Finite Element Analysis of Structures Through Unified Formulation -- Contents -- About the Authors -- Preface -- Nomenclature and Acronyms -- Symbols -- Acronyms -- 1 Introduction -- 1.1 What is in this Book -- 1.2 The Finite Element Method -- 1.2.1 Approximation of the Domain -- 1.2.2 The Numerical Approximation -- 1.3 Calculation of the Area of a Surface with a Complex Geometry via the FEM -- 1.4 Elasticity of a Bar -- 1.5 Stiffness Matrix of a Single Bar -- 1.6 Stiffness Matrix of a Bar via the PVD -- 1.7 Truss Structures and Their Automatic Calculation by Means of the FEM -- 1.8 Example of a Truss Structure -- 1.8.1 Element Matrices in the Local Reference System -- 1.8.2 Element Matrices in the Global Reference System -- 1.8.3 Global Structure Stiffness Matrix Assembly -- 1.8.4 Application of Boundary Conditions and the Numerical Solution -- 1.9 Outline of the Book Contents -- References -- 2 Fundamental Equations of 3D Elasticity -- 2.1 Equilibrium Conditions -- 2.2 Geometrical Relations -- 2.3 Hooke's Law -- 2.4 Displacement Formulation -- Further Reading -- 3 From 3D Problems to 2D and 1D Problems: Theories for Beams, Plates and Shells -- 3.1 Typical Structures -- 3.1.1 Three-Dimensional Structures (Solids) -- 3.1.2 Two-Dimensional Structures (Plates, Shells and Membranes) -- 3.1.3 One-Dimensional Structures (Beams and Bars) -- 3.2 Axiomatic Method -- 3.2.1 Two-Dimensional Case -- 3.2.2 One-Dimensional Case -- 3.3 Asymptotic Method -- Further Reading -- 4 Typical FE Governing Equations and Procedures -- 4.1 Static Response Analysis -- 4.2 Free Vibration Analysis -- 4.3 Dynamic Response Analysis -- References -- 5 Introduction to the Unified Formulation -- 5.1 Stiffness Matrix of a Bar and the Related FN -- 5.2 Case of a Bar Element with Internal Nodes -- 5.2.1 The Case of Bar with Three Nodes.

5.2.2 The Case of an Arbitrary Defined Number of Nodes -- 5.3 Combination of the FEM and the Theory of Structure Approximations: A Four-Index FN and the CUF -- 5.3.1 FN for a 1D Element with a Variable Axial Displacement over the Cross-section -- 5.3.2 FN for a 1D Structure with a Complete Displacement Field: The Case of a Refined Beam Model -- 5.4 CUF Assembly Technique -- 5.5 CUF as a Unique Approach for 1D, 2D and 3D Structures -- 5.6 Literature Review of the CUF -- References -- 6 The Displacement Approach via the PVD and FN for 1D, 2D and 3D Elements -- 6.1 Strong Form of the Equilibrium Equations via the PVD -- 6.1.1 The Two Fundamental Terms of the FN -- 6.2 Weak Form of the Solid Model Using the PVD -- 6.3 Weak Form of a Solid Element Using Index Notation -- 6.4 FN for 1D, 2D and 3D Problems in Unique Form -- 6.4.1 Three-Dimensional Models -- 6.4.2 Two-Dimensional Models -- 6.4.3 One-Dimensional Models -- 6.5 CUF at a Glance -- 6.5.1 Choice of Ni, Nj, F and Fs -- References -- 7 Three-Dimensional FEM Formulation (Solid Elements) -- 7.1 An Eight-Node Element Using Classical Matrix Notation -- 7.1.1 Stiffness Matrix -- 7.1.2 Load Vector -- 7.2 Derivation of the Stiffness Matrix Using the Index Notation -- 7.2.1 Governing Equations -- 7.2.2 FE Approximation in the CUF -- 7.2.3 Stiffness Matrix -- 7.2.4 Mass Matrix -- 7.2.5 Loading Vector -- 7.3 Three-Dimensional Numerical Integration -- 7.3.1 Three-Dimensional Gauss-Legendre Quadrature -- 7.3.2 Isoparametric Formulation -- 7.3.3 Reduced Integration: Shear Locking Correction -- 7.4 Shape Functions -- References -- 8 One-Dimensional Models with Nth-Order Displacement Field, the Taylor Expansion Class -- 8.1 Classical Models and the Complete Linear Expansion Case -- 8.1.1 The Euler-Bernoulli Beam Model -- 8.1.2 The Timoshenko Beam Theory (TBT) -- 8.1.3 The Complete Linear Expansion Case.

