Finite Element Analysis of Structures through Unified Formulation
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List of symbols and acronyms xvii
1 Introduction 1
1.1 What is in this book 1
1.2 The finite element method 2
1.2.1 Approximation of the domain 2
1.2.2 The numerical approximation 4
1.3 Calculation of the area of a surface with a complex geometry via FEM 5
1.4 Elasticity of a bar 6
1.5 Stiffness matrix of a single bar 8
1.6 Stiffness matrix of a bar via the Principle of Virtual Displacements 11
1.7 Truss structures and their automatic calculation by means of FEM 14
1.8 Example of a truss structure 17
1.8.1 Element matrices in the local reference system 18
1.8.2 Element matrices in the global reference system 18
1.8.3 Global structure stiffness matrix assembly 19
1.8.4 Application of boundary conditions and the numerical solution 20
1.9 Outline of the book contents 22
2 Fundamental equations of three-dimensional elasticity 25
2.1 Equilibrium conditions 25
2.2 Geometrical relations 27
2.3 Hooke's law 27
2.4 Displacement formulations 28
3 From 3D problems to 2D and 1D problems: theories for beams, plates and shells 31
3.1 Typical structures 31
3.1.1 Three-dimensional structures, 3D (solids) 32
3.1.2 Two-dimensional structures, 2D (plates, shells and membranes) 32
3.1.3 One-dimensional structures, 1D (beams and bars) 33
3.2 Axiomatic method 33
3.2.1 2D case 34
3.2.2 1D Case 37
3.3 Asymptotic method 39
4 Typical FE governing equations and procedures 41
4.1 Static response analysis 41
4.2 Free vibration analysis 42
4.3 Dynamic response analysis 43
5 Introduction to the unified formulation 47
5.1 Stiffness matrix of a bar and the related fundamental nucleus 47
5.2 Fundamental nucleus for the case of a bar element with internal nodes 49
5.2.1 The case of an arbitrary defined number of nodes 53
5.3 Combination of FEM and the theory of structure approximations: a four indices fundamental nucleus and the Carrera unified formulation 54
5.3.1 Fundamental nucleus for a 1D element with a variable axial displacement over the cross-section 55
5.3.2 Fundamental nucleus for a 1D structure with a complete displacement field: the case of a refined beam model 56
5.4 CUF assembly technique 58
5.5 CUF as a unique approach for one-, two- and three-dimensional structures 59
5.6 Literature review of the CUF 60
6 The displacement approach via the Principle of Virtual Displacements and FN for 1D, 2D and 3D elements 65
6.1 Strong form of the equilibrium equations via PVD 65
6.1.1 The two fundamental terms of the fundamental nucleus 69
6.2 Weak form of the solid model using the PVD 69
6.3 Weak form of a solid element using indicial notation 72
6.4 Fundamental nucleus for 1D, 2D and 3D problems in unique form 73
6.4.1 Three-dimensional models 74
6.4.2 Two-dimensional models 74
6.4.3 One-dimensional models 75
6.5 CUF at a glance 76
6.5.1 Choice of Ni, Nj, F and Fs 78
7 3D FEM formulation (solid elements) 81
7.1 An 8-node element using the classical matrix notation 81
7.1.1 Stiffness Matrix 83
7.1.2 Load Vector 84
7.2 Derivation of the stiffness matrix using the indicial notation 85
7.2.1 Governing equations 86
7.2.2 Finite element approximation in the CUF framework 86
7.2.3 Stiffness matrix 87
7.2.4 Mass matrix 89
7.2.5 Loading vector 90
7.3 3D numerical integration 91
7.3.1 3D Gauss-Legendre quadrature 91
7.3.2 Isoparametric formulation 92
7.3.3 Reduced integration: shear locking correction 93
7.4 Shape functions 95
8 1D models with N-order displacement field, the Taylor Expansion class (TE) 99
8.1 Classical models and the complete linear expansion case 99
8.1.1 The Euler-Bernoulli beam model (EBBT) 101
8.1.2 The Timoshenko beam theory (TBT) 102
8.1.3 The complete linear expansion case 105
8.1.4 A finite element based on N = 1 106
8.2 EBBT, TBT and N = 1 in unified form 107
8.2.1 Unified formulation of N = 1 108
8.2.2 EBBT and TBT as particular cases of N = 1 109
8.3 Carrera unified formulation for higher-order models 110
8.3.1 N = 3 and N = 4 112
8.3.2 N-order 113
8.4 Governing equations, finite element formulation and the fundamental nucleus 114
8.4.1 Governing equations 115
8.4.2 Finite element formulation 116
8.4.3 Stiffness matrix 117
8.4.4 Mass matrix 120
8.4.5 Loading vector 121
8.5 Locking phenomena 122
8.5.1 Poisson locking and its correction 123
8.5.2 Shear Locking 125
8.6 Numerical applications 126
8.6.1 Structural analysis of a thin-walled cylinder 128
8.6.2 Dynamic response of compact and thin-walled structures 132
9 1D models with a physical volume/surface-based geometry and pure displacement variables, the Lagrange Expansion class (LE) 143
9.1 Physical volume/surface approach 143
9.2 Lagrange polynomials and isoparametric formulation 145
9.2.1 Lagrange polynomials 147
9.2.2 Isoparametric formulation 150
9.3 LE displacement fields and cross-section elements 153
