Anisotropic Doubly-Curved Shells - Higher-Order Strong and Weak Formulations for Arbitrarily Shaped Shell Structures

The title, Anisotropic Doubly-Curved Shells, illustrates the themes followed in the present volume. The main aim of this book is to analyze the static and dynamic behavior of doubly-curved shells made of anisotropic materials applying the Differential Quadrature (DQ) and Integral Quadrature (IQ) techniques. The major structural theories for the analysis of the mechanical behavior of doubly-curved shell structures are presented in depth in the volume. In particular, the strong and weak formulations of the corresponding governing equations are discussed and illustrated. In addition, several numerical applications are developed to support the theoretical models. The book is made up of eight chapters, in which both the structural models and the numerical techniques are examined. The first chapter includes the mathematical fundamentals of the Differential Quadrature (DQ) method. In particular, the Generalized Differential Quadrature (GDQ) method developed by Shu is discussed. He had the aim of improving the DQ technique presented by Bellman at the beginning of the 1970s. In particular, the weighting coefficient computations, used in the derivative approximation of any order, will be shown. Thus, the differential quadrature method based on the Lagrange polynomials (Polynomial Differential Quadrature or PDQ) and the one based on the Fourier series (Harmonic Differential Quadrature or HDQ) are described. As it is well-known, differential quadrature methods, available in the literature, differ from the choice of the approximating functions (or basis functions) and for the type of grid distributions, which locate discrete points on the domain where the derivative of a function has to be evaluated. For this reason, a general approach to the differential quadrature is proposed. The weighting coefficients for different basis functions and grid distributions are determined. Furthermore, the expressions of the principal approximating polynomials and grid distributions, available in the literature, are shown. Besides the classic orthogonal polynomials, a new class of basis functions, which depend on the radial distance between the discretization points, is presented. They are known as Radial Basis Functions (or RBFs). The general expressions for the derivative evaluation can be utilized in the local form to reduce the computational cost. From this concept the Local Generalized Differential Quadrature (LGDQ) method is derived. Moreover, a procedure for numerical integration, based on the weighting coefficients utilized for the differential quadrature, is illustrated. This is done because several numerical applications require the evaluation of integrals in their formulation. Thus, the Generalized Integral Quadrature (GIQ) technique, introduced by Shu following the same concept of the GDQ method, is shown. This method can be used employing several basis functions, without any restriction on the point distributions for the given definition domain. To better underline these concepts some classical numerical integration schemes are reported, such as the trapezoidal rule or the Simpson method. The general approach to the integral and differential quadrature can be extended directly to the multidimensional case. For this reason, a simple and compact mathematical formulation, for the computation of the weighting coefficients and the numerical evaluation of derivatives and integrals for a two-dimensional space, is presented. An alternative approach based on Taylor series is also illustrated to approximate integrals. This technique is named as Generalized Taylor-based Integral Quadrature (GTIQ) method. Finally, the accuracy of the GDQ and GIQ methods is demonstrated through few numerical applications. The second chapter of the book, instead, introduces the bases of the Differential Geometry, a fundamental tool for the analysis of the doubly-curved structures at issue. In fact, it allows to obtain those geometric parameters that are involved in the writing of shell governing equations. Then, the chapter continues analyzing the main shell structures that are commonly employed in engineering. For each considered doubly-curved, singly-curved or degenerate shell structure, the main geometric features are presented. A peculiar procedure is also introduced to deal with distorted domains. For this purpose, a proper coordinate transformation is presented to define arbitrarily shaped curved surfaces. This technique is known as mapping procedure. In other words, this approach aims to convert a regular domain, described by the principal coordinates, into a distorted element. Therefore, the original structural problem is transferred in the computational domain, known as parent space, which is described by the so-called natural coordinates. In this book, a mapping procedure based on the use of Non-Uniform Rational Basis Splines (NURBS) is presented. Such techniques is known as Isogeometric Mapping. In the third chapter, the 3D Elasticity Equations in Principal Curvilinear Coordinates are presented for shell structures. They represent the basis for the development of several engineering theories, which allow to analyze the mechanical behavior of various doubly-curved shell structures. In this chapter, the kinematic equations are obtained for a generic three-dimensional solid described in a principal curvilinear coordinate reference system. Then, the strain components are related to the corresponding stress components by means of the generalized Hooke laws for several kinds of materials. The indefinite equilibrium equations, as well as the natural boundary conditions, are eventually worked out by the Hamilton principle. The kinematic, constitutive and dynamic equilibrium equations are modified through simple geometric considerations in order to obtain the whole set of three-dimensional equations which governs the mechanical behavior of different shell structures. By substituting the kinematic and the constitutive relations into the equilibrium equations, the behavior of a generic shell structure can be described through the indefinite equilibrium equations expressed in terms of generalized displacement components, which represent the degrees of freedom of the problem at issue. The relations that are deduced through all these substitutions are known as fundamental equations and include the three aspects of the elastic problem, that are kinematic, constitutive and dynamic equilibrium, of thick and moderately thick composite shells, within a unique fundamental system. These 3D Elasticity equations contemplate a three-dimensional structural model without taking into account any hypothesis about the displacement field. In the third chapter, the three-dimensional problem is reduced to a two-dimensional problem defined on the shell middle surface, according to the principles of the Equivalent Single Layer (ESL) approach, by means of a proper kinematic model. According to the choice of the kinematic model in hand, several Higher-order Shear Deformation Theories (HSDTs) are obtained. This approach, as it will be examined in depth in the following, represents a peculiar technique that allows to consider and study several higher-order kinematic models in a unified manner. In fact, the expansion order of the three dimensional displacements is