Structural Mechanics in Lightweight Engineering

Description

This book provides a comprehensive yet concise presentation of the analysis methods of lightweight engineering in the context of the statics of beam structures and is divided into four sections. Starting from very general remarks on the fundamentals of elasticity theory, the first section also addresses plane problems as well as strength criteria of isotropic materials. The second section is devoted to the analytical treatment of the statics of beam structures, addressing beams under bending, shear and torsion. The third section deals with the work and energy methods in lightweight construction, spanning classical methods and modern computational methods such as the finite element method. Finally, the fourth section addresses more advanced beam models, discussing hybrid structures as well as laminated and sandwich beams, in addition to shear field beams and shear deformable beams. This book is intended for students at technical colleges and universities, as well as for engineers in practice and researchers in engineering. Preface Contents 1 Introduction 1.1 Definition and Tasks of Lightweight Construction and Engineering 1.1.1 Introduction 1.1.2 What Is Lightweight Engineering? 1.2 Structural Analysis in Lightweight Engineering 1.3 Structural Optimization in Lightweight Engineering 1.4 Structural Elements in Lightweight Engineering 1.5 About the Functionality of an Aircraft Fuselage 1.5.1 Main Components of an Aircraft Fuselage 1.5.2 Loads and Classification into Structural Elements 1.6 Aim and Structure of This Book 1.7 Notes on Relevant Literature 1.8 A Few Notes on Nomenclature References Part I Fundamentals 2 Theory of Elasticity 2.1 Introduction 2.2 State of Stress 2.2.1 Stress Vector and Stress Tensor 2.2.2 Stress Transformation 2.2.3 Principal Stresses, Invariants, Mohr's Circles 2.2.4 Decomposition of the Stress Tensor 2.2.5 Equilibrium Conditions 2.3 Deformations and Strains 2.3.1 Introduction 2.3.2 Green-Lagrange Strain Tensor 2.3.3 Von-Kármán Strains 2.3.4 Infinitesimal Strain Tensor 2.3.5 Compatibility Conditions 2.3.6 Volume Strain 2.3.7 Decomposition of the Infinitesimal Strain Tensor 2.4 Constitutive Equations 2.4.1 Introduction 2.4.2 Hooke's Generalized Law 2.4.3 Strain Energy 2.4.4 Complementary Strain Energy 2.5 Boundary Value Problems 2.6 Material Symmetries 2.6.1 Full Anisotropy 2.6.2 Monotropy 2.6.3 Orthogonal Anisotropy/Orthotropy 2.6.4 Transverse Isotropy 2.6.5 Isotropy 2.6.6 Engineering Constants 2.6.7 Value Ranges for the Material Parameters 2.6.8 Alternative Representation of Isotropic Materials 2.7 Transformation Rules 2.8 Hygrothermal Problems 2.9 Cylindrical Coordinates References 3 Plane Problems 3.1 Introduction 3.2 Surface Structures 3.2.1 Plane Surface Structures: Disks and Plates 3.2.2 Curved Surface Structures: Shells 3.3 Plane State of Strain 3.4 Plane State of Stress 3.5 Stress Transformation 3.5.1 Introduction 3.5.2 Principal Stresses 3.5.3 Mohr's Circle 3.6 Strain Transformation 3.7 Formulation for Orthotropic Materials 3.7.1 Plane State of Stress 3.7.2 Plane State of Strain 3.8 Formulation for Polar Coordinates Bibliography 4 Strength Criteria for Isotropic Materials 4.1 Introduction 4.2 Principal Stress Hypothesis 4.3 Principal Strain Hypothesis 4.4 Beltrami Strain Energy Hypothesis 4.5 Von Mises Strain Energy Hypothesis 4.6 Tresca Yield Criterion 4.7 Coulomb-Mohr Hypothesis 4.8 Drucker-Prager Hypothesis 4.9 Cuntze's Failure Mode Concept