# Finite Element Analysis of Structures through Unified Formulation

# Finite Element Analysis of Structures through Unified Formulation

ISBN: 978-1-119-94121-7 September 2014 410 Pages

## Description

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 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