STATIC ANALYSIS OF CROSS - PLY LAMINATED COMPOSITE PLATE USING FINITE ELEMENT METHOD A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Technology in Mechanical Engineering (Machine Design and Analysis Specialization) By Venkata Sai Gopal . K Department of Mechanical Engineering National Institute of Technology, Rourkela Rourkela-769008(Orissa) May 2007 STATIC ANALYSIS OF CROSS-PLY LAMINATED COMPOSITE PLATE USING FINITE ELEMENT METHOD A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Technology in Mechanical Engineering (Machine Design and Analysis Specialization) BY Venkata Sai Gopal . K Under the esteemed guidance of Dr. N.Kavi Supervisor Professor Dept.of Mechanical Engg. N.I.T. ,Rourkela Shri. B.Rambabu Co-Supervisor Scientist SDSC-SHAR Sriharikota Department of Mechanical Engineering National Institute of Technology, Rourkela Rourkela-769008(Orissa) May 2007 National Institute of Technology Rourkela CERTIFICATE This is to certify that the thesis entitled “static analysis of Cross-Ply laminated composite plate using Finite Element Method” submitted by Mr. Venkata sai Gopal .K , in partial fulfillment of the requirements for the degree of Master of Technology in Mechanical Engineering with specialization in Machine Design and Analysis during session 2005-2007 in the department of Mechanical Engineering, National Institute of Technology, Rourkela (Deemed University) is an authentic work carried out by him under my supervision and guidance. To the best of my knowledge, the matter embodied in the thesis has not been submitted to any other University/Institute for the award of any Degree or Diploma. (Dr.Niranjan Kavi) Date: Professor Dept. of Mechanical Engg. National Institute of Technology Rourkela CERTIFICATE This is to certify that the work in this thesis entitled “static analysis of Cross-Ply laminated composite plate using Finite Element Method” by Mr. Venkata Sai Gopal .K, has been carried out under my supervision in partial fulfillment of the requirements for the degree of Master of Technology in Mechanical Engineering with specialization in Machine Design and Analysis during session 2006-2007 in the department of Solid Propellant Plant, Satish Dhawan Space Centre, SHAR, Sriharikota. (B.Rambabu) Date: Scientist Solid Propellant Space Booster Plant Satish Dhawan Space Centre Sriharikota ACKNOWLEDGEMENT The author expresses his sincere gratitude and indebtedness to the thesis guide Dr. Niranjan Kavi, Professor, Department of Mechanical Engineering, N.I.T., Rourkela for proposing this area for research, for his valuable guidance, encouragement and moral support for the successful completion of this work. His kind attitude always encouraged the author to carry out the present work firmly. The author is thankful to his co-supervisor and colleague at SDSC, SHAR, Shri B.Rambabu for his encouragement and invaluable suggestions in the enhancement of the present work. The author remains grateful to Dr.B.K.Nanda, Head of the Department, Department of Mechanical Engineering, for his kind approval to continue the 4th semester thesis work at Satish Dhawan Space Centre, ISRO. Thanks are due to all the friends of the author, who are involved directly or indirectly in successful completion of the present work. Rourkela Date: (Venkata Sai Gopal. K) ABSTRACT Finite element Analysis is carried out to perform static analysis on a cross-ply laminated composite square plate based on the First order Shear Deformation Theory (FSDT). The theory accounts for constant variation of transverse shear stresses across the thickness of the laminate; and it uses a shear correction factor .The element formulated is an 8-noded iso-parametric quadratic (Serendipity) element. In this analysis, the square plate is analyzed for transverse loading viz., sinusoidal varying load and uniformly distributed load under simply supported boundary conditions. A program is written in MATLAB to obtain the finite element solutions for transverse displacements, normal stresses and transverse shear stresses. Solutions are obtained for 3, 4 and 5 layers of the laminate with cross-ply orientation for different values of side to thickness ratios. Reduced integration scheme is adopted to alleviate shear locking effects. Stresses are found at Gauss points from constitutive relations. The solutions are compared with closed form solutions of FSDT, 3D elasticity solutions and Classical Laminate Plate Theory (CLPT) solutions. It is observed that the results are in close agreement with the available solutions. The element being a C0 continuous element, it ensures the continuity of generalized displacements only; not strains and thus stresses. The model is validated by its good convergence with the analytical results. Analysis can be done on thin as well as moderately thick plates satisfactorily by using this model. CONTENTS Page No Abstract i List of Figures iv List of Tables v Chapter-1 INTRODUCTION 1.1 Introduction 2 1.2 Objective of thesis 3 Chapter -2 LITERATURE SURVEY 2.1 Literature Survey 5 Chapter - 3 FIRST ORDER SHEAR DEFORMATION THEORY 3.1 Introduction 8 3.2 Kinematic relations 10 3.3 Constitutive relations 12 3.4 Virtual work statement 15 Chapter - 4 FINITE ELEMENT METHOD 4.1 Introduction 20 4.2 Principle of virtual displacement 21 4.3 Stiffness matrix derivation 22 4.4 Post computation of stresses and strains 29 ii Chapter – 5 RESULTS AND DISCUSSION 5.1 Results 32 5.2 Discussion 41 5.3 Observations 42 Chapter -6 CONCLUSIONS 45 Chapter -7 SCOPE FOR FUTURE WORK 46 REFERENCES 48 APPENDIX 51 iii List of Figures 1)Non dimensionalized central transverse deflection versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to uniformly distributed loading 2) Non dimensionalized normal stress sigma xx versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to uniformly distributed loading 3) Non dimensionalized normal stress sigma yy versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to uniformly distributed loading 4) Non dimensionalized transverse shear stress sigma xz versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to uniformly distributed loading 5) Non dimensionalized transverse shear stress sigma yz versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to uniformly distributed Loading 6) Non dimensionalized central transverse deflection versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to sinusoidal varying load 7) Non dimensionalized normal stress sigma xx versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to sinusoidal varying load 8) Non dimensionalized normal stress sigma yy versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to sinusoidal varying load 9) Non dimensionalized transverse shear stress sigma xz versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to sinusoidal varying load 10) Non dimensionalized transverse shear stress sigma yz versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to sinusoidal varying load iv List of Tables 1) Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/0) square plate subjected to uniformly distributed loading 2) Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/90/0) square plate subjected to uniformly distributed loading 3) Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/0/90/0) square plate subjected to uniformly distributed loading 4) Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/0) square plate subjected to sinusoidal loading 5) Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/90/0) square plate subjected to sinusoidal loading 6) Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/0/90/0) square plate subjected to sinusoidal loading v Chapter-1 INTRODUCTION 1.1 INTRODUCTION: Composite materials are increasingly used in aerospace, under water, and automotive structures and space structures. The application of composite materials to engineering components has spurred a major effort to analyze structural components made from them Composite materials provide unique advantages over their metallic counterparts, but they also present complex and