Most composite textbooks focus on the calculation of laminate stiffness and strength properties using laminated plate theory. While an important topic, this is only one small step in the design of laminates for real-world applications. And while it can be a challenging topic when first learned, it is relatively simple compared to the task of designing a laminate to meet specific requirements.

In "Design and Optimization of Laminated Composite Materials," Zafer Gürdal, Raphael Haftka and Prabhat Hajela have brought together the topics of composite analysis, mathematical optimization and laminate design into a single reference accessible to undergraduates. Laminate theory is covered in sufficient detail for a first year composites course. Design and optimization add important topics to the basic course and make this text stand out from others.

Laminate design can be stated as an optimization problem: given a design objective and constraints, find the combination of design variables that maximizes the objective. For a laminate, the objective might be to maximize longitudinal stiffness or panel frequency while minimizing weight. Constraints might include a minimum thickness and shear modulus. The design variables might include materials, ply angles, stacking sequence and ply thickness. Most likely the variables would be limited to common angles and commercially available thicknesses.

A number of general algorithms have been developed for solving optimization problems such as these. Until recently, composite engineers wanting to use these methods have had to resort to literature searches, first learning how to program the methods, then learning how to apply them to composite designs. What has been missing is a concise reference introducing the methods as used in composite design.

The authors consider two broad categories of optimization methods that have proven useful for composite design: graphical and integer linear programming (ILP). Enumeration is a type of ILP that takes a brute-force approach, calculating all solutions in the design space and comparing the results. Although computationally intensive, it is simple to implement and can be quite effective for relatively small design spaces.

Graphical methods developed specifically for composites can be simple yet powerful tools. The Miki diagram in particular can describe the entire family of [0/90/±θ₁/±θ₂/…]_S laminates in a single 2D plot. As shown in the section on laminate theory, the laminate engineering constants or A matrix can be expressed in terms of the traditional invariants (U_i) and two lamination parameters (V₁ and V₃) that are functions of the stacking sequence only. The axes of the Miki diagram are the lamination parameters; points on the diagram correspond to different stacking sequences. By superimposing engineering constant contours on the diagram, it is possible to find laminates that meet specific property constraints. The process sounds complex, but the authors provide clear instructions for developing the plots and numerous, fully-worked examples.

Integer programming methods can handle a broader class of problems than the Miki diagram, and genetic algorithms (GAs) in particular are well-suited to composite optimization. Laminates are coded as integer strings, analogous to DNA strands. A set of randomly generated strings provides the initial population. The strings are then recombined using methods such as crossover, mutation and permutation to generate a child population. After a number of generations, the population tends towards an optimum. Although GAs do not get stuck on local optima, their randomness also means they are unlikely to find the same solution if run multiple times. This can be an advantage, though, providing the engineer multiple designs to choose from.

The authors suggest using a GA from a commercial program or one of the publicly available math libraries, but they describe the algorithms in sufficient detail to write simple programs (and in fact they also provide Fortran source code for key subroutines). Guidance is also given for selecting parameters such as variable codings, penalty and fitness constants, and number of runs. As with the Mike diagrams, a number of examples show how to set up the problems and run them.

Most of the design section of the book focuses on optimization of in-plane stiffness. Bending, strength and other problems such as frequency and buckling are also covered, but their solution is more complex. The authors provide good introductions to those types of problems, but developing general solutions using only the materials presented in the book would be difficult. For these situations, readers should explore the references given at the end of each chapter.

The book's companion Web site offers additional resources for both students and instructors. For students, a series of Mathematica notebooks and program files offer tutorials and programming examples for the key concepts introduced in the book. These include not only basic laminate analysis, but also Miki diagrams, integer linear programming and genetic algorithms.

The instructor portion of the site is password protected. It offers two sets of materials: figures and lecture notes. The figures section contains all of the figures from the book in GIF format. The lecture notes includes 18 sets of viewgraphs in PDF format, used in classroom instruction. As with the student tutorials, they cover all of the topics in the book, and are suitable as supplemental classroom materials.

Details: "Design and Optimization of Laminated Composite Materials" by Gürdal, Haftka and Hajela, published by John Wiley, 1999, ISBN 0-471-25276-X.
Preface; 1. Introduction; 2. Mechanics of Laminated Composite Materials; 3. Hygrothermal Analysis of Laminated Composites; 4. Laminate In-Plane Stiffness Design; 5. Integer Programming; 6. Failure Criteria for Laminated Composites; 7. Strength Design of Laminates; 8. Laminate Design for Flexural and Combined Response; Index