Alan Morris has recently retired from his post at Cranfield University in the UK as Professor of Structural Analysis. His range of expertise covers the development and application of optimal design and analysis systems, with special emphasis on the finite element methods and structural optimisation programs. Previously, he was a Principal Research Officer in the Royal Aircraft Establishment (now QinetiQ) at Farnborough and was head of the Structures and Optimization Sections at the Advanced Concepts Section.

"The book is highly recommended for all engineering students, researchers and practicing engineers involved in developing/ utilising finite element techniques. I have confidence that even an experienced analyst would benefit from a review of this material!" (The Aeronautical Journal, March 2010)

• Preface xiii1 Introduction 11.1 Aim of the book 11.2 Finite element types – a brief overview 31.3 Finite element analysis and finite element representations 61.4 Multi-model analyses 81.5 Consistency, logic and error control 81.6 Chapter contents 111.6.1 Chapter 2 Overview of static finite element analysis 111.6.2 Chapter 3 Overview of dynamic analysis 121.6.3 Chapter 4 What’s energy got to do with it? 131.6.4 Chapter 5 Preliminary review of errors and error control 141.6.5 Chapter 6 Discretisation: elements and meshes or some ways to avoid generated error 141.6.6 Chapter 7 Idealisation error types and sources 151.6.7 Chapter 8 Error control 151.6.8 Chapter 9 Error-controlled analyses 161.6.9 Chapter 10 FEMEC walkthrough example 17References 172 Overview of Static Finite Element Analysis 192.1 Introduction 192.2 The direct method for static analyses 202.2.1 Element matrices 202.2.2 Assembled global stiffness matrix for static analyses 212.2.3 Global coordinates 262.2.4 Some typical elements 292.2.5 The analysis loop 292.3 Reducing the problem size 332.3.1 Symmetry 332.3.2 Condensation and superelements 402.3.3 Sub-structures 46References 493 Overview of Dynamic Analysis 513.1 Introduction 513.2 Element mass matrix 513.2.1 Free undamped vibrations 533.3 Additional information that can be extracted to support a dynamic finite element analysis 593.3.1 Sturm property check 593.3.2 Rayleigh Quotient 613.4 Forced responses 623.4.1 Modal analysis 623.4.2 Direct integration 653.5 Damped forced responses 663.5.1 Modal analysis with damping 663.5.2 Modal damping ratio 683.5.3 Direct integration 693.6 Reducing the problem size 703.6.1 Symmetry 713.6.2 Reducing the number of variables 723.6.3 Sub-structure analysis (component mode synthesis) 76References 784 What’s Energy Got to Do with It? 794.1 Introduction 794.2 Strain energy 814.3 Potential energy 824.4 Simple bar 834.5 General case 834.6 Minimum potential energy 844.7 The principle of minimum potential energy applied to a simple finite element problem 864.8 Finite element formulation 884.9 Direct application to an axial bar element 904.10 Convergence in energy and convergence in stress 924.10.1 Single bar element model 944.10.2 2-bar element model 964.10.3 4-bar element model 984.10.4 8-bar element model 1014.10.5 16-bar element model 1024.11 Results interpretation 1034.11.1 Potential energy convergence 1034.11.2 Stress improvement 1044.11.3 Displacement convergence 1064.12 Kinetic energy 1064.13 Final remarks 111References 1115 Preliminary Review of Errors and Error Control 1135.1 Introduction 1135.2 The finite element process 1145.3 Error and uncertainty 1175.4 Novelty, complexity and experience 1205.4.1 Analysis novelty 1205.4.2 Degree of complexity 1235.4.3 Experience 1255.5 Role of testing 1285.6 Initial steps 1295.6.1 Qualification process 1295.6.2 Acceptable magnitude of error or uncertainty 1325.7 Analysis Validation Plan (AVP) 1355.8 Applied common sense 1385.9 The process 140References 1416 Discretisation: Elements and Meshes or Some Ways to Avoid Generated Error 1436.1 Introduction 1436.2 Element delivery 1446.2.1 Two-dimensional elements 1456.2.2 Three-dimensional elements 1486.2.3 Why do this? 1516.2.4 Optimal stress points and making the most of them 1516.3 Mesh grading and mesh distortion 1556.3.1 Mesh grading 1556.3.2 Element distortions 1586.3.3 Group 3 distortions (quadratic distortions) 1616.3.4 Group 4 distortions (cubic and higher order distortions) 1626.3.5 Distortions of other element types 1626.3.6 General principles with respect to element distortions 1636.4 The accuracy ladder 1646.4.1 The stress ladder 1646.4.2 The mesh ladder 1656.4.3 Automatically moving up the accuracy ladder 165References 1747 Idealisation Error Types and Sources 1757.1 Design reduction and idealisation errors 1757.2 Analysis features 1787.3 The domain 1807.3.1 The domain of analysis 1807.3.2 Domain reduction 1837.4 Levels of abstraction 1847.5 Boundary conditions 1927.6 Material properties 1987.7 Loads and masses 2007.7.1 Loads 2007.7.2 Masses 201References 2018 Error Control 2038.1 Introduction 2038.2 Approach and techniques 2048.2.1 Approach 2048.2.2 Techniques 2078.3 Accumulation of errors and uncertainties 2418.4 The role of testing 2428.4.1 Analyst-requested tests 2448.4.2 Validation test 245References 2479 Error-Controlled Analyses 2499.1 Introduction 2499.2 Is the finite element system fit for purpose? 2519.2.1 What does it do? 2519.2.2 How do we know it does it? 2529.2.3 Is size important? 2549.3 Quality Report 2569.4 The error and uncertainty control method 2599.4.1 Introduction 2599.4.2 FEMEC 2609.4.3 FEMEC implementation process 26110 FEMEC Walkthrough Example 30310.1 Introduction 30310.2 FEMEC static analysis illustrative problem 30410.2.1 The design requirement 30410.2.2 Application of FEMEC via the Quality Report 30610.3 A brief look at the dynamic FEMEC world 34510.3.1 Modified design and qualification parameters 34510.3.2 FEMEC approach 346References 351Index 353

