Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UFE0044354/00001
## Material Information- Title:
- Mesh Independent Finite Element Analysis of Composite Plates
- Creator:
- Solai Rameshbabu, Vignesh
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2012
- Language:
- english
- Physical Description:
- 1 online resource (105 p.)
## Thesis/Dissertation Information- Degree:
- Master's ( M.S.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Aerospace Engineering
Mechanical and Aerospace Engineering - Committee Chair:
- Kumar, Ashok V
- Committee Members:
- Ifju, Peter
Sankar, Bhavani V - Graduation Date:
- 8/11/2012
## Subjects- Subjects / Keywords:
- Bending ( jstor )
Boundary conditions ( jstor ) Composite materials ( jstor ) Eggshells ( jstor ) Finite element method ( jstor ) Geometry ( jstor ) Laminates ( jstor ) Plate theory ( jstor ) Stiffness ( jstor ) Stiffness matrix ( jstor ) Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF composites - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Aerospace Engineering thesis, M.S.
## Notes- Abstract:
- Mesh Independent Finite Element Analysis uses geometry that is represented using equations or surface triangles and uses a background mesh to perform analysis. The geometry is independent of mesh and is not approximated with the elements during the analysis. The Implicit Boundary Finite Element Method (IBFEM) is a mesh independent approach where boundary conditions are imposed using solution structures constructed using approximate step functions. It generates a structured mesh automatically which does not have to conform to the geometry. Plate like structures can be modeled in IBFEM using 3D-shell elements or using 2D-SDPTShell elements which use shear deformable plate theory. The 3D-Shell elements are hexahedral elements that use B-spline interpolation. It uses 3D stress-strain relations and the principle of virtual work for formulating the weak form. The geometry is modeled as a surface that passes through a structured 3D mesh. The 2D-SDPTShell elements are 2D quadrilateral elements which uses a 2D structured mesh. The geometry is represented by the boundary curves that pass through the mesh. In thesis, we extend these two approaches for modeling plates to include the ability to model unidirectional composite laminates. The laminate is defined by the number of layers, orientation, material properties and thickness of each lamina. Using these, the effective properties of the laminate are usually computed as a relation between force/moment resultants and strain/curvature. This can be directly used for the 2D-SDPTShell because it uses the weak form that is derived from the equations of the Shear Deformable Plate Theory (SDPT). For the 3D-shell approach, a 3D stress-strain relation is needed. This relation is determined from the effective properties of the laminate including the relation between force resultants and strains, the coupling between bending and in-plane forces, moment resultants and curvatures as well as shears force resultants and shear strains. These two types of elements are developed and compared using some standard and practical examples in different test cases and using different boundary conditions. The results are compared with analytical solutions, when available, or with commercial finite element analysis software. The convergence rate of these two elements is compared. Finally, the advantages and disadvantages of the two elements and future applications/extensions are discussed. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (M.S.)--University of Florida, 2012.
- Local:
- Adviser: Kumar, Ashok V.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-02-28
- Statement of Responsibility:
- by Vignesh Solai Rameshbabu.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Solai Rameshbabu, Vignesh. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 2/28/2013
- Resource Identifier:
- 857912863 ( OCLC )
- Classification:
- LD1780 2012 ( lcc )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

PAGE 1 1 MESH INDEPENDENT FINITE ELEMENT ANALYSIS OF COMPOSITE PLATES By VIG NESH SOLAI RAMESHBABU A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2012 PAGE 2 2 2012 Vignesh Solai Rameshbabu PAGE 3 3 To my Mom, Dad and all my teachers PAGE 4 4 ACKNOWLEDGMENTS I express my sincere thanks to my advisor, Assoc. Prof. Dr. Ashok V. Kumar and it might have become impossible for me to proceed with such an intense research without him. His motivation and supp ort was so helpful not only for my thesis, but for the whole course of study. I owe my deepest gratitude to the members of my supervisory committee Prof. Dr. Bhavani V. Sankar and Prof Dr. Peter G. Ifju for the ir gui dance during my thesis. It is an honor for me to have such a team whose criticism throughout my thesis work made me to build it robust. I thank Prof. P.S. Venkatanarayanan for his advice and encouragement to develop my interest in the field of Aerospace Composites and Finite Element Method dur ing undergraduate study in India. I thank my friends and all the professors during my undergraduate study that encouraged and guided me to do my m aster University of Florida, USA. Finally, I would like to show my hearty thanks to my parents, Ra mes hbabu and Suryak ala Rameshbabu, who bored, raised loved and taught me various aspects of life. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 LIST OF ABBREVIATIONS ................................ ................................ ........................... 10 ABSTRACT ................................ ................................ ................................ ................... 12 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 14 Goals and Objectives ................................ ................................ .............................. 17 Outline ................................ ................................ ................................ .................... 18 2 ANALYSIS OF COMPOSITE MATERIALS ................................ ............................. 20 Overview ................................ ................................ ................................ ................. 20 Stress Strain Relations of a Composite Lamina ................................ ..................... 22 Stress Strain Relations of a Composite Laminate ................................ .................. 26 Classical Lamination Plate Theory (CLPT) ................................ ....................... 26 Shear Deformable Plate Theory (SDPT) ................................ .......................... 35 3 MESH INDEPENDENT FINITE ELEMENT METHOD ................................ ............ 39 Introdu ction ................................ ................................ ................................ ............. 39 Formulation of 3D Element (3D Shell) ................................ ................................ .... 42 B Spline Basis Functions ................................ ................................ .................. 46 Stiffness Matrix ................................ ................................ ................................ 48 Formulation of 2D Element (2D SDPTShell) ................................ ........................... 53 Strong Form ................................ ................................ ................................ ..... 53 Weak Form ................................ ................................ ................................ ....... 55 Stiffness Matrix ................................ ................................ ................................ 57 4 FINITE ELEMENT FORMULATION FOR COMPOSITE PLATES .......................... 58 Stiffness Matrix in 3D Shell Element ................................ ................................ ....... 59 Input Parameters ................................ ................................ .............................. 60 Stiffness Matrix of Each Laye r in Material Direction (1 2 direction) .................. 60 Stiffness Matrix of Each Layer in Global Co ordinates (X Y Direction) ............. 61 Combined Bendi ng Stiffness of the Laminate ................................ ................... 61 Approximation for Unsymmetrical Laminates ................................ ................... 63 PAGE 6 6 Stiffness Matrix in 2D SDPTShell ................................ ................................ ........... 69 5 RESULTS AND DISCUSSION ................................ ................................ ............... 72 Example 1: Composite Beam ................................ ................................ ................. 72 Example 2: Square Composite Plate ................................ ................................ ...... 82 Example 3: Composite Wing of Micro Air Vehicle ................................ ................... 90 Example 4: Composite Plate with a Hole ................................ ................................ 92 6 CONCLUSION ................................ ................................ ................................ ........ 97 Summary ................................ ................................ ................................ ................ 97 Scope of Future Research ................................ ................................ ...................... 98 APPENDIX A PROPERTIES OF MATRIX AND FIBERS ................................ ............................ 100 B MATERIAL PROPERTIES OF VARIOUS LAMINATES ................................ ........ 101 LIST OF REFE RENCES ................................ ................................ ............................. 102 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 105 PAGE 7 7 LIST OF TABLES Table page 5 1 Maximum transverse displacements of a HH beam ................................ ........... 75 5 2 Maximum transverse displacements of a CC beam ................................ ........... 75 5 3 Maximum transverse displacements of a CF beam ................................ ............ 76 5 4 Maximum transverse displacements of a S S square plate ................................ 84 5 5 Maximum transverse displacement of a composite wing ................................ .... 9 1 5 6 Maximum transverse displacement of a composite plate with hole .................... 94 A 1 Properties of fibers ................................ ................................ ........................... 100 A 2 Properties of matrix materials ................................ ................................ ........... 100 B 1 Pr operties of various laminates ................................ ................................ ........ 101 PAGE 8 8 LIST OF FIGURES Figure page 1 1 Mesh generation for analyzing circular plate using traditional FEA .................... 15 1 2 3D mesh generation for analyzing circular plate using IBFEM ........................... 