Mechanics of Laminated Composite Doubly-Curved Shell Structures. The Generalized Differential Quadrature Method and the Strong Formulation Finite Element Method PDF

Preview

Summary

Francesco Tornabene Nicholas Fantuzzi Mechanics of Laminated Composite Doubly-Curved Shell Structures The Generalized Differential Quadrature Method DiQuMASPAB Project and Software and the Strong Formulation Finite Element Method This manuscript comes from the experience gained over ten years of stu...

Description

Francesco Tornabene

Nicholas Fantuzzi

DiQuMASPAB Project and Software

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

Mechanics of Laminated Composite Doubly-Curved Shell Structures

This manuscript comes from the experience gained over ten years of study and research on shell structures and on the Generalized Differential 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 application 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, characterized by a continuous variation of the mechanical 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.

Mechanics of Laminated Composite Doubly-Curved Shell Structures

Francesco Tornabene Nicholas Fantuzzi Euro 95,00 www.editrice-esculapio.it

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

All rights reserved. The reader can photocopy this publication for his personal purpose within the limit of 15% of the total pages and after the payment to SIAE of the amount foreseen in the art. 68, comma 4, L. 22 April 1941, n. 663, that includes the agreement reached among SIAE, AIE, SNS and CNA, CONFARTIGIANATO, CASA, CLAAI, confcommercio, confesercenti on December 18, 2000. For purposes other than personal, this publication may be reproduced within the limit of 15% of the total pages with the prior and compulsory permission of AIDRO, via delle Erbe, n. 2, 20121 Milano, Telefax 02-80.95.06, e-mail: [email protected]

40131 Bologna - Via U. Terracini 30 - Tel. 051-63.40.113 - Fax 051-63.41.136 www.editrice-esculapio.it

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

About the Authors Francesco 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 Jodi for 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: Unified Formulation 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

Mechanics of Laminated Composite Doubly-Curved Shell Structures

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

F. Tornabene, N. Fantuzzi

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 Mechanics of Laminated Composite Doubly-Curved Shell Structures

III

Index 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 G -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 IV

F. Tornabene, N. Fantuzzi

Index 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 G -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

Mechanics of Laminated Composite Doubly-Curved Shell Structures

V

Index 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 ........................................................................................