“VIBRATION AND STABILITY OF LAMINATED COMPOSITE DOUBLY CURVED SHELLS BY HIGHER ORDER SHEAR DEFORMATION THEORY” A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Technology in Structural Engineering By NEMI SHARAN Roll No.-209CE2048 DEPARTMENT OF CIVIL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA ODISHA-769008 MAY 2011 “VIBRATION AND STABILITY OF LAMINATED COMPOSITE DOUBLY CURVED SHELLS BY HIGHER ORDER SHEAR DEFORMATION THEORY” A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Technology in Structural Engineering By NEMI SHARAN Roll no-209CE2048 Under the guidance of Dr. A.V. Asha DEPARTMENT OF CIVIL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA ODISHA-769008 MAY 2011 National Institute of Technology Rourkela CERTIFICATE This is to certify that the thesis entitled “VIBRATION AND STABILITY OF LAMINATED COMPOSITE DOUBLY CURVED SHELLS BY A HIGHER ORDER SHEAR DEFORMATION THEORY” submitted by Mr. NEMI SHARAN in partial fulfillment of the requirements for the award of Master of Technology Degree in Civil Engineering with specialization in Structural Engineering at the 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. Date: Dr. A.V.Asha Department of Civil Engineering National Institute of Technology Rourkela – 769008 ACKNOWLEDGEMENT The satisfaction and euphoria on the successful completion of any task would be incomplete without the mention of the people who made it possible whose constant guidance and encouragement crowned my effort with success. I am grateful to the Dept. of Civil Engineering, NIT ROURKELA, for giving me the opportunity to execute this project, which is an integral part of the curriculum in M.Tech programme at the National Institute of Technology, Rourkela. I would also like to take this opportunity to express heartfelt gratitude for my project guide Dr. A.V Asha, who provided me with valuable inputs at the critical stages of this project execution. My special thanks are due to Prof. M. Panda, Head of the Civil Engineering Department, for all the facilities provided to successfully complete this work I am also very thankful to all the faculty members of the department, especially Structural Engineering specialization for their constant encouragement, invaluable advice, encouragement, inspiration and blessings during the project. Submitting this thesis would have been a Herculean job, without the constant help, encouragement, support and suggestions from my friends, especially Alok, Pragyan, Javed, Trishanu, for their timely help. I will relish your memories for years to come. Last but not the least I would like to thank my parents, who taught me the value of hard work by their own example. I would like to share this bite of happiness with my mother and father. They rendered me enormous support during the whole tenure of my stay at NIT, Rourkela. Date: Nemi Sharan Roll No:-209CE2048 M.Tech (Structural Engineering) Department of Civil Engineering NIT, Rourkela,Odisha CONTENTS Page No ABSTRACT i LIST OF TABLES ii LIST OF FIGURE iv NOTATIONS v CHAPTER 1 : 1.1 1.2 1.3 CHAPTER 2 : INTRODUCTION 1-4 General Application of shells Vibration of composite Shells 1 3 4 LITERATURE REVIEW 5-10 2.1 Introduction 5 2.2 Objective and scope of present investigation 10 CHAPTER 3: THEORY AND FORMULATION 11-36 3.1 Introduction 12 3.2 Basic assumption 13 3.3 Geometry of Shell 13 3.4 Strain Displacement Relation 16 3.4.1 First Order Shear Deformation Theory 16 3.4.2 Higher Order Shear Deformation Theory 18 3.5 Stress-Strain Relations 20 3.6 Governing Equation 21 3.6.1 Strain Energy 21 3.6.2 Kinetic Energy 23 3.6.3 Hamilton’s Principle 24 3.7 Stress Resultants and Stress Couples 26 3.8 Boundary Conditions 33 CHAPTER 4: NUMERICAL RESULTS AND DISCUSSIONS 37-48 4.1 Introduction 38 4.2 The validation of the formulation and comparison of results 38 4.3 Numerical result 41 4.3.1 Vibration analysis 42 4.3.2 Stability analysis 46 CHAPTER 5: 5.1 CONCLUSION 5.2 Scope of future work 49-51 52 REFERENCES 53-57 APPENDIX 58-61 ABSTRACT The present study deals with a higher order shear deformation theory of laminated shells as suggested by Reddy and Liu. The theory is based on a displacement field in which the displacements of the middle surface are expanded as cubic functions of the thickness coordinate, and the transverse displacement is assumed to be constant through the thickness. This displacement field leads to the parabolic distribution of the transverse shear stresses (and zero transverse normal strain) and therefore no shear correction factors are used. The theory is also based on the assumption that the thickness to radius ratio of shell is small compared to unity and hence negligible. The governing equations are derived in orthogonal curvilinear coordinates. These equations are then reduced to those of doubly curved shell. All the quantities are suitably nondimensionalised. The Navier solution has been used which gives rise to a generalized eigenvalue problem in matrix formulation. The natural frequencies for vibration and buckling loads of laminated orthotropic doubly curved shells and panels with simply supported ends are obtained. The eigenvalues, and hence the frequency parameters are calculated by using a standard computer program. To check the derivation and computer program, the frequencies in HZ for different layer are compares with earlier results. The lowest value of frequency parameter and buckling load are computed for the laminated composite doubly curved shell. The effects of various parameters such as number of layers, aspect ratio, modular ratio, etc on the above are studied. Frequency also increases as number of layers of the shell increases for symmetric cross-ply layout. But when there is unsymmetrical cross-ply layout, then frequency decreases. With the increasing of modular ratio, non-dimensional frequency is also increasing. i LIST OF TABLES Table Number 4.1 Description Page Comparison of lowest non dimensional Frequency 39 for a simply supported laminated composite spherical shell (a/h=100) 4.2 Comparison of lowest non dimensional Frequency 40 for a simply supported laminated composite spherical shell (a/h=10) 4.3 Comparison of non-dimensional buckling load for a 41 simply supported laminated composite doubly curved shell (a/h=10) 4.4 4.5 Variations in non-dimensional frequency parameter for laminated composite paraboloid shell by higher order shear deformation theory Variations in non-dimensional frequency parameter 42 43 for laminated composite elliptical shell by higher order shear deformation theory. 4.6 Variations in non-dimensional frequency parameter 43 for laminated Composite doubly curved shell by higher order shear deformation theory with variation of modular ratio 4.7 Variations in non-dimensional frequency parameter 44 with different layers for laminated composite doubly curved shells a/h=100 4.8 Variations in non-dimensional buckling load 46 parameter for laminated composite paraboloidal shell by higher order shear deformation theory 4.9 Variations in non-dimensional buckling load parameter for laminated composite elliptical shell by higher order shear deformation theory ii 47 4.10 Variations in non-dimensional buckling load 47 parameter for laminated composite doubly curved shell by higher order shear deformation theory with variation of modular ratio of [0/90/90/0] and a/h=100 4.11 Variations in non-dimensional buckling load parameter with different layers for laminated composite doubly curved shells a/h=100 iii 48 LIST OF FIGURES FIGURE NUMBER DESCRIPTION PAGE 1.2.1 Elliptical paraboloid shell 04 1.2.2 Hyperbolic paraboloid shell 04 3.1 Geometry of a laminated shell 10 3.2 Stress and moment resultants 22 4.1 Variation in non-dimensional frequency of Hyper paraboloid shells with R2/a 4.2 Variation of non-dimensional frequency of cross-ply elliptical shell with R1/a 4.3 45 45 Variation of non-dimensional frequency of cross-ply spherical shell, hyperbolic and elliptical paraboloid shell with modular ratio iv 46 LIST OF NOTATIONS The principal symbols used in this thesis are presented for easy reference. A symbol is used for different meaning depending on the context and defined in the text as they occur. English Notation Description A1 , A2 Lames Parameter or Surface Metrics Aij , Bij , Dij Laminate Stiffnesses Eij , Fij , H ij ,C C The matrix notation described in appendix The distance between points , , z and d , d , z dz dS dV Volume of the shell element dt Time derivative E1 , E2 Longitudinal and transverse