Alan Morris has recently retired from his post at Cranfield University in the UK as Professor of Structural Analysis. His range of expertise covers the development and application of optimal design and analysis systems, with special emphasis on the finite element methods and structural optimisation programs. Previously, he was a Principal Research Officer in the Royal Aircraft Establishment (now QinetiQ) at Farnborough and was head of the Structures and Optimization Sections at the Advanced Concepts Section.

"The book is highly recommended for all engineering students, researchers and practicing engineers involved in developing/ utilising finite element techniques. I have confidence that even an experienced analyst would benefit from a review of this material!" (The Aeronautical Journal, March 2010)

• Preface xiii1 Introduction 11.1 Aim of the book 11.2 Finite element types – a brief overview 31.3 Finite element analysis and finite element representations 61.4 Multi-model analyses 81.5 Consistency, logic and error control 81.6 Chapter contents 111.6.1 Chapter 2 Overview of static finite element analysis 111.6.2 Chapter 3 Overview of dynamic analysis 121.6.3 Chapter 4 What’s energy got to do with it? 131.6.4 Chapter 5 Preliminary review of errors and error control 141.6.5 Chapter 6 Discretisation: elements and meshes or some ways to avoid generated error 141.6.6 Chapter 7 Idealisation error types and sources 151.6.7 Chapter 8 Error control 151.6.8 Chapter 9 Error-controlled analyses 161.6.9 Chapter 10 FEMEC walkthrough example 17References 172 Overview of Static Finite Element Analysis 192.1 Introduction 192.2 The direct method for static analyses 202.2.1 Element matrices 202.2.2 Assembled global stiffness matrix for static analyses 212.2.3 Global coordinates 262.2.4 Some typical elements 292.2.5 The analysis loop 292.3 Reducing the problem size 332.3.1 Symmetry 332.3.2 Condensation and superelements 402.3.3 Sub-structures 46References 493 Overview of Dynamic Analysis 513.1 Introduction 513.2 Element mass matrix 513.2.1 Free undamped vibrations 533.3 Additional information that can be extracted to support a dynamic finite element analysis 593.3.1 Sturm property check 593.3.2 Rayleigh Quotient 613.4 Forced responses 623.4.1 Modal analysis 623.4.2 Direct integration 653.5 Damped forced responses 663.5.1 Modal analysis with damping 663.5.2 Modal damping ratio 683.5.3 Direct integration 693.6 Reducing the problem size 703.6.1 Symmetry 713.6.2 Reducing the number of variables 723.6.3 Sub-structure analysis (component mode synthesis) 76References 784 What’s Energy Got to Do with It? 794.1 Introduction 794.2 Strain energy 814.3 Potential energy 824.4 Simple bar 834.5 General case 834.6 Minimum potential energy 844.7 The principle of minimum potential energy applied to a simple finite element problem 864.8 Finite element formulation 884.9 Direct application to an axial bar element 904.10 Convergence in energy and convergence in stress 924.10.1 Single bar element model 944.10.2 2-bar element model 964.10.3 4-bar element model 984.10.4 8-bar element model 1014.10.5 16-bar element model 1024.11 Results interpretation 1034.11.1 Potential energy convergence 1034.11.2 Stress improvement 1044.11.3 Displacement convergence 1064.12 Kinetic energy 1064.13 Final remarks 111References 