Computational Statics and Dynamics: An Introduction Based on the Finite Element Method

Description

This book is the 3rd edition of an introduction to modern computational mechanics based on the finite element method. This third edition is largely extended, adding many new examples to let the reader understand the principles better by performing calculations by hand, as well as numerical example to practice the finite element approach to engineering problems. The new edition comes together with a set of digital flash cards with questions and answers that improve learning success. Featuring over 100 more pages, the new edition will help students succeed in mechanics courses by showing them how to apply the fundamental knowledge they gained in the first years of their engineering education to more advanced topics. In order to deepen readers’ understanding of the equations and theories discussed, each chapter also includes supplementary problems. These problems start with fundamental knowledge questions on the theory presented in the respective chapter, followed by calculation problems. In total, over 80 such calculation problems are provided, along with brief solutions for each. Test your knowledge with questions and answers about the book in the Springer Nature Flashcards app. Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Acknowledgements Contents Symbols and Abbreviations Latin Symbols (Capital Letters) Latin Symbols (Small Letters) Greek Symbols (Capital Letters) Greek Symbols (Small Letters) Mathematical Symbols Indices, Superscripted Indices, Subscripted Abbreviations Some Standard Abbreviations 1 Introduction to the Finite Element Method References 2 Rods and Trusses 2.1 Introduction 2.2 Derivation of the Governing Differential Equation 2.2.1 Kinematics 2.2.2 Constitutive Equation 2.2.3 Equilibrium 2.2.4 Differential Equation 2.3 Finite Element Solution 2.3.1 Derivation of the Principal Finite Element Equation 2.3.2 Derivation of Interpolation Functions 2.3.3 Assembly of Elements and Consideration of Boundary Conditions 2.3.4 Post-computation: Determination of Strain, Stress and Further Quantities 2.3.5 Analogies to Other Field Problems 2.3.6 Solved Rod Problems 2.4 Assembly of Elements to Plane Truss Structures 2.4.1 Rotational Transformation in a Plane 2.4.2 Solved Truss Problems 2.5 Supplementary Problems References 3 Euler-Bernoulli Beams and Frames 3.1 Introduction 3.2 Derivation of the Governing Differential Equation 3.2.1 Kinematics 3.2.2 Constitutive Equation 3.2.3 Equilibrium 3.2.4 Differential Equation 3.3 Finite Element Solution 3.3.1 Derivation of the Principal Finite Element Equation 3.3.2 Derivation of Interpolation Functions 3.3.3 Assembly of Elements and Consideration of Boundary Conditions 3.3.4 Post-computation: Determination of Strain, Stress and Further Quantities 3.3.5 Solved Beam Problems 3.4 Assembly of Elements to Plane Frame Structures 3.4.1 Rotation of a Beam Element 3.4.2 Generalized Beam Element 3.4.3 Solved Problems 3.5 Supplementary Problems References 4 Timoshenko Beams 4.1 Introduction 4.2 Derivation of the Governing Differential Equation 4.2.1 Kinematics 4.2.2 Equilibrium 4.2.3 Constitutive Equation 4.2.4 Differential Equation 4.3 Finite Element Solution 4.3.1 Derivation of the Principal Finite Element Equation 4.3.2 Linear Interpolation Functions for the Displacement and Rotational Field 4.3.3 Higher-Order Interpolation Functions for the Beam with Shear Contribution 4.3.4 Solved Problems 4.4 Supplementary Problems References 5 Plane Elements 5.1 Introduction 5.2 Derivation of the Governing Differential Equation 5.2.1 Kinematics 5.2.2 Constitutive Equation 5.2.3 Equilibrium 5.2.4 Differential Equation 5.3 Finite Element Solution 5.3.1 Derivation of the Principal Finite Element