elastic moduli respectively G12 , G13 , G23 In plane and Transverse shear moduli h Total thickness of the shell hk Distance from the reference surface to the layer interface k211, k2 22 The shear correction factors. k1 , k2 , k6 Quantities whose expression are given in equation [11.b] 0 0 0 2 2 2 1 1 k1 , k 2 , k 6 , k1 k2 , k6 , k4 , k5 Quantities whose expressions are given in equation [11.b] v L1 , L2 , L3 lames coefficient n1 and n2 axial load applied in α and β direction a ,b size of the spherical shell N i , M i , Pi , Q1 Q2 , K 1 , K 2 Stress resultant and stress couples M,M Matrix notation described in appendix m Axial half wave number n Circumferential wave number Q ij (k) Reduced stiffness matrix of the constituent layer R Radius of the reference surface of spherical shell R1 , R2 Principal radii of curvature T Kinetic energy U, V, W, 1, 2 Amplitude of displacement Us Strain energy U , V , W , 1 , 2 Non dimensionalised amplitude of displacement u, v, w Displacement component at the reference surface u, v, w Non dimensionalised displacement component at any point in the shell Ii , I i , I i ' Inertia terms defined by equation {X} Column matrix of amplitude of vibration or eigenvector. vi {X} Non dimensionalised form of column matrix Greek Notation Description ,,z Shell coordinates I Strain components 1 , 2 , 4 0 0 0 50 , 60 Quantities whose expressions are given in equation[11.b] , Non dimensionalised axial and circumferential wave parameter m n Material density i Stress components 1 , 2 Rotation at z = 0 of normals to the mid-surface with respect to and axes Natural circular frequency parameter vii CHAPTER 1 INTRODUCTION 1 CHAPTER 1 INTRODUCTION 1.1 GENERAL: An increasing number of structural designs, especially in the aerospace, automobile, and petrochemical industries are extensively utilizing fiber composite laminated plates and shells as structural elements. The laminated orthotropic shell belongs to the composite shell category. One of the important factors in the analysis of the layered shells is its individual layer properties, which may be anisotropic, orthotropic or isotropic. A shell is a curved, thin walled structure. Two important classes of shells are plates (shells which are flat when un-deformed) and membranes (shells whose walls offer no resistance to bending). Shells may be made of a single homogeneous or anisotropic material or may be made of layers of different materials. The primary function of a shell may be to transfer loads from one of its edges to another, to support a surface load, to provide a covering, to contain a fluid, to please the eye or a combination of these. A thin shell is defined as a shell with a thickness which is small compared to its other dimensions and in which deformations are not large compared to thickness. A primary difference between a shell structure and a plate structure is that, in the unstressed state, the shell structure has curvature as opposed to plates. Membrane action in a shell is primarily caused by in-plane forces (plane stress), though there may be secondary forces resulting from flexural deformations. Where a flat plate acts similar to a beam with bending and shear stresses, shells are analogous to a cable which resists loads through tensile stresses. The ideal thin shell must be capable of developing both tension and compression. At the present time, many of the existing methods of analysis for multilayered laminated orthotropic plates and shells are direct extensions of those developed earlier for homogeneous isotropic and orthotropic plates and shells. In the classical lamination theory (CLT), the well known Kirchhoff-love hypothesis is assumed to be verified. The range of applicability of the C.L.T solution has been well established for laminated flat plates by Pagano (1969, 1970). These analyses have indicated that the hypothesis of no deformable 2 normal, while acceptable for isotropic plates and shells, is often quite unacceptable for multilayered anisotropic plates and shells with very large ratio of Young’s modulus to shear modulus, even if they are relatively thin. A Mindlin-type first-order transverse shear theory deformation theory (S.D.T) has been first developed by Dong and Tso (1972) for multilayered anisotropic shells. The present study deals with a higher-order but simple shear deformation theory of laminated shells as proposed by Reddy and Liu [24] for plates and shallow shells. The theory is based on a displacement field in which the displacements of the middle surface are expanded as cubic functions of the thickness coordinate, and the transverse displacement is assumed to be constant through the thickness. The latter assumption is equivalent to neglecting the stretching of the normal to the middle surface of the shell. This displacement field leads to the parabolic distribution of the transverse shear stresses (and zero transverse normal strain), and therefore no shear correction factors are used. The governing equations are derived in curvilinear orthogonal coordinates. These equations are then reduced to doubly curved shell. The free vibration and stability analysis of the doubly curved shell is carried out by computing lowest value of frequencies and buckling loads for various shell parameters. 1.2 APPLICATION OF SHELLS Thin shell structures are light weight constructions using shell elements. These elements are typically curved and are assembled to large structures. Typical applications are fuselages of aero planes, boat hulls and roof structures in some buildings. Shell structures are mainly used in industrial applications such as automobile, civil, aerospace and petrochemical engineering. Various types of shells are used in civil field such as conoid, hyperbolic paraboloid and elliptical paraboloid shell. All are used for roofing to cover large column-free areas. Laminated composites are such type of material which has high strength to weight and strength to stiffness ratios. The mechanical properties of the laminated composites depend on the degree of orthotropy of the layers, ratio of the transverse shear modulus to the in-plane shear modulus and stacking sequence of laminates. Many of the classical theories developed for thin elastic shells are based on the Love-Kirchhoff assumptions in which the 3 normal to the mid-plane before deformation is considered to be normal and straight after the deformation. Figure:1.2.1 Elliptical Paraboloidal Shell Figure:1.2.2 Hyperbolic Paraboloidal Shell 1.3 VIBRATION OF COMPOSITE SHELLS Shell vibration was first studied by Germaine in 1821.Aron (1874) gave a set of five equations which reduced to plate equations when curvatures were zero. Logical extensions of the beam and plate equations for both transverse and in-plane motion were introduced by Love (1863-1940) in 1888. Love’s equations brought the basic development of the theory of vibration of continuous structures, which have a thickness that is much less than any length or surface dimensions, to a satisfying end. 4 CHAPTER 2 LITERATURE REVIEW 5 CHAPTER 2 LITERATURE REVIEW 2.1 INTRODUCTION The increasing use of high performance ﬁbre reinforced laminated shell structures in aerospace and other applications are well documented. A reliable prediction of the response of these laminated shells or doubly-curved panels must account for transverse shear deformation. Since the matrix material is of relatively low shearing stiffness as compared to the ﬁbers, polymeric composite shell type structures are highly prone to transverse shear related fatigue failures. Additionally, a solution to the problem of deformation of laminated shells of ﬁnite dimensions must satisfy the prescribed boundary conditions, which introduce additional complexities into the analysis. Most of the investigations use (a) The classical lamination theory (CLT) or the ﬁrst-order shear deformation theory (FSDT). (A detailed review of CLT-based analyses is available in, e.g., Bert and Reddy [6], and Chaudhuri et al. [8], while that relating to their FSDT counterparts is available in, e.g., Abu Arja and Chaudhuri [1], Chaudhuri and Abu Arja [9], and Librescu et al. [17].) (b) Higher-order shear deformation theory(HSDT) Leissa and Kadi (1971) worked on the effect of curvature upon the vibration frequencies of shallow shells. For this purpose, the shell was chosen to have a rectangular boundary supported by shear diaphragms and yielded exact, closed form solutions for convenient comparison in the linear, small deflection