Optimization of Finite Dimensional Structures K11056_FM.indd 1 6/11/10 3:18:11 PM Optimization of Finite Dimensional Structures Makoto Ohsaki Hiroshima University Higashi-Hiroshima, Japan K11056_FM.indd 2 6/11/10 3:18:11 PM Optimization of Finite Dimensional Structures Makoto Ohsaki Hiroshima University Higashi-Hiroshima, Japan K11056_FM.indd 3 6/11/10 3:18:11 PM CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4398-2004-9 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources. 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For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Preface The attempt to find mechanically efficient structural designs and shapes was initiatedmainlyinthefieldsofmechanicalengineeringandaeronauticalengi- neering, which established the field known as structural optimization. Many practically acceptable results have been developed for application to auto- mobiles and aircraft. Some examples are structural components, including the wings of aircraft and engine mounts of automobiles, which can be fully optimized using efficient shape optimization techniques. Incontrast,regardingcivilengineeringandarchitecturalengineering,struc- tural optimization is difficult to apply because structures in these fields are not mass products: structures are designed in accordance with their specific designrequirements. Furthermore,thestructure’sshapeandgeometryarede- termined by a designer or an architect in view of nonstructural performance, includingtheaestheticperspective. Therefore, themainroleofstructuralen- gineersisoftenlimitedtoselectionofmaterials,determinationofmembersizes through structural analyses, planning details of the construction process, and so on. However, for special structures, such as shells, membrane structures, spatial long-span frames, and highrise buildings, the structural shape should be determined in view of the responses against static and dynamic loads. In truth, the beauty of the structural form is related closely to the mechanical performance of the structure. Therefore, cooperation between designers and structural engineers is very important in designing such structures. Even for building frames, because of the recent trend of performance-based design,optimizationhasbeenidentifiedasapowerfultoolfordesigningstruc- turesunderconstraintsimposedonpracticalperformancemeasures,including elastic/plastic stresses and displacements under static/dynamic design loads. Furthermore, recent rapid advancements in the areas of computer hardware and software enabled us to carry out structural analysis many times to ob- tain optimal or approximately optimal designs. Optimization of real-world structureswithrealisticobjectivefunctionandconstraintsispossiblethrough quantitative evaluation of nonstructural performance criteria, e.g., aesthetic properties, and life-cycle costs, including costs of construction, fabrication, and maintenance. Many books describing structural optimization have been published since the 1960s; e.g., Hemp (1973), Rozvany (1976), Haug and Cea (1981), Haftka, Gu¨rdal, and Kamat (1990), Papalambos and Wilde (2000), Bendsøe and Sig- mund (2003), Arora (2004), etc. These books are mainly classifiable into the following three categories: v vi Preface 1. Basic theories and methodologies for optimization with examples of small structural optimization problems. 2. Continuum-based approaches for application to mechanical and aero- nautical structures. 3. Theoretical and analytical results of structural optimization in earlier times without the assistance of computer technology. Using books of the first category, readers can learn only the concepts and somedifficulttheoriesofstructuraloptimizationwithoutapplicationtolarge- scale structures. On the other hand, for the books of the second category, a goodbackgroundinappliedmathematicsandcontinuummechanicsisneeded to fully understand the basic concepts and methods. Unfortunately, most researchers, practicing engineers, and graduate students in the field of civil engineering have no such background and are not strongly interested in the basic theories or methods of structural optimization. Also, in mechanical en- gineering, the finite element approach is used for practical applications, and complex practical design problems are solved in a finite dimensional formula- tion. Thederivativesofobjectiveandconstraintfunctions,calleddesignsensitiv- ity coefficients, should be computed if a gradient-based approach is used for structural optimization. However, most methods of design sensitivity analy- sis are