Bibliography Part II Thin-Walled Beam Structures 5 Beams Under Normal Forces and Bending Moments 5.1 Introduction 5.2 Basic Equations for an Arbitrary Reference System 5.3 First Cross-Sectional Normalization: Center of Gravity S 5.4 Second Cross-Sectional Normalization: Principal Axes 5.5 Selected Basic Cases 5.6 Analysis of Arbitrarily Segmented Cross-Sections 5.7 Calculation of Beam Deflections 5.8 Rod Structures Bibliography 6 Beams Under Transverse Shear Forces 6.1 Introduction 6.2 Shear Stresses in Open Cross-Sections 6.2.1 Basic Equations 6.2.2 Simplified Analysis of an I-Cross-Section 6.2.3 Unit Shear Flow 6.3 Shear Stresses in Closed Cross-Sections 6.3.1 Single-Cell Cross-Sections 6.3.2 Multi-Cell Cross-Sections 6.3.3 Mixed Cross-Sections 6.3.4 Use of Symmetry Properties 6.4 Shear Center 6.4.1 Open Cross-Sections 6.4.2 Closed Cross-Sections Bibliography 7 St. Venant Torsion 7.1 Introduction 7.2 Assumptions and Constitutive Law 7.3 Arbitrary Thin-Walled Hollow Cross-Sections 7.4 Open Thin-Walled Cross Sections 7.5 Comparison of Closed and Open Cross-Sections 7.6 Multi-cell Cross Sections 7.7 Assembled Cross-Sections 7.8 Effective Wall Thicknesses 7.8.1 Truss Girders 7.8.2 Stiffened Box Beams 7.9 Determination of Internal Forces Bibliography 8 Warping Torsion 8.1 Introduction 8.2 Warping of Open Cross-Sections 8.3 Warping of Closed Cross-Sections 8.3.1 Single Cell Cross-Sections 8.3.2 Multi-cell Cross-Sections 8.4 Unit Warping with Respect to the Shear Center 8.5 The First-Order Bending-Torsion Problem 8.6 Cross-Sectional Normalizations 8.6.1 First Cross-Sectional Normalization: Center of Gravity S 8.6.2 Second Cross-Sectional Normalization: Principal Axes 8.7 Example 8.8 Selected Basic Cases 8.8.1 Double Symmetrical I-Cross-Section 8.8.2 Single Symmetrical I-Cross-Section 8.8.3 C-Cross-Section 8.8.4 Z-Cross-Section 8.9 Determination of Internal Moments 8.9.1 Differential Equation of Warping Torsion 8.9.2 Selected Basic Cases 8.9.3 Influence of Normal Forces 8.10 Stress Analysis 8.11 Comparison of Open and Closed Cross-Sections Bibliography Part III Energy Methods 9 Work and Energy 9.1 Introduction 9.2 Work and Energy 9.2.1 Fundamentals 9.2.2 Internal and External Work 9.2.3 Principle of Work and Energy 9.3 Strain Energy and Complementary Strain Energy 9.3.1 The Rod 9.3.2 The Euler-Bernoulli Beam 9.3.3 Torsion 9.3.4 Combined Loading 9.3.5 Generalization for the Continuum 9.4 Application of the Principle of Work and Energy to Elastic Deformations 9.5 General Principle of Work and Energy of Elastostatics Bibliography 10 Principle of Virtual Displacements 10.1 Introduction 10.2 Virtual Displacements and Virtual Works 10.3 The Principle of Virtual Displacements 10.3.1 Determination of Forces and Moments in Statically Determinate Systems 10.3.2 Influence Lines for Forces and Moments in Statically Determinate Systems 10.4 Pole Plans and Kinematic Chains 10.5 The Variational Operator δ 10.6 Formulation for a Continuum 10.7 The Rod 10.8 The Euler-Bernoulli Beam 10.9 Beam Under Torsion 10.10 Beam Under Combined Loads Bibliography 11 Principle of Stationary Value of the Total Elastic Potential 11.1 Introduction 11.2 Fundamentals of Calculus of Variations 11.2.1 Functional with First Order Derivatives 11.2.2 Functional with Second Order Derivatives 11.2.3 Functional with n-th Order Derivatives 11.2.4 Functional with n Functions with First Order Derivatives 11.3 Principle of the Stationary Value of the Total Elastic Potential 11.3.1 The Rod 11.3.2 The