Equation 5.3.2 Four-Node Planar Element 5.3.3 Solved Plane Elasticity Problems 5.4 Supplementary Problems References 6 Classical Plate Elements 6.1 Introduction 6.2 Derivation of the Governing Differential Equation 6.2.1 Kinematics 6.2.2 Constitutive Equation 6.2.3 Equilibrium 6.2.4 Differential Equation 6.3 Finite Element Solution 6.3.1 Derivation of the Principal Finite Element Equation 6.3.2 Rectangular Four-Node Plate Element 6.3.3 Distorted Four-Node Plate Element 6.3.4 Solved Classical Plate Element Problems 6.4 Supplementary Problems References 7 Shear Deformable Plate Elements 7.1 Introduction 7.2 Derivation of the Governing Differential Equation 7.2.1 Kinematics 7.2.2 Constitutive Equation 7.2.3 Equilibrium 7.2.4 Differential Equation 7.3 Finite Element Solution 7.3.1 Derivation of the Principal Finite Element Equation 7.3.2 Rectangular Four-Node Plate Element 7.3.3 Solved Thick Plate Element Problems 7.4 Supplementary Problems References 8 Three-Dimensional Elements 8.1 Derivation of the Governing Differential Equation 8.1.1 Kinematics 8.1.2 Constitutive Equation 8.1.3 Equilibrium 8.1.4 Differential Equation 8.2 Finite Element Solution 8.2.1 Derivation of the Principal Finite Element Equation 8.2.2 Hexahedron Solid Elements 8.2.3 Solved Three-Dimensional Element Problems 8.3 Supplementary Problems References 9 Principles of Linear Dynamics 9.1 Newton's Laws of Motion 9.2 Relationship Between Displacement, Velocity and Acceleration 9.3 Solved Problems 9.4 Supplementary Problems References 10 Integration Methods for Transient Problems 10.1 Introduction 10.2 Derivation of the Governing Differential Equation 10.2.1 Kinematics 10.2.2 Constitutive Equation 10.2.3 Equilibrium 10.2.4 Differential Equation 10.3 Finite Element Solution 10.3.1 Derivation of the Principal Finite Element Equation 10.3.2 Consideration of Damping 10.3.3 Transient Solution Schemes 10.3.4 Solved Problems 10.4 Supplementary Problems References Appendix A Mathematics A.1 Greek Alphabet A.2 Frequently Used Constants A.3 Special Products A.4 Trigonometric Functions A.5 Derivatives A.6 Integrals A.7 Integration by Parts A.8 Integration and Coordinate Transformation A.9 Numerical Integration A.9.1 Simpson's Rule A.9.2 Gauss-Legendre Quadrature A.10 Taylor's Series Expansion A.11 Matrix Operations A.11.1 Matrix Multiplication A.11.2 Scalar Product A.11.3 Dyadic Product A.11.4 Inverse of Matrices A.12 Solution of Linear Systems of Equations A.12.1 Elimination of Variables A.12.2 Matrix Solution A.13 Elementary Geometry A.14 Analytical Geometry A.14.1 Straight-Line Equations A.14.2 Sign of Second Derivative of a Curve A.14.3 Area of a Polygon Appendix B Mechanics B.1 Centroids B.2 Second Moment of Area B.3 Parallel-Axis Theorem Appendix C Units and Conversion C.1 SI Base Units C.2 Coherent SI Derived Units C.3 Consistent Units C.4 Conversion of Important English Units to The Metric System Appendix D Triangular Elements D.1 Plane Elements D.2 Classical Plate Elements Appendix E Summary of Stiffness Matrices E.1 One-Dimensional Elements E.2 Two-Dimensional Elements E.3 Three-Dimensional Elements Appendix F Extrapolation from Integration Points to Nodes Appendix G Answers to Supplementary Problems G.1 Problems from Chap. 2摥映數爠eflinkchap:RodsandTrusses22 G.2 Problems from Chap. 3摥映數爠eflinkchap:EBspsBeamsspsFrames33 G.3 Problems from Chap. 4摥映數爠eflinkchap:Timoshenkospsbeams44 G.4 Problems from Chap. 5摥映數爠eflinkchapspsPlaneElements55 G.5 Problems from Chap. 6摥映數爠eflinkchap:FEspsclassicalspsplate66 G.6 Problems from Chapter 7摥映數爠eflinkchap:ShearDeformablePlateElements77 G.7 Problems from Chap. 8摥映數爠eflinkchap:ThreespsDimensionalspsElements88 G.8 Problems from Chapter 9摥映數爠eflinkchap:LinspsDyn99 G.9 Problems from Chap. 10摥映數爠eflinkchap:IntegrationspsTrans1010 Index