developed mainly for a continuum utilizing variational principles, for whichsensitivitycoefficientsaretobecomputedforafunctional,suchascom- pliance that can be formulated in an integral of a bilinear form of response. For finite dimensional structures, including trusses and frames, variational formulations are not needed, and sensitivity coefficients can be found simply by differentiating the governing equations in a matrix-vector form. Another important aspect of optimization in civil engineering is that the design variables often have discrete values: the frame members are usually selectedfromapre-assignedlistorcatalogofavailablesections. Furthermore, some traditional layouts are often used for plane and spatial trusses and for latticed domes. Therefore, the optimization problem often turns out to be a combinatorial problem, a fact that is not fully introduced into most books addressing the study of structural optimization. This book introduces methodologies and applications that are closely re- lated to design problems of finite dimensional structures, to serve thereby as a bridge between the communities of structural optimization in mechanical engineering and the researchers and engineers in civil engineering. The book provides readers with the basics of optimization of frame structures, such as trusses,buildingframes,andlong-spanstructures,withdescriptionsofvarious applications to real-world problems. Recently, many efficient techniques of optimization have been developed for convex programming problems, e.g., semidefinite programming and inte- rior point algorithms, which are extensions of the approaches used for linear Preface vii and quadratic programming problems. The book introduces application of these methods to optimization of finite-dimensional structures. Approximate methodsresemblingtheconventionaloptimalitycriteriaapproacheshavealso been developed with no reference to the pioneering papers in the 1960s and 1970s. Therefore,itisextremelyimportanttodescribetheirdevelopmenthis- tory to young researchers so that similar methods are not re-developed with no knowledge related to conventional approaches. For that reason, another purpose of this book is to present the historical development of the method- ologies and theorems on optimization of frame structures. The book is organized as follows: In Chapter 1, the basic concepts and methodologies of optimization of trussesandframesarepresentedwithillustrativeexamples. Traditionalprob- lemswithconstraintsonlimitloads, memberstresses, compliance, andeigen- values of vibration are described in detail. A brief introduction is also pre- sentedformultiobjectivestructuraloptimization,andtheshapeandtopology optimization of trusses. InChapter2,themethodofdesignsensitivityanalysis,whichisanecessary tool for optimization using nonlinear programming, is presented for various response quantities, including static response, eigenvalue of vibration, tran- sient response for dynamic load, and so on. All formulations are written in matrix-vectorformwithoutresorttovariationalformulationtosupportready comprehension by researchers and engineers. InChapter3,detailsoftrusstopologyoptimizationaredescribed,including historicaldevelopmentsanddifficultiesinproblemswithstressconstraintsand multiple eigenvalue constraints. Recently developed formulations by semidef- inite programming and mixed integer programming are introduced. Applica- tions to plane and spatial trusses are demonstrated. Chapter 4 presents methods for configuration optimization for simultane- ously optimizing the geometry and topology of trusses. Difficulties in opti- mization of regular trusses are extensively discussed, and an application to generating a link mechanism is presented. Chapter 5 summarizes various results of optimization of building frames. Uniqueness of the optimal solution of a regular frame is first investigated, and applications of parametric programming are presented. Multiobjective optimization problems are also presented for application to seismic design, and a simple heuristic method based on local search is presented. In Chapter 6, as a unique aspect of this book, optimization results are pre- sentedforspatialtrussesandlatticeddomes. Simpleapplicationsofnonlinear programmingandheuristicmethodsarefirstintroduced,andthespatialvaria- tion of seismic excitation is addressed in the following sections. The trade-off designs between geometrical properties and stiffness under static loads are shown for arch-type frames and latticed domes described using parametric curve and