16 1 3 2D mesh generation for analyzing circular plate using IBFEM ........................... 17 2 1 Composite lamina ................................ ................................ ............................... 22 2 2 ................................ ......................... 24 2 3 Classical Lamination Plate Theory ................................ ................................ ..... 27 2 4 Shear Deformable Pla te Theory ................................ ................................ ......... 35 3 1 Representation of the geometry ................................ ................................ ......... 40 3 2 Representation of a shell. A) Traditional FEA. B) IBFEM ................................ ... 42 3 3 Representation of volume of a shell ................................ ................................ ... 44 3 4 Clamped boundary condition of a plate/shell ................................ ...................... 44 3 5 Simply supported boundary condition of a plate/shell ................................ ........ 45 5 1 Composite beam ................................ ................................ ................................ 73 5 2 Mesh pattern for Composite beam ................................ ................................ ..... 73 5 3 Traditional FEA (abaqus) results of Composite beam ................................ ........ 73 5 4 3D Shell results of Composite beam ................................ ................................ .. 74 5 5 2D SDPTShell results of Composite beam ................................ ......................... 74 5 6 Convergence plots of Hinged Hinged beam ................................ ....................... 77 5 7 Convergence plots of Clamped Clamped beam ................................ ................. 78 5 8 Convergence plots of Clamped Free beam ................................ ........................ 80 5 9 Composite square plate ................................ ................................ ...................... 82 5 10 Mesh pattern for Composite square plate ................................ ........................... 83 PAGE 9 9 5 11 Analysis results of a Composite square plate ................................ ..................... 83 5 12 Co nvergence plots of S.S Composite Square Plate ................................ ........... 85 5 13 Composite wing ................................ ................................ ................................ .. 90 5 14 Mesh pattern for composite wing ................................ ................................ ........ 90 5 15 Analysis results of a composite wing ................................ ................................ .. 91 5 16 Convergence plot of a composite wing ................................ ............................... 91 5 17 Composite plate with a hole ................................ ................................ ............... 92 5 18 Me sh pattern for composite plate with a hole ................................ ..................... 93 5 19 Analysis results of Composite plate with a hole ................................ .................. 93 5 20 Convergence plots of S.S composite square plate with hole .............................. 95 PAGE 10 10 LIST OF ABBREVIATION S 2D SDPT Two Dimensional Shear Deformable Plate Theory Shell AS ME American Society of Mechanical Engineers ASTM American Society for Testing Materials ATSM2D Analysis Type Solid Mechanics Two Dimensional ATSM3D Analysis Type Solid Mechanics Three Dimensional CAD Computed Aided Design CLPT Classical Lamination Plate Th eory EBC Essential Boundary Condition EFGM Element Free Galerkin Method FDM Finite Difference Method FEA Finite Element Analysis FEM Finite Element Method FRP Fiber Reinforced Plastics FVM Finite Volume Method IBFEM Implicit Boundary Finite Element Method IBM Implicit Boundary Method LHS Left Hand Side MAV Micro Air Vehicle MLPG Meshless Local Petrov Galerkin MLS Moving Least Square NBC Natural Boundary Condition NURBS Non Uniform Rational Basis Spline PEEK Polyether Ether Ketone PRP Particle Reinforced Pla stics PAGE 11 11 RHS Right Hand Side SDPT Shear Deformable Plate Theory SSBC Simply Supported Boundary Condition UAV Unmanned Aerial Vehicle UDL Uniformly Distributed Load UTM Universal Testing Machine X FEM Extended Finite Element Method PAGE 12 12 Abstract of Thesis Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for th e Degree of Master of Science MESH INDEPENDENT FINITE ELEMENT ANALYSIS OF COMPOSITE PLATES By Vignesh Solai Rameshbabu August 2012 Chair: Ashok V. Kumar Major: Aerospace Engineering Mesh Independent Finite Elem ent Analysis uses geometry that is represented using equations or surface triangles and uses a background mesh to perform analysis The geometry is independent of mesh and is not approximated with the elements during the analysis. The Implicit Boundary Fin ite Element Method (IBFEM) is a mesh independent approach where boundary conditions are imposed using solution structures constructed using approximate step functions. It generates a structured mesh automatically which does not have to conform to the geome try. Plate like structures can be modeled in IBFEM using 3D shell elements or using 2D SDPTShell elements which use s hear deformable plate theory. The 3D Shell elements are hexahedral elements that use B spline interpolation. It uses 3D stress strain relat ions and the principle of virtual work for formulati ng the weak form The geometry is modeled as a surface that passes through a structured 3D mesh. The 2D SDPTShell elements are 2D quadrilateral elements which uses a 2D structured mesh. The geometry is re presented by the boundary curves that pass through the mesh. PAGE 13 13 In thesis we extend these two approaches for modeling plates to include the ability to model unidirectional composite laminates The laminate is defined by the number of layers, orientation, mat erial properties and thickness of each lamina. Using these, t he effective properties of the laminate are usually computed as a relation between force/moment resultants and strain/curvature. This can be directly used for the 2D SDPTShell because it uses the weak form that is derived from the equations of the Shear Deformable Plate Theory (SDPT). For the 3D shell approach, a 3D stress strain relation is needed This relation is determined from the effective properties of the laminate including the relation be tween force resultants and strains, the coupling between bending and in plane forces, moment resultants and curvatures as well as shears force resultants and shear strains. T he se two types of elements are developed and compared using some standard and prac tical examples in different test cases and using different boundary conditions The results are compared with analytical solutions when available or with commercial finite element analysis software. The convergence rate of these two elements is compared Finally the advantages and disadvantages of the two elements and future applications/extensions are discussed PAGE 14 14 CHAPTER 1 INTRODUCTION Traditional Finite Element Method ( [1] [2] ) uses a mesh to approximate the geometry of the analysis. The analysis results are also approxim ated by interpolation within the elements of the mesh. It is difficult to automatically generate a mesh for complex geometry. So methods for avoiding mesh generation have been explored including meshless and mesh independent methods, extended FEM (X FEM) ( [3] ) and iso geometric methods. Meshless methods use a scattered set of nodes for the analysis without connecting the nodes to form elements. Examples of meshless methods include Moving Least Square (MLS) method ( [4] ) Element Free Galerkin Method (EFGM) ( [5] ) and Meshless Loca l Petrov Galerkin (MLPG) Method ( [6] ) This method seems to be effective in many cases but had some difficulties. Firs tly, there is no connectivity between the nodes so it is necessary to search for neighboring nodes. Another disadvantage of these methods is that the approximations used for displacement does not have Kronecker Delta properties. Therefore it is difficult t o apply essential boundary conditions (EBCs). An alternative approach to eliminate some of the disadvantages of traditional, meshless and mesh f ree techniques is the Implicit Boundary Fini te Element Method (IBFEM) ( [7] [8] [9] [10] ) It use s step functions to construct test and trial functions in order to apply the EBCs. The equation s of the geometry or triangulated approximations of the b oundary are obtained from CAD model s A solution structure can be as follows. ( 1 1 1 ) In this solution structure, is the implicit equation of the boundary, is the variable function which is approximated as and is interpolated PAGE 15 15 piecewise throughout the mesh and is the prescribed value of the essential boundary condition. In this method, a background mesh is generated around the geometry and the integration is done by directly using t he implicit equations of the geometry. Therefore it is also known as Implicit Boundary Finite Element Method (IBFEM). The accuracy is improved because the geometry equations are directly used and is not approximated using elements The main goal of this th esis is applying this method for analyzing composite plates. Two different approaches for modeling plates were used implement composite material properties. One uses 3D elements and cubic B spline approximation functi ons while the other uses Shear Deformab le Plate theory and mixed formulation approach. Figure s 1 1 1 2 and 1 3 exp lain the advantage of Mesh Independent technique over the traditional FEA. A B Figure 1 1. Mesh generation for an alyzing circular plate using traditional FEA A ) High mesh d ensity. B ) L ow mesh density PAGE 16 16 Figure 1 1 shows the mesh generated for an analysis of circular plate using traditional FEA. From the figure, it is clear that the mesh generated is approximated f or the whole geometry and the mesh seems to be conforming when the mesh density is high. But, when the mesh density is poor, the mesh does not conform to the geometry. This problem can be overcome by implementing mesh free technique since it does not appro ximate the geometry by replacing it with meshes and it directly uses the equation of the geometry. So, the mesh density does not affect the accuracy of the geometry A B Figure 1 2 3D mesh ge neration for analyzing circular plate using IBFEM A) High mesh density. B) Low mesh density Figure 1 2 clearly shows the 3D mesh generation for the analysis of circular plate using IBFEM. As seen in the figure the geometry is not replaced by the mesh but it directly uses the equation of the geometry. Each element is 3D and the plate passes through the elements To overcome some of the difficulties in the 3D element a new element which uses Shear Deformable Plate Theory is developed Since, this