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 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. 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 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 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.

F. Tornabene, N. Fantuzzi (2014). Mechanics of Laminated Composite Doubly-Curved Shell Structures. The Generalized Differential Quadrature Method and the Strong Formulation Finite Element Method. Bologna : Esculapio.

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

#####
*TORNABENE, FRANCESCO;FANTUZZI, NICHOLAS*

##### 2014

#### Abstract

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 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. 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 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 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.