Citation
Mesh Independent Analysis of 3D Shell-Like Laminate Structures Using ABD-Equivalent Material Model

Material Information

Title:
Mesh Independent Analysis of 3D Shell-Like Laminate Structures Using ABD-Equivalent Material Model
Creator:
Liang, Li
Publisher:
University of Florida
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
KUMAR,ASHOK V
Committee Co-Chair:
KIM,NAM HO

Subjects

Subjects / Keywords:
3d-shell
abd-matrix
composite
fea
ibfem
laminate

Notes

General Note:
Mesh Independent Finite Element Analysis uses geometry imported from CAD software to perform analysis without generating a conforming mesh to approximate the geometry, as in the traditional Finite Element Method (FEA). The Implicit Boundary Finite Element Method (IBFEM) is a numerical approach, where approximate step functions are created to impose the boundary conditions by constraining the displacement field in the prescribed manner. It uses an automatically generated background mesh, which is independent of the geometry, to avoid the difficulty of mesh generation and the error introduced by the traditional mesh that have distorted elements to conform the geometry. For analysis of shell-like structures, it uses 3D stress-strain relationship and the general principle of virtual work define a 3D-shell element. B-spline basis functions are used to interpolate the displacement field within the shell so that tangent (C1) continuity is guaranteed. In this thesis, the 3D shell element in IBFEM is extend for modeling shell-like structures that are made of composite laminate using ABD-equivalent material model the for composite laminate. Laminate can be defined by specifying the sequence of laminas in the laminate, the fiber orientations, the material properties and the thickness of each lamina. The effective properties (ABD matrix) of the laminate can be determined by combining properties of each lamina. For 3D shell elements, we need an equivalent stress-strain relation for the laminate. An ABD-equivalent 3D stress-strain relation for an equivalent 3-layer composite laminate (ABD-equivalent Material model for laminate) can be built. This stress-strain relation for the laminate is then transformed into global coordinate system. Some practical plate and shell examples with different geometry and boundary conditions are analyzed and the results are compared with analytical solutions, if available, as well as results obtained from commercial FEA software. Finally, the advantages and limits of this method are discussed.

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UFRGP
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All applicable rights reserved by the source institution and holding location.
Embargo Date:
5/31/2018

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MESH INDEPENDENT ANALYSIS OF 3D SHEL L LIKE LAMINATE STRUCTURES USING ABD EQUIVALENT MATERIAL MODEL By LI LIANG 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 2016

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2016 Li Liang

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To my Mom Dad and all of the teachers

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4 ACKNOWLEDGMENTS I would like to express the most sincere gratitude to my advisor, Dr. Ashok V. Kumar. His support helps me not only for my thesis and research but also for my entire study life of the graduate school I t would be impossible for me to finish such a research witho ut him. I extend my greatest thanks to the member of my supervisory committee, Prof. Nam Ho Kim for his guidance during my thesis. It is an honor for me to work in such a team which de veloped m y critical thinking and research ability. I thank Prof. Peter G. Ifju and Prof. Bhavani V. Sankar for inciting interest in the field of c omposites material and Finite Element Method during graduate study at the University of Florida. F inally, I would like to thank my Mom and Dad who bore, raised me a nd taught me the meaning s of life.

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5 TABLE OF CONTENTS page ACKNOWL EDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 LIST OF ABBREVIATIONS ................................ ................................ ............................. 9 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 Goals and Objectives ................................ ................................ .............................. 13 Goals ................................ ................................ ................................ ................ 13 Objectives ................................ ................................ ................................ ......... 13 Outline ................................ ................................ ................................ .................... 14 2 MESH INDEPENDENT FINITE ELEMENT METHOD ................................ ............ 16 Overview ................................ ................................ ................................ ................. 16 Formulation of 3D Element (3D Shell) ................................ ................................ .... 18 Stiffnes s Matrix for 3D Shell Element ................................ ................................ ..... 20 Boundary Condition ................................ ................................ ................................ 24 Clamped ................................ ................................ ................................ ........... 24 Simply Supported ................................ ................................ ............................. 25 Symmetry Boundary Condition ................................ ................................ ......... 26 3 ANALYSIS OF COMPOSITE MATERIAL ................................ ............................... 27 Overview ................................ ................................ ................................ ................. 27 Stress Strain Relations of a Composite Lamina ................................ ..................... 28 Classical La minated Plate Theory (CLPT) ................................ .............................. 32 Shear Deformable Plate Theory (SDPT) ................................ ................................ 40 4 ABD EQUIVALENT MATER IAL MODEL OF COMPOSITE LAMINATE ................. 43 Overview ................................ ................................ ................................ ................. 43 The ABD Equivalent M aterial Model of Laminate ................................ ................... 44 Local and Global Stiffness Matrix ................................ ................................ ........... 48

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6 5 RESULTS AND DISCUSSION ................................ ................................ ............... 56 Ov erview ................................ ................................ ................................ ................. 56 Example of Square Plate ................................ ................................ ........................ 56 Example of Pressured Cylinder ................................ ................................ .............. 60 Example of Scordelis Lo Roof ................................ ................................ ................ 62 Example of Doubly Curved Shell ................................ ................................ ............ 65 6 CONCLUSION ................................ ................................ ................................ ........ 68 Summary ................................ ................................ ................................ ................ 68 Future Work ................................ ................................ ................................ ............ 69 LIST OF REFERENC ES ................................ ................................ ............................... 70 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 73

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7 LIST OF TABLES Table page 5 1 Maximum Displacement ( 10 7 inch) =1/10 ................................ ........................ 58 5 2 Maximum Displacement ( 10 4 inch) =1/100 ................................ ...................... 58 5 3 Maximum Displacement ( 10 1 inch) =1/1000 ................................ .................... 59 5 4 Maximum Radius Displacement of Cylinder Subjected to Internal Pressure (10 1 inch) ................................ ................................ ................................ ........... 62 5 5 Maximum Displacement of Scordelis Lo Roof (inch) =100 ............................... 64 5 6 Maximum Displacement of Scordelis Lo Roof (10 1 inch) =50 .......................... 64 5 7 Maximum Displacement of Scordelis Lo Roof (10 2 inch) =20 .......................... 65 5 8 Maximum Displacement (Non dimensionalized) [0/90] T ................................ ..... 67 5 9 Maximum Displacement (Non dimensionalized) [0/90] S ................................ ..... 67

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8 LIST OF FIGURES Figure page 2 1 Mesh Generation for A Circular ................................ ................................ .......... 17 2 2 A Shell Like Structure in IBFEM ................................ ................................ ......... 19 4 1 Coordinate Systems ................................ ................................ ........................... 48 5 1 Ge ometry and Load of Clamped Square Plate ................................ ................... 57 5 2 Clamped Square Plate in IBFEM ................................ ................................ ........ 58 5 3 Converge Plot of Strain Energy ................................ ................................ .......... 60 5 4 Converge Plot of Maximum Displacement ................................ .......................... 60 5 5 Geometry and Load of Pressured Cylinder ................................ ........................ 61 5 6 Pressured Cylinder in IBFEM ................................ ................................ ............. 62 5 7 Geometry and Load of Scordelis Lo Roof ................................ .......................... 63 5 8 Scordelis Lo Roof in IBFEM ................................ ................................ ............... 64 5 9 Geometry and Load of Doubly Curved Shell ................................ ...................... 66 5 10 Doubly Curved Shell in IBFEM ................................ ................................ ........... 67

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9 LIST OF ABBREVIATION S CLPT Classical Laminated Plate Theory EBCs Essential Boundary Conditions FEA Finite Element Analysis FEM Finite Element Method IBFEM Implicit Boundary Finite Element Method SDPT Shear Deformable Plate Theory SnS Scan and Solve SW SolidWorks

