CHAPTER Introduction 1.1 Background on Finite Element Method (FEM) The Brachistochrone problem was one of the earliest problems in calculus of variations posted by Johann Bernoulli in Acta Eroditorum in 1696. The problem is to find the minimum time trajectory described by an object moving from one point to another in a constant uniform gravitational field. In his paper, he remarked that “Nature always tends to act in the simplest way” when explaining the solution of the Brachistochrone problem. He obtained the solution by dividing the plane into strips and assumes that the particle follows a straight line in each strip. Hence, it is the nature of human mind to perceive the behavior of any complex mechanism in nature by first subdividing into smaller units (or elements) which can better represent the actual behavior and then reassemble the original system from such smaller units to study its overall behavior, in a natural way. There is a strong analogy between the nature of human mind and the concept of finite element method (FEM) which employs piecewise approximation functions to obtain solutions of physical problems. The finite element method (FEM) as it stands today has seen a gripping evolution since 1690. The idea of using piecewise approximating functions to solve the Brachistochrone problem posed by Bernoulli was given by Leibniz (1692). Introduction Schellbach (1851) presented the solution of the Plateau problem, by a piecewise linear function defined on a triangular mesh and the triangles were referred to as ‘elements’. Both the Brachistochrone and Plateau problems are regarded as problems involving calculus of variations. Leibniz and Schellbach used the concept of subdividing into smaller units to derive solutions for their respective problems. About fifty decades later, Ritz (1909) and Rayleigh (1911) used trial functions to approximate solutions for a minimization problem. Galerkin (1915) and Bubonov (Mikhlin, 1964) proposed an alternative form of the variational method that could be used to solve any differential equation unlike the Ritz method. In 1943, Courant proposed the use a piecewise linear function on a triangular mesh to approximate the solution of a variational problem. In early 1950s, there was a great thrust from Boeing to analyze aircraft structures. Turner (1956) presented a new approach which involved local approximations to partial differential equations and an assembly process. The modern FEM takes its point of departure in the research article of Turner. Later, the works of researchers like O.C. Zienkiewicz, KlausJürgen Bathe, Clough, T.R Hughes, M.A. Crisfield, T. Belytschko and J.N. Reddy in the area played an important role in the engineering development of FEM. It is rather difficult to point out an exact date of origin of FEM. The evolutionary path which FEM has traversed encompasses the joint efforts of mathematicians, physicists and engineers over a span of 150 years. In the present state, FEM has attained an unflinching pole position in analyzing complex engineering problems because of its versatility and mature development. In the FEM, the structural system is modeled by a set of appropriate finite elements that are interconnected at points called nodes. Introduction At this point, it would be appropriate to state the mathematical definition of finite element as given by Ciarlet (1978) A finite element is a triad defined by K = (W , R, S) where W is a domain in a real dimensional space Rd P is a space of polynomials on K having a dimension dim(P)=NP S = {L1 , L2 , … LNP} is a set of linear forms Li : P® Rd , (i=1,2 , NP) The domain of real dimensional space Rd may represent a one-dimensional, two dimensional or a three dimensional finite element based on the value of d (i.e. d = denotes one–dimensional elements, d = denotes triangles and quadrilaterals; d = denotes tetrahedra, bricks and prisms). The space of polynomials P has the degree of basis functions to be p and each basis function is composed of NP terms. The elements of S are used to derive shape or interpolation functions required for approximating the solution variables (field variables) inside a finite element (FE). By tuning the three main ingredients namely (W, R, S ) that constitute a finite element, we can have several versions of FE. In this thesis, we choose to tune P, which plays an auspicious role in the numerical accuracy of the finite element solution. Based on the degree of polynomial basis function used for approximating the field variables (e.g. displacements, stresses), there are two basic versions of finite elements. These two versions have been referred to as the h-version and the p-version (Babuska et al., 1981). In the h- version FEM, which has been the platform for the development of commercial software packages, the degree of polynomial basis function is kept low (say, linear or Introduction quadratic at most) and the mesh is refined until the prescribed accuracy of the solutions is achieved. Although the h- version of FEM seems to be computationally viable, the use of lower order shape functions in some problems involving singularities and high stress gradients may predispose the stress resultants to erroneous values which are unacceptable from a design standpoint. Even though one uses a highly refined mesh to model structures with free edges, reentrant corners, cutouts and highly skewed shapes where singularities manifest, the