Finite Element Analysis of Structures Through Unified Formulation

The finite element method (FEM) is a computational tool widely used to design and analyse complex structures. Currently, there are a number of different approaches to analysis using the FEM that vary according to the type of structure being analysed: beams and plates may use 1D or 2D approaches, shells and solids 2D or 3D approaches, and methods that work for one structure are typically not optimized to work for another.
 
Finite Element Analysis of Structures Through Unified Formulation deals with the FEM used for the analysis of the mechanics of structures in the case of linear elasticity. The novelty of this book is that the finite elements (FEs) are formulated on the basis of a class of theories of structures known as the Carrera Unified Formulation (CUF). It formulates 1D, 2D and 3D FEs on the basis of the same 'fundamental nucleus' that comes from geometrical relations and Hooke's law, and presents both 1D and 2D refined FEs that only have displacement variables as in 3D elements. It also covers 1D and 2D FEs that make use of 'real' physical surfaces rather than 'artificial' mathematical surfaces which are difficult to interface in CAD/CAE software.
 
Key features:
* Covers how the refined formulation can be easily and conveniently used to analyse laminated structures, such as sandwich and composite structures, and to deal with multifield problems
* Shows the performance of different FE models through the 'best theory diagram' which allows different models to be compared in terms of accuracy and computational cost
* Introduces an axiomatic/asymptotic approach that reduces the computational cost of the structural analysis without affecting the accuracy
* Introduces an innovative 'component-wise' approach to deal with complex structures
* Accompanied by a website hosting the dedicated software package MUL2 (www.mul2.com)
 
Finite Element Analysis of Structures Through Unified Formulation is a valuable reference for researchers and practitioners, and is also a useful source of information for graduate students in civil, mechanical and aerospace engineering.

List of contents

Preface xiii
 
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

About the author

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.

Summary

This book deals with the Finite Element Method for the analysis of elastic structures such as beams, plates, shells and solids. The modern approach of Unified Formulation (UF), as proposed by the lead author, deals with the consideration of one-dimensional (beams), two-dimensional (plates and shells) and three-dimensional (solids) elements.

Product details

• Authors: E Carrera, Erasm Carrera, Erasmo Carrera, Erasmo (Politecnico DI Torino) Cinefra Carrera, Erasmo Cinefra Carrera, Carrera Erasmo, Mari Cinefra, Maria Cinefra, Cinefra Maria, Marco Petrolo, Marco (Politecnico DI Torino) Carrera Petrolo, Marco et a Petrolo, Petrolo Marco, Enrico Zappino, Zappino Enrico
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 05.09.2014
• EAN: 9781119941217
• ISBN: 978-1-119-94121-7
• No. of pages: 410
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Maschinenbau, Mechanical Engineering, Solid Mechanics, Bauingenieur- u. Bauwesen, Civil Engineering & Construction, Baustatik u. Baumechanik, Festkörpermechanik, Structural Theory & Structural Mechanics, Computational / Numerical Methods, Rechnergestützte / Numerische Verfahren im Maschinenbau, Finite-Element-Methode