Mechanics of Laminated Composite Doubly-Curved Shell Structures

Mechanics of Laminated CompositeDoubly-Curved Shell Structures

The Generalized Differential Quadrature Methodand the Strong Formulation Finite Element Method

Francesco TornabeneNicholas Fantuzzi

Mechanics of Laminated Com

positeDoubly-Curved Shell Structures

www.editrice-esculapio.it

Francesco Tornabene

Nicholas Fantuzzi

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This manuscript comes from the experience gained over ten years of study and research on shell structures and on the Generalized Diffe-rential Quadrature method.

The title, Mechanics of Laminated Composite Doubly-Curved Shell Structures, illustrates the theme followed in the present volume. The present study aims to analyze the static and dynamic behavior of moderately thick shells made of composite materials through the ap-plication of the Differential Quadrature (DQ) technique. A particular attention is paid, other than fibrous and laminated composites, also to “Functionally Graded Materials” (FGMs). They are non-homogeneous materials, characteri-zed by a continuous variation of the mechani-cal properties through a particular direction.

The GDQ numerical solution is compared, not only with literature results, but also with the ones supplied and obtained through the use of different structural codes based on the Finite Element Method (FEM).

Furthermore, an advanced version of GDQ method is also presented. This methodology is termed Strong Formulation Finite Element Method (SFEM) because it employs the strong form of the differential system of equations at the master element level and the mapping technique, proper of FEM. The connectivity between two elements is enforced through compatibility conditions.

DiQuMASPAB Project and Software

Francesco Tornabene Nicholas Fantuzzi

Mechanics of Laminated Composite

Doubly-Curved Shell Structures

The Generalized Differential Quadrature Method and the Strong Formulation Finite Element Method

ISBN 978-88-7488-687-6

First edition: January 2014 Publishing Manager: Alessandro Parenti Editorial Staff: Giancarla Panigali, Carlotta Lenzi

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To my parents and relatives

in particular to my Father (1947-2011)

“Practice and Theory

are complementary,

but Practice without Theory

can only grope”.

Francesco Tornabene

To Ilaria for her support,

encouragement and constant love

“There are people in everyone's lives who make

success both possible and rewarding.”

Nicholas Fantuzzi

Mechanics of Laminated Composite Doubly-Curved Shell Structures

About the AuthorsFrancesco Tornabene was born in Bologna, January 13, 1978. School-leaving examination in a classical liceo achieved at Liceo Classico San Luigi in Bologna in 1997. Patent for Industrial Invention: Friction Clutch for High Performance Vehicles Question BO2001A00442 filed on 13/07/2001 in National Patent Bologna (Italy). Assignees: Alma Mater Studiorum - University of Bologna. Degree in Mechanical Engineering (Course of Studies in Structural Mechanics) obtained at the Alma Mater Studiorum - University of

Bologna on 23/07/2003. Thesis Title (in Italian): Dynamic Behavior of Cylindrical Shells: Formulation and Solution. First position obtained in the competition for admission to the PhD in Structural Mechanics at the Alma Mater Studiorum - University of Bologna in December 2003. Winner of the scholarship, Carlo Felice Jodifor a degree in Structural Mechanics in 2004. Adjunct Professor (Tutor Contract) for activities of supporting the teaching of Scienza delle Costruzioni (Structural Mechanics) L, for the course in Civil Engineering, at Alma Mater Studiorum - University of Bologna, a.a. 2005/2006. Adjunct Professor (Tutor Contract) for activities to support the teaching of Scienza delle Costruzioni (Structural Mechanics) L, for the course in Civil Engineering, at Alma Mater Studiorum - University of Bologna, a.a. 2007/2008. PhD in Structural Mechanics at the Alma Mater Studiorum - University of Bologna on 31/05/2007. PhD Thesis Title the (in Italian): Modeling and Solution of Shell Structures Made of Anisotropic Materials. Owner of the research grant entitled: UnifiedFormulation of Shell Structures Made of Anisotropic Materials. Numerical Analysis Using the Generalized Differential Quadrature Method and the Finite Element Method from January 2007 to January 2009 at the Alma Mater Studiorum - University of Bologna. Adjunct Professor (Tutor Contract) for activities to support the teaching of Mechanical Design and Laboratory T C.I., for the Degree in Mechanical Engineering, at Alma Mater Studiorum - University of Bologna, a.a. 2010/2011. Winner of the Senior research grant entitled: Design for Recycling Methodologies Applied to the Nautical Field from February 2011 to October 2011 at the Alma Mater Studiorum - University of Bologna. Junior researcher for the research program entitled: Advanced Numerical Schemes for Anisotropic Materials from December 2011 to January 2012 at the Alma Mater Studiorum - University of Bologna. Research Activities in collaboration with Foreign University Professors. Author of the book (in Italian) entitled: Mechanics of Shell Structures Made of Composite Materials. The Generalized Differential Quadrature Method, Esculapio, Bologna, 2012. Member of the Editorial Board of Journal of Computational Engineering and ISRN Mechanical Engineering since 2013. Member of Scientific Committee, Promoter and Secretary of CIMEST Center, Center for Studies and Research on the Identification of Materials and Structures - “Michele Capurso” - at the Department DICAM of the Alma Mater Studiorum - University of Bologna, since 2005. Professor of Dynamics of Structures since 2012 and of Computational Mechanics since 2013. Assistant Professor at the Alma Mater Studiorum - University of Bologna since 2012. Author of more than seventy research papers since 2005.