regime and, at the same time, represented a situation which could readily occur in practical application. He extended his analysis into non linear, large deflection regime by assuming mode shapes satisfying the non-linear field equations of motion and compatibility approximately by means of the Galerkin procedure. Bhimaraddi (1983) worked on the free, undamped vibration of an isotropic circular cylindrical shell with higher order displacement model, giving rise to a more realistic 6 parabolic variation of transverse shear strains. In this work in-plane inertia, rotatory inertia, and shear deformation effects on the dynamic response of cylindrical shells was studied. Bhimaraddi (1991) studied the free vibration analysis of homogeneous and laminated doubly curved shells on rectangular planform and made of an orthotropic material using the three- dimensional elasticity equations. A solution was obtained utilizing the asumption that the ratio of the shell thickness to its middle surface radius is negligible as compared to unity. However, it was shown that by dividing the shell thickness into layers of smaller thickness and matching the interface displacement and stress continuity conditions, very accurate results could be obtained even for very thick shells. Touratier (1991) generalized the geometrically linear shear deformation theories for small elastic strains for multilayered axisymmetric shells of general shape without any assumption other than neglecting the transverse normal strain. He used a certain sine shear function and no shear correction factor was used. He presented a new theory and compared with the classical theories for a simply-supported thick laminated cylindrical shell under an internal pressure. Leissa and Chang (1996) derived a rigorous theory which governs the linearly elastic deformation of shells made of laminated composite materials including shear deformation and rotary inertia effects. The equations derived are applicable for static and dynamic problems for shells of arbitrary curvature, but constant thickness. Qatu (1999) derived accurate equations of deformation for laminated composite deep, thick shells. These equations include shells with a pre- twist and accurate force and moment resultants which are considerably different from plates. He also derived consistent set of equations of motion, energy functionals and boundary conditions. Tabiei and Tanov(1999) worked on the finite element formulation of a higher order shear deformation shell element for non linear dynamic analysis with explicit time integration scheme. They combined co rotational approach with velocity strain equations of a general third order theory in the formulation of a four noded quadrilateral element with selectively reduced integration. 7 Jayasuriya et al (2002) worked on development of isoparametric, finite element for doubly curved anisotropic laminated shells of revolution. Variable-order Legendre polynomials were used in the derivation of interpolation functions. They also developed a user -friendly finiteelement code based on the higher-order theory and isoperimetric hierarchical finite-element formulation, and a numerical verification and application of the above model for the study of static and dynamic analysis with mechanical and hydrothermal loadings. Messina (2003) described the dynamics of freely vibrating multilayered doubly curved shells. The relevant governing differential equations, associated boundary conditions and constitutive equations were derived from one of Reissners mixed variational theorems. Both the governing differential equations and the related boundary conditions were presented in terms of resultant stresses and displacements. In spite of the multi-layer nature of the shell, the theory was developed as if the shell were virtually made of a single layer. Sudhakar et al (2003) formulated a degenerate shell element using higher order shear deformation theory taking the piezoelectric effect in account. An eight-noded element was used to derive global coupled electro elastic behavior of the overall structure. Amabili (2005) investigated vibrations of doubly curved shallow shells with rectangular base, simply supported at the four edges and subjected to harmonic excitation normal to the surface in the spectral neighborhood of the fundamental mode . In-plane inertia and geometric imperfections were taken into account. The solution was obtained by Lagrangian approach. Oktem and Chaudhuri (2006) worked on the Fourier series approach to solve a system of ﬁve highly coupled linear partial differential equations, generated by the HSDT-based laminated shell analysis, with the C4-type simply supported boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to the SS3-type constraint. The numerical accuracy of the solution was ascertained by studying the convergence characteristics of the deﬂection and moment of a cross-ply spherical panel, and also by comparison with the available FSDT (ﬁrst-order shear deformation theory) based analytical solution. 8 Adam (2006) studied the nonlinear ﬂexural vibrations of shallow shells composed of three thick layers with different shear ﬂexibility, which were symmetrically arranged with respect to the middle surface. He considered shell structures of polygonal platform were hard hinged simply supported (i.e. all in-plane rotations and the bending moment vanish) with the edges fully restrained against displacements in any direction. He formulated kinematic field equations layer wise by first order shear deformation theory. Numerical results of rectangular shallow shells in nonlinear steady-state vibration were presented for various ratios of shell rise to thickness, and non-dimensional load amplitude. Kulikov and Carrera (2008) worked on higher shell models by new concept of I surfaces inside the shell body. They introduced N (N≥3) I surfaces. The general 3N-parameter shell model was proposed in the framework of the Lagrangian description. Amabili and Reddy (2009) worked on the use of higher order shear deformation non linear theory for shells of generic shape, taking geometric imperfections into account. They found that results were obtained by keeping non-linear terms of the Von Karman type for amplitudes of about two times the shell thickness. Tripathy and Suryanarayan (2009) presented vibration and buckling of thin-walled tubular beam shells typical of automotive structures, which are fabricated by joining sheet metal stampings along the two longitudinal edges with periodic spot welds, adhesive bonding, or combination of spot welds and bonding, known as weld bonding. Solutions were obtained for such beam shells of rectangular cross section with two opposite ends simply supported. The beam shell was modeled as an assembly of the constituent walls and Levy-type formulation was used to obtain a series solution for the transverse displacement of each of the walls. 9 2.2 OBJECTIVE AND SCOPE OF PRESENT INVESTIGATION In this project, analytical solution of frequency characteristics and buckling loads of laminated composite doubly curved shells will be presented. Higher order shear deformation theory as proposed by Reddy and Liu will be used for the formulation. The governing equations have been developed. These equations are then reduced to the equations of motion for doubly curved panels and the Navier solution has been obtained for cross-ply laminated composite doubly curved panels. The resulting equations are suitably non-dimensionalized. The Eigen value problem is then solved to obtain the free vibration frequencies and buckling loads. 