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10 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science MESH INDEPENDENT ANALYSIS OF 3D SHEL L LIKE LAMINATE STRUCTURES USING ABD EQUIVALENT MATERIAL MODEL By Li Liang May 2016 Chair: Ashok V. Kumar Major: Mechanical Engineering Mesh Independent Finite Element Analysis uses geometry imported from CAD software to perfor m analysis without generating a conforming mesh to approximate the geometry as in the traditional Finite Element Method ( FEA ) The Implicit Boundary Finite Element Method (IBFEM) is a numerical approach where approximate step function s are created to impose the boundary conditions by constrain ing the dis placement field in the prescribed manner It uses an automatically generated background mesh which is independent of the geometry to avoid the difficulty of mesh generation and the error introduced by the traditional mesh that have distorted elements to conform the geometry For analysis of shell like structures, it uses 3D stress strain relation ship and the general principle of virtual work define a 3D shell element B spline basis functions are used to interpolate the displacement field within the shell so that tangent (C 1 ) continu i ty is guarantee d In this thesis, the 3D shell element in IBFEM is extend for modeling shell like structures that are made of composite laminate using ABD equivalent material model the for composite laminate. L aminat e can be defined by specifying the sequence of laminas in the laminate, the fiber orientation s the material properties and the thick ness

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11 of each lamina. T he effective properties (ABD matrix) of the laminate can be determi ned by combining properties of each lamina For 3D shell elements, we need an equivalent stress strain relation for the laminate. An ABD equivalent 3D stress strain relation for an equivalent 3 layer composite laminate (ABD equivalent Material model for laminate) can be built This stress strain relation for the laminate is then transformed into global coordinate system S ome pract ical plate and shell examples with different geometry and boundary conditions are analyzed and t he results are compared with analytical solutions, if avai lable, as well as results obtained from commercial FEA software. Finally, the advantag es and limits of this method are discussed.

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12 CHAPTER 1 INTRODUCTION Traditional ly, the Finite Element Method (FEM) [1, 2] uses a mesh to approximate the geometry of the structure to be analyzed Although this is an effective method, it has some difficulties as well For instance automatically generating a mesh for complex geometry is difficult. R egenerat ion of the mesh is n eeded for large deformation analysis, because the original elements are distorted significantly when the structure is loaded and also analysis of c rack propagation So me methods including meshless and mesh independent methods were invented to avoid mesh g eneration process Meshless methods use nodes scattered with in the geometry to perform analysis without connecting those nodes to form elements. Examples inc luding Moving Least Square method [ 3 ] Ele ment Free Galerkin Method [ 4 ] Meshl ess Local Petrov Galerkin Method [ 5 ] and so on Those methods are effective in many cases but they have their own difficulties as well An other alternative approach is Implicit Boundary Finite Element Method (IBFEM) [ 6 9 ] which imposes the Essential Boundary Condition s (EBCs) by employing step functions to construct test and trial functions It generate s the mesh automatically and the mesh is independent of the geometry so that it is not necessary to conform to the geometry. T he accuracy can be improved because accurate geometry will be directly used for the analysis Meanwhile the difficulties of traditional mesh can be overcome since the mesh is independent of the geometry especially for complex geometry. The composite mate rial can provide high strength to weigh t and stiffness to weight ratio along with lots of other advantages which give it wide usage in aerospace,

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13 sports and other fields where high performance materials is needed [ 10, 11 ]. In general, a composite l aminate is difficult to analyze using 3D elements since it can be formed by a large number of layers which could have different material properties Therefore e ach layer in a laminate will need to be analyzed independently. All of th e se reasons will make the num erical computation very computation expensive One way to overcome th is difficulty is to employ ABD equivalent model for laminate where original multi ply laminate is replaced by a new laminate that has a fewer number of plies (3 plies) which will behave similar to the original multi ply laminate base d on the effective laminate properties [ 1 2 ] By doing that, only three laminae, instead of the origin number of lamina e which can easily be over a hundred need to be analyzed Thus, the total time tak en for computation can be reduced dramatically and the macro behavior of the laminate can be captured as well. Meanwhile it can provide stress strain relationship for 3D shell elements in IBFEM which use material stress strain relationship directly, inste ad of the effective propriety (ABD matrix) of laminate Goals and Objectives Goals The main goal of this thesis is to extend the ABD equivalent material model of composite laminates and adapt it to 3D shell elements in Implicit Boundary F inite Element Method and use it to analyze shell like structure s which a re made of composite laminate Objectives Implement the ABD equivalent material model of composite laminates for 3 D shell elements to model flat composite laminates. T he geometry is modeled as a fla t surface which is made in CAD software.

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14 Implement ABD equivalent material model of composite laminates for 3D shell elements to model a curved shell that made of composite laminates. The geometry is modeled as a curved surface in CAD software. Compare res ul ts with analytical solution, if available and commercial FEA software where the geometry is approximated by mesh. Outline The remaining portion of the thesis is organized as follows: In Chapter 2, the Implicit Boundary Finite Element Method is discussed especially ho w it is applied to the 3D shell like structure. Details about how to derive the weak form and construct the stiffness matrix for 3D shell are also discussed. In Chapter 3, the properties and applications of composite structures are discussed followed by a detail ed discussion of the Classical Laminated Plate Theory (CLPT) or C lassical Lamination Theory as well as the Shear Deformable Plate Theory (SDPT) which are essential for set ting up ABD equivalent model of composite laminate In Chapter 4, the disadvantages of the traditional FEA laminate simulation and the motivation for construct ing ABD equivalent model o f composite laminate for 3D FEA are discussed. Then the details of how t he ABD e quivalent m aterial properties of composite l aminate ar e computed a nd how it been used for 3D shell like structure finite element analysis is discussed In addition, the transformation between material coordinate system, element coordinate system and the glob al coordinate system are discussed in detail. In Cha pter 5, examples of applying the ABD equivalent material model of composite laminates for 3D FEA in mesh independent finite element method are listed and results have been compared and discussed. The first example is a square composite p late which have all four edges fixed and a uniform pressure applied on the

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15 top of the plate uniformly. The second example is a composite cylinder has been fixed along it s outer edge and subjected to an internal pressure. The third example is the Barrel Vau lt Problem which is a vault been supported at it s edge s and loaded with a vertical pressure. And the fourth example is a Doubly C urved shell loaded with pressure that is acting outward.

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16 CHAPTER 2 MESH INDEPENDENT FINITE ELEMENT METHOD Overview The one of the most popular numerical method which is used to solve structural, thermal and fluid problems is the Finite Element Meth od (FEM) It can solve problems with arbitrary geometry combined with nonlinear, coupling and so on, for which it is almost impossible to find an analytical solution. A mesh is generate d to represent the arbitrary geometry with simple shaped elements like triangle s or tetrahedron s By using this approach, the arbitrary geometry can be split into pieces of simple geometry and analyzed However, there are som e draw backs of this mo thed. Fir st of all, the geometry will lose some accuracy when a mesh is used and it will introduce error Secondly, for complex geometry, it is very difficult and time consuming to generate a suitable mesh and even a fine mesh will still introduce error be cause the mesh is always an approximation of the geometry no matter how fine it is Although the error cause by geometry approximation will become smaller and smaller as the mesh goes finer and finer, it is very computation ally expensive to use a very dens e mesh In order to avoid the difficulties caused by the mesh and mesh generation process number s of meshless or mesh free techniques have been developed. Belytschko.T et al. [ 1 3 ] introduced Element Free Galerkin Method, a meshless method J.J. Monaghan [ 1 4 ] explained how mesh free methods can be used to solve astrophysical problems. In this approach, nodes are scattered all over the system which need s to be analyzed and the system can be solved based on the nodes. ( Figure 2 1 )