Assistant Professor at School of Engineering and Architecture

Department of Civil, Chemical, Environmental and Materials Engineering

(DICAM)

Alma Mater Studiorum - University of Bologna

Viale del Risorgimento 2, Bologna 40136, Italy

E-mail Address: [email protected]

web page: http://software.dicam.unibo.it/diqumaspab-project

F. Tornabene, N. Fantuzzi

Nicholas Fantuzzi was born in Bologna, June 16, 1984. School-leaving examination in a scientific liceo achieved at Liceo Scientifico Augusto Righi in Bologna in 2003. Bachelor’s degree in Civil Engineering (Course of Studies in Structural Engineering)obtained at the Alma Mater Studiorum - University of Bologna on 16/10/2006, grade 110/110 cum laude. Thesis title (in Italian): On the Behavior of Cylindrical Vaults. Master degree in Civil Engineering (Course of Studies in Structural Engineering) obtained at the Alma Mater

Studiorum - University of Bologna on 16/01/2009, grade 110/110 cum laude. Thesis title (in Italian): Curvature Effect on the Behavior of Shells with Anisotropic Material. First position obtained in the competition for admission to the PhD in Structural Engineering and Hydraulics at the Alma Mater Studiorum - University of Bologna in December 2009. PhD in Structural Engineering and Hydraulics at the Alma Mater Studiorum - University of Bologna on 31/05/2013. PhD Thesis title: Generalized Differential Quadrature Finite Element Method Applied to Advanced Structural Mechanics. Owner of the research grant entitled: About Shell Structures Made of Anisotropic Materials. Unified Formulation and Numerical Analysis since June 2013 at the Alma Mater Studiorum - University of Bologna. Winner of the “ICCS17 Ian Marshall's Award for Best Student Paper” with the work entitled: Static Analysis of Doubly-Curved Anisotropic Shells and Panels Using CUF Approach, Differential Geometry and Differential Quadrature Method by F. Tornabene, N. Fantuzzi, E. Viola and E. Carrera, published in Composite Structures 107, 675-697 (2014). Adjunct Professor (Tutor Contract) for activities to support the teaching of Second Level Master in “Design of Oil & Gas plants”, for ENI Corporate University, at Alma Mater Studiorum - University of Bologna, since 2010. Member of Scientific Committee, Promoter and Secretary of CIMEST Center, Center for Studies and Research on the Identification of Materials and Structures - “Michele Capurso” - at the Department DICAM of the Alma Mater Studiorum - University of Bologna, since 2011.

Research Assistant at School of Engineering and Architecture

Department of Civil, Chemical, Environmental and Materials Engineering

(DICAM)

Alma Mater Studiorum - University of Bologna

Viale del Risorgimento 2, Bologna 40136, Italy

E-mail Address: [email protected]

web page: http://software.dicam.unibo.it/diqumaspab-project

Mechanics of Laminated Composite Doubly-Curved Shell Structures III

Index

PREFACE ............................................................................................................................... XIII

1 THE GENERALIZED DIFFERENTIAL QUADRATURE METHOD

1.1 DIFFERENTIAL QUADRATURE: THE POLYNOMIAL VECTOR SPACE AND FUNCTIONAL

APPROXIMATION ............................................................................................................ 14

1.1.1 INTRODUCTION .......................................................................................................... 14

1.1.2 GENESIS OF THE DIFFERENTIAL QUADRATURE METHOD .......................................... 15

1.1.2.1 Preamble ............................................................................................................. 15

1.1.2.2 Bellman Differential Quadrature ........................................................................ 16

1.1.3 THE DIFFERENTIAL QUADRATURE LAW .................................................................... 18

1.1.3.1 Integral quadrature ............................................................................................. 18

1.1.3.2 Differential quadrature ....................................................................................... 19

1.1.4 POLYNOMIAL VECTOR SPACE ................................................................................... 20

1.1.4.1 Linear vector space definition ............................................................................ 21

1.1.4.2 Properties of a linear vector space ...................................................................... 23

1.1.5 FUNCTIONAL APPROXIMATION .................................................................................. 25

1.1.5.1 Polynomial approximation ................................................................................. 26

1.1.5.2 Fourier series expansion ..................................................................................... 32

1.1.5.2.1 Expansion of a generic function ................................................................... 32

1.1.5.2.2 Expansion of an even function ..................................................................... 34

1.1.5.2.3 Expansion of an odd function ....................................................................... 36

1.2 MATHEMATICAL FORMULATION ..................................................................................... 38

1.2.1 POLYNOMIAL DIFFERENTIAL QUADRATURE .............................................................. 38

1.2.1.1 Calculation of the coefficients for the derivatives of the first order .................. 39

1.2.1.1.1 Bellman approach ........................................................................................ 39

1.2.1.1.1.1 First Bellman approach ......................................................................... 40

1.2.1.1.1.2 Second Bellman approach ...................................................................... 41

1.2.1.1.2 Quan and Chang approach .......................................................................... 42

Index

F. Tornabene, N. Fantuzzi IV

1.2.1.1.3 Generalized Shu approach ........................................................................... 43

1.2.1.2 Calculation of the coefficients for the derivatives of higher order ................... 49

1.2.1.2.1 Weighting coefficients for the second order derivatives .............................. 49

1.2.1.2.1.1 Quan e Chang approach ........................................................................ 49

1.2.1.2.1.2 Generalized Shu approach ..................................................................... 50

1.2.1.2.2 Coefficients of the higher order derivatives: recursive formulae ................ 52

1.2.2 DIFFERENTIAL QUADRATURE BASED ON THE FOURIER EXPANSION SERIES .............. 56

1.2.3 MATRIX MULTIPLICATION APPROACH ...................................................................... 62

1.2.4 EXTENSION TO THE MULTIDIMENSIONAL CASE ......................................................... 64

1.2.5 TYPES OF DISCRETIZATIONS ...................................................................................... 71

1.2.5.1 -sampling points technique ............................................................................. 75