8.1.4 A Finite Element Based on -- 8.2 EBBT, TBT and N = 1 in Unified Form -- 8.2.1 Unified Formulation of -- 8.2.2 EBBT and TBT as Particular Cases of -- 8.3 CUF for Higher-Order Models -- 8.3.1 N = 3 and N = 4 -- 8.3.2 Nth-Order -- 8.4 Governing Equations, FE Formulation and the FN -- 8.4.1 Governing Equations -- 8.4.2 FE Formulation -- 8.4.3 Stiffness Matrix -- 8.4.4 Mass Matrix -- 8.4.5 Loading Vector -- 8.5 Locking Phenomena -- 8.5.1 Poisson Locking and its Correction -- 8.5.2 Shear Locking -- 8.6 Numerical Applications -- 8.6.1 Structural Analysis of a Thin-Walled Cylinder -- 8.6.2 Dynamic Response of Compact and Thin-Walled Structures -- References -- 9 One-Dimensional Models with a Physical Volume/Surface-Based Geometry and Pure Displacement Variables, the Lagrange Expansion Class -- 9.1 Physical Volume/Surface Approach -- 9.2 Lagrange Polynomials and Isoparametric Formulation -- 9.2.1 Lagrange Polynomials -- 9.2.2 Isoparametric Formulation -- 9.3 LE Displacement Fields and Cross-section Elements -- 9.3.1 FE Formulation and FN -- 9.4 Cross-section Multi-elements and Locally Refined Models -- 9.5 Numerical Examples -- 9.5.1 Mesh Refinement and Convergence Analysis -- 9.5.2 Considerations on PL -- 9.5.3 Thin-Walled Structures and Open Cross-Sections -- 9.5.4 Solid-like Geometrical BCs -- 9.6 The Component-Wise Approach for Aerospace and Civil Engineering Applications -- 9.6.1 CW Approach for Aeronautical Structures -- 9.6.2 CW Approach for Civil Engineering -- References -- 10 Two-Dimensional Plate Models with Nth-Order Displacement Field, the Taylor Expansion Class -- 10.1 Classical Models and the Complete Linear Expansion -- 10.1.1 Classical Plate Theory -- 10.1.2 First-Order Shear Deformation Theory -- 10.1.3 The Complete Linear Expansion Case -- 10.1.4 An FE Based on -- 10.2 CPT, FSDT and Model in Unified Form.

10.2.1 Unified Formulation of the Model -- 10.2.2 CPT and FSDT as Particular Cases of -- 10.3 CUF of Nth Order -- 10.3.1 N = 3 and N = 4 -- 10.4 Governing Equations, the FE Formulation and the FN -- 10.4.1 Governing Equations -- 10.4.2 FE Formulation -- 10.4.3 Stiffness Matrix -- 10.4.4 Mass Matrix -- 10.4.5 Loading Vector -- 10.4.6 Numerical Integration -- 10.5 Locking Phenomena -- 10.5.1 Poisson Locking and its Correction -- 10.5.2 Shear Locking and its Correction -- 10.6 Numerical Applications -- References -- 11 Two-Dimensional Shell Models with Nth-Order Displacement Field, the TE Class -- 11.1 Geometrical Description -- 11.2 Classical Models and Unified Formulation -- 11.3 Geometrical Relations for Cylindrical Shells -- 11.4 Governing Equations, FE Formulation and the FN -- 11.4.1 Governing Equations -- 11.4.2 FE Formulation -- 11.5 Membrane and Shear Locking Phenomenon -- 11.5.1 MITC9 Shell Element -- 11.5.2 Stiffness Matrix -- 11.6 Numerical Applications -- References -- 12 Two-Dimensional Models with Physical Volume/Surface-Based Geometry and Pure Displacement Variables, the LE Class -- 12.1 Physical Volume/Surface Approach -- 12.2 LE Model -- 12.3 Numerical Examples -- References -- 13 Discussion on Possible Best Beam, Plate and Shell Diagrams -- 13.1 The MAAA -- 13.2 Static Analysis of Beams -- 13.2.1 Influence of the Loading Conditions -- 13.2.2 Influence of the Cross-section Geometry -- 13.2.3 Reduced Models vs Accuracy -- 13.3 Modal Analysis of Beams -- 13.3.1 Influence of the Cross-section Geometry -- 13.3.2 Influence of BCs -- 13.4 Static Analysis of Plates and Shells -- 13.4.1 Influence of BCs -- 13.4.2 Influence of the Loading Conditions -- 13.4.3 Influence of the Loading and Thickness -- 13.4.4 Influence of the Thickness Ratio on Shells -- 13.5 The BTD -- References -- 14 Mixing Variable Kinematic Models.

14.1 Coupling Variable Kinematic Models via Shared Stiffness -- 14.1.1 Application of the Shared Stiffness Method -- 14.2 Coupling Variable Kinematic Models via the LM Method -- 14.2.1 Application of the LM Method to Variable Kinematic Models -- 14.3 Coupling Variable Kinematic Models via the Arlequin Method -- 14.3.1 Application of the Arlequin Method -- References -- 15 Extension to Multilayered Structures -- 15.1 Multilayered Structures -- 15.2 Theories for Multilayered Structures -- 15.2.1 Requirements -- 15.2.2 Refined Theories -- 15.2.3 Zigzag Theories -- 15.2.4 Layer-Wise Theories -- 15.2.5 Mixed Theories -- 15.3 Unified Formulation for Multilayered Structures -- 15.3.1 ESLMs -- 15.3.2 Inclusion of Murakami's ZZ Function -- 15.3.3 LW Theory and Legendre Expansion -- 15.3.4 Mixed Models with Displacement and Transverse Stress Variables -- 15.4 FE Formulation -- 15.4.1 Assembly at Multilayer Level -- 15.4.2 Selected Results -- 15.5 Literature on the CUF Extended to Multilayered Structures -- References -- 16 Extension to Multifield Problems -- 16.1 Mechanical vs Field Loadings -- 16.2 The Need for Second-Generation FEs for Multifield Loadings -- 16.3 Constitutive Equations for MFPs -- 16.4 Variational Statements for MFPs -- 16.4.1 PVD -- 16.4.2 RMVT -- 16.5 Use of Variational Statements to Obtain FE equations in Terms of 'Fundamental Nuclei' -- 16.5.1 PVD - Applications -- 16.5.2 RMVT - Applications -- 16.6 Selected Results -- 16.6.1 Mechanical-Electrical Coupling: Static Analysis of an Actuator Plate -- 16.6.2 Mechanical-Electrical Coupling: Comparison between RMVT Analyses -- 16.7 Literature on the CUF Extended to MFPs -- References -- Appendix A Numerical Integration -- A.1 Gauss-Legendre Quadrature -- References -- Appendix B CUF FE Models: Programming and Implementation Guidelines -- B.1 Preprocessing and Input Descriptions.

B.1.1 General FE Inputs.