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17 Figure 2 1 Mesh G eneration for A Circular. A) Traditional FE M ; B) Meshless FEM ; C) Mesh Independent FEM Another alternate approach of the mesh free method solve s the system based on a structured, non conforming mesh, in other word, the mesh is independent of the geometry. Because the mesh does not depend on the geometry of the structure there is no geometry approximation during meshing process and the error cause d by using a mesh to represent the geometry will goes to zero Meanwhile since the mesh does not necessary need to conform the geometry, it is very easy to generate uniform mesh with regular shape such as rectangle or c uboid whi ch will further redu ce the error cause by Jacobi a n transformation To satisfy the essential boundary conditions Hollig [ 1 5 ] constructed B spline finite elements by using a function shows as ( 2 1 ) is a fi eld variable in the equation and must be satisfied at along a given bou ndary By defining the function for any when there is an essential boundary condition present the boundary condition s can be guaranteed to satisfy along a certain boundary . . . . . . . . A B C

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18 Formulation of 3D E lement (3D Shell) The implicit boundary finite element method uses implicit equation of geometry in its solution structure as follows. ( 2 2 ) The definition of the variables in the equation above are list ed as below : the trial function : the piecewise approximation of the element of the structured grid derived from the implicit equation of the boundary. : the boundary value function. : the step function which that has a unit value inside the domain of analysis and on any free boundaries whereas it equal to 0 at the boundaries where an essential boundary condition is specified. The function can be guaranteed to satisfy the boundary condition since the step function value is set equal to zero at the boundary. The main application here is to impl ement this method for shell like structure s For a shell like structure the mid plane surface and the thickness of the shell are two things that used to define the structure. A parametric surface can be used for representing the mid plane of a shell structure as and its boundaries be defined using a set of oriented edges that are defined parametrically as ( 2 3 ) W here is the number of boundaries. The i th boundary curve has a domain of and its location in parameter space of the shell mid surface is

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19 The boundaries of the mid plane surfaces are the edges of the plane and t he vector s can be de fined and a typical shell can be shown as Figure 2 2 Figure 2 2 A Shell Like Structure in IBFEM The is the normal on the boundary and it can be defined as ( 2 4 ) The is the tangent vector on the boundary which can be defined as ( 2 5 ) The is the b inormal on the boundary which can be calculated as ( 2 6 ) So, a ny point in the shell like structure are well defined, and the coordinate can be denoted as

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20 ( 2 7 ) W here and h is the thickness of the shell U nder the edge coordinate system, points in the vicinity of oriented edges can be express ed easily as the equation ( 2 8 ) Stiffness Matrix for 3D Shell Element The displacement field within the shell must satisfy the weak form for elastostatics which can be written in the following form for shell like geometry ( 2 9 ) S ince the shell is defined as its mid plane surface, the volume integration of the weak form can be modified as integral through area of the mid plane (domain ) and integral through the thickness of the shell. is the virtual strain, is the virtual displacement, is the traction acting on the edge is the body force and is pressure load per unit area acting normal to the shell. The stresses and strains are separated into two parts, the homogeneous p art and the boundary value part and they can be shown as ( 2 10 ) ( 2 11 ) Where the homogeneous part and boundary value part strains can be calculated based on the displacement field as

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21 ( 2 12 ) ( 2 13 ) Submit the modified stresses and strains equation back to the weak form of the elastostatics, equ ation ( 2 9 ) will become ( 2 14 ) The traction T can be calculated as ( 2 15 ) Where tension in the negative binormal direction, shear force in the normal direction bending moment about the tangent axis, and torque about the binormal axis can be defined with respe ct to the edge coordinate system. The displacement field trial and test functions within an element are approximated using a B spline shape functions which can be shown as ( 2 16 ) ( 2 17 )

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22 So the strain s field can be derived by combining the definition of strains and shape functions The strain displacement function can be split into two parts. Th ey are which contains only the derivatives of the shape function s, and which contains derivatives of the approximate step functions and it can be shown as ( 2 18 ) Plug in the terms inside and rewrite the matrix as ( 2 19 ) Using these definitions of the strain displacement function the weak form above (equation ( 2 14 ) ) can be reformed as the standard discrete form as the traditional finite element method as ( 2 20 ) Where each terms can be computed as ( 2 21 )

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23 ( 2 22 ) ( 2 23 ) ( 2 24 ) ( 2 25 ) For the internal elements in other words the element is completely inside and there is no edges passing through the element all t he terms that relate to the boundary will vanish and the strain displacement matrix can be simplified as Therefore, equation used for computed the stiffness (equation ( 2 21 ) ) can be simplified as ( 2 26 ) For elements that contain edges with specified displacements boundary conditions, the derivation of the approximate step functions are exist but only in a small transition width. By making this transition width very narrow say magnitude of 10 5 it is reasonable to suppos e that tran sition area is entirely inside the elements which edges pass through Because only contain the derivation of the step functions so its value will go to zero outside the transition width, the stiffness matrix computation al equation (equation ( 2 21 ) ) can be simplified as ( 2 27 ) Where eac h term can be compute d individual ly as

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24 ( 2 28 ) ( 2 29 ) ( 2 30 ) Boundary Condit ion In the implicit boundary method, the trial function and the test function are defined as the equation ( 2 31 ) ( 2 32 ) is t he boundary value function and i ts value at the boundary is equal to the essential boundary condition that applied at that boundary. T he value of the step function should goes zero at the boundaries where an essential bounda ry condition is specified so that the boundary condition will be satisfied The definition of approximate step functions depends on the type of boundary conditions. Typical essential boundary conditions for shells include three types : t hey are clamped, simply supported and symmetric boundary condition Clamped T he edge face of the shell can be defined as ( 2 33 ) A point near the boundary can be defined based on the definition of the edge face coordinate system and its coordinate can be written as

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25 ( 2 34 ) Since the displacement field are equal to zero at edges where the fixed boundary applied the step function can be define as ( 2 35 ) Where is the binormal component of the position vector of the point of interest in the edge coordi nate system and is a small distance The is set to be small to make that step function can transition from 0 to 1 within a small distance Various step function can be used but the slop should not equal to zero around the edge to make sure the derivation is existed Simply Supported The essential boundary condition on a simply supported edge can be satisfied with the following conditions : Th e displacement towards b inormal direc tion equal to zero at the edge and t he shell i s free to rotate about the edge. In the same time, t here are no external moments applied on the same edge. Define the radial distance from the edge as ( 2 36 ) T he step function for this kind of boundary condition can be defined as ( 2 37 )

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26 Again, the is a small distance and set small to make that step function can transition from 0 to 1 within a small distance Symmetry Boundary C ondition Symmetry boundary condition can be used to reduce the size of the mold effectively for instance, only half of a structure will need to be model if one symmetry face (edge) exist s in the structure Since in the IBFEM the boundary condition s on shell are defined as displacement and rotation for each edge the s ymmetry condition can be easy to set up by setting displacement in specific direction or rotation around specific axis equal to zero base d on the symmetry type (symmetry or anti symmetry) and the face (edge) o f symmetry Take a shell structure with one symmetry edge for instance, the nodes on edge should not have displacement a long the bi normal direction of the symmetry edge. T he rotation about the normal and tang ent direction of that edge will also need to be set equal to zero.