1.2.5.2 Linear domain discretization .............................................................................. 77

1.2.6 APPLICATION TO SIMPLE FUNCTIONS ........................................................................ 87

1.2.6.1 Power function ................................................................................................... 89

1.2.6.2 Square root function ........................................................................................... 90

1.2.6.3 Approximation of the first four derivatives of some functions and the local

derivative notion (Local GDQ) .......................................................................... 94

1.3 A GENERAL VIEW ON DIFFERENTIAL QUADRATURE .................................................... 106

1.3.1 BASIS FUNCTIONS OR BASIS OF ORTHOGONAL POLYNOMIALS ................................ 108

1.3.1.1 Lagrange basis functions .................................................................................. 109

1.3.1.2 Lagrange trigonometric basis functions ........................................................... 109

1.3.1.3 Jacobi basis functions ....................................................................................... 110

1.3.1.3.1 Legendre basis functions ............................................................................ 112

1.3.1.3.2 Chebyshev basis basis functions (first kind) .............................................. 112

1.3.1.3.3 Chebyshev basis basis functions (second kind) .......................................... 113

1.3.1.4 Chebyshev basis functions (third kind) ............................................................ 113

1.3.1.5 Chebyshev basis functions (fourth kind) .......................................................... 114

1.3.1.6 Power or monomial basis functions ................................................................. 114

1.3.1.7 Exponential basis functions .............................................................................. 115

1.3.1.8 Hermite basis functions .................................................................................... 115

1.3.1.9 Laguerre basis functions ................................................................................... 115

1.3.1.10 Bernstein basis functions ................................................................................ 116

Index

Mechanics of Laminated Composite Doubly-Curved Shell Structures V

1.3.1.11 Fourier basis functions ................................................................................... 116

1.3.1.12 Lobatto basis functions ................................................................................... 117

1.3.1.13 Radial basis functions ..................................................................................... 117

1.3.2 GRID DISTRIBUTIONS .............................................................................................. 119

1.3.2.1 Coordinate transformation ................................................................................ 119

1.3.2.2 -point distribution ......................................................................................... 119

1.3.2.3 Stretching formulation ...................................................................................... 120

1.3.2.4 Several types of discretization .......................................................................... 120

1.4 GENERALIZED INTEGRAL QUADRATURE ....................................................................... 124

2 THEORY OF COMPOSITE SHELL STRUCTURES

2.1 ELEMENTS OF DIFFERENTIAL GEOMETRY .................................................................... 134

2.1.1 CURVES IN SPACE .................................................................................................... 134

2.1.1.1 Parametric representation of a curve ................................................................ 134

2.1.1.2 Tangent unit vector ........................................................................................... 134

2.1.1.3 Osculating plane and main normal ................................................................... 137

2.1.1.4 Curvature .......................................................................................................... 137

2.1.2 SURFACES IN SPACE ................................................................................................ 139

2.1.2.1 Parametric curves: first fundamental form ....................................................... 139

2.1.2.2 Normal to the surface ....................................................................................... 141

2.1.2.3 Second fundamental form ................................................................................ 142

2.1.2.4 Principal curvatures and principal directions ................................................... 144

2.1.2.5 Derivatives of the unit vectors along the parametric lines ............................... 147

2.1.2.6 Fundamental theorem of the theory of surfaces ............................................... 150

2.1.2.7 Gaussian curvature ........................................................................................... 152

2.1.2.8 Classifications of surfaces ................................................................................ 153

2.1.2.8.1 Classification based on shape .................................................................... 153

2.1.2.8.2 Classification based on the curvature ........................................................ 155

2.1.2.8.3 Classification based on the developability ................................................. 155

2.1.2.9 Definition of a surface of revolution .............................................................. 156

Index

F. Tornabene, N. Fantuzzi VI

2.1.2.10 Definition of a cylindrical surface of translation ........................................... 162

2.2 REISSNER-MINDLIN THEORY ......................................................................................... 164

2.2.1 FUNDAMENTAL ASSUMPTIONS ................................................................................ 164

2.2.2 COORDINATES OF A GENERIC SHELL ....................................................................... 166

2.2.3 KINEMATIC EQUATIONS .......................................................................................... 170

2.2.3.1 Kinematic model .............................................................................................. 170

2.2.3.2 Strain characteristics ........................................................................................ 171

2.2.4 CONSTITUTIVE EQUATIONS ..................................................................................... 182

2.2.4.1 Generalized Hooke laws ................................................................................... 182

2.2.4.1.1 Anisotropic materials ................................................................................. 185

2.2.4.1.2 Material symmetry ...................................................................................... 186

2.2.4.1.3 Monoclinic material ................................................................................... 186

2.2.4.1.4 Orthotropic materials ................................................................................. 187

2.2.4.1.5 Transversely isotropic materials ................................................................ 189

2.2.4.1.6 Isotropic materials ..................................................................................... 190

2.2.4.2 Composite materials ......................................................................................... 191