10 CHAPTER 3 THEORY AND FORMULATION 11 CHAPTER-3 THEORY AND FORMULATION 3.1 INTRODUCTION In the present study, laminated composite doubly curved shells are considered. A number of theories exist for layered composite shells. Many of these theories were developed originally for thin shells, and are based on the Kirchhoff-Love kinematic hypothesis that straight lines normal to the undeformed midsurface remain straight and normal to the middle surface after deformation and undergo no thickness stretching. The higher order theory is based on a displacement field as proposed by Reddy and Liu in which the displacements of the middle surface are expanded as cubic functions of the thickness coordinate and the transverse displacement is assumed to be constant through the thickness. The additional dependent unknowns introduced with the higher order powers of the thickness coordinate are evaluated in terms of the derivatives of the transverse displacement and the rotation of the normal at the middle surface. This displacement field leads to the parabolic distribution of the transverse shear strains and hence shear stresses and zero transverse normal strain, and therefore no shear correction factors are used. In first order shear deformation theory, shear correction factors are used. The present study deals with the free vibration and buckling characteristics of thin laminated composite cross-ply doubly curved panels. The displacement components u, v, and w in the α, β, and z directions in a laminate element can be expressed in terms of the corresponding mid-plane displacement components u°, v° , w°, and the rotations ɸ1 , ɸ2 of the mid-plane normal along α and β axes. The higher order shear deformation theory is based on a displacement field in which the displacements of the middle surface are expanded as cubic functions of the thickness coordinate and the transverse displacement is assumed to be constant through the thickness. This displacement field leads to parabolic distribution of the transverse shear stresses and zero transverse normal strain and hence no shear correction factors are used. 12 The governing equations including the effect of shear deformation are presented in orthogonal curvilinear co-ordinates for laminated composite shells. These equations are then reduced to the governing equations for free vibration of laminated composite cross-ply doubly curved shells. The equations are suitably non-dimensionalised. The Navier solution has been used and the generalized eigen value problem so obtained in matrix formulation is solved to obtain the eigen values which are the natural frequencies and the buckling loads. 3.2 BASIC ASSUMPTIONS A set of simplifying assumptions that provide a reasonable description of the behavior of thin elastic shells is used to derive the equilibrium equations that are consistent with the assumed displacement field. 1. No slippage takes place between the layers. 2. The effect of transverse normal stress on the gross response of the laminate is assumed to be negligible. 3. The line elements of the shell normal to the reference surface do not change their length after deformation. 4. The thickness coordinate of the shell is small compared to the principal radii of curvature (z/R1, z/R2 <<<1). 5. Normal to the reference surface of the shell before deformation remains straight, but not necessarily normal, after deformation (a relaxed Kirchhoff -Love hypothesis). 3.3 GEOMETRY OF SHELL Figure 1 shows an element of a doubly curved shell. Here (α, β, z) denote the orthogonal curvilinear coordinates (shell coordinates) such that α and β curves are lines of curvature on the mid surface, z = 0, and z-curves are straight lines perpendicular to the surface, z = 0. For the doubly curved shells discussed here, the lines of principal curvature coincide with the coordinate lines. The values of the principal curvature of the middle surface are denoted by K 1 and K2. 13 Figure.3.1 Geometry of a laminated shell The position vector of a point on the middle surface is denoted by r and the position of a point at distance, z, from the middle surface is denoted by R. The distance, ds, between points (α, β, z) and α dα, β d β, z dzis determined by (ds) ² dr dr dr (1) r r d d (2) The magnitude ds of d r is given in equation (2), the vectors r r and are tangent to the and β coordinate lines. Then equation (1) can be modified as ds 2 r r r r r r (d ) 2 (d ) 2 2 d .d (3) The following derivation is limited to orthogonal curvilinear coordinates which coincide with the lines of principal curvature of the neutral surface. The third term in equation (3) thus becomes 14 2 r r r r d .d 2 . cos d .d 0 2 (4) Where we define r r r r r r 2 A1 2 (5) 2 A2 2 Now the equation (3) becomes ds 2 A12 (d ) 2 A2 2 (d ) 2 (6) This equation is called the fundamental form and A1 and A2 are the fundamental form parameters, Lame parameters, or surface metrics. The distance, dS, between points (α, β, z) and (α dα, β d β, z dzis given by ds 2 d R.d R. L12 (d ) 2 L2 2 (d ) 2 L3 2 (dz) 2 In which d R R R R d d dz and L1, L2, and L3 are the Lame’s coefficients. z z z L1 A1 1 ; L2 A2 1 ; L3 =1 R2 R1 The vectors (7) (8) R r R r and are parallel to the vectors and . From the figure the elements of area of the cross section are z da1 L1 d .dz A1 1 d .dz R1 (9) 15 z da 2 L2 d .dz A2 1 d .dz R2 The strain displacement equations of a shell are an approximation, within the assumptions made previously, of the strain displacement relations referred to orthogonal curvilinear coordinates. In addition, it is assumed that the transverse displacement, w, does not vary with z. 3.4 STRAIN DISPLACEMENT RELATIONS 3.4.1 FIRST ORDER SHEAR DEFORMATION THEORY According to the first order shear deformation theory, the displacement field is given by: z u 1 u z1 R1 (10) z v z 2 v 1 R2 ww Here (u,v,w) = the displacement of a point (α, β, z) along the (α, β, z) coordinates; and (u, v, w) = the displacements of a point (α, β, 0). Now substituting equation [10] in the strain displacement relations referred to an orthogonal curvilinear coordinate system we get 1 1 0 zk1 2 2 0 zk 2 4 40 (11.a) 5 50 6 6 0 zk 6 Where, 16 10 1 u w . ; A1 R1 20 1 v w . ; A2 R2 60 v 1 u ; A1 A2 1 . 4 0 2 1 w u ; A2 R1 5 0 1 1 w v ; A1 R2 …………………… k1 1 1 . ; A1 k2 1 2 . ; A2 k6 1 2 1 1 1 1 1 1 v 1 u ; . A1 A2 2 R1 R2 A1 A2 11.b 1 and 2 are the rotations of the normals to the reference surface, z = 0, about the α and β coordinate axes, respectively. The displacement field in equation [10] can be used to derive the general theory of laminated shells. 17 3.4.2 HIGHER ORDER SHEAR DEFORMATION THEORY The higher order displacement field relations as proposed by Reddy and Liu [24] are z 1 w 4 u 1 u z1 z 3 2 1 R1 A1 3h z 1 w 4 v z 2 z 3 2 2 v 1 R A 3 h 2 2 (12) ww Here (u,v,w) = the displacement of a point (α, β, z) along the (α, β, z) coordinates; and (u, v, w) = the displacements of a point (α, β, 0). Now substituting equation [12] in strain displacement relations referred to an orthogonal curvilinear coordinate system, we get z k 1 1 0 z k 1 0 z 2 k1 2 2 20 0 2 z 2k2 2 4 4 0 z 2 k 41 5 5 0 z 2 k 51 6 6 0 z k6 0 z 2 k6 2 ………………………. Where, 18 (13) 10 1 u w . ; A1 R1 20 1 v w . ; A2 R2 60 v 1 u ; A1 A2 1 . 4 0 2 1 w ; A2 5 0 1 1 w ; A1 k1 1 1 . ; A1 k2 1 2 . ; A2 0 0 1 2 1 1 1 1 1 1 v 1 u ; . A1 A2 2 R1 R2 A1 A2 w 4 2 2 h k6 0 k4 1 w 4 1 k 5 2 1 h 2w 4 2 k1 2 1 2 3h k2 2 4 2 3h 4 2 k 6 2 3h ……………. (14) 2 2 w 2 2w 2 1 2 Where 1 and 2 are the rotations of the normals to the reference surface, z = 0, about α and β coordinate axes, respectively. The stress-strain relations are as given in equation (15). 19 3.5 STRESS-STRAIN RELATIONS: The stress-strain relation for the kth orthotropic layer takes the following form: 1k k k k Q11 Q12 0 0 Q16 k k k k 2 0 0 Q26 Q12 Q22 k k k 0 Q44 Q45 0 4 0 k k 0 Q45 Q55 0 0 5k Q k Q k k 0 0 Q66 26 16 k 6 1k k 2 k 4 5k k 6 (15) For special orthotropic material, in which the principal axis direction coincides with the axis of the material direction, Q16K Q26K Q45K 0 Then, 1k k k Q11 Q12 0 0 0 k k k 2 Q12 Q22 0 0 0 k k 0 Q44 0 0 4 0 k 0 0 Q55 0 0 5k 0 k 0 0 0 Q66 k 6 1k k 2 k 4 5k k 6 (16) For generalized plane stress condition, the above elastic moduli are related to the engineering constants as follows: Q11 E1 1 12 21 Q12 E1 21 E 2 12 1 12 21 1 12 21 Q22 E2 1 12 21 (17) 20 Q44 = G13 Q55 = G23 Q66 = G12 E1 12 E2 21 3.6 GOVERNING EQUATIONS The governing differential equations, the strain energy due to loads, kinetic energy and formulations of the general problem are derived on the basis of Hamilton’s principle. The equation of motion is obtained by taking a differential element of the shell as shown in Figure 3.1. The figure 3.1 shows an element with internal forces like membrane (N1, N2, and N6), shearing forces (Q1, and Q2) and the moment resultants (M1, M2 and M6). 