application of maximum stress and maximum strain criteria is straightforward, these criteria fail to represent interactions of different stress components in failure mechanisms. Experimental observations for both isotropic materials and orthotropic, fiber-reinforced, materials show that such interactions can affect the failure of material. Quadratic failure criteria similar to the von Mises criterion can account for the interaction of the stresses. There are various quadratic failure criteria for composite layers. The most frequently used ones are the Tsai-Hill, Hoffman, and Tsai-

2000

2000

1750

1750

1500

~

1250

'0~

1000 shear failure

I

:--:: :-:- -::5. 15

= (sin2 0 + r cos 2 0)'

30

=

s l(r- 1) sin 0 cos 01

.

45

e. (deg)

(a) r =

(jx

~ 1250


0)' 250

fiber ..--failure

1500

'fiber "'-..._failure

750

yt (jx

6.2.2 Tsai-Hill Criterion

60

crJcrx = 0.05

75

90

15

30

45 60 9, (deg)

75

90

(b) r = crjcrx = 0.15

Figure 6.5. Biaxial strength of unidirectional Boron/Epoxy.

1!:1

I~

1111111

238


Wu criteria (see Tsai, 1968 and Wu, 1974). In this subsection introduce the Tsai-Hill and Hoffman criteria. Hill (1948) proposed an extension of the von Mises yield to anisotropic materials with equal strengths in tension and compression. For a three-dimensional stress state, Hill's criterion is given F(cr 1 - cr2)2 + G(cr2 - cr3 ) 2 + H(cr3 - cr 1) 2 + L'r:~ 3 + Mti3 + Nr:i2 < 1, (6.2. where coefficients for the stresses are related to the yield strengths different directions. Tsai (1968) extended Hill's criterion to composites by relating those coefficients to the longitudinal, verse, and shear failure strengths, X,Y, and S, respectively. For a tension test specimen with fibers running along the length of the men, at failure (cr 1=X1, cr2 = cr3 ='tu = 0), Hill's criterion yields, (F+H)

and

(G+H)Y~=l,

since the failure stress in the 2 and 3 directions are both equal Yr The solution of Eqs. (6.2.8) and (6.2.9) for the coefficients produces 1

F=

2x;·

1 I G= y2- 2X 2 t

F(cr 1- cr2)2 + G(cr 2 - cr3)2 + H(cr3 - cr 1? + Pcr 1 + Qcr 2 + Ra3 + L't~ 3 + M'ti 3 + N'ti 2 < 1,

we evaluate the coefficients by matching the criterion to results obtained under simple load conditions as described for the Tsai-Hill criterion. In this case, for each normal stress direction we use results from two experiments, one for compression and one for tension. Evaluation of the coefficients (see Exercise 2 at the end of the chapter) yields the following failure criterion, which is referred to as the Hoffman criterion: (JI

(JI(J2

t

and

I

H=

2X;·

1 2 1 2 I 1 2 x2 cri + y2 0'2- x2 crlcr2 + s2 't12 < 1, t

(J2

1

1

X X2 - X X + Y Y2 + [ X - X tc

tc

tc

t

) c

1 1 't12 2 < 1. cr 1 + [ Y- y ) cr2 + Sl t

(6.2.13)

c

Example 6.2.2 Repeat Example 6.2.1 to determine the strength of Boron/Epoxy layers using the Tsai-Hill criterion as a function of the fiber orientation angle 0° ~ e ~ 90°. Substituting Eq. (b) of Example 6.2.1 into the Tsai-Hill failure criterion, Eq. (6.2.11), we obtain the following single equation for the maximum stress ax that can be applied to the layer: {(r 2X t2 - rYt2 + Y t2 )SZm4 + (r 2 Y 2t - rY t2 + X t2 )S 2n4

Under the plane stress assumption, Eq. (6.2.7) reduces to what is generally referred to as the Tsai-Hill criterion:

t

(6.2.12)

x; = 1.

Similarly, uniaxial tests in the 2 and 3 directions yield (F +G) Y~ =I

239

6.2 FAILURE OF FIBER·REINFORCED ORTHOTROPIC LAYERS

t

where the final coefficient N in Eq. (6.2.7) is obtained from a shear test with a pure 't 12 stress causing failure at S. If one of the normal stress components, cr 1 or cr2, is compressive then we replace the corresponding strength quantity, X1 or Y1, with its compressive counterpart. A generalization of the Hill's criterion that allows for different tensile and compressive strengths is offered by Hoffman (1967). Starting with the following quadratic form for the failure criterion:

+ [(r 2 - 2r + l)(x;r;- r;s 2 )]m2n 2 } cr; ~

x;r;s;.

For the pure axial tension (i) and pure axial compression (ii), the stress ratio r = 0; therefore, the safe level of stress is given by

(Jx

~

--.IX 2S2 t

sin S4 +

Y2t S2

XtYrSt cos 84 + (X t2 Yt2 - Yt2S 2) cos 82 sine"'

where, in the case of compression, X, and Y1 are replaced by Xc and Yc, respectively. Strength as a function of orientation is plotted in Fig. 6.6a and 6.6b for tensile and compressive stresses, respectively. In-

240


1400r:-~ 1200~1 I I

~ 1000 \ 1

~ 8o0 \ t:>

3000rl

\_ I I I I I I I

""'fiber failure

I I I

1 shear 1 1• /failure

•..•~ I\

I

~2ooo

~~

'7.1500

transverse I matrix 1 failure I

\ \ 200

I

2500~-

,._

'~ '

---.----~

30 (a)

45 9, (deg)

60

tensile loading

/

t:>

~-~

"' 1 'V!_ber

\

:

1 1 1

shear Jailure

1

1


\. J '

90

1

'

', -

~

o.;;

woo~''II

~ 800 • t:>~

.\

Figure 6.6. Axial strength of Boron/Epoxy layer based on Tsai-Hill criterion.

eluded in the figure as dashed lines are the stress limits from Fig. 6.4 for comparison. The strength predicted by Tsai-Hill is almost identical to the ones predicted by the maximum stress criteria used in Example 6.2.1 except in the neighborhood of the fiber-orientation angles 0, where two of the maximum stress criteria intersect. For example, the largest difference between the Tsai-Hill and a maximum stress criterion prediction is for the fiber-orientation angle of about 3°, where maximum stress criterion caused by fiber tensile failure and matrix shear failure is predicted at a tensile stress level of ax= 1264 MPa; see Fig. 6.6. For the same fiber orientation, Tsai-Hill predicts the failure to be at about ax= 893 MPa, an almost 30% lower strength than the maximum stress criterion. For cases of uniaxial tension or compression, the strengths predicted by the Tsai-Hill and maximum stress criteria are identical for the oo and 90° layers. In the case of biaxial loading, however, existence of transverse stresses can cause transverse matrix failure even for the 0° layer. For example, it was shown in Example 6.2.1 that for r =a/ax= 0.05, the failure of the layer is due to transverse matrix failure. Use of an interactive failure criterion, such as the Tsai-Hill, heightens the sensitivity of the failure load to the existence of biaxial stress state with small transverse stress. Failure predictions are plotted in Fig. 6.7a and

y

200f

\

~

1

d"

30 (a)

45

a, (deg)

75

60

~ 800 600

200

-

15

1000

400

~~\' / I /._~,{---"' Tsai-Hill

90

I1 I I

transverse,' matrix 1 failure 1

\\

.

I

45 60 75 9, (deg) (b) compressive loading

shear failure

\

I

1

30

I I I I 1 1

1

11

I ---=----"'

1400 I / 1-11._ ""-... jibe r 1200 1 1 failure

,.

1200t-l .... "'"'-.fiber \ failure

I

failure

1

~~ I

1400r 1

I I I

I

\

I

75

I

241

6.2 FAILURE OF FII!R·RI!INFORCED ORTHOTROPIC LAYERS

I I II

I I I I I

~-r

I/

failure

1

transverse matrix I

I I I

-~~'-- ?> __-__

90

r = O"/O"x = 0.05

..::::...,._-___,-/-;..,

15

30 (b)

45

60

a. (deg) r =O"/O"x =0.15

75

90

Figure 6.7. Biaxial tensile strength of Boron/Epoxy layer based on Tsai-Hill criterion.

6.7b for two levels of stress ratio, r = 0.05 and r = 0.15, considered in the previous example. For a 0° layer, the failure stress predicted by the Tsai-Hill criterion is substantially lower than the one predicted by the maximum stress criterion based on shear failure.

6.2.3 Tsai-Wu Criterion A more general form of the failure criterion for orthotropic materials under plane stress assumption is expressed as

F 11 af + F22 a~ + F66 -cf2 + 2F 1p 1a 2 + 2F16a 1-c 12

+ 2Fz6az'tu + F,al + Fzaz + F6-c12 < 1.

(6.2.14)

Evaluation of the strength coefficients F 11 , F 22 , Fw F 12 , F 16 , F2e Fl' F2, and F6 requires using the results of experiments on unidirectional fiber-reinforced specimens under simple load conditions. For example, if the failure stresses of a specimen loaded along the fiber direction in tension and compression are X1 and Xc, respectively, then we get F11 X~ + F 1 X1 = 1,

(6.2.15)

242


F 11 X~- F 1Xc= 1.

(6.2.16)

6.2

and, since the value of scomb is independent of the shear direction 2

F11r S~omb + F66S~omb- 2F,6rS~omb +FirS comb= 1.

Equations (6.2.15) and (6.2.16) are a system of two equations in two unknowns, solution of which yields F =__!_X x1 1

and

c

1

1

Fll

= Xtxc·

(6.2.17)

Another set of equations similar to Eqs. (6.2.15) and (6.2.16) is obtained by considering tension and compression tests in a direction transverse to the fiber direction. These equations are F 22 Y; + F 2 Y1 = 1,

(6.2.18)

= 1,

(6.2.19)

F22 Y~- F 2 Yc

243

FAILURE OF FIBER·REINFORCED ORTHOTROPIC LAYERS

(6.2.24)

The last two equations are almost identical except for the sign change in front of the F 16 term. Subtracting Eq. (6.2.24) from Eq. (6.2.23) yields (6.2.25)

F 16 =0.

A similar reasoning for an experiment that involves a combination of a 2 and 't 12 leads to the determination of

Fz6 = 0.

(6.2.26)

The only coefficient left to be determined, F 12, is the one that reflects the effect of interaction of the two normal stresses on failure:

and their solution is 1 1 Fz =y--yc

and

t

1 Fzz = ytyc·

(6.2.20)

and

F 66S 2 - F6 S = 1.

(6.2.21)

Solution of these equations leads to the determination of two more coefficients, which are given as F 6 =0

and

1 F66= S2'

(6.2.22)

In order to determine the coefficients that reflect the interaction of the normal and shearing stresses, namely F 16 and F26 , we need experiments that will generate combined stress states involving a 1 and -r 12 and a 2 and 't 12. For example, consider an experiment with l-r 121> 0 and a 1 > 0 with a 2 = 0 and a/'t 12 = ±r. If, for the experiment under consideration, the specimen fails at 't 12 =±Scomb' then from Eq. (6.2.14) we have

Fli?S~omb + F66S~omb + 2F,6rS~omb + F,rScomb = 1,

t

c

t

c

t

t

(6.2.27)

We next consider pure positive and negative shear tests where the specimens fail at a stress level of 't 12 = ±S. Therefore, F 66S 2 + F 6S = 1

X-X1) a,+ [1 y--y-1) az
1 a,+yy 2 1 az+s2't,z+2F,p,az+ 2 1 2 [1 XX

(6.2.23)

There are a number of different approaches one can devise to establish a failure criterion from this equation. For example, Narayanaswami and Adelman (1977) suggested F 12 = 0, which ignores the interaction of the two normal stresses. Usually, however, F 12 is assumed to have a nonzero value and the value is generally selected to make the criterion agree with one of the simple criteria discussed earlier. For example, F 12 =- 112x; with X1 = Xc and Y1 = Yc reduces to Tsai-Hill's criterion, and F 12 =- 112X1Xc to Hoffman's criterion. The criterion generally referred to as the Tsai-Wu criterion is a case where Eq. (6.2.27) is made to resemble the von Mises criterion of Eq. (6.1.14) as explained in the following. We define nondimensional stresses as

a~= ~a"

a;= ...JFzzaz,

't~z = ...JF66't,z,

(6.2.28)

and normalized strength coefficients as

F,

F~ = ...JFII'

Fz

F; = ...JFzz'

F12

F*--~ 12- 'IF,IF22

(6.2.29)

244


6.3 FAILURE OF LAMINATED COMPOSITES

and rewrite Eq. (6.2.27) as

cr*I2+ cr*22+ 't*122+ 2F*12cr*cr* I 2+ F*cr* I I + F*cr* 2 2< 1·

245

1400

1400

/200

1200

(6.2.30)

Note that in the case of isotropic materials with X1 =Xc = Y1 = Yc, we have F~ = F; = 0. Also the principal stress state will have 't~ 2 = 't = 0. 12 Therefore, Eq. (6.2.30) reduces to the von Mises criterion if we chose

/000

~

~ 800

J

J' Tsai-Wu

Tsai-Hill 400

1 F~2 =-2,

(6.2.31)

200

or

200

30

F12=

1

(6.2.32)

2.YXIXCYIYC .

cri

cr~

+ 'tiz_ 2

s

crlcr2 .Vxtxcytyc

+(_!_ _ _!_Jcr 1 +(_!_ _ _!_Jcr <1. XC

xt

yt

yc

2

(6.2.33)

Example 6.2.3 Repeat parts (i) and (ii) of Example 6.2.1 to determine the uniaxial tensile and compressive strengths of a Boron/Epoxy layer as a function of the fiber-orientation angle, 0° :::; e : :; 90°' using Tsai-Wu criterion F~ 2 =- ~For the uniaxial loading, the stress ratio, r = 0'/0'x is zero. Substituting Eq. (a) of Example 6.2.1 into the Tsai-Wu failure criterion, Eq. (6.2.33), we have the following quadratic equation to solve for the maximum allowable stress level: m4 n4 m2 n2 ( xx+yy+----:51 c

t

c

t

r==m=2=n=z=J cr2 .YxC xt YC yt

t

c

75

90

/5

30

45

60

75

90

9, (deg) (b) compressive loading

to the tensile failure of the layer, and the other one corresponds to the compressive stresses. Strength values as a function of orientation are plotted in Fig. 6.8a and 6.8b for tensile and compressive uniaxial loadings along with the strengths predicted by the Tsai-Hill criterion. Compared to the Tsai-Hill, prediction of the Tsai-Wu criterion is slightly more conservative for tensile failure and slightly less conservative for compressive failure, but the differences are relatively minor. A comparison of the Tsai-Hill and Tsai-Wu criteria presented in the previous example is for uniaxial tensile and compressive loads. It is expected, however, that the biggest difference between the two criteria would be for biaxial loads. Comparison of the two criteria under combined ax- cry and ax- Txy loads is left as an exercise; see Exercise 3 at the end of the chapter.

X

m2 n2 m2 n2J + (x-+y--x--y- ax= 1. t

60

Figure 6.8. Strength of Boron/Epoxy layer based on Tsai-Wu criterion.

For this choice the form of the Tsai-Wu criterion is

xtxc + ytyc

45

9, (deg) (a) tensile loading

c

There are two roots of this equation. The root with a plus sign in front of the terms under a radical is the stress that corresponds

6.3

FAILURE OF LAMINATED COMPOSITES

Once the strength of a single layer under a combined state of stress is established, we can proceed with the strength prediction of a laminate. Laminate strength prediction is carried out by evaluating the stress state within each layer based on the classical lamination theory

246


described in Chapter 2 and implementing one of the criteria described in Section 6.2. As mentioned in the introduction to this chapter, however, the important issue is to establish what constitutes laminate failure. In a laminate, stresses in layers with different orientations or different material properties are generally different. Therefore, some of the layers are likely to reach their limiting stresses before the rest of the layers and fail first. This is generally referred to as first-ply failure. For laminates with ductile layers (layers made of conventional ductile metals such as aluminum and steel), such failures are in the form of local yielding of the layer upon which the loads around the region of local failure are gradually transferred to the adjacent layers without a catastrophic failure. Although yielding implies failure initiation, the material is usually capable of sustaining nearly constant loads in a plastic state as the loading is increased. A fiber-reinforced laminate, on the other hand, may or may not be able to carry loads beyond failure initiation, depending on the nature of the first failure and to some extent on the sign of the stresses. There are two factors that contribute to this behavior. First, the constituent materials for most composite laminates are brittle in nature and do not tolerate local failures well. Lack of local yielding, which . helps release local stress concentrations around the failure initiation ·.• points, can cause unstable brittle fracture leading to catastrophic failure of a laminate. This behavior is further complicated because of the layered nature of composite laminates with weak interlaminar normal strength. Under compressive loads, for example, failure of a loadcarrying layer may in.duce through-the-thickness imperfections and damage between the layers of a laminate that can create local threedimensional stress states. The interlaminar strength or the interlaminar fracture toughness (which is a measure of the resistance of a material to crack growth) of laminates may not be sufficient to resist rapid crack growth between the layers, which further destabilizes the laminate. The second factor is the large difference of stiffness and strength between the two principal material (fiber and transverse to the fiber) directions in a layer. In a laminate made of a mix of various fiber orientation angles and loaded along the x-coordinate axis, 0° layers (if there are any), which are oriented along the x axis, owing to their large stiffness carry most of the applied loads. For example, for a

6.3 FAILURE OF LAMINATED COMPOSITES

247

Graphite/Epoxy laminate E/£2 = 15.4 with 50% oo and 50% 90° layers, almost 94% of the loads will be carried by the oo layers. In a quasi-isotropic laminate (with 0°, ±45°, and 90° layers), although the oo layers make up only 25% of the total thickness, they carry 58% of the applied loads. Under these circumstances, failure of a loadcarrying layer may dictate the failure of the laminate. This is especially true for laminates tested under load-controlled rather than displacement-controlled loading, as there will be a sudden jump in the stresses in the remaining layers. This is precisely the reason for placing the load-carrying 0° layers in the interior of laminates for damage tolerance considerations. By making the ±45° layers the outermost layers, which are susceptible to experiencing damage due to external impact, possible loss of load-carrying capability is minimized. Although the layers oriented along the load axis carry a major portion of the applied loads, their strength properties along the fiber direction are also substantially higher than their transverse strength. It is, therefore, common that the lightly loaded layers may experience the first failures in a laminate. For example, for a cross-ply laminate [0/90], loaded in compression along the x axis, the 90° layer might carry less than 6% of the applied loads. However, because of its low strength, matrix crushing failure is the likely local failure in that layer. Upon crushing, the stiffness of the layer will be locally reduced and the other layers in the laminate will carry a larger portion of the applied loads, raising the stresses in them. However, since the low stiffness matrix carries only a small fraction of the applied loads, the increased stresses in the other layers may not cause failure and overall laminate integrity may be maintained. This may also be true in case of transverse matrix tensile failure of a layer placed at a 90° orientation from the major load axis. The transverse tensile failure of the layer appears in the form of matrix cracks between the fibers. Although the stiffness properties of the laminate as a whole may change noticeably, transverse cracking of a layer may simply result in internal redistribution of the stresses within the laminate so that still larger loads can be sustained. A highly schematic representation of the load versus deflection diagram of a laminate with such a failure is shown in Fig. 6.9 (in an actual test, the kinks in the plot are also associated with discrete jumps in load or deflection or both, depending on the testing conditions). Note that upon failure, the slope of the initial load path changes, demonstrating the stiffness reduction. As stated in the

248


249

6.3 FAILUAE OF LAMINATED COMPOSITES

6.3.1

Failure under In-Plane Loads

Load

For plane stress, implementation of failure criteria is straightforward. Strains are constant through the thickness, and stresses in layers with the same orientations are identical. Therefore, layer-by-layer application of either a strain-based or a stress-based criterion requires calculations to be performed only for layers with different properties or fiber orientations. As will be discussed briefly in Section 6.3.2, prediction of failure under bending loads and deformations is more involved. Deflection

Figure 6.9. Progressive failure of a composite laminate.

introduction, the number of failures will keep increasing progressively (therefore referred to as progressive failure) until the failure of an additional layer will cause failure of all the remaining layers. An example of this kind of a failure will be discussed in Section 6.3.1. In some cases it is not desirable to have local damage, even in the form of transverse matrix cracks, that changes laminate elastic properties. Also such local failures are likely to create weak links in the laminate that may initiate future failures. Therefore, it is often assumed that initial failure of a layer implies laminate failure. This is referred to as first-ply failure criterion. The first-ply failure criterion provides a conservative estimate of the laminate failure load since first failure in the form of matrix cracks may not lead to laminate failure. Even in those situations where one may be willing to accept local failures in the form of transverse matrix cracking, a first-ply failure criterion might still be useful. Layer-by-layer application of either the maximum stress or maximum strain failure criteria with only X1 and Xc (ignoring the ~and Yc strengths) would serve for the purpose. Finally, it is important that any residual stresses or strains that may exist in a composite laminate are accounted for during the strength analysis. As explained in Chapter 3, the residual stress levels introduced during manufacturing depend not only on the operating temperature and humidity, but also on the stacking sequence of the laminate. These residual strains may either help or worsen the load-carrying capacity of a ply, depending upon the direction of the applied loading.