Euler-Bernoulli Beam 11.3.3 Beam Under Torsion 11.3.4 Beam Under Combined Load 11.4 First Theorem of Castiglianio 11.5 Theorem of Clapeyron Bibliography 12 Principle of Virtual Forces 12.1 Introduction 12.2 Virtual Forces and Complementary Virtual Work 12.3 The Principle of Virtual Forces 12.4 The Unit Load Theorem 12.5 The Principle of the Stationary Value of the Elastic Complementary Potential 12.6 Second Theorem of Castigliano 12.7 Theorem of Menabrea 12.8 The Force Method 12.8.1 Calculation of Deformations of Statically Determinate Systems 12.8.2 Analysis of Simply Statically Indeterminate Systems 12.9 Reciprocity Theorems 12.9.1 Theorem of Betti 12.9.2 Theorem of Maxwell 12.10 Calculation of Multiple Statically Indeterminate Systems 12.11 Influence Lines for Deformations of Statically Determinate Systems 12.12 The Reduction Theorem of Statics 12.13 Analysis of Continuous Beams Bibliography 13 Energy-Based Approximation Methods 13.1 Introduction 13.2 The Ritz Method 13.2.1 The Euler-Bernoulli Beam 13.2.2 The Rod 13.2.3 Torsion 13.3 The Galerkin Method Bibliography 14 The Finite Element Method 14.1 Introduction 14.2 Finite Elements for Plane Trusses 14.2.1 Element Formulation and Calculation Steps 14.2.2 Statically Indeterminate Trusses 14.2.3 Examples 14.3 Finite Elements for Plane Systems of Straight Rods 14.3.1 The Two-Noded Rod Element 14.3.2 The Three-Noded Rod Element 14.4 Finite Elements for Plane Systems of Straight Beams 14.4.1 The Two-Noded Beam Element 14.4.2 Quality of the Solution, Convergence Behavior 14.4.3 The Three-Noded Beam Element 14.4.4 Comparison Ritz / FEM 14.5 Finite Elements for Torsion Bibliography Part IV Advanced Beam Models 15 Shear Field Beams 15.1 Introduction 15.2 Rectangular Skin Fields 15.2.1 Determination of Stiffener Forces and Shear Flows 15.2.2 Determination of Deformations 15.3 Parallelogram Skin Fields 15.4 Trapezoidal Skin Fields 15.5 Statically Indeterminate Shear Field Beams 15.6 Applications of the Shear Field Beam Model 15.6.1 Flexurally Rigid Beam Connections 15.6.2 Large Area Stiffened Structures 15.6.3 Load Introductions 15.6.4 Adhesive Overlap Joints References 16 The Timoshenko Beam 16.1 Introduction 16.2 Kinematics and Constitutive Law 16.3 Displacements and Stresses 16.4 Elementary Examples 16.5 Shear Correction Factor K 16.6 Energetic Consideration 16.7 The Force Method 16.7.1 Determination of Displacements 16.7.2 Statically Indeterminate Systems 16.8 The Ritz Method 16.9 Finite Beam Element References 17 Hybrid Beams 17.1 Introduction 17.2 Beams Under Normal Forces and Bending Moments 17.2.1 Basic Equations for an Arbitrary Reference System 17.2.2 First Cross-Sectional Normalization: Elastic Center of Gravity S 17.2.3 Second Cross-Sectional Normalization: Principal Axes 17.2.4 Selected Basic Cases 17.3 Beams Under Transverse Shear Forces 17.3.1 Shear Flow in Open Cross-Sections 17.3.2 Shear Flow in Closed Cross-Sections 17.3.3 Elastic Shear Center 17.4 Torsion of Thin-Walled Hybrid Beams References 18 Laminated and Sandwich Beams 18.1 Classical Laminate Theory 18.1.1 Constitutive Law and Equilibrium Conditions 18.1.2 Calculation of Stresses 18.1.3 Deformations 18.1.4 Energetic Consideration 18.1.5 The Ritz Method 18.2 First-Order Shear Deformation Theory 18.2.1 Constitutive Law and Equilibrium Conditions 18.2.2 Calculation of Stresses 18.2.3 Deformations 18.2.4 Energetic Consideration 18.3 Sandwich Beams 18.3.1 Introduction 18.3.2 Constitutive Law 18.3.3 Calculation of Stresses References Index