3.6.1 STRAIN ENERGY The strain energy of a differential shell element can be written as, U 1 2 1 1 z 2 2 6 6 4 4 5 5 .dV (18) dV = Volume of shell element z z .d .d .dz. dV A1 1 A2 1 R1 R2 (19) The variation of strain energy U is given by U 1 2 z 1 1 2 2 6 6 4 4 5 5 .dV (20) The equation [20] is independent of the material property. Substituting the variation of strain function and dV in equation (20), 21 U 1 2 [ N M 1 k1 N 2 2 M 2 k 2 N 6 w1 M 6 1 N 6 w2 M 6 2 0 0 1 1 z z z 5 ] A1 A2 dd . Q1 1 4 Q2 1 R1 R2 (21) The variation of strain energy is given as: U 1 2 [ N 1 z 0 1 M 1k1 N 2 2 M 2k 2 N 6w1 M 6 1 N 6w2 M 6 2 0 z z 5 ] A1 A2 dd . Q1 1 4 Q2 1 R1 R2 (22) Substituting for 10 , 2 0 , k1 , k 2 .... .in equation (22) and integrating the resulting expression, the equation becomes, U { N2 ( N1 A2 ) A N A A A (M 1 A2 ) A u N1v 1 1 1 2 w M 1 2 1 1 N 2u 2 R1 A1 A2 ( N 2 A1 ) A (M 2 A1 ) A ( N 6 A2 ) A w v M 21 2 2 N 6u 1 v M 61 1 R2 ( M 6 A2 ) A ( N 6 A1 ) A ( M 6 A1 ) u 2 N 6v 2 u M 6 2 2 1 Q1 A1 A2 1 R1 (Q1 A2 ) u (Q2 A1 ) w Q2 A1 A2 1 w d .d R 2 N u N v Q w M M A d N v N u Q w M M A d 1 6 2 1 6 1 2 1 2 6 2 2 2 6 1 22 1 (23) 3.6.2 KINETIC ENERGY If U be the displacement vector, the kinetic energy of the shell element is given by, T 1 U . U dV 2 v (24) where is the mass density and U represents differentiation with respect to time. U U1 n1 U 2 n2 W n1 u 2 v 2 w 2 h 2 2 2 h 1 T 2 A1 A2 .d .d 2 t t t 12 t t (25) The variation of kinetic energy is given as T h h2 u u v v w w 12 (1 1 2 2 )A1 A2 .d .d (26) This equation contains time derivatives of the variations, ie u etc. To eliminate these terms, integrating equation (26) by parts (the variations of limits t = t 1 and t = t2 must vanish) and neglecting the terms h 3 12 1 and h 3 12 2 which represent rotatory inertia, the above equation reduces to, t2 t2 T .dt h t1 t1 u u v v w w A1 A2 .d .d .dt (27) If the shell is subjected to both body and surface forces and if q 1,q2 and q n are the components of body and surface forces along the parametric lines, then the variation of work done by the external loads are, Ws q u q v q wA .A .d .d 1 2 n 1 2 We1 N1u N 6v Q1w M 11 M 6 2 A2 .d We 2 N6u N 2v Q2w M 61 M 2 2 A1.d 23 (28) 3.6.3 HAMILTON’S PRINCIPLE The equations of equilibrium are derived by applying the dynamic version of the principle of virtual work, which is the Hamilton’s Principle. It states that among the set of all admissible configurations of system, the actual motion t2 makes the quantity L.dt stationary, provided the configuration is known at the limits t = t 1 t1 t2 and t = t2. Mathematically this means L.dt t1 Here, L is called Lagrangian and is equal to L= T - (U –V) (29) Where, T = Kinetic energy, U= Strain energy, V= potential of all applied loads, = Mathematical operation called variation. It is analogous to partial differentiation. It is clear from equation [29] that the Lagrangian consists of kinetic, strain energy and potential of applied loads. By applying the dynamic version of the principle of virtual work (Hamilton’s principle), integrating the displacement gradients by parts in the resulting equation and setting the coefficients of δu, δv, δw, δφ 1 and δφ2 to zero separately, the following equations of equilibrium are obtained: u : ( A2 M 1 ) ( A1M 6 ) ( A2 N 2 ) ( A1 N 6 ) A A A A 1 w N 2 2 N 6 1 k1 M6 1 M2 2 A1 A2 q1 A1 A2 I 1u I 21 I 3 A2 (30.a) v : ' ' ' 1 w ( A2 N 6 ) ( A1 N 2 ) ( A2 M 6 ) ( A2 M 6 ) A A A A N1 1 N 6 2 k 2 M1 1 M 2 1 A1 A2 q 2 A1 A2 I 1v I 2 2 I 3 A 2 (30.b) 24 w : ( A2 Q1 ) ( A1Q2 ) A1 A2 k1 N1 A1 A2 k 2 N 2 4 3h 2 1 ( A2 P1 ) 1 ( A2 P2 ) 1 ( A1P6 ) 1 ( A2 P6 ) P1 A1 P2 A2 P6 A1 P6 A2 A A1 A2 A2 A1 A1 A2 1 A2 4 ( A2 K1 ) ( A1K 2 ) 2w 2w A1 A2 n2 I1w q n A1 A2 n1 2 2 h 2 ( A u) 2 I ( A21 ) I ( A1v) I ( A1 2 ) 16 I 7 5 5 I 3 3 9 h4 A1 w A2 w A1 A2 (30.c) 1 : ( M 1 A2 ) ( M 6 A1 ) A A A A A 4 A2 P1 A1 P6 M2 2 M6 1 M6 1 P2 2 P6 1 2 3h Q1 A1 A2 4 h2 1 w K 1 A1 A2 I 2 u I 41 I 5 A1 A2 A 1 (30d) 2 : ( M 6 A2 ) ( M 2 A1 ) A A A A 4 A2 P6 A1 P2 M1 2 M 6 2 P1 1 P6 2 3h 2 Q1 A1 A2 1 w K 2 A1 A2 I 2 v I 42 I 5 A1 A2 A 2 h 4 2 (30e) q1, q2, qn can be defined as the transverse loads ' The inertias I 1 and I 1 (i=1, 2, 3, 4, 5) are defined by the equations, I 1 I1 2k1I 2 ' I 1 I1 2k2 I 2 I 2 I 2 k1 I 3 4 4k I 4 12 I 5 2 3h 3h 25 ' I 2 I 2 k2 I 3 I3 ' I3 4 4k I 12 I 5 2 4 3h 3h 4 4k I 12 I 5 2 4 3h 3h I 4 I3 I5 where 4 4k I 22 I 5 2 4 3h 3h (31) 8 16 I 2 I7 2 5 3h 9h 4 16k I 41 I 7 2 5 3h 9h I , I , I , I , I , I 1, , , , , d n 1 1 1 1 1 1 hk 1 k 2 3 4 6 k 1 hk Where 𝝆(k) is the density of the material of the kth layer. 3.7 STRESS RESULTANTS AND STRESS COUPLES Figure 3.2: STRESS AND MOMENT RESULTANTS 26 (32) Let N1 be the tensile force, measured per unit length along β coordinate line, on a cross section perpendicular to α coordinate line. Then the total tensile force on the differential element in the α direction is h 2 N1. A2 1 da2 dz h 2 (33) In which h = the thickness of the shell (z = -h/2 and z = h/2 denote the bottom and top surfaces of the shell) and da2 is the area of cross section. Using equation (9), the remaining stress resultants per unit length are given by: z 1 1 R2 2 1 z R1 z N1 6 1 R N 2 2 N12 z 5 1 R2 N 21 h 2 z Q1 1 4 R1 dz Q2 h 2 M 1 z 1 1 z R2 M 2 M z 12 z 1 2 M 21 R1 z z 6 1 R2 z 1 z 6 R1 …………………… 34 In contrast to the plate theory (where 1/R1 =0,1/R2 = 0), the shear stress resultants, N12 and N21, and the twisting moments, M12 and M21, are, in general, not equal. For shallow shells the terms z/R1 and z/R2 can be neglected in comparison with unity. Hence N12 = N21 = N6 and M12 = M21 = M6. 27 . The shell under consideration is composed of finite number of orthotropic layers of uniform thickness, as shown in Figure 2. In view of assumption 1, the stress resultant in equation [34] can be expressed as N hk M i k 1 i 1, z .dz...................i 1,2,6 n i, k hk 1 hk Qi k 1 i .dz...................i 4,5 n …………………… k (35) hk 1 In which n = the number of layers in the shell; hk and hk-1 is the top and bottom z coordinates of the kth lamina. Substitution of equation [11] and [15] into equation [35] leads to the following expression for the stress resultants and stress couples in the first order shear deformation theory: N i Aij . j Bij .K j ; 0 0 M i Bij j Dij K j ; 0 0 ……………….. Q1 A45 . 4 A55 5 ; 0 0 (36) Q2 A44 . 4 A45 5 ; 0 0 Here Aij, Bij and Dij denote the extensional, flexural-extensional coupling, and flexural stiffness. They may be defined as: A ij , Bij , Dij hk n Q k 1 hk i ij k (1, z , z 2 ).dz. (37) For i, j = 1, 2, 4, 5, 6. hk and hk-1 are the distances measured as shown in figure -1 According to the higher order theory, similarly, N hk 3 i , M i , Pi k 1 i 1, z , z .dz...................i 1,2,6 n k hk 1 Q1 , K1 k 1 4 k 1, z 2 .dz n hk hk 1 …………………… 28 (38) Q2 , K 2 nk 1 5 k 1, z 2 .dz hk hk 1 In which n = the number of layers in the shell; hk and hk-1 is the top and bottom z coordinates of the kth lamina. Substituting of equation [13] and [15] into equation [38] leads to the following expression for the stress resultants and stress couples N i Aij . j Bij .k j Eij k j ; 0 0 2 M i Bij j Dij k j Fij k j ; 0 0 2 Pi Eij j Fij k j H ij k j ; 0 0 2 i, j 1,2,6 Q1 A5 j . j D5 j k j ; 0 1 Q2 A4 j . j D4 j k j ; 0 1 K1 D4 j j F4 j k j ; 0 1 K 2 D5 j j F5 j k j ; 0 1 ………………...... (39) j 4,5 where Aij, Bij , etc. are the laminate stiffnesses expressed as A , B ij ij , Dij , Eij , Fij , H ij n hk Q k 1 hk i k ij (1, z , z 2 , z 3 , z 4 , z 5 ).dz . (40) The strain displacement relations (13) are substituted in the equations for the stress resultants and stress couples given in equation (39). Since the solution for the equations of motion is done by using the Navier solution, therefore such a solution exists only for specially antisymmetric cross ply laminate for which the following laminate stiffnesses are zero. 29 The expression for the stress resultants and stress couples so obtained are then substituted into the equation of motion (30). The equation of motion in terms of the displacements for doubly curved shells hence reduces to ( A11 2 B11k1 D11k12 ) 2u 2u 2 ( A 66 2 B66 k1 D 66k 1 ) 2 2 A12 B12k 2 A66 B66k 2 B12k1 D12k 2 k1 k1 B66 D66k 2 k1 2v w 4 3w 4 3w 4 3w A11k1 A12k 2 B11k1 B12k 2 k1 2 E11 3 2 F11k1 3 2 E12 2 E66 F12k1 2 F66 3h 3h 3h 2 2 2 2 4 4 4 4 1 1 B11 2 E11 D11k1 2 F11k1 B66 2 E66 D66 k1 2 F66 2 2 3h 3h 3h 3h 2 4 4 4 4 2 B E B E D k F k D k F k 66 66 12 1 12 1 66 1 66 1 12 3h 2 12 3h 2 3h 2 3h 2 w I 1u I 21 I 3 A66 B66k1 B21k 2 D21k 2 k1 k 2 B66 D66k 2 k1 B21k1 A21 A66 2 B66 k 2 D66 k 22 2v 2v 2 A 2 B k D k 22 22 2 22 2 2 2 A21k1 A22k 2 k1k 2 B21 k 2u B22 2 2 w 4 3w 4 3w 2 E 22 F 22 3h 3 3h 2 3 4 3w 2 E E F F 66 21 66 21 3h 2 2 2 21 21 4 4 1 B66 2 E 66 B21 2 E 21 3h 3h 21 21 21 21 4 4 D66 k 2 2 F66 k 2 D21k 2 2 F21k 2 3h 3h 2 2 4 4 4 4 2 2 B66 3h 2 E 66 3h 2 F66 k 2 D66 k 2 2 B22 3h 2 E 22 3h 2 F22 k 2 D22 k 2 2 ' w I 1v I 22 I 3 30 u k1 A11k1 A12 k 2 k 2 A21k1 A22 k 2 w v k1 A12 B12 k 2 k 2 A22 B22 k 2 k1 A11 B11k1 k 2 A21 B21k1 4 4 4 k1 B11 2 E11 k 2 2 E 21k 2 B21 A55 2 D55 1 3h h 3h 4 4 4 k1 