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27 CHAPTER 3 ANALYSIS OF COMPOSITE MATERIAL Overview Composites, which consist of two or more separate materials combined in a macroscopic structural unit, are made from various combinations of the other materials such as metals, polymer and ceramic [ 10 ]. Although many man made material have two or more cons tituents, they are generally not considered as composites if the structural unit is formed at the microscopic level rather than the macroscopic level. For example, alloys and ceramics are made of many constituents, but they are no considered as composite u nder this definition. The composite material can provide a high strength to weight and stiffness to weight ratio. In addition they can provide a variety of other advantages such as corrosion resistant, friction and wear resistant, vibration damping, fire resistant, acoustical insulation, etc. The pa rticle reinforced composite fiber reinforced composite and composite laminate are the most frequently used type s of composites One important advantage that composite s have over other material s is the composite material itself is designable, in other words the material properties of compo site can be design ed according to the requirement A deep understanding of the material however, is needed to design the composite properly. Experimental methods can be used to determine the properties of composite by performing various tests on the testing machine according to the standards. Because experimental methods can only be applie d on simple geometries mostly a bar subjected to sim p l e load, the usage of finite e lement method (FEM) [ 1 6 1 8 ] which can perform anal ysis of complex structure s is increasing. C omposite laminate with plate and shell form are the most common types

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28 of composite laminate that are used in the industry So it is important to derive the equations of composite laminate that can be used for plate or shell. T his chapter contains the theory and equations that are commonly used for analyzing composite laminate in a plate or a shell form. Stress Strain Relations of a Com posite Lamina Strain relationship of a material is defined as: ( 3 1 ) Where the and is the stress and strain vector respectively and is the material stiffness matrix The equation above gives the Stress Strain relationship for an an isotropic material Because of the symmetric of the stiffness matrix, there are only 21 independent materi al stiffness coefficients If there is one symmetric plane for material properties some t erms in the material sti ffness matrix will reduce to zero and t he Stress Strain relationship can be simplified as ( 3 2 )

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29 The number of independent materia l stiffness coefficients will reduce to 13 and this material is called monoclinic material I f there are two symmetric plane s exist in one material in the same time the third plane of material properties will become symmetric plane automatically. Four m ore terms in the material stiffness matrix will reduce to zero and the Stress Strain relationship will become ( 3 3 ) This type of material is called orthotropic material and it only has 9 independent stiffness coefficients. If there is a material such that the prope rties are equal in all the directions at any point in a particular plane, which means the 2 and the 3 in the stiffness coefficients are interchangeable, the Stress Strain relation can be given by ( 3 4 ) The material is called transversely isotropic material and the number of independent stiffness coefficients are 6.

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30 Finally if all the planes are s ymmetry plane for the material, the Stress Strain relation becomes: ( 3 5 ) The material is called isotropic material and there are only 2 independent material stiffness coefficients. The composite plate and shell that people are dealing with in most cases are made of laminate which behalves as orthotrop ic material. If the fiber s are only aligned in the 1 and 2 direction, that is, no fiber s are aligned in the thickness direction, t he discussion can be limited to rotation of the coordinate system only about the 3 axis. Appl y ing p lane stress condition the Strain Stress relation ship will becomes ( 3 6 ) Where the is the material constants and they can be determine by the equations below

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31 ( 3 7 ) A transformed stiffness matrix is used to show the Strain Stress relation ship in the global x y coordinate system instead of the 1 2 material coordinate system. Strain Stress relationship in the global x y coordinate system can be show as ( 3 8 ) By applying the transformation matrix for the stresses and s trains, the transformed stiffness matrix in the global x y coordinate system can be obtained. ( 3 9 ) Where the transformation matrix is ( 3 10 ) So, according to the equation ( 3 8 ) ( 3 9 ) and ( 3 10 ) the transformed stiffness ma trix can be obtained as

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32 ( 3 11 ) Or each terms can be written explicitly as ( 3 12 ) Classical Laminated Plate Theory (CLPT) Classical laminate theory (CLPT) was apparently developed in the 1950s and 1960s i nvestig ated by investigators such as S mith [ 1 9 ] Pister and Dong [ 20 ] R eissner and S tavsky [ 21 ] S tavsky [ 2 2 ] S tavsky and H off [ 2 3 ] etc The CLPT is used to analyze thin plate s by ignoring transverse shear stress es The basic a ssumptions of this theory are: 1. The x y plane is the middle plane of the plate and the z axi s is in the thickness direction; 2. The plate contain several layers which bond perfectly and the 3 rd di rection (thickness direction) of the material principal coordinate system coincides with the z axis ; 3. the x y plane ; 4. D isplacements , and ; 5. The in plane strains and are small compared to unity ; 6. The transverse shear strains and are negligible and so is the transverse normal strain 7. The transverse normal a nd shear stresses and are negligibly small compared to the in plane normal and shear stresses and ;

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33 Using the assumptions above, the displacement fields and can be written as Taylor series expansion in terms of as ( 3 13 ) ( 3 14 ) ( 3 15 ) Since the will very s mall because the ply is very thin according to the basic assumption the linear term s will dominate the equation s ( 3 13 ) ( 3 14 ) and ( 3 15 ) So, i f only the linear terms are taken in to consideration the equation ( 3 13 ) and ( 3 14 ) above will reduce to ( 3 16 ) ( 3 17 ) Where and is known as mid plane displacement. It means the displacement s of a point located in the middle plane of the plate T he transverse displacement will be independent of the z coordinate which means the displacement is a function only of and according to the assumption, hence the equation ( 3 15 ) will reduce to ( 3 18 ) However if the structure is loaded with large transverse load which make become non negligible the n equation ( 3 18 ) will not be valid and other assumpti on s must be made. But the in the most cases where the thin composite laminate are employed, the transverse load will be small and the equation ( 3 18 ) will wor k. From the definition of the strain,

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34 ( 3 19 ) T he transverse shear strains can be computed as ( 3 20 ) By plug ging in the equations ( 3 16 ) and ( 3 17 ) into the equation s ( 3 20 ) it will become ( 3 21 ) Since the transverse shear stress are negligible in the plate based on the assumptions that is, both the and are equal to zero at any point, equation s ( 3 21 ) can be simplified as ( 3 22 ) Submitting the equation ( 3 22 ) in to equation ( 3 16 ) and ( 3 17 ) to calculate the in plane displacement the displacement can be computed as ( 3 23 ) By using the definition of the strain (equation ( 3 19 ) ) the in plate strains can be obtained as

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35 ( 3 24 ) It can be shown explicitly as ( 3 25 ) Where and are the mid plane s train and each of them can be computed as ( 3 26 ) and are th e mid plane curvature and each of them can be computed as ( 3 27 ) Equation ( 3 24 ) can be put in to matrix form as ( 3 28 )

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36 The in plane force resultants and are defin e d as the forces per unit length along the edge T hey can calculated by integrating the stresses on the edge through the thickness direction as ( 3 29 ) Equation ( 3 29 ) can be rewritten in matrix form as ( 3 30 ) Submitting the plane stress constitutive relation ship (Stress Strain relationship) equation ( 3 8 ) in to the equation ( 3 30 ) the in plane force resultants can be computed as ( 3 31 ) By s ubmi t t ing the relati onship between strains for any point and mid plane strain and curvature equation ( 3 28 ) the in plane force resultants can be determined by the equation ( 3 32 ) show as ( 3 32 )

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37 Equation ( 3 32 ) can be rewritten in index notation as ( 3 33 ) Since the mid plane strain and curvature are independent of the thickness, in other words they remain constant in the thickness direction, hence they can be take n out of the integral in the equation ( 3 33 ) as ( 3 34 ) And the equation ( 3 34 ) can be further simplified as ( 3 35 ) Where and are 3 by 3 matrix defined as ( 3 36 ) ( 3 37 ) Similarly to the in plane force resultants, the in plane moment resultants and are defined as the moments per uni t length and they can be obtained by ( 3 38 ) Equations ( 3 38 ) can be put in to matrix as

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38 ( 3 39 ) Submitting the plane stress constitutive relation ship equation ( 3 8 ) and the relationship between strain at any point and in plane strain and curvature ( equation ( 3 28 ) ) the equation ( 3 39 ) above will become ( 3 40 ) ( 3 41 ) Equation ( 3 41 ) can be rewritten in index notation as ( 3 42 ) And by taking out the in plane strain and curvature which are independent of the thickness, equation ( 3 42 ) wi ll become ( 3 43 ) T he equation ( 3 43 ) can be rewritten as ( 3 44 ) In equation ( 3 44 ) matrix is the same as it derived from force resultant s and matrix is a 3 by 3 mat rix defined as