9.3.1 Finite element formulation and fundamental nucleus 156
9.4 Cross-section multi-elements and locally refined models 159
9.5 Numerical examples 160
9.5.1 Mesh refinement and convergence analysis 160
9.5.2 Considerations on Poisson's locking 165
9.5.3 Thin-walled structures and open cross-sections 167
9.5.4 Solid-like geometrical boundary conditions 174
9.6 The Component-Wise approach for aerospace and civil engineering applications 184
9.6.1 CW for aeronautical structures 184
9.6.2 CW for civil engineering 197
10 2D plate models with N-order displacement field, the Taylor expansion class 201
10.1 Classical models and the complete linear expansion 201
10.1.1 Classical plate theory 203
10.1.2 First-order shear deformation theory 205
10.1.3 The complete linear expansion case 207
10.1.4 A finite element based on N = 1 207
10.2 CPT, FSDT and N = 1 model in unified form 209
10.2.1 Unified formulation of N = 1 model 209
10.2.2 CPT and FSDT as particular cases of N = 1 211
10.3 Carrera unified formulation of N-order 211
10.3.1 N = 3 and N = 4 213
10.4 Governing equations, finite element formulation and the fundamental nucleus 213
10.4.1 Governing equations 214
10.4.2 Finite element formulation 215
10.4.3 Stiffness matrix 216
10.4.4 Mass matrix 217
10.4.5 Loading vector 218
10.4.6 Numerical integration 218
10.5 Locking phenomena 220
10.5.1 Poisson locking and its correction 220
10.5.2 Shear locking and its correction 221
10.6 Numerical Applications 226
11 2D shell models with N-order displacement field, the Taylor expansion class 231
11.1 Geometry description 231
11.2 Classical models and unified formulation 234
11.3 Geometrical relations for cylindrical shells 235
11.4 Governing equations, finite element formulation and the fundamental nucleus 238
11.4.1 Governing equations 238
11.4.2 Finite element formulation 238
11.5 Membrane and shear locking phenomenon 239
11.5.1 MITC9 shell element 240
11.5.2 Stiffness matrix 244
11.6 Numerical applications 247
12 2D models with physical volume/surface-based geometry and pure displacement variables, the Lagrange Expansion class (LE) 255
12.1 Physical volume/surface approach 255
12.2 Lagrange expansion model 258
12.3 Numerical examples 259
13 Discussion on possible best beam, plate and shell diagrams 263
13.1 The Mixed Axiomatic/Asymptotic Method 263
13.2 Static analysis of beams 267
13.2.1 Influence of the loading conditions 267
13.2.2 Influence of the cross-section geometry 268
13.2.3 Reduced models vs accuracy 269
13.3 Modal analysis of beams 271
13.3.1 Influence of the cross-section geometry 271
13.3.2 Influence of the boundary conditions 276
13.4 Static analysis of plates and shells 276
13.4.1 Influence of the boundary conditions 279
13.4.2 Influence of the loading conditions 280
13.4.3 Influence of the loading and thickness 283
13.4.4 Influence of the thickness ratio on shells 287
13.5 The best theory diagram 290
14 Mixing variable kinematic models 295
14.1 Coupling variable kinematic models via shared stiffness 296
14.1.1 Application of the shared stiffness method 298
14.2 Coupling variable kinematic models via the Lagrange multiplier method 299
14.2.1 Application of the Lagrange multiplier method to variable kinematics models 302
14.3 Coupling variable kinematic models via the Arlequin method 303
14.3.1 Application of the Arlequin method 305
15 Extension to multilayered structures 307
15.1 Multilayered structures 307
15.2 Theories on multilayered structures 311
15.2.1 C0z-requirements 312
15.2.2 Refined theories 312
15.2.3 Zig-Zag theories 313
15.2.4 Layer-Wise theories 314
15.2.5 Mixed theories 315
15.3 Unified formulation for multilayered structures 315
15.3.1 ESL models 316
15.3.2 Inclusion of Murakami's Zig-Zag function 316
15.3.3 Layer-Wise theory and Legendre expansion 317
15.3.4 Mixed models with displacement an transverse stress variables 318
15.4 Finite element formulation 319
15.4.1 Assemblage at multi-layer level 320
15.4.2 Selected results 320
15.5 Literature on CUF extended to multilayered structures 323
16 Extension to multifield problems 329
16.1 Mechanical vs field loadings 329
16.2 The need for second generation FEs for multifaced cases 330
16.3 Constitutive equations for multifield problems 331
16.4 Variational statements for multifield problems 334
16.4.1 PVD - Principle of Virtual Displacements 335
16.4.2 RMVT - Reissner Mixed Variational Theorem 338
16.5 Use of variational statements to obtained FE equations in terms of "Fundamental Nuclei" 340
16.5.1 PVD - applications 341
16.5.2 RMVT - applications 343
16.6 Selected results 346
16.6.1 Mechanical-Electrical coupling: static analysis of an actuator plate 347
16.6.2 Mechanical-Electrical coupling: comparison between RMVT analyses 349
16.7 Literature on CUF extended to multifield problems 349
A Numerical integration 357