element i s exclusively developed for composite plates; it is enough if only 2D meshes are generated. Hence PAGE 17 17 the grids generated for this analysis is 2D which is shown in figure 1 3. In this method, the equations from the Shear Deformable Plate Theory (SDPT) are used for the formulation of weak form. Hence, the in plane stiffness [A], coupling stiffness [B] and bending stiffness [D] of the laminate and is directly used in the formulation which reduces the error percentage of the result. A B Figure 1 3 2D mesh generation for analyzing circular plate using IBFEM A) High mesh density. B) Low mesh density Goals a nd Objectives Goal The main goal of this thesis is to develop composite plate elements which i mplement the idea of mesh independent Finite Element Method Objectives Implement composite material properties for 3D shell elements to model flat composite laminates where the geometry is modeled as a planar surface that is imported from CAD software and mesh is a background structured mesh consisting of uniform 3D cubic/ cuboid elements. PAGE 18 18 Implement composite material properties for 2D Mindlin plate elements This method uses the equations of Shear Deformable Plate Theory (SDPT) directly for formulation. Mo reover, the stiffness matrix is defined from the in plane stiffness, coupling stiffness and bending stiffness matrix. The geometry is again a planar surface and the mesh is a 2D background mesh with uniform square/rectangular elements. Compare the performa nce of these t wo elements for modeling plates. Perform convergence studies for some standard examples that have analytical solutions and also perform convergence studies for some practical examples. Compare results with commercial FEA software. Outline The remaining portion of the thesis is systematized as follows: In Chapter 2, the introduction of composite materials, its types and its application in engineering is explained. It also discloses the history of finite element analysis of composite plates and the past work on composite FEA. This chapter also explains the methods used for structural analysis of composite plates. It also gives the equation used for structural analysis and determination of strength. Moreover, it also explains the two major plate t heories and its equations used in the Finite Element formulation. In Chapter 3, explains the need for mesh free finite element analysis and the past work of researchers in this area It also clarifies the concept of Implicit Boundary Finite Element Method (IBFEM) and the use of step function in its formulation. It also explains the calculation of stiffness matrix in this formulation. In Chapter 4 the implementation of IBFEM for composite plates is explained It also explains the determination of equivalen t stiffness of the composite laminate from PAGE 19 19 the material properties of the individual lamina which can be used in the formulation of 3D element. T he constitutive model used in plate theory formulation is also explained In Chapter 5, various examples have b een used to test the composite plates and mixed formulation in IBFEM of composite plates. The first example explains it through the 1 D analysis of composite beam. The second example proves the concept in 2 D analysis in Composite Square Plate. The third e xample is a practical example which proves the concept in the Composite Wing of a Micro Air Vehicle (MAV). It shows clearly that IBFEM is effective in such cases as it is a good example of non conforming mesh type. The fourth example involves the analysis of a composite square plate with a hole at the center. It also contains convergence plots for each example in various cases such as different boundary conditions, various possible stacking sequences which clearly shows that this method is the most cheap an d efficient method for Finite Element Method of Composite Plates. In Chapter 6, summary of the work and the conclusions drawn from the results are provided The advantages and disadvantages of the two approaches are explained and the appropriate condition to use the two element s which is formulated from two different methods is deduced. This chapter also clearly explains the research work which can be done in the future in extension to the current research. PAGE 20 20 CHAPTER 2 ANALYSIS OF COMPOSITE MATERIALS Overvi ew Composite materials, as per the definition are obtained from the combination of two or more distinct materials. They are designed to satisfy design criteria such that the properties of the whole are better than the sum of the properties of each constituent taken separately It is different from isotropic materials as its property var ies according to the location and direction. As they give a very high strength to weight ratio, they are also known as high performance materials and are used in aerospace and automotive applications. Some of the secondary advantages of composite materials are, they are corrosion resistant, friction and wear resistant, vibration damping, fire resistant, acoustical insulation, piezoelectric etc. There are various types of composites such as particle reinforced plastics (PRP) fiber reinforced plastics (FRP) laminated plastics etc. Among these, the most frequently used type of composites are FRPs and laminated composites as the mechanical properties of the material can be calculated from the location and the orientation of the fiber. The frequently used FRPs in engineering applications are unidirectional composites as it gives an enhanced strength in a particular direction which is aligned with the direction of loads acting on the component. The design of components using these materials is different from tho se from isotropic materials because, in this case, the material is made at the same time as the component. As the material is fabricated according to the design specifications of the component, designers/engineers should pay attention to the methods used t o fabricate the mater ial which is a big factor which can affect the design of the component. Hence there should be a better communication between the design and manufacturing to PAGE 21 21 ensure optimal result. Several methods are used for predicting and study the p roperties of composites including a nalytical/classical methods, experimental methods and finite element methods. Analytical methods which include macro mechanics and micro mechanics are the one which uses equilibrium, constitutive and compatibility equatio ns to derive the governing equations which in turn are solved to find the stresses and displacements. Experimental methods are used to find the properties and strength of the material by performing various tests in the Universal Testing Machine (UTM) accor ding to ASTM standards. Since these methods are limited to simple geometries, finite element method (FEM) [11] [12] [13] [14] is increasing being used by designers/engineers to model and analyze composite structures J.N. Reddy [15] proposed a method to implement Shear Deformable Plate Theory (SDPT) in FEM Automatic mesh generation is difficult when the geometry is complex. Several meshless and meshfree methods have been developed that try to avoid using a mesh for the analysis. R.J. Razzaq, A. El Zafrany [16] explained how to use reduced mesh for the non linear analysis of composite p lates and shells. J. Sladek, V. Sladek and S.N. Atluri [17] introduced Meshless Local Petrov Galerkin Method (MLPG) in anisotropic Elasticity. Similarly, J. Belinha and L.M.J.S. Dinis [18] came up w ith the idea of Element Free Galerkin Method (EFGM) in analyzing composite laminates. The next stage of development in the FEA research in composites leads to development of Higher order Displacement model [19] and the introduc tion of Higher Order Shear and Normal Deformable Plate Theory (HOSNDPT) and the Meshless Method with Radial Basis Functions by J.R. Xiao, D.F. Gilhooley, R.C. Batra, J.W. Gillespie Jr., M.A. McCarthy PAGE 22 22 [19] This chapter contains the derivation of the equations and the relations used for the formulation in Shear deformable plate theory Stress Strain Relations of a Composite Lamina Composite materials are obtained by the combination of two or more materials at macro scale and its behavior is describable by continuum mechanics. As it contains a continuous phase called matrix and a discontinuous phase called fibers, its properties are different at different point and direction. Figure 2 1 Composite lamina Figure 2 1 shows that the direction along the fiber is 1 and the direction perpendicular to the fiber is 2. he general form of Stress Strain relation [21] [22] [23] [24] [25] is given by, ( 2 2 1 ) The above equation refers to the Stress Strain relation of anisot ropic materials (or) otherwise known as triclinic materials (i.e.) the three axes of the material are 1 2 Fiber cross section PAGE 23 23 oblique to one another. Because it is assumed to be hyper elastic, the [C] matrix becomes symmetric and hence C ij = C ji Hence we have 21 stiffness co eff ecients. If there is only one pair of symmetry of the material property, the Stress Strain relation becomes, ( 2 2 2 ) In this equation, the number of stiffness co effecients has reduced to 13 and the material is termed as monoclinic material. If there are two planes of material prope rty symmetry which is orthogonal, the Stress Strain relation becomes, ( 2 2 3 ) This type of material is known as orthotropic material and it has only 9 stiffness co effecients. If there is a material such that the properties are equal in all the directions at any point in a particular plane, then it is called transversely isotropic. In case of unidirectional composite Laminate, the direction 1 represents the fiber direction, 2 and 3 represents the transverse directions and the properties are same in 2 and 3 direction. Therefore, 2 and 3 in the stiffness co effecients are interchangeable. Hence the Stress Strain relation is given by, PAGE 24 24 ( 2 2 4 ) The above equation gives the Stress Strain relation of a composite laminate, the stiffness matrix in this equation is represented by [Q]. It gives the stresses and strains al ong the material directions (i.e.) in terms of fiber matrix directions. It holds well only if the fiber is aligned along the global x y direction. If the fiber is oriented at from the x axis, as shown in figure 2 2, the stresses in 1 2 directions has to be transformed to x y directions by multiplying it with the transformation matrix. Figure 2 2 Composite l amina with y x z 1 2 PAGE 25 25 ( 2 2 5 ) W here, [ ] = [ T] [Q] [T ] T and the Transformation matrix is given by, ( 2 2 6 ) When the fibers are oriented along the direction which i axis, the Stress Strain relation in x y direction is given by, ( 2 2 7 ) Where PAGE 26 26 For Plane stress condition, the above str ess strain relation can be written as, ( 2 2 8 ) Stress Strain Relations of a Composite Laminate