Maximum Strain Failure Criterion. For specified values of laminate stress resultants (Nx, Ny, and Nxy) or average laminate stresses (
the implementation of this failure criterion calls for the evaluation of strains along the principal material axes for each fiber orientation. The laminate strains are calculated from

{ £~}£~ = A- lNxl:Y = hA- lcrxl~Y = A*- lcrxl~Y 1

1

1;y

xy

(6.3.1)

1

xy

,

xy

where A* = A/h, with h being the laminate thickness (see Chapter 2). With the laminate strains known, strains for each layer in the principal material system can be calculated using the engineering strain transformation matrix of Eq. (2.4.16), 0

1

£ {'Y12

= T~k) {££~

f2 }

}

YxY

(k)

=

l

n2

m2 n2

mn -mn

m2

-2mn 2mn

1 {£~}

m2 - n2 (k)

£

0

(6.3.2)

YxY '

where m=cos9k,n=sin9k. We then use Eqs. (6.2.4)-(6.2.6) for each layer to check for failure. Maximum Stress Failure Criterion. Given the strains in the principal material directions for each layer, stresses in the kth layer are calculated from

crl} cr2

{'t"

12 (k)

-lQu - Ql2 0

Ql2 Q22 0

00 Q66

j{£1} £2

"{12

. (k)

(6.3.3)

250


Substituting Eqs. (6.3.1) and (6.3.2) into the above equation and rearranging we have

{~:}

= K(k) {

12 (k)

:J} =K
Yxy

K(k)

=

r

-2mnQ66

2mnQ66

(Oo) -

45

E) ol = 3.302

(6.3.5)

Example 6.3.1 A (Ozl±45)s Kevlar/Epoxy laminate is loaded by biaxial loads with N/Nx = 0.5. Draw the strength envelope for the laminate in the crx-crY plane based on maximum strain criterion and determine the maximum value of Nx that can be applied. Following is a set of typical elastic and strength properties for Kevlar/Epoxy: E 1 = 76.00 GPa, E 2 = 5.50 GPa, G 12 = 2.30 GPa, 3 J.112 = 0.34, and t = 0.125 X 10- m. f~ = 18.42 X 10-3, f~= 3.09 X 3 3 10- , f~ = 2.18 X 10- , f~ = 9.64 X 10-3, and y·; 2 = 14.78 X 10-3 . For the (Ozl±45)s Kevlar/Epoxy laminate, the in-plane stiffness matrix is

and

-

91 · 31 'txy' -

crx + 31.72 cry+ 45.65 ::rxy'

E~45 ol = 3.302

crx + 31.72 cry- 45.651xy'

y)~o) = -40.30

crx + 97.12 cry.

Equating the strains to their appropriate limiting values specified in the problem statement produces the equations for the failure envelopes. The failure envelope for the 0° layer is given by the four lines (no shear failure since iJ~ol = 0) 23.45 crx-16.85 cry= 18.42,

23.45

crx- 16.85 cry= -3.09

and -16.85

crx + 80.28 cry= 2.18,

-16.85

ax+ 80.28 cry= -9 .64.

These four lines constituting the failure envelope for the 0° layer are shown in Fig. 6.1 Oa. For the 45° layers, the failure envelope would also normally be defined by six equations. However, due to the lack of applied shear, we have £ 1 = £ 2 and, therefore, the normal strain component with the smaller limiting value controls the failure. In this case, failure is controlled by transverse tension failure 3.302 crx + 31.72 cry= 2.18,

[ 0.02345 -{).01685 0.0 ] 1 A*- 1 = -0.01685 0.08028 0.0 GPa 0.0 0.0 0.09131

ax+ 80.28 cry,

where all average stresses are in GPa and the strains are in millistrains (10- 3 m/m). Similarly, strains in the ±45° layer in the principal material directions are calculated from Eq. (6.3.2) as

The following example demonstrates the use of the maximum strain criterion for a laminate under combined loads.

l

Eioo) = Ey = -16.85

'¥12 - 'Yxy-

(m2- n2)Q66

[ 50.216 10.538 0.0 A= 10.538 14.668 0.0 MN/m 0.0 0.0 10.952

23 .45 -ax - 16 . 85 cry, -

(6.3.4)

where the matrix K is given by m2QII + n2QI2 n2Q,I + m2QI2 mn(Q 11 - Q 12)] m2QI2 + n2Q22 n2QI2 + m2Q22 mn(Q 12 - Q22 ) •

so that from Eq. (6.3.1) we have f (Oo) 1 -Ex-

y

251

6.3 FAILUFI! OF LAMINATED COMPOSITES

longitudinal compressive failure 3.302 crx + 31.72 cry= -3.09,

252


253

6.3 FAILURE 0111' LAMINATI!D COMPOSITES

Oy (GPa)

(a) 0° layers

0"- e~

0.4

,

,

~

;_.~/

, Ox(GPa)

1

= ;,:S(]y. ~-c::_""G'"'

-0.4 t compression

(b) +45° or -45° layer

Figure 6.11. Failure envelope for the laminate.

Oy (GPa) 0.4 0.3

to the failure of the 0° layer due to transverse tension. Values of the maximum average axial stress ax that can be applied are determined by picking the appropriate equation corresponding to either point A or point B and substituting cry= 0.5 ax. At point A, failure is due to longitudinal compressive strain in the 45° layer, longitudinal compression

Figure 6.10. Failure envelopes for the

oo and 45o layers.

3.302 ax+ 31.72 cry= -3.09

or

(3.302 + 31.72 X 0.5) ax= -3.09,

which yields, crx=-3.09/19.16=-0.161GPa or Nx=hax=-161xl0 3 N/m. Transverse tensile failure of the 0° fibers governs the failure at point B:

and shear failure -40.30 ax+ 97.12 cry= 14.78,

-40.30 ax+ 97.12 cry= -14.78.

The failure envelope is drawn in Fig. 6.10b. Superposition of the failure envelopes for each layer gives the failure envelope for the laminate shown in Fig. 6.11. The thin line marked 0'/0'x = 0.5 (N/Nx = 0.5) shows the specified combination of the biaxial loads. This line intersects the failure envelope at two points, A and B, which mark the load limits. Keeping the same 0'/0'x ratio, the safe region of the stress plane will be any point located between A and B. Point A corresponds to compressive failure of the 45° layer in the fiber direction. Point B corresponds

-16.85 ax+ 80.28 cry= 2.18

and

(-16.85 + 80.28 X 0.5) ax= 2.18,

ax= 2.18/23.3 = 0.0936 GPa or Nx = h ax= 93.6 x 103 N/m. Altogether -161

X

103 N/m ~ Nx ~ 93.6 X 103 N/m.

254


Progressive Failure of Laminates. In the context of classical lamination theory, a prediction of the additional load capacity of the laminate is based on modeling of the laminate's stiffnesses subsequent to failure of a single ply or a group of plies. Such modeling was, and continues to be, an active area of research. In order to define the constraints for preliminary design, a simplistic and conservative approach can be adopted, where the elastic constants for failed plies are set to very low numerical values. These failed plies are assumed to occupy the same geometric location in the laminate as prior to failure, and the laminate analysis is used to predict finite strains and curvatures in these plies. It is important to emphasize that these strains are somewhat meaningless and should not be the basis of any post-failure analysis; the stress levels in these plies will be vanishingly small quantities because of the choice of elastic constants for the failed plies. A less conservative approximation used in the modeling of reduced effective stiffness of laminates is based on a selective reduction of the elastic constants E" E2, and G12 • Such approximations are derived from a more detailed understanding of the failure mechanism of a particular ply. As an example, if the ply failure is due to matrix cracking, it is assumed to have lost the ability to carry any shear loads. Also, if the applied load produces a tensile load that further opens the cracks in the matrix material, the elastic constant E 2 can also be assumed to be zero. However, the fiber direction will continue to carry loads as before. The reader should note, however, that the failed matrix material can be responsible for introducing local fiber instabilities, particularly if a compressive load is applied in the fiber direction. In such instances, there would be a need to modify the elastic constant, E" of the failed ply, and a straightforward approximation for this case is more difficult to obtain. Finally, failure of the fibers in such an approach would suggest that each of the constants El' E2, and G12 for that particular ply be set to zero. A more refined version of a progressive failure model has recently been developed by Tsai (1992), who proposed two types of material property degradation. These are the fiber and matrix modes of degradations, which are based on micromechanics equations. The matrix mode of degradation assumes formation of microcracks in the matrix material under transverse tension and involves a set of empirical constants. The fiber degradation model also uses an empirical fiber degradation factor. In this case, both the fiber and matrix properties are degraded, leading to changes in every stiffness property of the layer (£" E 2 , G 12, and v12 ) through mi-

255

6.3 FAILUFIE OF LAMINATED COMPOSITES

cromechanics relations. A more detailed description of the model is provided in Section 9 of Tsai (1992). Laminate strength analysis is illustrated with the help of the following example problem, where the approximations described in the preceding discussion are used. Example 6.3.2

Consider a symmetric cross-ply laminate with a layup sequence of [90/0i90 3Jr where each ply is 0.25 x 10-3 m thick. For the T300/5208 Graphite/Epoxy system, the longitudinal, transverse, and shear stress to failure and the stiffness components are given as follows: xt =XC= 1500 MPa,

E 1 =181GPa,

yt = 40 MPa,

E 2 =10.3GPa,

yc = 246 MPa,

G 12 =7.17GPa,

s = 68 MPa, v12 =0.28.

The laminate is subjected to a tensile uniaxial stress resultant Nx in the direction of the fibers of the oo layer. Determine the value of Nx, at which the first ply fails, and the ultimate loadcarrying capacity of the laminate N:rlt· Use the maximum strain criterion for the calculations. Solution

The maximum allowable strains for fracture under tension loading are first computed as follows: r

El

xt = E = 0.00828,

t

E2

yt = E = 0.00388.

(a)

2

1

For the in-plane loading as defined and the laminate in-plane stiffness computed from the given ply characteristics, the laminate strains can be obtained in terms of Nx from the following equations:

\

Nx~ r288.2 0 = 8.691 0

0.0

8.691 288.2 0.0

0.0 ] 0.0 21.51

{Ex1 MN Ey ---;-· Yxy

(b)

256


In the cross-ply laminate considered in this example, it is possible to determine the order of ply failure by inspection. Note that Ex is the larger of the two extensional strains, and since Ex corresponds to the transverse direction strain for the 90° layer, this layer would fail when Ex= E~ = 0.00388. Substituting the value of Ex to the right-hand side of Eq. (b), we first solve for the transverse strain EY = 0.117 x 10-4 from the second row, then using the first row of Eq. (b) we can determine the load corresponding to failure as ~pf = 111.7KN/m where the superscript "fpf' denotes "first-ply failure." The shear strain component 'Yxy = 0.0 in this case. To determine if the laminate has any residual load-carrying capacity, a new in-plane stiffness matrix must be evaluated. A conservative approximation is to assume that all stiffness contributions from the 90° layers can be neglected, which yields the following modified in-plane stiffness matrix: (c)

272.7 4.345 4.345 15.52 [

0.0

0.0

j

0 0

=

[288.2 8.691 8.691 288.2 0.0

0.0

23

246 x wE~ = 10.3

272.7 4.345 4.345 287.0

(6.3.6)

[

0.0

0.0

0.0 ] 0.0

~N ·

(6.3.8)

10.76

For this assumption, the incremental load is computed as

10.76

0.0 ] {0.00441 0.0 &Y MN_ L\ m 21.51 'Yxy

=o.02388

and this confirms that fiber failure in tension will precede matrix failure for the 0° layers. If a less conservative approximation for the failure load is desired, one can selectively reduce the stiffness coefficients of the failed 90° layers on the basis of the mode of failure. Once again, for a cross-ply laminate, matrix failure in the 90° layers is most likely to be a series of cracks in the matrix material that run parallel to the fiber direction. The layer stiffness in the fiber directions still contributes to the laminate stiffness, and hence one chooses E 1 = 181 GPa and E 2 = G 12 = 0.0 for this layer to obtain a modified laminate in-plane stiffness matrix as follows:

0.0] MN 0.0 m ·

Wx = 1199.7 KN/m, and is somewhat higher than that obtained

The remaining oo plies now carry all of the applied load and would fail when the strain Ex in the direction of the applied loading reaches a value E11 = 0.00828, which implies an incremental strain ~Ex= 0.0044. Hence, the incremental load necessary to reach these strain levels is computed from the following equations:

/1NXJ

257

6.3 FAILUAI! OF LAMINATED COMPOSITES

(d)

(6.3.7)

as 11Nx = 1194.6 KN/m. The ultimate load is thus obtained as

N~it=Nr+mx= 1306.3KN/m.

Note that the compressive strain in the transverse direction for the oo layers at this load is EY = -0.00134; the allowable strains in compression in the transverse direction are obtained as

with the more conservative approximation. The ultimate load is obtained as N:,Jt = ~pf + 11Nx = 1311.4 KN/m.

6.3.2

Failure under Bending Loads

Under bending loads, the strains vary linearly through the thickness of the laminate. The implication of the linear variation of the strains is that layers with the same properties and orientations will not reach their limiting stresses or strains simultaneously, as they do in the case of plane stress. Therefore, through-the-thickness location of failures will be more localized in the case of bending-related failures, and hence, the degree of conservatism in first-ply failure criterion will be increased. For isotropic monolithic plates in bending, failure occurs near the surface where the strains are highest. For composite laminates this is not necessarily the case. Under combined loads it is possible that, out of two layers with the same orientations located on the same side of

258


the mid-plane of the laminate, the one closer to the mid-plane may fail earlier than the one closer to the surface. For example, if the laminate is under loads that generate constant shear stresses through the thickness but linearly varying normal stresses in each layer, the existence of the larger normal stresses close to the laminate surface may dissipate the effect of shear stress in those layers preventing shear failure. Yet the same shear stress may be high enough in the layer closer to the laminate mid-plane to cause failure. Furthermore, even if the laminate is symmetric, strength criterion needs to be checked on both sides of the laminate because of the different allowables in tension and compression. Therefore, to be on the safe side, evaluation of the strength criteria in every layer is advisable.

REFERENCES

4.

For the unidirectional Boron/Epoxy layer of Examples 6.2.1 and 6.2.3, plot the failure envelopes for e = 0°' 45°' and 90° fiber orientations as a function of (a) r= ajax under combined ax-ay stresses and (b) s = :rx;ax under combined ax-1:xy stresses in the range 0 ::::; r::::; 0.25 and 0 ::::; s ::::; 0.25, respectively, using the TsaiHill and Tsai-Wu failure criteria.

5.

An off-axis unidirectional Glass/Epoxy layer (£ 1 = 18.6 GPa, £ = 8.7 GPa, G12 = 4.14 GPa, and v 12 = 0.45) with its principal ma2 terial axis rotated from the laminate X axis by 8 degrees is loaded in shear resulting in a pure shear stress state 'txy· Determine the maximum value of the shear stress 'txy that can be applied to this layer as a function of the fiber-orientation angle, oo < e < 90°, using (a) Maximum stress criterion (b) Tsai-Hill criterion and plot and compare the results. The strength properties of the unidirectional Glass/Epoxy layer are X 1 = 1062 M P a, Xc = 610 MPa, Y1 =31 MPa, Yc= 118 MPa, and S = 72 MPa.

6.

A Boron/Epoxy laminate (see properties in Exercise 1) with (0/±45/902)zs stacking sequence is loaded by biaxial loads. Draw the strength envelope for the laminate in the ax-ay plane based on Tsai-Hill criterion. (a) Determine the maximum value of pure axial tensile and compressive average stresses that the laminate can carry without failure. (b) Determine the maximum axial average tensile stress that the laminate can carry if it is loaded such that ajax= 0.5.

EXERCISES 1.

2.

The unidirectional Boron/Epoxy layer of Example 6.2.1 (E 1 = 204GPa,E2 =18.5GPa, G 12 =5.59GPa, v 12 =0.23,and t = 0.125 x 1o-3 m) is to be loaded by combined axial and shear stresses with :rx/ax = r. Determine the strength of this layer based on the Tsai-Hill failure criterion as a function of the fiber orientation in the range oo : : ; e : : ; 90°, and plot the strength for r = 0.05 and r = 0.2. Typical strength properties of Boron/Epoxy layers are X 1 = 1260 MPa, Xc = 2500 MPa, Y1 = 61 MPa, Yc = 202 MPa, and S = 67 MPa. Derive the strength coefficients F, G, H, P, Q, R, L, M, and N for Hoffman's strength criterion of Eq. (6.2.13). Plot the failure envelope and show the load limits under biaxial load a/ax= 0.1 for Glass/Epoxy with the following properties. Elastic properties: £ 1 = 5.60 x 106 psi, £ 2 = 1.20 x 106 psi, G 12 =0.60 x 106 psi, v 12 = 0.26, and t = 0.1 in. Strength properties: X 1 = 154 x 103 psi, Xc = 88.5 X 103 psi, Y1 = 4.50 X 103 psi, Yc = 17.1 X 103 psi, and S = 10.4 x 103 psi.

3. For the unidirectional Boron/Epoxy layer of Example 6.2.1 and Example 6.2.3, plot the failure envelopes as a function of the fiber orientation angle e under (a) combined stresses of r=a/ax= 0.15, and (b) s = :rx/ax= 0.15 using the Tsai-Hill and Tsai-Wu failure criteria.

259

REFERENCES Beer, F. P. and Johnston, E. R. Jr. (1992). Mechanics of Materials. Second Edition. McGraw-Hill, New York. Chamis, C. C. (1974a). "Micromechanics Strength Theories." In Composite Materials, Vol. 5, pp. 93-151. Broutman, L. J. and Krock, R. H. (eds.) Academic Press, New York. Chamis, C. C. (1974b). "Mechanics of Load Transfer at the Interface." In Composite Materials, Vol. 6, pp. 31-77. Broutman, L. J. and Krock, R. H. (eds.) Academic Press, New York.

280


Hashin, Z. (1980). "Failure Criteria for Unidirectional Composites." Journal of Applied Mechanics 47, 329-334. Hill, R. ( 1948). "A Theory of the Yielding and Plastic Flow of Anisotropic Metals." Proceedings of the Royal Society A(l93) 281-297. Hoffman, 0. (1967). "The Brittle Strength of Orthotropic Materials." Journal of Composite Materials 1(2), 200-297.

7

Narayanaswami, R. and Adelman, H. M. (1977). "Evaluation of the Tensor Polynomial and Hoffman Strength Theories for Composite Materials." Journal of Composite Materials 11(4), 366-377. Schultheisz, C. R. and Waas, A.M. (1996). "Compressive Failure of Composites. Part I: Testing and Micromechanical Theories." Prog. Aerospace Sci. 32, 1-42. Soni, S. R. (1983). "A Comparative Study of Failure Envelopes in Composite Laminates." Journal of Reinforced Composites and Plastics 2(1), 34-42. Tsai, S. W. ( 1968). "Strength Theories of Filamentary Structures." In Fundamental Aspects of Fiber Reinforced Plastic Composites, pp. 3-11. Schwartz, R. T., and Schwartz H. S. (eds.) Wiley Interscience, New York.

STRENGTH DESIGN OF LAMINATES

Tsai, S. W. (1992). Theory of Composites Design. pp. 9-8-9-14,7-9-7-11,7-137-15. Think Composites, Dayton Ohio. Wu, E. M. (1974). "Strength and Fracture of Composites." In Composite Materials Vol. 5, pp. 191-247. Broutman, L. J. and Krock, R. H. (eds.) Academic Press, New York.