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39 ( 3 45 ) To sum up, and are all 3 by 3 and called laminate stiffness matrices, which known as follows: : In plane stiffness matrix : Coupling stiffness matrix : Bending Stiffness matrix And the definition of those matrixes are ( 3 46 ) ( 3 47 ) ( 3 48 ) Because both the force resultants and moment resultants are defined in a similarly form so t he equation ( 3 35 ) and equation ( 3 44 ) can be combined together and rewritten as matrix form as ( 3 49 ) It can be written explicitly as:

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40 ( 3 50 ) Shear Deformable Plate Theory (SDPT) The CLPT is set up based on assumptions that transverse normal and shear stresses are neglected. Although it is a good approach for a thin structure, the answers will become in accuracy when the structure goes thicker. In order capture the effects of the transverse stresses, Mindlin [ 2 4 ] and Reissner [ 2 5 ] developed the Shear Deformable Plate Theory (SDPT) for a thick plate, in which the transverse stresses are also taken into consideration. In the shear deformable plate t heory the out of plane shear strains are not be ignored, that is and are no longer assumed to be zero. The displacement fields of equation s ( 3 16 ) and ( 3 17 ) will become: ( 3 51 ) Where the and are the rotation of the edge surface. The in plane strains will be the same as those in the CLPT but the equations in plane curvature will become

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41 ( 3 52 ) I n addition to the in plane force and moment resultants which are same as those in the CLPT, the shear force resultants also exist and can be defined similarly as follow s : ( 3 53 ) T ransverse shear stre sses can be computed as ( 3 54 ) S tiffness matrix for the out of plane shearing in the x y z global coordinate system can be derived as ( 3 55 ) Applying Stress Strain relationship for the out of plane shearing stiffness matrix for the out of plane shearing in the x y z global coordinate system can calculated as ( 3 56 ) Submitting the equation ( 3 54 ) in to equation ( 3 53 ) the shear force resultants can be obtained as

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42 ( 3 57 ) In the matrix form, equation ( 3 57 ) can be written as ( 3 58 ) Where ( 3 59 ) E quation ( 3 35 ) ( 3 44 ) and ( 3 58 ) can be combined together to form consti tutive relation of the shear deformable laminate as ( 3 60 ) Equation ( 3 60 ) can be written explicitly as ( 3 61 ) Instead of effective laminate properties (ABD matrix) in plate/shell theory the IBFEM use s 3D shell element formulation which requires an effective material stress strain relationship. So, the effective laminate properties need to be modified to a new form that can be implemented into the IBFEM.

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43 CHAPTER 4 ABD EQUIVALENT MATERIAL MODEL OF COMPOSITE LAMINATE Overview In general, a composite laminate is difficult to analyze using 3D shell elements in IBFEM since it can be formed by a large number of layers When the structure is loaded different layer will perform differently because different layers of the composite laminate have different material properties I ntegration in the thickness direction is needed for volume integration That will make the numerical integration required for v olume integration present within each element extremely computation ally expensive. To fully integrate a linear brick element with constant material coefficients, 8 integration points are needed [ 2 6 ] Say there is a laminate with 100 layer s and 8 point s are used for integration of the thickness direction for each layer s all in all there are 800 integration point s are used for the thickness direction for each element. It is very computational expensive for a structure which involved large number of element s [ 1 2 ]. Another approach is replacing the original multi ply laminate by a n equivalent laminate that has fewer number of plies and behaves similar to the original laminate and results in the same stiff ness matrices. An ABD equivalent material model for laminate can be derived according to that assumption [ 1 2 ]. By doing that the time taken for integration of the thickness direction can be reduced dramaticall y Although the laminate will p er form slightly di fferent ly due to the replacement and the stresses and strains distribution in the laminate might change differences will be sma ll and acceptable. Also, if a study is mainly focus ed on the behavior of the laminate in the global sense instead of focusing on stresses and strain s distribution with in the laminate this approach will be

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44 good enough as long as the new laminate is guarantee d to have similar behavior in the global sense as original laminate The ABD Equivalent Material Model of Laminate Accord ing to the classical laminate theory, for a give laminate, equation ( 3 49 ) can be used to determin e the force and moment resultants ( 4 1 ) Where and matrix are In plane stiffness matrix, couplin g stiffness matrix and bending s tiffness matrix defined as equation ( 3 46 ) ( 3 47 ) and ( 3 48 ) respectively, and they can be calculate d as equation s below ( 4 2 ) One way to ensure that a new 3 ply laminate will have similar global behavior is to assume that the new 3 ply laminate has the same effective properties or and matrix as the original multi ply laminate. By this defi nition, t he equation ( 4 2 ) hold its validity for the new 3 ply laminate ( 4 3 )

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45 In these equation s ( 4 2 ) ( 4 3 ) denote s the properties of the original plie s and denote s the properties of the 3 new plies. Assuming that the matrix is constant th r ough each ply for both original multi ply and the new 3 ply laminate the integration in the equations ( 4 2 ) and ( 4 3 ) will become a summation. As there are only 3 plies in the new laminate, equat ions ( 4 3 ) can be written explicitly as ( 4 4 ) E quations ( 4 4 ) can be put into matrix format as ( 4 5 ) Also equation ( 4 5 ) can be written explicitly as ( 4 6 ) By solvi ng the equation s above, the ( ; ) for the new 3 ply laminate can be obtained. Out of plane shear stresses also have to be taken into consideration T he shear force resultants can be determined by the equation ( 3 58 )

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46 ( 4 7 ) Where is a constant and and can be determined by ( 4 8 ) ( 4 9 ) As it has been discussed the in plane stress es are much larger tha n the out of plane stresses in most case for plane and shell like structure s Also, because the matrix of the composite laminate will dominate the laminate properties in the thickness direction, the modulus and are relatively small and to plane in most cases So it is good enough to assume that the ou t of plane shear properties are the same for all 3 plies for the new 3 ply laminate. Under this assumption equation ( 4 9 ) will simplified as ( 4 10 ) And it can be further simplified as ( 4 11 ) O ut of plane properties ( ; ) of the new 3 ply laminate can be obtained as ( 4 12 ) Where is the total thickness of the laminate. The modulu s of the thickness direction of the new 3 ply laminate is assumed to be constant as the out of plane shear stiffness and it can be approximated as the

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47 s of the thickness direction of each ply of the original multi ply laminate Reasons for this assumption is similar to the reasons for setting the out of pla n e shear stiffness as constants for the new 3 ply laminate, that is, the stress in the thickness is negligible and modulus of thickness direction is small and will not vary much from ply to ply. So, it can be computed as ( 4 13 ) By solving the equation s ( 4 13 ) the modulus of thickness direction of the new 3 ply laminate can be determined. As out of plane shear properties, the composite matrix material will dominate the properties of the laminate in the thickness direction the out o f plane stresses are negligible and the modulus change much fro m plane to plane in most cases So, it is good enough to assume that the properties normal to the plane are same for all 3 ply for the new 3 ply and all the plies in the original multi ply laminate Under this assumption equation ( 4 13 ) can be simplified as ( 4 14 ) in most case s ( 4 14 ) To sum up, for a typical laminate : Solving equation ( 4 6 ) the in plane properties of the new 3 ply laminate can be determined Solving equation ( 4 12 ) the out of plane properties of the new 3 ply laminate can be determined Finally, solving equation ( 4 13 ) the can be determined

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48 All in all, all the material properties of the 3 plies of the new 3 ply laminate can be obtain ed. Local and Global Stiffness Matrix The material properties discussed above are all with respect to the coordinate system attached to the surface passing through an element such that its z axis is normal to the surface In a real problem, lots of element will be involved the element coordinate system attached to the surface may be translated or rotated with respect to the global coordinate system Since the variable s like displacement, force are defined with respect to the global coordinate system, it is important to set up the transformation matrix between the global and the local or element coordinate system Figure 4 1 Coordinate Systems T here are three coordinate systems involved and the r elationship between each of them are shown in Figure 4 1 1 2 3 coordinate system : material coordinate system; x y z coordinate system : element coordinate system; X Y Z coordinate system : global coordinate system; (3)

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49 As equation in ( 4 15 ) and ( 4 16 ) if the stress strain relationship are set up in a x y z element coordinate system and an X Y Z global coordinate systems as ( 4 15 ) ( 4 16 ) To transform a stress tensor from X Y Z coordinate system (global coordinate system in most cases) to a given x y z coordinate system (element coordinate system in most cases ) equation ( 4 17 ) can be used. ( 4 17 ) Where the stress tensor are defined as ( 4 18 ) ( 4 19 ) The transformation matrix are defined as ( 4 20 ) In the equations above is the cosine of the x axis with respect to X axis, is the cosine of the x axis with respect to Y axis and is the cosine of the x axis with respect to Y axis. T he rest of variables are defined similarly and are the cosines of the y axis with respect to X, Y and Z axis, respectively, and and are the cosines of the z axis wi th respect to X, Y and Z axis, respectively.