Classical Lamination Plate Theory (CLPT) In aerospace and auto motive applications, unidirectional fiber composites are widely used. The properties of unidirectional composites are close to transversely isotropic materials. There are several other types of composites such as woven and braided composites whose properti es are close to orthotropic material s. These unidirectional composites are using mats which are laminated layer by layer to form plates. Here we introduce the classical lamination plate theory (CLPT) which makes the following assumptions Stiffness and Str esses are averaged through the thickness. x y plane is the middle plane of the plate and z axis is the thickness direction. PAGE 27 27 The thickness of the plate is so small when compared to the lateral dimensions of the plate. The displacements u, v and w are very s mall when compared to the plate thickness. This theory is known as Small Displacement Theory. The in plane strains are very small when compared to unity. The transverse shear strains are negligible. ( i.e. ) .The thickness change is also very small and so the transverse normal strain can be neglected. ( i.e. ) The transverse normal and shear stresses are negligibly small when compared to in plane stresses. Figure 2 3 Classical Lamination Plate Theory Let t he displacement fields be expressed in Taylor's series as follows. ( 2 3 1 ) X Z PAGE 28 28 Eliminating the higher order terms and assuming the transverse displacement independent of z coordinate, eqn s (2.3.1) becomes, ( 2 3 2 ) From the Small Strain Theory, the transverse shear strains can be derived as: ( 2 3 3 ) Substituting eqns (2.3.2) in eqns ( 2.3.3), ( 2 3 4 ) In CLPT, as the transver se shear strains are assumed to be negligible, (i.e.) eqns (2.3.4) becomes, ( 2 3 5 ) Substituting eqns (2.3.5) in eqns (2.3. 2 ) we get ( 2 3 6 ) From the above equation, the in plane strains can be derived as, PAGE 29 29 ( 2 3 7 ) Where and are the mid plane strains and are the curvatures which can be defined as follows: ( 2 3 8 ) Eqn. (2.3.7) can be written in Matrix form as follows: ( 2 3 9 ) ( 2 3 10 ) The three dimensional equations of motion for a body considering the large deformations are, From the se equations of equilibrium, without considering body forces and large deformations and neglecting the product of transverse stresses and in plane strains in the non Linear terms which are small, we get, ( 2 3 11 ) PAGE 30 30 Since, in CLPT, we are deriving the equations in an average sense, we obtain the equations of CLPT by integrating the above equations throughout the thickness. Hence, by integrating the first equation of Eqn.(2.3.11), Neglecting the rotary inertia term the above equation can be written as, ( 2 3 12 ) Where and a re the force resultants and is the total axial force in x direction, which can be defined as, Similarly, by integrating the second equation of Eqn. (2.3.11), we get, ( 2 3 13 ) Where is the force resultant a nd is the total axial force in y direction, which can be defined as, By integrating the third equation of Eqn. (2.3.11), we get, PAGE 31 31 ( 2 3 14 ) Where and are the transverse shear force resultants and is the transverse force acting in z direction which can be defined as follows: Expanding Eqn. (2.3.14), substituting Eqn. (2.3.12) and Eqn. (2.3.13) and neglecti ng the in plane inertia terms, we get ( 2 3 15 ) Multiplying the first and second equations of Eqn. (2.3.11) by z on both sides and integrating through the thickness, we get ( 2 3 16 ) The above equation can be written as, ( 2 3 17 ) PAGE 32 32 Where the moment resultants are and are distributed couples wh ich can be defined as, Eliminating and from Eqn. (2.3.15) using Eqn. (2.3.17), we get, ( 2 3 18 ) Therefore, from Eqn. (2.3.12), Eqn. (2.3.13) and Eqn. (2.3.18), the 3 equations of motion of CLPT are as follo ws: ( 2 3 19 ) Constituti ve rel ations of the l aminate from CLPT From the definition of in plane stress and moment resultants, it can be written in column matrix form as follows: ( 2 3 20 ) PAGE 33 33 ( 2 3 21 ) Substituting Eqn. (2.2.8) in Eqn (2.3.20), we ge t ( 2 3 22 ) Substituting Eqn. (2.3.7) in Eqn (2.3.21), ( 2 3 23 ) ( 2 3 24 ) The above equation can be further simplified as, ( 2 3 25 ) PAGE 34 34 ( 2 3 26 ) Where [A] [B] and [D] are called laminate stiffness matrices which are individually known as follows: [A]: In plane stiffness matrix [B]: Coupling stiffness matrix [D]: Bending Stiffness matrix These matrices are defined as, ( 2 3 27 ) Eqn. (2.3.23) and Eqn. (2.3.24) can be combined and written in one matrix equation as follows: ( 2 3 28 ) PAGE 35 35 Shear Deformable Plate Theory (SDPT) This theory explains that one of the assumptions of CLPT that says the plane sections normal to the mid plane of the plate remain plane and normal after the plate deforms is valid only in case of thin plates. This further explains that the transverse shear strains and does not vanish and remain constant throug hout the thickness of the plate (i.e.) the plane sections normal to the mid section remain plane but not necessarily remain normal to the mid surface after deformation. Figure 2 4 Shear Deformable Plate Theory T he displacement fields can be expressed as, ( 2 4 1 ) Where and are rotations of the cross section about y axis and x axis respectively. The in plane strains can be derived as, X Z x PAGE 36 36 ( 2 4 2 ) Where the mid plane strains and curvatures are defined as, ( 2 4 3 ) Similarly, the equations of SDPT can be derived from the equilibrium equations which are as follows: ( 2 4 4 ) Where, PAGE 37 37 Constituti ve relations of the l aminate from SDPT The in plane stress a n d moment resultants are same as those derived using CLPT. In addition to that, there are shear force resultants and which can be w ritten in the column matrix as follows: ( 2 4 5 ) Where and are the transverse shear stresses which can be defined from Eqn (2.2.7) as follows ( 2 4 6 ) Where, ( 2 4 7 ) Substituting Eqn (2.4.6) in Eqn (2.4.5), we get, ( 2 4 8 ) The above equation can also be written as, ( 2 4 9 ) Where PAGE 38 38 Eqn (2.3.23), Eqn. (2.3.24) and Eqn. (2.4. 9) can be combined and written in the form of one matrix equation to get the constitutive relation of the laminate which is as follows: ( 2 4 10 ) The equation above, can be further written as, ( 2 4 11 ) Eqn (2.4.11) gives the constitutive relation of the laminate using SDPT. PAGE 39 39 CHAPTER 3 M ESH INDEPENDENT FINI TE ELEMENT METHO D Introduction The most successful methods whi ch are used to solve structural, thermal and fluid flow problems are Finite Difference Method (FDM), Finite Element Method (FEM) and Spectral Method. For s tructural analysis FEM is the most popular numerical method because it can be used for arbitrary geom etry as long as a mesh can be generated for the analysis. Figure 3 1(a) shows the approximation of the whole geometry into a finite element mesh Automatic mesh generation is difficult for 3D problems To avoid such difficulties, a number of meshless or me sh free analysis techniques have been proposed in the past two decades Belytschko.T Y. Krongauz, D. Organ, M. Fleming, P. Krysl [26] introduced meshless method and explained its implementation. J.J. Monaghan [27] explained how mesh free methods can be used to solve astrophysical problems. This approach uses a scatter ed set of nodes all over the sys tem for analysis (figure 3 1(b)). One of the most widely used basis functions for scattered meshless m ethods is Moving Least square ( MLS) approximation which was clearly explained by P.Lancaster and K.Salkauskas [4] Followed by MLS, several other techniques such as Element Free Galerkin Method ( EFGM) [5] Meshless local P etrov Galerkin Method (MLPG) [6] were consecutively developed. One of the main features of meshless methods is, there is no connectivity between the nodes An alternate approach to the mesh free method i s using structured, non conforming mesh, in which the geometry is represented using implicit equations and therefore is i ndependent of the mesh. Figure 3 1(c) shows the generation of non conforming uniform, structured mesh. PAGE 40 40 Figure 3 1. Representation of the geometry A ) Traditional FEA. B ) Meshless approach. C ) Non conforming mesh approach Since the mesh does not depend on the geometry of the system, there is no approximation during mesh generation. Moreover, uniform elements are generated which makes the mesh generation and integration easier and computational less expe nsive. The main difficulty faced when this approach is used is the non conformity of the mesh to the boundary. There are no nodes on the boundary of the analysis domain This condition make s it different from traditional FEA approach when the essential boundary conditions (EBC) are applied. One of the methods used for the application of EBC was propose d by Kantorovich and Krylov [28] The solution structure for s olving such problems given by them was in the form as follows. ( 3 1 1 ) A B C PAGE 41 41 In the above equation (Eqn. (3.1.1)), is the equation of the boundary of the domain and is the essential boundary condition. The variable part of the solution st ructure is the function which is approximated as which is piecewise interpolated over the grid. Several other techniques such as R function technique were developed by Shapiro and Tsukanov [29] and it was extended to non linear vibrations of thin plates by Lidia Kurpa, Galina Pilgun and Eduard Ventsel [30] C. Daux, N. Mos, J. Dolbow, N. Sukumar and T. Belytschko [3] pr oposed a similar method known as X FEM which is based on structured, uniform mesh and IBM. They introduced it to find the stress distribution in crack tips and it was further extended to 2 D and 3 D crack modeling Another approach used to solve such diffi cultie s in boundary value problems is the Implicit Boundary FEM (IBFEM) propos ed by Kumar A. V. [7] [8] [9] [10] As discussed in chapter 1, th is method constructs a solution structure for the displacement and virtual displacement field using approximate step functions of the boundaries Figure 3 2 shows a shell subjected to gravity pressure. The structural analysis is performed and the displac em ent contour is obtained from A baqus which follows traditional finite element method and IBFEM which uses uniform n on conformal mesh The figure clearly shows the type of mesh used in traditional FEA and IBFEM. In traditional FEA, the geometry is replaced b y the meshes. But, in IBFEM, the geometry is immersed in the structured mesh For shell like structures IBFEM uses B spline basis functions which are C 2 continuous (i.e.) the functions used are curvature continuous, whereas the Lagrange interpolation funct ions are C 0 continuous (i.e.) their magnitudes are continuous PAGE 42 42 A B Figure 3 2 Representation of a shell A ) Traditional FEA. B ) IB F E M This thesis involves devel opment of two type s of elements for analyzing