challenging problems to analysts and designers. To take advantage of full potential of composite materials, accurate models and design methods are required the most common structural elements are plates and shells. An accurate modeling of stress fields is of paramount importance in the design of such components. Laminated composites are one of the classifications of the composites which are used in structural elements like leaf springs, automobile drive shafts, and gears, and axles. The Navier, Levy, and Rayleigh-Ritz developed solutions to composite beams and plate problems. However, exact analytical or variational solutions to these problems cannot be developed when complex geometries, arbitrary boundary conditions or nonlinearities ar involved. Therefore one must resort to approximate methods of analysis that are capable of solving such problems. There are several theories available to describe the kinematics of the laminates. Classical Laminate Plate Theory and First order Shear Deformation Theory are some among them. The Finite element Method is such an approximate method and powerful numerical technique for the solution of differential and integral equations that arise in various fields of engineering and applied science. FEM is an effective method of obtaining numerical solutions to boundary value, initial value and eigen value problems. 2 1.2Objective of the thesis: The objective of the present work is to determine transverse displacements, normal stresses, and transverse shear stresses of 3, 4 and 5 layered cross ply laminated composite square plate subjected to transverse loading viz., sinusoidal varying load and uniformly distributed load when it is simply supported at the edges. In this present work, a displacement based finite element model is formulated based on First order Shear Deformation Theory. It is an iso parametric element with 8 nodes and 3 degrees of freedom at each node. The 3 degrees of freedom are; transverse displacement, rotation about x and rotation about y axes. 3 Chapter- 2 LITERATURE SURVEY 4 2.1Literature Survey: Closed form solutions to 3D elasticity problem of laminated structures are scarce and limited in scope. Pagano developed solutions for simply supported rectangular plates with symmetric lamination undergoing cylindrical and bidirectional bending; the fiber orientation is 0° and 90°. Ren (1987) has extended the cylindrical bending solution to infinitely long cylindrical shells. Noor and Burton (1990a), (1992) have provided solutions for the bending, buckling and vibration of anti-symmetrically laminated rectangular plates, periodic in the in-plane directions. Savithri and Varadan (1992) studied plates under uniformly distributed and concentrated loads. All these approaches use Fourier expansions in the in-plane directions resulting in sets of ordinary differential equations with constant coefficients, which can be solved exactly. The unknown coefficients of the solutions are determined by boundary and interface conditions in thickness direction. J.N.Reddy and W.C.Chao studied and derived Closed form solutions and Finite Element Solutions for Laminated Anisotropic Rectangular Plates.[1]. AA.Khedier and J.N.Reddy derived Exact solutions for bending of thin and thick cross-ply laminated beams[2]. T.J.R.Hughes and T.E.Tezduyar developed Four-Node Bilinear Isoparametric Element based upon Mindlin Plate Theory [3]. The state-space concept in conjunction with the Jordan canonical form is presented to solve the governing equations for the bending of cross-ply laminated composite beams by J.N.Reddy1997[4].An elastic-plastic stress analysis was carried out on simply supported and clamped aluminum metal-matrix laminated plates. The thin plate model is also used for the study of rectangular plates with practically important mixed edge constraints by Liew, Hua & Lim and Laura & Gutierrez.A composite material model is presented to analyze progressive failure in composite structures by Raimondo Luciano,Raffaele Zinno in 2000. Song Cen, Zhen-Han Yao developed a new 4-node quadrilateral finite element for the analysis of composite plates in 2002. A new analytical method was developed by M.R.Khalili and R.K.Mittal to analyze the response of laminated composite plates subjected to static and dynamic loading.(2005).B.N.Pandya and T.Kant worked on Finite Element analysis of Laminated Composite Plates Using a Higher – Order Displacement Model [5].B.R.Somashekar,G.Pratap and C.Ramesh Babu developed a simple and efficient Four nodded, Laminated Anisotropic Plate Element[6]. Xiao-Ping Shu,Kostas P.Soldatos determined Stress distributions in angle-ply laminated plates subjected to cylindrical bending[7]. 5 Exact solutions for rectangular bidirectional composites and sandwich plates were developed by Pagano,N.J.,[8].Reddy, J.N., Khdeir, A.A. developed Levy type solutions for symmetrically laminated rectangular plates using first order shear deformation theory[9].An exact approach to the elastic stat of stress of shear deformable antisymmetic angle ply laminated plates was developed by Khedeir A. A., [10]. He also compared shear deformable and kirchoff theories for bending buckling and vibration of antisymmetric angle ply laminated plates.[11].Srinivas and Jogarao got some results from exact analysis of thick laminates in vibration and buckling[12]. A review is made on plate bending finite elements by Hrabok, M.M. and Hrudey [13].Fraeijis de veubeke developed a conforming finite element for plate bending [14]. A triangular refined plate bending element was suggested by Bell K. [15]. Irons B.M. developed a conforming quartic triangular element for plate bending [16].Strcklin, J.a. haisler, w developed a rapidly converging triangular plate element [17]. A study was made on 3 node triangular plate bending element by Batoz, J.L and Bathe, K.J. [18]. 6 Chapter-3 FIRST ORDER SHEAR DEFORMATION THEORY 7 3.1Introduction: The use of composite materials in structural components is increasing due to their attractive properties such as high strength-to-weight ratio, ability to tailor the structural properties, etc. Plate structures find numerous applications in the aerospace, military and automotive industries. The effects of transverse shear deformation are considerable for composite structures, because of their high ratio of extensional modulus to transverse shear modulus. Most of the structural theories used to characterize the behavior of composite laminates fall into the category of equivalent single layer (ESL) theories. In these theories, the material properties of the constituent layers are combined to form a hypothetical single layer whose properties are equivalent to through the thickness integrated sum of its constituents. This category of theories has been found to be adequate in predicting global response characteristics of laminates, like maximum deflections, maximum stresses, and fundamental frequencies, or critical buckling loads In the context of ESL theories, the simplest one is the CLT which neglects the shear contribution in the laminate thickness. However, flat structures made of fiber-reinforced composite materials are characterized by non negligible shear deformations in the thickness direction, since the longitudinal elastic modulus of the lamina can much higher than the shear and the transversal moduli; hence the use of a shear deformation laminate theory is recommended. The extension of the Reissner and Mindlin model to the case of laminated anisotropic plates, i.e. FSDT ,accounts for shear deformation along the thickness in the simplest way. It gives satisfactory results for a wide class of structural problems, even for moderately thick laminates, requiring only C0-continuity for the displacement field. The transverse shearing strains (stresses) are assumed to be constant along the plate thickness so that stress boundary conditions on the top and the bottom of the plate are violated; shear correction factors must be introduced. The determination of shear correction factors is not a trivial task, since they depend both on the lamination sequence and on the state of deformation . 