Structural Optimization, 1985 Volume 1: Optimality Criteria Mathematical Concepts and Methods in Science and Engineering Series, Vol. 34

After the IUTAM Symposium on Optimization in Structural Design held in Warsaw in 1973, it was clear to me that the time had come for organizing into a consistent body of thought the enormous quantity of results obtained in this domain, studied from so many different points of view, with so many different methods, and at so many levels of practical applicability. My colleague and friend Gianantonnio Sacchi from Milan and I met with Professor Prager in Savognin in July 1974, where I submitted to them my first ideas for a treatise on structural optimization: It should cover the whole domain from basic theory to practical applications, and deal with various materials, various types of structures, various functions required of the structures, and various types of cost . . Obviously, this was to be a team effort, to total three or four volumes, to be written in a balanced manner as textbooks and handbooks. Nothing similar existed at that time, and, indeed, nothing has been published to date. Professor Prager was immedi­ ately in favor of such a project. He agreed to write a first part on optimality criteria with me and to help me in the general organization of the series. Since Professor Sacchi was willing to write the text on variational methods, it remained to find authors for parts on the mathematical programming approach to structural optimization (and, more generally, on numerical methods) and on practical optimal design procedures in metal and concrete.

1. Review of Some Basic Concepts and Theorems of Structural Analysis.- 1.1. Static and Kinematic Variables.- 1.2. Linearly Elastic Structures.- 1.3. Rigid, Perfectly Plastic Structures.- 1.4. Elastic, Perfectly Plastic Structures.- 2. Problem Formulation and Optimality Criteria.- 2.1. Terminology.- 2.2. Problem Types.- 2.3. Elementary Problems of Structural Optimization.- 2.4. Empirical Optimality Criteria.- 2.5. Ill-Posed Problems.- 3. Optimal Design of Beams and Frames.- 3.1. Optimality Condition for Prescribed Load Factor at Plastic Collapse.- 3.2. Optimal Plastic Design for Given Collapse Load.- 3.3. Optimality Condition for Prescribed Elastic Compliance.- 3.4. Optimal Elastic Design for Given Compliance.- 3.5. Various Behavioral Constraints: Common Features and Optimality Conditions.- 3.6. Examples of Optimal Single-Purpose Structures for Various Behavioral Constraints.- 3.7. General Formulation of Optimality Conditions for Multipurpose Beams and Frames.- 3.8. Examples of Optimal Design of Multipurpose Structures.- 3.9. Optimization of the Layout.- 4. Optimal Design of Trusses.- 4.1. Optimal Plastic Design of a Truss of Given Layout for a Single Loading.- 4.2. Optimal Layout of a Truss in Plastic Design for a Single Loading.- 4.3. Michell’s Problem.- 4.4. Michell Fields.- 4.5. Use of Michell Fields.- 4.6. Optimal Design of an Elastic Truss for Given Compliance.- 4.7. Nearly Optimal Layout of Trusses with a Finite Number of Joints.- 4.8. Some Remarks on Optimal Trusses.- 4.9. Appendix.- 5. Optimal Design of Grillages.- 5.1. Optimal Plastic Design of Rectangular Grillages of Orthogonal Prismatic Beams.- 5.2. Optimal Layout of a Grillage in Plastic Design for a Single Loading.- 5.3. Morley Fields.- 5.4. Matching of Morley Fields.- 5.5. Optimal Polygonal Grillages.- 5.6. Partially Discretized Grillages.- 6. Optimal Design of Plates, Shells, and Disks.- 6.1. Introduction.- 6.2. Circular Plates with Axisymmetric Loading.- 6.3. Other Plate Problems.- 6.4. Cylindrical Shell under Axisymmetric Loads.- 6.5. Other Shell Problems.- 6.6. Circular Disks in Plane Stress.- 6.7. Other Optimal Disk Problems.- 7. Lagrange Multiplier Methods for Optimization with Constraints.- 7.1. Introduction.- 7.2. Beams in Bending.- 7.3. Plates in Bending.- 7.4. Optimal Segmentation and Optimal Location of Supports.- 7.5. General Discussion.- 8. Review of Some Extensions of Design Procedures.- 8.1. Introduction.- 8.2. Extensions to Other Structures.- 8.3. Extensions to Other Behavioral Constraints.- 8.4. Extensions to Other Cost Functions.- 9. Solutions to Exercises.- References.