2.2.4.2.1 Fibrous composites: unidirectional lamina ............................................... 192

2.2.4.2.2 Granular composites: functionally graded materials ................................ 196

2.2.4.2.3 Trasformation of the stress and strain components ................................... 201

2.2.4.2.4 Trasformation of the elastic coefficients .................................................... 205

2.2.4.2.5 Composite laminates .................................................................................. 208

2.2.4.2.6 Constitutive equations for the first order theory (FSDT) ........................... 211

2.2.4.3 Stress resultants ................................................................................................ 215

2.2.4.4 Laminates and laminations schemes ................................................................ 221

2.2.4.4.1 Composites made of a single lamina .......................................................... 225

2.2.4.4.2 Symmetric laminates .................................................................................. 230

2.2.4.4.3 Antisymmetric laminates ............................................................................ 233

2.2.4.4.4 Balanced and almost isotropic laminates .................................................. 237

2.2.5 INDEFINITE EQUILIBRIUM EQUATIONS .................................................................... 238

2.2.5.1 Generalized vector of external actions ............................................................. 239

2.2.5.1.1 External actions on the bottom and top shell surfaces ............................... 239

2.2.5.2 Equations of motion via Hamilton principle .................................................... 241

Index

Mechanics of Laminated Composite Doubly-Curved Shell Structures VII

2.2.6 FUNDAMENTAL EQUATIONS .................................................................................... 256

2.2.7 SPECIALIZATION FOR SHELLS OF REVOLUTION ....................................................... 268

2.2.7.1 Kinematic equations ......................................................................................... 270

2.2.7.2 Constitutive equations ...................................................................................... 271

2.2.7.3 Indefinite equilibrium equations ...................................................................... 272

2.2.7.4 Fundamental equations ..................................................................................... 273

2.2.8 SOME SPECIAL CASES ............................................................................................. 278

2.2.8.1 Rotating shells .................................................................................................. 279

2.2.8.2 Shells resting on elastic foundation and subjected to viscous forces ............... 282

2.2.8.3 Shells subjected to seismic actions ................................................................... 288

3 MAIN SHELL STRUCTURES

3.1 GENERIC DOUBLY-CURVED AND SINGLY-CURVED SHELLS AND DEGENERATE

SHELLS ............................................................................................................................ 295

3.1.1 ENNEPER SURFACE .................................................................................................. 295

3.1.2 DOUBLY-CURVED TRANSLATIONAL SURFACES ...................................................... 297

3.1.3 ELLIPTIC CYLINDER, ELLIPTIC CONE AND ELLIPTIC PLATE ..................................... 303

3.1.4 DEGENERATE PLATES: PARABOLIC, ELLIPTIC AND BIPOLAR ................................... 306

3.2 DOUBLY-CURVED SHELLS OF REVOLUTION ................................................................. 311

3.2.1 HYPERBOLIC MERIDIAN SHELL ............................................................................... 318

3.2.2 SHELL WITH A TRACTRIX MERIDIAN (PSEUDOSPHERE) ........................................... 325

3.2.3 SHELL WITH A CATENARY MERIDIAN ..................................................................... 329

3.2.4 SHELL WITH A CYCLOIDAL MERIDIAN .................................................................... 335

3.2.5 SHELL WITH A PARABOLIC MERIDIAN ..................................................................... 338

3.2.6 SHELL WITH AN ELLIPTIC AND CIRCULAR MERIDIAN .............................................. 342

3.3 SINGLY-CURVED SHELLS OF REVOLUTION ................................................................... 353

3.3.1 CONICAL SHELL ...................................................................................................... 353

3.3.2 CIRCULAR CYLINDRICAL SHELL ............................................................................. 361

3.4 SINGLY-CURVED SHELLS OF TRANSLATION ................................................................. 366

3.5 DEGENERATE SHELLS ..................................................................................................... 377

Index

F. Tornabene, N. Fantuzzi VIII

3.5.1 CIRCULAR PLATE .................................................................................................... 377

3.5.2 RECTANGULAR PLATE ............................................................................................. 382

4 3D ELASTICITY EQUATIONS IN ORTHOGONAL CURVILINEAR

COORDINATES

4.1 EQUATIONS OF THE THREE-DIMENSIONAL SOLID ........................................................ 390

4.1.1 HAMILTON PRINCIPLE ............................................................................................. 390

4.1.2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS ......................................... 393

4.2 THREE-DIMENSIONAL EQUATIONS FOR A SHELL.......................................................... 399

4.2.1 DOUBLY-CURVED SHELLS OF REVOLUTION ............................................................ 401

4.2.2 CONICAL SHELL ...................................................................................................... 406

4.2.3 CIRCULAR CILINDRICAL SHELL ............................................................................... 407

4.2.4 SINGLY-CURVED SHELLS OF TRANSLATION ............................................................ 407

4.2.5 CIRCULAR PLATE .................................................................................................... 408

4.2.6 RECTANGULAR PLATE ............................................................................................. 409

5 DYNAMIC ANALYSIS: FREE VIBRATIONS

5.1 FREE VIBRATION ANALYSIS ........................................................................................... 412

5.1.1 MOTION EQUATIONS AND BOUNDARY CONDITIONS ............................................... 412

5.1.1.1 Discretization of the equations of motion ........................................................ 426

5.1.2 SOLUTION OF THE EIGENVALUE PROBLEM .............................................................. 439

5.2 NUMERICAL APPLICATIONS ........................................................................................... 441

6 STATIC ANALYSIS: STRESS AND STRAIN RECOVERY

6.1 STATIC ANALYSIS ........................................................................................................... 485