8 Assumptions: 1) The layers are perfectly bonded 2) The material of each layer is linearly elastic and has two planes of material symmetry 3) The strains and displacements are small 4) Deflection is wholly due to bending strains only 5) Plane sections originally perpendicular to the longitudinal plane of the plate remain plane, but not necessarily perpendicular to longitudinal plane 6) The transverse shearing strains (stresses) are assumed to be constant along the plate thickness 9 3.2Kinematic relations: The displacement field of the first-order theory is of the form u ( x , y , z , t ) = u 0 ( x , y , t ) + zφ x ( x , y , t ) v ( x , y , z , t ) = v 0 ( x , y , t ) + z φ y ( x , y , t ) ----------------------- (1) w (x, y, z,t) = w0 (x, y,t) Where (u0 , v0 , w0 , φ x , φ y ) are unknown functions, called the generalized displacements. (u0 , v0 , w0 ) , denote the displacements of a point on the plane z = 0. ∂u ∂v = φx, = φy ∂z ∂z Indicate that φx and φ y are the rotations of the transverse normal about the y- and x- axes, respectively, owing to bending only. The strains associated with the displacement field (1) are given by: ε xx = ⎛ ∂u ∂v ⎞ ∂v ∂w ∂u ∂w ∂w , ε yy = , ε zz = , γ xy = ⎜ + ⎟ , γ xz = + φ y -------(2) + φx , γ yz = ∂y ∂y ∂x ∂z ∂x ⎝ ∂y ∂x ⎠ Substituting the expressions for u,v,w from eq. (1) in equation (2) gives: ε xx = ∂φ y ∂v ∂u0 ∂φ , ε zz = 0 + z x , ε yy = 0 + z ∂y ∂y ∂x ∂x ⎛ ∂u ∂v ⎞ ⎛ ∂φ ∂φ ⎞ ∂w ∂w γ xy = ⎜ 0 + 0 ⎟ + z ⎜ x + y ⎟ , γ xz = 0 + φx , γ yz = 0 + φ y -------------------- (3) ∂y ∂x ∂x ⎠ ⎝ ∂y ∂x ⎠ ⎝ ∂y In Matrix form the above equations are given by 10 ⎧ ∂u0 ⎫ ⎪ ∂x ⎪ ⎧ ∂φx ⎪ ⎪ ⎪ ∂x ∂v0 ⎪ ⎪ ⎧ε xx ⎫ ⎪ ∂φ y ⎪ ⎪ ⎪ε ⎪ ⎪ ∂y ⎪ ⎪ ⎪⎪ yy ⎪⎪ ⎪ ∂u ∂v ⎪ ⎪⎪ ∂y 0 + 0 ⎬+ z⎨ ⎨γ xy ⎬ = ⎨ ∂φ ⎪γ ⎪ ⎪ ∂y ∂x ⎪ ⎪⎜⎛ ∂φx + y xz ⎪ ⎪ ⎪ ∂w0 ⎪ ⎪ ∂y ∂x + φx ⎪ ⎪⎝ ⎪⎩γ yz ⎪⎭ ⎪ 0 ⎪ ∂x ⎪ ⎪ ∂ w ⎪ 0 ⎪ ⎪ 0 ⎪ ∂y + φ y ⎪ ⎩ ⎩ ⎭ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎞⎪ ⎟⎪ ⎠ ⎪ ⎪ ⎪⎭ The above matrix is in the form: ⎧ε xx ⎫ ⎧ε xx0 ⎫ ⎧ kx ⎫ ⎪ε ⎪ ⎪ 0 ⎪ ⎪k ⎪ y ⎪⎪ yy ⎪⎪ ⎪⎪ε xx ⎪⎪ ⎪⎪ ⎪⎪ 0 γ ⎨ xy ⎬ = ⎨γ xy ⎬ + z ⎨k xy ⎬ ⎪γ ⎪ ⎪γ 0 ⎪ ⎪0⎪ ⎪ xz ⎪ ⎪ xz0 ⎪ ⎪ ⎪ ⎪⎩γ yz ⎪⎭ ⎪⎩γ yz ⎪⎭ ⎪⎩ 0 ⎪⎭ For plate bending problem, the in- plane displacements (u,v) are uncoupled from ( w0 , φ x , φ y ) . Hence, the equations reduce as follows: ∂φ ⎧ z x ⎪ ∂x ⎧ zk x ⎫ ⎪ ∂ φ ⎪ z y ⎧ε xx ⎫ ⎪ zk y ⎪ ⎪ ⎪ ∂y ⎪ε ⎪ ⎪ ⎪ ⎪ ⎪ yy zk ⎪⎪ ⎪⎪ ⎪ xy ⎪ ⎪ ⎛ ∂φx ∂φ y + ⎨γ xy ⎬ = ⎨ ∂w0 ⎬ = ⎨ z ⎜ ∂x ⎪γ ⎪ ⎪φx + ∂x ⎪ ⎪ ⎝ ∂y xz ⎪ ⎪ ⎪ ⎪ ⎪ ∂w0 ⎪ ⎪ φx + ∂w0 ⎪⎩γ yz ⎪⎭ ⎪ ∂x ⎪φ y + ∂y ⎪ ⎪ ⎩ ⎭ ⎪ ∂w0 ⎪ φy + ∂y ⎩ 11 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎞⎪ ⎟ ⎬ ------------------(4) ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ 3.3Constitutive Relations: The Stress-Strain relations for a typical lamina k with reference to the lamina co-ordinate axes (1-2-3) are given by k k k ⎧σ 1 ⎫ ⎡ Q11 Q12 0 ⎤ ⎧ ε1 ⎫ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎨σ 2 ⎬ = ⎢Q21 Q22 0 ⎥ ⎨ ε 2 ⎬ ⎪τ ⎪ ⎢ 0 0 Q33 ⎥⎦ ⎩⎪γ 12 ⎭⎪ ⎩ 12 ⎭ ⎣ k ⎧τ 23 ⎫ ⎡Q44 ⎨ ⎬ =K⎢ ⎩τ 13 ⎭ ⎣ 0 In which (σ 1 , σ 2 ,τ12 ,τ 23 ,τ13 ) k k 0 ⎤ ⎧γ 23 ⎫ ⎨ ⎬ ---------------------(5) Q55 ⎥⎦ ⎩γ 13 ⎭ are the stress and (ε1 , ε 2 , γ 12 , γ 23 , γ 13 ) are the linear strain components referred to the lamina co-ordinate axes (1-2-3). The Qij’s are the plane stress reduced elastic constants of the kth lamina and the following relations hold between these and the engineering constants. Q11 = E1 1 −ν 12ν 21 Q22 = , E2 1 − ν 12ν 21 Q33 = G12 , Q12 = , ν 12 E2 1 − ν 12ν 21 = ν 21 E1 1 − ν 12ν 21 , Q44 = G23 , Q55 = G13 . The stress-strain relations are the basis for the stiffness and stress analysis of an individual lamina subjected to forces in its own plane. The relations are therefore indispensable in the analysis of laminates. There are 4 independent material properties, E1 , E2 , G12 , and ν 12 the reciprocal relation is given by ν 12 E1 = ν 21 E2 Stress – Strain relations for a lamina of arbitrary orientation From elementary mechanics of materials the transformation equations for expressing stresses in 1-2 coordinate system(principal coordinate system) in terms of stresses in x-y coordinate system. 2 ⎧σ 1 ⎫ ⎡ cos θ ⎪ ⎪ ⎢ 2 ⎨σ 2 ⎬ = ⎢ sin θ ⎪τ ⎪ ⎢ − sin θ cos θ ⎩ 12 ⎭ ⎣ sin 2 θ cos 2 θ sin θ cos θ 2sin θ cos θ ⎤ ⎧σ X ⎫ ⎥⎪ ⎪ −2sin θ cos θ ⎥ ⎨σ y ⎬ -------------------(6) cos 2 θ − sin 2 θ ⎥⎦ ⎪τ xy ⎪ ⎩ ⎭ 12 Where θ is the angle from the x-axis to the axis 1. Following the usual transformation of Stress-Strain between the lamina and laminate coordinate systems, the Stress-Strain relations for the kth lamina in the laminate coordinates (x,y,z) are written as: k ⎧σ x ⎫ ⎡ Q11 Q12 ⎪ ⎪ ⎢ ⎨σ y ⎬ = ⎢Q21 Q22 ⎪ ⎪ ⎢ ⎩τ xy ⎭ ⎢⎣Q31 Q32 k ⎡Q44 ⎧τ yz ⎫ ⎨ ⎬ =K⎢ ⎩τ xz ⎭ ⎢⎣Q54 Q13 ⎤ ⎥ Q23 ⎥ ⎥ Q33 ⎦⎥ k ⎧εx ⎫ ⎪ ⎪ ⎨ε y ⎬ ⎪γ ⎪ ⎩ xy ⎭ k k k Q45 ⎤ ⎧γ yz ⎫ ⎥ ⎨ ⎬ -----------------------(7) Q55 ⎥⎦ ⎩γ xz ⎭ in which , σ = (σ x , σ y , τ xy , τ yz , τ xz ) and ε = ( ε x , ε y , γ xy , γ yz , γ xz ) are the stress and linear t t strain vectors with respect to the laminate axes and Qij s are the plane stress reduced elastic constants in the plate (laminate) axes of the kth lamina given in Appendix. The superscript t denotes the transpose of a matrix. K refers to the Shear Correction Factor used in FSDT. Normally its value is 5/6. Stress Resultants: The resultant forces and moments acting on a laminate are obtained by integration of the stresses in each layer or lamina through the laminate thickness, for example, t /2 N x = ∫ t/2 σ xdz M x = ∫ σ x zd z −t / 2 −t / 2 Actually, N x is a force per unit length (width) of the cross section of the laminate. Similarly, M x is a moment per unit length. The entire collection of force and moment resultants for an N-layered laminate is shown and is defined as k k ⎧ Nx ⎫ ⎧σ x ⎫ ⎧σ x ⎫ zk t/2 N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Ny ⎬ = ∫ ⎨σ y ⎬ dz = ∑ ∫ ⎨σ y ⎬ dz k =1 zk−1 ⎪ ⎪ ⎪ −t / 2 ⎪ ⎪ ⎪ N τ τ ⎩ xy ⎭ ⎩ xy ⎭ ⎩ xy ⎭ 13 k k ⎧M x ⎫ ⎧σ x ⎫ ⎧σ x ⎫ zk t/2 N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ M zdz σ = = ⎨ y ⎬ ∫ ⎨ y⎬ ⎨σ y ⎬ zdz -----------------(8) ∑ ∫ k =1 zk −1 ⎪ ⎪ ⎪ −t / 2 ⎪ ⎪ ⎪ M τ τ ⎩ xy ⎭ ⎩ xy ⎭ ⎩ xy ⎭ Where, Z k and Z k-1 are defined in figure shown in appendix. Z 0 = - t/2.These force and moment resultants do not depend on Z after integration, but are functions of x and y, the coordinates in the plane of the laminate middle surface. k ⎡Q11 Q12 Q13 ⎤ ⎧ ⎧ε 0 x ⎫ ⎧Nx ⎫ ⎧kx ⎫ ⎫ z z k k N ⎢ ⎥ ⎪⎪ ⎪⎪ 0 ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ N Q Q Q dz = + ε ⎨ y ⎬ ∑ ⎢ 21 ⎨k y ⎬zdz ⎬ 22 23 ⎥ ⎨ ∫ ⎨ y ⎬ ∫ ⎥ ⎪ zk−1 ⎪ 0 ⎪ zk −1 ⎪ ⎪ ⎪ k =1 ⎢ ⎪ ⎪ N k Q Q Q γ xy xy 32 33 ⎥ ⎢⎣ 31 ⎩ ⎭ ⎩ ⎭ ⎪⎭ ⎦ ⎪⎩ ⎪⎩ xy ⎪⎭ k ⎫ ⎡Q11 Q12 Q13 ⎤ ⎧ ⎧ε 0 x ⎫ ⎧M x ⎫ ⎧kx ⎫ z z k k N ⎢ ⎥ ⎪⎪ ⎪⎪ 0 ⎪⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪⎪ = + M Q Q Q zdz ε ⎨ y ⎬ ∑ ⎢ 21 22 ⎨ky ⎬z dz ⎬ 23 ⎥ ⎨ ∫ ⎨ y ⎬ ∫ ⎥ ⎪zk−1 ⎪ 0 ⎪ zk −1 ⎪ ⎪ ⎪ k =1 ⎢ ⎪ ⎪ M k Q Q Q γ xy xy 31 32 33 ⎢⎣ ⎥⎦ ⎪⎩ ⎩⎪ xy ⎭⎪ ⎩ ⎭ ⎩ ⎭ ⎪⎭ Thus the above equations can be written as ⎧Nx ⎫ ⎡ A11 ⎪ ⎪ N ⎢ ⎨ N y ⎬ = ∑ ⎢ A12 ⎪ ⎪ k =1 ⎢ A ⎣ 31 ⎩ N xy ⎭ A12 A22 A32 ⎧M x ⎫ ⎡ B11 B12 ⎪ ⎪ N ⎢ ⎨M y ⎬ = ∑ ⎢ B12 B22 ⎪ ⎪ k =1 ⎢ B B M 32 ⎣ 31 ⎩ xy ⎭ 0 A13 ⎤ ⎧ε x ⎫ ⎡ B11 B12 ⎪⎪ ⎪⎪ A23 ⎥⎥ ⎨ε 0 y ⎬ + ⎢⎢ B12 B22 A33 ⎥⎦ ⎪γ 0 ⎪ ⎢⎣ B31 B32 ⎩⎪ xy ⎭⎪ B13 ⎤ ⎧kx ⎫ ⎪ ⎪ B23 ⎥⎥ ⎨k y ⎬ B33 ⎥⎦ ⎪kxy ⎪ ⎩ ⎭ 0 B13 ⎤ ⎧ε x ⎫ ⎡ D11 D12 ⎪⎪ ⎪⎪ B23 ⎥⎥ ⎨ε 0 y ⎬ + ⎢⎢ D12 D22 B33 ⎥⎦ ⎪γ 0 ⎪ ⎢⎣ D31 D32 ⎪⎩ xy ⎪⎭ D13 ⎤ ⎧kx ⎫ ⎪ ⎪ D23 ⎥⎥ ⎨k y ⎬ D33 ⎥⎦ ⎪kxy ⎪ ⎩ ⎭ Where N ( ) Aij = ∑ Qij k =1 ( Z k − Z k −1 ) k Bij = ( ) (Z 1 N ∑ Qij 2 k =1 2 k − Z 2 k −1 ) Dij = k Where Aij Coefficients are called extensional stiff nesses, Bij Coefficients are called coupling stiff nesses, 14 ( ) (Z 1 N ∑ Qij 3 k =1 k 3 k − Z 3k −1 ) ---(9) Dij Coefficients are called bending stiff nesses. In this present work , as only cross ply laminates only analyzed, the Bij terms get vanished. D16=D26=0. As there present no inplane forces, they are uncoupled from the equations. So only D matrix is used in this plate bending analysis. 