The simple nature of the in-plane stiffness design problems considered in Chapter 4 lends itself to graphical approaches or other approaches that do not require computerized solution techniques. However, the complex nature of the strength criteria discussed in the previous chapter rarely permits us to incorporate strength constraints into design problems without the aid of computer-based techniques. This is particularly true for problems involving combined bending and in-plane response of laminates. However, in order to provide an insight into the design of laminates with strength constraints, we limit the discussion of this chapter to the in-plane response of laminates. The first section of this chapter extends the use of Miki's graphical technique, introduced earlier for in-plane stiffness design, to designing laminates with strength constraints (Miki 1986; Miki and Sugiyama, 1991). Laminates with fiber-orientation variables that are either continuous or discrete valued are considered. Section 7.2 discusses the general aspects of the use of traditional mathematical optimization techniques, which treat the design variables as continuously varying quantities, for problems with strength constraints. Such traditional optimization techniques commonly require derivatives of the problem constraints as well as objective function.

Jl

I,

liill

261

IIIII 1

1

111 1

262


Therefore, derivation of the derivatives of the strength constraints with respect to the thickness variables is demonstrated. In Section 7.3 treatment of stacking sequence design, which requires use of discrete variables within the context of strength design, is discussed. A linearization scheme for forming integer linear programming is introduced. Finally, examples of a more versatile approach for nonlinear discrete optimization problems, namely genetic algorithms introduced in Chapter 5, is demonstrated.

7.1

263

GRAPHICAL STRENGTH DESIGN

Given a set of specified stress resultants or average stress loading cry, and txy at a point in the laminate, strains are calculated from Eq. (6.3.1):

ax,

j:j} = A*-~~~:].

hxy

(7 .1.1) .

xy

A*

A

where = A/h. From the definition of the matrix in Table 2.1, the matrix for a balanced laminate is given as

7.1


In this section we use Miki's graphical procedure to design laminates with stress or strain constraints. Equations needed for calculation of stresses and strains in the layers of a laminate were derived in Chapter 2. In the following, we make use of in-plane lamination parameters to express laminate strains in the general laminate coordinate system and ply level strains in the principal material directions for graphical representation on Miki's in-plane lamination diagram.

7.1.1

Design for Specified Laminate Strain Limits

Although it is not common practice, the easiest application of strength constraints is at the laminate level where we may limit the magnitude of strains along the laminate x-y axes. The reader should be warned that this approach is more of a habit inherited from designing conventional monolithic materials and should be avoided for laminated composites. However, equations developed in this section form the basis for the discussions in Sections 7 .1.2 and 7 .1.3 and, hence, justify retaining this section. The convenience of this approach is because calculation of the laminate strains does not require explicit knowledge about the orientation of the individual plies in the laminate. All the information about the laminate is contained in the in-plane lamination parameters, V~ and v;, defined in Section 4.2.1. In order to demonstrate this aspect, we revisit the basic relations, starting with the stress-strain relation for a laminate defined in Chapter 6.

A*

A* = UU

1

4

[ 0

U U1

oo1+ lu2o -Uo ool ~ + l-Uu3

0

U5

4

3

2

0

0

0

0

-u3 U3 0

l

~ v;, -u3

(7.1.2)

where the U/s are the material invariants defined by Eq. (2.4.20), and V~ and v; are the in-plane lamination parameters. For a balanced symmetric laminate made up of I ply groups with orientations ±ei, where i = 1, ... , I, the in-plane lamination parameters are given as

v~

=I vk cos 2ek

and

v; =I vk cos 4ek.

(7 .1.3)

k=l

k=l

where vi is the fraction of the total thickness occupied by the ply group ±ei. Clearly, for the purpose of laminate strain calculations, all the information about the stacking sequence of the laminate is contained in the two in-plane lamination parameters. Inverting the from Eq. (7.1.2) and substituting into Eq. (7.1.1) we obtain

A*

(U 1 - U2 V~ + U3 v;) ax- (U4 - U3 v;)

2 2 2 •2

cry

E~ = (U -U )-U V 1 +2U3(U 1 +U4 )V 3* 2 4 1

o _ EY-

-(U4 - U3 v;) ax+ (U 1 + U2 V~ + U3 v;) (Ui-

U~)- U~V? + 2U/U 1 + U4 )

'

cry

v; '

(7.1.4)

(7 .1.5)

264


t

(} -----.:2:

v·· 3 3

(7.1.6)

Yxy- U - U 5

If limits on the laminate strains, E~, E;, andy~, are prescribed, then Eqs. (7 .1.4 )-(7 .1.6) can be used to find the regions of the in-plane lamination diagram that correspond to laminates satisfying the strain limits for a specified load combination. Contours of the boundaries of these regions are obtained by solving Eqs. (7 .1.4 )-(7 .1.6) for v; in terms of the v; with the strains set to their limiting values 2 aV* 2 U4cryUlcrx - + (U 21 - U 42) Ex+ a U2crx - V*1 - U 2Ex 1 Ulax +a)- 2U/U1 + U4 ) E~

(7.1.7)

V3=--~--~~--------~--~~~--~~-

2 aV* 2 U4crxUlay+ (U 2I - U 42) Eya U2cry - V*I - U 2Ey I Uicrx +cry)- 2UiU1 + U4) E; '

(7.1.8)

V:3-- -txy + U5yaxy

(7.1.9)

v·

3

*

uya 3 xy

7.1

265


The laminate should also have the largest possible stiffness in the x direction. From Eq. (2.4.20) we calculate U1 = 76.3682GPa, U 2 = 85.7325 GPa, U3 = 19.7104 GPa, U4 = 22.6074 GPa, and U5 = 26.8804 GPa. Substituting the U/s into Eqs. (7.1.7)-(7.1.9), bounds for the regions of the lamination diagram that satisfy the strain limits are obtained:

v; = -0.6108- 2.6089 v~ + 3.5787 v? v; = -16.3785 + 26.0115 V~ + 35.6805 V?

for the E~ constraint,

v; =0.7296

for the Y'zy constraint.

for theE~ constraint,

These are plotted in Fig. 7 .1. A check of the strains at v~ = v; = 0 establishes this point as feasible and helps us decide the feasible side of each constraint. The in-plane lamination parameters of a laminate that has the largest axial stiffness correspond to point A on the diagram,

Vj

However, there are infinitely many laminates with various ply orientation combinations that correspond to the values of the in-plane lamination parameters inside the feasible region, and there is no guarantee that a selected laminate will not fail because of excessive strains in the principal material direction of the individual layers. That is, the laminate level strain limits must be used with extreme caution (if not completely avoided) since they do not represent a type of formal failure criterion we presented in the preceding chapter. The following example demonstrates the use of the in-plane lamination diagram to design laminates with constraints on laminate strains.

Example 7.1.1 The material properties of a Graphite/Epoxy material are given as E 1 = 181 GPa, E2 =10.3 GPa, G 12 =7.17 GPa, and 1/12 =0.28. Design a laminate that will withstand an average combined stress of ax= 500 MPa, cry= 250 MPa, and txy = 100 MPa, without exceeding laminate strains of E~ = 0.008, E~ = 0.004, and y~ 2 = 0.008.

Figure 7.1. Graphical representation of laminate strain limits.

266


which is the intersection of the shear strain and the transverse strain limits, with coordinates of~= 0.4180 and v; = 0.7296. If we assume that the laminate has two different fiber orientations with one of them being 0° layers, Eq. (4.2.10) can be solved for the fraction of the laminate with oo layers and the second fiber orientation, yielding v2 = 0.3292 and 92 = ±70.07°. As stated earlier, it is not guaranteed that the individual layers will satisfy strain limits in the principal material directions. In this case, assuming the ply strain limits (E~, E~, and y~ 2 ) to be same as the specified laminate strain limits, the oo layer of the laminate passes the test automatically. For oo layers, strains in the principal material directions are same as the laminate strains. For the ±70.07° layer, we can calculate the strain in the principal material directions from Eq. (6.3.2). For this problem, strains in the ±70.07 layer are E 1 = 6.5276 X IQ- 3, E2 = 1.1623 X IQ- 3, and y12 = -5.9424 x I0-3 . Therefore the laminate is safe against individual ply failures. As demonstrated in Example 7 .1.1, in order to design laminates subject to stress or strain constraints in the principal material directions of individual layers, we need to know the orientations of the individual layers, the very same quantity that we are trying to design. If the ply orientation is oo degrees, the ply strains are same as the laminate strains. For ply orientations other than 0°, the ply strains in the principal material directions need to be calculated from Eq. (6.3.2) and the stresses from Eq. (6.3.4) [see Section 2.4.2 and equations (2.4.12)(2.4.16) for more detail]. In the next section we discuss the implementation of a graphical design procedure for a special class of laminates with ply-level strain constraints.

7.1.2

Design of Laminates with Two-Fiber Orientations

With the laminate strains determined, calculation of the ply level strains requires the use of transformation for layers with fiber orientations other than 0° plies. From Eq. (6.3.2) the strains in the principal material directions of the kth layer are EkI

= m2Eox + n2Eoy + mnyoxy'

7.1

267


Ek2

= n2E"x + m2E"y -

mny"xy'

2 Yk12 = -2mnE + 2mnEy + (m 0

0

X

0 n 2)yxy'

(7 .1.10)

where m = cos 9k, n = sin 9k, and the laminate strains are obtained in terms of the in-plane stiffness parameters and the applied average stresses in Eqs. (7.1.4)-(7.1.6). In order to show graphically the plylevel strains on the lamination diagram, regardless of the orientation of the ply, we need to express m and n in terms of the lamination parameters v~ and v;. We start with the definitions of the lamination parameters of Eq. (7.1.3). The simplest general case is a laminate with only two different stacks of fiber orientations 9 1 and 92 • With these two-fiber orientation angles, it is possible to generate values of the in-plane lamination parameters that cover the entire design space of Miki's diagram and are given by ~

= v 1 cos 29 1 + v 2 cos 292

and

v; = v 1 cos 49 1 + v2 cos 492 .

(7 .1.11) For given volume fractions v 1 and v2 = 1 - v" we can solve for the cosines of the ply orientation angles in terms of the in-plane lamination parameters. These equations are given in Eq. (4.2.11): cos 29 1 = T 1 and

(7.1.12)

cos 29 2 = T2,

where v2(1-

T~=~±

2~2

+ v;)

and

2v 1

T2 =~+

v 1(1- 2~ 2

+ v;)

2v2

(7 .1.13) Then, using basic trigonometric identities, Eq. (7 .1.1 0) can be written as e,_l+T1 0 l-T1 o sin(cos- 1 T1) El 2 Ex+ 2 Ey + 2

Yxy'

e,_l-T1 0 l+T1 o sin(cos- 1 T1) E2 2 Ex+ 2 EY2

Yxy'

o

o

1 1 0 0 r12 = -sin(cos- T)I E + sin(cos- T) 1 Ey + T l yoxy'

,f;J,

X

(7.1.14)

Similar expressions can be written for the layers of the second orientation. Having the strains in the principal material direction, stresses can be calculated by multiplying the strains with the Q matrix. There-

268


fore, we can now show the limits on stresses and strains in the individual layers on the in-plane stiffness diagram. The next example demonstrates the use of the lamination diagram with a strain-based failure criterion implemented for each layer of a two-angle laminate.

Example 7.1.2 Using the graphical procedure and plies with the following material properties; E 1 = 181 GPa, E 2 = 10.3 GPa, G12 = 7.17 GPa, and v 12 = 0.28, design a laminate that will stand an average combined stress of ax= 500 MPa, cry= 250 MPa, and ~xy = 0, without exceeding ply strains of E~ = 8.29 X 10-3, E~ =3.88 X 10-3, and 1'~2 = 9.48 X 10-3. The laminate should also have the largest possible stiffness in the x direction, and the shear stiffness should be larger than 20 GPa. Using the given material properties and Eqs. (7 .1.4 )-(7 .1.6), the laminate strains are calculated as 3.2532 X 10 19 -4.2866 X 10 19 V; + 1.4783 X 10 19 V;

£0=--------~--------~~--------~~

5.3210 x 10 21 -7.3501 x 1021 V? + 3.9017 x 1021 v;·

x

Eo= y

l'~y

7.7884 X 10 18 + 2.1433 X 10 19 V~ + 1.4783 X 10 19 V; 5.3210 X 1021 -7.3501 X 1021 V? + 3.9017 X 1021 v;'

= 0.0.

Substituting these strains into Eq. (7 .1.14 ), ply level strains in the layers with el orientations are -5.8275 (V 3'- 1.3638

a, E1

a, E2

2

)

+vi (4.2246 v;- 5.7613) + T1 (Vi (12.674

v 3- 17.284)- 4.8771 v; + 6.6513)

(14.700 Vi 2 -7.8034 V)-10.642)(197.10 V)-268.80) =

-5.8275 (V 32 - 1.3638

2

)

+vi (4.2246 v;- 5.7613) + T1 (Vi (12.674

at

(2.4744- 6.4299 Vi)~ (7.3501 vi'- 3.9017

288

GRAPHICAL. STRENGTH DESIGN

replaced by T2, which is also defined in Eq. (7 .1.13). Because of the presence of the ± sign in the equation for T1 and T2 there are two sets of equations for each ply orietnation, resulting in a total of 12 equations for the strains. For a specified value of the volume fraction v1 (v2 = 1.0- v1), either the first set of six equations (say, + sign for T 1 and - sign for T2 ) or the second set (with - sign for T1 and + sign for T2 ) can be used to plot the lines corresponding to the limiting values of the strains in the V~-v; plane. These lines divide the in-plane stiffness diagram into feasible regions where the strains are below the limiting ply-level strains and into infeasible regions where the strain limits are exceeded. For example, limiting strains for both ply orientations (+ sign for T 1 and - sign for T2) are used to generate the in-plane lamination diagram for a laminate with v1 = 0.4 (v 2 = 0.6). The design space generated for this case is shown in Fig. 7 .2. The objective is to find the laminate with the largest stiffness in the x direction. Contours of laminate stiffness Ex are also shown on Fig. 7.2 along with the shear stiffness constraint. It was shown in Section 4.2.1 that the laminate with the largest stiffness is on the upper right corner of the in-plane lamination diagram, which corresponds to an all 0° laminate. Because of the constraints, the largest stiffness in the x direction is obtained at the upper right corner (point A) of the feasible design space where the transverse strain in the plies with the first orientation (second of the strian equations shown above) and the shear stiffness constraints are active. The in-plane stiffness parameters that correspond to point A are v~

= 0.395,

and

v; = 0.349,

and the values of the T 1 and T2 corresponding to these in-plane stiffness parameters are

v;- 17.284)- 4.8771 v; + 6.6513)

(14.700 v? -7.8034 v;- 10.642) (197. 10 v;- 268.80)

~z-

7.1

v;- 5.3210) 10

2

'

where T 1 is defined in Eq. (7.1.13). Equations for the strains in the principal material direction of the layers with the second fiber orientation 82 are same as the ones above except that the T 1 is

T 1 = 1.277

and

T2 = 0.1936.

Note, however, that the T1 value is larger than 1.0 indicating that no real value of the orientation angle 8 1 exists for this case. A plot of the feasible design space for the second sign of T and T2 is shown in Fig. 7 .3, and the optimal design is marked 1 as point B. In this case the in-plane stiffness parameters are

270


7.1

Vj

·.,

\"l~

. . y·

Vj

, I , ·1·\·11' , {, , ' v; 1

Figure 7.2. Graphical representation of the laminate strain limits.

v; = 0.397,

and

and T2 =0.9838.

The fiber orientation angles for the optimal laminate are

el = 59.48°

and

;

-1

I

,

I

I

1

v;

Figure 7.3. Graphical representation of the laminate strain limits.

v; = 0.349,

and the values of the T 1 and T2 corresponding to these in-plane stiffness parameters are T1 =-0.4842

271


e2 = 5.157°.

Active constraints for the laminate are the transverse strain in the layer with the second fiber orientation angle and the shear strain limit. The results presented here correspond to a selected value of v1 = 0.4. It is also possible to draw the limiting values of the ply-level strains for other combinations of v1 and v • Although 2 some of these additional cases may yield no feasible regions in the in-plane stiffness diagram, in general there will be other laminates that will satisfy the requirements of the problem.

7.1.3 Design of Multiple-Ply Laminates with Discrete Fiber Orientations The use of the graphical procedure is most suitable for designing laminates with discrete fiber orientations. In this case, since the ply orientation angles are known, we can draw the failure envelopes of plies with the specified angles in the in-plane lamination diagram. We then superimpose the polygons for the laminates with prescribed angles (see Section 4.3, Fig. 4.8) on this in-plane lamination diagram and identify points with various ply orientation combinations that fall within the appropriate feasible region of the diagram. If the design space bounded by the limiting strain curves does not include a discrete point, we can assume one or more of the prescribed angles to be absent from the laminate, thereby increasing the area of the design space. We also have to make sure that we do not consider discrete points associated with the omitted fiber orientation. The following example demonstrates the use of the procedure.

272


Example 7.1.3 Using the graphical procedure and layers with the following material properties: E 1 = 181 GPa, E 2 = 10.3 GPa, G12 = 7.17 GPa, and V 12 = 0.28, design a 16-layer symmetric laminate with discrete ply orientations of 0°, ±45°, and 90° which will stand an average combined stress of ax= 100 MPa, cry= 100 MPa, and 1xy = 100 MPa, without violating the Tsai-Hill failure criterion. Strength properties of the layers are X1 = 1500 MPa, Y1 = 40 MPa, and S = 68 MPa. The laminate should also have the largest possible elastic modulus in the x direction. Using the given material properties in Eqs. (7 .1.4 )-(7 .1.6), the laminate mid-plane strains are calculated as 5.376- 8.573v; + 3.942v; E~ = 5321. -7350.V~ 2 + 3902. v;' 5.376 + 8.573V~ + 3.942v; E~ = 5321. -7350. V~ 2 + 3902. v;' 0 1 Yxy-268.8-197.1V;

With laminate strains calculated, we can next use the equations for the transformed stiffness matrices for the 0°, ±45°, and 90° layers to calculate the stresses in those layers in the laminate coordinates, x-y. The transformed reduced stiffness matrices for the 0°, +45°, and 90° layers are

r81.8 2.897

Q0'l = 2.897 10.35 0

90 Q< 'l =

r2.897 10.35 0

Q<+4S"l =

r56.66 42.32

0

0 0 GPa, 7.170

1

2.897 0 181.8 0 GPa, 0 7.170

1

42.32 42.87 56.66 42.87 GPa. 42.87 42.87 46.59

1

7.1


273

For the -45° layer, the sign of the Q16 andQ26 terms is reversed. Multiplying the strains with the appropriate transformed reduced stiffness matrices, we find the stresses. For example, for the oo layer we have O') _

~

-

cr(O') _ Y

-

(O'J

=

'txy

993.0- 1534. v~ + 728.1 v; • * MPa, 5.321 -7.350V 12 + 3.902V3 71.20 + 63.86V~ + 52.21 v; 5.321 -7. 35ov,*2 + 3.902 V 3* MPa, 71.70 MPa. 2.688- 1.971 v;

Stresses in the 90° layers have the same equations, except the ax and crY are interchanged. Equations for the stresses in the 45° layers are slightly lengthier and for brevity are not shown. Having the stresses in the laminate coordinates, stress transformation equations can be used to calculate the stresses in the principal material directions. Subsequently, the principal material direction stresses can be substituted into the Tsai-Hill criterion, which is now in terms of the V~ and v; only. Parametric plots of such equations corresponding to the right-hand side value of 1.0 can be generated using graphical tools. These curves, superimposed on Miki's in-plane lamination diagram, make up the boundaries of the region with feasible laminate stacking sequences. Tsai-Hill constraints for the 0°, ±45°, and 90° layers for the present problem are shown in Fig. 7 .4. Also shown in the figure are the contours of the elastic modulus of the laminate in the x direction. Because of the discrete nature of the problem, not all points on the diagram are acceptable. As discussed in Chapter 4, for problems with oo, ±45°, and 90° layers, there is a triangular pattern of acceptable discrete points. Some of those points for 16layer laminates are also shown in the figure. One has to be careful in choosing the best design, as selection of the applicable Tsai-Hill constraint boundaries depends on the orientation angles in a given laminate. For example, for the present problem, the most restrictive Tsai-Hill criterion is the one for the -45° layer.

274


V*3

Figure 7.4. Graphical representation of Tsai-Hill constraints for discrete angles.

If the laminate being designed does not include a -45° layer, then we should not be using that boundary. For the present loading, the Tsai-Hill constraints clearly eliminate the regions of the lamination diagram that does not include 45° layers (the horizontal line at v; = 1.0). At this point, there are two possibilities. If the laminate to be designed is to have a balanced stacking sequence, then not all the points shown on the diagram are possible. By inspection, the best laminate is at point A, which corresponds to [Oi90i±45]s. If, on the other hand, we relax the balanced condition and design a laminate with only +45° layers, then we can ignore the Tsai-Hill failure boundary for the -45° layer and choose the [Osf90/+45 2 ]s laminate at point B, which has a higher stiffness than the balanced design.