accuracy of stresses obtained by using lower-order finite elements is rather poor (Rank, et al. 1998). To cite an instance of the poor quality of stress resultant distributions obtained using the commonly used quadrilateral shell element S8R in ABAQUS, we present the stress resultants for Morley’s acute skew plate with simply supported edges and under a uniformly distributed load (Fig. 1.1). It can be seen that the stress resultant distributions are not smooth even with a very fine finite element mesh (Fig. 1.2) involving 26,166 degrees of freedom (d.o.f). It can be seen that the sharp corners of the Morley’s skew plate cause the mesh refinement algorithm to induce severe distortion to the S8R elements in the vicinity of the corners A and D. The distortion of the elements lead to non-smooth distribution of the stress resultants in the plate. Bending Moment Myy Twisting Moment Mxy Fig. 1.1 Stress resultant distributions obtained from ABAQUS–S8R shell element (degrees of freedom = 26,166) for the be uniformly loaded, Morley’s skew plate (Interior angle = 300 , thickness/length = 0.001, material properties: E1 =2.5´107 psi ; E2 =1´106 psi ; G12 = G13 =0.5 E2 ; G23 = 0.2 E2 ; n12 = 0.25) Introduction C A D B Fig. 1.2 S8R mesh design used in bending analysis of Morley’s skew plate In contrast to the h-version, the p-version of FEM which was initiated in the early 1980s relies on enhancing the degree of basis function to achieve the desired accuracy. Because of the higher order shape functions, there is continuity of the higher order derivatives of the displacement field, thereby enabling the p-version FEM to furnish better prediction of the stress-resultants and stresses. There exists a slight difference between the h and p versions of FEM and the h and p versions of convergence. While the two versions are interrelated, it can be said that in the former (i.e h and p versions of FEM) we categorize based on the maximum diameter h of the element tending to zero (h - version) and the degree of approximating polynomial p tending to infinity (i.e taking to high values). In the latter version (i.e h and p convergence versions), we refer to the asymptotic behaviour of the finite elements (h and p version FE) based on uniform or quasi uniform mesh refinements. There are two ways of developing a p-version finite element. They are the modal/hierarchical basis approach and the nodal basis approach. While the formal accuracy of solution obtained for a given degree ‘p’ of any polynomial basis employed in the nodal or modal approach is the same, the nodal basis approach renders straightforward visualization and interpretation of computed results. Introduction 1.2 Lite rature survey on applications of p-version elements The quest for a robust finite element, with the ability to predict stress resultants accurately for a wide class of practical problems involving stress singularities and be free of numerical problems, has triggered researchers to develop higher-order finite elements, both as separate elements or in the framework of the “p- version of the finite element method (p-FEM)”. Mathematical justifications showing the advantages of using higher-order approximations of the field variables (namely high accuracy, high convergence rate with coarser meshes and improved performance in handling stress singularity problems) have been reported by Babuska et al., 1981, Babuska and Szabo (1982), Babuska and Suri (1987). From the computational point of view, the choice of the basis function used for the approximation of field variables influences the stability and efficiency of the numerical procedures used to calculate the approximated solution. The inception of mathematical proofs demonstrating the exceptional performance of p-version elements led researchers to focus on the development of efficient basis functions for higher order elements or p-version elements. Peano (1976) proposed new families of C0 and C1 hierarchical interpolations over triangles which were complete up to any polynomial of degree p. Each new d.o.f for p ³ represents the pth derivative of the field variable. The classical hierarchical functions for quadrilaterals and hexahedra were introduced by Szabó and Babuška (1991) who used the Legendre polynomials due to their excellent sparsity and conditioning properties. However, it was not easier to formulate basis functions for triangles and tetrahedral that can have better condition numbers for increasing degree of polynomial basis p. Research related to the Introduction formulation of efficient basis functions for triangles and tetrahedral that improve the condition numbers and sparsity of the global matrices can be found in the works of Carnevali (1993), Sherwin and Karniadakis (1995), Webb and Abouchacra (1995) and Adjerid et al. (2001). A unified approach to construct h- and p-shape functions for quadrilaterals, hexahedral, triangles and tetrahedral based on the tensorial product of one-dimensional bases was presented by Bittencourt (2007) The formulation of efficient basis functions paved the way for the evolution of a wide class of p–version elements that were applied to solve several problems in structural mechanics. Following the work of Peano (1976), Wang et al. (1984) formulated a family of triangular finite elements of degree p ³ having C1 continuity and analyzed simply supported and clamped isotropic plates subjected to a central point