Fundamentals of Structural Optimization: Stability and Contact Mechanics (Mathematical Engineering)

✍ Scribed by Vladimir Kobelev

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 368
• Category: Library

✦ Synopsis

This book serves as a complementary resource to the courses "Advanced structural optimization" and "Structural optimization in automotive engineering" taught by the author at the University of Siegen, North-Rhine-Westphalia, Germany since 2001. Focusing on optimization problems in the field of structural engineering, this book offers a rigorous and analytical approach to problem-solving.

Each chapter of the book begins with a brief overview of classical results and the derivation of governing equations. The solutions to optimization problems are then presented in a closed form, with the author guiding readers through several analytical methods for solving stability and contact tasks. Throughout the book, the author takes care to ensure that even readers without extensive experience in numerical computations can understand the conclusion of each relation.

The book features several basic optimization problems, selected from a large pool of previously solvedproblems, with a particular emphasis on the unique features of optimization problems. By presenting analytical solutions, readers can better understand other known optimization problems and gain the skills needed to independently set and solve new problems. With its comprehensive and rigorous approach to problem-solving, this book is sure to enhance the reader's understanding of the field and equip them with the skills needed to tackle new challenges.

✦ Table of Contents

Foreword
Preface
About This Book
Concept and Main Ideas of the Manuscript
Structure of the Book
Part I: Optimization in Contact Problems
Part II: Optimization in Stability Problems
Target Audience of the Book
References
Contents
Part I Optimization in Contact Problems
1 Optimization and Inverse Solutions for Plane Contacts
1.1 Plane Elasticity Problems
1.1.1 Plane Stress Problem
1.1.2 Plane Strain Problem
1.1.3 Airy Stress Function
1.1.4 Equations in Polar Coordinates
1.1.5 Boundary Forces on Half-Space
1.2 Direct and Inverse Plane Contact Problems
1.2.1 Equilibrium Equations
1.2.2 Solutions with Chebyshev Polynomials, or Base Functions of First Type
1.2.3 Solutions with Base Functions of Second Type
1.2.4 Solutions with Base Functions of Third Type
1.2.5 Integral Formulations of Inverse Problem with Given Normal Stress and Friction Free Contact
1.2.6 Optimal Shape of Rigid Punch Penetrating into Elastic Layer
1.3 Optimal Shapes of Periodically Arranged Indenters
1.3.1 Normal Displacement Under the Action of Periodical Forces
1.3.2 Optimal Shapes of Periodically Spaced Penetrators
1.4 Conclusions
1.5 Summary of Principal Results
1.6 List of Symbols
References
2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks
2.1 Axisymmetric Elastostatics
2.1.1 Boussinesq-Papkovich Solution
2.1.2 Integral Equation of Axisymmetric Contact Problem
2.1.3 Green and Collins Solution
2.2 Series Solutions of Contact Equation
2.2.1 Form Factor and Shape Function
2.2.2 Direct Integration
2.2.3 Reaction Force as Function of Form Factor
2.2.4 Elastic Energy as Function of Form Factor
2.2.5 Dependence of Reaction Force and Elastic Energy Upon the Indenter Radius
2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular Contact Region
2.3.1 Solutions with Form Factor p = 0
2.3.2 Solutions with Form Factor p = 1/2
2.3.3 Solutions with Form Factor p=1
2.3.4 Optimization of Total Force and Contact Stress
2.3.5 Optimization of Stiffness of Contact Region
2.4 Optimization of Ring-Shape Indenters
2.4.1 Stored Elastic Energy, Spring Rate and Contact Force of Concentric Ring-Shaped Indenters
2.4.2 Multiple Concentric Indenters
2.4.3 Optimization of Ring-Shaped Indenters
2.5 Electromagnetic Potentials of Disk with Radially-Variable Charge or Current
2.6 Conclusions
2.7 Summary of Principal Results
2.8 List of Symbols
Appendix
References
3 Optimization of Needle-Shaped Stiffeners
3.1 Load Diffusion and Load Transfer
3.2 Stringer with Variable Cross-Section
3.3 Compliance and Deformation Energy of Stiffened Elastic Body
3.4 Minimization of the Maximum Stress
3.5 Mass Optimization of a Stiffener
3.5.1 Method of Lagrange Multipliers
3.5.2 Alternative Solution
3.6 Conclusions
3.7 Summary of Principal Results
3.8 List of Symbols
References
4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners
4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners
4.1.1 Equilibrium Equations for Periodic Array of Inclusions or Stiffeners
4.1.2 Lagrange Multipliers Method for Optimality Conditions
4.1.3 Optimal Forms of Periodic Rows of Coaxial Stiffeners
4.1.4 Optimal Forms of Periodically Located, Parallel Stiffeners
4.1.5 Character of Boundary Value Problems for Periodically Located Optimal Stiffeners