7.2 NUMERICAL STRENGTH OPTIMIZATION USING CONTINUOUS VARIABLES

275

tation angles. However, the majority of the commercially available optimization software is for continuous-valued design variables, and most past work on laminate optimization is based on the use of such variables. For example, the total thicknesses of contiguous layers with the same orientation, referred to as the thickness variables, were commonly used as design variables. Orientation angles were also occasionally used as design variables, with orientations taking any value between 0° and 90°. The less frequent use of ply orientation angles was, in part, because most optimization problems with strength constraints were formulated to minimize the laminate weight, which was explicitly expressed in terms of the thickness design variables. In weight minimization problems, orientation variables influence only the strength constraints and, therefore, their convergence behavior is slow compared to the thickness variables, requiring robust algorithms for full convergence to an optimal solution. Once the design is optimized using continuous variables, the final thicknesses (or orientations) can be rounded off to integer multiples of the commercially available layer thickness (or conventional orientation angles).

7.2.1

Strength Design with Thickness Design Variables

Schmit and Farshi (1973) were among the first to consider the optimal design of a symmetric balanced laminate with fixed orientation angles. Because of the laminate symmetry, only the thicknesses, tk, k = 1, ... , I, of one-half of the total number of layers, I = N/2, are used as design variables. The laminate is considered to be under the action of combined uniform in-plane stress resultants, Nx, NY, and Nxy· The optimization problem is formulated in the following form:

minimize

W=

L2

(7.2.1) Pk tk

k=i

7.2 NUMERICAL STRENGTH OPTIMIZATION USING CONTINUOUS VARIABLES As emphasized in earlier chapters, the design of a composite laminate is a combinatorial problem with discrete layer thicknesses and orien-

subject to

gkj= (PJkl Elk+ QYl E2k + RYl 'Y12k)- 1 ~ 0,

All

~Af{,

A 22 ~A~~.

(7.2.2)

A 66 ~A6~, (7.2.3)

276


and

-tk:::;

O,

(7.2.4)

p
for k

= 1, ... , I,

j

= 1, ... , J,

(7.2.5)

where pk and tk are the density and the thickness, respectively, of the kth layer, Pjkl, Qyl, Ryl, are coefficients that define the jth boundary of a failure envelope for each layer (k) in the strain space, and the Ew £ 2k, andy12k are the strains in the principal material direction in the kth layer. Also specified are the limits on the in-plane stiffness terms with the following allowable values·. Aan Aa 11 andAan II' 22• 66' For a simple maximum strain criterion, which puts bounds on the values of the strains in the principal material directions, the failure envelope has six facets with P and Q defined as the inverse of the normal failure strains in the longitudinal and transverse directions to the fibers, respectively, once in tension and once compression. The coefficient R is the inverse of the shear failure strain written twice (once for positive shear and once for negative shear). For example, the maximum strain criteria of Eqs. (6.2.4)-(6.2.6) results in the following values of the coefficients: For tensile normal stresses, 1

p
t'

£1

QCkl = 0 I

'

R
3

c' £1

6

i=]

' .

yl2

ac

l

a

l

ack _I

at;

d£o aco dyo =mz_x +n2~+mn~ dt;

at;

at; '

'

R
at;

(7.2.7)

ai,

aco ac" dyo =nz_x +m2~-mn-=2

at; a£ at;

at,

at, '

a£ at;

0

ayo at;

0

~=-2mn-x +2mn-Y +(m 2 -n 2 )~.

at;

QCkl = 0 3

'

R
J

l

(7.2.12) where d£ 1/dt;, d£ 2/dt;, and d£ 12klat; are the derivatives of the principal material direction strains in the kth layer with respect to the thickness of the ith layer. These derivatives are related to the derivatives of the laminate mid-plane strains with respect to the ith layer thickness. Differentiating Eq. (7 .1.1 0), we have

(7.2.8) '

-1 R
(7.2.9)

(7.2.13)

In matrix notation, these equations take the form a~:(k)

P4(k)_O '

(7.2.11)

-1

R
'

ac

(7.2.6)

For compressive normal stresses, p
6

(t. - t .) [ p\kl ____!! + Q
ack = _!_t' Q
Q
'

Schmit and Farshi transformed the nonlinear programming problem described in Eqs. (7.2.1)-(7.2.5) into a sequence of linear programs, a procedure commonly referred to as sequential linear programming; see Haftka and Giirdal (1993). The strength constraint of Eq. (7.2.2) is a nonlinear function of the thickness variables and, therefore, is linearized as

_2

P
277


~ = Te(k) I

dEo dt. '

(7.2.14)

I

2

For shear stresses, P 5(k)_O '

Q5
R
-

Yi2

'

(7.2.10)

where E(k) = (Ew £ 2k, y12kf, E 0 =(Ex, £Y, Yxyf, and Te(k) is the engineering strain transformation matrix for the kth layer defined by Eq. (2.4.16). For a specified in-plane loading N = (Nx, NY, Nxy?' the derivative of the laminate strains with respect to the thickness variables can be deter-

278


mined by differentiating the in-plane portion (zero B matrix) of the stress-strain relation of Eq. (2.4.31), N =At", to obtain

dN dA d£ 0 . - = - t 0 +A-=0. • dt; dt; dt;

(7.2.15)

Rearranging, the derivative of the mid-plane strain with respect to the thickness variables may be written as

dto =-A-1 ()A dt; dt;

Eo.

(7.2.16)

a;=Q(i)'

(7.2.17)

Substituting this relation into Eq. (7.2.15), we can determine the derivatives of the mid-plane strains needed in equation Eq. (7.2.14) as :1 o

dt;

U2 = 8.481 Msi,

U4 = 2.414 Msi,

U3 = 1.973 Msi,

and U5 = 2.783 Msi.

For a balanced symmetric laminate of [Ox /±45/90 2 ]s, the terms of the A matrix can be computed in terms of the thickness-related variable x as A 11 = 10.45 (2x- 0.02) + 7.979 (2x + 0.06),

I

~=-A-

279

The maximum strain failure limits for the material are ci = £~ = 0.0115, £~ = £~ = 0.00535,and y~ 2 = 0.02. Formulate the linear approximations for the strains in the principal material directions in each layer with respect to the change in the thickness of the oo layer,[0/±45 2 /902],, and plot the strains in the individual layers as the thickness of the 0° layers change in the range 0 ::; x ::; 0.02, where x = t ·:X. With the provided elastic stiffness properties, the following U;'s are computed: U1 = 7.979 Msi,

As defined in Eq. (2.4.28), the in-plane stiffness matrix A is a linear function of the thickness variables. Therefore, the derivative of the A matrix with respect to the ith thickness variable is simply equal to the transformed reduced stiffnesses of the ith layer,

aA _


1Q

o (i)t ·

(7.2.18)

Once the derivatives of the strains in the fiber and transverse to the fiber directions are calculated from Eq. (7.2.14), linear approximations to the strain constraints can be constructed using Eq. (7.2.12) at any step of the sequential linearizations. Example 7.2.1

Design of a laminate with 0°, ±45°, and 90° layers is to be started from an initial design that corresponds to a 16-ply quasi-isotropic laminate, [0/±45/902 ],. The laminate is under loading that induces Nx = 4000 lb/in, NY= 1000 lb/in, and Nxy = 0.0. Properties of the layers correspond to that of AS4/3501-6 Carbon/Epoxy material: £ 1 =18.3 Msi, £ 2 =1.46 Msi, G12 =0.81 Msi, and V 12 = 0.3. The thickness of the individual layers is t = 0.005 in.

A 12 = -1.973 (2x- 0.02) + 2.414 (2x + 0.06),

A 22 = -6.508 (2x- 0.02) + 7.979 (2x + 0.06), A 66

= -1.973 (2x- 0.02) + 2.783 (2x + 0.06).

Note that the 2 in front of the variable x stems from the fact that the laminate is mid-plane symmetric and the x is related to the thickness of the total oo layers in the laminate. The linear approximation is to be performed for a nominal point x = 2t, which corresponds to the initial quasi-isotropic laminate. The in-plane stiffness matrix for the nominal design is 269.7 184.3 0.0 ] A= 184.3 608.9 0.0 kip/in. [ 0.0 0.0 206.4 For the specified stress resultants, the laminate mid-plane strains corresponding to the nominal design are

280

1.724 t

0

-0.5217

1

=A- N= --D.5217

r

1.724

0.0

0.0

0.0 o.o

l


{4} 1 x 4.492 0

j

0.638 ) %. w-3 = -0.0362

Strain 0.02

0.0 0.015

Strains in the individual layers can be computed by using the engineering strain transformation matrix Eq. (2.4.16):

Te(Oo)

Te(45o)

281


=

=

1

001

0 1 0 , 0 0 1

r

0.5 0.5 0.5] 0.5 0.5 -0.5 , -1 1 0

r

Te(goo)

l

1 o =o 1 0 0

r

,

0 0 -1

Te(-45o)

=

0.5 0.5 -0.5] 0.5 0.5 0.5 . 1 -1 0

r

0.01

0.005 ~----~--~~==~~==------~--====~x

0.01

0.015

-0.005

Therefore,

j

0.638 )

t(o') = -0.0362 %,

0.0

t('45 o) =

j

0.02

Figure 7.5. Principal material direction strains in the

oo and goo layers.

0.301 ) 0.301 %, -0.337

These computations may be performed for each value of the thickness variable, 0 ::::; x ::::; 4t, and plotted as shown in Figs. 7.5 and 7.6. The filled circles in the figures correspond to the strains for the quasi-isotropic laminate. Since the laminate is balanced and remains balanced as the thickness of the oo layers change, and there is no applied shear stress resultant Nxy' there is no shear strain in the 0° layer. Also, the magnitude of the shear strain y12 is the same in the +45° and -45° layer with only the sign reversing between the two layers. Clearly, as the thickness of the 0° layers decrease the strains in principal material directions in all the layers increase. In order to construct the linear approximations to the principal material direction strains, we need the derivative of the laminate mid-plane strains with respect to the thickness variable. We first compute the derivative of the in-plane stiffness matrix with respect to the thickness variable, which is given by the reduced transformed stiffness matrix of the layer being changed, Eq.

Strain

0.01

0.005

(45°) (-45°) (-45°) £2 £1 £2

-r--------~--------~--------~--------~x 0.005 0.01 0.015 0.02

-0.005

-0.01

-0.015 Figure 7.6. Principal material direction strains in the 45° and -45° layers.

282


(7 .2.17). In this particular problem, only the thickness of the 0° layers is being changed. Therefore,

dA dX

0

Jr0.4412 18.43

EC45o)(x)

0.0

0.4412 0.0 1.471 0.0 0.0 0.81

l

0.638! -0.0362 ' 0.0

dE -~-0.4024) dX - 0.1146 . 0

0.0 Derivatives of the principal material direction strains in the individual layers are found from the derivatives of the mid-plane strains using Eq. (7.2.14), which produces

dE

dX

~-0.4024) 0.1146 ,

dE

~= dX

0.0

{-0.1439) -0.1439 , 0.2585

dE

~=

!

dX

0.1146 -0.4024 0.0

j

Using these derivatives, the linear approximations for the principal material direction strains in the individual layers are oo layer, L

(

) _

(

E(0°) X - t(O'') X=

Etlo)

=

2t) + (x- 2t)

dE(oo) a.;-•

0.638 ) ~-0.4024) -0.0362 + (x- 2t) 0.1146 . 0.0 0.0

l

283

45° layer,

l18.43 0.4412 o.ol Msi = Q(0°) = 0.4412 1.471 0.0 0.0 0.0 0.81

[ 1.724 -0.5217 0.0 2 =- --().5217 1.724 0.0 0.0 0.0 4.492

~=

NUMERICAL STRENGTH OPTIMIZATION USING CONTINUOUS VARIABLES

-

Note, however, that the variable x used in the laminate stacking sequence description controls the thickness of the 0° layers on both sides of the laminate simultaneously. The derivative term computed above is the derivative of the A matrix with respect to change in the single layer. Therefore, for the derivative of the mid-plane strain, we multiply Eq. (7.2.18) by 2:

dE dX

7.2

E(45 ol

=

t(45°/x

=

= 2t) + (x- 2t)

a~;o)'

0 301) ~-0.1439} 0.301 + (x- 2t) -0.1439 . -0.337 0.2585

1

90° layer,

d~:Oo).

EC90o)(x)

= E(goo/X = 2t) +(X- 2t)

E(9oo) =

-0.0362) 0.1146! 0.638 + (x- 2t) -0.4024 . 0.0 0.0

I

1

Two examples of the plot of the linear strain approximations along with the actual strains are shown in Figs. 7.7 and 7.8. In Fig. 7. 7 actual and approximate longitudinal strain in the oo layer and the transverse strain in the 90° layer are shown along with the corresponding failure strains. Figure 7.8 shows the shear strains in the principal material direction in the +45° layer along with the corresponding shear failure strain. Although the approximations are reasonably good within the immediate vicinity of the nominal point, their accuracy degrades, as expected, especially for small values of the thickness variable x. In particular, the approximate longitudinal strain in the oo layer e)0l indicates that the thickness of the layer may be reduced to zero without failure. The actual strain, however, reaches its critical level at around x = 0.003. Note, on the other hand, that the strain to failure in the transverse to the fiber direction, £~, is much lower than the fiber direction failure strain. In fact, transverse strain in the 90° layer for the nominal quasi-isotropic laminate exceeds the failure strain. Therefore, if the first-ply failure criterion is used in designing the laminate, this would indicate a constraint violation, and the variable x would be pushed to a higher value of about 0.013 to prevent the violation. The magnitude of the shear strain

284

STAENQTH DESIGN OF LAMINATES

strain


28!5

in the +45° layer for the nominal design is smaller than the failure strain and would not be critical at this particular design point.

0.02 0.0175 0.015

ef -----------

0.0125 0.01 0.0075 0.005 0.0025

~~------~--------~------~~--~~~x

0.005

0.01

Figure 7.7. Linear strain approximations in the

0.015

0.02

oo and 90° layers.

strain

0.005

0.01

-0.0025 -0.005 -0.0075 -0.01 -0.0125 -0.015 -0.0175 -0.02

-----------~~--------s

Figure 7.8. Linear shear strain approximation in the 45° layer.

-

Although derivative-based optimization methods are quite powerful search procedures, one has to be careful in the case of composite laminate design problems with continuous layer thickness design variables. One of the difficulties with such problems is the choice of the initial laminate stacking sequence, as this may influence the final design optimization result. This is best exemplified by Schmit and Farshi (1973 ), who presented the results of an optimization study using conventional laminates with 0°, ±45°, and 90° orientation angles under various combinations of in-plane normal and shear loads. For example, for a laminate under uniaxial stress and limits on shear stiffness, they have demonstrated that it does make a difference whether we initially select a [0/±45/90]. laminate or a [0/±45], laminate even though at the end of the design iterations the thickness of the 90° plies of the first laminate vanishes. Results of these laminates obtained from Schmit and Farshi ( 1973) are summarized in Table 7 .1. The final design of the first laminate has a critical strength constraint for the 90° layer, even though the thickness of the layer is vanishingly small. Compared to the second laminate, [0/±45]., it is about 9% thicker due to an additional 0° layer required for the first laminate to prevent violation of the strength constraint in the 90° layer. In order to achieve a true optimal solution, therefore, the designer has to repeat the optimization process with different laminate definitions, especially by removing the layer(s) that converge to their lower bounds. However, the fact that a layer assumes a value different from its lower bound may not mean that the particular layer is essential for the optimal design. That is, it is quite possible that once a layer with a thickness different from its lower bound is removed, the optimization procedure can resize the remaining layers to achieve a weight lower than the one achieved before. This can make the design procedure difficult, because of the need to try all possible combinations of preselected angles. However, for most practical applications the presence of plies with fibers running in prescribed directions (such as fibers transverse to the load direction) is desirable. Therefore, lower limits, which are generally different than zero, are imposed, and the removal of such layers is not an option. Multiple load conditions also tend to produce designs where ply removal may not be possible.

286


Table 7.1. Minimum weight laminates with stiffness constraints loaded In axial compression Layup and Layer Number

Orientation Angle Initial Design Final Design (deg) t; (in) t1 (in)

Final Design

% .

Number of Plies (rounded)

[01±45!90],

l 2 3 4

oo +45° -45° 90° L. t;

0.032281 0.032281 0.032281 0.032281

0.018793 0.023048 0.023048 0.000000

0.129124

0.064890

0.034194 0.034194 0.034194

0.012555 0.023441 0.023441

0.102583

0.059438

28.96 35.52 35.52 0

0

21.12 39.44 39.44

3 6 6

4 6 6

[0/±45Js 1

oo

2

+45° +450

3

L. t; Source:

Schmit and Farshi (1973).

7.2.2 Strength Design with Orientation Angle Design Variables In order to find a laminate stacking sequence that is best suited to a specified loading, the orientation angles of the layers as well as their thicknesses may be used as design variables. Indeed, many design codes can treat both as design variables. In order to demonstrate the use of ply orientations as design variables, however, we concentrate in this section on examples with orientation variables only. As mentioned earlier, for optimization problems formulated as minimum weight designs, the objective function is independent of the orientation of the layers and, therefore, some optimization algorithms may experience convergence difficulties. An alternative to the weight objective function minimization is the maximization of the laminate strength as demonstrated by Park (1982) and Massard (1984). A quadratic first-ply failure criterion based on an approximate failure envelope in the strain space is used by Park for laminates under


287

various in-plane loading conditions Nx, NY, Nxy This approximate failure envelope is given by E2 x

+ E2y + .!.2 ~12hy = b2o•

where b0 is defined solely in terms of the stiffness and strength properties in the principal material directions. Note that compared to the quadratic failure criteria discussed in Chapter 6, such as the Tsai-Hill criterion or the Tsai-Wu criterion, this failure envelope is in the strain space and does not include interactive strain terms such as ExEy or Ex"fxy The objective function to be minimized is defined as 2 f = Ex2 + Ey2 + 2I '¥xy'

which represents the square of the norm of the strain vector. The smaller the objective function value, the larger the loads that can be applied to the laminate before the failure envelope is violated, therefore, the stronger the laminate against first-ply failure. One key feature of this approximate strain failure envelope is that it applies to laminate strains and does not require ply-level strain calculations. Only balanced symmetric laminates are considered by Park, and six different laminates were studied, five of which were the following conventional layups: [-e/+eJ.,, [-e/0/+eJ,, [-e/90/+eJ,, [-e/0/90/+e]s, and [-e/-45/+45/+e]s. The sixth laminate was called a continuous laminate and was assumed to have a fiber orientation changing linearly from the top surface to the mid-plane of the laminate, covering a range from -e 0 to eo orientations. Results by Park showed that under combined loading the best laminate, according to the first-ply failure criterion, for large longitudinal loading without shear was the [-e/0/+eJ, type, and for large shear loading without the longitudinal load, the best was the [-e/-45/ +45/+SJ, laminate. The optimum angle for the [-e/0/+e],. laminate depended on the magnitude of the transverse load NY, and was equal to oo for NY= 0. As the transverse load was increased, the optimum angle reached 45° for NY= NJ2, and was equal to 60° for NY= Nx. Similarly, for the [-e/-45/+45/+e]s laminate (with shear loading and no axial loading), the optimal angle was 45° for NY= 0. As the transverse load NY increased, the optimal angle increased and reached a value of about 73 o for NY= Nxy· The continuous laminate proved to

288


have the best overall performance under combined longitudinal and shear loadings with a range ±65° for NY= 0. The above results were intuitively appealing in that the fibers were mostly placed in a direction parallel to the applied loads, but such intuition may not always lead to optimal designs when working with composite materials. Consider, for example, the Tsai-Hill criterion of Eq. (6.2.11): 2

!=

(xcr~J - (cricrzJ + (crzJ "Y + ('stiz) : :; 2

)(2

2

1•

used for the strength prediction of a unidirectional composite (Tsai, 1968). The quantities X, Yare the normal strengths in directions parallel and transverse to the fibers, and S is the shear strength of a ply. Brandmaier (1970) showed that if the transverse normal strength Y is less than the shear strength S, optimal placement of the fibers is not along the principal stress directions, but depends on the values of the strength quantities as well as the applied stresses. This can be demonstrated (see Exercise 1) by expressing the stresses in the principal material directions in terms of the applied stresses ax, crY, 'Cxy and the fiber orientation 9, and equating the derivative of the failure index of Eq. (7.2.2) with respect to the fiber orientation to zero.