A.1 Gauss-Legendre quadrature 357
B CUF finite element models: programming and implementation guidelines 361
B.1 Preprocessing and input descriptions 361
B.1.1 General FE inputs 362
B.1.2 Specific CUF inputs 367
B.2 FEM code 371
B.2.1 Stiffness and mass matrix 372
B.2.2 Stiffness and mass matrix numerical examples 377
B.2.3 Constraints and reduced models 379
B.2.4 Load vector 382
B.3 Postprocessing 384
B.3.1 Stresses and strains 385
References 386
About the Authors
Erasmo Carrera
Erasmo Carrera graduated in Aeronautics in 1986 and in Space Engineering in 1988 from the Politecnico di Torino. He obtained a PhD in Aerospace Engineering in 1991 within the framework of a joint PhD programme between the Politecnico di Milano, the Politecnico di Torino and the Università di Pisa. He became assistant professor in 1992. He has continuously held courses at Bachelor, Master and PhD levels on Fundamentals of Theory of Structures, Aerospace Structures, Nonlinear Problems, Plates and Shells, Thermal Stress, Composite Materials, Multifield Problems and Computational Aeroelasticity. Currently he is a full professor in the Department of Mechanical and Aerospace Engineering. He has also been a visiting professor at the University of Stuttgart, Virginia Tech, Supmeca and the Centre of Research Public Henri Tudor.
His research topics cover: composite materials, nonlinear problems and the stability of structures, contact mechanics, multibody dynamics, finite elements, path-following methods in nonlinear finite element (FE) analysis, meshless methods, unconventional lifting systems, smart structures, thermal stress for coupled and uncoupled problems, multifield interaction, aeroelasticity, panel flutter, wind blades, explosion effects on flying aircraft, advanced theories for beams, plates and shells, mixed variational methods; zigzag, mixed and layer-wise modellings for multilayered beams, plates and shells; local--global methods and the Arlequin-type approach; advanced structural models for wings, fuselage and complete aircraft/spacecraft through the introduction of the so-called component-wise approach; failure and progressive failure analysis of laminated structures; inflatable structures for manned and unmanned space applications; and the design and analysis of full composite aircraft, including trikes and unmanned aerial vehicles (UAVs).
Professor Carrera developed the Reissner mixed variational theorem (RMVT) as a natural extension of the principle of virtual displacements to layered structure analysis. He 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. The CUF has been applied extensively to both strong and weak forms (FE and meshless solutions). The main feature of the CUF is that it permits any expansion of the unknown variables over the thickness/cross-section domain to be handled in a compact manner. Governing equations are in fact obtained in terms of a few fundamental nuclei whose forms do not depend on either the order of the expansion or the base functions used. As a result, the CUF allows the so-called best theory diagram (BTD) (which shows the minimum number of unknown variables vs the error on an assigned parameter) to be computed for a given problem. The BTD is a way of enhancing axiomatic and asymptotic approaches in the theory of structures.
Professor Carrera is the author and coauthor of about 500 papers on the above topics, most of which have been published in primary international journals, as well as of two recent books published by John Wiley & Sons, Ltd. His papers have received about 5000 citations with an h-index=39 (data from Scopus). He has held invited seminars in various European and North American universities, as well as plenary talks at international conferences. Professor Carrera serves as the Associate Editor for Composite Structures, Journal of Thermal Stress, Mechanics of Advanced Structures, Computer and Structures and the International Journal of Aeronautical and Space Sciences. He is founder and Editor-in-Chief of Advances in Aircraft and Spacecraft Science; acts as a reviewer for about 80 journals; and is on the Editorial Board of many international conferences. He is also in charge of the chapter on `Shells' in the Encyclopedia of Thermal Stress, published by Springer. Professor Carrera is the founder of the non-profit international conference DeMEASS and the main organizer of ICMNMMCS (Turin, June 2012, co-chaired by Professor A. Ferreira), the ECCOMASS SMART 13 conference (Turin, June 2013) and ISVCS IX (Courmayeur, July 2013). He is member of the Distinguished Professor Board at King Abdulaziz University (Saudi Arabia). He has been a member of PhD and Habilitation committees in Germany, France, the Netherlands and Portugal. He is president of the Piedmont Section of AIDAA (Associazione Italiana di Aeronautica ed Astronautica).