composite plates and a comparative study of these two elements is done One element named 3D Shell uses B spline interpolation functions with 3D non conformal structured grid As shown in figure3 2(b), the geometry is surrounded by 3D mesh which is uniform throughout the domain The basis function used in this element is cubic in nature and hence it is c 2 continuous (i.e.) it is tangent and curvature continuous. Another element named 2D SDPTShell consists of 2D element with a two dimensional background mesh. The geometry is surrounded by 2D mesh and is non conformal in nature. As the analysis involves flat composite plates, the mesh generated is 2D. It involves interpolation functions that are not tangent continuous which involve Mindlin Plate formulation as it uses the equations from Shear Deformable Plate Theory (SDPT) The formulation of these two elements is explained as follows. Formulation of 3D E lement ( 3D Shell ) The implicit boundary finite element method uses implicit equation in its solution structure which is as follows. PAGE 43 43 ( 3 1 2 ) Where, Trail function Piecewise approximation of the element of the structured grid derived from the implicit equation of the boundary. Boundary value function which contains the essential boundary condition Diagonal matrix whose components are the step functions. In this research, as the discussion is about composite plates and shells, the formulation is sim ilar to isotropic plates and shells. First, the plate is integra ted over the area of the mid section and then the integration through the thickness is performed. The volume of the shell can be represented as ( 3 1 3 ) Where Parametric equation of the surf ace representing the mid plane. Total thickness of the shell Unit normal to the sur face The volume of the shell can also be represented in terms of unit normal and bi no rmal as ( 3 1 4 ) Where, PAGE 44 44 Parametric equation of the i th boundary Unit normal to the surface at the respective boundary Unit bi normal to the surface at the respective boundary which is the cross p roduct of the unit normal and unit tangent of the boundary, where the unit tangent is the first derivative of the equation of the boundary for which the tangent is to be determined. Figure 3 3 Representation of volume of a shell Clamped boundary conditi on Figure 3 4 Clamped boundary condition of a plate/shell PAGE 45 45 From figure 3 4 the essential boundary condition of a clamped plate/shell is at the edge But, since the angle is not the nodal degree of freedom, the boundary condition s can be defined as The approximate step function H for this type of boundary condition is in the form ( 3 1 5 ) Where, Transition width where the step function transitions from 0 to 1 Distance from the edge face. The value of gi ves the distance from the edge and hence the step function is also known as Distance function. The step function represents a quadratic function and fits a quadratic curve in the transition width. At th e point the slope of the curve goes to zero. Simply supported boundary condition Figure 3 5 Simply supported boundary condition of a plate/shell PAGE 46 46 From the figure 3 5 the essential boundary condition of a clamped plate/shell is at the edge. Hence the approximate step function is defined as, ( 3 1 6 ) Where, Radial distance from the edge which is given by Symmetric boundary condition When a symmetric structure is to be analyzed; o nly a part of it is m odeled to reduce the model size ( total number of elements) and therefore the computational time is greatly reduced. The essential boundary condition of this type of boundary condition is Hence the step functi on of this bounda ry condition is same as Eqn. 3.1.6 but can be applied only for B Spline Basis Functions Basis functions are used to map between the local and the global co ordinate system. In traditional FEA, Lagrange interpola tion functions are used as basis functions which are c 0 continuous. But in the IBFEM, B spline basis functions are used which gives a higher degree of up to c 2 continuity. Since the structured, uniform grids are used, uniform B spline basis functions are u sed in this case. There are 2 types of basis functions used in two different types of elements. They are quadratic B spline basis functions which are C 1 continuous and cubic B spline basis functions [31] which are C 2 continuou s The concept of recursion (i.e.) calling a function PAGE 47 47 or a polynomial again and again is used to construct this basis function. Here is an example for 1 D cubic B spline basis function. Figure 3 6 1 D cubic B spline approximation In the figure, it is c lear that the element E1 goes from node 2 to 3. The element is actually controlled by four nodes starting from 1 through 4. Hence these nodes are also known as support nodes. The approximated values such as displacement at a specific node are not equal to the nodal value T he cubic B spline basis functions can be represented as follows. ( 3 1 7 ) Since the functions are cubic in nature (i.e.) the order of the function is 3, there are 4 basis functions and 4 support nodes, in which 2 nodes lie outside the element 1 2 3 4 E 1 Nodal Values Approximated Function Nodes Elements PAGE 48 48 The basis functions of 2 D cubic B spline basis functions can be constructed from the product of two 1 D cubic basis functions. ( 3 1 8 ) Figure 3 7 2 D cubic B spline elements From figure 3 7, it is shown that e ach 2 D element is supported by 16 support nodes and 16 basis functions. Hence, a 2 D cubic basis element has 16 basis functions and a 3 D cubic basis element has 64 functions. Stiffness M atrix In the 3D element approach the weak form used is the 3D Principle of Virtual Work, which can be expressed as follows, ( 3 1 9 ) Where Virtual Strain Cauchy Stress tensor Edges on which the traction is applied Traction vector 13 14 15 16 9 10 11 12 1 2 3 4 5 6 7 8 PAGE 49 49 Virtual displacement Body force Pressure load per unit area For plates and shells, the weak form (Eqn. 3.1 .1 2 ) can be written as ( 3 1 10 ) The strains and stresses can be broke down into homogenous and boundary value part as, ( 3 1 11 ) Substituting Eqn. (3. 1.14) in Eqn. (3.1.13 ), we get, ( 3 1 12 ) From the above equation, it is clear that the right hand side o f the equation has an additional term which is due to the boundary value stress The trail and test functions of the displacement field can be discretized using B spline basis functions as follows. ( 3 1 13 ) From the above equation, the strain displacement equ ation can be derived a s PAGE 50 50 ( 3 1 14 ) Where, B Strain displacement matrix The Strain displacement matrix is a combined matrix of B 1 and B 2 which is as follows B 1 matrix contains step functions and the derivatives of shape functions, whereas B 2 matrix contains s hape functions and the derivatives of the step functions and B 3 matrix contains only the derivatives of shape functions which are as follows. ( 3 1 15 ) ( 3 1 16 ) ( 3 1 17 ) Therefore, using these equations, the weak form can be discretized and can be written as, ( 3 1 18 ) PAGE 51 51 Where, ( 3 1 19 ) ( 3 1 20 ) ( 3 1 21 ) ( 3 1 22 ) ( 3 1 23 ) Where, Stiffness Matrix Domain of integration Set of elements whose edge is with traction boundary condition For the elements which is completely inside the boundary, that is, for internal elements, the strain displacement matrix Therefore, the stiffness matrix becomes, ( 3 1 24 ) Where, Strain displacement matrix which contains only the derivatives of shape functions PAGE 52 52 Number of triangles which result from the triangulation of the surface integration with the element Area of the i th triangle For boundary elements, the strain displa cement matrix Therefore, the stiffness matrix becomes, ( 3 1 25 ) ( 3 1 26 ) Since in most of the boundary except the region and is used only in the elements where the essential boundary condition is applied, for t he integrant which contains Hence, PAGE 53 53 ( 3 1 27 ) ( 3 1 28 ) ( 3 1 29 ) Substituting (Eqn. 3.1.29) to (Eqn. 3.1.32 ) in (Eqn.3. 1.28 ), ( 3 1 30 ) Formulation of 2D E lement ( 2D SDPTShell ) Strong Form The formulation for analyzing co mposite plates using 2D element is different from that of 3D elements. Here, the equations obtained from Shear Deformable Pl ate Theory PAGE 54 54 (SDPT) are used. T he inertial terms in Eqn (2.4.4) are eliminated and can be written in Matrix form. The first two equati ons of Eqn (2.4.4 ) can be written as The last two equations of Eqn ( 2.4.4 ) can be written as The third equation of Eqn (2.4.4 ) can be written as The above Matrix equation s can be simplified as ( 3 2 1 ) ( 3 2 2 ) ( 3 2 3 ) Where, And PAGE 55 55 ( 3 2 4 ) ( 3 2 5 ) ( 3 2 6 ) Substituting Eqns (3.2.4) and (3.2.5 ) in Eqns (3.2. 1) and (3.2.2 ) r espectively, ( 3 2 7 ) ( 3 2 8 ) From Eqn (3.2.3 ), ( 3 2 9 ) From Eqn (3.2.6 ), ( 3 2 10 ) Weak Form The above four equati ons Eqn (3.2.7) to Eqn (3.2.10 ) are the governing equations which are used for further formulation. To these equations, the weighted residual form. After a pplying weig hted residual m ethod the weak form obtained is as follows. PAGE 56 56 ( 3 2 11 ) Where, Known traction (natural boundary condition) on the boundary Known moment (natural boundary condition) on the boundary Known shear (natural boundary condition) on the boundary To solve these equations, a three field mixed formulation is used. The following substitutions are made to solve it using mixed formulation. ( 3 2 12 ) The shape functions are formulated based on the number of parameters Using the above expression and simplifying the weak form, the matrix equation of the final weak form becomes ( 3 2 13 ) Where, PAGE 57 57 St iffness Matrix From the above equation, it is clear that the stiffness matrix of the system of equations depends upon matrices and takes the form, ( 3 2 14 ) PAGE 58 58 CHAPTER 4 FINITE ELEMENT FORMU LATION FOR COMPOSITE PLATES I n the composite Laminate, the property of the laminate differ layer wise. E ach layer has a different material property which depends upon the The property of the laminate also depends upon the thickness of the lamina. Hence, the calculation of stiffness matrix is different from th ose of Isotropic materials. The stiffness of the laminate can be determined by the Laminate constitutive relations (Eqn. 2.3.26 and Eqn. 2.4.11). The first equation (Eqn. 2.3.26) gives the laminate constitutive relation derived using Classical Lamination Plate Theory, in which the transverse normal and shear forces are neglected and hence the transverse shear strains are negligible. As this theory holds good for only thinner plates and shells, another theory was developed which considers the transverse no rmal and shear stresses into consideration and takes shear strains into account while calculating the maximum deflection of the plate for applied load, which is known as Shear Deformable