Finite Element Analysis of Structures through Unified Formulation

Erasmo Carrera is currently a full professor at the Department of Mechanical and Aerospace Engineering at Politecnico di Torino.  He is the founder and leader of the MUL2 group at the university, which has acquired a significant international reputation in the field of multilayered structures subjected to multifield loadings, see also www.mul2.com. He has introduced the Unified Formulation, or CUF (Carrera Unified Formulation), as a tool to establish a new framework in which beam, plate and shell theories can be developed for metallic and composite multilayered structures under mechanical, thermal electrical and magnetic loadings. CUF has been applied extensively to both strong and weak forms (FE and meshless solutions). Carrera has been author and co-author of about 500 papers on structural mechanics and aerospace engineering topics. Most of these works have been published in first rate international journals, as well as of two recent books published by J Wiley & Sons. Carrera’s papers have had about 500 citations with h-index=34 (data taken from Scopus).Maria Cinefra is currently a research assistant at the Politecnico di Torino. Since 2010, she has worked as a teaching assistant on the "Non-linear analysis of structures", "Structures for spatial vehicles" and "Fundamentals of structural mechanics" courses. She is currently collaborating with the Department of Mathematics at Pavia University in order to develop a mixed shell finite element based on the Carrera Unified Formulation for the analysis of composite structures. She is currently working in the STEPS regional project, in collaboration with Thales Alenia Space. M. Cinefra is also working on the extension of the shell finite element, based on the CUF, to the analysis of multi-field problems. Marco Petrolo is a Post-Doc fellow at the Politecnico di Torino (Italy). He works in Professor Carrera's research group on various research topics related to the development of refined structural models of composite structures. His research activity is connected to the structural analysis of composite lifting surfaces; refined beam, plate and shell models; component-wise approaches and axiomatic/asymptotic analyses. He is author and coauthor of some 50 publications, including 2 books and 25 articles that have been published in peer-reviewed journals. Marco has recently been appointed Adjunct Professor in Fundamentals of Strength of Materials (BSc in Mechanical Engineering at the Turin Polytechnic University in Tashkent, Uzbekistan).Enrico Zappino is a Ph.D student at the Politecnico di Torino (Italy). He has worked in Professor Erasmo Carrera's research group since 2010. His research activities concern structural analysis using classical and advanced models, multi-field analysis, composite materials and FEM advanced models. He is co-author of many works that have been published in international peer-reviewed journals. Enrico was employed as a research assistant in Professor Erasmo Carrera's group from September 2010 to January 2011, where his research, in cooperation with Tales Alenia Space (TASI), was about the panel flutter phenomena of composite panels in supersonic flows.

• Preface xiii List of symbols and acronyms xvii1 Introduction 11.1 What is in this book 11.2 The finite element method 21.2.1 Approximation of the domain 21.2.2 The numerical approximation 41.3 Calculation of the area of a surface with a complex geometry via FEM 51.4 Elasticity of a bar 61.5 Stiffness matrix of a single bar 81.6 Stiffness matrix of a bar via the Principle of Virtual Displacements 111.7 Truss structures and their automatic calculation by means of FEM 141.8 Example of a truss structure 171.8.1 Element matrices in the local reference system 181.8.2 Element matrices in the global reference system 181.8.3 Global structure stiffness matrix assembly 191.8.4 Application of boundary conditions and the numerical solution 201.9 Outline of the book contents 222 Fundamental equations of three-dimensional elasticity 252.1 Equilibrium conditions 252.2 Geometrical relations 272.3 Hooke's law 272.4 Displacement formulations 283 From 3D problems to 2D and 1D problems: theories for beams, plates and shells 313.1 Typical structures 313.1.1 Three-dimensional structures, 3D (solids) 323.1.2 