7.3 NUMERICAL STRENGTH OPTIMIZATION USING DISCRETE VARIABLES Practical stacking sequence design of a composite laminate requires the use of the methods of integer programming discussed in Chapter 5. For those design problems that can be cast in the form of linear programming, powerful integer linear programming techniques may be used. For nonlinear problems, however, the designer has to either use a nonlinear programming approach that can handle discrete variables, such as genetic algorithms, or apply linearization schemes to those nonlinear response quantities so that integer linear programming may be used. The strength of a composite laminate, which may be used either as a design constraint or the objective function, is clearly a nonlinear function of both the thickness and orientation variables and must be

7.3 NUMERICAL STRENGTH OPTIMIZATION USING DISCRETE VARIABLES

289

treated accordingly. In the following sections, first a linearization scheme which is suitable for the stacking sequence design of laminates is presented. Next, implementation of the genetic algorithm to the design of laminates with strength constraints is demonstrated.

7.3.1

Integer Linear Programming for Strength Design

Traditionally, constraint linearization is one of the useful techniques in handling difficult nonlinear functions in powerful linear optimization algorithms. In Example 7 .2.1 one such linearization is demonstrated for constructing strain approximations with respect to a thickness design variable. The same approach may also be used to form linear approximations with respect to the orientation of the individual layers in a laminate. However, for stacking sequence design, which involves discrete values of both the thickness and the orientation angle variables, a different form of linearization may be more appropriate. In this section a linearization that can be used to cast the nonlinear strain quantities in terms of the elements of the in-plane stiffness matrix is presented. The advantage of linearization with respect to the in-plane stiffness terms is their form which lends itself to integer linear programming for the laminate. As introduced in Chapter 2, the inplane stiffness parameters capture all the information about the thicknesses and orientations of the layers of a laminate. For a set of prescribed orientation angles, the in-plane stiffness parameters can easily be turned into linear expressions in terms of the number of layers of the prescribed orientations. For example, Eq. (5 .2.7) shows the expressions for the in-plane stiffnesses for a balanced symmetric laminate made of only oo, ±45°, and 90° layers. A strength constraint linearized in terms of the in-plane stiffnesses, therefore, can be used directly in an integer linear programming problem. The linear approximation to the mid-plane strains of a laminate is given by

1,2,6(at

E~(x) = E (X) + ~ 0

0

l

aAij " [Aij- Au(xo)],

(7.3.1)

290


where Au(x,) and £ 0 (xJ are the values of the in-plane stiffness terms and the mid-plane laminate strains, respectively, at the nominal design point X0 • In order to set up the linearization, we need to compute the values of the derivatives of the mid-plane strains with respect to the in-plane stiffness terms at the nominal design point. The expressions for the derivative terms can be obtained from the in-plane constitutive relations connecting mid-plane strains to the stress resultants. For example, for a balanced laminate, A 16 =A 26 = 0, the mid-plane strains are computed from o- A22Nx- A12Ny £X-

H

'

o_ A11Ny-Al2Nx EyH ,

and

y: = Nxy xy

(7.3.2)

A ' 66

where H = A 11 A 22 - Ai2. Therefore, the derivatives with respect to the in-plane stiffness terms can be shown to be OEX0 A 22 £X0 CJA11 =-~,

(7.3.3)

CJE~ _ 2 A 12 E~ ~ CJA 12 H - H'

(7.3.4)

CJE~ A E~ Nx - - = - -11 - + CJA22 H H'

(7.3.5)

7.3 NUMERICAL STRENGTH OPTIMIZATION USING DISCRETE VARIABLES

291

with the rest of the derivative terms, such as CJE~/CJA 66 , being zero. Derivatives of the strains in the principal material directions of the individual layers can be computed using the engineering strain transformation relation of Eq. (2.4.16). For specified discrete angles, further manipulation of the derivative terms is possible. For example, with the discrete orientation angles limited to 0°, ±45°, and 90°, the strain derivatives in the 45° layer are computed by using {J£(45°) I

1 [ CJ£ox CJ£oy v~,o] f xy CJAij =2 CJAu + CJAu + CJAu '

(7.3.10)

CJE~45o) 1 [ dE~ CJ£o CJy~y] - - - - -+-=...=L CJAu - 2 ()Au CJAu - CJAu '

(7.3.11)

iJy\4,] = _ oE~ + 0

)

CJAu

CJAu

oE;

(7.3.12)

CJAu.

Further, it can be shown that iJE(450J

(JAil CJE"

= _ A 22 E; + ~

CJAll

H

(7.3.6)

H'

0

_

-

2 A 12 E; H

_

Nx

(7.3.7)

H'

CJA11

CJE _ All E; CJAz2-- H ' CJy~Y Nxy CJA66 =- A~6'

(7.3.8)

(7.3.9)

12

22

2H

(7.3.13)

£0

2H

CJA22

x'

Eo.

(7.3.14)

Y

Also note that some of the terms which are zero in the x-y coordinate system are not zero in the principal material directions. For example, CJE(450J

0

(A -A )

CJE\450) = CJE~450J =-(All- Alz)

CJA2z OE CJA 12

CJE(450J

_1_=_2_=

N

xy A66

1 --=--2.

CJA66

(7.3.15)

An example of the use of linearized strain constrains in buckling load maximization design of laminated composites is discussed in Section 8.3.3, Example 8.3.2.

292

7.3.2

STFIENQTH DESIGN OF LAMINATES

Genetic Algorithms for Strength Design

Genetic algorithms are a natural tool for strength design of composite laminates. Because the calculation of ply strains or stresses takes very little computer time, the large number of analyses required for genetic optimization is easily affordable. For in-plane loads and symmetric laminates, the design variables can be the number of layers of each orientation. However, for bending loads the stacking sequence itself needs to be designed. Even when there are no bending loads, designers often introduce constraints that require the design of the stacking sequence. For example, in order to minimize matrix cracking problems, we often require that there be no more than four contiguous plies of the same orientation. The following example illustrates the use of a genetic algorithm for design subject to strength constraints; the same example will be used in Chapter 8 with additional buckling constraints, which depend strongly on the stacking sequence.

7.3 NUMEFIICAL STRENGTH OPTIMIZATION USING DISCRETE VARIABLES

293

able except for very thin laminates, where the use of odd numbers of 0° or 90° plies can result in substantial relative reduction of the optimal thickness. With this coding, for example, the laminate [(±45)i0i90 L is coded as [E E 45 0 0 90], where E denotes 2 an empty stack. Based on previous experience, the population size was set at 8, the probability of crossover at 1.0, the probabilities of mutating, deleting, or adding a stack all set at 0.1, and the probability of a stack swap set at 1.0. The objective function <1> was the number of plies, n, with penalties for violating the strength and contiguity constraints and a bonus for constraint margin. The strain constraint is expressed in terms of a single parameter, A,cr' which is the load multiplier at which one of the strain limits will be exceeded. That is,

E~

Y'h]

1 . { mm . [- E2 - A, =-mm cr J i IE!il' 1~;1' ly12;1

'

where i is taken over all the layers. Then the strain constraint is

Example 7.3.1

expressed as Design the lightest symmetric and balanced Graphite/Epoxy laminate made of 0°, ±45 o, and 90° layers to carry loads of Nx =-10,000 lb/in, NY= -2,000 lb/in, and Nxy = 1,000 lb/in. Material properties are E 1 = 18.5 Msi, E 2 = 1.89 Msi, G 12 = 0.93 Msi, v12 = 0.3, and t = 0.005 in. Use the maximum strain criterion, with IE 11:::::; E~ = 0.008,

1- A,cr ~ 0.

Lumping all the strain constraints together in a single one does not work well for derivative-based optimization algorithms, because the minimum function is not smooth. However, for genetic algorithms it is a convenient way to represent the constraints. Using A, the objective function is given as

1£21: : :; E~ = 0.029, lyl21 ~ y~2 = 0.015,

with a safety factor off= 1.5. The stacking sequence is required to have no more than four contiguous layers of the same orientation. The problem was simplified by assuming that the layers came in stacks of 02 , ±45, or 902 , so that a laminate can be coded using a four-letter alphabet including the three stacks and an empty stack. The maximum number of stacks was taken to be 20, in order to be on the safe side. This simplification is reason-

p[n- E(Acr- 1)],

if Acr ~ 1,

!!!!:.. A, + 0,

otherwise,

= cr

where p is a penalty parameter for violations of contiguity, E is a bonus parameter for designs with a margin, and () is a penalty parameter. In the application here, E =0.5, and () = 0.1. The optimization was repeated 100 times, and the reliability as a function of the number of generations is shown in Fig. 7.9. It can be seen that about 300 generations (2400 analyses) are

294

STRE!NQTH DESIGN OF LAMINATES

295

REFERENCES

0.9~------~---------r--------.---------r--------,

and

a sin 28- cos 28 = b, where

a=

2'txy _ 1 and b= ay+O'x 1-a? cry- ax ~2- a}- 2' crY

and

a=XIY and

~=XIS,

where X, Y are the normal strengths in parallel and transverse directions to the fibers and S is the shear strength. 0.1

2. 100

150

200 Generations

250

300

Figure 7.9. Reliability as a function of number of generations for Example 7.3.1.

needed for about 80% reliability. The optimum design obtained was [±45/90i0i±4510i±45L, with the fiber strain constraint in the oo layers being the most critical constraint.

EXERCISES 1.

Use a genetic algorithm to design a maximum strength 16-ply balanced symmetric laminate with 0°, ±45°, and 90° plies under combined loadings of (a) Mx=A-andMxy=aA, where a= 0.1, 0.2, 0.4, 0.6, 1.0. (b) Mx =A, Mxy= 0.2aA-, and MY= aA,, where a = 0.2, 0.4, 0.6. Use Hoffman's failure criteria for the strength maximization with the following material properties and determine the value of A, in each case: E 1 = 134 GPa, E 2 = 8.90 GPa, G 12 = 5.10 GPa, v12 = 0.28, xt = 2130 MPa, XC= 1100 MPa, yt = 80 MPa, yc = 200 MPa, S = 160 MPa, and t = 0.125 mm.

REFERENCES

For a unidirectional laminate under uniform applied stresses, ax, crY, and 'txy• show that the stationary values of the Tsai-Hill function

J~(i)-(i~)+(i)+(~) are achieved for a

cos 28 + sin 28 = 0

Brandmaier, H. E. ( 1970). "Optimum Filament Orientation Criteria." J. Composite Materials 4, 422-425. Haftka, R. T., and Gi.irdal, z. (1993). Elements of Structural Optimization Third Revised and Expanded Edition. Kluwer Academic Publishers, Boston. Massard, T. N. (1984). "Computer Sizing of Composite Laminates for Strength." J. Reinf. Plastics and Composites 3, 300-345. Miki, M. (1986). "Optimum Design of Laminated Composite Plates Subject to Axial Compression." Composites' 86: Recent Advances in Japan and the United States. pp. 673-680. Kawata, K., Umekawa, S., and Kobayashi, A. (eds.). Proc. Japan-U.S. CCM-III, Tokyo.

STPII!NQTH DESIGN OF LAMINATES

Miki, M., and Sugiyama, Y. ( 1991 ). "Optimum Design of Laminated Composite Plates Using Lamination Parameters." Proceedings of the AIAAIASME! ASCE!AHS!ASC 32th Structures, Structural Dynamics, and Materials Conference, Part I, 275-283. Baltimore, MD. Nagendra, S., Haftka, R. T., and Giirdal, Z. (1992). "Stacking Sequence Optimization of Simply Supported Laminates with Stability and Strain Constraints." AIM J. 30(8), 2132-2137.

8

Park, W. J. (1982). "An Optimal Design of Simple Symmetric Laminates Under the First Ply Failure Criterion." J. Camp. Matis. 16,341-355. Schmit, L. A., and Farshi, B. (1973). "Optimum Design of Laminated Fibre Composite Plates." Int. J. Num. Meth. Engng. 11, 623-640. Tsai, S. W. (1968). "Strength Theories of Filamentary Structures." In Fundamental Aspects of Fiber Reinforced Plastic. pp. 3-11. Schwartz, R. T. and Schwartz, H. S. (eds.). Wiley Interscience, New York.

LAMINATE DESIGN FOR FLEXURAL AND COMBINED RESPONSE

The previous chapters have laid the foundation for graphical and analytical methods of laminate design. These methods have been applied to problems of laminate design for in-plane response. The present chapter is concerned with the application of these methods to flexural response and combined in-plane and flexural response. The major difference between design for in-plane response and flexural response is in the importance of the stacking sequence. For in-plane response, only the total thickness of all layers of a given orientation matters. For flexural response, the order of the layers with different orientations is as important as their total thickness. Layers that are located on the outside of the laminate contribute much more to the flexural stiffness of the laminate than identical layers that are located near the center plane. Unlike in-plane response, flexural stiffness response often depends not only on the thickness of the laminate but also on its other dimensions. In this chapter we limit ourselves to the flexural response of rectangular plates. We start this chapter with a summary of the equations governing some of the important response characteristics that 297

298


are commonly used in design. These include the out-of-plane displacement of a simply supported laminate loaded by transverse loads and its natural vibration frequencies and the buckling response of a simply supported laminate under in-plane loads. These equations are limited to the special case of balanced symmetric laminates.

8.1

FLEXURAL RESPONSE EQUATIONS

8.1

terms D and D26 , because the contribution of the positive and negative 16 angle plies also depends on their distance from the laminate center plane. However, we will assume that these terms are made negligible by proper arrangement of layers in the laminate, so that the contribution to D and D 26 of the layers with positive angles is balanced ex16 actly by layers of negative angles. Except for very thin laminates, this can be accomplished by simply stacking a +8 layer always next to a -8 layer. The relative magnitude of D 16 and D 26 can be assessed by the following two nondimensional terms

The equations governing the out-of-plane displacement w of a symmetric and balanced laminate subjected to a pressure loading q (see Fig. 8.1) are ()4

()4w

w

()4w

()4w

Dtt ()x4 + 4Dt6 ()x3()y + 2(Dt2 + 2D66) ()x2()y2 + 4D26 ()x()y3 + D22

()4 w

()l =q.

(8.1.1) The bending-twisting coupling terms D 16 and D 26 in the equation complicate its solution, making it impossible to obtain closed form solutions that are useful for simple design problems. Furthermore, the bending-twisting coupling that they represent is sometimes undesirable. The source of the D 16 and D 26 terms are the off-axis layers that have nonzero Q 16 and Q26 terms. For balanced laminates the contributions of the positive angle plies cancels the contribution of the negative angle plies to the in-plane shear-extension coupling terms A 16 and A 26 • The same does not hold for the bending-twisting coupling stiffness q(x,y)

a

r•!fr

------------- __L - -

' ''

/,''

''

'

X

D16

y= (D3 D 11

)114

and

D26

0= D3 D ( 22

22

11

(8.1.2) )114·

Nemeth (1985) suggests that the effects of D 16 and D 26 for buckling are small for y and 0 below 0.2. We will use this criterion in the more general setting of flexural response. For a simply supported plate under sinusoidally varying pressure,

q(x, y)

.nx.ny

= q0 sm-;; sm b'

(8.1.3)

the solution of Eq. (8.1.1) yields

w=

a4 q 0 sin(nx/a) sin(ny/b) 1t

4

·

(8.1.4)

2

4

[D 11 + 2(D 12 + 2D66) (alb) + D 22 (alb) ]

(see Kedward and Whitney, 1990). For more general out-of-plane loadings, the pressure distribution is expanded in a double Fourier series, and the solution for each term is obtained in a form similar to Eq. (8.1.4). As an example, for a uniform pressure distribution of magnitude q0 , the transverse displacement field is obtained as 16q0

w=--61t

+z Figure 8.1. Laminate under transverse loading.

299

FLEXURAL RESPONSE EQUATIONS

nny .L .L wmnsm--sm-b, a .

m=1,3, ... =I ,3, ...

where

m1t.X

.

(8.1.5)

il

300

LAMINATE DESIGN FOR FLEXURAL AND COMBINED RESPONSE 4

~ ~n [o., (:)

w=

2

+ 2(D, +2D") (::) +

4

D22 ~)

r

(8.16)

For the transverse vibrations of a laminated plate, we replace q in Eq. (8.1.1) with the inertia load

q=-ph

a2w

(8.1.7)

ar'

where p denotes the mass density of the plate, and t denotes time. For a simply supported plate, the vibration modes are sinusoidal in both the x and y directions and the natural vibration frequencies are given as 2

ffimn=

-~ "--D 11 (mla) 4 + 2(D 12 + 2D66 ) (m/a) 2 (nlb) 2 + D2z(nlb)4 ,

d4w d4w d4w d2w d2w d2w Dll dx4 + 2(DI2 + 2D66) axZal + Dzz dy4 =Nx dx2 + 2Nxy dxdy +NY al·

(8.1.9) For a simply supported plate with no shear load, Nxy is zero and the buckling mode is sinusoidal. If the plate is loaded such that Nx = -Wxo and NY = -ANyo' then the critical value of 'A corresponding to a buckling load with m half waves in the x direction and n half waves in the y direction is 'Acr' and is determined as "~

(m

cr '

the Du's (in particular, large D 11 /D 22 such as 3 or more), the critical values of m and n are small.

8.2 STIFFNESS DESIGN BY MIKI'S GRAPHICAL PROCEDURE Miki (1985) showed that a diagram analogous to the in-plane lamination diagram discussed earlier can be constructed for designing laminates for flexural response. This diagram is referred to as the flexural lamination diagram, and we will discuss the use of this diagram for maximizing flexural stiffness under lateral loads and for maximizing the fundamental frequency of a laminated plate. Applications to the maximization of the buckling load of a laminate are somewhat cumbersome with Miki's graphical procedure (see Miki, 1986) and will be solved using integer linear programming in the next section.

'I ph

(8.1.8) where m, n are the number of half waves in the x and y directions, respectively. The fundamental frequency is associated with m = n = 1 (see Vinson and Sierakowski, 1987). When the plate is loaded by in-plane loads that introduce uniform in-plane stress resultants Nx, NY, and Nxy' the equation of equilibrium in the direction normal to the plate (neglecting D 16 and D26 ) is

~

301

8.2 STIFFNESS DESIGN BY MIKI'S GRAPHICAL PROCEDURE

2 4 2(D 1t [D 11 (mla) 2D66 )(m/a) 2 (n/b) 2 D 22 (n/b) 4] 12 n)---~~------~~--=------------=----(m/a) 2 Nxo (n/b) 2 Nyo

+

-

+

+

+

(8.1.10) The buckling load multiplier is obtained by finding the lowest value of \r for all combinations of m and n. Unless the plate has a very high aspect ratio (such as an alb of 3 or larger), or extreme ratios of

8.2.1

Linear Problems

For a symmetric laminate we define flexural lamination parameters as I

l

* 12Vm Wt= h3

=""'£.....

sk cos 29k

_ 12V3b = h3

and

w;-

k=l

L sk cos 49k, k=l

(8.2.1)

where Vm and V3n are given in Eq. (2.4.35), I is the number of different ±9 groups, and

.=(2z;1_ (2z;_

s,

h)

h

3

1

J.

(8.2.2)

Miki (1985) shows that a relation of the same form as Eq. (4.2.8) is obtained for the flexural lamination parameters

w; ~ 2W~

2

(8.2.3)

-1.

Therefore, any balanced symmetric angle-ply laminate with multiple orientations can be represented as a point in a region bounded by 2 W*3 - 2W*1 -1

-1::;;

w~:::;

1.

(8.2.4)

302


Points on the parabolic boundaries correspond to designs with unique angle-ply laminates with only one lamination angle, [±8]ns• with

w; = cos 28

and

w; = cos 48

w; = s 1 COS 28 1 + s2 COS 282,

w; = s 1 COS 48 1 + Sz COS 482,

(8.2.6) where s2 = 1 - s 1• If we specify s 1, the two angles can be obtained as 81 = 0.5 cos-l T,,

82 = 0.5

COS-I

T2,

W~U2

(h3)

w;u3), :

D,2 = 12 (U4-

D 66 =

(~;) (U

5 -

+ w;u3),

(8.2.10)

w;u3).

Therefore, S is a linear function of W~ and w;, and contours of constant S are straight lines on Miki's flexural diagram. This means that the laminate with the highest fundamental frequency corresponds to a boundary point on Miki's parabola, and therefore to an [±8]ns angle-ply laminate. However, if we specify constraints on several frequencies, we may get flexural lamination parameters inside the boundary.