B12 2 E12 k 2 2 E 22 B22 A44 2 D44 2 3h h 3h 2 4 4 4 w 2 E11k1 2 k 2 E 21 A55 2 D55 2 3h h 3h 2 4 4 4 w 3h 2 k1 E12 3h 2 k 2 E 22 k 2 E 21 k 2 E 22 A44 h 2 D44 2 3 3 2 u w 4 3v 1 E F k 2 E E11 F11k1 3 k1 k 2 2 F11 2 H 11 12 12 2 66 3 3h 2 3 2 4 4w 3v H E F k F k E F12 11 12 21 2 22 2 22 2 2 4 3 3h 3 4 4 4 2 F H F H 21 21 22 22 3 3h 2 3h 2 3h 2 4 3 u 4 H 4 H w 2 E 66 3h 2 21 3h 2 22 4 2 3 3 4 v 4 w 2 k F F H H 66 2 2 2 66 2 66 3h 2 66 2 2 2w 2 w ' v 1 2 4 2 I 7 2 I1 w I I I 3 5 5 2 3h 31 2 4 4 4 4 4 u w B D k E F k B11k1 B12 k 2 2 E11k1 2 E12 k 2 A55 2 D55 11 1 11 11 1 2 2 2 11 3h 3h 3h 3h 3h 2 4 4 v B12 D12 k 2 B66 D66 k 2 2 E12 2 F12 k 2 E 66 F66 k 2 3h 3h 2 8 16 1 D F H 11 3h 2 11 9h 4 11 2 2 4 4 4 16 4 16 2 D12 2 F12 D66 2 F66 2 F12 4 H 12 2 F66 4 H 66 3h 3h 3h 9h 3h 9h 3 4 3 w 16 3w 4 4 16 w 2 F11 3 4 H 11 3 2 F12 2 F66 4 H 12 2 3h 9h 3h 9h 3h 2 4 4 u B66 D66 k1 2 E 66 2 F66 k1 2 3h 3h 2 8 16 2w 4 1 16 D F H H 66 3h 2 66 9h 4 66 2 9h 4 66 A55 3h 2 D55 1 4 4 4 4 w 2 D55 2 F55 1 2 D55 2 F55 h 3h h 3h w I 2 u I 41 I 5 32 2 2 4 4 4 4 v u B66 D66 k 2 3h 2 E 66 3h 2 F66 k 2 2 B66 k1 D66 B21 k1 D21 3h 2 E 66 3h 2 F66 k1 2 2 4 4 4 16 4 4 16 1 2 D66 3h 2 F66 D21 3h 2 F21 3h 2 F66 9h 4 H 66 D66 3h 2 F66 3h 2 F66 9h 4 H 66 2 3 2 4 4 4 8 w v 2 F66 2 F21 2 H 66 B D k E F k F k E 22 22 2 21 21 2 22 2 22 2 2 3h 3h 3h 2 3h 2 4 4 4 4 4 w 2 B k B k E k E k D F F H F H 21 1 21 2 22 2 22 22 22 3h 2 22 3h 2 21 3h 2 21 2 22 2 3h 2 3h 2 3 16 16 4 4 4 w F H H A D 22 21 22 44 44 2 4 4 3 2 3h 9h 9h h h2 4 D44 2 F44 2 h 4 4 4 w A44 h 2 D44 h 2 D44 h 2 F44 w I 2 v I 42 I 5 3.8 BOUNDARY CONDITIONS Up to now, the analysis has been general without reference to the boundary conditions. For reasons of simplicity, only simply supported boundary conditions are considered along all edges for the shell. The boundary conditions for the simply supported doubly curved shell are obtained as given below N1 = 0, v = 0, w = 0, Following the Navier solution procedure, the following solution form which satisfies the boundary conditions in the above equation is assumed: 33 u U cos m . sin n e i t v V sin m . cos n e it W sin m . sin n e it 1 1 cos m . sin n e it …………… (42) 2 2 sin m . cos n e it where , m m n and U, V, W, 1 and 2 are the maximum amplitudes, m and , n a b n are known as the axial half wave number and circumferential wave number respectively. Introducing the expressions [42] into the governing equations of motion in terms of displacements [41], the following equation in matrix form is obtained, which is a general eigen value problem. C X 2 M X C X BX (43) (43.a) Where, 2 is the eigenvalue is the critical buckling load {X} is a column matrix of amplitude of vibration or eigenvector. [C], [M] and [B] are 5 x 5 matrices. For convenience, the elements of the above matrices are suitably non-dimensionalised as follows 34 U Uh V Vh W Wh 1 1 …………………. 2 2 Aij Aij Q2 h (44) Eij E ij Q2 h 4 Bij B ij Q2 h 2 Fij F ij Q2 h 5 Dij D ij Q2 h 3 H ij H ij Q2 h 7 (- (bar) on top indicates non-dimensionalised quantities) And I1 I1 1h I 2 I 2 1h 2 I 3 I 3 1h 3 I 4 I 4 1h 4 I 5 I 5 1h 5 I 7 I 7 1h 7 (45) I , I , I ,.......... are the non-dimensionalised quantities 1 2 3 Also ~ I 1 I 1 1 h I 1 1 h 4 ~ I 2 I 2 I 4 1 h 2 I 2 1 h 2 3 8 16 ~ I 3 I 3 I 5 I 7 1 h 3 I 3 1 h 3 3 9 4 ~ I 5 I 5 I 7 1 h 5 I 5 1 h 5 3 ………………. (46) After non–dimensionalising the terms, the equation [43] in matrix form can be written as given below 35 H X X 2 H X X (47) (47.a) Where, 2 b 2 2 Q 2 4 a n b h (1 ) 1 2 (48) 2 12 21 A non-trivial solution for the column matrix X will give the required eigenvalues, which are the values of the square of the frequency parameter in the present case. The lowest value of is of particular interest. will give the critical buckling load. 36 CHAPTER 4 RESULTS&DISCUSSION 37 CHAPTER 4 NUMERICAL RESULTS AND DISCUSSIONS 4.1 INTRODUCTION The frequency parameters and buckling loads are calculated by using a computer program for laminated composite doubly curved shells. The results obtained using the present theory are compared to earlier results and are tabulated. The numerical values of the lowest value of frequency parameter and non-dimensional buckling load are presented for various shell parameters in this chapter to study the effect of number of layers, the orientation of layers and the ‘a/h’ ratio (side by thickness ratio). SOLUTION OF EIGEN VALUE PROBLEM AND COMPUTER PROGRAM A standard subroutine in the computer program to find the eigenvalue of matrices has been used, which consists of root power method of iteration with Wielandt’s deflection technique. The program is called RTPM, which is capable of finding the required number of roots in descending order. The change of the sign of the determinant value is checked for values of one percent on either side of the root to verify the convergence. The RTPM program gives H then the highest the highest value of eigenvalue first, so if the M matrix is taken as C 1 value of 2 1 , 2 is obtained, that is the lowest value of and which is of particular interest. These programs are written in FORTRAN language. 4.2 The validation of the formulation and comparison of results Using the formulation developed above, numerical studies are carried out. The lowest value of the frequencies has been calculated at first for two layer, three layer and four layer laminated composite spherical shells for various values of R/a and a/h by higher order shear deformation theory. These results are compared with earlier available results. This also serves to check on the validity of the present theory, as the results are mostly agreeable. Table 4. 1 and Table 4.2 shows the comparison of present results and those of J .N. Reddy [24] for the 38 a 2 2 of [0/90], [0/90/0] and [0/90/90/0] non-dimensional frequency parameter 2 Q22 2 2 b h E2 simply supported spherical shell. The geometrical and material properties used are E11 25E 22 , G12 G13 0.5E 22 , G 23 0.2E 22 , 12 0.25, 1 For all the panels, a/b =1 and R1 R2 R . Table 4.1: Comparison of lowest non-dimensional frequency for a simply supported laminated composite spherical shell (a/h=100) R/a 0 / 90 / 90 / 0 0 / 90 / 0 0 / 90 Theory [24] Present Difference [24] value Present Difference [24] value Present Difference value 5 HSDT 28.840 28.886 0.046 31.020 30.927 -0.093 31.100 31.026 -0.074 10 HSDT 16.710 16.811 0.101 20.350 20.245 -0.105 20.380 20.300 -0.08 20 HSDT 11.840 11.911 0.071 16.620 16.511 -0.109 16.63 16.540 -0.09 50 HSDT 10.060 10.230 0.17 15.420 15.291 -0.129 15.42 15.320 -0.1 100 HSDT 9.784 9.964 0.18 15.240 15.110 -0.13 15.23 15.140 -0.09 Plate HSDT 9.688 9.687 -0.001 15.170 15.171 0.001 15.170 15.169 -0.001 39 Table 4.2: Comparison of lowest non-dimensional frequency for a simply supported laminated composite spherical Shell (a/h=10) R/a 0 / 90 / 90 / 0 0 / 90 / 0 0 / 90 Theory [24] Present Difference [24] value Present Differen value ce [24] Present Difference value 5 HSDT 9.337 9.332 -0.005 12.060 12.110 0.05 12.040 12.649 0.609 10 HSDT 9.068 9.095 0.027 11.860 11.948 0.088 11.840 12.498 0.658 20 HSDT 8.999 9.035 0.036 11.810 11.907 0.097 11.790 12.460 0.67 50 HSDT 8.980 9.019 0.039 11.790 11.896 0.106 11.780 12.450 0.67 100 HSDT 8.977 9.017 0.04 11.790 11.894 0.104 11.780 12.448 0.668 Plate HSDT 8.976 8.877 -0.099 11.790 11.890 0.100 11.780 12.445 0.665 As can be seen from the Table 4.1 and Table 4.2, the values are quite agreeable. The validation of the formulation for buckling load is similarly done by comparing the results with that of Librescu et al [17]. It can be seen from Table 4.3 that the results are in agreement. 40 Table 4.3: Comparison of non-dimensional buckling load for a simply supported laminated composite doubly curved shell (a/h=10) E11 40E 22 , G12 G13 0.6E 22 , Curvature 12 0.25, G 23 0.5E 22 , Present work 1 Theory FSDT [17] HSDT [17] Rx/a=5, Ry/a=5 12.530 12.214 12.431 Rx/a=10, Ry/a=5 12.236 11.822 12.007 Rx/a=10, Ry/a=20 11.861 11.479 11.697 Rx/a=20, Ry/a=20 11.797 11.406 11.610 Plate 11.753 11.353 11.555 4.3 NUMERICAL RESULTS: The governing equations are solved for doubly curved shells (spherical, parabolic, and elliptical paraboloid shells). Analysis is carried out for simply supported boundary condition. The material properties for the parametric study are assumed as: E11 25 109 , G23 0.2 109 E22 01109 , G12 0.5 109 , G13 0.5 109 a 100 h 12 0.25 The geometry for various shells unless otherwise specified is For spherical shell: R1 R 5, 2 5 a a For hyperbolic paraboloid shell: R1 R 5 , 2 5 a a For elliptical paraboloidal shell: R R1 5 , 2 7.5 a a 41 4.3.1 VIBRATION ANALYSIS The first study is done to investigate the variation in R/a ratio on the non-dimensional frequency parameter. Table 4.4 shows the variation of R/a ratio on the non-dimensional frequency parameters of the hyperbolic paraboloid shell. Table 4.4: Variation in non-dimensional frequency parameter with change in curvature ratio (R/a) for laminated composite hyperbolic paraboloidal shell R1 a R2 a -1 0/90 0/90/90/0 a/h=200 a/h=100 a/h=10 a/h=200 a/h=100 a/h=10 1 45.590 32.826 12.582 12.352 12.330 10.166 -2 2 19.494 15.377 9.402 14.354 14.328 11.826 -3 3 13.533 11.798 9.028 14.764 14.737 12.167 -4 4 11.567 10.721 8.958 14.910 14.883 12.269 -5 5 10.774 10.308 8.945 14.979 14.952 12.346 -10 10 9.990 9.919 8.966 15.071 15.044 12.422 Here it is seen that as R/a ratio increases, then non-dimensional frequencies are decreasing. 