Two-dimensional structures, 2D (plates, shells and membranes) 323.1.3 One-dimensional structures, 1D (beams and bars) 333.2 Axiomatic method 333.2.1 2D case 343.2.2 1D Case 373.3 Asymptotic method 394 Typical FE governing equations and procedures 414.1 Static response analysis 414.2 Free vibration analysis 424.3 Dynamic response analysis 435 Introduction to the unified formulation 475.1 Stiffness matrix of a bar and the related fundamental nucleus 475.2 Fundamental nucleus for the case of a bar element with internal nodes 495.2.1 The case of an arbitrary defined number of nodes 535.3 Combination of FEM and the theory of structure approximations: a four indices fundamental nucleus and the Carrera unified formulation 545.3.1 Fundamental nucleus for a 1D element with a variable axial displacement over the cross-section 555.3.2 Fundamental nucleus for a 1D structure with a complete displacement field: the case of a refined beam model 565.4 CUF assembly technique 585.5 CUF as a unique approach for one-, two- and three-dimensional structures 595.6 Literature review of the CUF 606 The displacement approach via the Principle of Virtual Displacements and FN for 1D, 2D and 3D elements 656.1 Strong form of the equilibrium equations via PVD 656.1.1 The two fundamental terms of the fundamental nucleus 696.2 Weak form of the solid model using the PVD 696.3 Weak form of a solid element using indicial notation 726.4 Fundamental nucleus for 1D, 2D and 3D problems in unique form 736.4.1 Three-dimensional models 746.4.2 Two-dimensional models 746.4.3 One-dimensional models 756.5 CUF at a glance 766.5.1 Choice of Ni, Nj, F and Fs 787 3D FEM formulation (solid elements) 817.1 An 8-node element using the classical matrix notation 817.1.1 Stiffness Matrix 837.1.2 Load Vector 847.2 Derivation of the stiffness matrix using the indicial notation 857.2.1 Governing equations 867.2.2 Finite element approximation in the CUF framework 867.2.3 Stiffness matrix 877.2.4 Mass matrix 897.2.5 Loading vector 907.3 3D numerical integration 917.3.1 3D Gauss-Legendre quadrature 917.3.2 Isoparametric formulation 927.3.3 Reduced integration: shear locking correction 937.4 Shape functions 958 1D models with N-order displacement field, the Taylor Expansion class (TE) 998.1 Classical models and the complete linear expansion case 998.1.1 The Euler-Bernoulli beam model (EBBT) 1018.1.2 The Timoshenko beam theory (TBT) 1028.1.3 The complete linear expansion case 1058.1.4 A finite element based on N = 1 1068.2 EBBT, TBT and N = 1 in unified form 1078.2.1 Unified formulation of N = 1 1088.2.2 EBBT and TBT as particular cases of N = 1 1098.3 Carrera unified formulation for higher-order models 1108.3.1 N = 3 and N = 4 1128.3.2 N-order 1138.4 Governing equations, finite element formulation and the fundamental nucleus 1148.4.1 Governing equations 1158.4.2 Finite element formulation 1168.4.3 Stiffness matrix 1178.4.4 Mass matrix 1208.4.5 Loading vector 1218.5 Locking phenomena 1228.5.1 Poisson locking and its correction 1238.5.2 Shear Locking 1258.6 Numerical applications 1268.6.1 Structural analysis of a thin-walled cylinder 1288.6.2 Dynamic response of compact and thin-walled structures 1329 1D models with a physical volume/surface-based geometry and pure displacement variables, the Lagrange Expansion class (LE) 1439.1 Physical volume/surface approach 1439.2 Lagrange polynomials and isoparametric formulation 1459.2.1 Lagrange polynomials 1479.2.2 Isoparametric formulation 1509.3 LE displacement fields and cross-section elements 1539.3.1 Finite element formulation and fundamental nucleus 1569.4 Cross-section multi-elements and locally refined models 1599.5 Numerical examples 1609.5.1 Mesh refinement and convergence analysis 1609.5.2 Considerations on Poisson’s locking 1659.5.3 Thin-walled structures and open cross-sections 1679.5.4 Solid-like geometrical boundary conditions 1749.6 The Component-Wise approach for aerospace and civil engineering applications 1849.6.1 CW for aeronautical structures 1849.6.2 CW for civil engineering 19710 2D plate models with N-order displacement field, the Taylor expansion class 20110.1 Classical models and the complete linear expansion 20110.1.1 Classical plate theory 20310.1.2 First-order shear deformation theory 20510.1.3 The complete linear expansion case 20710.1.4 A finite element based on