surface. Mathematicalpreliminariesandbasicmethodologiesaresummarizedinthe Appendix,sothatreaderscanunderstandthedetails,ifnecessary,withoutthe viii Preface exposition of tedious mathematics presented in the main chapters. Various methodologies specifically utilized in some of the sections, e.g., the response spectrum approach for seismic response analysis, are also explained in the Appendix. Also, many small examples that can be solved by hand or using a simple program are presented in the main chapters. Therefore, this book is self-contained, and easily used as a textbook or sub-textbook in a graduate course. TheauthorwouldliketodeliverhissincereappreciationtoProf. Tsuneyoshi Nakamura, Prof. Emeritus of Kyoto University, Japan, for supervising the author’s study for master’s degree and Ph.D. dissertation on structural op- timization. Supervision by Prof. Jasbir S. Arora of The University of Iowa during the author’s sabbatical leave is also acknowledged. The numerical examples in this book are a compilation of the author’s workonstructuraloptimizationatKyotoUniversity,Japan,duringtheperiod 1985–2010. Theauthorwouldliketoextendhisappreciationtoresearchersfor collaborations on the studies that appear as valuable contents in this book, namely, Prof. Naoki Katoh of the Dept. of Architecture and Architectural Engineering, Kyoto University; Prof. Shinji Nishiwaki of the Dept. of Me- chanical Engineering and Science, Kyoto University; Prof. Hiroshi Tagawa of theDept.ofEnvironmentalEngineeringandArchitecture,NagoyaUniversity; Prof. Yoshihiro Kanno of the Dept. of Mathematical Informatics, University of Tokyo; Prof. Peng Pan of the Dept. of Civil Engineering, Tsinghua Uni- versity,P.R.China; Dr.TakaoHagishitaofMitsubishiHeavyIndustries; Mr. Yuji Kato of JSOL Corporation; Mr. Takuya Kinoshita, Mr. Shinnosuke Fu- jita, and Mr. Ryo Watada, graduate students in the Dept. of Architecture and Architectural Engineering, Kyoto University. The author would also like tothankagainProf. YoshihiroKannoofUniversityofTokyoforcheckingthe details of the manuscript. The assistance of Ms. Kari Budyk and Ms. Leong Li-Ming of CRC Press and Taylor & Francis in bringing the manuscript to its final form is heartily acknowledged. January 2010 Makoto Ohsaki Contents Preface v 1 Various Formulations of Structural Optimization 1 1.1 Overview of structural optimization . . . . . . . . . . . . . . 1 1.2 History of structural optimization . . . . . . . . . . . . . . . 3 1.3 Structural optimization problem . . . . . . . . . . . . . . . . 5 1.3.1 Continuous problem . . . . . . . . . . . . . . . . . . . 5 1.3.2 Discrete problem . . . . . . . . . . . . . . . . . . . . . 10 1.4 Plastic design . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Stress constraints . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Fully-stressed design . . . . . . . . . . . . . . . . . . . . . . . 17 1.6.1 Stress-ratio approach . . . . . . . . . . . . . . . . . . . 17 1.6.2 Single loading condition . . . . . . . . . . . . . . . . . 20 1.6.3 Multiple loading conditions . . . . . . . . . . . . . . . 23 1.7 Optimality criteria approach . . . . . . . . . . . . . . . . . . 25 1.8 Compliance constraint . . . . . . . . . . . . . . . . . . . . . . 29 1.8.1 Problem formulation and sensitivity analysis . . . . . 29 1.8.2 Optimality conditions . . . . . . . . . . . . . . . . . . 31 1.8.3 Reformulation of the optimization problem . . . . . . 34 1.8.4 Convexity of compliance . . . . . . . . . . . . . . . . . 39 1.8.5 Other topics on compliance optimization. . . . . . . . 42 1.9 Frequency constraints . . . . . . . . . . . . . . . . . . . . . . 43 1.10 Configuration optimization of trusses . . . . . . . . . . . . . 48 1.11 Multiobjective structural optimization . . . . . . . . . . . . . 50 1.11.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . 50 1.11.2 Problem formulation . . . . . . . . . . . . . . . . . . . 51 1.12 Heuristic approach . . . . . . . . . . . . . . . . . . . . . . . . 52 1.13 Simultaneous analysis and design . . . . . . . . . . . . . . . 55 2 Design Sensitivity Analysis 59 2.1 Overview of design sensitivity analysis . . . . . . . . . . . . . 59 2.2 Static responses . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.2.1 Direct differentiation method . . . . . . . . . . . . . . 62 2.2.2 Adjoint variable method . . . . . . . . . . . . . . . . . 66 2.3 Eigenvalues of free vibration . . . . . . . . . . . . . . . . . . 69 2.3.1 Simple eigenvalue . . . . . . . . . . . . . . . . . . . . . 69 2.3.2 Multiple eigenvalues . . . . . . . . . . . . . . . . . . . 73 ix