Plate Theory. From this Theory, the constitutive equation of the lam inate is derived which is given in Eqn 2.4.11. From this relation, the stiffness matrix of the laminate is found. When the two laminate constitutive relations are observed, the stiffness matrix of the laminate depends upon 3 matrices, In plane Stiffness Ma trix [A], Bending Stiffness matrix [D] and coupling Stiffness Matrix [B]. Hence, to calculate the laminate stiffness, we need to determine all these three matrices. After the 3 matrices are calculated, the equivalent stiffness matrix of the laminate is cal culated in two different ways. This is shown i n two different elements used in IBFEM. They are 3D Shell and 2D SDPTShell PAGE 59 59 In the 3D Shell the weak form is derived from the stress strain relation and hence the stiffness matrix is obtained from the stress s train relation of the laminate. Since, the stiffness matrix of the laminate is not directly available from the stress strain relation of the laminate; the equivalent stif fness of the laminate is obtained from the Reduced Bending Stiffness Matrix [D r ] which is derived from the in plane stiffness matrix [A], coupling stiffness matrix [B] and bending stiffness matrix [D] This may increase the percentage of error as well as the computational time. To overcome these difficulties, another element is developed wh ich is known as 2D SDPTShell in which the weak form is formulated from the equations obtained from Shear Deformable Plate Theory (SDPT). Since the weak form is formulated from the Force/Moment Strain/curvature relation, from the in plane stiffness matrix [ A], coupling stiffness matrix [B] and bending stiffness matrix [D] can be directly used in the formulation. Then the error percentages of the results obtained from these two elements are compared and the best element used for composite plates is concluded. This shows that the error percentage and computational time is better than the other element. It also converges faster, (i.e.) in a very low mesh density than the other elements which is clearly shown from the convergence plots. The following chapter expl ains the computation of the stiffness matrix for each element. S tiffness M atrix in 3D Shell E lement The explanation of calculation of stiffness matrix which is used in 3D Shell element involves first the determination of bending stiffness matrix for only s ymmetric laminates followed by generalized determination of bending stiffness matrix for laminates. First the stiffness matrix of each layer in the material direction is calculated. Then it is transformed to global co ordinates. It is followed by combinin g the transformed stiffness of each layer to obtain the stiffness of the whole laminate. PAGE 60 60 Input Parameters The parameters which we need to calculate the stiffness matrix are collected from the user through the graphical interface which is used in IBFEM. The properties are different for different lamina and these properties are collected for each and every layer. The properties which we need for each layer are as follows. 1. Young's Modulus along the direction of the fibe r E 1 2. Young's Modulus perpendicular to the direction of the fiber E 2 3. Young's Modulus along the transverse d irection of the fiber E 3 4. Poisson's Ratio of transverse strain in 2 direction to axial strain in 1 direction 12 5. Poisson's Ratio of transverse strain in 3 direction to axial strain in 2 direction 23 6. Poisson's Ratio of transverse strain in 3 direction to axial strain in 1 direction 13 7. Shear Modulus in 1 2 plane G 12 8. Shear Modulus in 2 3 plane G 23 9. Shear Modulus in 1 3 plane G 13 Stiffness Matrix of Each Layer in Material D irection (1 2 direction) Here, each lamina is considered as an orthotropic material whose material properties are collected. The stiffness Matrix of each layer is calculated from the stress strain relation as follows. ( 4 1 1 ) From this relation, the compliance matrix [ S ] is given as ( 4 1 2 ) PAGE 61 61 Where From [S] matrix, the stiffness matrix [Q] is determined by, Stiffness Matrix of Each Layer in Global Co ordinates ( X Y D irection) Since the fiber direction is different for each layer, the stiffness matrix of each layer is different and it depends upon the direction of the fiber. To calculate the total stiffness of the laminate, the stiffness matrix of each layer shou ld be transformed to a global co ordinate system For convenience, they are all transformed to global direction in which the plate or shell is aligned. Hence, the stiffness matrix of each layer is multiplied with the transformation matrix to align it with the global co ordinate system. The transformation matrix is given by ( 4 1 3 ) And th e transformed stiffness matrix of a single layer [ ] = [T] [Q] [T ] T is given by ( 4 1 4 ) Combined Bending Stiffness of the Laminate To find the stiffness of the whole laminate, the stiffness matrix of each layer should be combined together. This can be done by finding the combined bending stiffness of the laminate and convert it i nto stiffness matrix. From the moment equation, PAGE 62 62 For pure bending, for only transverse load, ( 4 1 5 ) Where [D] is the laminate Bending stiffness matrix as explained in chapter 2, which can be calculated as follows. W PAGE 63 63 ( 4 1 6 ) where Therefore [D] matrix takes the form, ( 4 1 7 ) Approximation for Unsymmetrical Laminates Unsymmetrical laminates are the one in which the in plane stresses are developed when the plate is subjected to only transverse load. This is because of the anisotropic property of the laminate which can be seen from the emergence of [B] matrix for unsymmetrical laminates. From Eqn 2.3.23 and Eqn. 2.3.24, the in plane stress resultants and moment resultants depends on all the three matrices (i.e.) [A], [B], and [D] matrices. But for symmetric laminates, as the [B] matrix vanishes, the two equations uncouple and it becomes easy to fin d the stiffness matrix of the laminate. But, for unsymmetrical laminates, as the two equations are coupled to each other by [B] matrix, it becomes important to consider all the three matrices when the stiffness of the laminate is determined. Due to the exi stence of this coupling effect, (i.e.) existence of in plane stresses when the plane is subjected to only bending loads, the magnitude of bending displacement gets reduced. This is because of the presence of in plane stiffness which reduces the bending sti ffness. So the material gets stiffer due to the reduction in the overall bending stiffness. This is known as Stress Stiffening effect. So PAGE 64 64 the overall bending stiffness is reduced and is called Reduced Bending Stiffness Matrix [D r ]. It can be calculated as follows. From the first equation of the laminate constitutive equation, Where [A] and [B] are called laminate stiffness matrices as explained in chapter 2, which are individua lly known as follows: [A]: In plane stiffness matrix [B]: Coupling stiffness matrix These matrices are defined as, PAGE 65 65 ( 4 1 8 ) From the second equation of the laminate constitutive equation, ( 4 1 9 ) Substituting eqn (4.2.1) in eqn (4.2.2), ( 4 1 10 ) Where, The strain energy per unit area of the laminate is given by PAGE 66 66 ( 4 1 11 ) Substituting Eqn (4.2.3) in Eqn (4.2.4), we get the Strain Energy per unit area of the unsymmetrical laminate. ( 4 1 12 ) For symmetric laminate, [B] = 0, hence the constitutive equations become, ( 4 1 13 ) ( 4 1 14 ) Substituting Eqn (4.2.6) and Eqn (4.2.7) in Eqn (4.2.4), we get the Strain Energy per unit area of symmetric laminate ( 4 1 15 ) Comparing Eqn (4.2.5) and Eqn (4.2.8), the bending stiffness of the unsymmetrical laminate is ap proximately given as, ( 4 1 16 ) Due to in plane bending coupling, the bending stiffness of unsymmetrical laminates get reduced due to stress stiffening effect. This equivalent bending stiffness of the laminate can be used to find the equivalent stiffness of the laminate and its equ ivalent material properties. PAGE 67 67 Therefore [D r ] matrix takes the form, ( 4 1 17 ) Therefore, this reduced stiffness [D r ] can be used instead of the bending stiffness [D] from which the stiffness matrix of the laminate can be determined. Stiffness Matrix of the Laminate If the whole laminate is considered as a single layer, becomes zero and h k becomes the total thickness t. Therefore, ( 4 1 18 ) Hence, [Q ] is a 3 x 3 matrix from which [S ] is obtained which is as follows Hence, the equivalent material propert ies of the laminate is determined as, From SDPT, the Shear stiffness matrix [ A s ] is as follows. PAGE 68 68 Where, for each layer can be calculated as follows, If the whole laminate is considered as a single layer, [ A s ] becomes, From is obtained as, As t he stiffness matrix is obt ained from the 3D weak form, it should be 3 dimensional and the following equivalent properties are determined. PAGE 69 69 To complete the equivalent 3D compliance matrix of the laminate the following terms are determined Therefore the equivalent 3D compliance matrix takes the form, The equivalent stiffness matrix of the laminate is obtained as, ( 4 1 19 ) Stiffness M atrix in 2D SDPTShell The equivalent stiffness matrix used in 2D SDPTShell element is directly obtained from in plane, coupling and bending stiffness matrices. Since this element is formulated using Shear Deformable Plate Theory, it also considers transverse n ormal and shear PAGE 70 70 forces and the transverse shear strains are also considered. Hence, the formulation of stiffness matrix includes another 3 new terms A 44 A 4 5 and A 55 which is obtained from the constitutive relations of the s hear stress and strains. The formulation of equivalent stiffness matrix used in 2D SDPTShell is as follows. Equivalent Stiffness Matrix From the constitutive relation of the laminate using Shear Deformable Plate Theory, ( 4 2 1 ) Where [A] [B] [D] and [A s ] are known as follows: [A]: I n plane stiffness matrix [B]: Coupling stiffness matrix [D]: Bending stiffness matrix [A s ]: Shear stiffness matrix These matrices are defined as, PAGE 71 71 Hence, the stiffness matrix used in 2D SDPTShell element is an 8x8 matrix which is in the form ( 4 2 2 ) Hence the equivalent stiffness matrix of the laminate [Q eq ] is a 8X8 matrix and is in the form ( 4 2 3 ) This stiffness matrix [Q eq ] can be used directly in the formulation of the weak form. PAGE 72 72 CHAPTER 5 RESULTS AND DISCUSSI ON In order to check the method which have been used to analyze composite plates and shells and to authenticate the application of B spline Finite Elemen t Method and SDPT Composite Plates and shells, few standard examples wh ich include composite beams, square plates Composite Wing and Composite plate with a hole have been used. In the following chapter, the structural analysis of composite plates is done using two types of elements which have been formulated using two