(8.2.7) Example 8.2.1

where Tl=

(12h3) (U 1-

D22 =

(8.2.5)

and the points on the straight portion of the boundary corresponding to w; = 1 are cross-ply laminates [On 190n,L with various combinations of n 1 and n2 • For desired values of the flexural lamination parameters w~ and w; that correspond to the interior of the flexural lamination diagram, we can usually find a large number of laminates with two or more distinct orientation angles. For a two-angle (/ = 2) laminate, Eq. (8.2.1) becomes

303


2s 1

w; ± 'V4siW?- 22s

1

(2W?- s2w;- s2)

sl

T = W*1 -s 1T 1 2

s2 (8.2.8)

Consider now the design for minimum displacement under sinusoidal load, Eq. (8.1.4), or the design for maximum fundamental frequency, Eq. (8.1.8) with m = n = 1. In both cases we want to maximize the stiffness parameter S given by

S = D, + 2(D, + W,)

2 )4 (~J+ D, (~ .

(8.2.9)

However, the Dij's are linear functions of W~ and w;. Indeed, from Table 2.1 we have

D,

= u~ }u, + w;u, + w;u,).

Consider the design of a 16-layer 20 x 15 in laminated Graphite/Epoxy plate. (a) Design the plate to maximize its fundamental vibration frequency. (b) Design the plate so that its fundamental frequency is 60Hz, and the second frequency is 120Hz. Assume that this second frequency corresponds to two half waves in the longer dimension. The material properties are £ 1= 18.5 x 106 psi, E 2 = 1.89 X 106 psi, G 12 = 0.93 x 106 psi, v 12 = 0.3, t = 0.005 in, and p = 0.057 lb/in 3 . The total thickness of the laminate is 0.08 in. Note that the weight density has to be divided by g = 386.4 in/sec 2 to obtain the mass density. For convenience in dealing with the frequencies, we rewrite Eq. (8.1.8) as

amn

42

()2

()4

• * 2 2 Q_ * na - 12pa (Omn- * 4 = rt4h2 -Dum + 2(Dl2 + 2D66)m n b + Dz2 b '

where 12DiJ D*.=-h3 . lj

(a)

r~

304


We first calculate the material constants U1 from Eq. (2.4.20), with the aid of Eq. (2.4.8), to obtain U1= 8.3253

X

6

10 psi,

U2 = 8.3821

U4 = 2.5366 X 10 psi, 6

106 psi,

X

U5

U3 = 1.9643 X 106 psi,

= 2.8943 X 106 psi.

a 11

= (6.4238- 1.8109W~- 1.2708W;) X 107,

a 2, = (2.7790 + 1.0762W~- 0.4617W;) x 108 •

-1.8109 1.2708 Solving for

W~

and calculating

W~opt = -0.3542,

Substituting these expressions into the equations for the D~ and then to Eq. (a) for a 11 and Uz 1, we get

w;.

W? -

0. 12 =120Hz

w;opt

=4w;.

w;,

a"' and ro"' we get

= -0.7490,

a 11

= 8.0226 X 107,

which corresponds to a frequency of ro11 = 419.2 rad/sec, or 66.724 Hz. The fiber-orientation angle at the optimum can be found as 8 = 0.5 cos- 1 w~opt = 55.37°.

(b)

To maximize the fundamental frequency ro11 we need to maximize a," which is a linear function of W~ and Contours of equal fundamental frequency, which are contours of equal a," are straight lines on Miki's diagram as shown in Figure 8.2. The maximum value of the frequency can be obtained graphically. However, for a precise value we note that, at the maximum, the line a,, = constant is tangent to the parabola so that its slope is 1. That is, equal to the slope of the parabola w; = 2

305


So the optimum design is a [±55.37]8s laminate. The reader can check that replacing the calculated angle with a more conventional angle of 60° would reduce the fundamental frequency by only a fraction of a percent, to 66.5 Hz. This, however, does not indicate that the frequency is completely insensitive to the angle. As can be seen from Fig. 8.2, for example, an all-zero laminate has a frequency of only 43.0 Hz. Next, we consider the problem of designing a laminate with the first two frequencies of 60 Hz and 120 Hz (377 rad/sec and 754 rad/sec, respectively). The corresponding values of a 11 and aw from Eq. (a), are

a = 12(0.057 /386.4) X 204 X 3772 = 6 .457 X 107, II

a2, .

-1

\

'.n ~

I''{

11

\
\.

t: I

rI

, • wj O.n =60Hz

Figure 8.2. Fundamental vibration frequency on flexural lamination diagram.

1t4 X

=

0.082

12(0.057 /386.4) X 204 X 7542 1t4 x 0.08 2

=2.58 X 108.

Equating these values to the expressions in Eq. (b) we get two straight lines, shown in Fig. 8.2 as the line for ro11 = 60 Hz and the line for ro21 = 120 Hz. The desired design is found at the intersection of these two lines, at w~ = -0.1197 and w; = 0.1202. Finally, we need to find a laminate that will have the flexural parameters calculated in the preceding. There are many possible laminates; for illustration we will consider a laminate with only two angles, 8 1 and 8 2. For a 16-layer symmetric and balanced laminate, this gives us two possibilities, [(±8 1)/(±82) 21, or

I

;I~

1~ 1 Iii I Iii I

306


[±9 1 /(±92)3].. If we select the second laminate, then z = 0.5h, 2

r--

---layer 1

z1 =0.375h, z0 =0.0, and Eq. (8.2.2) gives us s1 =0.421875and

s 2 = 0.578125. Substituting these values into Eq. (8.2.8) we obtain

1

T 1 = 0.7451,

T2 = -0.7500

or

T 1 = -0.9845,

T2 = 0.5113.

Then, using Eq. (8.2.7) we calculate the fiber orientation angles to be 91 = 20.92°,

92 = 69.33°

or

91 = 84.96°,

8.2.2

As can be seen from Example 8.2.1, the composition of a laminate with given flexural stiffness is not unique, except when W~ anctw; lie on the boundary of Miki's parabola. Shin et al. (1989) have shown that, given a laminate with more than one orientation angle, the stacking order of the layers with different orientation angles can be permuted arbitrarily without changing the Du's of the laminate. However, a change in stacking sequence of a laminate requires corresponding changes in the thickness fraction of the layers that are permuted. The process of permuting the layers is illustrated in Fig. 8.3. The figure shows the bottom half of a laminate [(±e1)NI(±91_ 1)N I· · · /(±9 )N L 1 with I different orientation angles. In the figure, we consider permu'ting the order of the angle in the 'tth stack with that of the adjacent ('t + 1)th stack. We assume that all the other stacks remain unchanged, and we want to find a new boundary between the two stacks, z~ = z*, such that W~ and w; remain unchanged. From Eqs. (8.2.1) and (8.2.2) we see that one can satisfy this condition if the value of si associated with each angle does not change. For the angle associated with stack 't in the laminate on the left in Fig. 8.3 and stack 't + 1 in the laminate on the right, this condition is

( ;J-r;·J~r;·J-(~·J

or

layer t+1

]T

, r-1

~-,-1 layer 1

-

z,.J

..

~J

layer t+l

J ~· =-_!

layert

=-_j

r

z1 +I

-

layer I

layer I

Figure 8.3. Ply-permutation with constant flexural stiffness.

Changes in Stacking Sequence

2

layert

92 = 29.62°.

That is, the two laminates are [±20.9/(±69.3)31, and [±85.0/ (±29.6) 3]s.

307


(8.2.11)

z•3 --

3

3

ZH 1 - z~

3 + z~_ 1•

(8.2.12)

It is easy to check that we get the same condition for the other permuted stack. Furthermore, z* always lies between z~_ 1 and zH 1 (see

Exercise 3), so that we always get a physically realizable result provided that the layer thicknesses are allowed to vary continuously. We have thus proved that we can permute any two adjacent stacks. Obviously we can change the stacking sequence in any order by doing a sequence of binary permutations. Table 8.1 shows results obtained for six laminates by Shin et al. (1989). The noninteger number of layers ni was obtained by maximizing the buckling load of a Graphite/Epoxy plate with an aspect ratio of 1.2 under uniaxial compression. All six laminates have the same Du's and therefore the same buckling load. The results reported by Shin et al. (1989) are in terms of volume fractions; these have been converted here to numbers of layers for a laminate with a total of 50 layers. The numbers in parentheses correspond to rounding of the exact number of layers for the purpose of obtaining an integer number of layers for a 50-ply laminate. The rounding introduces variations in the flexural properties. However, these are small enough that all six buckling loads are within 2% of one another. Thus, when we solve a stacking sequence design problem, we may have many designs that are very close in performance to the optimum one. Often it is desirable to obtain all of these designs rather than just the optimum. For exam-

309


308

LAMINATE DESIGN FOR FLEXURAL AND COMBINED RESPONSE -

w= ""' £...

Table8.1. Optimum designs with equivalent D matrix Stacking Sequence

n,

(n,)a

nz

(nz)"

n3

(n3)a

[On,f90n/ ± 45n)s [On,! ±45n,f90n)s [±45n, /On/90,)s [±45n, /90n/On)s [90n,1±45n,f0n,ls

0.915 0.915 2.785 2.785 3.500 3.500

(1) (1) (3) (3) (3) (3)

3.848 3.120 1.585 7.610 4.840 1.265

(4) (3)

10.119 17.847 17.848 11.823 11.823 10.119

(10) (18) (18) (12) (12) (10)

[90n, /On/±45n,ls

(1)

(7) (5) (2)

"Layer numbers are rounded such that each laminate has an integer number of layers with a total of 50 layers.

ple, the designer may want to sacrifice a percent or two of performance if one of the near-optimal laminates has some desirable properties that were not included in the formulation of the original design problem. Genetic algorithms, discussed in Chapter 5, can be used to yield such multiple optimal and near-optimal designs.

8.2.3

Nonlinear Problems

In Example 8.2.1 we could get linear equations of the frequency contours and plot them on Miki's flexure diagram. However, it is not necessary that we have a closed-form equation for the contour in order to be able to plot it. As long as we can calculate the value of the response for pairs of W~ and w; we can generate contours numerically and plot them. This approach is useful when the contour equations are nonlinear. As an example of this approach, we consider the design of a plate for minimum displacement.

(a)

. m1t . n1t £... wmnsm2sm 2,

""'

m=l,3, ... n=l,3, ...

where w mn is given by Eq. (8.1.6). Our objective is to find the values of W~ and w; that minimize winside the Miki's parabola. For that purpose, a subroutine was written that, for given values of W~ and w;, calculates first the Dij's using Eq. (8.2.10) and then calculates w using Eq. (a). In Eq. (a) values of m and n from 1 to 19 were used, for a total of 100 terms. Most of the contribution was due to the first term in the series, and with 100 terms the series has converged to six significant figures. Contours of equal w were plotted in the W~-w; plane as shown in Figure 8.4. The contours for a given value W = c were obtained by varying W~ and solving for the value of w; that satisfies the equation w =c. The solution was performed with subroutine NEQNF of the IMSL library. The figure indicates that the optimum is obtained on the boundary of the parabola that corresponds to an angle-ply laminate. The optimal fiber orientation is calculated to be 8* = 56.SO, which yields a value of

V3

1v;

Example 8.2.2 Design the plate of Example 8.2.1 for minimum displacement at its center under uniform pressure. We use Eq. (8.1.5) for the displacement. Since the value of the applied pressure % does not influence the choice of optimum plate, we minimize W = (1t6 /16)(w I q 0 ) instead of w. At the center of the plate (x = a/2, y = b/2), W is given as

Figure 8.4. Flexural lamination diagram for Example 8.2.2.

310


w == 44.89 in 3/lb. As in Example 8.2.1, the loss of performance .-

when using a somewhat similar but more conventional laminate of [±60°] 4s is minimal, yielding a value of w == 45.05 in 3/lb. The closeness of the optimal angles for the maximum frequency and minimum deflection laminates is no coincidence. It is because the first term in Eq. (a) dominates the series, and this term is proportional to the square of the frequency [see Eq. (8.1.8)].

8.3 FLEXURAL STIFFNESS DESIGN BY INTEGER LINEAR PROGRAMMING


As we will see later, it is convenient to number the layers so that the first one is nearest to the plane of symmetry of the laminate, rather than in accordance with the convention that lists the layers from the outside in. Thus, the laminate [0/±45], will be described by f ~ == fi =o3 == o4 = 1, with the other twelve variables associated with the laminate being zero. ~ith these zero-one ply-identity variables, both the in-plane integrals VoA• VIA' and v3A and the flexural integrals Von• VID, and V3n [see Eq. (2.4.35)] are linear functions of the design variables. Therefore, the A and D matrices are linear functions of these variables. The integrals V0A, Vw and V3A are given in terms of the ply-identity variables and the thickness of a single layer t is given as

In many applications, the layer orientation angles that we can use are limited to a discrete set. While Miki's method is applicable to such problems, the points on the diagram corresponding to the discrete set of angles are not as simple to obtain as in the case of in-plane stiffness design. Furthermore, for buckling design, the change in critical buckling mode with changes in lamination sequence [that is, the change in m and n in Eq. (8.1.1 0)] creates further complications; see Miki ( 1986). For this reason, we treat stiffness and strength design problems with discrete angles by the integer programming techniques described in Chapter 5. In this section we focus on problems that can be formulated as linear integer programming problems so that they are readily solved by commonly available software.

N/2

h/2

VOA

Ply-Identity and Stack-Identity Design Variables

While the in-plane stiffness matrix A is a linear function of the number of layers of a given orientation, the flexural stiffness matrix D is not. However, the matrix D can be made a linear function of the design variables if we use zero-one ply-identity design variables, Haftka and Walsh (1992). For example, if the laminate is made up of 0°, 90°, and ±45° layers, the stacking sequence can be defined in terms of a set of four ply-orientation identity variables, o;, n;,Jf, and/~. where i = 1 , . . . , N/2, that are zero-one integer variables. A variable o;, n;,ff, or is equal to 1 if there is a 0°, 90°, 45°, or -45° layer, respectively, in the ith layer. Of course, we will need a constraint that will prevent more than one of these variables from being equal to 1 simultaneously.

fr

(8.3.1)

n

= Jdz = 2 t L (ok + nk + + f'k). -h/2

k=l

h/2

N/2

(8.3.2)

v,A =Jcos 29 dz = 2 t L (ok- nk), k=l

-h/2

h/2

N/2

v3A = Jcos 49 dz = 2 t L (ok + nk - f ~ - f 'k).

(8.3.3)

k=l

-h/2

8.3.1

311

For the flexural response, the integrals Von• Vm, and V3n are expressed as 3

3 3

2 t N/2 Von=3

LPk

[( zk ]

t

('~') ~ 2;'I [k'- (k-1) ](o, + n, +fl +J;), ]

_

N/2

k=l

2t vm =- 3-

N/2

~ p, coo ze,

(8.3.4)

k-1

3

3

3

[

(

N/2

3

t) _[zk;') ]~ ~

2 ;'

I k=l

[k' _ (k

-l)'](o,- n,), (8.3.5)

312

LAMINATE DESIGN FOR FLEXURAL AND COMBINED RESPONSE 3

3

2t N/2 [ v3D = -3- LPk cos 4ek k=l

t IW-

3

t J - ('~· J

( 4

8.3 FLEXURAL STIFFNESS DESIGN BY INTEGER LINEAR PROGRAMMING N/4

l

VoA = 4 t L (ok + nk + fk),

=

(8.3.8)

k=l

N/4

N/2

2

313

(k-l?](ok+nk-

n -JT),

V1A

(8.3.6)

= 4 t L (ok- nk), k=l

k=l

where ff andfT do not appear in the expression for V1A and Vm since the cosine of 90° is equal to zero. The variable pk in Eq. (8.3.6) is unity if the kth layer is occupied and is zero if it is empty: Pk = 0 k + nk +if+ IT

(8.3.9)

-:;, 1.

(8.3.7)

N/4

v3A

= 4 t L (ok + nk- fk),

(8.3.10)

k=l

and N/4

As we will demonstrate in Section 8.3.4, empty layers will be used to reduce the laminate thickness during optimization. Constraints must be applied during the optimization to ensure that pk can be zero only for the outermost layers. The use of ply-identity design variables has the disadvantage that it creates a very large number of design variables, especially for laminates with large number of layers. One strategy that may be used to reduce that number is to replace the ply-identity design variables with stack-identity design variables that control a stack of two or more layers instead of a single layer. The most natural stack for balanced laminates is a stack of two layers, ±e. It is also convenient to have all stacks the same thickness. Therefore, we use two-layer stacks also for 0° and 90° layers. With two-layer stacks, we need only N/4 sets of stack-identity design variables and each set has fewer variables because positive and negative angles are lumped in a single stack. For a laminate made up of 0°, 90°, and ±45° layers, the stacks are ±45°, and 90;, and the corresponding stack-identity design variables associated with the ith stack are denoted by o;.f;, and n;, respectively. For example, the laminate [0i±45L will now be described by 6 design variables instead of 16, with ! 1 = o 2 = 1 and o 1 = n 1 = f 2 = n 2 = 0. The use of stack variables as just described simplifies somewhat the equations for the in-plane and flexural integrals, so that Eqs. (8.3.4)-(8.3.6) become

16t3 ~ 3 3 Von= - - £.. [k - (k- 1) ](ok + nk + fk), 3

(8.3.11)

k=l

VID = 1~t3

N/4

L W- (k- 1)3](ok- nk),

(8.3.12)

k=l

N/4

- 16t3 ~ k3 3 V3D- 3 £..[ -(k-1)](ok+nk-jk).

(8.3.13)

k=l

The simplification associated with the use of stacks is not without a price. The use of stacks constrains the solution, and so we may obtain a suboptimal design. However, the difference for the case of two-layer stacks is usually small.

o;,

8.3.2 When layers empty dition

Stiffness Design with Fixed Thickness the total thickness of the laminate and hence the number of are given, we do not have to worry about the possibility of layers. The ply-identity design variables have to satisfy the conthat for each layer at least one of the variables is equal to 1

314


and the others are equal to 0. For our laminate, made up of and ±45° layers, this condition is o;+n;+Jf+f~=

1,

i = 1, ... , N/2.

oo,

90°,

Next, we write Eq. (8.2.10) for the U;'s in Example 8.2.1:

To demonstrate how we use the stack-identity design variables, we consider an example of frequency maximization.

Solve Example 8.2.1 with the added condition that the orientation angles are multiples of 45°. Then add the condition that the laminate is stiff enough in the x direction by requiring A 11 to be at least 50% of its value for an all-zero laminate. As in Example 8.2.1, we maximize the frequency parameter a 11 , given by Eq. (a) in that example, instead of the fundamental frequency ro11 • For the plate considered in the example, alb = 4/3 so that our objective function is

3

D~~ + [ ~ )cn~2 + 2D~6)

i )n;2,

2 6 +[ 1

12Dij

Dij=J;J· With a 16-ply laminate, N = 16, so that we have N/4 = 4 sets of stack-identity design variables, ok,fk, and nk, k = 1, ... , 4. Instead of V 10 and V30 , it is convenient to work with *

using the values for the

V; 0 , 5 10 V~ 0 + 0.30692 X 10 V; 0 , 5

D~ 1 = 8.3253

X

106 + 0.13097

X

D; 2 = 8.3253

X

106 - 0.13097

X

3Vm * t =64 W 1 16 3

and

* _ 3 v3o

v3n- 16t 3

D~ 6 = 2.8943

106 V~ 0 + 0.30692 X 10 6

= 64 w;,

where we have used h = 16t to establish the relation between v~D and w~ and between v;D and w;; see Eq. (8.2.1). For our problem, Eqs. (8.3.12) and (8.3.13) give

X

106 -0.30692

X

105

(c)

V;0 .

Using Eqs. (a), (b), and (c), the following mixed integer linear programming problem can be formulated: maximize 81D~ 1 + 288(D~ 2 + 2D~ 6 ) + 256D; 2 such that o 1 - n 1 + 7(o2 - n2) + 19(o3 - n3) + 37(o4 - n4 ) - V~n = 0, o1 -

(a)

where

Vm=

Dij's

D~ 2 = 2.5366 x 106 -0.30692 x 1osv;0 ,

Example 8.3.1

all=

(b)

(8.3.14)

(8.3.15)

i=l, ... ,N/4.

V ~D = o 1 - n 1 + 7(o 2 - n2) + 19(o3 - n 3) + 37(o4 - n4),

v;D = 01- j 1 + nl + 7(02- j 2+ n2) + 1~0 3 - j3 + n3) + 37(04- j3 + n4).