6.1.1 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS ......................................... 485

6.1.1.1 Discretization of the equilibrium equations ..................................................... 494

Index

Mechanics of Laminated Composite Doubly-Curved Shell Structures IX

6.1.2 STATIC PROBLEM SOLUTION ................................................................................... 508

6.2 STRAIN AND STRESS RECOVERY .................................................................................... 510

6.2.1 RECOVERY OF THE SHEAR STRESSES ....................................................................... 510

6.2.2 RECOVERY OF THE NORMAL STRESS ....................................................................... 510

6.2.3 RECOVERY OF THE STRAINS .................................................................................... 515

6.2.4 CORRECTION OF THE MEMBRANE STRESSES: REMOVAL OF THE STRESS

PLANE STATE ......................................................................................................... 520

6.3 NUMERICAL APPLICATIONS ........................................................................................... 521

7 THEORY OF COMPOSITE THIN SHELLS

7.1 KIRCHHOFF-LOVE THEORY ........................................................................................... 564

7.1.1 FUNDAMENTAL ASSUMPTIONS ................................................................................ 564

7.1.2 KINEMATIC MODEL ................................................................................................. 565

7.1.3 STRAIN CHARACTERISTICS ...................................................................................... 566

7.1.4 STRESS RESULTANTS ............................................................................................... 568


7.1.6 SHELLS OF REVOLUTION ......................................................................................... 571




7.1.7 MAIN SHELL STRUCTURES ...................................................................................... 573

7.1.7.1 Doubly-curved shells of revolution .................................................................. 574

7.1.7.2 Singly-curved shells of revolution ................................................................... 575

7.1.7.2.1 Conical shell ............................................................................................... 575

7.1.7.2.2 Circular cylindrical shell ........................................................................... 577

7.1.7.3 Singly-curved shell of translation .................................................................... 577

7.1.7.4 Degenerate shells .............................................................................................. 579

7.1.7.4.1 Circular plate ............................................................................................. 579

7.1.7.4.2 Rectangular plate ....................................................................................... 579

7.2 MEMBRANE THEORY ...................................................................................................... 581

Index

F. Tornabene, N. Fantuzzi X

7.2.1 KINEMATIC MODEL ................................................................................................. 581




7.2.5 MEMBRANES OF REVOLUTION ................................................................................ 584




7.2.6 MAIN SHELL STRUCTURES ACCORDING TO MEMBRANE THEORY ........................... 586

7.2.6.1 Doubly-curved membranes of revolution ........................................................ 586

7.2.6.2 Singly-curved membranes of revolution .......................................................... 587

7.2.6.2.1 Conical membrane ..................................................................................... 587

7.2.6.2.2 Circular cylindrical membrane .................................................................. 588

7.2.6.3 Singly-curved membranes of translation .......................................................... 589

7.2.6.4 Degenerate membranes .................................................................................... 590

7.2.6.4.1 Circular membrane .................................................................................... 590

7.2.6.4.2 Rectangular membrane .............................................................................. 590

7.2.7 SPHERICAL CAP SUBJECTED TO SELF-WEIGHT LOAD ............................................. 591

8 THEORY OF COMPOSITE ARCHES AND BEAMS

8.1 TIMOSHENKO THEORY ................................................................................................... 594

8.1.1 KINEMATIC MODEL OF THE ARCH ........................................................................... 594




8.1.4.1 Timoshenko straight beams .............................................................................. 598

8.2 EULER-BERNOULLI THEORY ......................................................................................... 599

8.2.1 KINEMATIC MODEL OF THE ARCH ........................................................................... 599



Index

Mechanics of Laminated Composite Doubly-Curved Shell Structures XI


8.2.4.1 Euler-Bernoulli straight beams ......................................................................... 602

9 GENERAL SHELL THEORY

9.1 SHELL THEORY: CURVATURE EFFECTS ........................................................................ 606

9.1.1 KINEMATIC MODEL ................................................................................................. 606

9.1.2 STRAIN COMPONENTS ............................................................................................. 607



9.1.4.1 Generilized vector of external actions .............................................................. 618

9.1.4.1.1 External actions at the bottom and top shell surfaces ............................... 619

9.1.4.2 Equations of motion via Hamilton principle .................................................... 621

9.1.4.3 Fundamental equations ..................................................................................... 634

9.1.4.4 Shells resting on a nonlinear elastic foundation and subjected to viscous

forces ................................................................................................................ 637

9.1.4.5 Shells subjected to seismic action .................................................................... 642

9.2 WINKLER-PASTERNAK FOUNDATION FFFECT .............................................................. 644

9.2.1 FREE VIBRATION ANALYSIS OF DOUBLY-CURVED SHELLS ..................................... 645

9.2.2 STATIC ANALYSIS OF DOUBLY-CURVED SHELLS ON NONLINEAR FOUNDATIONS ... 656

10 LAMINATED COMPOSITE PLATES OF ARBITRARY SHAPE

10.1 STRONG FORMULATION FINITE ELEMENT METHOD .................................................. 673

10.1.1 GENERAL ASPECTS OF THE TECHNIQUE ................................................................ 676

10.1.2 MAPPING TECHNIQUE FOR IRREGULAR DOMAINS ................................................. 680

10.1.3 BOUNDARY CONDITIONS IMPLEMENTATION ......................................................... 689

10.1.4 ASSEMBLY PROCEDURE ........................................................................................ 697

10.2 NUMERICAL APPLICATIONS ......................................................................................... 700