different methods. In the discussion, the element named 3D Shell is a 3D element and is formulated using B Spline Finite Element Method. The element named 2D SDPTShell is a 2D element and is formulated using equation from Shear Deformable Plate Theory (SDPT) considering the concept of constant transverse shear force throughout the thickness of the plate. This chapter is mainly to find the difference among the elements developed and to conclude which element is the best for which cases as well as for the future research. Example 1: Composite Beam A 1 D composite plate [23] whose width is small when compared to the length and the applied load varies in only one direct ion is shown in Figure 5 1 The plate has dimension 1 X 0.1 m 2 and is loaded with an unifor mly distributed load (UDL) of 1000 Pa tra nsversely. T he material properties used are : E 1 = 225 GPa G12 = G 1 3 = 6.9 GPa, G 23 = 3 GPa. The beam is analyzed with 3 types of boundary conditions, Hinged Hinged (HH) Clamped Clamped (CC) and Clamped Free (CF) and the results obtained using Traditional Finite Element Method is compared with the analytical results. PAGE 73 73 Figure 5 1 Composite b eam A B C Figure 5 2 Mesh pattern for Composite b eam A ) T raditional FEA (a baqus). B ) 3D Shell C ) 2D SDPTShell A B C Figure 5 3 Tradit ional FEA (aba qus) results of Composite b eam A ) Hinged Hinged B) Clamped Clamped. C ) Clamped Free 0.1 1 PAGE 74 74 A B C Figure 5 4 3D Shell results of Composite beam A ) Hinged Hinged. B) Clamped Clamped. C ) Clamped Free A B C Figure 5 5 2D S DPTShell results of Composite beam A) Hinged Hinged. B) Clamped Clamped. C ) Clamped Free PAGE 75 75 Table 5 1 Maximum transverse displacements of a HH beam Solution (10 4 m) Traditional FEA (abaqus) (10 4 m) 3D Shell (10 4 m) 5x1 10x1 20x2 33x3 40x4 5x1 10x1 2 0x2 33x3 40x4 0 6.98 6.989 6.992 6.991 6.991 6.991 6.981 7.036 7.086 7.086 7.084 90 173.7 173.7 173.7 173.7 173.7 173.7 175.9 179.2 180.5 180.5 180.6 (+45/ 45) s 99.63 95.89 95.98 94.55 94.55 94.56 93.36 94.57 95.56 95.68 95.66 Table 5 1. Continue d Solution (10 4 m) 2D SDPTShell (10 4 m) 5x1 10x1 20x2 33x3 40x4 0 6.98 7.046 7.006 6.995 6.993 6.992 90 173.7 175.1 174.1 173.8 173.7 173.7 (+45/ 45) s 99.63 94.17 93.61 93.87 94.05 94.12 Table 5 2 Maximum transverse displacements of a C C beam Solution (10 4 m) Traditional FEA (abaqus) (10 4 m) 3D Shell (10 4 m) 5x1 10x1 20x2 33x3 40x4 5x1 10x1 20x2 33x3 40x4 0 1.42 1.423 1.426 1.426 1.426 1.426 1.946 1.813 1.696 1.595 1.247 90 34.8 34.73 34.8 34.8 34.8 34.8 32.03 34.34 34.77 35.13 34.77 (+45/ 45) s 19.92 13.99 15.76 15.04 15.15 15.2 18.59 16.68 16.34 16.01 15.88 PAGE 76 76 Table 5 2 Continued Solution (10 4 m) 2D SDPTShell (10 4 m) 5x1 10x1 20x2 33x3 40x4 0 1.42 1.48 1.44 1.43 1.427 1.427 90 34.8 36.15 35.18 34.9 34.85 34 .83 (+45/ 45) s 19.92 13.85 14.43 14.75 14.92 14.98 Table 5 3 Maximum transverse displacements of a CF beam Table 5 3. Continued Solution (10 4 m) Traditional FEA (abaqus) (10 4 m) 3D Shell (10 4 m) 5x1 10x1 20x2 33x3 40x4 5x1 10x1 20x2 33x3 40x4 0 67 66.8 66.8 66.8 66.8 66.8 79.68 76.18 72.83 67.84 62.68 90 1670 1666 1667 1667 1667 1667 1667 1667 1667 1674 1667 (+45/ 45) s 954 805.3 845.7 829.7 832.7 834 921.2 871.6 861.2 853.6 850.5 Solution (10 4 m) 2D SDPTShell (10 4 m) 5x1 10x1 20x2 33x3 40x4 0 67 67.01 66.83 66.79 66.78 66.78 90 1670 1673 1668 1667 1667 1667 (+45/ 45) s 954 795.7 812.8 821.4 826.3 828.2 PAGE 77 77 A B Figure 5 6. Convergence plots of Hinged Hinged beam. A) 0. B) 90. C) (+45/ 45) s PAGE 78 78 C Figure 5 6 Continued A Figure 5 7. Convergence plots of Clamped Clamped beam. A) 0. B) 90. C) (+45/ 45) s PAGE 79 79 B C Figure 5 7 Continued PAGE 80 80 A B Figure 5 8. Convergence plots of Clamped Free beam. A) 0. B) 90. C) (+45/ 45) s PAGE 81 81 C Figure 5 8 Continued PAGE 82 82 Example 2: Square Composite P late A square composite plate [23] of dimensi on 1X1 m 2 with thickness 0.01 m is subjected to uniform pressure of 1000 Pa and its maximum deflection at the center is determined. The material properties used are : E 1 = 225 GPa, E 2 = 9 GPa 12 = 0.25, G 12 = G 13 = 4. 5 GPa, G 23 = 1.8 GPa. It is simply supported on all its edges (i.e.) at x = 0.5 and 0.5, v 0 = w 0 = M xx = 0 & at y = 0.5 and 0.5, u 0 = w o = M yy = 0. Using these boundary conditions and material properties, results are obtained using traditional FEA (abaqus) and are compared with the analytical solutions. Figure 5 9 Composite s quare p late 1 1 PAGE 83 83 A B C Figure 5 10 Mesh pattern for Composite square p late A ) Traditional FEA (aba qus). B ) 3D Shell C ) 2D SDPTShell A B C Figure 5 11 A nalysis results of a Composite square p late A ) Tradi tional FEA (abaqus ) B ) 3D Shell C ) 2D SDPTShell PAGE 84 84 Table 5 4 Maximum t ransverse displacements of a S S square pla te Soln. (10 4 m) Traditional FEA (abaqus) (10 4 m) 3D Shell (10 4 m) 2x2 5x5 10x10 15x15 2x2 5x5 10x10 15x15 0 7.22 7.384 7.097 7.267 7.246 6. 856 7.336 7.306 7.331 0/90 18.84 17.86 18.18 18.96 18.86 15.64 18.47 18.82 18.91 0/90/0 7.4 7.7 7.457 7.702 7.673 6.722 7.387 7.434 7.462 (0/90)s 7.55 7.518 7.33 7.626 7.591 6.651 7.452 7.56 7.591 (0/90)4 7.94 7.798 7.649 8.001 7.959 6.848 7.787 7.933 7.967 0/90/90/0/90/90/0 7.63 7.497 7.345 7.677 7.638 6.608 7.494 7.632 7.665 0/90/0/90/0 7.6 7.516 7.351 7.667 7.629 6.644 7.484 7.609 7.641 ( 45/45) 11.45 8.95 7.947 8.35 8.304 10.12 11.39 11.67 11.77 ( 45/45)4 4.31 4.802 4.307 4.493 4.472 3.951 4.366 4.407 4.5 Table 5 4. Continued 2D SDPTShell (10 4 m) 2x2 5x5 10x10 15x15 0 8.076 7.304 7.269 7.265 0/90 20.6 19.03 18.93 18.93 0/90/0 8.254 7.519 7.48 7.475 (0/90)s 8.33 7.652 7.611 7.605 (0/90)4 8.695 8.038 7.994 7.987 0/90/90/0/90/90/0 8.346 7.712 7.67 7.663 0/90/0/90/0 8.353 7.699 7.657 7.651 ( 45/45) 9.469 7.928 7.838 7.825 ( 45/45)4 4.781 4.39 4.418 4.436 PAGE 85 85 A B Figure 5 12. Convergence plots of S.S Composite Square Plate. A) 0. B) (0/90). C) (0/90/0). D) (0/90) s E) (0/90) 4 F) (0/90/90/0/90/90/0). G) (0/90/0/90/0). H) (+45/ 45). I) (+45/ 45) 4 PAGE 86 86 C D Figure 5 12. Continued PAGE 87 87 E F Figure 5 12. Continued PAGE 88 88 G H Figure 5 12. Continued PAGE 89 89 I Figure 5 1 2 Continued PAGE 90 90 Example 3: Composite Wing of Micro A ir V ehicle Micro Air Vehicles ( MAV's) are UAV's of comparatively small size which have been used for several applications such as surveillance, bird flight or insect flight study and aerial photography In this case, a MAV wing is taken for analysis which is made up of Ca rbon Epoxy Composite Laminate whose material properties are: The wing is clamped at one edge. Therefore the boundary conditions are: At one edge,(x=0); It is made up of (26.5) s laminate with lamina th ickness of 0. 125mm. It is subjected to uniform pressure of 1Pa and the maximum deflection is determined. Figure 5 1 3 Composite w ing A B C Figure 5 14 Mesh pattern for composite w ing A) Traditional FEA (abaqus). B ) 3D Shell C ) 2D SDPTSh ell PAGE 91 91 A B C Figure 5 15 Analysis results of a composite wing A ) Tradi tional FEA (abaqus). B ) 3D Shell C ) 2D SDPTShell T able 5 5 Maximum t ransverse d isplacement of a composite w ing Mesh Density Traditional FEA (10 3 m) 3D Shell (10 3 m) 2D SDPTShell (10 3 m) 4x2 2.708 2.346 2.581 6x3 2.608 2.394 2.585 10x4 2.59 2.396 2.591 21x7 2.582 2.028 2.591 Figu re 5 16 Convergence plot of a composite w ing PAGE 92 92 Example 4: Composite Plate with a Hole A composite plate of size 2 X 2 m 2 with a hol e of diameter 1 m at the center is subjected to uniform pressure load of 100 Pa and its maximum displacement is determined. This plate is simply supported at all the 4 edges (i.e.) at x = 1 and 1, v 0 = w 0 = 0 & at y = 1 and 1, u 0 = w o = 0. The mechanical properties of the laminate of thickness 0.012 m are as follows: E 1 = 130 GPa, E 2 = E 3 = 10 12 = 0.3 5, G 12 = G 13 = 5 GPa, G 23 = 3.7 GPa Figure 5 17 Composite plate with a h ole Two types of stacking sequences which are (i) [0/90] 2s and (ii) [45 2 / 90/0] s are tested and analyzed The Traditional FEA results using Abaqus are extracted and compared with the IBFEM results from two types of elements developed and the method explained in this thesis is validated. 2 2 1 PAGE 93 93 A B C Figure 5 18 Mesh patte rn for c omposite plate with a hole A ) Traditional FEA (abaqus), B ) 3D Shell C ) 2D SDPTShell A B C Figure 5 19 Analysis results of Composite plate with a hole A ) Tradi tional FEA (abaqus) results. B ) 3D Shell results. C ) 2D SDPTShell results PAGE 94 94 Table 5 6 Maximum t ransverse displacement of a c omposite p late with hole Element Size (m) Traditional FEA (10 2 m) 3D Shell (10 2 m) 2D SDPTShell (10 2 m) 0 [0/90] 2s [45 2 /90/0] s 0 [0/90] 2s [45 2 /90/0] s 0 [0/90] 2s [45 2 /90/0] s 2x2 1.282 1.26 0.6815 0 .7849 0.7391 0. 50 43 1.207 1.201 0.8494 4x4 1.394 1.375 0.6977 1.092 1.082 0.71 74 1.452 1.446 0.8146 8x8 1.431 1.41 0.7924 1.432 1.415 0.8311 1.483 1.443 0.8354 10x10 1.432 1.412 0.8148 1.451 1.431 0.83 41 1.443 1.42 0.8358 PAGE 95 95 A B Figure 5 20. Convergence plots of S.S composite square plate with hole. A) 0. B) (0/90) 2s C) (45 2 /90/0) s PAGE 96 96 C Figure 5 20 Continued PAGE 97 97 CHAPTER 6 CONCLUSION Su mmary In this thesis, the use of IBFEM to model composite plates is studied Initially the equations requi red for general formulation of Finite Element Method in laminated composites is explained. It includes the derivation of equations used in FEM formulation using Classical Lamination Plate Theory (CLPT) and Shear Deformable Plate Theory (SDPT) This is foll owed by an introduction to the Implicit Boundary Finite Element Method (IBFEM) and mixed formulation. The main advantage of this method is that it uses uniform structured mesh for the analysis and such a mesh can be generated automatically. As the mesh doe s not have to conform to the geometry, it is easier to generate the mesh. The elements in the mesh are undistorted and therefore integration is more accurate. The convergence is also quicker with a low mesh density. T he motive of this thesis is to develop two types of elements namely 3D shell element formulation and 2D plate element using mixed formulation The results from the se two methods were computed and a comparative study was done using its results and its convergence plots. The main advantage of 2D elements is that it takes much less computational time. The computational time ta ken by 2D element is nearly 5 to 10 times lesser that of 3D element. The 2D element is a 9 node biquadratic element while the 3D element is a 64 node cubic B spline element The 2D plate element considers the transverse shear strain to be constant This 2D element avoid s shear locking and volumetric locking phenomena using mixed formulation. These type of locking are solved using special methods such as reduced integration i n case of traditional FEA. PAGE 98 98 The main advantage of developing 3D elements is that it can be used in mixed assemblies which contain both plates