If we use two-layer stacks, the corresponding condition is

o;+n;+.f;= 1,

315


f 1 + n 1 + 7(o 2 -

j 2 + n2) + 19(o3 -

f 3 + n3)

+ 37(o4 - j 3+ n4)- v;D = Q, 5

8.3253

X

106 + 0.13097

X

10 6 V~ 0 + 0.30692 X l0 V; 0

8.3253

X

106 -0.13097

X

106 V~ 0 + 0.30692

2.5366

X

106 - 0.30692 X l05

2.8943

X

106 - 0.30692

01+!1 +n1

= 1,

o2 + n2 + !2 = 1,

o3 +!3 + n3 = 1,

X

V;0 -

l05V; 0

-

D ~ 2 = 0, D ~6 = 0,

X

l0

5

-

D~ 1 = 0,

V;0 -

D;2 = 0,

I

316


317

8.3 FLEXURAL STIFFNESS DESIGN BY INTEGER LINEAR PROGRAMMING 04

+!4 + n4 = I.

This problem was solved with the LINDO program (Schrage, 1989). The LINDO program, like many other linear and integer linear programming codes, assumes that all variables are positive. However, for the present problem, v; 0 and lf;0 can be negative. To circumvent this difficulty, the standard approach is to replace each variable of an indefinite sign with the difference of two positive variables (see Section 5.1). That is, we replace v; by 0 v;Dp- ~Dm and v;D by v;Dp- v;Dm· The solution obtained by LINDO is / 1 = J; = f 3 = f 4 = I, which corresponds to an all ±45° laminate. The fundamental frequency for this laminate is 65.377 Hz, which is about 2% lower than the optimal design with unrestricted angles found in Example 8.2.1. The all ±45° laminate is rather soft with an All of only about a third of an all-zero laminate. To formulate the condition that All is at least 50% of that of an all-zero laminate, we use Table 2.1 to write A 11

= u,h + U2 v,A + U3 V.w

From Eqs. (8.3.9) and (8.3.10) we get, using h VIA=

0.25 h (0 1 + 0 2 + 0 3 + 0 4 - n, -

= I6t,

o1

o1 + o 2 + o 3 + o 4 -

n2 - n3 - n4 -

4V; = 0,

f3 - f4 + n 1 + n 2 + n 3 + n 4 -

4V; = 0.

+o2 +o3 + o4 -

/ 1 - /2 -

n1 -

These additional constraints were added to the original problem and solved with LINDO. Again, the variables v; and v;, being of indefinite sign, had to be replaced by the difference of positive variables. The solution obtained was o 1 = f 2 =f 3 = f 4 = 1, with the other stack-identity design variables being zero. That is, the laminate is [(±45)/0 2 ],. The fundamental frequency ro11 of the laminate was reduced by another half a percent to 65.086 Hz. Note that the addition of a zero stack near the centerline changes A 11 substantially, but has very little influence on the Dij's (because the contribution of a layer to the flexural terms depends on the square of its distance from the centerline). Therefore, by adding the zero layer near the centerline we are able to change the inplane stiffness with minimal effect on the fundamental frequency.

8.3.3 Buckling Load Maximization with Stiffness and Strength Constraints

n2- n3- n4)

and v3A

To solve the constrained problem, we need to add the constraint of Eq. (d) along with two equality constraints that come from the definition of v; and v;,

=0.25 h (o, + 02 + 03 + 04- fz- J;- A-h + n, + n2 + n3 + n4).

For an all-zero laminate we have o 1 = o 2 = o 3 = o = 1, with the 4 other stack-identity variables being zero, so that V A = V A = h, 1 3 and All= h(U1 + U2 + U3 ) = 18.672x 106 h lb/in. It is convenient to USe v; = V1A/h and v; = V3A/h, and then the COnStraint On All becomes

Optimization to maximize the buckling load of a laminated plate is more complicated than optimization to maximize the fundamental frequency. The reason for the difficulty is that the wave numbers m and n in Eq. (8.1.10) that correspond to the lowest buckling load are not known a priori and depend on the design. We want to maximize A.* given as A.*= min Acr (m, n),

(8.3.16)

m,n

A 11 1h = u, + U2 v; + U3 v; ~ 9.336 x 106 or 8.3821

v; + 1.9643 v; ~ 1.011.

(d)

where Acr(m,n) is given by Eq. (8.1.10). However, A.* is not a linear function of the ply-identity or stack-identity design variables, even though \r (m, n) is a linear function. Fortunately, we have a standard approach for resolving this difficulty. It consists of adding A.* as an additional design variable to the problem. For example, if we use the

318


o;,

two-layer stack design variables corresponding to the 90;, and ±45° layers, the maximization of the buckling load will be formulated as (Nagendra et al., 1992) find

A.*

to maximize

A*

and

oi, ni,J;,

such that /...* ~ Acr (m, n), oi + ni +J; = 1,

m

i

i= 1, ... ,N/4,

= 1, ... , m1 ,

= 1, ... , N/4.

n


respect to one another (say, N/ Nx = 0.5). The actual values of the stresses are dependent on the buckling load factor, as we would prefer the panel not to fail due to strength before the buckling load, the quantity being formulated for maximization, is reached. That is, if the value of a strain component corresponding to the reference values of the in-plane stress resultant is Ei for the ith constraint, and its limiting value is Ea11, then the constraint is expressed as,

/...* Ei ~fan.

= 1, ... , n1, (8.3.17)

The minimization over m and n is performed by checking for all values of m between 1 and m1, and all values of n between 1 and np where m1 and n1 are selected to be large enough to include all possible critical buckling modes. Equation (8.1.10) for \r is linear in the Dij's, which are linear functions of the stack-identity design variables. So the optimization problem of Eq. (8.3.17) is an integer linear programming problem, and can be solved by using software such as LINDO. The formulation presented in Eq. (8.3.17) also allows us to incorporate the stiffness constraints discussed in the previous section. This is simply accomplished by expanding the elements of the in-plane stiffness matrix in terms of the ply-identity, or stack-identity variables. Moreover, it is also possible to incorporate strength constraints into the present formulation. It was demonstrated in Chapter 7 that strength constraints expressed in terms of laminate mid-plane strains may be approximated by linearizing them; see Eq. (7 .3 .1 ). The linearization is performed in terms of the elements of the in-plane stiffness matrix, which are already linear in the ply- and stack-identity variables. Once the laminate mid-plane strains are expressed as a linear function of the design variables, constraints on principal material direction strains may be formed by using the strain transformation matrices. If desired, stress constraints can be obtained from the principal material direction strains by multiplying them with the transformed reduced stiffness matrices of the appropriate layers. The incorporation of the strength constraints into the buckling maximization problem, however, is still incomplete. The magnitude of the strains and stresses in the laminate depend on the magnitude of the applied loads. The stress resultants used in the buckling load equation, Eq. (8.1.10), are simply initial reference values, which are mostly used to specify their ratio with

319

(8.3.18)

Since Ei is a function of the design variables and A.* is used as one of the design variables in the formulation described in this section, the above constraint is again nonlinear. By moving /...* to the right-hand side of Eq. (8.3.18), and expanding 1/A.* in a linear Taylor's series,

1 (-1]

_!__ /...*- Ao + A~ (A*- Ao),

(8.3.19)

the constraint equation of Eq. (8.3.18) can be written as,

Aof . +A.* A- fall<- 2fall,

(8.3.20)

1

0

where A is the buckling load factor for the nominal design. 0 The linear strain constraint of Eq. (8.3.20) can now be added to the problem formulation of Eq. (8.3.17) for designing laminates that have maximum buckling load and are strength-failure resistant. A final implementation note is for the method of solution. Since the formulation involves a local approximation for the strength constraint, a straight use of integer linear programming is not advisable as the quality of the approximation may deteriorate away from the nominal point. The standard procedure recommended in this case is the use of sequential linear programming. In using sequential linear programming, move limits on design variables are generally invoked so that designs generated based on approximate constraints remain in or near the feasible design space. In the case of zero-one ply-identity variables, imposing move limits on the design variables is not practical. An alternative approach is to apply bounds on the extensional stiffnesses Aij' expressed in terms of the ply-identity variables. This requires addition of six more constraints to the problem at each step of the sequential integer linear programming solutions,

320


At-Au$0

and

A lj.. - A ll!j< 0' -

i,j=I,2.

(8.3.21)

"' "' "' .~

.~

Ol Ql

Ol Ql

Ol Ql

"9"9"9 0"' 0> "d'

~11

Example 8.3.2 A a = 20-in long rectangular Graphite/Epoxy laminate with an aspect ratio of alb = 4 is to be designed to maximize its buckling load under combined longitudinal and transverse loads. The laminate is to have a 48-layer balanced symmetric layup with oo, ±45°, and 90° orientation angles. The elastic properties for the layers are £ 1 = 18.5 x 106 psi, £ 2 = 1.89 X 106 psi, G 12 = 0.93 x 106 psi, v12 = 0.3, and t = 0.005 in. The laminate is to be designed for two combinations of in-plane stress resultant ratios, N/Nx = 0.25 and N/Nx = 0.5 with strain constraints in the individual layers. The failure strains in the principal material directions are E~n=0.008,~n=0.029,and 1:~=0.015. Design results for the 48-ply laminates under two different combination of biaxial loads, N/Nx = 0.25 and N/Nx = 0.5, are presented in Fig. 8.5. The reference value used for the longitudinal stress resultant is Nxo = 1.0 lb/in so that the computed value of the A at any step corresponds to the buckling value. Since the method used involves local approximations, the final design may be a locally optimal design. Designs with a higher confidence of being globally optimum can be generated by using the genetic search algorithm discussed in Chapter 5. For each of the biaxial load combinations considered, three stacking sequences are shown in Fig. 8.5 schematically. The first one represents the laminate for maximum buckling load obtained by sequential use of integer linear programming with no constraints. The second one is the maximum buckling load design with the strain constraint. Finally, the last design is generated using the genetic algorithm discussed earlier and verified to be actually the global optimum design for each load case. Compared to the design without strength failure constraint, for NY= 0.25 the failure load factor decreased from 13441.85 to 12622.44 (by about 6.0%) because of the inclusion of the strain constraint. Inclusion of four stacks of oo layers was responsible

.~

"C. C.. C..

The results of a design study formulated using the approach described in this section are summarized in the following example taken from Nagendra et al. (1992).

0

C\1 (j) (j)

c"'

~ 0

-~

N

Ul C

cOl

~

g-~

en

c"' .@

Cii

to

tiic

8 -~

Q5

:;