10.2.1 ISOTROPIC SQUARE PLATE: STATIC PROBLEM ....................................................... 701

10.2.2 ISOTROPIC SQUARE PLATE: FREE VIBRATION ....................................................... 705

Index

F. Tornabene, N. Fantuzzi XII

10.2.3 ARBITRARILY SHAPED ISOTROPIC PLATES ............................................................ 714

10.2.4 COMPOSITE PLATES OF ARBITRARY SHAPE ........................................................... 718

10.2.5 ARBITRARILY SHAPED CRACKED COMPOSITE PLATES .......................................... 725

10.2.6 FUNCTIONALLY GRADED DEGENERATE PLATES ................................................... 731

10.2.7 FUNCTIONALLY GRADED ARBITRARILY SHAPED CRACKED PLATES ..................... 737

10.2.8 STRESS RECOVERY OF ARBITRARILY SHAPED PLATES .......................................... 742

BIBLIOGRAPHY ...................................................................................................................... 763

Mechanics of Laminated Composite Doubly-Curved Shell Structures XIII

Preface

This manuscript comes from the experience gained over ten years (2003-2013) of study

and research on Laminated Composite Doubly-Curved Shell Structures and on the

Generalized Differential Quadrature Method. The present book was born when Professor

Viola gave to Tornabene the book by Kraus (Thin Elastic Shells, 1967) and the book by

Markuš (The Mechanics of Vibrations of Cylindrical Shells, 1988). After that life episode,

Tornabene started to study the interesting world of shell structures, concluding his studies at

the University of Bologna in 2003 with the Master Thesis (in Italian) entitled: Dynamic

Behavior of Cylindrical Shells: Formulation and Solution. After that, he finished in 2007 his

PhD in Structural Mechanics at the same University with the PhD Thesis (in Italian) entitled:

Modelling and Solution of Shell Structures Made of Anisotropic Material. During these years

Tornabene has met Dr. Nicholas Fantuzzi in occasion of his three level degrees at the

University of Bologna. In fact, Tornabene was the co-advisor for the three theses that Dr.

Fantuzzi has discussed in his student carrier. Finally, Tornabene became Assistant Professor

at the University of Bologna in 2012 and he published the book (in Italian) entitled:

Mechanics of Shell Structures Made of Composite Materials. The Generalized Differential

Quadrature Method.

The book by Tornabene represents the first manuscript in Italian language that treats the

theoretical aspects about laminated composite shell structures using the differential geometry

and that exposes the recovery procedure that allows to evaluate the stresses and the strains

through the thickness of a doubly-curved shell structure. It is also the first book that presents

to the Italian audience the Differential Quadrature Method and uses this methodology to

solve the governing equations of laminated composite doubly-curved shell structures.

Furthermore, the three fundamental aspects that characterize the book by Tornabene are two

theoretical and one numeric. The first one is the theory considered for studying shell

structures: the First-Order Shear Deformation Theory (FSDT). The second one is the use of

the Differential Geometry as a powerful tool for describing the shell reference surface. In fact,

a huge number of reference surfaces, useful to analyze various shell structures, has been

collected by Tornabene in his book. Finally, the third aspect is the numerical technique, called

Preface

F. Tornabene, N. FantuzziXIV

Generalized Differential Quadrature Method. This method allows to approximate the

derivatives of geometrical quantities and to solve the system of differential shell equations.

After the previous historical events, it is possible to introduce the present book that

represents the translation and the generalization of the book by Tornabene for the worldwide

audience. In particular, the present manuscript was written as an attempt to show, in an easy

way, the theoretical aspects of doubly-curved composite shell structures. The present volume

is aimed at Master degree and PhD students in structural and applied mechanics, as well as

experts in these fields. Furthermore, it shows some basic and advanced computational aspects

using non-standard numerical techniques.

The title, Mechanics of Laminated Composite Doubly-Curved Shell Structures. The

Generalized Differential Quadrature Method and the Strong Formulation Finite Element

Method, illustrates the themes followed in the present volume. The main aim of this book is to

analyze the static and dynamic behavior of moderately thick doubly-curved shells made of

composite materials applying the Differential Quadrature (DQ) technique and a new

numerical decomposition technique based on the strong formulation of the shell problem.

In fact, this book presents a general approach for studying doubly-curved laminated

composite shell structures solved using a numerical methodology based on the strong

formulation. The main reason for presenting this book to the engineering community is to

review and extend the literature, about shell theories, that appeared in the last seventy years.

Furthermore, the innovative aspects solved in this volume are the free vibration analysis and

the stress recovery procedure applied to doubly-curved shell structures.

The present volume is divided into ten chapters, in which static and dynamic analyses of

several structural elements are provided in detail. Furthermore, the results of the adopted

numerical technique are presented for several problems such as different loading and

boundary conditions.

In the first chapter the mathematical fundamentals regarding the Generalized Differential

Quadrature (GDQ) Method are exposed. In particular, the weighting coefficient calculations

and the most used discretizations are illustrated. The differential quadrature based on

Lagrange polynomials (Polynomial Differential Quadrature) and the one based on the

expansion in Fourier series (Harmonic Differential Quadrature) are described together with

other kinds of collocation techniques.

Preface

Mechanics of Laminated Composite Doubly-Curved Shell Structures XV

Starting from the Differential Geometry, fundamental tool for the analysis of the structures

at issue, the second chapter presents the Theory of Composite Laminated Shell Structures. In

the theoretical discussion the displacement field associated to the Reissner-Mindlin theory,

also known as “First-order Shear Deformation Theory” (FSDT), is considered. Once the

kinematic equations and the constitutive equations are introduced, the indefinite equilibrium

equations and the natural boundary conditions are deducted through the Hamilton principle.