load and uniformly distributed load. A brief survey until 1988 on various aspects of the p and hp version of the finite element method was given by Babuska (1988). Akhtar and Basu (1991) developed a p version plate finite element based on Reissner-Mindlin theory and analysed flat plates and membranes. Scapolla and Croce (1992) demonstrated the robustness of hierarchic quadrilateral higher order elements in overcoming shear locking problems in Reissner-Mindlin plates. Research pertaining to the study of bending analyses of isotropic and laminated composite plates using p- version finite elements can be found in the works of Ahmed and Basu (1993), Croce and Scapolla (1992), Pereira and Freitas (1996), Xenophontos, Kurtz et al. (2006). Bardell, Dunsdon et al. (1995), Beslin and Nicolas (1997), Houmat (2005), Ribeiro (2003; 2004; 2005; 2006; 2009), Smith (1995) , Reddy and Arcineiga (2006) employed quadrilateral p- Introduction version elements to study the vibration and buckling of plates and shells. Woo and Basu (1989) proposed a new hierarchic p-version quadrilateral cylindrical shell element and assessed its performance in a cylinder having symmetrically located rectangular cutouts. Liu and Surana (1992) presented a p-version geometrically nonlinear (GNL) formulation based on total Lagrangian approach for a three- node axisymmetric curved shell element and studied the load-deflection behaviour of a clamped spherical shell cap. They considered a hierarchical element displacement field with arbitrary polynomial orders in both longitudinal and transverse directions of the element. The approximation functions and the nodal variables for the element were derived directly from the Lagrange family of interpolation functions. Fish and Guttal (1997) developed an assumed strain formulation of p- version quadrilateral shell elements up to degree and proposed special quadrature schemes and adaptive selection of higher order modes for optimising the computational time. The development and application of p-version shell finite elements were made by Düster (2001), Basar and Hanskotter (2001; 2003), Arciniega and Reddy (2007). The application of p-version finite elements in solving elasto-plastic and dynamic nonlinear elasticity problems can be seen in the works of Düster and Rank (2001), Holzer and Yosibash (1996), Jeremic and Xenophontos (1999) and Woo et al. (2004). 1.3 Objectives and scope of research Although the aforementioned studies cover a wide range of engineering problems, there has been limited research showing the performance of pversion elements in the stress analysis of plated and shell structures using Introduction triangular elements. While the performance of most of the p-version plate and shell finite elements have been reported for well defined boundary conditions (e.g. clamped and simply supported edges), convergence of deflections and strain energy error norms, there are hardly any results available to provide a true representation of stress resultants for boundary value problems such as transverse shear force and twisting moment distributions along the free edge boundary and stress contours in the vicinity of a cutout or sharp corners. Furthermore, existing p-version finite elements are based on a modal/hierarchical approach and applied to a quadrilateral shape rather than a triangular shape. The reason is that the quadrilateral shape allows simple implementation of the shape functions and numerical integration in the FEM code. However, the quadrilateral element has some setbacks. For example, it cannot model arbitrarily shaped plated and shell structures (especially with sharp corners) very well and also it does not allow for easy meshing. Triangular elements are hence more adaptable for meshing complicated shapes of plates and shells. Although, many researchers have pointed out that quadrilateral elements have better accuracy as compared to triangular elements, their accuracies have not been compared in the context of higher order elements. One of the reasons for the poor performance of lower order triangular elements is due to the difficulty in achieving good interelement continuity of field variables using lower order shape functions. This problem can be circumvented by using enriched higher order shape functions that shows better interelement continuity. So far, relatively few studies have been made on p-version triangular elements (Wang et al. 1984; Houmat 2005). These studies are mainly confined to free vibration analysis of plates and Introduction shells and employ the modal basis approach. There are hardly any validation studies to establish the p-version triangular element for stress analysis. Although there can be different types of finite element formulations such as displacement based, stress based and mixed (hybrid) form, we shall adopt a displacement based formulation in our research due to the following reasons: 1. In a stress based formulation, the field variables (unknowns) are in terms of stresses. Even though one achieves accurate stresses using lower order finite elements, it is not easy to obtain accurate displacements (that are of much practical importance from serviceability standpoint) because the have to be determined from stresses through post-processing computations. 