and 3D structures. For example, if there is a need to perform structural analysis in some structures like an aircr aft wing structure with stiffeners, or a cantilever plate supported by pillars, or a circular plate surrounded with frame, 3D element can be used. This is because, in this case, a 3D mesh is used for both the shell like regions and the solid regions This is not possible in case of 2D elements because it can only be used to model thin plate like structures and are not compatible with 3D elements Scope of Future Research The IBFEM was extended to model composite plates. B spline elements have been successfu lly formulated for composite plates for all possible kinds of stacking sequences by transforming the stiffness matrix with different orientations in a single plane to a global co ordinate. This concept can be further extended to 3D transformation of the st iffness matrix and can be developed for analyzing composite curved shells. This method can also be used for analyzing composite curved shells which uses SDPT approach. A 3D element can be developed which uses mixed formulation using SDPT and can be used to analyze composite curved shells. In this research a constant shear correction factor of 5/6 is used. But the actual shear correction factor of the laminate can be calculated and can be used in the formulation. This can further reduce the error percentage of the results obtained from IBFEM. This procedure is effective for structural analysis of composite laminates. This can be extended to thermal stress analysis as well. This can be further extended to coupled field analysis such as when a system is subjec ted to both stresses and heat flux PAGE 99 99 The current research deals only with the structural analysis to find the maximum displacement and stress distribution in the whole laminate. It can be further elaborated such that the stresses in each layer can be found and the exact location of failure of a laminate can be determined. A thorough failure analysis can be done by taking maximum tensile, compressive and shear strength as inputs from the user. Hence, all these steps can be taken into consideration for the fu ture research in the same field. PAGE 100 100 APPENDIX A PROPERTIES OF MATRIX AND FIBERS Table A 1 Properties of fibers Material (GPa) Tensile Strength (G Pa) Density (g/cc) Glass E glass 69 2.4 2.5 Glass S glass 86 3.5 2.5 Carbon HS 160 270 3.5 1.8 Carbon HM 325 440 2 .5 1.8 Carbon UHM 440 + 2.0 2.0 Aramid LM 60 3.6 1.45 Aramid HM 120 3.1 1.45 Aramid UHM 180 3.4 1.47 Alumina 380 1.7 2.7 Boron 400 420 3.5 2.5 Quartz 69 0.89 2.2 Table A 2 Properties of matrix material s Material (GPa) Tensile Strength (GPa) Density (g/cc) Nylon 2.0 3.6 0.082 1.15 Polyethylene 0.18 1.6 0.015 0.9 1.4 Polypropylene 1.4 0.033 0.9 1.24 Epoxy 3.5 0.069 1.25 Phenolic 3.0 0.006 1.35 PAGE 101 101 APPENDIX B MATERIAL PROPERTIES OF VARIO US LAMINATES Table B 1 Properties of various laminates Material E 1 (GPa) E 2 (GPa) G 12 (GPa) 12 (no units) Density (g/cc) C arbon/PEEK (unidirectional) 134 8.9 5.1 0.28 1.6 Carbon/Epoxy ( Woven) 56.3 55.2 2.85 0.042 1.45 Carbon/Epoxy (unidirectional) 300 6.5 5.0 0.23 1.59 E Glass/Epoxy (unidirectional) 39 8.6 3.8 0.28 2.10 E Glass/Epoxy(woven) 25 25 4.0 0.20 1.90 S Glass/Epoxy (unidirectional) 43 8.9 4.5 0.27 2.00 S Glass/Epoxy (Woven) 29.7 29.7 5.3 0.17 2.20 Kevlar/Epoxy ( unidirectional) 87 5.5 2.2 0.34 1.38 Kevlar/Epoxy (Woven) 30 30 5.0 0.30 1.4 Boron/ Epoxy (unidirectional) 200 15 5.0 0.23 2.00 PAGE 102 102 LIST OF REFERENCES [1] Jacob Fish and Ted Belytschko, A First Course in Finite Elements, JOHN WILEY & SONS LTD., 2007 [2] K.J. Bathe, Finite Element Procedures, PRENTICE HALL, 1996 [3] Christophe Dauxz, Nicolas Moes, John Dolbow,Natarajan Sukumar k and T ed Belytschko Arbitrary branched and intersecting cracks with the extended finite element method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2000; 48:1741 1760 [4] P. Lancaster and K. Salkauskas Surfaces Generated by Moving Least Squares Meth ods MATHEMATICS OF COMPUTATION, VOL. 37, NO. 155, July 1981 [5] Y.Y.Lu, T.Belytschko, L.Gu, A new implementation of Element Free Galerkin Method (EFGM), COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 113, 1994, 397 414 [6] Satya N. Atluri, The Meshless Local Petrov Galerkin (MLPG)Method for Domain & BIE Discretizations INTERNATIONAL WORKSHOP ON MESH FREE METHODS 2003 [7] Ashok V. Kumar and Jongho Lee, Step Function Representation of Solid Models and Application to Mesh Free Engineering Analysis, AS ME, 128( 1). Pp. 46 56. [8] Ashok V. Kumar and Premdheepak S. Periyasamy, Mesh Independent Analysis of Shell Like Structures, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS AND ENGINEERING, 2011 [9] Ravi K. Burla and Ashok V. Kumar, Implicit boundary method for analysis using uniform B spline basis and structured grid, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 2008 [10] Ashok V. Kumar, Ravi K. Burla, Sanjeev Padmanaban and Linxia Gu, Finite Element Analysis using Nonconforming Mesh, JOURNAL OF COMPUTING AND INFORM ATION SCIENCE IN ENGINEERING, Sep. 2008 [11] E. Carrera, Theories and Finite Elements for Multilayered, Anisotropic Composite Plates and Shells, ARCHIVES IN COMPUTATIONAL METHODS IN ENGINEERING, VOL. 9, 2, 87 140, 2002 [12] Y.X. Zhang and C.H. Yang, Recent Developm ents in Finite Element Analysis for Laminated Composite Plates, COMPOSITE STRUCTURES, 88, 2009 [13] M. Nurhaniza, M.K.A. Ariffin, Aidy Ali, F. Mustapha and A. W. Noraini Finite Element Analysis of Composite Materials for Aerospace Applications, 9 TH NATIONAL S YMPOSIUM ON POLYMERIC MATERIALS, 2009 PAGE 103 103 [14] P.F. Liu J.Y. Zheng Recent Developments on Damage Modeling and Finite Element Analysis for Composite Plates: A review, MATERIALS AND DESIGN, 2010 [15] J.N. Reddy, Bending of Laminated Anisotropic Shells by a Shear Deforma ble Finite Element, FIBER SCIENCE AND TECHNOLOGY 17, 1982 [16] R.J. Razzaq, A. El Zafrany Non linear stress analysis of composite layered plates and shells using a mesh reduction method, ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 29, 2005 [17] J. Sladek, V. Slad ek and S.N. Atluri Meshless Local Petrov Galerkin (MLPG) Method in Anisotropic Elasticity, COMPUTER MODELING IN ENGINEERING AND SCIENCES (CMES) VOL.6, NO. 5, 2004 [18] J. Belinha and L.M.J.S. Dinis Non linear analysis of plates and laminates using the Elemen t Free Galerkin Method (EFGM), COMPOSITE STRUCTURES 78, 2007 [19] B.N. Pandya and T. Kant, Finite Element Analysis of Composite Plates using a High order displacement model, COMPOSITES SCIENCE AND TECHNOLOGY 32, 1988 [20] J.R. Xiao, R.C. Batra, D.F. Gilhooley, J.W. Gillespie Jr., M.A. McCarthy Analysi s of thick plates by using a Higher Order Shear and Normal Deformable Plate Theory (HOSNDPT) and a Meshless Local Petrov Galerkin (MLPG) Method with Radial Basis Functions, COMPUTER METHODS IN APPLIED MECHANICS AND ENGI NEERING 196, 2007 [21] Zafer Gurdal, Rapheal T. Haftka and Prabhat Hajela, Design and Optimization of Laminated Composite Materials, JOHN WILEY & SONS LTD., 1999 [22] Robert M. Jones, Mechanics of Composite Materials, TAYLOR & FRANCIS GROUP, 1999 [23] J.N. Reddy, Mechani cs of Laminated Composite Plates and Shells: Theory and Analysis, CRC PRESS, 1997 [24] Autar K. Kaw, Mechanics of Composite Materials, TAYLOR & FRANCIS GROUP, 2006 [25] Ronald F. Gibson, Principles of Composite Material Mechanics, CRC PRESS, 2007 [26] T. Bely tschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Meshless Methods: An overview and recent developments COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 1996 PAGE 104 104 [27] J.J.Monaghan, An Introduction to SPH COMPUTER PHYSICS COMMUNICATIONS 48 (1988) 89 96 [28] Kantorovich LV, Krylov VI, Approximate Methods for Higher Analysis INTERSCIENCE PUBLISHERS INC., 1958 [29] V.Shapiro, I.Tsukanov Meshfree simulation of deforming domains COMPUTER AIDED DESIGN 31 (1999) 459 471 [30] Lidia Kurpa, Galina Pilgun, Eduard Ventsel, Application of the R function method to nonlinear vibrations of thin plates of arbitrary shape JOURNAL OF SOUND AND VIBRATION 284 (2005) 379 392 [31] Tae Kyoung Uhm and Sung Kie Youn T Spline Finite Element Method for the analysis of shell structures, INTERNA TIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, June 2009 [32] James D. Lee, Three Dimensional Finite Element Analysis of Layered Fiber reinforced Composite Materials, COMPUTERS AND STRUCTURES, 1980 [33] Tara pada Roy, P.Manikandan Debabrata Chakraborty Impro ved Shell Finite Element for Piezothermoelastic Analysis of Smart Fiber Reinforced Composite Structures, FINITE ELEMENTS IN ANALYSIS AND DESIGN, 2010 [34] Y. Basar, M. Itskov, A. Eckstein Composite laminates : nonlinear interlaminar stress analysis by multi lay er shell elements, COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 2000 [35] R.L. Spilker, S.C. Chou, O. Orringer, Alternate Hybrid Stress Elements for Analysis of Multilayered Composite Plates, JOURNAL OF COMPOSITE MATERIALS, 1977 [36] Juan Miquel, Salvador Botello and Eugenio Onate, A finite element formulation for analysis of composite laminate shells, INTERNATIONAL CENTER FOR NUMERICAL METHODS IN ENGINEERING [37] Sun C.T., Mechanics of Aircraft Structures, JOHN WILEY & SONS LTD., 1998 [38] O.O. Ochoa and J.N. Reddy, Finite Element Analysis of Composite Laminates, KLUWER ACADEMIC PUBLISHERS, 1992 [39] Premdheepak Salem Periyasamy, Finite Element Analysis of shell like structures using Implicit Boundary Method, UNIVERSITY OF FLORIDA, 2012 [40] Hailong Chen, Mixed formulation us ing Implicit Boundary Finite Element Method UNIVERSITY OF FLORIDA, 2012 PAGE 105 105 BIOGRAPHICAL SKETCH Vignesh Solai Rameshbabu was born and brought up in a small industrial town called Neyveli, in the state of Tamil Nadu, India. He did his high school in Jawahar H igher Secondary School in Neyveli and graduated in 2005. Then, he joined Hindustan College of Engineering, Anna University in the yea r of 2005 and graduated with a ngineering with the specialization in Aircraft Structures Aerodynamics and Aircraft Propulsion in 2009. Meanwhile, he gained some industrial experience by undergoing a 1 month industrial training in June 2007 at the Ground Test Centre, Rotary Wing Research and Development Centre, Hindustan Aeronautics Ltd, Bang alore, India. Then, h e worked as a Design Engineer in the Design Department, in a manufacturing industry called DMW CNC Solutions Pvt. Ltd., which is located in a small town called Perundurai, Tamil Nadu India. Then, he received a pace e ngineering in the summer of 2012 from the University of Florida, Gainesville, Florida, USA. His areas of specialization include Finite Element Method, Aeros pace Composites, Computational M ethods in Design and Manufacturing and developing softwares us ed in Mechanical/Aerospace/Automotive/Civil Engineering using Object Orientation Programming Techniques |