The equations of doubly-curved shells are worked out and summarized in the scheme of

physical theories and specialized to structures of revolution.

As far as the constitutive equations are concerned, particular attention is given to

composite materials due to the increasing development in several structural engineering areas.

The scientific interest in these materials, that have the high makings of application, suggested

the static and dynamic analysis of composite shell structures. A new class of composite

materials, recently introduced in literature, is also taken into account. As it is well-known,

laminated composite materials are affected by inevitable problems of delamination due to the

presence of interfaces where different materials are in contact. On the contrary, “Functionally

Graded Materials” (FGMs) are characterized by a continuous variation of the mechanical

properties, such as the elastic modulus, material density and Poisson ratio, along a particular

direction. This feature is obtained by varying gradually, along a preferential direction, the

volume fraction of the constituent materials with appropriate industrial processes. Therefore,

FGMs are non-homogeneous materials, typically composed of metal and ceramic.

Starting from the analysis of shells of translation and doubly-curved shells of revolution, in

the third chapter the fundamental equations of Main Shell Structures are presented. In this

chapter it is shown how to carry out, through simple geometric relations, the governing

equations of the elastic problem of conical and cylindrical shells, circular and rectangular

plates and translational shells with a generic profile from the equations of doubly-curved

shells of revolution.

In the fourth chapter the 3D Elasticity Equations in Orthogonal Curvilinear Coordinates

are presented. They are the basis for a correct recovery of the stress and strain states through

the shell thickness. The recovery procedure is necessary because certain effects, due to the

transition from a three-dimensional theory to a two-dimensional one, are neglected and this is

done to reduce the computational cost of the structural analysis. This simplification of the

three-dimensional theory to an engineering theory is due to the introduction of suitable

Preface

F. Tornabene, N. FantuzziXVI

assumptions that limit the applicability of these theories within an appropriate validity range.

The three-dimensional equations in curvilinear orthogonal coordinates are worked out through

the Hamilton principle.

The book continues with the fifth and the sixth chapter, that show the numerical results

obtained for different types of structures. The results of the Dynamic Analysis (Free

Vibrations) and of the Static Analysis (Stress and Strain Recovery) of main composite shell

structures are presented. The effects of the mechanical properties on the vibration frequencies

and the stress and strain fields are illustrated.

In addition, for different structures and lamination schemes used in the numerical analyses,

the characteristics of convergence and stability of GDQ method are presented. Finally, the

GDQ numerical solution is compared, not only with literature results, but also with the ones

supplied and obtained through the use of different structural codes.

The sixth chapter includes a special emphasis on the a posteriori shear and normal stress

recovery procedure. Using the GDQ method, these quantities are computed from the two-

dimensional engineering solution, through numerical integration along the thickness of the

three-dimensional elasticity equations, carried out in the fourth chapter. Several examples

show the results of the recovery procedure of the stress state. Finally, also the strain state is

recovered through the use of the constitutive equations. Thus, the final results are useful for

structural design in order to avoid delamination problems in composite structures.

In the seventh chapter the Theory of Composite Thin Shells is derived in a simple and

intuitive manner from the theory of moderately thick shells developed in the second chapter.

In particular the Kirchhoff-Love Theory and the Membrane Theory for composite shells are

shown.

The eighth chapter exposes the Theory of Composite Arches and Beams. In particular, the

equations of the Timoshenko Theory and the Euler-Bernoulli Theory, with and without

curvature, are directly deducted from the equations of singly-curved shells of translation and

of plates.

The ninth chapter presents the so-called General Shell Theory in which the curvature

effect is embedded into the FSDT kinematic model. This effect is reflected into the stress

resultants and strain characteristics of the model. Due to these considerations the stress

resultants directly depend on the geometry of the structure in terms of curvature coefficients

and the hypothesis of the symmetry of the in-plane shearing force resultants and the torsional

Preface

Mechanics of Laminated Composite Doubly-Curved Shell Structures XVII

or twisting moments is not valid. Furthermore, several numerical applications are presented in

the chapter at hand for the sake of completeness.

The volume is completed by the tenth chapter in which an advanced version of the GDQ

method is presented in order to analyze Laminated Composite Plates of Arbitrary Shape.

Since the GDQ method can be applied to single domains without material and geometric

discontinuities, a domain decomposition technique has to be employed in order to study

arbitrarily shaped composite structures. The physical domain is divided into several sub-

domains according to the problem geometry and the mapping technique is applied in order to

map a generic element in Cartesian coordinates into a master element, where the GDQ

method can be applied. Moreover, the connection among these elements has been dealt with

continuity (compatibility) conditions. For the sake of generality this method has been termed

Strong Formulation Finite Element Method (SFEM), because it joins the high accuracy of the

strong formulation with the versatility of the domain decomposition, typical of the finite

element method.

This book is intended to be a reference for experts in structural, applied and computational

mechanics. It can be also used as a text book, or a reference book, for a graduate or PhD

courses on plates and shells, composite materials, vibration of continuous systems and stress

recovery of the previous structures. Finally, the present book also have the same audience of

the book by Professor Harry Kraus (1967). Thus, using his words: “The” present “book is

aimed primarily at graduate students at the intermediate level in engineering mechanics,

aerospace engineering, mechanical engineering and civil engineering, whose field of

specialization is solid mechanics. Stress analysts in industry will find the” present “book a

useful introduction that will equip them to read further in the literature of solutions to

technically important shell problems, while research specialists will find it useful as an

introduction to current theoretical work. This volume is not intended to be an exhaustive

treatise on the theory of thin” and thick “elastic shells but, rather, a broad introduction from

which each reader can follow his own interests further”. In addition, it is opinion of the

authors that the present volume represents the continuation and the generalization of the work

begun by Kraus in 1967.