4.2 Optimization of Double-Periodic Array of Inclusions or Stiffeners
4.2.1 Necessary Optimality Conditions for Chess-Board Lattices of Elastic Stiffeners
4.2.2 Rectangular and Upright Square Lattice
4.2.3 Optimization Problem for Double Periodic Arrays of Inclusions
4.2.4 Shapes of Double-Periodic Arrays of Inclusions
4.3 Conclusions
4.4 Summary of Principal Results
4.5 List of Symbols
References
Part II Optimization in Stability Problems
5 Optimization of Compressed Rods with Sturm Boundary Conditions
5.1 Stability of Axially Compressed Rod
5.2 Boundary Conditions
5.3 Efficiency Approach for Optimization in Stability Problems
5.4 Optimization Problem of Sturm Type
5.5 Auxiliary Solution of Generalized Emden–Fowler Equation
5.6 Closed-Form Solution of Optimization Problem
5.7 Isoperimetric Inequalities
5.8 Conclusions
5.9 Summary of Principal Results
5.10 List of Symbols
References
6 Optimization of Axially Compressed Rods with Mixed Boundary Conditions
6.1 Optimization of Compressed Rods with Mixed Type Boundary Conditions
6.2 Optimality Conditions for Mixed Type Boundary Conditions
6.3 Isoperimetric Inequality for Mixed Type Boundary Conditions
6.4 Equations of Optimization Problem with Mixed Type Boundary Conditions
6.5 Shape of Optimal Column
6.6 Length, Volume and Total Stiffness of Optimal Column
6.7 Fundamental Functions for Buckling Moments
6.8 Fundamental Functions for Buckling Displacements
6.9 Asymptotic Solutions
6.10 Isoperimetric Inequalities
6.11 Conclusions
6.12 Summary of Principal Results
6.13 List of Symbols
References
7 Stability Optimization of Twisted Rods
7.1 Isoperimetric Inequality for Twisted Rod with Arbitrary Convex, Simply-Connected Cross-Section
7.2 Optimization Problem and Isoperimetric Inequality for Stability
7.3 Closed-Form Solution of Optimization Problem
7.4 Effectiveness of Optimal Designs
7.5 Conclusions
7.6 Summary of Principal Results
7.7 List of Symbols
References
8 Periodic Greenhill’s Problem for Twisted Elastic Rod
8.1 Periodic Greenhill’s Problem
8.2 Periodic Conditions
8.3 Stability of Twisted, Periodically Supported Rod with Varying Stiffness
8.4 Optimization Problem for Periodically Supported Twisted Rod
8.5 Isoperimetric Inequality for Periodically Supported Twisted Rod
8.6 Conclusions
8.7 Summary of Principal Results
8.8 List of Symbols
References
9 Optimization of Concurrently Compressed and Torqued Rod
9.1 Twisted and Axially Compressed Shafts with Convex Simply-Connected Cross-Sections
9.2 Optimization Problem and Isoperimetric Inequality for Stability
9.3 Closed-Form Solution of Optimization Task
9.4 Special Cases
9.4.1 Optimal Rod for Greenhill Torsion
9.4.2 Optimal Strut for Euler Compression
9.5 Arbitrary Relation Compression to Torque
9.5.1 Optimal Rod for Shape Exponent α = 1
9.5.2 Optimal Rod for Shape Exponent α = 2
9.5.3 Optimal Rod for Shape Exponent α = 3
9.6 Mass Comparisons of Optimal Shafts to Constant-Cross-Section Shafts
9.7 Conclusions
9.8 Summary of Principal Results
9.9 List of Symbols
References
10 Optimization for Buckling of Conservative Systems of Second Kind
10.1 Pfluger Column
10.2 Stability Optimization for “Pfluger Column”
10.3 Auxiliary Conservative System of First Kind: “Generalized Euler Column”
10.4 Isoperimetric Inequality
10.5 Optimal Shapes of “Generalized Pfluger Columns”
10.6 Conclusions
10.7 Summary of Principal Results
10.8 List of Symbols
References
11 Structural Optimization for Stability of Circular Rings
11.1 Stability of Circular Rings and Arches
11.2 Optimization for Stability of Circular Rings and Arches
11.3 Basic Equations and Formulation of Optimization
11.4 Transforming of Variational Formulation and General Properties of Boundary Value Problem
11.5 The Proof of Isoperimetric Inequality
11.6 Conclusions
11.7 Summary of Principal Results
11.8 List of Symbols
References
12 Stability Optimization of Axially Compressed Rods on Elastic Foundations
12.1 Stability for Axially Compressed Rods on Elastic Foundation
12.2 Deformation of an Infinite Elastic Layer
12.2.1 Elastic Layer Under the Surface Load
12.2.2 Elastic Layer of Intermediate Thickness
12.2.3 Limit Case of Half-Infinite Elastic Medium
12.2.4 Limit Case of Thin Elastic Layer
12.3 Stability of Infinitely Long, Homogeneous Struts
12.3.1 Stability of Homogeneous Infinite Strut on Winkler Foundation
12.3.2 Stability of Homogeneous Strut on Semi-infinite Elastic Foundation
12.3.3 Stability of Homogeneous Strut on Elastic Layer
12.4 Optimal Strut on Elastic Foundations
12.4.1 Formulation of Optimization Problem
12.4.2 Optimality Conditions
12.4.3 Optimal Strut on Winkler Foundation
12.4.4 Optimal Strut on Reissner Foundation
12.4.5 Optimal Strut on Half-Infinite Elastic Space
12.5 Optimal Strut on an Elastic Layer
12.5.1 Optimization of Compressed Strut on Elastic Layer
12.6 Appendix. Direct Calculation of Hilbert Integrals
12.7 Conclusions
12.8 Summary of Principal Results
12.9 List of Symbols
References
Index