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50 The strain tensor is transformed in the same manner. ( 4 21 ) Where strain tensor are defined as ( 4 22 ) ( 4 23 ) T he stress es ar e always put in to a vector fo rm instead of a matrix form in finite element analysis T he stress vector can be written as ( 4 24 ) ( 4 25 ) So t he translation matrix should be formed as ( 4 26 ) By doing the matrix multiplication in equation ( 4 17 ) and rearranging the terms to satisf y the form of equatio n ( 4 26 ) the transformation matrix for stress vector can be obtained as ( 4 27 )

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51 S imilarly, strain s are also used in a vector (or column matrix) form instead of the square matrix form for finite element method most often as ( 4 28 ) ( 4 29 ) And the transformation matrix are defined as ( 4 30 ) By doing the matrix multiplication, it can be shown that transformation matrix for the strain tensor is same as the transformation matrix used for stre ss tensor defined as equation ( 4 27 ) It is also very important to derive the equation that is used to transform the stresses and strain from local x y z co ordinate system back to global X Y Z coordinate system, which means ( 4 31 ) From equation ( 4 17 ) it can be obtained that ( 4 32 ) By using the definition of transformation matrix in equation ( 4 20 ) it can be concluded that the inverse and the transverse of it are equa l that is ( 4 33 ) So equation ( 4 32 ) can be simplified as ( 4 34 )

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52 By doing the matrix multiplication in equation ( 4 34 ) and rearranging the terms to satisfied the form of equation ( 4 31 ) the inverse of the transformation matrix for stress tensor can be obta ined as ( 4 35 ) The s ame process can be used for the strain tensor to g et inverse of transformation matrix that used for strain tensor in the equation ( 4 36 ) ( 4 36 ) It can be shown that t ransformation matrix is the same as the transformation matrix for the stress vector defined as equation ( 4 35 ) E ngineering strain instead of true strain are used more frequently in FEM The normal st r ain s are the same in both cases but the shear strains are differen t and the relationship between engineering shear strain and true strain can be written as: ( 4 37 ) If the strain are written in the vector form the relationship can be shown as ( 4 38 ) Where ( 4 39 )

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53 ( 4 40 ) S train stress relation ship in X Y Z coordinate system can be written as ( 4 41 ) W here the transformed stiffness matrix can be computed as ( 4 42 ) This can be derived as ( 4 43 ) T he transformed matrix can be put in the equation ( 4 44 ) to calculat e the virtual strain energy in the system according to the global x y z coordinate sy stem as ( 4 44 ) Another approach to calculate the virtual strain energy in the system, can be started with directly deriv ing the transformation matrix for stress vector and strain vector respective ly shown as ( 4 45 )

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54 ( 4 46 ) Where ( 4 47 ) ( 4 48 ) So the virtual strain in the x y z coordinate system can be obtained as ( 4 49 ) And the virtual strain energy can be computed as ( 4 50 ) W here ( 4 51 ) The s ame transformed stiffness matrix can be obtained from both approach. In the IBFEM, the element stiffness matrix can be calcul ated as in equation ( 2 21 ) ( 4 52 )

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55 Instead of having a constant matrix inside one element as for isotropic materials, the properties of composite laminate are typically orthotropic and the material property matrix will be different from layer to layer. For n ply composite, the stiffness matrix for one element should be rewritten as ( 4 53 ) Where the and denote the upper and bottom location in thickness direction of the i th layer. In ABD e quivalent material model of laminate, the number of ply are 3 instead of so equat ion ( 4 53 ) will become ( 4 54 ) As the matrix is constant in the thickness direction and the material properties are assumed to be constant through each layer for the new 3 ply laminate, the integration of the thickness direction can be separate in to 3 parts and equation ( 4 54 ) will become ( 4 55 ) As can be seen, the integration in the thickness direction will reduce to 3 instead of n for a multi ply laminate if the new 3 ply laminate is used. By employing the ABD e quivalent material model for the laminate, the compu tation will become much faster especially for a laminate containing a large number of layers.

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56 CHAPTER 5 RESULTS AND DISCUSSION Overview It is very important to implement the laminate ABD equivalent model in to program s and test it with examples to prove its validly By comparing the answers with the analytical solution if available, or with answers from commercialized FEM software, the laminate ABD equivalent material model can be verified. The commercialized software that has been used is Solidworks. SolidWorks is a solid modeling C omputer A ided D esign (CAD) and C omputer A ided E ngineering (CAE) software program that been widely used i n engineering SolidWorks has been marketed by the Dassault Syst e mes since 1997. It is a very popular software that can be used for solid modeling and drawing generation I t also contain s function ality for FEA which allows user s to perform structural analy sis. Implicit Boundary Finite Element Method ( IBFEM ) can directly use the geometry created in CAD software for analysis without generating a mesh to approxi mate the geometry so that the accuracy of the geometry can be guaranteed. It a llows loads and boundary conditions to be applied directly on the solid and provide s a platform for implement ing the ABD equivalent material model of composite laminate for 3D an alysis Several examples have been analyzed both in Solidworks and IBFEM a nswers are been compared with each other and analytical solutions, if available and discussed in this chapter Example of Square Plate The first example is a square plate clamped on all fo ur edges, and it is subject to a uniform pressure normal to the plate as show n in F ig ure 5 1

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57 Figure 5 1 Geometry and Load of Clamped Square Plate A plate under these boundary conditions shows pure bending, with the maximum displacement at the center. The plate is 10 inch by 10 inch and the uniform pressure applied is 0.01 psi. Various thickness (1 inch, 0.1 inch and 0.01 inch) are used and the ratio r is defined as ( 5 1 ) The ratio is defined to indicate wh e ther the plate is thick or not. A typi cal thin plate should have a ratio smaller than 0.1. The material properties are used are: ( 5 2 ) Two type of laminate angle d ply and cross ed ply are used in th is example and the maximum displacement are calculated T he mesh and the displacement of 10 layer angled ply when =1/100 are plot ted ( Figure 5 2 ). In addition the results are listed in T able 5 1, T able 5 2 and T able 5 3 (The SW, ANSYS and SnS

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58 results are reported by Kumar and Shapiro [ 1 2 ] and the analytical solution is report by Reddy [ 2 7 ] ) Figure 5 2 Clamped Square Plate in IBFEM A) Geometry and Mesh; B ) Displacement of 10 layer angled ply ( =1/1 0) Table 5 1 Maximum Displacement ( 10 7 inch) =1/10 Reddy IBFEM SW ANSYS SnS Element N umber 225 1k 10k 1k 3k 2 angle d 3.891 7.846 6.984 2 crossed 3.814 7.848 6.515 10 angle d 4.286 4.36 5.094 4.057 4.152 10 crossed 3.981 4.058 4.748 3.855 3.762 Table 5 2 Maximum Displacement ( 10 4 inch) =1/100 Reddy IBFEM SW ANSYS SnS Element Number 225 1k 10k 1k 3k 2 angled 3.891 4.122 4.110 2 crossed 3.814 3.987 3.978 10 angled 1.621 1.62 1.629 1.597 1.611 10 crossed 1.55 1.55 1.543 1.532 1.543 B A