3.4Virtual Work Statement The variational formuations form a powerful basis for obtaining approximate solutions to real world/practical problems. The variational method uses the variational principles, such as the Principle of Virtual displacements, to determine approximate displacements as continuous functions of position in the domain. In the Classical sense, variational principle has to do with the minimization of a functional, which includes all the intrinsic features of the problem, such as the governing equations, boundary and /or initial conditions, and constraint conditions. One of the concepts of Variational formulation is Principle of virtual work. It is the work done on a particle or a deformable body by actual forces in displacing the particle or the body through a hypothetical displacement that is consistent with the geometric constraints. The applied forces are kept constant during the virtual displacement. The Principle of virtual displacement states that the virtual work done by actual forces in moving through virtual displacements is zero if the body is in equilibrium. The principle of virtual work states that “a continuous body is in equilibrium if the virtual work of all forces acting on the body is zero in a virtual displacement”. The principle of virtual work is independent of any constitutive law and applies to elastic (linear and non linear) and in elastic continuum problems. The plate to be analyzed may have curved or straight boundaries as well as different boundary conditions. The principle of virtual work statement for the plate can be stated as δ WI + δ WE = 0 Where δWI = Virtual work resulting from internal forces 15 δWE = Virtual work resulting from external forces The governing equations of the first order shear deformation theory are derived using the dynamic version of the principle of virtual displacements for the displacements (w, φx , φ y ): T 0 = ∫ (δ K − ( δ U + δ V ) )d t --------------(10) 0 δWI = δU and Here δWE = δK+δV Where δU is virtual strain energy, δV is virtual work done by applied forces and δK is virtual kinetic energy. On substituting the expressions for δU, δV and δK in the equation noA. we get ⎛ ⎪⎧ ∂ 2φ y ∂ 2φ x ∂ 2 w ⎪⎫ 2 2 + ρ z δφ y + ρδ w ⎜ ⎨ ρ z δφ x ⎬ ∂t ∂t ∂ t ⎭⎪ ⎜ ⎩⎪ ⎜ 0 = ∫ ⎜ + {δ ε x x σ x x + δ ε y y σ y y + 2 δ ε x y σ x y + 2 δ ε x z σ x z + 2 δ ε y z σ Ve ⎜ ⎜ − ⎧ δ w qdxdy ⎫ ⎬ ⎜ ⎨∫ e Ω ⎩ ⎭ ⎝ Carrying out integration with respect to z, we get 0= ∫[ h/2 ∫ Ωe − h / 2 ρ z 2δφx ∂ 2φ y h / 2 ∂ 2φx h / 2 2 ∂2w z w + ρ δφ + ρδ y ∂t 2 − h∫/ 2 ∂t 2 − h∫/ 2 ∂t 2 h/2 h/2 ∂δφ y ⎛ ∂δφx ∂δφ y ∂δφx z z σ σ + + + ⎜ xx yy ∫ ∫ ∫ ∂x ∂y ∂y ∂x −h / 2 −h / 2 −h/ 2 ⎝ h/2 + ⎞ ⎟σ xy z ⎠ h/2 h/2 ⎛ ∂δ w ⎞ ∂δ w ⎞ ⎛ + ∫ ⎜ δφx + ⎟σ yz z − ∫ qδ w]dxdydz ⎟σ xz z + ∫ ⎜ δφ y + ∂x ⎠ ∂y ⎠ −h / 2 ⎝ −h / 2 ⎝ −h / 2 h/2 On simplification the above equation yields to 16 ⎞ ⎟ ⎟ ⎟ yz } ⎟ ⎟ ⎟ ⎟ ⎠ ⎡ ⎤ ⎛ ∂ 2φ y ⎞ ∂δφ y ∂ 2φx ∂δφx ∂2w + M yy ⎢ I 0δ w 2 + I 2 ⎜⎜ δφx 2 + δφ y 2 ⎟⎟ + M xx ⎥ ∂t ∂t ∂t ⎠ ∂x ∂y ⎢ ⎥ ⎝ 0= ∫ ⎢ ⎥dxdy δφ ∂ ⎛ ∂δφx ⎛ Ωe ⎢ ∂δ w ⎞ ∂δ w ⎞ ⎛ y ⎞ ⎥ + M xy ⎜ + ⎟ + Qx ⎜ δφx + ⎟ − qδ w⎥ ⎟ + Qy ⎜ δφ y + ⎢ y x x y ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ Where h/2 I0 = ∫ h/2 ρ dz , I 2 = −h / 2 ρ z dz , Qx = K −h / 2 h/2 M xx = ∫ ∫ ∫ σ xz dz , Qy = K −h / 2 h/2 σ xx zdz , ∫ M yy = −h / 2 Where, h/2 2 σ yy zdz , h/2 ∫ σ yz dz , −h / 2 h/2 M xy = −h / 2 ∫ σ xy zdz −h / 2 I0 is the mass moment of inertia term, I2 is the rotary inertia term and K is the shear correction factor. For Static case, the above virtual displacement equation becomes, 0= ⎡ ∫ ⎣⎢ M Ωe xx ∂δφ y ⎛ ∂δφx ∂δφ y ∂δφx + M yy + M xy ⎜ + ∂x ∂y ∂x ⎝ ∂y ⎤ ⎞ ⎛ ∂δ w ⎞ ∂δ w ⎞ ⎛ ⎟ + Qx ⎜ δφx + ⎟ − qδ w⎥dxdy ⎟ + Qy ⎜ δφ y + ∂x ⎠ ∂y ⎠ ⎝ ⎝ ⎠ ⎦ The Virtual Work Statement contains 3 weak forms for the 3 displacements (w, φx , φ y ).They are identified by collecting the terms involving δw , δ φx and δ φ y separately and equating them to zero: 0= ⎛ ∫ ⎜⎝ Q x Ωe 0= ⎛ ∫ ⎜⎝ M xx Ωe 0= Ωe ⎞ ∂δφx ∂δφ x + M xy + Qxδφx ⎟dxdy ∂x ∂y ⎠ ∂δφ y ⎛ ∫ ⎜⎝ M ⎞ ∂δ w ∂δ w + Qy − qδ w ⎟dxdy ∂x ∂y ⎠ xy ∂x + M yy ∂δφ y ⎞ + Qyδφ y ⎟dxdy ∂y ⎠ The governing equations of FSDT are obtained from the weak forms given above 17 ∂Qx ∂Qy + +q =0 ∂x ∂y ∂M xx ∂M xy + − Qx = 0 -----------------(11) ∂x ∂y ∂M xy ∂M yy + − Qy = 0 ∂x ∂y 18 Chapter-4 FINITE ELEMENT METHOD 19 4.1Introduction: The finite Element Method is a powerful computational technique for the solution of differential and integral equations that arise in various fields of engineering and applied science. The method is a generalization of the Classical Variational (i.e., the Rayleith-Ritz) and weighted – residual (Galerkin, Least-squares etc.)methods. Since most real-world problems are defined on domains that are geometrically complex and may have different typers of boundary conditions on different portions of the boundary of the domain, it is difficult to generate approximation functions required in the traditional variational methods. The basic idea of the finite element method is to view a given domain as an assemblage of simple geometric shapes called finite elements, for which it is possible to systematically generate the approximation functions needed in the solution methods. The ability to represent domains with irregular geometries by a collection of finite elements makes the method a valuable practical tool for the solution of boundary, initial, and eigen value problems arising in various fields of engineering. The approximation functions are often constructed using ideas from interpolation theory, and hence they are also called interpolation functions. Thus, the finite element method is a piecewise application of the variational and weighted –residual methods. Finite Element Modelling In the FEM, the total solution domain is discretized in to N elements (sub-domains).Then, the finite element model of the problem is developed using variational method .The variational formuations form a powerful basis for obtaining approximate solutions to practical problems. The variational method uses the variational principles, such as the Principle of Virtual displacements, to determine approximate displacements as continuous functions of position in the domain. In the Classical sense, variational principle has to do with the minimization of a functional, which includes all the intrinsic features of the problem, such as the governing equations, boundary and /or initial conditions, and constraint conditions. 20 4.2Principle of Virtual Displacements: It states that “a deformable body is in equilibrium if the total external virtual work is equal to the total internal virtual work for every virtual displacement satisfying the kinematic boundary conditions”. δ WI = δ WE Where δWI = Virtual work resulting from internal forces δWE = Virtual work resulting from external forces The Principle of Virtual work for the plate can be stated as ∫ δε V T σ dv + ∫ δγ Tτ dv = ∫ qδ ddA ------------(12) V V The integration over the thickness reduces eq 12 as follows: ∫ {δε σ xx + δε Tyyσ yy + δγ xyT σ xy + δγ xzT σ xz + δγ Tyzσ yz }dA -------------------(13) T xx A Where A is the cross-sectional area and V is the volume of the plate. Using the lamina constitutive relation eq 12 leads to the following form: ∫ ⎡⎣ ε {M } + Φ {Q }⎤⎦ d A t t A Replacing the stress resultants by the product of rigidity matrix and strains in the strain energy expression in equation-------------, we get ∫ {δε T xx DBε xx + δε Tyy DBε yy + δγ xyT DBγ xy + δγ xzT Dsγ xz + δγ Tyz Dsγ yz }dA -----------------(14) A Which is the final form of the virtual work principle as it is required for finite element calculations. 