1115 Preliminary Review of Errors and Error Control 1135.1 Introduction 1135.2 The finite element process 1145.3 Error and uncertainty 1175.4 Novelty, complexity and experience 1205.4.1 Analysis novelty 1205.4.2 Degree of complexity 1235.4.3 Experience 1255.5 Role of testing 1285.6 Initial steps 1295.6.1 Qualification process 1295.6.2 Acceptable magnitude of error or uncertainty 1325.7 Analysis Validation Plan (AVP) 1355.8 Applied common sense 1385.9 The process 140References 1416 Discretisation: Elements and Meshes or Some Ways to Avoid Generated Error 1436.1 Introduction 1436.2 Element delivery 1446.2.1 Two-dimensional elements 1456.2.2 Three-dimensional elements 1486.2.3 Why do this? 1516.2.4 Optimal stress points and making the most of them 1516.3 Mesh grading and mesh distortion 1556.3.1 Mesh grading 1556.3.2 Element distortions 1586.3.3 Group 3 distortions (quadratic distortions) 1616.3.4 Group 4 distortions (cubic and higher order distortions) 1626.3.5 Distortions of other element types 1626.3.6 General principles with respect to element distortions 1636.4 The accuracy ladder 1646.4.1 The stress ladder 1646.4.2 The mesh ladder 1656.4.3 Automatically moving up the accuracy ladder 165References 1747 Idealisation Error Types and Sources 1757.1 Design reduction and idealisation errors 1757.2 Analysis features 1787.3 The domain 1807.3.1 The domain of analysis 1807.3.2 Domain reduction 1837.4 Levels of abstraction 1847.5 Boundary conditions 1927.6 Material properties 1987.7 Loads and masses 2007.7.1 Loads 2007.7.2 Masses 201References 2018 Error Control 2038.1 Introduction 2038.2 Approach and techniques 2048.2.1 Approach 2048.2.2 Techniques 2078.3 Accumulation of errors and uncertainties 2418.4 The role of testing 2428.4.1 Analyst-requested tests 2448.4.2 Validation test 245References 2479 Error-Controlled Analyses 2499.1 Introduction 2499.2 Is the finite element system fit for purpose? 2519.2.1 What does it do? 2519.2.2 How do we know it does it? 2529.2.3 Is size important? 2549.3 Quality Report 2569.4 The error and uncertainty control method 2599.4.1 Introduction 2599.4.2 FEMEC 2609.4.3 FEMEC implementation process 26110 FEMEC Walkthrough Example 30310.1 Introduction 30310.2 FEMEC static analysis illustrative problem 30410.2.1 The design requirement 30410.2.2 Application of FEMEC via the Quality Report 30610.3 A brief look at the dynamic FEMEC world 34510.3.1 Modified design and qualification parameters 34510.3.2 FEMEC approach 346References 351Index 353

Alan Morris has recently retired from his post at Cranfield University in the UK as Professor of Structural Analysis. His range of expertise covers the development and application of optimal design and analysis systems, with special emphasis on the finite element methods and structural optimisation programs. Previously, he was a Principal Research Officer in the Royal Aircraft Establishment (now QinetiQ) at Farnborough and was head of the Structures and Optimization Sections at the Advanced Concepts Section.

"The book is highly recommended for all engineering students, researchers and practicing engineers involved in developing/ utilising finite element techniques. I have confidence that even an experienced analyst would benefit from a review of this material!" (The Aeronautical Journal, March 2010)