42 The study is extended to elliptical paraboloidal shells by increasing the R/a ratio, keeping the R1/R2 ratio constant, as given in Table 4.5. Table 4.5: Variation in non-dimensional frequency parameter with change in curvature ratio (R/a) for laminated composite elliptical paraboloidal shell a/h =100 R1/a a/h=10 R2/a 0/90 0/90/90/0 0/90/0/90/0 0/90 1 1.5 107.105 107.763 107.634 12.928 15.095 15.093 2 3.0 56.682 57.788 57.730 10.270 13.216 13.180 3 4.5 38.821 40.438 40.398 9.613 12.800 12.755 4 6.0 29.924 31.996 31.978 9.363 12.648 12.599 5 7.5 24.692 27.175 27.166 9.242 12.577 12.526 10 15 15.034 18.855 18.868 9.076 12.480 12.427 0/90/90/0 0/90/0/90/0 In case of elliptical paraboloidal shell, with the increase of R/a ratio, non-dimensional frequency decreases. With the increase of thickness parameter (a/h), i.e., as thickness decreases, then frequency parameter is also increasing. Table 4.6 shows the variation of modular ratio on the non-dimensional frequency parameter. Four layer cross-ply [0/90/90/0] and a/h = 100 value is chosen for the study. Table 4.6: Variation in non-dimensional frequency parameter with modular ratio for laminated composite doubly curved shells of [0/90/90/0] and a/h=100 Hyperbolic Elliptical paraboloidal shell paraboloidal shell R1/a= -5,R2/a=5 R1/a= 5,R2/a=7.5 R1/a=R2/a=5 25 14.952 27.175 31.026 40 18.432 29.465 33.131 60 22.220 32.101 35.538 E 11 E 22 Spherical shell With the increasing modular ratio, it is seen that the non-dimensional frequency parameter is also increasing. But hyperbolic paraboloidal shell has less non-dimensional frequency; hence it is more stable than the other two doubly curved shells. 43 Table 4.7 shows the variation of the non-dimensional frequency with number of layers for all shell geometries. Table 4.7: Variation in non-dimensional frequency parameter with number of layers for laminated composite doubly curved shells a/h=100 Shell Curvature 0/90 0/90/0 0/90/90/0 0/90/0/90/0 Spherical R1/a=R2/a=5 28.886 30.928 31.026 31.009 Elliptical R1/a=5,R2/a=7.5 24.692 27.090 27.175 27.166 Hyper R1/a= -5,R2/a=5 10.308 14.924 14.952 14.977 paraboloid Above table shows that as number of layers changes, non-dimensional frequencies generally increase. Some of the above results are also presented in graphical form as shown in Figures 4.1-4.3. Figure 4.1: Variation in non-dimensional frequency with R2/a of hyperbolic paraboloid shells for various a/h ratios 44 Figure 4.2: Variation of non-dimensional frequency with R1/a of elliptical paraboloid shell for various a/h ratios Figure 4.3: Variation of non-dimensional frequency of cross-ply spherical shell, hyperbolic and elliptical paraboloid shell with modular ratio. 45 4.3.2 STABILITY ANALYSIS For determination of non-dimensional buckling load, the geometrical and material properties used are E11 25E 22 , G12 G13 0.5E 22 , G 23 0.2E 22 , 12 0.25, 1 Table 4.8 shows the variation in non-dimensional buckling load with R/a ratio for a hyperbolic paraboloidal shell. It is observed that here as R/a ratio increases, non-dimensional buckling load decreases. Table 4.8: Variation in non-dimensional buckling load with curvature ratio (R/a) for laminated composite hyperbolic paraboloidal shell R1 a R2 a -1 0/90 0/90/90/0 a/h=200 a/h=100 a/h=10 a/h=200 a/h=100 a/h=10 1 254.088 132.165 20.560 18.594 18.529 12.720 -2 2 40.545 25.285 9.925 21.935 21.859 15.040 -3 3 19.003 14.468 8.828 22.584 22.505 15.490 -4 4 13.745 11.822 8.563 22.813 22.734 15.650 -5 5 11.869 10.876 8.471 22.919 22.840 15.724 -10 10 10.138 10.001 8.398 23.061 22.982 15.822 46 Similar study is done next for the laminated composite elliptical paraboloid shell in Table 4.9 and the same variation is observed. Table 4.9: Variation in non-dimensional buckling load with curvature ratio (R/a) for laminated composite elliptical paraboloid shell a/h=100 R1/a R2/a 0/90 a/h=10 0/90/90/0 0/90/0/90/0 0/90 0/90/90/0 0/90/0/90/0 1 1.5 1310.338 1327.928 1324.926 19.186 26.322 26.301 2 3.0 335.768 349.220 348.530 11.169 18.449 18.338 3 4.5 154.832 168.000 167.737 9.644 17.006 16.877 4 6.0 91.445 104.575 104.461 9.106 16.502 16.367 5 7.5 62.092 75.218 75.173 8.856 16.269 16.131 10 15 22.935 36.076 36.123 8.521 15.959 15.816 Table 4.10 shows the variation of modular ratio on the non-dimensional buckling load for all shell geometries. Four layer cross-ply was taken with a/h ratio of 100. The nondimensional buckling load is found to increase with increase in modular ratio for all shell geometries. Table 4.10: Variation in non-dimensional buckling load with modular ratio for laminated composite doubly curved shells [0/90/90/0] and a/h=100. Hyperbolic Elliptical paraboloid shell shell R1/a= -5,R2/a=5 R1/a= 5,R2/a=7.5 R1/a=R2/a=5 25 98.235 75.218 22.840 40 112.059 88.452 34.710 60 128.961 105.006 50.442 E 11 E 22 47 paraboloid Spherical shell Table 4.11 shows the variation of the non-dimensional frequency with number of layers for all shell geometries. Table 4.11: Variation in non-dimensional buckling load with number of layers for laminated composite doubly curved shells a/h=100 Shell Curvature 0/90 0/90/0 0/90/90/0 0/90/0/90/0 Spherical R1/a=R2/a=5 85.161 97.606 98.235 98.126 Elliptical R1/a=5,R2/a=7.5 62.092 74.748 75.218 75.173 Hyper R1/a= -5,R2/a=5 10.876 22.756 22.840 22.918 paraboloid Above table shows that with increase in number of layers, the non-dimensional buckling load generally increases. 48 CHAPTER 5 CONCLUSION 49 CHAPTER 5 5.1 CONCLUSION: The governing equations including the effect of shear deformations have been presented in orthogonal curvilinear coordinates for laminated orthotropic doubly curved shells. The theory is based on a displacement field as proposed by Reddy and Liu [24] in which the displacements of the middle surface are expanded as cubic functions of the thickness coordinate and transverse displacement is assumed to be constant through the thickness. This displacement field leads to the parabolic distribution of the transverse shear strain and hence shear stresses and therefore no shear correction factors are used. The governing equations of motion are derived by integrating the displacement gradients by parts and setting the coefficients of u , v , w , 1 and 2 to zero separately. In the present study, for first two equations the moment terms are also considered. The governing equations are then specialized for a doubly curved shell. These equations have been solved for simply supported doubly covered shells and the associated eigenvalue problem has been solved by means of a computer program. Lowest frequencies and buckling loads have been considered throughout. The results of the present theory have been compared with the earlier results available by Reddy and Liu [24]. Results are compared and found that it matches that of Reddy and Liu for spherical shells. The study is extended to vibration of hyperbolic and elliptical paraboloid shells and influence of various parameters like aspect ratio, number of layers, modular ratio,etc on the same are studied. The non-dimensional buckling load for axial compression in one direction is also studied for various shell geometries and variation in other parameters. Following conclusions were observed and are summarized again below: 1) With the increasing modular ratio, non-dimensional frequency is also increasing. But hyperbolic paraboloid shell has less non-dimensional frequency as compared to elliptical paraboloid shell and spherical shell. 2) With the increase of curvature ratio R/a frequency decreases. 3) With the increase of a/h ratio, frequency also increases. 