Finally, in order to help the reader the following flow chart shows relations between the

book chapters and represents a useful scheme for reading the present manuscript.

Preface

F. Tornabene, N. FantuzziXVIII

Acknowledgements by the first author

I take this opportunity to thank to Professor Erasmo Viola, who always transmitted

enthusiasm, knowledge and intellectual curiosity, without them this work would not be

possible. I want to extend a special thanks to Nicholas Fantuzzi for his patient and precious

work of translation and proofreading and for his contribution in the first, ninth and tenth

chapters.

I extend a special thanks to all the students who have contributed to create the present

manuscript, over the years: Ilaria Ancona, Silvia Mercuri, Andrea Benedetti, Alessia Fusaro,

Filippo Fanti, Paolo Pastore, Ilaria Ricci, Nicholas Fantuzzi, Giancarlo De Vittorio, Luca Fini

and Santa Tornabene (my sister). To them I intend to extend my gratitude for the enthusiasm

and dedication shown during the development of their thesis.

Finally, I thank my family for always supported me in all these years of study and research.

I thank my sisters Agata and Santa, and in particular I dedicate this work to my father

Rosario, who prematurely died, and to my mother Giuseppa.

Finally, I would like to conclude by recalling an incident from the life of Saint Thomas

Aquinas, who, after having written several volumes of his Summa Theologiae (unfinished)

and near the end of his earthly life, he said to his fellow:

‘‘Reginaldo, I cannot’’ (write anymore), ‘‘because everything I’ve written is like straw for

me, in comparison to what now is revealed to me by God ’’.

Chap. 1

Chap. 2

Chap. 3

Chap. 5 Chap. 4

Chap. 6

Chap. 7

Chap. 8

Chap. 9 Chap. 10

Preface

Mechanics of Laminated Composite Doubly-Curved Shell Structures XIX

Due to the fact that no one owns the truth, as indicated by the words of Saint Thomas

Aquinas, but everyone must do his best to find out at least a part, I would like to quote the

following passage from the Divine Comedy:

‘‘Credette Cimabue nella pintura

tener lo campo, ed ora ha Giotto il grido,

sì che la fama di colui è oscura’’.

‘‘Once Cimabùe thought that he would hold

the filed in painting, yet the cry is all

for Giotto now, hence that one’s fame is dark’’.

Dante Alighieri, Purgatorio XI

Acknowledgements by the second author

When Francesco asked me for help in his book’s translation, I immediately agreed with

enthusiasm, because I realized that it would be a great opportunity to enrich my personal

knowledge. Indeed it was. I could focus more on some topics that I did not have enough time

to study during my PhD. It was immediately clear to me that the original version was given

birth after years of passionate and deep study on the topic. Only now I can merely understand

that writing a book is an all-encompassing and time-consuming project and I hope that this

book will have the success that deserves.

All those people, who change the lives of others by giving their best on what they believed,

deserve a special mention. In particular, I thank first Professor Erasmo Viola whose guidance

and encouragement was vital for addressing my studies as graduate student and my PhD

research. I also thank Professor Francesco Tornabene who represents an example for me to

follow and increases my interests in computational and applied mechanics. To all my

extended family who gave me their support in a way or another. Finally, to the ever-faithful

Ilaria who always steadfastly supported and encouraged me all these years together.

Francesco Tornabene

Nicholas Fantuzzi

Bologna, December 31, 2013

Preface

F. Tornabene, N. FantuzziXX

Di.Qu.M.A.S.P.A.B. Project and Software

Recently, after more than ten years of publications produced by their colleagues and

themselves, the authors decided to make available on line a summary of all their work

entitled: Differential Quadrature for Mechanics of Anisotropic Shells, Plates, Arches and

Beams (the acronyms of which is Di.Qu.M.A.S.P.A.B.). The project explanation and the

software are available at the following site: http://software.dicam.unibo.it/diqumaspab-

project, where the reader can find a free application that can solve laminated composite

doubly-curved shell structures using the Generalized Differential Quadrature method, the

Differential Geometry tool and different kinds of shell theories: from the First-Order Shear

Deformation Theory (FSDT) to various Higher-Order Shear Deformation Theories.

Moreover, the main background of this code is also presented by the book (in Italian) by

Tornabene entitled: Mechanics of Shell Structures Made of Composite Materials. The

Generalized Differential Quadrature Method, and more than fifty papers published on

international journals. All the results presented in the book by Tornabene and in the present

manuscript are completely repeatable using the software at hand.

The authors consider this project particularly significant, because it took a whole year of

hard work in order to come out with the first public version of the software.

Mechanics of Laminated Composite Doubly-Curved Shell Structures 763

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Carried out at DICAM Department

Headquarter of ‘Scienza delle Costruzioni’ (Structural Mechanics)

of Alma Mater Studiorum - University of Bologna, Italy.

Bologna, 31 Dicembre 2013

© F. Tornabene, N. Fantuzzi, 2013

[email protected]

[email protected]

DICAM

Mechanics of Laminated Composite Doubly-Curved Shell Structures

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