N = 1 20710.2 CPT, FSDT and N = 1 model in unified form 20910.2.1 Unified formulation of N = 1 model 20910.2.2 CPT and FSDT as particular cases of N = 1 21110.3 Carrera unified formulation of N-order 21110.3.1 N = 3 and N = 4 21310.4 Governing equations, finite element formulation and the fundamental nucleus 21310.4.1 Governing equations 21410.4.2 Finite element formulation 21510.4.3 Stiffness matrix 21610.4.4 Mass matrix 21710.4.5 Loading vector 21810.4.6 Numerical integration 21810.5 Locking phenomena 22010.5.1 Poisson locking and its correction 22010.5.2 Shear locking and its correction 22110.6 Numerical Applications 22611 2D shell models with N-order displacement field, the Taylor expansion class 23111.1 Geometry description 23111.2 Classical models and unified formulation 23411.3 Geometrical relations for cylindrical shells 23511.4 Governing equations, finite element formulation and the fundamental nucleus 23811.4.1 Governing equations 23811.4.2 Finite element formulation 23811.5 Membrane and shear locking phenomenon 23911.5.1 MITC9 shell element 24011.5.2 Stiffness matrix 24411.6 Numerical applications 24712 2D models with physical volume/surface-based geometry and pure displacement variables, the Lagrange Expansion class (LE) 25512.1 Physical volume/surface approach 25512.2 Lagrange expansion model 25812.3 Numerical examples 25913 Discussion on possible best beam, plate and shell diagrams 26313.1 The Mixed Axiomatic/Asymptotic Method 26313.2 Static analysis of beams 26713.2.1 Influence of the loading conditions 26713.2.2 Influence of the cross-section geometry 26813.2.3 Reduced models vs accuracy 26913.3 Modal analysis of beams 27113.3.1 Influence of the cross-section geometry 27113.3.2 Influence of the boundary conditions 27613.4 Static analysis of plates and shells 27613.4.1 Influence of the boundary conditions 27913.4.2 Influence of the loading conditions 28013.4.3 Influence of the loading and thickness 28313.4.4 Influence of the thickness ratio on shells 28713.5 The best theory diagram 29014 Mixing variable kinematic models 29514.1 Coupling variable kinematic models via shared stiffness 29614.1.1 Application of the shared stiffness method 29814.2 Coupling variable kinematic models via the Lagrange multiplier method 29914.2.1 Application of the Lagrange multiplier method to variable kinematics models 30214.3 Coupling variable kinematic models via the Arlequin method 30314.3.1 Application of the Arlequin method 30515 Extension to multilayered structures 30715.1 Multilayered structures 30715.2 Theories on multilayered structures 31115.2.1 C0z–requirements 31215.2.2 Refined theories 31215.2.3 Zig-Zag theories 31315.2.4 Layer-Wise theories 31415.2.5 Mixed theories 31515.3 Unified formulation for multilayered structures 31515.3.1 ESL models 31615.3.2 Inclusion of Murakami’s Zig-Zag function 31615.3.3 Layer-Wise theory and Legendre expansion 31715.3.4 Mixed models with displacement an transverse stress variables 31815.4 Finite element formulation 31915.4.1 Assemblage at multi-layer level 32015.4.2 Selected results 32015.5 Literature on CUF extended to multilayered structures 32316 Extension to multifield problems 32916.1 Mechanical vs field loadings 32916.2 The need for second generation FEs for multifaced cases 33016.3 Constitutive equations for multifield problems 33116.4 Variational statements for multifield problems 33416.4.1 PVD - Principle of Virtual Displacements 33516.4.2 RMVT - Reissner Mixed Variational Theorem 33816.5 Use of variational statements to obtained FE equations in terms of ”Fundamental Nuclei” 34016.5.1 PVD - applications 34116.5.2 RMVT - applications 34316.6 Selected results 34616.6.1 Mechanical-Electrical coupling: static analysis of an actuator plate 34716.6.2 Mechanical-Electrical coupling: comparison between RMVT analyses 34916.7 Literature on CUF extended to multifield problems 349A Numerical integration 357A.1 Gauss-Legendre quadrature 357B CUF finite element models: programming and implementation guidelines 361B.1 Preprocessing and input descriptions 361B.1.1 General FE inputs 362B.1.2 Specific CUF inputs 367B.2 FEM code 371B.2.1 Stiffness and mass matrix 372B.2.2 Stiffness and mass matrix numerical examples 377B.2.3 Constraints and reduced models 379B.2.4 Load vector 382B.3 Postprocessing 384B.3.1 Stresses and strains 385References 386