21 4.3Stiffness matrix derivation : In this work, an eight noded isoparametric element (Serendipity Element) is chosen to discretize the plate domain. The variation of displacement u is expressed by the polynomial in natural coordinates as: u = α1 + α 2 r + α 3 s + α 4 r 2 + α 5 rs + α 6 s 2 + α 7 r 2 s + α 8 rs 2 ----------(15) In the above polynomial, cubic terms have been omitted. The nodal displacement vector {d } is obtained by substituting the coordinates for the nodes as: ⎧ u1 ⎫ ⎡1 −1 −1 ⎪u ⎪ ⎢1 1 −1 ⎪ 2⎪ ⎢ ⎪u3 ⎪ ⎢1 1 1 ⎪ ⎪ ⎢ u 1 −1 1 {d } = ⎨⎪ 4 ⎬⎪ = ⎢⎢ ⎪u5 ⎪ ⎢1 0 −1 ⎪u6 ⎪ ⎢1 1 0 ⎪ ⎪ ⎢ ⎪u7 ⎪ ⎢1 0 1 ⎪u ⎪ ⎢1 −1 0 ⎩ 8⎭ ⎣ 1 −1 −1⎤ ⎧α1 ⎫ 1 −1 1 −1 1 ⎥⎥ ⎪⎪α 2 ⎪⎪ 1 1 1 1 1 ⎥ ⎪α 3 ⎪ ⎥⎪ ⎪ 1 −1 1 1 −1⎥ ⎪α 4 ⎪ ⎨ ⎬ 0 0 1 0 0 ⎥ ⎪α 5 ⎪ ⎥ 1 0 0 0 0 ⎥ ⎪α 6 ⎪ ⎪ ⎪ 0 0 1 0 0 ⎥ ⎪α 7 ⎪ ⎥ 1 0 0 0 0 ⎥⎦ ⎪⎩α 8 ⎪⎭ 1 1 {α } = [ A]−1 {d } where [ A]−1 is as given below: ⎡ −1 −1 −1 −1 2 2 2 2 ⎤ ⎢ 0 0 0 0 0 2 0 −2 ⎥ ⎢ ⎥ ⎢ 0 0 0 0 −2 0 2 0 ⎥ ⎢ ⎥ 1 ⎢ 1 1 1 1 −2 0 −2 0 ⎥ −1 And [ A] = 4 ⎢ 1 −1 1 −1 0 0 0 0 ⎥ ⎢ ⎥ ⎢ 1 1 1 1 0 − 2 0 −2 ⎥ ⎢ −1 −1 1 1 2 0 −2 0 ⎥ ⎢ ⎥ ⎢⎣ −1 1 1 −1 0 −2 0 2 ⎥⎦ {δ } T = [1, r , s, r 2 , rs, s 2 , r 2 s, rs 2 ] 22 ⎡ −1 −1 −1 −1 2 2 2 2 ⎤ ⎢ ⎥ ⎢ 0 0 0 0 0 2 0 −2 ⎥ ⎢ 0 0 0 0 −2 0 2 0 ⎥ ⎢ ⎥ 1 ⎢ 1 1 1 1 −2 0 −2 0 ⎥ −1 [ A] = 4 ⎢ 1 −1 1 −1 0 0 0 0 ⎥ ⎢ ⎥ ⎢ 1 1 1 1 0 −2 0 −2 ⎥ ⎢ −1 −1 1 1 2 0 −2 0 ⎥ ⎢ ⎥ ⎢⎣ −1 1 1 −1 0 −2 0 2 ⎥⎦ Let δ be expressed as {δ } = [1, r , s, r 2 , rs, s 2 , r 2 s, rs 2 ] then, the shape functions for this T element are given by [N ] T = {δ } [ A]−1 which is given as T ⎧(1 − r )(1 − s )(−r − s − 1) ⎫ ⎪(1 + r )(1 − s )(r − s − 1) ⎪ ⎪ ⎪ ⎪(1 + r )(1 + s )(r + s − 1) ⎪ ⎪ ⎪ 1 ⎪(1 − r )(1 + s )(− r + s − 1) ⎪ [N ] = ⎨ ⎬ ------------------(16) 4 ⎪2(1 + r )(1 − r )(1 − s) ⎪ ⎪2(1 + r )(1 + s )(1 − s) ⎪ ⎪ ⎪ ⎪2(1 + r )(1 − r )(1 + s) ⎪ ⎪2(1 − r )(1 + s )(1 − s) ⎪ ⎩ ⎭ The shape functions can be expressed in concise form as follows [N ] T 1 = [ N1 N 2 N 3 N 4 N 5 N 6 N 7 N8 ] 4 In the present work, of laminated plate bending, the transverse displacement w, the rotation about x-axis φx and the rotation about y-axis φ y are considered as the only 3 degrees of freedom at each node of the element. The in- plane displacements (u, v) are neglected in this study. The displacement vector d is given as d = ( w, φ x , φ y ) The generalized displacements at any point (x, y) in the element are expressed in terms of the nodal values of displacements and shape functions as given below: 23 8 8 w = ∑ N i wi 8 φx = ∑ N iφxi i =1 φ y = ∑ N iφ yi i =1 i =1 Adopting the same shape function ‘N’ to define all the components of the generalized displacement vector, d, we can write N d = ∑ N i d i -----------(17) i =1 The nodal displacements are given by {d } T = ⎡⎣ w1θ x1θ y1w2θ x 2θ y 2 w3θ x 3θ y 3 w4θ x 4θ y 4 w5θ x 5θ y 5 w6θ x 6θ y 6 w7θ x 7θ y 7 w8θ x8θ y 8 ⎤⎦ In which, N is the number of nodes in the element. Now, referring to the expressions in equation (4), the bending curvatures and the shear strains can be written in terms of nodal displacements d using the matrix notation as follows: {Φ} = LS d {ε } = LB d In which the subscripts B and S refer to bending and shear respectively and the matrices Lb And Ls attain the following form ⎡ ∂ ⎢0 ∂x ⎢ ⎢ LB = ⎢0 0 ⎢ ⎢ ∂ ⎢0 ∂y ⎣ ⎡∂ ⎢ ∂x LS = ⎢ ⎢∂ ⎢⎣ ∂y ⎤ 0⎥ ⎥ ∂⎥ ∂y ⎥⎥ ∂⎥ ⎥ ∂x ⎦ ⎤ 1 0⎥ ⎥ -------(18) 0 1⎥ ⎥⎦ Knowing the generalized displacement vector, d, at all points within the element, the generalized strain vectors at any point are determined with the aid of equations (17) and (18) as follows: 24 N N i =1 i =1 N N i =1 i =1 {ε } = LB d = LB ∑ Ni di = ∑ BiB di = BB a {Φ} = LS d = LS ∑ Ni di = ∑ BiS di = BS a ----------------------------(19) In which The B matrix for the i th node can be written as ⎡B ⎤ Bi = ⎢ iB ⎥ ⎣ BiS ⎦ ⎡ ⎢0 ⎢ ⎢ [ BiB ] = [ LB ][ Ni ] = ⎢0 ⎢ ⎢ ⎢0 ⎣ ⎤ 0 ⎥ ⎥ ∂N i ⎥ ⎥ ∂y ⎥ ∂N i ⎥ ⎥ ∂x ⎦ ∂N i ∂x 0 ∂N i ∂y ⎡ ∂N i ⎢ ∂x [ BiS ] = [ LS ][ N i ] = ⎢ ⎢ ∂N i ⎢⎣ ∂y ⎤ 1 0⎥ ⎥ 0 1⎥ ⎥⎦ N , BB = ∑ BiB i =1 N , BS = ∑ BiS i =1 and a = (d1T , d 2T , d 3T ,............, d NT ) Substituting the above strain-displacement matrix B, in the virtual work statement derived above results in ∫ ⎡⎣a B D t A t B B BB a + a t Bst D s Bs a ⎤⎦dA or ∫ (a K a)dA t e A e In which K is the element stiffness matrix and is expressed as K e = ∫ ⎡⎣ BBt D B BB + Bst D s Bs ⎤⎦dA A Because of the symmetry of the stiffness matrix, only the blocks Kij lying on one side of the main diagonal are formed for simplification. The integral is evaluated numerically using the Gauss quadrature rule, in the limits of -1 to +1 Kije = ∫ 1 ∫ 1 −1 −1 Bit DB j J drds 25 g g K ije = ∑∑ WaWb Bit DB j J a =1 b =1 We = a t Fc + a t ∫ ( N iT q + N iT P)dA A mπ x nπ y ⎞ T ⎛ sin Pi = ∑∑ WaWb J N iT {100} ⎜ q + Pmn sin ⎟ a b ⎠ ⎝ a =1 b =1 g g Where, Wa and Wb are the weights of the Gauss points determined using Legendre Polynomials. g’s are the Gauss sampling points at which numerical integration is carried out. J is the determinant of the Jacobian matrix [J].Subscripts i and j vary from one to the number of odes per element. The matrices Bi and D are given above and Bj is obtained by replacing i by j. For this flexural analysis , the total external work done by the applied external loads for an element e, is given by We = a t Fc + at ∫ ( NiT q + NiT P)dA A In which suffix, i , varies from one to number of nodes per element. Fc is the vector of concentrated nodal loads corresponding to nodal degrees-of-freedom. q and P are the uniform and sinusoidal distributed load intensities acting over an element e in the z-direction. The integral of the above equation is evaluated numerically using the Gauss quadrature rule as follows mπ x nπ y ⎞ T ⎛ Pi = ∑∑ WaWb J N iT {100} ⎜ q + Pmn sin sin ⎟ in which a and b are the plate a b ⎠ ⎝ a =1 b =1 g g dimensions, x and y are the Gauss point coordinates and m and n ar the usual harmonic numbers. 26 Present work: In the present study of Cross-Ply laminated composite plate, using finite element method, a square laminated composite plate is taken as focus of study. Let, the side of the square plate is ‘a’ unit. And the Laminate consists of ‘N’ number of laminas. The laminas have either 0 degree or 90 degree orientation with respect to the material coordinates, i.e., the lamina’s are cross-plied. Here, the laminated square plate is considered as the domain. The domain is discretized in to sub-domains/finite elements using 8- noded isoparametric quadratic element (Serendipity Element). As the Plate is symmetric about both the axes in its plane, a quarter of the plate is considered for the study. This quarter plate model is again discretized with 2×2 mesh. So, there are 4 numbers of elements in the quarter plate. The 2×2 mesh in quarter plate model is equivalent to full plate model with 4×4 mesh. After discretizing the domain into sub-domains, the finite element model of the problem is developed using classical variational method as explained above. Finally, the element stiffness matrix is obtained as: ⎡ K 11 ⎢ [K e ] = ⎢ ⎢ ⎣ K 12 K 22 K 13 ⎤ ⎥ K 23 ⎥ K 33 ⎥⎦ Where, ⎛ ∂N ∂N j ∂N ∂N j ⎞ Kij11 = ∫ ⎜ A55 i + A44 i ⎟dxdy A ∂x ∂x ∂y ∂y ⎠ ⎝ ∂N ⎛ ⎞ K ij12 = ∫ ⎜ A55 i N j ⎟dxdy A ∂x ⎝ ⎠ ⎛ ⎞ ∂N K ij13 = ∫ ⎜ A44 i N j ⎟dxdy A ∂y ⎝ ⎠ ⎛ ⎞ ∂N ∂N j ∂N ∂N j Kij22 = ∫ ⎜ D11 i + D33 i + A55 Ni N j ⎟dxdy A ∂x ∂x ∂y ∂y ⎝ ⎠ 27 ⎛ ∂N ∂N j ∂N ∂N j Kij23 = ∫ ⎜ D12 i + D33 i A ∂x ∂y ∂y ∂x ⎝ ⎞ ⎟dxdy ⎠ ⎛ ⎞ ∂N ∂N j ∂N ∂N j K ij33 = ∫ ⎜ D33 i + D22 i + A44 Ni N j ⎟dxdy A ∂x ∂x ∂y ∂y ⎝ ⎠ Because of the symmetry of the stiffness matrix, only the blocks Kij lying on one side of the main diagonal are formed for simplification. Subscripts i and j vary from one to the number of nodes per element. The integral is evaluated numerically using the Gauss quadrature rule, in the limits of -1 to +1. Kije = ∫ 1 ∫ 1 −1 −1 g Bit DB j J drds g K ije = ∑∑ WaWb Bit DB j J a =1 b =1 σ yz = 0 φx = 0 φy = 0 Here, the bending stiffness and shear stiffness values are evaluated separately, to avoid shear locking of problem. The bending terms are evaluated using 3×3 order of integration i.e. at 9 sampling points and the shear terms are evaluated using 2×2 order of integration i.e., at 4 sampling points for each element. Reduced integration scheme is adopted for shear terms. The values of sampling points and weights for each order are given as below: For 3×3 order, First weight is 0.8888888889 and the corresponding sampling point is 0 The other weight is 0.555555555 and the corresponding sampling point is +/- 0.7745966692 For 2×2 order, the weights are 1 and sampling points are +/- 0.5773502692 Since there are 3 degrees of freedom per node of an element