• Preface xiii1 Introduction 11.1 Aim of the book 11.2 Finite element types – a brief overview 31.3 Finite element analysis and finite element representations 61.4 Multi-model analyses 81.5 Consistency, logic and error control 81.6 Chapter contents 111.6.1 Chapter 2 Overview of static finite element analysis 111.6.2 Chapter 3 Overview of dynamic analysis 121.6.3 Chapter 4 What’s energy got to do with it? 131.6.4 Chapter 5 Preliminary review of errors and error control 141.6.5 Chapter 6 Discretisation: elements and meshes or some ways to avoid generated error 141.6.6 Chapter 7 Idealisation error types and sources 151.6.7 Chapter 8 Error control 151.6.8 Chapter 9 Error-controlled analyses 161.6.9 Chapter 10 FEMEC walkthrough example 17References 172 Overview of Static Finite Element Analysis 192.1 Introduction 192.2 The direct method for static analyses 202.2.1 Element matrices 202.2.2 Assembled global stiffness matrix for static analyses 212.2.3 Global coordinates 262.2.4 Some typical elements 292.2.5 The analysis loop 292.3 Reducing the problem size 332.3.1 Symmetry 332.3.2 Condensation and superelements 402.3.3 Sub-structures 46References 493 Overview of Dynamic Analysis 513.1 Introduction 513.2 Element mass matrix 513.2.1 Free undamped vibrations 533.3 Additional information that can be extracted to support a dynamic finite element analysis 593.3.1 Sturm property check 593.3.2 Rayleigh Quotient 613.4 Forced responses 623.4.1 Modal analysis 623.4.2 Direct integration 653.5 Damped forced responses 663.5.1 Modal analysis with damping 663.5.2 Modal damping ratio 683.5.3 Direct integration 693.6 Reducing the problem size 703.6.1 Symmetry 713.6.2 Reducing the number of variables 723.6.3 Sub-structure analysis (component mode synthesis) 76References 784 What’s Energy Got to Do with It? 794.1 Introduction 794.2 Strain energy 814.3 Potential energy 824.4 Simple bar 834.5 General case 834.6 Minimum potential energy 844.7 The principle of minimum potential energy applied to a simple finite element problem 864.8 Finite element formulation 884.9 Direct application to an axial bar element 904.10 Convergence in energy and convergence in stress 924.10.1 Single bar element model 944.10.2 2-bar element model 964.10.3 4-bar element model 984.10.4 8-bar element model 1014.10.5 16-bar element model 1024.11 Results interpretation 1034.11.1 Potential energy convergence 1034.11.2 Stress improvement 1044.11.3 Displacement convergence 1064.12 Kinetic energy 1064.13 Final remarks 111References 1115 Preliminary Review of Errors and Error Control 1135.1 Introduction 1135.2 The finite element process 1145.3 Error and uncertainty 1175.4 Novelty, complexity and experience 1205.4.1 Analysis novelty 1205.4.2 Degree of complexity 1235.4.3 Experience 1255.5 Role of testing 1285.6 Initial steps 1295.6.1 Qualification process 1295.6.2 Acceptable magnitude of error or uncertainty 1325.7 Analysis Validation Plan (AVP) 1355.8 Applied common sense 1385.9 The process 140References 1416 Discretisation: Elements and Meshes or Some Ways to Avoid Generated Error 1436.1 Introduction 1436.2 Element delivery 1446.2.1 Two-dimensional elements 1456.2.2 Three-dimensional elements 1486.2.3 Why do this? 1516.2.4 Optimal stress points and making the most of them 1516.3 Mesh grading and mesh distortion 1556.3.1 Mesh grading 1556.3.2 Element distortions 1586.3.3 Group 3 distortions (quadratic distortions) 1616.3.4 Group 4 distortions (cubic and higher order distortions) 1626.3.5 Distortions of other element types 1626.3.6 General principles with respect to element distortions 1636.4 The accuracy ladder 1646.4.1 The stress ladder 1646.4.2 The mesh ladder 1656.4.3 Automatically moving up the accuracy ladder 165References 1747 Idealisation Error Types and Sources 1757.1 Design reduction and idealisation errors 1757.2 Analysis features 1787.3 The domain 1807.3.1 The domain of analysis 1807.3.2 Domain reduction 1837.4 Levels of abstraction 1847.5 Boundary conditions 1927.6 Material properties 1987.7 Loads and masses 2007.7.1 Loads 2007.7.2 Masses 201References 2018 Error Control 2038.1 Introduction 2038.2 Approach and techniques 2048.2.1 Approach 2048.2.2 Techniques 2078.3 Accumulation of errors and uncertainties 2418.4 The role of testing 2428.4.1 Analyst-requested tests 2448.4.2 Validation test 245References 2479 Error-Controlled Analyses 2499.1 Introduction 2499.2 Is the finite element system fit for purpose? 2519.2.1 What does it do? 2519.2.2 How do we know it does it? 2529.2.3 Is size important? 2549.3 Quality Report 2569.4 The error and uncertainty control method 2599.4.1 Introduction 2599.4.2 FEMEC 2609.4.3 FEMEC implementation process 26110 FEMEC Walkthrough Example 30310.1 Introduction 30310.2 FEMEC static analysis illustrative problem 30410.2.1 The design requirement 30410.2.2 Application of FEMEC via the Quality Report 30610.3 A brief look at the dynamic FEMEC world 34510.3.1 Modified design and qualification parameters 34510.3.2 FEMEC approach 346References 351Index 353