lO

+I -.::t ~0

COl

8ro~o

a~~ E

ctl

-oc

~~~E

a>"":2E LO co ·ca ::s va>-E

Ill c. 011~0

ou.:-o

~

1, ~ ~

'E

C\1.!: ·.;:::.

+I

~85-ffia

'OJ~

~..2~-ca

....,_j;"''Cil ~ c .0 +I ro ..Q 'OJ Ol(!J 0 c 0') == -"'

-

~

-~(9

e. ~

g

.c

Q)

Ol

ctl

Cll

-~

iii c 0

.c

(.)

.c

~ -~"' 0 ~

LO

USc

'N

0~

11

5 .Ql

E

LOII'iilO 'OJ

~

-

c

Ol

roz

Ol-

Cll

Ol '(ij

~~-iij·a

"0

-

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for the decrease of the buckling load factor. Compared to the genetic algorithm design, which was verified to be the global optimum stacking sequence, the load factor was fractionally smaller. Although the total number of stacks with different orientations was identical between the global optimal genetic algorithm design and the integer linear programming design, a slight variation of the locations of the 0° and the 90° stacks was responsible for the small difference of the buckling load factor. For the load ratio of NY= 0.5, the design without the strain constraint violated the shear strength in the 45° layers by 7%. The design obtained from the sequential integer linear programming approach was identical to the genetic algorithm design and, therefore, was globally optimum.

8.3.4

Stiffness Design for Minimum Thickness

Instead of maximizing the frequency or buckling load of a laminate of given thickness, we often need to solve the problem of minimizing the thickness of a laminate subject to a limit on the frequency or buckling load. The thickness minimization problem is more complicated because we do not know the value of the number of layers, N, to use in equations such as Eq. (8.3.4). What we have to do is to select N as an upper bound for the number of layers (for example by finding a design that satisfies all the constraints). We can no longer use equations such as Eq. (8.3.15), which require every layer to be occupied, but instead replace this equation by oi+ni+,h5, 1,

i= 1, ... ,N/4,

(8.3.22)

thus allowing all the stack-identity variables to assume zero value for a given layer. This equation has to be supplemented with oi+ni+f;5,oi-l +ni-l +,h_"

i=2, ... ,N/4,

(8.3.23)

which require that if the ith stack is occupied, so is the (i - 1)th. These latter conditions eliminate the possibility of ending with designs with empty stacks in the interior of the laminate. For design subject to constraints on frequencies, we also have to be careful in formulating the constraints so as to preserve linearity of the constraint in term of stack-identity design variables. Because of

8.3 PL!XUFIAL STIFFNESS DESIGN BY INTEGER LINEAR PROGRAMMING

323

the dependence on h, the square of romn in Eq. (8.1.8) is no longer a linear function of these design variables. However, we can replace a constraint on the frequency such as (!)mn

~ (l)req

(8.3.24)

by N/4

h ro~n?. h ro~eq =4 t (!)~

L (oi +h + n).

(8.3.25)

i=l

The left side of Eq. (8.3.25) is a linear function of the Du's [see Eq. (8.1.8)] and therefore of the stack-identity design variables. Another problem associated with minimum thickness design has to do with nonuniqueness of the solution. That is, we may have multiple designs of the same thickness that satisfy all the constraints. In such a circumstance, we may prefer the design that satisfies the constraints with the largest margin. Consider, for example, the case of minimizing the thickness of a laminated plate subject to the requirement that it does not buckle under an Nx = 0.8 MN/m. Assume, furthermore, that if we optimize a 16-ply laminate for maximum buckling load, the optimum buckling load is 0.75 MN/m. This means that we need the next higher available thickness, which, with two-layer stack design variables, will be a 20-ply laminate. The buckling load of the laminate, being linear in the Du's, is approximately proportional to the cube of the thickness (it will be exactly proportional to the thickness if we can preserve the value of the lamination parameters W~ and w;). Therefore, with a 20-ply lami3 nate we can expect a maximum buckling load of about 0.75 x 1.25 = 1.46 MN/m. With such a large margin between the maximum buckling load and the required buckling load, we can expect many 20-ply laminates to be feasible. If we do not have some additional requirements that will help us choose between all the possible laminates, we should at least try to obtain the 20-ply laminate with the highest buckling load. Such a laminate will satisfy the constraint with the highest margin. This can be achieved by subtracting from the objective function (thickness) a small constant times the buckling load. The following example demonstrates the use of this technique.

324


Example 8.3.3 Design a minimum thickness laminate for a 20 x 15 in Graphite/Epoxy plate so that its fundamental frequency is at least 50 Hz. The material properties are £ 1 = 18.5 x 106 psi, £ 2 = 1.89 x 106 psi, G 12 = 0.93 x 106 psi, v12 = 0.3, t = 0.005 in, and p = 0.057 lb/in 3 . As a first step, we need to estimate an upper bound on thickness of the laminate by finding one laminate that satisfies that 50 Hz requirement. We can do that by selecting a single orientation angle and determining how thick this angle-ply laminate needs to be in order to satisfy the 50 Hz requirement. The optimum laminate is guaranteed to be no thicker. For convenience, we choose an all-zero laminate, so that W~ = w; = 1. We calculate the material constants Ui from Eqs. (2.4.8) and (2.4.20) to obtain U 1 = 8.3253

X

106 psi,

U 2 = 8.3821

X

106 psi,

U 3 = 1.9643

X

Next, we specialize Eq. (8.3.25) to our case. We rewrite Eq. (8.1.8) as 2

h roj, With

=:a: [D.,+ 2(D, + 2D (~) 00)

(~; }u, + U + U 2

3)

= 1.5560 x 106 h3 lb-in,

u~ }u,- V, + V,) = 1.5896 X)(}' h' lb-in,

D,z =

(~;

D 66

}u

4 -

U 3) = 4.7689 X 104 h3 lb-in,

=(~;}us- U

3)

= 7.7500 x 1Q4 h3 lb-in.

Substituting these and p = 0.057/386.4 into Eq. (8.1.8) we get C011

= 537.74 h Hz.

So, for a frequency of 50 Hz, we need h = 0.093 in or 18.6 layers. For a symmetric and balanced laminate, this translates to 20 layers, or with stack design variables to N/4 = 5 sets of stack design variables.

(~)

l (a)

Next, we write the equation for V0n from Eq. (8.3.4) as o 1 + n 1 + f 1 + 7(o2 + n 2 + f 2) + 19(o3 + n 3 + f 3) + 37(o4 + n 4 + ! 4) (b)

+ 61(os + ns + fs)- 1.5 x 106 Von= 0,

with similar equations for V 1n and V3n. Next, we use Table 2.1 to express D 11 as D 11

= U 1V0n + U2 Vm + U 3 V 30

or 8.3253

D, =

+ D,

__ z h-4.127 _ [ D + (32) (D +2D )+ (256) D 22] ~98,696h. WI 66 12 11 1 81 9

Then, from Eq. (8.2.10), we calculate

D 11 =

4

ro0 =50 Hz = 314.16 rad/s, Eq. (8.3.25) may be written as

106 psi,

Us= 2.8943 X 106 psi.

U4 = 2.5366 X 106 psi,

325


X

106 V0n + 8.3821

X

106 Vm + 1.9643

X

106 V3n- D 11 = 0. (c)

Using Eq. (a) and equations of the type of (b) and (c), and defining ~ = roj 1 h, the optimization problem is formulated as minimize

h

such that

o, + n, + !, + Oz + n2 + !2 + o3 + n3 + f, + o4 + n4 + !4 + Os + n 5 + f 5 - 50h = 0, 0.05095 [81D 11 + 288(D 12 + 2D66 ) + 256D 22] - ~ = 0, ~~

98696h,

o 1 + n 1 + f 1 + 7(o2 + n2 + f 2) + 19(o3 + n3 + ! 3) 6

+ 37(o4 + n4 + f 4 ) + 61(o5 + n 5 + f 5 ) - 1.5 x 10 V0n = 0,

326


o 1- n 1+ 7(o2 - n 2) + 19(o3 - n 3) + 37(o4 - n4) + 61 (o 5 - n5) - 1.5 x 106 V10 = o, o1 + n 1 -

/ 1

+ 7(o2 + n2 -

/ 2)

+ 19(o3 + n 3 -

8.3253

X

10 V0n + 8.3821 x

106

V 10

+ 1.9643 x 106 V3n- D 11 = o, 8.3253

X

106 V0n- 8.3821

X

106 V 10

+ 1.9643 x 106 V3n - D 22 = o, 6

2.5366 X 10 Von- 1.9643

X

106 V3n- D12 = 0,

106 V0n- 1.9643

X

106 V3n- D 66 = 0,

2.8943

X

01

+ nl +!1

~ 1,

02

+ nz + fz

~

0 3 + n3

01

+ nl + f1,

V!D := V!Dp - V!Dm'

:=

V3np - V3Dm·

For the problem of thickness minimization of a laminate capable of sustaining a specified loading Nx and NY without buckling, the formulation is similar. The buckling constraint will take the form of (8.3.26)

min Acr (m, n) ~ 1. m,n

If we want to get the minimum thickness laminate with the largest margin, the formulation takes the form

o5 + n5 + fs ~ o4 + n4 + f4,

o;, n;,f; are binary integers,

V3n

The solution obtained was o 1 = o2 = o3 = n4 = 1, corresponding to a [90/06 ]. laminate with a frequency of 55.0 Hz. To check whether we can obtain a solution with the same thickness but higher frequency, the objective function was changed to h- 0.0001fj. The second term gives designs with higher frequencies a lower objective function, but it is small enough so that increasing h by adding another layer increases the objective function even though it increases the frequency. This time the solution from LINDO was / 1 =f 2 =f 1 =f 4 =I or [±45]4s laminate with a fundamental frequency of 65.4 Hz.

+!3 ~ 0 2 + nz + fz,

04 + n4 +!4 ~ 0 3 + n3 + f3,

find i

= 1, ... , 5.

When we attempted to solve this optimization problem with the LINDO program (Schrage, 1989), two problems surfaced. The first was associated with the scaling of the variables. The integrals such as V0n were of the order of 105 , the Du's were of the order of 10, and ~ = roi 1 h was of the order of 10 5 • LINDO warned about possible ill conditioning associated with such scale differences. The problem was solved by using scaled vari= 1Cf ViD, i = 0, 1' 3, and II= 1o-5 ~. The second problem ables

vlD

was associated, as in Example 8.3.1, with the assumption by LINDO that all variables are positive. Since V 10 and V3n can take negative values, they had to be replaced by the difference of two positive variables

/ ) 3

+ 37(o4 + n4 - / 4) + 61(o5 + n5 - fs)- 1.5 X 106 V3D = 0, 6

327


o;, n;,f;, and')...*,

i

= 1, ... , N/4,

(8.3.27)

N/4

to minimize

L (o; + n; +/;)- E'A*

(8.3.28)

i=1

such that Acr (m, n)

~'A*,

m = 1, ... , mf'

n

= 1, ... 'nf, (8.3.29)

'A*~

1,

328


oi + ni +!; ~ 1,

i

= 1, ... , N/4,

and oi + ni +!;~oi-l +ni-l +.f;_ 1.

(8.3.30)

(8.3.31)

Here, E is a small number (of the order of 0.01 or 0.001) used to make the optimization procedure favor laminates with high buckling load over laminates of the same thickness with lower buckling load.

EXERCISES 1.

The plate of Example 8.2.1 is to be designed so that the fundamental frequency is at least 10 Hz, and the second frequency is as high as possible. Use Miki's diagram to find the stacking sequence of the optimum laminate.

2.

Repeat Example 8.2.2 for a 15 in square plate.

3.

Prove that the solution of Eq. (8.2.12) satisfies

4.

Design the plate of Example 8.2.1 for maximum buckling load under biaxial loading with N/Nx = 0.25.

5.

Find the optimum design for Example 8.3.3 without the constraint of using two-layer stack design variables.

z~_ 1 ~z* ~zH 1 •

REFERENCES Haftka, R. T. and Walsh, J. L. (1992). "Stacking Sequence Optimization for Buckling of Laminated Plates by Integer Programming." AIAA J. 30(3), 814819. Kedward, K. T. and Whitney J. M. (1990). Design Studies, Vol. 5. Delaware Composites Design Encyclopedia, Carlsson, L. A. and Gillespie, J. W. (eds.) Technomic Publishing Co, Lancaster, PA. Miki, M. (1985). "Design of Laminated Fibrous Composite Plates with Required Flexural Thickness." Recent Advances in Composites in the United States and Japan, pp. 387-400. ASTM STP 864. Vinson, J. R. and Taya, M. (eds.) American Society for Testing and Materials, Philadelphia. Miki, M. (1986). "Optimum Design of Laminated Composite Plates Subject to Axial Compression." Composites '86: Recent Advances in Japan and the United

REFERENCES

329

States, pp. 673-680. Kawata, K., Umekawa, S., and Kobayashi, A. (eds.) Proc. Japan-U.S. CCM-III, Tokyo. Nagendra, S., Haftka, R. T., and Gtirdal, Z. (1992). "Stacking Sequence Optimization of Simply Supported Laminates with Stability and Strain Constraints." AIAA J. 30(8), 2132-2137. Nemeth, M. P. (1986). "Importance of Anisotropy on Buckling of Compression. Loaded Symmetric Composite Plates." AIAA J. 24(11), 1831-1835. Schrage, L. (1989). User's Manual for LINDO. Fourth Edition. Scientific Press, Redwood City, CA. Shin, Y. S., Haftka, R. T., Watson, L. T., and Plaut, R. H. (1989). "Design of Laminated Plates for Maximum Buckling Load." J. Composite Materials 23, 348-369. Vinson, J. R. and Sierakowski, R. L. (1987). The Behavior of Structures Composed of Composite Materials. Martin us Nijhoff Publishers, Dordrecht.

INDEX

Active constraint, 25 Aluminum, 2, 20, 47, 58, 87, 246 Angle-ply laminate, 18, 20, 77, 125, 138, 146, 162, 301, 309, 324 Angle-ply stacks, 156 Anisotropic materials, 9 Antisymmetric laminate, 18, 87, 119-120 Aramid/Epoxy, 11 AS4/3501 Graphite/Epoxy, 11 AS4/350 l-6 Carbon/Epoxy, 279 Autoclave, 91 Average laminate stresses, 45 Averaging crossover, 204 Balanced laminate, 18, 74, 83, 128, 298, 301 Balanced symmetric laminate, 172, 287 Bending of laminates, 49, 52 Bending stiffness matrix, 20, 74, 86, 103, 208 Bending-extension coupling, 54, 85, 103, 118-119, 128, 214 Bending-extension coupling matrix, 55 Bending -extension stiffness, 106 Bending-twisting coupling, 74, 78, 87, 89, 298 Binary coding, 207, 214 Binary design variables, 135 Binary string, 185, 203 Binary variables, 172 Bonus parameter, 192-193, 195, 212, 293-294 Boron fibers, 8 Boron/Epoxy, 11, 91, 234, 244 B( 4)/5505, 11 Branch-and-bound, 169, 179, 180, 182, 220

Brittle fracture, 246 Brittle materials, 223, 226, 246 Buckling constraint, 128, 189, 210, 214, 292, 327 Buckling load, 33, 300, 318, 328 Buckling load maximization, 317 Buckling mode, 300, 310, 318 Buckling of a laminate, 298 Carbon fibers, 8 Carbon/Epoxy, 124 AS4/3501-6, 279 Classical lamination theory, 49, 69, 102, 127, 245, 254 Coefficient of thermal expansion, 90, 94, 153 Combined stresses, 242 Compliance coefficients, 41 Composite materials, 4 Composite strength, 233 Composites fiber reinforced, 7 fibrous, 7 flake, 6 laminated, 8 particulate, 6 short fiber, 8 Compressive failure, 232, 252, 260 Compressive strain, 50, 68, 256 Compressive strength, 236, 238 ultimate, 227 Constituent materials, 4-5, 11, 90, 232, 246 Constituent properties, 12 Constitutive behavior, 43 Constitutive equations, 21, 61, 70, 79, 129, 290 331

332 Constraint shear stiffness, 182 strength, 289 Constraint derivatives, 261 Constraint functions, 2, 28, 170 Constraint margin, 191, 193, 195 Constraints, 24, 134 buckling, 189, 210, 214 elastic properties, 138 engineering stiffnesses, 157 equality, 24, 123, 135 frequency, 322 inequality, 24-25, 135, 174 layer percentage, 175 normalized, 191 notation, 24 Poisson's ratio, 153, 161, 174 shear, 161 shear stiffness, 174, 270 stiffness, 25, 208 strain, 134, 137, 190, 211 strength, 123, 189, 261 Continuous optimization, 177 Coupling bending-extension, 85, 103, 118-119, 128, 214 bending-twisting, 74, 78, 87, 89, 298 extension-shear, 74, 122 extension-twisting, 73, 89, 120, 214, 218 shear-extension, 225, 298 twisting-entension, 118 Coupling response, 73, 81, 89 hygrothermal, I 05 Cross-linking, 91 Cross-ply laminate, 18, 97, 111, 143, 165, 247, 255, 302 Crossover averaging, 204 multiple point, 202 probability, 202 single-point, 201 Cure temperature, 91 Curvature, 51-52, 54, 56, 58, 60, 69, 73, 79, 100, 103 hygrothermal, 102

INDEX

twisting, 73-74, 120 Curvature-stable laminates, 100, 120, 122, 214 Curvatures, 60 Damage tolerance, 247 Delamination damage, 225 Delaminations, 23 Derivatives of constraints, 261 Design variables, 24, 134 binary, 28, 135, 172 bounds, 171 continuous, 4, 24, 134, 171, 261, 274 discrete, 4, 24, 183, 261, 310 integer, 24, 28, 134, 185 notation, 24 orientation, 30, 216, 274, 286 ply-identity, 310, 313 rounding off, 29 stack-identity, 312, 317, 323 thickness, 156, 275 Discrete design variables, 4, 183, 261, 310 Discrete variables, 155 Displacement minimization, 308 Distortional energy (von Mises) criterion, 230 Ductile materials, 224, 226, 230 Effective engineering properties, 46-47, 81, 128, 138, 140, 149, 174 Effective engineering stiffnesses, 149 Effective Poisson's ratio, 150 Effective elastic properties of a laminate, 82 Effective shear modulus, 47 Elastic modulus Aluminum, 19 effective, 46 Graphite/Epoxy, 19 Elastic properties, 10, 22, 233 Electrical conductivity, 16 Engineering strain transformation, 65, 249, 291

INDEX

'

l

·..~,:.·:.'...·:.

.i(

::~~

Engineering strain transformation matrix., 278 Engineering strains, 64 Enumeration, 134, 160, 169, 177, 179, 181, 183 Enumeration tree, 178 Environmental effects, 5, 16 Epoxy, 26 Equality constraints, 24, 135 Equilibrium equations, 36 Evolutionary strategies, 186 Extension-shear coupling, 74, 122 Extension-twisting coupling, 89, 120, 214, 218 Extensional stiffness matrix, 46 Fabrication of laminates, 90 Failure brittle materials, 223-224 ductile materials, 224 Failure criteria, 223-224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, 254, 256, 258, 260 distortion energy, 230 first-ply, 287 first-ply failure, 224, 248, 285 Hoffman, 237, 239 isotropic, 226-227, 229 maximum normal stress, 226 maximum shear stress, 229 maximum strain, 227, 249 maximum stress, 249 orthotropic, 241 progressive failure, 224, 248 quadratic, 237 Tresca, 229 Tsai-Hill, 237-238, 272, 287 Tsai-Wu, 238, 241, 244, 287 von Mises, 238, 243 Failure envelope, 226 normal stresses, 233 Failure mechanisms, 232 Failure model, 23 Failure stress, 232 Feasible design, 191

333 Feasible domain, 25, 139, 149 Fiber direction, I 0, 225 Fiber failure, 16 Fiber microbuckling, 232 Fiber radius, 8 Fiber-reinforced composites, 7 Fiber strength, 225, 231 Fiber tows/bundles, 8 Fiber volume fraction, 11, 26, 231 Fiberglass, 8 Filament winding, 21 First-ply failure, 224, 246, 285, 287 Flake composites, 6 Flexural lamination diagram, 301 Flexural lamination parameters, 301, 303 Flexural stiffness, 301 Flexural stiffness matrix, 310 Fracture, 226 Fracture toughness, 246 Frequency constraints, 322 Frequency maximization, 314 Fundamental frequency, 301-303, 317 Generally orthotropic, 61 Genetic algorithms, 29, 183-185, 187, 189, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 262, 289, 292, 320 coding, 185 fitness, 191 initial population, 188 mutation, 206 practical reliability, 210 ranking of designs, 198 reliability, 209 roulette wheel, 195 seeding, 189 Genetic operators crossover, 184, 201 gene swap, 184 inversion, 184 mutation, 184, 206 permutation, 184 reproduction, 184 selection, 191

334 specialized operators, 208 Glass/Epoxy, 2, II, 74, 91, 151, 258-259 Scotchply-1002, 151 Global optimum, 183 Graphical optimization, 261 Graphical strength design, 262-263, 265, 267, 269, 271, 273 Graphite/Epoxy, 2, 20, 67, 91, 124, 130, 139, 154, 211, 247, 264, 292, 303, 307, 320, 324 AS4/3501, 11 HT-S/4617, 215 T300/5208, 11, 19, 111, 116, 151, 255 Hoffman criterion, 239 Hooke's law, 34, 39 Hygral strains, 99 Hygrothermal curvatures, I 02 Hygrothermal isotropy, 122 Hygrothermal loads, 103 Hygrothermal moment resultants, 102 Hygrothermal response, 89 Hygrothermal strains, 100, 102 Hygrothermal stress resultants, 102 Hygrothermally curvature-stable laminates, 100, 120, 122 Hygrothermally stable laminates, 120, 214 In-plane isotropy, 83 In-plane lamination parameters, 140-141, 162, 172, 262 In-plane stiffness design, 261 In-plane stiffness matrix, 103, 173, 278, 310 In-plane stresses, 226 Inequality constraints, 24-25, 135, 157, 174 Inorganic composites, 5 Inorganic materials, 5 Integer design variables, 185 Integer linear programming, 262, 288 flexural stiffness design, 310-311, 313, 315, 317, 319, 321, 323, 325, 327

INDEX

Integer programming, 30, 160, 169-170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 288 Integer programming problem, 29 Integer variables, 134 Interlaminar stresses, 225 Isotropic hygrothermal, 122 in-plane, 83 Isotropic materials, 9, 10, 231 Kevlar, 26 Kevlar/Epoxy, 250 Kevlar49/Ep, 11 Kirchhoff-Love assumptions, 49-50, 54 Laminate angle-ply, 18, 77, 125, 138, 146, 301, 309 antisymmetric, 18, 87, 119 balanced, 18, 74, 298 balanced, symmetric, 172, 287 coefficient of thermal expansion, 153 cross-ply, 18, 97,111,143, 165,247, 255, 302 curvature-stable, 120, 122, 214-215 fabrication, 90 hygrothermally stable, 120 orthotropic, 107, 129 quasi-isotropic, 19, 83, 144, 146, 247 symmetric, 17, 43, 106 unsymmetric, 54, 74, 118-119 Laminate buckling, 298 Laminate failure, progressive, 254 Laminate in-plane stiffness design, 127-128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168 Laminate shear stiffness, 151 Laminate strain limits, 264 Laminate strength, 22

y

INDEX

Laminate stresses, average, 45 Laminate weight, 275 Laminated composites, 8 Laminates-quasi-isotropic, 132 Lamination diagram, in-plane, 262 Lamination parameter diagram, 141 Lamination parameters, 141, 169 flexural, 301 in-plane, 140, 172, 262 Lamination point, 142 quasi-isotropic, 145 Lamination theory, 49 Layer volume fraction, 173 LINDO, 316, 326 Linear integer programming, 310 Linear programming, 28, 170, 176, 288 integer, 171 mixed, 171 sequential, 319 zero-one, 172 Local failures, 223 Local optimum, 183 Longitudinal modulus, 27 Macromechanics, 21 Manufacturing induced stresses, 248 Mass density, 300 Material constants, orthotropic, 62 Materials anisotropic, 9 isotropic, 9 orthotropic, 9 Mathematical optimization problem, 128 Matrix constituent, 98 Matrix material, 5, 10, 224, 231 Matrix shear strength, 225 Matrix strength, 231 Maximum buckling load, 317 Maximum frequency, 314 Maximum fundamental frequency, 302-303 Maximum normal stress criterion, 226 Maximum shear stress (Tresca) criterion, 229 Maximum strain criterion, 227, 276

335 orthotropic materials, 233 Maximum strain failure criteria, 249 Maximum stress criterion, orthotropic materials, 233 Maximum stress failure criteria, 249 Microbuckling, 232 Micromechanics, 21-22 Mid-plane strains, 69 Miki's lamination diagram, 140, 155, 160, 162, 169, 261-262, 267, 301 flexural, 303, 310 flexure, 308 Minimum displacement, 302 Minimum thickness design, 322 Minimum thickness laminates, 192 Mixed integer linear programming, 171 Moisture, 89 Moisture diffusion coefficient, 92 Moisture effects, 21 Moment resultants, 51, 53, 57, 69, 73, 102, 109, 119 hygrothermal, 102 Multiple minima, 183 Mutation, 206 probability, 206 Natural vibration frequency, 298, 300 Nonlinear programming, 28 Nonmechanical strains, 100 Normal strain, 34 Objective function, 24, 134-135, 191, 289 notation, 24 Off-axis layer, 17-18, 61, 67, 73-74, 187, 235, 259, 298 specimen, 231 strength, 231-232 On-axis layer, 17, 68, 81 Optimization, 133 Organic materials, 5 Orientation angle, 17, 72 Orientation design variables, 275, 286 Orthotropic, 61 Orthotropic laminate, 107, 129

336 Orthotropic layers, 46, 64, 225 Orthotropic material constrants, 62 Orthotropic materials, 9 Orthotropic properties, 10, 224 Out-of-plane displacement, 298 Particulate composites, 6 PEEK, 6 Penalty parameters, 195 Permutation of stacking sequence, 306 Plane stress, 39, 229, 238, 249 Ply orientations, 128 Ply-identity variables, 313 Ply-level strains, 267 Poisson's effect, 224 Poisson's ratio, 10, 13, 36, 136, 174, 229 Poisson's ratio maximization, 219 Practical reliability, 210 Prepreg, 8, 90 Principal material axes, 61, 231, 233, 249 Principal material directions, I 0, 246, 291 Principal strains, 228 Principal stresses, 226, 231 Probability of mutation, 206 Progressive failure, 224, 248 Progressive failure of laminates, 254 Quadratic failure criteria, 237 Quasi-isotropic, 247 Quasi-isotropic laminate, 19, 83, 132, 144, 146, 279 Random search, 183 Ranking of fitnesses, 198 Reduced effective stiffness, 254 Reduced stiffness matrix, 41, 47 Reduced transformed stiffness matrix, 282 Reliability of genetic algorithms, 209 Residual strains, I 01 Residual stresses, 91, 97, 10 I, 248 Roulette wheel, 195 Rule of mixtures, 22

INDEX

Sequential linear programming, 277, 319 Shear coupling, 121 Shear modulus, 10, 36 effective, 47 Shear stiffness, 174 Shear strain, 34 Shear strength, 288 Shear-extension coupling, 74, 225, 298 Short fiber composites, 8 Simplex method, 171 Simply supported plate, 299 Simulated annealing, 29, 183, 219 Singular optimum, 189 Specific heat, 92 Specific moisture content, 92-93 Specific stiffness, 2, 20 Graphite/Epoxy, 20 Specific strength, 2 Stack thickness, 156 Stack-identity design variables, 312, 317, 323 Stacking sequence, 17, 20, 248, 292, 297 Stacking sequence design, 30, 208, 288 Stacking sequence permutation, 306 Stiffness matrix bending-extension, 55 extensional, 46 flexural, 310 in-plane, 103, 173, 278 reduced, 41 three-dimensional, 35 transformed reduced, 65, 84, 278, 318 two-dimensional, 40 Stiffness matrix transformed reduced, 113 Stiffness maximization, 175 Stochastic search, 29 Strain hygral, 99 hygrothermal, I 02 nonmechanical, 100 normal, 34 residual, 10 I shear, 34

337

INDEX

thermal, 99 Strain approximation, 283 Strain constraints, 137 Strain energy density, 230 Strain space, 227 Strain transformation, 64, 291 Strain transformation matrix, engineering, 249 Strain-displacement relation, 34 Strength constraint, 123, 214, 261, 275, 289 Strength design, 289 Strength of a laminate, 22 Strength properties, 233 Stress concentrations, 223 Stress couples, 51, 69 Stress resultants, 45, 49, 53, 56, 58, 60, 69, 73, 102, 119, 129, 249, 263, 275, 290, 300, 318 hygrothermal, I 02 Stress space, 228 Stress transformation, 64 Stresses residual, 91 through-the-thickness, 23 Symmetric laminates, 17, 43, 83, 106, 128, 153, 164, 287 genetic coding, 202 genetic representation, 187

f,

)·

I f~.

.

'

.,,

Tensile strength, 16 ultimate, 227 Tensor strains, 64 Thermal bending, 122 Thermal diffusivity, 92 Thermal effects, 89 Thermal isotropy, 154 Thermal strains, 99 Thermoforming, 21 Thermoplastic composites, 6 Thermoset composites, 6 Thickness design variables, 275 Thickness variable, 262 Tow-placement, 21 Transformation matrix, 64

engineering strain, 65, 278 Transformed reduced stiffness matrix, 65, 84, 101, 113, 278, 318 Transverse cracking, 247 Transverse direction, 10 Transverse failure, 248 Transverse loads, 298 Transverse modulus, 28 Transverse normal strength, 288 Tresca failure criterion, 229 Tsai-Hill criterion, 238 Tsai-Hill failure criterion, 237, 272, 287 Tsai-Wu criterion, 244 Tsai-Wu failure criterion, 241, 287 Twisting curvature, 74, 120 Twisting deformations, 73 Twisting-entension coupling, 118 Ultimate strains, 227 Ultimate strength, 227, 234 Unbalanced laminates, 192 Unidirectional tape, 8 Unsymmetric laminates, 54, 74, 118-119 bending-extension coupling, 54 coupling, 74 Vibration frequency, 300 Volume fraction, 11, 142, 145 Von Mises criterion, 237, 243-244 Von Mises failure criterion, 230 Warp-stable laminates, 118 Warping, 118 Weight density, 2 Aluminum, 20 Graphite/Epoxy, 20 Weight minimization, 175 Wood, 5 Woven fabric composite, 8 Yield strength, 226, 229 Yielding, 226, 229, 246 Young's modulus, 10, 12, 33, 35

Design and Optimization of Laminated Composite Materials

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