50 4) Increase of thickness parameter (a/h) ratio then non-dimensional frequency of the doubly curved shell increases. 5) It is also observed that as number of layers of the shell increases, buckling load increases. 6) In comparison among spherical, hyperbolic paraboloid shell and ellipitical paraboloid shell, the hyperbolic paraboloidal shell has least non-dimensional buckling load. 7) In case of both shell geometries, with the increase of curvature ratio(R/a) buckling load decreases. 8) As a/h ratio increases, the non-dimensional buckling load also increases. 9) As number of layers increases, in general the non-dimensional frequency and the nondimensional buckling load increases. Thus by the above study it can be seen that by suitably changing the orientation of the layers or the number of layers, the properties of the laminated composite doubly curved shells can be tailored to suit the particular needs. 51 5.2 SCOPE FOR FUTURE WORK 1) HSDT can be used to study the vibration and buckling characteristics of thick laminated composite shells. 2) HSDT is more important for laminated composite plates and shells because of greater shear effect 3) Incorporation of finite element techniques to take care of higher shear deformation theory instead of analytical method used in this work. 52 REFERENCES 53 REFERENCES 1. Abu-Arja, K.R., Chaudhuri, R.A. (1989): ‘Moderately-thick angle-ply cylindrical shells under internal pressure,’ ASME Journal of Applied Mechanics, 56: pp.652–7. 2. Adam, C. (2007): ‘Moderately large vibrations of doubly curved shallow open shells composed of thick layers,’ Journal of Sound and Vibration, 299, pp.854–86. 3. Amabili, M., Reddy, J.N. (2009): ‘A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells’, International Journal of Non-Linear Mechanics, 45:pp. 409–418. 4. Amabili, M. (2005): ‘Non-linear vibrations of doubly curved shallow shells’, International Journal of Non-Linear Mechanics, 40: pp.683 – 710 5. Bhimaraddi, A. (1991): ‘Free vibration analysis of doubly curved shallow shells on rectangular planform using three- dimensional elasticity theory,’ International Journal of Solids and structures, 27(7), pp.897-913 6. Bert, C.W., Reddy, V.S. (1982): ‘Cylindrical shells of bimodulus composite materials’, ASCE Journal of Engineering Mechanics, 108: pp.675–88 7. Bhimaraddi, A. (1984): ‘A higher order theory for free vibration analysis of circular cylindrical shells,’ International Journal of Solids and Structures, 20(7): pp.623-630. 8. Chaudhuri, R.A., Balaraman, K., Kunukkasseril, V.X. (1986): ‘Arbitrarily laminated anisotropic cylindrical shells under uniform pressure,’ AIAA Journal, 24, pp.1851–8. 9. Chaudhuri, R.A., Abu-Arja, K.R. (1989): ‘Closed-form solutions for arbitrarily laminated anisotropic cylindrical shells (tubes) including shear deformation,’ AIAA Journal, 27, pp.1597–605. 54 10. Cho, M., Kim, K. and Kim, M. (1996): ‘Efficient higher-order shell theory for laminated composites,’ Composite Structures, 34(2), pp.197-212. 11. Jayasuriya, Dwivedi. N, Sivaneri.T, and Lyons (2002): ‘Doubly Curved Laminated Composite Shells with Hygrothermal Conditioning and Dynamic Loads’, Part 2: FEA and Numerical Results of Shells of Revolution, Mechanics of Advanced Materials and Structures, 9: pp.69 –97 12. Kant, T. and Khare, R.K. (1997): ‘A higher-order facet quadrilateral composite shell element', International Journal for Numerical Methods in Engineering, 40: pp 44774499. 13. Kant, T. and Khare, R.K. (2004): ‘Free vibration of composite and sandwich laminates with a higher-order facet shell element', Composite Structures, 65, pp.405–418. 14. Kulikov G.M., Carrera E. (2008): ‘Finite deformation higher-order shell models and rigidbody motions’, International Journal of Solids and Structures, 45, pp.3153–3172 15. Leissa, A. W., Kadi, A.S. (1971): ‘Curvature Effects On Shallow Shell Vibrations,’ Journal of Sound and Vibration, 16 (2), pp.173-187 16. Leissa, A.W., Chang, J.D. (1996): ‘Elastic deformation of thick, laminated composite shells,’ Composite Sructures, 35: pp.153-170 17. Librescu, L, Khdeir AA, Frederick D. (1989) ‘A shear deformable theory of laminated composite shallow shell-type panels and their response analysis’, I. Free vibration and buckling’, Acta Mechanica, 76: pp.1–33. 18. Maurice Touratier (1992): ‘A Generalization of shear deformation theories for axisymmetric multilayered shells’, International Journal of Solids and structures, 29(11), pp.1379-1399. 55 19. Mindlin, R.D. (1951): ‘Influence of rotary inertia and shear in flexural motions of isotropic elastic plates’, ASME Journal of Applied mechanics, 18(2):pp.31-38. 20. Messina Arcangelo (2003): ‘Free vibrations of multilayered doubly curved shells based on a mixed variational approach and global piecewise-smooth function,’ International Journal of Solids and Structures, 40: pp.3069–3088. 21. Oktem A.S, Chaudhuri R.A.(2007). ‘Fourier analysis of thick cross-ply Levy type clamped doubly-curved panels’, Composite Structures, 80: pp.489–503. 22. Qatu, S.M(1999).’Accurate equations for laminated composite deep thick shells,’ International journal of solids and structures, 36: pp.2917-2941. 23. Reissner, E. (1945). ‘The effect of transverse shear deformation on the bending of elastic plates,’ ASME Journal of Applied mechanics, 12(2): pp.69-77. 24. Reddy, J.N. and Liu, C.F. (1985). ‘A higher-order shear deformation theory of laminated elastic shells,’Int. J. Engng. Sci., 23(3):pp.319-330. 25. Sudhakar A. Kulkarni, Kamal M. Bajoria (2003).’Finite element modeling of smart plates/shells using higher order shear deformation theory’, Composite Structures, 62: pp. 41–55 26. Tabiei and Tanov (1999): ‘New Nonlinear Higher Order Shear Deformation Shell Element For Metal Forming And Crashworthiness Analysis: Part 1. Formulation And Finite Element Equations’, Centre Of Exellence In Dyna3D Analysis. 27. Tripathy, B., Suryanarayan, S. (2009): ‘Analytical studies for buckling and vibration of weld-bonded beam shells of rectangular cross-section,’ International Journal of Mechanical Sciences, 51, pp.77–88. 28. Topal U, ‘Mode-Frequency Analysis of Laminated Spherical Shell’ Session ENG pp 501001 56 29. Yang, T.Y. (1973): ‘High order rectangular shallow shell finite element’, Journal of the Engineering Mechanics Division, 99(EM1): pp.157-181. 57 APPENDIX 58 The non-dimensionalised coefficients of the [ Cij ] and [ Mij ] matrices of Higher order shear deformation theory are given below, C (1,1) A11 2 KHB11 K 2 H 2 D11 S 2 m 2 A66 2 KHB66 K 2 H 2 D66 2 n S C (1,2) A12 A66 H B12 B66 KH B12 B66 KH 2 D12 D66 m n J 4H E12 2E66 KH F12 2F66 m 2 n C (1,3) KHA11 HA12 K 2 H 2 B11 KH 2 B11 m SH 3SJ 4 mH 3 E11 KHF11 S 3 J 3 2 4 4 4 4 m C (1,4) B11 E11 KHD11 KHF11 2 B66 E66 KHD66 KHF66 2 n 3 3 3 3 S 4 4 C (1,5) B12 B66 E12 E66 KH D12 D66 KH F12 F66 m n 3 3 S S C (2,1) A66 A21 KH B21 B66 H B21 B66 KH 2 D12 D66 m n 2m C (2,2) A66 2 HB66 H 2 D66 2 A22 2 HB22 H 2 D22 2 n S n J 4 2 m n H 2 2 2 E66 E21 2 HF66 HF21 KHA21 HA22 KH B21 H B22 H 3 S 2J C (2,3) 3 4 n H E22 F22 3 J 4 4 C (2,4) B21 B66 H D21 D66 E21 E66 H F21 F66 m n 3 3 S 2 4 4 4 4 m C (2,5) B66 HD66 E66 HF66 2 B22 HD22 E22 HF22 2 n 3 3 3 3 S 2 2 8 4 4 m 2 K H A K A K A A J A 8 D KH E H E H E 16 F 11 11 12 21 22 55 11 12 21 55 2 3 3 3 S 4 8 4 16 16 C (3,3) A44 8 D44 KHE21 HE22 KHE12 16 F44 2 n 4 2 H 11H 2 4 m 2 H 11H 2 4 n 3 3 3 9S J 9J 16 H 2 2 2 H 12 H 21 4 H 66 2 m 2 n 9S J 59 3 4 4 m J 4 mH A 8 D 16 F KH B KH E H B H E F H 11 55 55 11 11 21 21 11 55 3 3 3 SH 3 S J C (3,4) 4 F 4 H 2 F 8 H m 2 n H 66 66 3 21 3 21 3 SJ 4 4 4 8 n J 4 F12 H 12 2 F66 H 66 A44 8D44 16 F44 KHB12 KHE12 HB22 HF22 3 3 3 3 3 H C (3,5) 2 m n H 4 F 4 H 3 n H 22 22 S 2 J 3 3 J 2 4 4 4 4 J2 m C (4,4) D11 2 F11 H 11 2 D66 2 F66 H 66 2 n A55 8D55 16 F55 2 3 3 3 3 H S 4 16 C (4,5) D12 D66 2 F12 2 F66 H 12 H 66 m n 3 9 S C (1,3) C (3,1) C (2,3) C (3,2) C (1,4) C (4,1) C (1,5) C (5,1) C (2,4) C (4,2) C (2,5) C (5,2) C (3,5) C (5,3) C (5,4) C (4,5) 60 M (1,1) ( I1' 2 K1hI 2' ) M (1.2) M (2,1) 0 4 4 m M (1,3) M (3,1) ( I 4' K1hI 5' ) h 3 3 a M (1,4) M (4,1) ( I 2' K1hI 3' 4 ' 4 I 4 K1hI 5' ) 3 3 M (1,5) M (5,1) 0 M (2,2) ( I1' 2 K 2 hI 2' ) 4 4 n M (2,3) M (3,2) ( I 4' K 2 hI 5' ) h 3 3 b M (2,4) M (4,2) 0 M (2,5) ( I 2' K 2 hI 3' M (3,3) 4 ' 4 I 4 K 2 hI 5' ) 3 3 2 h 16 ' 2 h 2 I 7 ( m ( ) n ( ) 2 ) I1' 9 a b 4 16 h M (3,4) M (4,3) ( I 5' I 7' ) m 3 9 a 4 16 h M (3,5) M (5,3) ( I 5' I 7' ) n 3 9 b 8 16 M (4,4) M (5,5) ( I 3' I 5' I 7' ) 3 9 B (3,3) 2 61

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