Professor Carrera has been responsible for various research contracts granted by public and private national (including regional) and international institutions such as IVECO, the Italian Ministry of Education, the European Community, the European Space Agency, Alenia Spazio, Thales Alenia Space and Regione Piemonte. Among other projects, he has been responsible for the structural design and analysis of a full composite aircraft, named Sky-Y, by Alenia Aeronautica Torino, the first fully composite UAV made in Europe.
Professor Carrera is founder and leader of the MUL2 Group at the Politecnico di Torino. This group is considered one of the most active research teams in the Politecnico; it has acquired a significant international reputation in the field of multilayered structures subjected to multifield loadings; see also www.mul2.com. He is one of the Highly Cited Researchers by Thomson Reuters in both the Engineering and Materials Sections.
Maria Cinefra
Maria Cinefra is a research assistant at the Politecnico di Torino. She gained a BSc in Aerospace Engineering at the Politecnico di Torino in March 2007 with a thesis on the finite element method (FEM) in elliptic differential equations. Afterwards, she undertook an MSc in Aerospace Engineering at the Politecnico di Torino and gained her Master's degree, summa cum laude, in December 2008 from her work on the thermomechanical analysis of functionally graded material (FGM) shells. She began her PhD in January 2009, under the supervision of Professor Erasmo Carrera, on a research project related to the thermomechanical design of multilayered plates and shells embedding FGM layers. She was enrolled in a PhD with a foreign co-advisor, Professor Olivier Polit, at the University of Paris Ouest Nanterre. Her research project was funded by the Fonds National de la Recherche of Luxembourg and was performed in collaboration with the CRP Henri Tudor of Esch (Luxembourg). She was given the award for the best PhD paper (Ian Marshall's Award) at the 16th International Conference on Composite Structures (28--30 June 2011, Porto, Portugal). In January 2012, she was admitted to the final exam of her PhD and presented the defence of her thesis in April 2012. Since 2010, she has worked as a teaching assistant at the Politecnico di Torino on the courses Nonlinear Analysis of Structures, Structures for Space Vehicles and Fundamentals of Structural Mechanics. She is currently collaborating with the Department of Mathematics at Pavia University in order to develop a mixed shell FE based on the CUF for analysing composite structures. She has collaborated with Professor Ferreira, Editor of the Composite Structures Journal, on the radial basis functions method combined with the CUF. Dr Cinefra works as a reviewer for international journals such as Composite Structures and Mechanics of Advanced Materials and Structures. She is currently working on the STEPS regional project, in collaboration with Thales Alenia Space, and is also working on an extension of the shell FE, based on the CUF, to the analysis of multifield problems.
Marco Petrolo
Marco Petrolo is a Research Fellow at the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Melbourne, Australia. He was Post-Doc fellow at the Politecnico di Torino, Italy. He works in Professor Carrera's research group on various topics related to the development of refined structural models of composite structures. His research activity is connected with the structural analysis of composite lifting surfaces; refined beam, plate and shell models; component-wise approaches; and axiomatic/asymptotic analyses. He is the author and coauthor of some 50 publications, including 2 books and 25 articles that have been published in peer-reviewed journals.
Dr Petrolo gained his PhD in Aerospace Engineering at the Politecnico di Torino in April 2012, presenting a thesis on advanced aeroelastic models for the analysis of lifting surfaces made of composite materials. He also has an MSc in Aerospace Engineering from the Politecnico di Torino, an MSc in Aerospace Engineering from TU Delft (the Netherlands) and a BSc in Aerospace Engineering from the Politecnico di Torino. He has worked as an intern at EADS (Germany) and, as a Fulbright scholar, spent research periods at San Diego State University and the University of Michigan (USA). Dr Petrolo was appointed Adjunct Professor in Fundamentals of Strength of Materials (part of the BSc in Mechanical Engineering at the Turin Polytechnic University in Tashkent, Uzbekistan).
Enrico Zappino
Enrico Zappino is a post-doctoral fellow at the Politecnico di Torino. He has been in Professor Carrera's research group since 2010. His research activities concern structural analysis using classical and advanced models, multi-field analysis, and composite materials analysis. He is the coauthor of many works published in several international peer-reviewed journals. He obtained his PhD in April 2014, presenting a thesis on variable kinematic 1D,...