taken as a free parameter and it can include both several thickness functions and the so-called Zig-Zag effect (known also as Murakami’s function). Inserting this general displacement field into the three-dimensional kinematic equations, these relations can be expressed in terms of generalized strain characteristics, which are defined on the shell reference surface. As far as the constitutive equations are concerned, particular attention is given to anisotrpic 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. Their overall mechanical properties are obtained by means of different approaches. In particular, in the present work the classic theory of mixtures is compared to the Mori-Tanaka scheme for the computation of these mechanical properties. Other classes of advanced and innovative constituents, such as Carbon Nanotubes (CNTs) reinforced media and Variable Angle Tow (VAT) composites are illustrated. The relations which express the dynamic equilibrium of shell structures are obtained once again through the Hamilton variational principle. In order to transform the initial three-dimensional elastic problem in a two-dimensional one, the internal actions in terms of stress resultants for each order of kinematic expansion are introduced. It should be noticed that the Hamilton principle allows to consider also the effect of non-conservative forces. Thus, it is possible to introduce the structural damping into the dynamic equilibrium equations. The procedure for evaluating the damping coefficients is performed according to the Rayleigh proportional damping technique. In addition, the current approach allows to take into account different kinds of external loads. The surface forces applied on the external shell surfaces and the volume forces which are applied in each point of the three-dimensional solid, as well as the seismic excitation and the effect of the non-linear elastic foundation, are converted in statically equivalent loads acting on the middle surface of the shell. Finally, the case of rotating shells is also included in the treatise, by introducing a general approach for arbitrary rotations. Generally speaking, two formulations of the same system of governing equations can be developed, which are respectively the strong and weak (or variational) formulations. Once the governing equations that rule a generic structural problem are obtained, together with the corresponding boundary conditions, a differential system is written. In particular, the Strong Formulation (SF) of the governing equations is presented in the fourth chapter. The differentiability requirement, instead, is reduced through a weighted integral statement if the corresponding Weak Formulation (WF) of the governing equations is developed. Thus, an equivalent integral formulation is presented in the fifth chapter, starting directly from the previous one. In particular, the formulation in hand is obtained by introducing a higher-order Lagrangian approximation of the degrees of freedom of the problem, which consists in the nodal displacements defined on the shell middle surface. The theoretical framework is based on a higher-order kinematic expansion of the displacement field, as shown in the fourth chapter. On the other hand, simpler structural models are considered and illustrated in the sixth chapter. In particular, the strong and weak forms are presented for one-dimensional and two-dimensional problems. As far as one-dimensional models are concerned, a unified approach is developed to deal with several kinds of mechanical problems. The cases of the rod and the Euler-Bernoulli beam are illustrated in depth. On the other hand, the membrane, the Kirchhoff-Love plate, the plane elasticity and the Reissner-Mindlin plate are the two-dimensional problems considered in the chapter. All these structural models are described through the same analytical formulation, highlighting the fact that the various matrices and operators assume different meanings according to the structure under consideration. The book continues with the seventh chapter, wherein the obtained numerical results are illustrated for several structural components. The dynamic analysis (free vibrations) and static analysis of composite shell structures are presented. The effect of the variation of the mechanical properties on the vibration frequencies and the stress field, for several structural theories shown in the previous chapters, are illustrated. Besides different geometries and different lamination schemes used in the numerical analyses, the convergence and stability trends of this technique are presented. Finally, the DQ and IQ numerical solutions are compared to results from the literature and the same obtained from structural code programs. In this chapter, a posteriori recovery procedure is introduced to compute the shear and normal stresses from the two-dimensional solution through a numerical integration along the thickness via DQ method of the three-dimensional elasticity equations shown in the previous chapters. A lot of examples show the results of this recovery procedure of the stress state. All these results are useful in the structural design to avoid delamination problems in composite materials. All the results presented in the book are obtained through the DiQuMASPAB software, acronym of “Differential Quadrature for Mechanics of Anisotropic Shells, Plates, Arches and Beams”. This code is generated using MATLAB environment which implement the considered theoretical formulations and aims to analyze the static and dynamic analyses of various shell structures. The need of studying arbitrarily shaped domains or characterized by mechanical and geometrical discontinuities leads to the development of new numerical approach that divide the structure in finite elements. Then, the strong form or the weak form of the fundamental equations are solved inside each element. The fundamental aspects of this technique, which the authors defined respectively Strong Formulation Finite Element Method (SFEM) and Weak Formulation Finite Element Method (WFEM), are presented in the eighth chapter. The starting points of these approaches are the differential and integral quadrature methods, introduced in the first chapter. The mapping technique, used also in the finite element methods, transforms an arbitrarily shaped element into a regular element (computational element). In other words, the SFEM and WFEM denote two numerical procedures that divide the physical domain in finite elements and use the differential or integral quadrature methods to solve the strong or weak forms of the equations inside each element, mapped into the computational space. Several numerical applications regarding the static and dynamic behavior of arches, beams, plates, membranes, plane stress and strain states are reported to study the convergence and stability characteristics and reliability of these numerical techniques. Finally, a recap of the main operations that can be performed between matrices and vectors is presented in the appendix, in which some concepts, operations and results of matrix algebra employed in the previous chapters are discussed. 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, anisotropic and composite materials, vibration of continuous systems and stress recovery of the previous structures. Finally, the present book has also 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.