viz., transverse displacement w, rotation about x- axis φx and rotation about y- axis φ y and the domain is discretized in to 2×2 meshes i.e., there are 4 elements present. And there are a total of 21 nodes and correspondingly, 21×3=63 degrees of freedom will be present. After getting the 28 Element stiffness matrix for all the elements, they are assembled to obtain the Global stiffness matrix. Therefore, there present 63 simultaneous equations in [K][d]=[Q] form. When the assembly of the element stiffness matrices is over, the boundary conditions are applied at the boundaries of the plate. Initially the plate is simply supported on all the edges of the plate. Since, a quarter plate model is considered for the analysis, only two of the edges are simply supported and the remaining are free. The applied boundary conditions are described as below: w = 0 at x=0 and y=0 φx = 0 at x=0 and y= 0 φ y = 0 at x=0 and y=0 After applying the boundary conditions, the plate is transversely loaded. The loading can be with a uniformly distributed load of magnitude ‘q’ units and a sinusoidal varying load on the plate surface acting individually. So, the stiffness matrix after applying boundary conditions is reduced from 63×63 matrix to 44×44 matrix. So, there are a total of 44 simultaneous equations present that are to be solved in the reduced stiffness matrix. These equations are solved to obtain the displacement vector matrix [d] like, [d] = [K]-1[Q].The inverse of the reduced stiffness matrix and the solution to obtain generalized displacement vector is carried out using the program written in MATLAB. 4.4Post computation of Stresses and Strains: Once the generalized displacements at the nodes are determined by solving the assembled equations of the problem , the transverse displacements and slopes at any (x,y) can be calculated using eq-----------.Strains at any point (x , y, z) in a typical element ‘e’ can be computed from the strain-displacement relations stated above. It is to be noted here that, only displacements are continuous across the element boundaries. Strain continuity across the boundaries is not ensured as we are using a C0 continuous element. That is, along a boundary common to two elements, the strains and hence stresses take different values on the two sides 29 of the interface. However, strains and hence stresses are continuous within an element. The stesses can be calculated using the constitutive relations stated above. Since the displacements in the finite element models are referred to the global coordinates (x,y,z), the stresses are computed in the global coordinates using the relations at the sampling points ; not at the nodes. The stresses can be transformed to principal material coordinates using the stress transformation relations. Similarly, strains can also be transformed to principle material coordinates. 30 Chapter-5 RESULTS AND DISCUSSION 31 5.1Results : Numerical results are obtained for a specific problem whose data is given below: Material: Graphite-Epoxy composite with material properties E1=175 GPa E2=7 GPa G12=G13=0.5 E2=3.5 GPa G23=0.2 E2= 1.4 GPa ν 12=0.25 Shear correction factor K=5/6 Boundary conditions: Simply supported on all edges Loading : a)Sinusiodal varying load and b) Uniformly distributed load acting individually The Non-dimensionalized displacement, and stresses results are tabulated and given below. Side to Thickness ratio a/h 10 Type of solution w σ xx σ yy σ xy σ xz σ yz FEM 100.35 0.7568 0.2865 0.0487 0.7157 0.2869 Closed 102.19 0.7719 0.3072 0.0514 0.7548 0.3107 FEM 69.25 0.7138 0.218 0.0423 0.7869 0.2654 Closed 75.72 0.7983 0.227 0.0453 0.7697 0.2902 FEM 62.78 0.7895 0.1856 0.0409 0.7496 0.2687 Closed 66.97 0.8072 0.1925 0.0426 0.7744 0.2842 66.00 0.8075 0.1912 0.0425 0.7191 0.3791 form 20 form 100 form CLPT Table 1: Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/0) square plate subjected to uniformly distributed loading 32 Side to Thickness ratio a/h 10 20 100 Type of solution w σ xx σ yy σ xy σ xz σ yz FEM 101.80 0.7493 0.4989 0.0438 0.7896 0.3406 Closed form 102.50 0.7577 0.5006 0.0470 0.7986 0.3499 FEM 76.77 0.7959 0.3905 0.0408 0.8298 0.3184 Closed form 76.94 0.8045 0.3968 0.0420 0.8305 0.3228 FEM 65.19 0.8394 0.3527 0.0315 0.8395 0.3092 Closed form 68.33 0.8420 0.3558 0.0396 0.8420 0.3140 CLPT 67.96 0.8236 0.3540 0.0395 0.6404 0.4548 Table 2: Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/90/0) square plate subjected to uniformly distributed loading Side to Thickness ratio a/h 10 20 100 Type of solution w σ xx σ yy σ xy σ xz σ yz FEM 93.5 0.7459 0.5268 0.0394 0.6751 0.3658 Closed form 97.27 0.7649 0.5525 0.0436 0.6901 0.4410 FEM 71.59 0.7952 0.4685 0.0401 0.6927 0.4089 Closed form 75.81 0.8080 0.4844 0.0403 0.7166 0.4188 FEM 65.35 0.7995 0.4227 0.0351 0.7186 0.3982 Closed form 68.74 0.8264 0.4559 0.0386 0.7267 0.4108 CLPT 68.44 0.8272 0.4546 0.0385 ---------- ---------- Table 3: Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/0/90/0) square plate subjected to uniformly distributed loading 33 Side to Thickness ratio a/h 10 20 100 Type of solution w σ yy σ xx σ xy σ xz σ yz FEM 66.92 0.5098 0.2518 0.0250 0.4060 0.0908 Closed form 66.93 0.5134 0.2536 0.0252 0.4089 0.0915 3D-Elasticity --------- 0.590 0.288 0.029 0.357 0.123 FEM 49.21 0.5281 0.1983 0.0222 0.4176 0.0754 Closed form 49.21 0.5318 0.1997 0.0223 0.4205 0.0759 3D-Elasticity ---------- 0.552 0.210 0.234 0.385 0.092 FEM 43.36 0.5346 0.1791 0.0212 0.4215 0.0699 Closed form 43.37 0.5384 0.1804 0.0213 0.4247 0.0703 3D-Elasticity --------- 0.539 0.181 0.0213 0.395 0.083 43.13 0.5387 0.1796 0.0213 0.3951 0.0823 CLPT Table 4: Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/0) square plate subjected to sinusoidal loading Side to Thickness ratio a/h 10 20 100 Type of w σ xx σ yy σ xy σ xz σ yz FEM 66.75 0.4906 0.3512 0.0258 0.398 0.112 Closed form 66.27 0.4989 0.3614 0.0241 0.416 0.129 3D-Elasticity 73.70 0.5590 0.4010 0.0276 0.301 0.196 FEM 49.25 0.5198 0.2906 0.0228 0.430 0.126 Closed form 49.12 0.5273 0.2956 0.0221 0.437 0.109 3D-Elasticity 51.28 0.5430 0.3080 0.0230 0.328 0.156 FEM 43.18 0.5215 0.2598 0.0208 0.448 0.118 Closed form 43.37 0.5382 0.2704 0.0213 0.445 0.101 3D-Elasticity 43.47 0.5390 0.2710 0.0214 0.339 0.139 CLPT 43.13 0.5387 0.2667 0.0213 0.339 0.138 solution Table 5 : Non dimensionalized maximum deflection and stresses of simply supported crossply (0/90/90/0) square plate subjected to sinusoidal loading 34 Side to Thickness ratio a/h 10 20 100 Type of solution w σ xx σ yy σ xy σ xz σ yz FEM 62.12 0.4986 0.4078 0.0219 0.3435 0.1984 Closed form 62.13 0.5021 0.4107 0.0221 0.3459 0.1998 3D-Elasticity 67.71 0.545 0.430 0.0247 0.258 0.223 FEM 47.96 0.5239 0.3722 0.0214 0.3592 0.1827 Closed form 47.96 0.5276 0.3748 0.0215 0.3617 0.1840 3D-Elasticity 49.38 0.539 0.380 0.0222 0.268 0.212 FEM 43.31 0.5345 0.3573 0.0211 0.3655 0.1761 Closed form 43.32 0.532 0.3598 0.0213 0.3683 0.1774 3D-Elasticity 43.38 0.539 0.360 0.0213 0.272 0.205 CLPT 43.13 0.5387 0.3591 0.0213 0.2722 0.2052 Table 6 :Non dimensionalized maximum deflection and stresses of simply supported cross-ply (0/90/0/90/0) square plate subjected to sinusoidal loading 35 Graphs: 120 Deflection w MaximumTransverse Maximum Deflection Vs a/h ratio 100 80 FEM 60 CFS 40 CLPT 20 0 0 50 100 150 Side to thickness ratio a/h Graph 1: Non dimensionalized central transverse deflection versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to uniformly distributed loading Normal Stress Sigma xx VsThickness FEM Sigma xx Normal stress 1 FSDT CLPT 0.5 0 -1 -0.5 0 -0.5 0.5 1 -1 Thickness z/h Graph 2: Non dimensionalized normal stress sigma xx versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to uniformly distributed loading 36 Normal Stress Sigma yy Normal Stress Sigma yy Vs z/h 0.8 0.6 FEM 0.4 FSDT CLPT 0.2 0 -1 -0.5 -0.2 0 0.5 1 -0.4 -0.6 -0.8 Thickness z/h Graph 3: Non dimensionalized normal stress sigma yy versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to uniformly distributed Sigma xz loading -1 Sigma xz Vs Thickness z/h 