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59 Table 5 3 Maximum Displacement ( 10 1 inch) =1/1000 Reddy IBFEM SW ANSYS SnS Element Number 225 1k 10k 1k 3k 2 angled 3.891 4.061 4.072 2 crossed 3.814 3.954 3.951 10 angled 1.58 1.578 1.581 1.163 1.684 10 crossed 1.525 1.552 1.51 1.145 1.661 As is shows in the table s the IBFEM shows good agreement with SolidWorks using a lower density mesh because cubic B spline elements are for modeling shells in IBFEM The result also shows good agreement with ANSYS when the ratio equal to 100 and 1000 but not the 10. That is because of the element type used in ANSYS is SHELL 181 which is designed for mo derately thick shell and it is not suitable for the case =10 (thick structure). In addition, the analytical solution reported by Reddy are calculated based on the K irchhoff thin shell theory which is also only good for thin shell not for thick shell S o it is not suitable to analyze the structure with the ratio equal to 0.1. The answers from IBFEM are better than the answers given by the SnS (Scan and Solve) with much less mesh density because the results for SnS were obtained using quadratic elements The converge study are also performed in I BFEM, to verify the convergence and converge rate of this method The relative error (in log scale) and the maximum displacement of different mes h density are plotted and t he relative error is computed based on the strai n energy of the entire structure

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60 Figure 5 3 Converge Plot of Strain Energy Figure 5 4 Converge Plot of Maximum Displacement IBFEM converge s very fast with respect to size of the elements because the elements are cubic and a good answer can be achieve d with a mesh density which is not too high. However the computational time for the analysis is higher than the typical shell elements in commercial software because of the size of the element stiffness matrix and cost of computing and assembling it. Example of Pressured Cylinder The second example is a thin shell like cylind er subjected to internal pressure and fixed a long the edges at both ends The geometry of the cylinder is shown as

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61 F igure 5 5 and its radius is 20 in ch height is 20 in ch thickness is 1 in ch and the internal pressure is 2.04 ks i. Figure 5 5 Geometry and Load of Pressured Cylinder The material properties are: ( 5 3 ) Using symmetry one eighth of the structure is modeled and meshed in IBFEM as shown in Figure 5 6. The maximum radial displacement are listed in the Table 5 4 (The SW, ANSYS and SnS results are reported by Kumar and Shapiro [ 1 2 ] and the Reddy solution is report by Reddy [ 2 7 ] )

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62 Figure 5 6 Pressured Cylinder in IBFEM A) Geometry and Mesh; B) Displacement of 10 angled ply Table 5 4 Maximum Radius Displacement of Cylinder Subjected to Internal Pressure (10 1 inch) Reddy IBFEM SW ANSYS SnS Element Number 190 1.2k 15k 1k 3k [0] T 3.754 3.763 3.752 2 crossed 1.870 1.763 1.848 1.706 1.820 1.773 2 angled 2.287 2.350 2.204 2.356 2.291 10 crossed 1.759 1.830 1.719 1.814 1.776 10 angled 2.271 2.340 2.21 2.334 2.282 It can be seen that answers obtained by IBFEM show good agreement with SW and analytical solutions with lower mesh density. Example of Scordelis Lo R oof Scordelis Lo roof is a thin shell like structure subjected to gravity loads and supported at both sides by diaphragms while the side edges are free as shown in the F igure 5 7 The geometry of the structure are the radius equal to 300 inch length is 600 A B

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63 in ch and the pressure equal to 0.625 ps i The two edges are constrained such that they cannot move in the z and x axis direction and cannot rotate about the y axis. Figure 5 7 Geometry and Load of Scordelis Lo R oof The material properties are: ( 5 4 ) Various thickness (3 inch, 6 inch and 15 inch) were tested T he ratio between the radius and shell thickness is defined as ( 5 5 ) Figure 5 8 shows the model and the mesh used for analysis using IBFEM The maximum displacement of Scordelis Lo R oof with different thicknesses has been computed and t he results are listed in Table 5 5 Table 5 6 and Table 5 7 (The SW, z 80

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64 ANSYS and SnS results are reported by Kumar and Shapiro [ 1 2 ] and the Reddy solution is report by Reddy [ 2 7 ] ) Figure 5 8 Scordelis Lo R oof in IBFEM. A) Geometry and Mesh of Scordelis Lo R oof ; B ) Displacement of 10 C rossed P ly ( =100) Table 5 5 Maximum Displacement of Scordelis Lo R oof (inch) =100 Reddy IBFEM SW ANSYS Sn S Element Number 16 1160 1.2K 10K 1K 3K 10 crossed 1.415 1.473 1.564 1.434 1.542 1.593 2 crossed 2.339 2.396 2.46 2.407 2.307 2.415 10 angled 1.818 1.82 1.955 1.836 1.821 1.912 2 angled 3.597 3.482 3.866 3.871 3.411 3.743 Table 5 6 Maximum Displacement of Scordelis Lo R oof (10 1 inch) =50 Reddy IBFEM SW ANSYS Sn S Element Number 16 1160 1.2K 10K 1K 3K 10 crossed 2.94 3.233 3.27 2.979 3.335 3.412 2 crossed 5.082 5.463 5.659 5.291 5.48 5.81 10 an gled 4.096 4.16 4.089 4 082 3.796 3.94 2 a ngled 6.76 6.557 7.17 7.652 6.675 7.157 B A

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65 Table 5 7 Maximum Displacement of Scordelis Lo R oof (10 2 inch) =20 Reddy IBFEM SW ANSYS S n S Element Number 16 1160 1.2K 10K 1K 3K 10 crossed 5.234 5.969 5.37 5.246 5.361 5.398 2 crossed 7.292 8.304 7.56 7.449 7.877 8.067 10 angled 10.04 10.27 9.594 9 727 7.856 8.009 2 angled 12.05 13.30 8.959 13.97 1.061 11.27 The IBFE M shows good agreement with both the analytical solutions and the numerical results given by the commercial software SW and ANSYS But the mesh density used in IBFEM are much less that the mesh used in SW, ANSYS and S n S Again, the element type in IBFEM is 3D cubi c B spline in this example, t he answers match better with analytical solutions based on thin shell theory when the structure is thinner Example of Doubly Curved Shell Doubly Curved Shell is a thin shell like structure subjected to internal pressure loads and simple supported at all of the four edges as shown in the F igure 5 9 The geometry of the structure is defined by the two radius of two curve s and a nd the length of the two edge s and In this case the radi i are assumed to be the same ( ) equal to 100 inch and The pressure are applied on all of the structure and acting outward. The value of it is set to be 10 psi in this example.

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66 Figure 5 9 Geometry and Load of Doubly Curved Shell The material properties are: ( 5 6 ) Various ratio and ratio are te sted and the center deflation values are non dimensionalized as in the equation below ( 5 7 ) The structure a s modeled and analyzed in IBFEM is shown in Figure 5 10. The m aximum displacement will occur at the center of the str ucture The results have been non dimensionalized and listed in the Table 5 8 and Table 5 9 (The Reddy solution is report ed by Reddy [ 2 7 ] )

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67 Figure 5 10 Doubly Curved Shell in IBFEM A) Geometry and Mesh Doubly Curved Shell ; B ) Displacement of [0/90] S P ly ( ) Table 5 8 Maximum Displacement (Non dimensionalized) [0/90] T Reddy SW IBFEM 2 0.2855 0.2816 0.2952 3 0.6441 0.6393 0. 6564 4 1.1412 1.1360 1.1564 5 1.7535 1.7487 1.7725 10 5.5428 5.544 5.583 10 30 16.98 17.06 17.10 Ta ble 5 9 Maximum Displacement (Non dimensionalized) [0/90] S Reddy SW IBFEM 2 0.2844 0.2796 0.2938 3 0.6246 0.6201 0. 6366 4 1.0559 1.0516 1.07 5 1.5358 1.5315 1.5525 10 3.7208 3.72 3.74 10 30 6.8331 6.846 6.85 IBFEM shows good agreement with both the analytical solutions and the numerical results given by the SolidWorks with much less mesh density. B A