2. The implementation of boundary conditions in a stress based formulation and mixed formulation (which involves both stresses and displacements as unknowns) becomes difficult when dealing with nonlinear problems. Hence displacement based higher order formulations are more efficient due to the fact that one achieves both accurate displacements as well as stresses and it is easier to implement boundary conditions in problem involving nonlinearity. The objectives and scope of the present research are therefore to · Develop a higher order triangular plate/shell displacement element based on the nodal basis approach that is capable of predicting accurate stress resultants in structural problems involving high stress gradients and singularities in loading and boundary conditions. · Develop a family of higher order elements namely HT-M21, HT-TSDR45 and HT-LT-R45 based on shear deformable plate theories and 10 Introduction layer-wise plate theory for better prediction of stress resultants and interlaminar stress distributions in isotropic and laminated composite plates. These elements have equally spaced nodes inside its triangular domain. · Develop a spectral higher order triangular element with optimal location of nodes inside its domain and having an improved basis function for defining shape functions. · Develop a total Lagrangian continuum shell formulation of the two versions of higher order elements HT-CS and HT-CS-X; ‘X’ is associated with the optimal nodal configuration and improved shape functions of the spectral higher order element. · Compare the convergence characteristics of HT-CS and HT-CS-X elements in several challenging shell benchmark problems that exhibit asymptotic behaviors and finally employ the best performing HT-CS-X elements to further study the nonlinear behaviour of shells undergoing large deflections and moderate rotations. 1.4 Layout of the thesis This thesis is organized into six chapters. Chapter presents a prologue to the research topic considered in this thesis work. In Chapter 2, the formulation of a first order shear deformable higher order triangular element called HT-M45 is presented. The finite element geometry and the nature of shape functions obtained for equidistant nodal configurations are discussed. The optimal degree of polynomial basis required for obtaining accurate stress distributions is established and associated shape functions are 11 Introduction presented. Several numerical examples are presented to verify the validity, convergence and accuracy of the proposed element. The last section of the chapter tests the performance of HT-M45 elements in the presence of singularities such as point supports and re-entrant corners. It should be remarked that finite elements based on first order shear deformable plate theory cannot furnish accurate interlaminar transverse shear stress distributions when the aspect ratio t/L (thickness/length) of plates is greater than 0.1. To this end, in Chapter 3, we extend the formulation of higher order triangular element HT-M45 to account for transverse shear stress variations in the vicinity of stress concentrations. This is accomplished by developing two elements namely, HT-TSD-R45 and HT- LT-R45 which are based on a third order shear deformable theory of Reddy (1984) and a layerwise theory of Reddy (1987). The former requires no shear correction factors in its formulation and accurately captures the transverse shear stress resultants for thin and moderately thick plates (0.01< t/L < 0.25). Numerical examples of laminated composite plates involving circular cutouts are presented. The distributions of stress resultants obtained by employing a coarse mesh of HT- TSD-R45 elements in the vicinity of a cutout are given. Further the results are compared for its numerical efficiency and accuracy with some of the widely used lower order finite elements in ABAQUS package. HT-LT-R45 element specializes in capturing accurate transverse shear stresses for very thick plates (t/L > 0.25) and has been applied to analyze interlaminar stresses of thick antisymmetric cross ply skew plates and corner supported thick plates. 12 Introduction For problems involving linear analysis, the nodal distribution has negligible effect on the accuracy of numerical solution. When the nonlinear behavior of a system has to be studied, the distribution of nodes inside a finite element plays a significant role in the accuracy and convergence of a solution. This prompts one to employ spectral finite elements that were originally applied to solve problems in computational fluid mechanics and magnetostatics (Sherwin and Karniadakis, 1995). Chapter begins with an introduction to the spectral finite element method and presents a detailed explanation on the choice of optimized spectral nodes, nodal basis function and derivation of shape functions for a higher order triangular spectral element. The main objective of this chapter is to develop a linear finite element model for continuum shell finite elements called HT-CS and HT-CS-X that have equidistant distributions of geometric nodes and optimized spectral nodal distributions respectively. Having developed the linear finite element model for shells, the accuracy and convergence characteristics of the two continuum shell elements are presented for several challenging linear shell benchmark problems. The convergence trends of maximum displacements and strain energy in the “Discriminating and Revealing Tests” confirm the robustness of HT-CS-X elements. Hence, a nonlinear finite element formulation of HT-CS-X