Structural Mechanics in Lightweight 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