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.5 0 0.5 1 Thickness FEM FSDT CLPT Graph 4: Non dimensionalized transverse shear stress sigma xz versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to uniformly distributed loading 37 Sigma yz Vs Thickness z/h 0.6 FEM 0.5 FSDT Sigma yz 0.4 CLPT 0.3 0.2 0.1 0 -1 -0.5 0 0.5 Thickness z/h 1 Graph 5: Non dimensionalized transverse shear stress sigma yz versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to uniformly distributed Loading Maximum Deflection Vs a/h ratio 100 Deflection w Maximum Transverse 120 80 60 FEM CFS 40 CLPT 20 0 0 50 100 150 Side to thickness ratio a/h Graph 6: Non dimensionalized central transverse deflection versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to sinusoidal varying load 38 Normal Stress Sigma xx VsThickness 1 Normal stress Sigma xx FEM FSDT 0.5 CLPT 0 -1 -0.5 0 -0.5 0.5 1 -1 Thickness z/h Graph 7: Non dimensionalized normal stress sigma xx versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to sinusoidal varying load Normal Stress Sigma yy Normal Stress Sigma yy Vs z/h 0.8 0.6 FEM 0.4 FSDT CLPT 0.2 0 -1 -0.5 -0.2 0 0.5 1 -0.4 -0.6 -0.8 Thickness z/h Graph 8: Non dimensionalized normal stress sigma yy versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to sinusoidal varying load 39 Sigma xz Vs z/h 0.45 0.44 Sigma xz 0.43 FEM 0.42 FSDT 0.41 CLPT 0.4 0.39 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Thickness z/h Graph 9: Non dimensionalized transverse shear stress sigma xz versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to sinusoidal varying load Sigma yz Vs z/h 1.2 FEM 1 FSDT CLPT Sigma yz 0.8 0.6 0.4 0.2 0 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Thickness z/h Graph 10: Non dimensionalized transverse shear stress sigma yz versus side to thickness ratio for simply supported cross ply (0/90/90/0) square laminate subjected to sinusoidal varying load 40 5.2Discussion: The following non-dimensional quantities are used to get the non dimensionalized stresses and deflections from the actual ones. w = w0 (0, 0) E2 h 2 a 4 q0 σ xy = σ xy (a / 2, a / 2, −h / 2) h2 σ xx = σ xx (0, 0, h / 2) 2 a q0 σ xz = σ xz (a / 2, 0, k = 1, 4) σ yy h aq0 h2 a 2 q0 h2 = σ yy (0, 0, h / 4) 2 a q0 σ yz = σ yz (0, a / 2, k = 1, 4) h aq0 The origin of the coordinate system is taken at the centre f the plate, -a/2<(x,y)<a/2 and –h/2<z<h/2. As mentioned earlier, the stresses are computed at the reduced Gauss points. The Gauss coordinates are mentioned like (A, A). The finite element solutions of the present study is compared with the closed form solutions obtained using FSDT and that of Classical Laminated Plate Theory (CLPT) for uniformly distributed loading case when the edges of the plate are simply supported. Similarly, the results obtained from present finite element model are compared with the closed form solutions of FSDT and CLPT as well as 3-D elasticity solutions for sinusoidal variation of loading case when all the edges of the plate are simply supported. Comparison is made, between non dimensional quantities of transverse displacements, for different values of side to thickness ratios. Comparison is also made with respect to the lamina orientation in the laminate for different side to thickness values between the transverse displacements. The results of non dimensional quantities of normal stresses, in-plane shear stresses and transverse shear stresses are compared, at various thickness co -ordinates for different values of side to thickness ratios. The results are obtained for different lamina orientation schemes (0 or 90 degrees i.e., cross – ply orientation with symmetry) in the laminate as well as for varying number of layers i.e., for 3 layer (0/90/0), 4 layer (0/90/90/0) and 5 layer (0/90/0/90/0) orientations. 41 5.3Observations: It is observed from the results presented in the tables that, the FEM results obtained using 8noded isoparametric Serendipity element are giving near approximations for a/h ratios <20, i.e., for thick plates. And as the ratio increases, the values are not that much satisfactory. That is, the present finite element model using First order shear deformation theory can well predict the results for a thick plate. As the plate a/h ratio is increasing i.e. when the plate is becoming thin, the results are not that much in good comparison as that for the thick plates. Reduced integration alleviated this phenomenon called shear locking to some extent. From the results it can be observed that the Finite Element Solutions are in well agreement with the results of closed form solutions of FSDT. The displacements converge faster than stresses. This is expected because; the rate of convergence of gradients of the solution is one order less than the rate of convergence of the solution. However, the results based on the finite element solutions should not be expected to agree well with the 3-D elasticity solutions. The finite element solutions should only converge to the closed form solutions of the FSDT. It is also observed that, the normal stresses are varying non-linearly across the thickness of the laminate. However, they are varying linearly for an individual lamina. The stresses are discontinuous across the thickness of the laminate. That means, there exist different values of stresses at the interfaces of the laminate. The stress at the bottom surface of a lamina zk, is different from that at the top surface of the adjacent lamina zk-1.This is obvious from the First order shear deformation theory. The element is a C0 continuous element. The generalized displacements only are continuous across the thickness of the laminate. But, the strains and thus the stresses are not continuous at the boundaries. The stress values are maximum at the top and bottom of the laminate with + ve and – ve signs respectively. The transverse shear stresses are constant throughout the thickness. It is because of the use of a constant in calculating the shear stresses, the shear correction factor. Its value 42 varies with lamina orientation and stacking sequence. In the present study it’s value is taken as 5/6. Since the stresses in the finite element analysis are computed at locations different from the analytical solutions, they are expected to be different. 43 Chapter -6 CONCLUSIONS 44 Conclusions Finite element analysis of cross ply laminated composite square plate is carried out, using a 8 – noded isoparametric quadratic element to predict the transverse displacements , normal stresses and transverse shear stresses, when it is subjected to transverse loading under simply supported boundary conditions. The present model is developed based on the First order Shear Deformation Theory (FSDT). This theory uses a shear correction factor to approximate the transverse shear stresses. A computer program is written in MATLAB to get various results. The accuracy of results obtained using the present formulation is demonstrated by comparing the results with three-dimensional elasticity solution ,closed form solutions of FSDT and Classical Laminated Plate Theory. The present analysis gives accurate values for displacements and stresses compared to Classical Laminated Plate Theory. It is observed that the results are in close agreement with closed form solutions of FSDT and 3-D elasticity solutions. It is found that, the transverse shear stresses vary constantly through the thickness. This is attributed to the use of shear correction factor in the theory. But, the actual variation of the transverse shear stresses is parabolic according to 3D elasticity using equilibrium relations in predicting the same. More over, the results of stresses are calculated at Gauss points and they are expected to differ from the analytical solutions. Adoption of reduced integration scheme alleviated the shear locking effects. The present model accurately predicts the transverse displacements and various stresses for thin as well as thick laminated composite plates. As the present model is developed using a non-conforming element, the results can be further improved using a conforming element with improved mesh size thereby increased no of elements. Infact , the FEM results approach the true solutions, with the increase in the number of elements. . 