elements is developed under the framework of Total Lagrangian approach to deal with shells undergoing large deflections and moderate rotations. Chapter deals with the development of a nonlinear finite element model of HT-CS-X elements based on Total Lagrangian approach. Two nonlinear solution procedures adopted in this work namely, the Newton-Raphson method and the arc length method are discussed. The former method of 13 Introduction solution works efficiently when the nonlinear load versus deflection response of the structure has no limit points. The latter method is employed to trace complex load versus deflection response of a structure that contains multiple limit points along its equilibrium path. The accuracy of the developed nonlinear finite element model for HT-CS-X elements and the associated nonlinear solution methods are verified in nonlinear shell benchmark problems. Having verified the correctness of the nonlinear finite element formulation, the capability of HT-CS-X elements in handling stresses in nonlinear regime efficiently as compared to conventional lower order elements is demonstrated by considering shell problems involving singularities and steep stress gradients. Finally, numerical simulations involving multilayered shells having negative Gaussian curvatures (hyperboloid shells) are performed. Details pertaining to stress distributions within the shell structures and the localization of internal stresses in the nonlinear regime are presented. Chapter presents the closure of our research. It contains a summary of the work, concluding remarks and recommendations for future studies on higher order triangular elements. 14 [...]...Introduction layer-wise plate theory for better prediction of stress resultants and interlaminar stress distributions in isotropic and laminated composite plates These elements have equally spaced nodes inside its triangular domain · Develop a spectral higher order triangular element with optimal location of nodes inside its domain and having an improved basis function for defining shape... model for HT-CS-X elements and the associated nonlinear solution methods are verified in nonlinear shell benchmark problems Having verified the correctness of the nonlinear finite element formulation, the capability of HT-CS-X elements in handling stresses in nonlinear regime efficiently as compared to conventional lower order elements is demonstrated by considering shell problems involving singularities... formulation of higher order triangular element HT-M45 to account for transverse shear stress variations in the vicinity of stress concentrations This is accomplished by developing two elements namely, HT-TSD-R45 and HT- LT-R45 which are based on a third order shear deformable theory of Reddy (19 84) and a layerwise theory of Reddy (19 87) The former requires no shear correction factors in its formulation and accurately... distributions of geometric nodes and optimized spectral nodal distributions respectively Having developed the linear finite element model for shells, the accuracy and convergence characteristics of the two continuum shell elements are presented for several challenging linear shell benchmark problems The convergence trends of maximum displacements and strain energy in the “Discriminating and Revealing Tests”... singularities and steep stress gradients Finally, numerical simulations involving multilayered shells having negative Gaussian curvatures (hyperboloid shells) are performed Details pertaining to stress distributions within the shell structures and the localization of internal stresses in the nonlinear regime are presented Chapter 6 presents the closure of our research It contains a summary of the work, concluding... continuum shell formulation of the two versions of higher order elements HT-CS and HT-CS-X; ‘X’ is associated with the optimal nodal configuration and improved shape functions of the spectral higher order element · Compare the convergence characteristics of HT-CS and HT-CS-X elements in several challenging shell benchmark problems that exhibit asymptotic behaviors and finally employ the best performing... Chapter 4 begins with an introduction to the spectral finite element method and presents a detailed explanation on the choice of optimized spectral nodes, nodal basis function and derivation of shape functions for a higher order triangular spectral element The main objective of this chapter is to develop a linear finite element model for continuum shell finite elements called HT-CS and HT-CS-X that have... transverse shear stress resultants for thin and moderately thick plates (0. 01 0.25) and has been applied to analyze interlaminar stresses of thick antisymmetric cross ply skew plates and corner supported thick plates 12 Introduction For problems involving linear analysis, the nodal distribution . capable of predicting accurate stress resultants in structural problems involving high stress gradients and singularities in loading and boundary conditions. · Develop a family of higher order elements. verified in nonlinear shell benchmark problems. Having verified the correctness of the nonlinear finite element formulation, the capability of HT-CS-X elements in handling stresses in nonlinear. derivation of shape functions for a higher order triangular spectral element. The main objective of this chapter is to develop a linear finite element model for continuum shell finite elements