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68 CHAPTER 6 CONCLUSION Summary In this thesis, ABD equivalent material model for laminate for which the original multi ply laminate is replaced by a new 3 ply laminate with the same laminate stiffness matrices, has been implement ed in the Implicit Boundary Finite Element Method (IBFEM). Initially the concepts and equations that required for the Implicit Boundary Fin ite Element Method are discussed followed by the discussion of the p roperties of composite laminate, Classical Lamination Plate Theory (CLPT) and Shear Deformable Plate Theory (SDPT). In addition the formation of ABD equivalent material model for the lami nate has been discussed in detail along with the description of the transformation s required In addition, the model has been tested with examples in IBFEM and the answers are compared with analytical solution s if available, as well as answers from other FEA software The main adva ntage of ABD equivalent material model is that it can reduce time taken by the numerical integratio n th r ough thickness direction by reducing the origin al multi ply laminate to a 3 ply laminate without losing the ability to catch the macro behavior of the laminate structure The examples tested using ABD equivalent material model for laminate shows validity A n accurate solution can be obtained using i n IBFEM with a lower mesh density t ha n SolidWorks, ANSYS and SnS. T he computational time taken by the integration through thickness direction is reduced significantly. A s in any approximation the ABD equivalent material mode l for laminate may sacrifice so me accuracy but the advantages of this mothed are significant with no observed loss of accuracy.

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69 Future Work In this thesis, ABD equivalent material model for laminate has been implemented in the Implicit Boundary Finite Element Method (IBFEM) and it sho however, more work is needed to improve this model First of all, the results presented here is using 3D cubic B spline elements that use a 3D stress strain formulation which requires computing the equivalent 3 ply model. It would be advan tageous to develop a 3D shell element that is based on the Kirchoff or Midlin shell theory so that the effective ABD matrix can be directly used. Secondly, the ABD equivalent material model for laminate can be further improved to endow the capability of mo deling thick laminate, honey cone laminate, etc. Furthermore the ABD equivalent material model for laminate is not capable of catch the micr o behavior of each layer. The method can be endowed with the ability to analyze the stress and strain distribution within each lamina

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70 LIST OF REFERENCES [1] Bathe KJ. Finite Element Procedures. PRENTICE HALL, 1996 [2] Fish J, Belytschko T. A First Course in Finite Elements. JOHN WILEY & SONS LTD : New York, 2007. [3] Lancaster P, Salkauskas K. Surfaces generated by moving least squares methods. Mathematics of Computation 1981; 37 (155):141 58. [4] Lu YY, Belytschko T, Gu L. A new implementation of the element free Galerkin method. Computer Methods in Applied Mechanics and Engineering 1994; 113 (3 4):397 414. [5] Atluri SN. The meshless method (MLPG) for domain & BIE discretizations. Forsyth: Tech Science Press; 2004. [6] Kumar AV, Lee J. Step function representation of sol id models and application to mesh free engineering analysis. Journal of Mechanical Design 2006; 128 (1):46 56. [7] Kumar AV, Periyasamy PS. Mesh independent analysis of shell like structures. International Journal for Numerical Methods in Engineering 2012 ; 91 (5):472 90. [8] Burla RK, Kumar AV. Implicit boundary method for analysis using uniform B spline basis and structured grid. International Journal for Numerical Methods in Engineering 2008; 76 (13):1993 2028. [9] Kumar AV, Burla R, Padmanabhan S, Gu L Finite element analysis using nonconforming mesh. Journal of Computing and Informat ion Science in Engineering 2008 ; 8 (3):031005. [10] Gibson RF. Principles of composite ma terial mechanics. CRC press, 2011 [11] Campbell Jr FC, editor. Manufacturing processes fo r advanced composites. Elsevier, 2003 [12] Kumar G, Shapiro V. Efficient 3D analysis of laminate structures using ABD equivalent material models. Finite Elements in Analysis and Design 2015; 106 :41 55. [13] Belytschko T, Krongauz Y, Organ D Fleming M, Krysl P. Meshless methods: an overview and recent developments. Computer M ethods in A pplied Mechanics and Engineering 1996 ; 139 (1):3 47.

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71 [14] Monaghan JJ. An introduction to SPH. Computer Physics Communications 1988; 48 (1):89 96. [15] Hll ig K. Finite eleme nt methods with B splines. Siam, 2003. [16] Carrera E. Theories and finite elements for multilayered, anisotropic, composite plates and shells. Archives of Computational Methods in Engineering 2002; 9 (2):87 140. [17] Zhang YX, Yang CH. Recent developments in finite element analysis for laminated composite plates. Composite Structures. 2009; 88 (1):147 57. [18] Nurhaniza M, Ariffin MK, Ali A, Mustapha F, Noraini AW. Finite element analysis of composites materials for aerospace applicatio ns. InIOP Conference Series: Materials Science and Engineering 20 10; 11 (1):012010. [19] Simth CB, S ome new type of orthotropic plates laminated of orthotr o pi c material Journal of Applied Mechanics 1953; 20 :286 288. [20] Pister KS, Dong SB. Elastic bendi ng of layered plates. Journal of the Engineer ing Mechanics Division 1959 ; 85 (4):1 0. [21] Reissner E, Stavsky Y. Bending and stretching of certain types of heterogeneous aeolotropic elastic plates. Journal of A pplied Mechanics 1961 ; 28 (3):402 8. [22] Sta vsky Y. On the general theory of heterogeneous aeolotropic plates(Deflection theory established for bending and stretching of elastic anisotropic plates with material heterogeneity). Aeronautical Quarterly 19 64 ; 15 :29 38. [23] Stavsky Y, Hoff NJ. Mechanic s of composite structures. Composite engineering laminates. 1969:5 9. [24] Mindlin RD. Influence of rotatory inertia and shear in flexural motion of isotropic, elastic plates. Journal of Applied Mechanics 1951; 18 :31 38. [25] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 1945; 12 :69 76. [26] Robert D Cook et al. Concepts and applications of finite element analysis. Wiley. com, 2007. [27] Reddy JN. Me chanics of laminated composite plates and shells: theor y and analysis. CRC press; 2004.

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72 [28] Kumar G, Shapiro V. Reduced Material Model of Composite Laminates fo r 3D Finite Element Analysis. ASME 2014 International Design Engineering Technical Conference s and Computers and Information in Engineering Conference. American Society of Mechanical Engineers. 2014. [29] Timoshenko SP, Woinowsky Krieger S. Theory of plates and shells. McGraw hill; 1959.

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73 BIOGRAPHICAL SKETCH Li Liang was grew up in Guangzhou, Guangdong province China He did his high school in No.4 Guangzhou High School located in Guangzhou and graduated in 2010 He got his b achelor degree of e ngineering m echanics and f inance at 2014 after 4 years of studyi ng beginning in 2010 in Southwest Jiaotong University, Chengdu, China Meanwhile, he gained some research experience s by working on two research Study of Micro Mechanism of Metallic Glass and Metallic Glass Matrix Composites and Study of C yclic D eformation Behavior of Polycarbonate P olymer in Hydrothermal E nvironment Also he did i ntern ship in Guangdong Hydropo wer Planning & Design Institute, Guangdong, China, from July to August 2013 After that he studied in the University of Florida, G ainesville, Florida, USA for 2 years and received a m d egree in m echanical e ngineering in May of 2016. His areas of specialization include s Finite Element Method, Composites Material and Computational Method During his m studies he did research on implementing Composite Laminate for 3D Shell Like Structure in Implicit Boundary Finite Element Method and work ed as a Teaching Assistant for the graduate course on Finite Element Analysis and Application in the S pring and F all semester s of 2015 and the undergraduate class on Finite Element Analysis and Application in S pring 2016.