45 Chapter-7 SCOPE FOR FUTURE WORK 46 Scope for future work: Results can be expected with excellent agreement with the analytical/experimental solutions by using a conforming element with increased mesh size. Analysis can be done for different loading conditions with various boundary conditions. There is a need to develop a mechanics based theory to find out the optimal stacking sequence, which will significantly help the designers. Study can be made on real life problems pertaining to stress concentration, non-linearity and complicated geometries 47 References 1) Reddy J.N., Chao W.C., “A Comparision of Closed Form and Finite Element Solutions of Anisotropic Rectangular Plates” Nuclear Engineering and Design Vol. 64, (1981) pp153-167 2) Khedier A.A., Reddy J.N., “An exact solution for the bending of thin and thick cross-ply laminated beams” Composite Structures 37 (1997) pp 195-203 3)Hughes T.J.R., Tezduyar T.E., “Finite Elements Based Upon Mindlin Plate Thoery With Particular Reference to the Four-Node Bilinear Isoparmetric Element”. Journal of Applied Mechanics, Sep 1981, Vol. 48, pp 587-597. 4)Reddy J.N. “state-space concept in conjunction with the Jordan canonical form for bending of cross-ply laminated composite beams” Journal of Numerical methods , Oct 1997 vol pp 168-179 5)Pandya B.N. ,.Kant T., ”Finite Element Analysis of Laminated Composite Plates using a Higher-Order Displacement Model”. Composites Science and Technology Vol 32 (1988) pp 137-155. 6)Somashekar B.R., Pratap G.,Ramesh Babu .C, “A Field – Consitent , Four-Node,Laminated ,Anisotropic Plate/Shell Element” , Computers &Structures Vol. 25, No. 3. pp. 345-353 ,1987. 7)Xiao-Ping Shu, Kostas P. Soldatos “Cylindrical bending of angle-ply laminates subjected to different sets of edge boundary conditions”. International Journal of Solids and Structures Vol 37 (2000) pp 4289-4307. 8) Pagano, N.J., “Exact solutions for Rectangular Bidirectional Composites and sandwich plates,” Journal of composite materials, Vol 4 , 20-34 (1970). 48 9) Reddy J. N. , Khdeir A. A. and Librescu, L., “ Levy type solutions for symmetrically laminated rectangular plates using first order shear deformation theory” , Journal of Applied mechanics, Vol54, 740-752(1987) 10)Khedeir, A.A., “An exact approach to the elastic state of stress of shear deformable antisymmetric angle ply laminated plates,” composite structures, Vol11,245-258 (1989) 11)Khdeir, A.A. “comparison between shear deformable and Kirchoff theory for bending, buckling, and vibration of antisymmetric angle ply laminated plates, : , Composite structures , Vol 13,359-472 12) Srinivas, S. and Rao A.K., “bending, vibration, and buckling of simply supported thick orthotropic rectangular plates and laminates “, International Journal of Solids and Structures, Vol 6, 1463-1481(1970) 13) Hrabok, M. M. and Hrudey, T.M. ‘A review and catalog of plate bending finite elements”, Computers and structures vol 19, (3), 479-495 (1984) 14)Fraeijis de veubeke, B., “ A conforming Finite element for plate Bending, “ International Journal of solids and structures, 4(1), 95-108 (1968) 15)Bell K., “A refined triangular plate bending finite element “ International journal for numerical methods for engineering vol 1,101-122, (1969) 16) Irons, B.M. A conforming quartic triangular element for plate bending international journal for numerical methods in engineering vol 1, 29-45 (1969) 17)Stricklin , J. A Haisler W, “A rapidly converging triangular plate element “ AIAA Journal , 7(1) , 180-181 (1969) 49 18) Batoz J.L Bathe K.J. “A study of Three node triangular plate bending elements” international journal for numerical methods in engineering Vol 15 (12) 1771-1812 (1980) 19) Reddy J.N.,”Mechanics of Laminated Composite Plates: Theory and Analysis”, CRC Press 1996 20)Jones R.M.,”Mechanics of Composite Materials” McGraw-Hill Book Company 21)Krishnamoorthy C.S.,”Finite Element Analysis: Theory and Programming” , Tata McGraw-Hill Publishing Company Limited 1994 22)Reddy J.N., “ An Introduction to the Finite Element Method” ,Tata McGraw-Hill Publishing Company Limited ,III edition 2006 23)Rudra Pratap, ”Getting Started with MATLAB”, Oxford University Press 2004 50 APPENDIX Composite: A combination of two or more materials on a macroscopic scale which are physically distinct. Classification: 1) Fiber reinforced composites which consists of fibers in a matrix 2) Laminated composites which consists of layers of various materials 3) Particulate composites which consists of particles in a matrix Advantages of composites: They usually exhibit the best qualities of their constituents and often some qualities that neither constituent possesses. The properties that can be improved by forming a composite material include: Strength Fatigue life Temperature-dependent behavior stiffness Wear resistance Corrosion resistance Weight Mechanical behavior of composite materials Composite materials unlike isotropic materials are often both inhomogeneous and nonisotropic Anisotropic body has material properties that are different in all directions at a point in the body. There are no planes of material property symmetry. Again, the properties are a function of orientation at a point in the body. Orthotropic body has material properties that are different in three mutually perpendicular planes of material symmetry. Thus, the properties are a function of orientation at a point in the body. Micromechanics are the study of composite material behavior where in the interaction of the constituent materials is examined on a microscopic scale. 51 Macro-mechanics is the study of composite material behavior wherein the material is presumed homogeneous and the effects of the constituent materials are detected only as averaged apparent properties of the composite. Basic terminology of laminated fiber-reinforced composite materials: Lamina: It is a flat (sometimes curved as in a shell) arrangement of unidirectional l fibers or woven fibers in a matrix. Laminate: It is a stack of lamina with various orientations of principal material direction in the lamina as shown .fig. The major purpose of lamination is to tailor the directional dependence of strength and stiffness of a material to match the loading environment of the structural element. Laminates are uniquely suited to this objective since the principal material direction of each layer can be oriented according to need. Geometry of the N- layered laminate is as shown: 52 The transformed reduced stiffness matrix terms can be expressed as Q11 = Q11 cos 4 θ + 2( Q12 +2 Q33 ) sin 2 θ cos 2 θ + Q22 sin 4 θ Q12 = ( Q11 + Q22 -4 Q33 ) sin 2 θ cos 2 θ + Q12 ( sin 4 θ + cos 4 θ ) Q22 = Q11 sin 4 θ + 2( Q12 +2 Q33 ) sin 2 θ cos 2 θ + Q22 cos 4 θ Q13 = ( Q11 - Q12 -2 Q33 ) sin θ cos3 θ + ( Q12 - Q22 +2 Q33 ) sin 3 θ cos θ Q23 = ( Q11 - Q12 -2 Q33 ) sin 3 θ cos θ + ( Q12 - Q22 +2 Q33 ) sin θ cos3 θ Q33 = ( Q11 + Q22 -2 Q12 -2 Q33 ) sin 2 θ cos 2 θ + Q33 ( sin 4 θ + cos 4 θ ) Jacobian Matrix: The derivatives of the shape functions with regard to x and y can be obtained by transformation from natural coordinate’s r and s, using Jacobian matrix. The Jacobian matrix is given by ⎡ ∂x ⎢ [ J ] = ⎢ ∂r ⎢ ∂x ⎣⎢ ∂r ∂y ⎤ ∂s ⎥ ⎥ ∂y ⎥ ∂s ⎦⎥ The transformation from (r, s) coordinates to (x, y) coordinate system is done by ⎧∂⎫ ⎧∂⎫ ⎪⎪ ∂x ⎪⎪ ⎪ ∂r ⎪⎪ −1 ⎪ ⎨ ∂ ⎬ = [J ] ⎨ ⎬ ⎪ ⎪ ⎪∂ ⎪ ⎩⎪ ∂s ⎭⎪ ⎩⎪ ∂y ⎭⎪ For the present problem with 8-noded Serendipity element, the transformation is as follows: ⎡ ∂N1 ⎢ ∂x ⎢ ⎢ ∂N1 ⎣⎢ ∂y ∂N 2 ∂x ∂N 2 ∂y ∂N 3 ∂N ⎤ ⎡ ∂N1 .......................... 8 ⎥ ⎢ ∂x ∂x −1 ⎥ = [ J ] ⎢ ∂r ∂N3 ∂N ⎢ ∂N1 .......................... 8 ⎥ ⎥ ⎢⎣ ∂s ∂y ∂y ⎦ 53 ∂N 2 ∂r ∂N 2 ∂s ∂N3 ∂N ⎤ .......................... 8 ⎥ ∂r ∂r ⎥ ∂N3 ∂N8 ⎥ .......................... ∂s ∂s ⎥⎦

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement