Closed-Form Methodology for the Structural Analysis of Composite Plates With Cutouts

Universidad de Sevilla Escuela Técnica Superior de Ingeniería PhD Dissertation: Closed-form Methodology for the Structural Analysis of Composite Plates with Cutouts A Dissertation presented by Daniel Pastorino Junquero In partial fulfilment of the requirements for the degree of Doctor of Philosophy Supervised by Antonio Blázquez Gámez Federico París Carballo 2020 Abstract The use of composite materials in the aerospace industry shows an increasing tendency due to its excellent specific mechanical properties. The long fibre reinforced polymers are well suited for its application in plate and shell structures. These structures are commonly weakened by cutouts of different shapes, which provide access to the internal systems and structures, allow system integration, or simply reduce the weight of the structure. These cutouts lead normally to a concentration of stresses around them, that requires a thoroughly analysis in order to provide an optimum design of the component. To this end, it is usual to perform a structural analysis on the basis of the methodologies in the existing literature: FEM, BEM and closed-form formulations. The closed-form formulations provide a simple and powerful tool for the first phases of the component’s design, considering that they are well suited for the resolution of parametric analyses with a competitive amount of computational power compared to FE methodologies. However, these formulations have a limited application, since they rely on a set of hypotheses that should be taken into account, e.g. analytical formulations are solved by assuming that the plate under study is infinite in comparison with the cutout. The main objective of the present dissertation is to develop a closed-form formulation that allows the stress analysis of composite panels with cutouts, and that simulates as detailed as possible the real features that are found in aeronautical parts. In this regard, the formulation must allow, among other features, the introduction of local reinforcements, non-uniform lay-ups over the panel, and the inclusion of stiffeners. The first part of the document is merely introductory and covers the first two chapters. Chapter 1 introduces the motivations, objectives and scope of the investigation, including a preliminary state of the art. Subsequently, iv Chapter 2 lays the foundations of the linear anisotropic elasticity, that is required for the development of the methodology. The following block of the manuscript comprises Chapters 3 to 6. Most of the innovative methods and procedures are exposed in these chapters, where the closed-form formulation is enhanced on the basis of the current state of the art. During this block, it is developed a formulation that considers the stretching, bending, and stretching-bending coupling performances of the plate. In addition, the development of a formulation for the analysis of stiffened plates is also conducted. The formulations developed are programmed and validated by means of direct comparison with respect to FEM results, for a set of benchmark exercises. Lastly, Chapter 7 collects some practical exercises that are solved by using the programmed tool, including the analysis of parametric exercises and the resolution of a real aeronautic component. The final conclusions and way-forward are drawn in Chapter 8. Acknowledgements First of all, I would like to express my sincere gratitude to my supervisors, Antonio Blázquez and Federico París, for their personal and technical guidance, their tireless support during the research, and their thorough revision of the manuscript. I am particularly grateful for the support provided by FIDAMC that has provided the funding for this research. In particular, I would like to thank Bernardo López for its diligent supervision of the project. And in general, I wish to thank all my FIDAMC’s colleagues, who have been the best company I could have during these years. I would like to acknowledge the contribution of Airbus Operations S.L. in the definition of the objectives, and the direct application of the research. In addition, I wish to acknowledge the help provided by Chyanbin Hwu during my stay at the NCKU. And naturally, thanks also to my colleagues at the NCKU department. Last but not least, I would like to offer my special thanks to my parents Antonio and Cecilia, my sister Cecilia, and my future wife Grisel for their unconditional support, patience and encouragement. Contents List of Figures xiii List of Tables xxi List of Acronyms xxiii Notation xxv 1 Introduction 1 1.1 Background and motivation . . . . . . . . . . . . . . . . . . 2 1.2 Introduction to the topic . . . . . . . . . . . . . . . . . . . . 3 1.3 Introduction to the state of the art . . . . . . . . . . . . . . 5 1.4 Scope and objectives . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Document structure . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Scientific dissemination . . . . . . . . . . . . . . . . . . . . 10 2 Linear anisotropic elasticity 13 2.1 Elasticity of three-dimensional bodies . . . . . . . . . . . . 14 2.1.1 Stress and equilibrium . . . . . . . . . . . . . . . . . 14 2.1.2 Compatibility . . . . . . . . . . . . . . . . . . . . . . 15 2.1.3 Material behaviour . . . . . . . . . . . . . . . . . . . 16 2.1.4 Boundary conditions . . . . . . . . . . . . . . . . . . 20 Acknowledgements viii 2.2 Elasticity of two-dimensional bodies . . . . . . . . . . . . . 21 2.2.1 Stress and equilibrium . . . . . . . . . . . . . . . . . 23 2.2.2 Compatibility . . . . . . . . . . . . . . . . . . . . . . 23 2.2.3 Material behaviour . . . . . . . . . . . . . . . . . . . 24 2.3 Classical Laminate Theory . . . . . . . . . . . . . . . . . . . 25 2.3.1 Resultant forces and moments . . . . . . . . . . . . . 28 2.3.2 Material behaviour . . . . . . . . . . . . . . . . . . . 29 3 Plate stretching analysis 33 3.1 State of the art for stretching analysis . . . . . . . . . . . . 34 3.1.1 Analysis of infinite plates . . . . . . . . . . . . . . . 34 3.1.2 Analysis of finite plates . . . . . . . . . . . . . . . . 36 3.1.3 General aspects . . . . . . . . . . . . . . . . . . . . . 38 3.2 Lekhnitskii formalism for plate stretching . . . . . . . . . . 39 3.2.1 Complex variable formalism . . . . . . . . . . . . . . 39 3.2.2 Displacements and resultant forces from Φj. . . . . 42 3.3 Cutoutgeometry........................ 43 3.3.1 Geometrical aspects . . . . . . . . . . . . . . . . . . 43 3.3.2 Conformal mapping . . . . . . . . . . . . . . . . . . 45 3.4 General methodology for plate stretching . . . . . . . . . . 46 3.4.1 Approximation function for Φj............ 47 3.4.2 Boundary conditions . . . . . . . . . . . . . . . . . . 49 3.4.3 Collocation points and least squares . . . . . . . . . 54 3.5 Extended methodology for plate stretching . . . . . . . . . 56 3.5.1 Boundary conditions on a shared boundary . . . . . 57 3.5.2 Reinforced cutout . . . . . . . . . . . . . . . . . . . 59 3.6 Stretching benchmark exercises . . . . . . . . . . . . . . . . 60 3.6.1 Benchmark exercise 1 . . . . . . . . . . . . . . . . . 61 3.6.2 Benchmark exercise 2 . . . . . . . . . . . . . . . . . 67 ix Acknowledgements 3.6.3 Benchmark exercise 3 . . . . . . . . . . . . . . . . . 69 3.7 Discussion............................ 72 4 Plate bending analysis 75 4.1 State of the art for bending analysis . . . . . . . . . . . . . 76 4.1.1 Analysis of infinite plates . . . . . . . . . . . . . . . 76 4.1.2 Analysis of finite plates . . . . . . . . . . . . . . . . 77 4.1.3 General aspects . . . . . . . . . . . . . . . . . . . . . 78 4.2 Lekhnitskii formalism for plate bending . . . . . . . . . . . 78 4.2.1 Complex variable formalism . . . . . . . . . . . . . . 78 4.2.2 Deflections and moments . . . . . . . . . . . . . . . 81 4.3 General methodology for plate bending . . . . . . . . . . . . 82 4.3.1 Approximation function for w′ j............ 83 4.3.2 Boundary conditions . . . . . . . . . . . . . . . . . . 83 4.3.3 Collocation points and least squares . . . . . . . . . 90 4.4 Extended methodology for plate bending . . . . . . . . . . . 91 4.4.1 Boundary conditions on a shared boundary . . . . . 91 4.4.2 Reinforced cutout . . . . . . . . . . . . . . . . . . . 93 4.4.3 Particular solution . . . . . . . . . . . . . . . . . . . 94 4.5 Bending benchmark exercises . . . . . . . . . . . . . . . . . 95 4.5.1 Benchmark exercise 4 . . . . . . . . . . . . . . . . . 96 4.5.2 Benchmark exercise 5 . . . . . . . . . . . . . . . . . 103 4.5.3 Benchmark exercise 6 . . . . . . . . . . . . . . . . . 106 4.6 Discussion............................ 109 5 Plate coupled stretching-bending analysis 111 5.1 State of the art for coupled analysis . . . . . . . . . . . . . 112 5.1.1 Analysis of infinite plates . . . . . . . . . . . . . . . 112 5.1.2 Analysis of finite plates . . . . . . . . . . . . . . . . 113 Acknowledgements xvi 4.16 Resultant moments around the boundaries of cutouts (1) and (2). Present method & FEM (CLT). N= 20 and Ru= 2. Benchmark exercise 6. . . . . . . . . . . . . . . . . . . . . . 107 4.17 Resultant moments in global coordinates. Benchmark exercise6. .............................. 108 5.1 Geometry and loads of benchmark exercise 7. . . . . . . . . 127 5.2 Resultant forces around the cutout boundary. Present method & FEM. N= 20 and Ru= 2. Benchmark exercise 7. . . . . 128 5.3 Resultant moments around the cutout boundary. Present method & FEM. N= 20 and Ru= 2. Benchmark exercise 7. 128 5.4 Resultant forces in global coordinates. Benchmark exercise 7. 129 5.5 Resultant moments in global coordinates. Benchmark exercise7. .............................. 130 5.6 Geometry and loads of benchmark exercise 8. . . . . . . . . 131 5.7 Resultant forces around the cutout boundary. Present method & FEM. N= 20 and Ru= 2. Benchmark exercise 8. . . . . 132 5.8 Resultant moments around the cutout boundary. Present method & FEM. N= 20 and Ru= 2. Benchmark exercise 8. 132 5.9 Resultant forces in global coordinates. Benchmark exercise 8. 133 5.10 Resultant moments in global coordinates. Benchmark exercise8. .............................. 134 5.11 Geometry and loads of benchmark exercise 9. . . . . . . . . 135 5.12 Resultant forces around the cutout boundary. Present method & FEM. N= 20 and Ru= 2. Benchmark exercise 9. . . . . 136 5.13 Resultant moments around the cutout boundary. Present method & FEM. N= 20 and Ru= 2. Benchmark exercise 9. 136 5.14 Resultant forces in global coordinates. Benchmark exercise 9. 137 5.15 Resultant moments in global coordinates. Benchmark exercise9. .............................. 138 6.1 Thermoplastic composite panel stiffened by five T-section longitudinal stringers. . . . . . . . . . . . . . . . . . . . . . 141 xvii Acknowledgements 6.2 Euler-Bernoulli beam model. . . . . . . . . . . . . . . . . . 145 6.3 Decomposition of the axial displacement field into axial translation, x2-axis and x3-axis rotations. . . . . . . . . . . . . . 146 6.4 Sign convention for beam section axial forces, shear forces andmoments........................... 149 6.5 Diagram of forces and moments acting on an infinitesimal beam slice in x1−x3(left) and x1−x2(right) planes. . . . 150 6.6 Kinematic description of a circular-section beam. The section twists as a result of the torsion loads. . . . . . . . . . . 153 6.7 Diagram of torsional moments acting on an infinitesimal beamslice. ........................... 156 6.8 Section twist induced by shear forces applied at different points of the beam section. . . . . . . . . . . . . . . . . . . 157 6.9 System of coordinates at the shared edge by a generic stringer and two adjacent plates. Notation for the stiffened plate. . 161 6.10 Embedded stiffened plate. The plate mid-plane contains the centroid and the shear centre. . . . . . . . . . . . . . . . . . 163 6.11 General geometry of a non-embedded stiffened plate. . . . . 163 6.12 Diagram of forces and moments on a stiffened plate (embedded stringer) subjected to in-plane loads and moments. . . . 164 6.13 Diagram of in-plane displacements on a stiffened plate (embedded stringers). . . . . . . . . . . . . . . . . . . . . . . . 166 6.14 Diagram of forces and moments on a stiffened plate (embedded stringer) subjected to out-of-plane loads and moments. 168 6.15 Diagram of out-of-plane displacements on a stiffened plate (embedded stringers). . . . . . . . . . . . . . . . . . . . . . 169 6.16 Geometry and loads of benchmark exercise 10. . . . . . . . 175 6.17 Relative difference Rd(%) along the stringer with respect to the reference (Nb= 100,127 collocation points along the stringer). Order of approximation assessment for a constant Ru= 2. Benchmark exercise 10. . . . . . . . . . . . . . . . . 176 Acknowledgements xviii 6.18 Distribution of axial forces along the stringer for different truncation orders Nb. Present method. N= 20, and Ru= 2. Benchmark exercise 10. . . . . . . . . . . . . . . . . . . . 177 6.19 Distribution of in-plane bending moments along the stringer for different truncation orders Nb. Present method. N= 20, and Ru= 2. Benchmark exercise 10. . . . . . . . . . . . . . 177 6.20 Resultant forces around the boundaries of cutouts (1) and (2). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 10. . . . . . . . . . . . . . . . . . . . . 178 6.21 Resultant forces in global coordinates. Benchmark exercise 10................................. 179 6.22 Distribution of axial forces along the stringer. Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 10................................. 180 6.23 Distribution of in-plane bending moments along the stringer. Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 10. . . . . . . . . . . . . . . . . . . . . 180 6.24 Geometry and loads of benchmark exercise 11. . . . . . . . 181 6.25 Resultant moments around the boundaries of cutouts (1), (2) and (3). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 11. . . . . . . . . . . . . . . . 182 6.26 Resultant moments in global coordinates. Benchmark exercise11............................... 183 6.27 Distribution of torsion moments along the stringers (A) and (B). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 11. . . . . . . . . . . . . . . . . . . . . 184 6.28 Distribution of out-of-plane bending moments along the stringers (A) and (B). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 11. . . . . . . . . . . . . . . . 184 6.29 Geometry and loads of benchmark exercise 12. . . . . . . . 185 6.30 Resultant forces around the boundaries of cutouts (1) and (2). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 12. . . . . . . . . . . . . . . . . . . . . 186 6.31 Resultant forces in global coordinates. Benchmark exercise 12................................. 187 xix Acknowledgements 6.32 Distribution of axial forces along the stringers (A) and (B). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 12. . . . . . . . . . . . . . . . . . . . . 188 6.33 Distribution of in-plane bending moments along the stringers (A) and (B). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 12. . . . . . . . . . . . . . . . 188 6.34 Resultant forces around the boundaries of cutouts (1) and (2). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 13. . . . . . . . . . . . . . . . . . . . . 190 6.35 Distribution of axial forces along the stringers (A) and (B). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 13. . . . . . . . . . . . . . . . . . . . . 191 6.36 Distribution of in-plane bending moments along the stringers (A) and (B). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 13. . . . . . . . . . . . . . . . 192 6.37 Resultant moments around the boundaries of cutouts (1) and (2). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 13. . . . . . . . . . . . . . . . . . . . . 192 6.38 Distribution of torsion moments along the stringers (A) and (B). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 13. . . . . . . . . . . . . . . . . . . . . 193 6.39 Distribution of out-of-plane bending moments along the stringers (A) and (B). Present method & FEM. N= 20,Nb= 40 and Ru= 2. Benchmark exercise 13. . . . . . . . . . . . . . . . 194 7.1 A350 XWB composite lower wing cover manufactured by Airbus in Illescas (Spain) factory. (Source: Airbus, 2011). . 198 7.2 Geometry and loads of parametric exercise 1. . . . . . . . . 199 7.3 Maximum resultant forces around the cutout boundary as a function of the major semiaxis (a) and the cutout orientation (α). N= 20 and Ru= 2. Parametric exercise 1. . . . . . . 200 7.4 Maximum resultant forces around the cutout boundary as a function of the ellipse eccentricity (e) and the cutout orientation (α). N= 20 and Ru= 2. Parametric exercise 1.................................. 201 Acknowledgements xx 7.5 Maximum resultant moments around the cutout boundaries as a function of the ply orientation (±η◦). N= 20 and Ru= 2. Parametric exercise 2. . . . . . . . . . . . . . . . . 202 7.6 Maximum and minimum resultant forces around the boundaries of cutouts (1) and (2) as a function of the modifiable plies stacking sequence [0◦(%) ,±45◦(%) ,90◦(%)].N= 20,Nb = 40 and Ru= 2. Parametric exercise 3. . . . . . . . 204 7.7 Distribution of axial forces along the stringer (A) for different stacking sequences. Present method. N= 20,Nb = 40 and Ru= 2. Parametric exercise 3. . . . . . . . . . . . . . . . . 205 7.8 Distribution of in-plane bending moments along the stringer (A) for different stacking sequences. Present method. N= 20,Nb = 40 and Ru= 2. Parametric exercise 3. . . . . . . . 205 A.1 Circular hollow section geometry. . . . . . . . . . . . . . . . 219 A.2 I-section geometry. . . . . . . . . . . . . . . . . . . . . . . . 221 A.3 T-section geometry. . . . . . . . . . . . . . . . . . . . . . . . 222 A.4 L-section geometry. . . . . . . . . . . . . . . . . . . . . . . . 224 List of Tables 2.1 Number of independent elastic constants for each material type................................ 20 3.1 Commonly used stretching boundary conditions for external edges. .............................. 54 3.2 Mechanical properties of a lamina. . . . . . . . . . . . . . . 60 3.3 Lay-ups used in the benchmark exercises. . . . . . . . . . . 61 3.4 Results of the benchmark exercises. Stretching analysis. . . 74 4.1 Commonly used bending boundary conditions for external edges. .............................. 90 4.2 Results of the benchmark exercises. Bending analysis. . . . 110 5.1 Unsymmetrical lay-ups used in the benchmark exercises. . . 127 5.2 Results of the benchmark exercises. Coupled stretchingbendinganalysis......................... 140 6.1 Boundary conditions for three dimensional beams. . . . . . 152 6.2 Boundary conditions for torsion of beams. . . . . . . . . . . 156 6.3 Stringer lay-ups used in the benchmark exercises. . . . . . . 174 6.4 Results of the benchmark exercises. Stiffened plates analysis. 196 7.1 Lay-ups used in the parametric exercise 3. . . . . . . . . . . 203 List of Acronyms 2D Two-Dimensional. 3D Three-Dimensional. ASC/PSC Average / Point Stress Criterion. BCM Boundary Collocation Method. BEM Boundary Element Method. CBT Classical Beam Theory. CFRP Carbon Fibre Reinforced Plastic. CLT Classical Laminate Theory. CPT Classical Plate Theory. DOF Degrees of Freedom. FCV Fracture Characteristic Volume. FE Finite Elements. FEM Finite Elements Method. FFM Finite Fracture Mechanics. FSDT First-order Shear Deformation Theory. FWC Finite-Width Correction. HSDT Higher-order Shear Deformation Theory. HTP Horizontal Tail Plane. JCR Journal Citation Reports. LEFM Linear Elastic Fracture Mechanics. LSE Linear System of Equations. LSQ Least Squares Method. MPE Minimum Potential Energy. PDE Partial Differential Equation. QI Quasi-Isotropic. SoA State of the Art. UD Unidirectional. Notation CStiffness matrix. SCompliance matrix. E1Elastic modulus in longitudinal direction. E2Elastic modulus in transverse direction. G12 In-plane shear modulus. ν12 In-plane Poisson ratio. hTotal thickness of the plate. fjComponent of the volumetric forces acting on a body. NTruncation order of the function to simulate plates. NbTruncation order of the function to simulate stiffeners. tjComponent of the stress vector in xjdirection. ˜ tjComponent of a prescribed traction in xjdirection. TjResultant force projected in xjdirection. ˜ TjPrescribed resultant force in xjdirection. ujComponent of the displacement. u0 jMid-plane displacement. ˜ujComponent of a prescribed displacement. xjComponents of local rectangular coordinates. x,y,zComponents of global rectangular coordinates. 0Origin of coordinates. εij Component of the strain tensor. γij Crossed term of the strain tensor. σij Component of the stress tensor. κij Component of the curvature tensor. Nij Component of the resultant forces by unit length. Mij Component of the resultant moments by unit length. QiShear force. ViEffective transverse shear force. TRotation matrix. CkStiffness matrix of a lamina k. Introduction 4 the presence of cutouts, which are required to give access to internal systems and structures’ maintenance, to allow system integration, or simply to decrease the structure’s weight. Figure 1.2: A350 XWB composite wing lower cover manufactured by Airbus. These cutouts lead normally to a stress concentration that can induce the failure of the component. Hence, it is of great importance to calculate accurately the stresses around the cutouts in order to provide a safe design of the structure. To this end, the mechanics of composites arise as an instrument for the determination of the stress state of the structure. There is a large variety of cutout topologies in the targeted structures, such as circular, elliptical, triangular, squared-shaped or oval. However, the present investigation is focused on the cutouts with circular or elliptical shapes, which are the most common topologies. In addition, it is intended to reproduce as accurate as possible the particular features characteristic of the aerospace composite structures (see Figures 1.2 and 1.3), e.g. change of thickness, reinforcements, stringers, etc. In the next section, the SoA on the topic will be briefly introduced in order to address properly to a gap in the literature. 5Introduction to the state of the art Figure 1.3: A400M wing assembly. The internal structure is comprised by stiffened panels with many accessibility cutouts (some of them are reinforced). (Source: CompositesWorld, 2012) 2. 1.3 Introduction to the state of the art The mechanics of composites have advanced hand in hand with the development of composite materials for specific applications, as argued by Herakovich [10]. One of the first investigations carried out in this topic was the determination of the elastic constitutive relations that, based on the Hooke’s law [11], were generalized to consider anisotropic materials in the early XIXth century by Cauchy [12]. Subsequently, a comprehensive analysis of the constitutive relations was conducted by Lekhnitskii [1] in 1968 in order to determine the number of independent constants for different types of elastic materials investigated (anisotropic, monoclinic, orthotropic, transversely isotropic and isotropic). In parallel, the formulation of the Classical Lamination Theory (CLT) was conceived in the early 1960s, and it is generally attributed to Reissner and Stavsky [13]. The CLT implies a great step forward in the mechanics of laminated composite materials, since it considers that a set of individual 2Retrieved from https://www.compositesworld.com/articles/a400m-wing- assembly-challenge-of-integrating-composites Introduction 6 plies can be assembled together to form an homogeneous equivalent laminate. As aforementioned, the area of interest is limited to composite plates with circular or elliptical cutouts. In this field, most of the references mention the analytical complex variable formulations (or closed-form formulations), known as Lekhnitskii and Stroh formalisms. On the one hand, the Lekhnitskii formalism [1] operates with Airy stress functions in order to fulfil equilibrium, and it is referred as a stress-based formulation. On the other hand, the Stroh formalism, that was developed by Ting [14] after the works from Stroh [15, 16], introduces a matrix structure by applying a displacement-based formulation. The advantages and disadvantages of using each one of the formalisms will be discussed in depth in Chapter 3. Many other researchers have contributed to this topic on the basis of both the Lekhnitskii and the Stroh complex variable formalisms. First, Lin and Ko [17] devise a methodology based on the Lekhnitskii formalism in order to consider non-infinite plates by using the boundary collocation method (BCM). To this same end, Xiong [18] develops a closed-form methodology to account for the finite boundaries by means of the minimum potential energy theorem (MPE), which is an energetic procedure. A simpler method is conceived by Tan [2], that consists on applying correction factors to the solution of the Lekhnitskii formalism for infinite plates. The efficiency of this method is excellent but at the expense of a lower versatility and accuracy. In a different area of research, Becker [19] develops a method to deal with unsymmetric laminates, that leads to the coupling of the stretching and bending performances of the laminate. The formulation is developed on the basis of the Lekhnitskii formalism. Additionally, Ukadgaonker and Rao [20] make use of customizable transformations in the complex plane in order to allow the calculation of nonelliptical cutouts. It involves a tedious formulation but allows the resolution of any topology of the cutout. On the basis of the Stroh formalism, Hwu [7] identifies and adapts the methodology for considering many applications, e.g. the calculation of hygrothermal stresses, non-symmetric lay-ups, filled hole conditions, etc. 7Scope and objectives Apart from the aforementioned complex variable formulations, the problem can also be solved by means of the finite element method (FEM) [21]. This method allows a numerical resolution of the problem, whose accuracy will be dependent on the mesh type, size and shape. Other numerical method that could be used is the boundary element method (BEM) [22]. It is worth to recall that a detailed SoA will be included at the beginning of each one of the technical chapters, with the relevant information of the topic under investigation in the chapter itself. 1.4 Scope and objectives An overall picture of the current SoA have been presented. In summary, the existing literature provides different formulations that can be used to analyse composite structures with cutouts, each one with its particular limitations. At this point, it is important to recall that the present research aims to be useful for the first phases of the component’s design. Generally, the analytical or closed form complex variable formulations involve a reduction of set-up and computational time with respect to the numerical approach, as concluded by Sevenois and Koussios [23]. Due to this fact, the analytical or closed-form approach will be pursued during the dissertation. In this context, the present dissertation aims to develop new features that enhance the applicability of the current complex variable formulations. In particular, it is intended to simulate more realistic situations by originally developing the following aspects: • The modelling of a large variety of boundary conditions. The developed methodology should allow the application of non-uniform load and displacement boundary conditions at the edges. Most of the researches in this field only consider remote uniform loads. • The simulation of ply drop-offs and local reinforcements is one of the most innovative features of the dissertation. It enables the possibility of having more than one region with different laminate properties in the component. It is important to note that, to the knowledge of the author, the closed form methodologies on the SoA are restricted to components with a single laminate. • The inclusion of stringers into the model. In this regard, the last Introduction 8 part of the thesis is focused on this original topic that will allow the simulation of stiffened plates. To sum up, the research is mainly focused on those specific aspects of the problem that are not covered by the existing literature. However, the SoA provide a wide basis that should not be ignored. It should be noted that the closed-form methodology is not intended to compete with FE formulations, but to be a useful complement that could be applied at the first stages of the design of components. The closed-form formulations are excellent for the resolution of parametric analysis since they are able to produce an accurate solution in a narrow execution time, with the possibility of being programmed in any programming language (and thus no licensed software is required). It is important to note that there will be three different formulations depending on the loads applied to the component. In particular, the stretching, the bending and the coupled stretching-bending performances are covered by slightly different formulations. As it will be shown below, each one of these formulations will be introduced and developed in separated chapters. Finally, the application of this formulation covers a wide range of components, from standard test coupons to large aeronautic parts. The matter under study could be considered of great interest for the aeronautic industry, which is usually contemplated as a spearhead sector concerning materials and structures. For instance, among many others, some suitable targets in the aerospace sector could be the wing covers as shown in Figure 1.2, or the frames, ribs and spars of the internal structure of every component of the aircraft as in Figure 1.3. The morphology of these components is compatible with the formulation developed in this document. 1.5 Document structure The complete outline of the dissertation is given in this section. In the first place, during Chapter 1 the background, motivations, objectives and scope of the research has been introduced briefly. In addition, a relation of the scientific dissemination planned by the author is also presented. Chapter 2 introduces the basis of the anisotropic elasticity, which is essential to settle the foundations of the research performed in subsequent chapters. In addition, it presents the nomenclature used throughout the 9Document structure whole document. Subsequently, Chapter 3 contains the analysis of composite plates under stretching or membrane conditions. In this chapter, the Lekhnitskii formalism [1] is introduced, since the formulation is based on it. In addition, an innovative method is built and applied to allow the inclusion of interesting functionalities, such as ply drop-offs and local reinforcements. To the knowledge of the author, no developments addressing these features can be found in the literature. Finally, a set of benchmark exercises is solved in order to validate the formulation. Similarly, Chapter 4 addresses to the out-of-plane bending performance of composite plates. The structure of this chapter is similar to the previous one. The functionalities added to the formulation enhance the applicability of the methodology, and are original of this dissertation. At the end of the chapter, some benchmark exercises are solved to validate the tool developed. As a combination of the last two chapters, Chapter 5 describes a different formulation to account for non-symmetric lay-ups. In this case, the stretching and the out-of-plane bending performances will be coupled and must be considered at once. This formulation is faced by making use of a modification of the Lekhnitskii formalism. Once again, some innovative features are added to the formulation in order to increase the applicability of the tool. The tool is validated by solving a set of benchmark exercises at the end of the chapter. Later, Chapter 6 introduces an innovative procedure to consider stiffened composite plates. To the knowledge of the author, there are not similar developments in the peer-reviewed literature. The formulation presented during the previous chapters is enhanced to allow the inclusion of stringers. The model is validated by solving a set of benchmark exercises. Chapter 7 does not present any modification on the formulation developed during the previous chapters. On the contrary, it is conceived to present the results of some parametric analysis launched by using the developed tool. In this way, the reader would be allowed to understand the real capabilities of the formulation developed. At the end of the chapter, the tool is used to analyse a real aeronautic component according to information provided by Airbus. Due to this fact, this chapter is tagged as confidential. Finally, Chapter 8 includes the conclusions from the dissertation and the proposed new developments and researches. Introduction 10 1.6 Scientific dissemination An important part of the research is the dissemination of the results and conclusions, in order to spread the knowledge to the scientific community. To this end, the dissemination activities carried out by the author during these years is collected in this section. In the first place, an article has been published in Composite Structures, which is a JCR indexed peer-reviewed journal ranked in quartile Q1 in the area of Materials science, composites, which is the closest area to the topic of the dissertation. This article by Pastorino et al. [24] presents an innovative methodology for considering composite plates with cutouts and non-uniform lay-ups, by making use of a collocation of points technique together with the least squares method. In addition, the author has presented in several national and international conferences of composite materials, as detailed below: • An article, abstract and presentation were prepared by Pastorino et al. [25] in order to attend to the XI Congreso Nacional de Materiales Compuestos (MATCOMP15), held in July 2015 in Móstoles, Spain. The topic of the research was the analysis of composite plates with cutouts and complicated displacement boundary conditions. • An abstract and a presentation were elaborated by Pastorino et al. [26] for the 19th International Conference on Composite Structures (ICCS19), held in September 2016 in Porto, Portugal. The presentation was focused on the methodology developed for analysing the effect of local reinforcements around the cutouts. • An abstract and a presentation were elaborated by Pastorino et al. [27] for the 3th International Conference on Mechanics of Composites (MECHCOMP3), held in July 2017 in Bologna, Italy. In this case, the topic of the research was to present a closed-form formulation that allows the resolution of panels with non-uniform lay-ups. The author has also presented at an internal conference organised by Airbus: • The PhD was introduced with a poster and a three-minutes format presentation in the Airbus PhD Day 2016, held in October 2016 in Getafe, Spain. 11 Scientific dissemination Apart from the aforementioned publications and conferences, the submission of novel articles is an on-going process. In particular, it is expected to publish at least three additional articles in the near future with the following tentative titles: • “Closed-form methodology for the structural analysis of composite plates with cutouts and non-uniform layup under bending loads”. This article will be the natural extension of the previous one (Pastorino et al. [24]), and will collect the developments, results and conclusions that are exposed in Chapter 4. • “Closed-form methodology for the structural analysis of composite plates with cutouts and non-uniform layup for unsymmetric laminates”. This publication will be focused on the analysis of unsymmetric laminates that is developed in Chapter 5. • “Closed-form methodology for the structural analysis of stiffened composite plates with cutouts”. The article will result in an innovative contribution to the current SoA, by allowing the structural analysis of plates with stringers. It will be based on the findings presented in Chapter 6. Last but not least, a research stay of three months has been undertaken at the National Cheng Kung University (NCKU), under the guidance of Professor Chyanbin Hwu. The department is one of the most active research groups in the field of interest. During this stay, a part of the investigation presented in Chapter 6 was developed. Chapter 2 Linear anisotropic elasticity The objective of the present chapter is to introduce the basic notation and fundamental equations of the linear anisotropic elasticity of three dimensional elastic bodies, which will be the basis of the formulation developed in this dissertation. In this regard, outstanding references on this topic can be found in literature, for instance in Sokolnikoff [28], Timoshenko and Goodier [29], Jones [30], Vinson and Sierakowski [31], Vasiliev and Morozov [32], Hwu [7], and Gould [33] among many others. By definition, the structural bodies possess the property of elasticity, which causes the deformation of the material under external forces [29]. During the present investigation it is assumed a perfect linear elastic behaviour of the material. Or in other words, the elastic body returns to its original undeformed state when the forces are removed. In addition, the elastic body under study is assumed to be completely homogeneous. Although materials are normally composed by perfectly identifiable micro-constituents (specifically in composite materials as shown in Figure 2.1), it is a common hypothesis to consider them as homogeneous from a macroscopic point of view. This chapter starts with the formulation of the three-dimensional elastic problem in Section 2.1. The Section 2.2 simplifies the model to account for two-dimensional bodies, which is finally used to determine the laws of the elasticity of layered composites, exposed in Section 2.3. Linear anisotropic elasticity 20 Summary Finally, a summary of the material types as a function of its stiffness matrix is found in Table 2.1. Table 2.1: Number of independent elastic constants for each material type. Material type Independent elastic constants Non-zero terms of C Anisotropic 21 36 Monoclinic 13 20 Orthotropic 9 12 Transversely isotropic 5 12 Isotropic 2 12 2.1.4 Boundary conditions The elastic three-dimensional problem is governed by the equilibrium equations derived in (2.2), the strain-displacement relation for small strains, as presented in (2.4), together with the constitutive law, that relates the stresses and strains as exposed in (2.6). Finally, a solution to the three dimensional elastic problem is reached by the application of the boundary conditions into the set of differential equations. The elastic direct boundary conditions are usually divided into three groups, depending on its nature: (1) load, (2) displacement, or (3) mixed boundary conditions. (1) Load (or Neumann) boundary conditions. A boundary ΓTis subjected to a load condition provided that a prescribed traction is applied. Hence, load boundary conditions require that t=   t1 t2 t3    =   ˜ t1 ˜ t2 ˜ t3    =˜ t, along boundary ΓT.(2.16) (2) Displacement (or Dirichlet) boundary conditions. On the con- 21 Elasticity of two-dimensional bodies trary, if a prescribed displacement at the boundary is given, u=   u1 u2 u3    =   ˜u1 ˜u2 ˜u3    =˜u, along boundary ΓU.(2.17) (3) Mixed boundary conditions. It takes place if a point belonging to a region ΓMcombines both Dirichlet and Neumann boundary conditions in orthogonal directions. Logically, conditions on tand ucannot be set at the same point and direction, since it will give rise to an overconstrained structure. However, it is possible to apply a load condition together with a displacement boundary condition at the same point with different direction. Without loss of generality, it can be considered that the boundary conditions at a point on the external surface of a control volume can be given in terms of loads (Neumann condition) and/or in terms of displacements (Dirichlet condition). 2.2 Elasticity of two-dimensional bodies The consideration of the elastic problem as two dimensional implies some simplifications in the constitutive relations. The following scenarios can be considered: (1) Plane strain. The plane strain condition is used to simplify the solution of infinite three-dimensional prism-shaped bodies with constant section. It can be assumed that the longitudinal displacement u3van- ishes, and that the displacement’s components are independent of the x3coordinate, ui=ui(x1, x2), and u3= 0,for i = 1,2, (2.18) which implies that the out-of-plane strains are strictly zero, γ13 =γ23 =ε33 = 0.(2.19) (2) Generalized plane stress. The plane stress condition considers a two dimensional approach equivalent to the plane strain but assuming that the out-of-plane stresses σi3vanish (instead of the strains εi3). Linear anisotropic elasticity 22 Normally, the plane stress condition is hardly applicable to a physical problem. However, a less restrictive hypotheses result from the generalized plane stress condition, which allows a simplified formulation for thin plates in which all the loads act normal to the thickness direction, see Figure 2.5. x3 x2 x1 h 0 tn Figure 2.5: Generalized plane stress. The mid-plane x3= 0 is a plane of symmetry. The plane stress hypotheses can be alleviated by introducing the notion of the mean values of stresses σij, strains εij, and displacements ui, σij =1 h∫h/2 −h/2 σijdx3.(2.20a) εij =1 h∫h/2 −h/2 εijdx3.(2.20b) ui=1 h∫h/2 −h/2 uidx3.(2.20c) Hence, the dependence with the x3coordinate is suppressed. And given that the plate mid-plane x3= 0 is a plane of symmetry, ui=ui(x1, x2), and u3= 0,for i = 1,2.(2.21) With regard to the stresses, the distribution of the out-of-plane shear σi3will be antisymmetric in the thickness direction due to the symmetry of x3= 0, and thus σi3= 0,for i = 1,2.(2.22) In addition, the normal out-of-plane shear stress σ3will vanish at the two external faces due to the boundary conditions. And, if the plate is thin enough, σ33 can be considered negligible as well inside the plate, σ33 ≃0.(2.23) 23 Elasticity of two-dimensional bodies 2.2.1 Stress and equilibrium It is interesting to recall the definition of the stress vector by the Cauchy’s formula (2.1). In this case, the x3coordinate can be disregarded from the calculations, and thus the stress vector diagram gets cleaner, as shown in Figure 2.6. Note that the angle formed by x1and sis denoted by θ. x2 x1 0 n t t1 t2 22 12 12 11 P -ts s tn  Figure 2.6: Stress vector acting on an arbitrary plane with outer normal n in 2D. With regard to the equilibrium equations, the expression of (2.2) can be used, since it remains unchanged. The same applies to the boundary conditions, whose expressions are given by (2.16) and (2.17). 2.2.2 Compatibility The strains are given by the three-dimensional relations (2.4) and (2.5) under the hypothesis of small strains. In the condition of plane strain, the out-of-plane strains will vanish according to (2.19). The in-plane strains are thus given by, ε11 =∂u1 ∂x1 , ε22 =∂u2 ∂x2 , ε12 =1 2(∂u1 ∂x2 +∂u2 ∂x1),(2.24) Hence, the compatibility equation for the two-dimensional plane strain condition can be reduced to, ∂2ε11 ∂x2 2 +∂2ε22 ∂x2 1−2∂2ε12 ∂x1∂x2 = 0.(2.25) Linear anisotropic elasticity 24 In the case of generalized plane stress, the out-of-plane strains do not vanish. However, the two-dimensional approach is interested in the in-plane strains only, and hence making use of the mean strains through thickness of (2.20b), the in-plane compatibility equation for generalized plane stress is defined as ∂2ε11 ∂x2 2 +∂2ε22 ∂x2 1−2∂2ε12 ∂x1∂x2 = 0.(2.26) 2.2.3 Material behaviour The constitutive relations can be simplified under the aforementioned twodimensional assumptions. The present dissertation is conceived specifically to deal with layered composite materials which are composed by a set of plies, and a single ply can be simulated as two-dimensional orthotropic material under the assumption of generalized plane stress condition. Thus, let the development of the two-dimensional formulation to start directly on the assumption of a generalized plane stress condition for an orthotropic material, skipping this way its development for the most general case (anisotropic material). Orthotropic material As from the generalized plane stress condition σ13 =σ23 = 0,σ33 ≃0, the out-of-plane strains for an orthotropic material will be ε33 =S13σ11 +S23σ22,ε13 = 0,ε23 = 0.(2.27) In addition, the stress-strain relation (2.13) can be reduced to    σ11 σ22 σ12    =  C11 C12 0 C12 C22 0 0 0 C66     ε11 ε22 ε12    .(2.28) Thus, Cbecomes a 3×3matrix for this particular case. The engineering constants are used to determine C, and are usually obtained from mechanical testing of the material. Without going into detail, the stiffness matrix for an orthotropic material under generalized plane stress condition as a function of its engineering constants is given by [7, 30], C=S−1=  1/E1−ν12/E10 −ν12/E11/E20 00 1/G12   −1 .(2.29) 25 Classical Laminate Theory Where E1and E2are the elastic modulus in longitudinal and transversal directions respectively. G12 is the in-plane shear modulus. ν12 is the Poisson’s ratio. Isotropic material Moreover, it is interesting to describe the two dimensional behaviour for isotropic materials under the hypotheses of generalized plane stress. In this case, the compliance matrix becomes C=S−1=  1/E −ν/E 0 −ν/E 1/E 0 0 0 2 (1 + ν)/E   −1 .(2.30) 2.3 Classical Laminate Theory The most general formulation for anisotropic materials have been presented in the last sections. However, it is required to direct the research towards the field of interest, which is the application to laminated composite plates. One of the most extended formulations to describe the behaviour of laminated composite plates is called the Classical Laminate Theory (or just CLT), see [7, 30, 31]. The CLT is based on the Kirchhoff-Love theory for thin plates, whose assumptions are listed below [34]: (1) Straight lines normal to the mid-surface remain straight after deformation. (2) Straight lines normal to the mid-surface remain normal to the midsurface after deformation. (3) The thickness of the plate does not change during deformation. The aforementioned two-dimensional plate model was developed by Love [35] using the assumptions proposed by Kirchhoff, and it is equivalent to the Euler-Bernoulli beam model. The listed hypotheses are represented in Figure 2.7, where ψ2=∂u3 ∂x2 , and ψ1=∂u3 ∂x1 .(2.31) Linear anisotropic elasticity 26 u3 x2 x3 2 u3 x1 x3 1 x3 x2 x1 00 Figure 2.7: Kirchhoff-Love two-dimensional plate model. Under these hypotheses the in-plane displacements become u1=u0 1−x3 ∂u3 ∂x1 , u2=u0 2−x3 ∂u3 ∂x2 ,(2.32) where u0 1=u0 1(x1, x2)and u0 2=u0 2(x1, x2)are the mid-plane displacements (x3= 0). From expression (2.32), in-plane strains are defined by ε1=∂u0 1 ∂x1−x3 ∂2u3 ∂x2 1 =∂u0 1 ∂x1−x3κ1,(2.33a) ε2=∂u0 2 ∂x2−x3 ∂2u3 ∂x2 2 =∂u0 2 ∂x2−x3κ2,(2.33b) γ12 =∂u0 2 ∂x1 +∂u0 1 ∂x2−2x3 ∂2u3 ∂x1∂x2 =∂u0 2 ∂x1 +∂u0 1 ∂x2−x3κ12,(2.33c) where κij are the components of the curvature vector of the deformed plate. Moreover, (2.33) can be written in matrix notation as, ε ε ε=ε0 ε0 ε0−x3κ κ κ, (2.34) where, ε ε ε0=   ε0 1 ε0 2 γ0 12    =   ∂u0 1/∂x1 ∂u0 2/∂x2 ∂u0 2/∂x1+∂u0 1/∂x2    ,(2.35a) κ κ κ=   κ1 κ2 κ12    =   ∂2u3/∂x2 1 ∂2u3/∂x2 2 2∂2u3/∂x1∂x2    .(2.35b) 27 Classical Laminate Theory In addition, the assumptions of u1(x1, x2, x3)and u2(x1, x2, x3)of (2.32) prove that the transverse shear strains are null, i.e. γ13 =∂u1 ∂x3 +∂u3 ∂x1 = 0, γ23 =∂u2 ∂x3 +∂u3 ∂x2 = 0.(2.36) Finally, from the third hypothesis it is clear that the out-of-plane normal strain will also vanish, ε3=∂u3 ∂x3 = 0.(2.37) As a summary, Kirchhoff-Love’s plate theory does not consider out-of-plane normal and shear strains. This assumption is accurate for thin plates, i.e. the ratio of the thickness to the characteristic length shall be below 1/20. In laminated composites, Kirchhoff-Love assumptions are usually addressed also as the Classical Laminate Theory (CLT). At this point, it is important to consider the particularities of the layered composite materials. Thus, a multi-layered plate whose mid-plane is coincident with the x3= 0 plane is considered. The plate is composed by K stacked plies of a given thickness, with a total thickness of h, as shown in Figure 2.8. This diagram will be useful subsequently for the definition of the material behaviour. midplane Ply K Ply k Ply 1 x3 h (K) x3 (K-1) x3 (k) x3 (k-1) x3 x1 x3 (0) Figure 2.8: Stacking sequence of a laminated composite plate. On the basis of the multilayered composite material of Figure 2.8, another hypothesis must be accomplished according to the CLT: Each ply of the laminate is perfectly bonded to its adjacent plies. Thus, no sliding nor separation between adjacent plies is permitted. Linear anisotropic elasticity 28 2.3.1 Resultant forces and moments Let Nij be the resultant forces through thickness associated to the stress component σij, Nij =∫h/2 −h/2 σijdx3= K ∑ k=1 (∫x(k) 3 x(k−1) 3 σk ijdx3).(2.38) Similarly, the resultant moments per unit length through thickness are denoted by Mij: Mij =∫h/2 −h/2 σijx3dx3= K ∑ k=1 (∫x(k) 3 x(k−1) 3 σk ijx3dx3).(2.39) x2 Q1+dQ1 M1+dM1 M12+dM12 M12+dM12 M2+dM2 Q2+dQ2 q M12 Q2 M2 Q1 M12 M1 dx1dx2 x3 x1 N2+dN2 N12+dN12 N12+dN12 N1+dN1 N12 N1 N12 N2 Figure 2.9: Diagram of equilibrium of forces and moments on an infinitesimal plate. The diagram of forces and moments at an infinitesimal plate is shown in Figure 2.9. Proceeding as in Section 2.1.1, the equilibrium equations in terms of resultant forces and moments can be achieved. In the first place, 29 Classical Laminate Theory the equilibrium of forces in x1and x2directions gives rise to, ∂N1 ∂x1 +∂N12 ∂x2 = 0,(2.40a) ∂N12 ∂x1 +∂N2 ∂x2 = 0,(2.40b) by considering that no body forces fiact in the infinitesimal domain. Secondly, following the same procedure with the equilibrium of forces in x3 direction, plus the equilibrium of moments, ∂Q1 ∂x1 +∂Q2 ∂x2 +q= 0,(2.41a) ∂M1 ∂x1 +∂M12 ∂x2−Q1= 0,(2.41b) ∂M12 ∂x1 +∂M2 ∂x2−Q2= 0.(2.41c) Thus, the equilibrium equations for stretching (2.40) and bending (2.41) of a plate have been obtained. These expressions will be used hereinafter. 2.3.2 Material behaviour Single lamina behaviour A single lamina (k) could be considered as a 2D orthotropic material under the hypotheses of plane stress, and hence, its behaviour is given by (2.28). Normally, the stiffness matrix of each single lamina is denoted by Ck. Rewriting the stiffness matrix (2.29) using the engineering constants for the case of plane stress, the expression of Ckis found to be Ck=  1/E1−ν21/E20 −ν12/E11/E20 0 0 1/G12   −1 .(2.42) Ckis given in lamina (or local) coordinates, which are denoted by xj′. However, the local coordinates might not be coincident with the geometrical coordinates, which will be denoted by xj(see Figure 2.10). Subsequently, it is required to perform a transformation of coordinates in order to get the stiffness matrix in geometrical coordinates (xj), which will Plate stretching analysis 36 nite composite plate under biaxial loading. Unlike the analytical solution presented by Lekhnitskii [1], a eight order polynomial defines the stress distribution as a function of the distance to the hole, but just in a straight line in radial direction, perpendicular to one principal load direction. 3.1.2 Analysis of finite plates Thus far, only formulations assuming an infinite medium have been analysed. Actually, these models are conceived to assume that the plate size is infinite with respect to the cutout size, which is an appropriate assumption whenever the ratio between the characteristic length of the plate and the cutout remains above 10, as stated by Chauhan and Sharma [43]. Hence, there are many investigations that provide a solution for anisotropic finite plates with holes. The most simple approach to include this effect is to apply a correction factor to the model under the infinite medium assumption. Naturally, the computational time remains barely unaffected, although the accuracy obtained with this method is limited. The most extended formulation using a finite-width correction factor (FWC) was proposed by Tan [2]. On this basis, Xiong [44] proposed another FWC model for problems considering an elastic inclusion. An alternative procedure to consider a finite medium was conceived originally by Ogonowski [36]. In this case, a Laurent series is used to account for the external boundary on the definition of the stress function. Subsequently, a set of boundary collocation points is distributed along the external boundaries of the domain, and in conjunction with the least squares method (LSQ), the best fit solution is obtained. This method is hereinafter referred as the boundary collocation method (BCM). Logically, it is sensitive to the selection of points. As a counterpart, this method delivers important advantages, given that it is quite straightforward to implement, versatile, and computationally efficient. From then on, many authors have made use of the BCM introduced by Ogonowski [36] in order to consider a finite medium on the basis of the Lekhnitskii formulation. For instance, Lin and Ko [17] applied the Laurent series approximation to predict the failure of notched plates, and subsequently compared the results with respect to experimental ones. Xu et al. [45] introduced the Faber polynomial expansion in order to allow a smooth outer contour of the plate, and consequently a more accurate 37 State of the art for stretching analysis solution at the external boundaries. However, the paper warns the reader about the lengthy and complicated process required to implement the Faber series, and finally uses the approach given by Ogonowski [36]. In this paper, the effect of the hole size, layup and a variety of loading conditions on a finite orthotropic medium with an elliptical cutout are analysed. In a latter paper, Xu et al. [46] adapted the methodology to account for several unloaded cutouts under stretching conditions. And a few years later, Xu et al. [47] used the same formulation for loaded cutouts, and exercises of plates with several pin-loaded holes were analysed. On this basis, Zheng and Xu [48] simulated the problem of a finite plate with an elastic elliptical inclusion. In a similar manner, Hufenbach and Kroll [49] applied the BCM to solve a finite plate model under mechanical and hygrothermal loads. The results obtained are successfully compared with respect to the FE results. Moreover, Hufenbach and Zhou [50] developed a formulation, likewise based on the BCM, in order to consider finite plates with elliptical cutouts and an elastic inclusion. The solution is found by assuming the conditions of continuous normal stress and displacements at the boundary. The Lekhnitskii formalism together with the BCM have been using by many other authors to consider finite plates with cutouts under specific conditions. In this regard, another contribution is given by Chauhan and Sharma [43], who derived the solution for quasi-rectangular shaped holes under in-plane loading condition, by using a complex mapping function, based on the works from Ukadgaonker and Rao [20] and Sharma [51] (for isotropic plates). Recently, a summarised version of the formulation developed throughout the present chapter have been published in peer-reviewed literature, see Pastorino et al. [24]. In this paper, the Lekhnitskii formalism together with the BCM is used to consider a domain composed by several regions with different laminates. The model is validated through the resolution of benchmark exercises compared with respect to FE results. In addition, some parametric analysis are launched and investigated in order to exhibit the possibilities of the formulation. Finally, Xiong [18] presented a model for solving multi-fastener joints under uniaxial loading conditions based on Lekhnitskii formalism. The fasteners are modelled as elastic beams. In contrast to the latter ones, the minimum potential energy theorem (MPE) is used to consider the plate finite boundaries. Basically, the total potential energy (the strain energy plus the potential energy) over the whole domain is derived to get the mini- Plate stretching analysis 38 mum as a function of the model unknowns. In this way, the energy of the system is minimised and the correct solution is found. Subsequently, an enhanced formulation was provided by Xiong [52] in order to introduce biaxial loads. In this case, the author highlights that it is required to use the superposition principle if a biaxial loading is considered. 3.1.3 General aspects At this point, it is necessary to undertake a reflection about the meaning of analytical. The formulations assuming an infinite plate, i.e. Lekhnitskii [1] and Stroh [14] formalisms, are pure analytical for most of the conventional cases of loading, naturally under their particular hypotheses, i.e. the solution obtained is strictly exact under the assumptions made. In contrast, all the methodologies considering that the plate is finite (e.g. Ogonowski [36], Lin and Ko [17], Xu et al. [45], and Xiong [18]), require the use of infinite series to solve the problem. As no infinite terms can be used, the series must be truncated to the Nth-term, and the solution should not be called analytical. That is the reason why throughout this document, the formulation developed is usually referred as semi-analytical or simply closed-form. Another important aspect is which formalism, the Lekhnitskii [1] or the Stroh [14], will be used for the methodology development. A thought on this topic is given at the conclusion of the review from Sevenois and Koussios [23], according to which Lekhnitskii and Stroh formalisms are more suitable for static and dynamic analyses respectively. Although the foregoing conclusion could be questioned, a compelling reason to use Lekhnitskii over Stroh is that the analysis of finite plates has been successfully overcome in many articles on the basis of the first: Lin and Ko [17], Xiong [18], Ogonowski [36], Xu et al. [45], Hufenbach and Kroll [49]; but not with the latter formulation, at least to the knowledge of the author. Even if the development of a finite plate theory on the basis of Stroh formalism would be straightforward, it does not justify the effort, since the elegant notation will become as tedious as the Lekhnitskii formulation due to the use of truncated series. Thus, inasmuch as most of the previous works relevant to the topic are developed by using the Lekhnitskii formalism, and given that no major improvements are expected by using Stroh,Lekhnitskii is selected to be used as a basis of the methodology in this dissertation. 39 Lekhnitskii formalism for plate stretching To conclude on this point, there have been presented three different methodologies to deal with the finite plate problem. In the first place, using an experimental correction factor is not conceived, since the accuracy of the solution can be highly reduced, and experimental data is required. The other two options are to use the BCM or the MPE. Although the MPE, attributed to Xiong [18], is specially elegant and does not require collocation, the BCM is far more widespread than the MPE. This is evidenced in the present section, where most of the references given are based on the work from Ogonowski [36]. In addition, the BCM provides a highly versatile solution. Thus, the BCM is selected as the approach to consider finite plates. 3.2 Lekhnitskii formalism for plate stretching In this section the Lekhnitskii formalism will be introduced on the basis of the existing literature, see Lekhnitskii [1], Hwu [7]. First of all, some particularities of the formulation are mentioned: 1. The Lekhnitskii formalism is not only valid for anisotropic plates with cutouts, but it is also used to solve many different problems associated with anisotropic materials, e.g. concentrated forces in infinite medium, analysis of cracks, buckling of plates, curved beams, etc. 2. The solution for plate structures is obtained under the hypotheses of generalised plane stress, whose implications were explained in Chapter 2. 3. The formulation is initially conceived for infinite anisotropic plates, i.e. the plate size is assumed to be infinite with respect to the distinctive feature to be analysed, whether it is a crack, a concentrated load, or a cutout. 4. In this chapter, the plate is assumed to work on a pure stretching stress state. It means that the prescribed loads are strictly contained by the plate middle plane, and that no stretching-bending coupling takes place, i.e. B= 0. 3.2.1 Complex variable formalism Now, the mathematical development is exposed. It should be noted that some of the expressions of Chapter 2 will be rewritten in order to consider Plate stretching analysis 40 a stretching stress state, according to which Mij = 0 and B= 0. First, an Airy stress function φ=φ(x1, x2)is introduced into the equilibrium of forces (2.40). In this way, equilibrium is automatically fulfilled for any φfunction provided that, N1=∂2φ ∂x2 2 ,(3.1a) N2=∂2φ ∂x2 1 ,(3.1b) N12 =−∂2φ ∂x1∂x2 .(3.1c) Where the body forces fihave been intentionally dropped. Subsequently, the strains are given from (2.33). Due to the stretching nature of the problem, the displacements will be constant through the thickness, and hence κi= 0. Therefore, without loss of generality it is assumed that u0 j=uj. Combining the three equations (2.33), it is obtained that, ∂2ε1 ∂x2 2 +∂2ε2 ∂x2 1−∂2γ12 ∂x1∂x2 = 0.(3.2) The constitutive equation is given by (2.48). Let aij be the terms of the inverse of the extensional stiffness matrix A. Assuming that both stretching and bending are decoupled and that ε0 ij =εij, (2.48) could be reduced to,    ε1 ε2 γ12    =  a11 a12 a16 a12 a22 a26 a16 a26 a66     N1 N2 N12    .(3.3) Substituting strains by resultant forces in (3.2) by making use of (3.3), and at the same time replacing the resultant forces by the Airy function according to (3.1), the characteristic equation is obtained, a22 ∂4φ ∂x4 1−2a26 ∂4φ ∂x3 1∂x2 + (2a12 +a66)∂4φ ∂x2 1∂x2 2− 2a16 ∂4φ ∂x1∂x3 2 +a11 ∂4φ ∂x4 2 = 0.(3.4) Which is a fourth-order homogeneous partial derivative equation. This equation in partial derivatives can be integrated by using four first-order operators as follows (see [1, 7]), F1(F2(F3(F4(φ)))) = 0,(3.5) 41 Lekhnitskii formalism for plate stretching where Fjdesignate the operation Fj=∂ ∂x2−µj ∂ ∂x1 ,(3.6) and µjare the roots of the characteristic equation a11µ4 j−2a16µ3 j+ (2a12 +a66)µ2 j−2a26µ+a22 = 0.(3.7) A total of four µjroots are obtained from (3.7), which are complex and conjugated two-by-two, or in other words µ3=µ1and µ4=µ2(see [1]). Thus, µj+2 =µj,j= 1,2.(3.8) Dealing with isotropic elastic materials is usually a particular solution of the anisotropic elastic theory with large simplifications. However, in this case, the isotropy does not simplify the formulation, and implies that a degenerate solution arises, in which µ1=µ2=iand µ1=µ2=−i. This issue is usually avoided by conferring the isotropic material a quite small perturbation, in order to turn it into an anisotropic material (at least mathematically). Proceeding as in [1], integrating with the operators one at a time on the basis of (3.5), F4(φ) = g3,F3(g3) = g2,F2(g2) = g1.(3.9) And finally, F1(g1) = ∂g1 ∂x2−µ1 ∂g1 ∂x1 = 0.(3.10) By integration of (3.10) it is clear that the solution must fulfil, g1=f1(x1+µ1x2).(3.11) Where f1is an arbitrary function dependent on x1+µ1x2. Similarly, the step-by-step integration of (3.9) results in the general solution for (3.4) (see [1, 7]), φ(x1, x2) = 4 ∑ j=1 φj(zj),(3.12) where φjare arbitrary functions dependent on zj, which is a complex variable defined as, zj=x1+µjx2.(3.13) Plate stretching analysis 42 Now, let Φj=dφj dzjbe the derivative of each one of the φjfunctions. In addition, considering that µjare conjugated two-by-two, (3.12) can be rewritten into Φ (x1, x2) = 4 ∑ j=1 Φj(zj) = 2Re {2 ∑ j=1 Φj(zj)},(3.14) where Re represents the real part of a complex number. Note that Φjhave been intentionally defined as Φ3(z3) = Φ1(z1)and Φ4(z4) = Φ2(z2). In this way, Φ (x1, x2)will always be real. 3.2.2 Displacements and resultant forces from Φj The resultant forces are extracted by substituting (3.14) into (3.1), see [1], N1= 2Re {2 ∑ j=1 (µ2 j dΦj dzj)},(3.15a) N2= 2Re {2 ∑ j=1 (dΦj dzj)},(3.15b) N12 = 2Re {2 ∑ j=1 (−µj dΦj dzj)}.(3.15c) Furthermore, strains are obtained through substitution of the resultant forces obtained in (3.15) into the constitutive law (3.3). And the calculated strains are substituted into (2.33) in order to get the expression of the displacements, u1= 2Re {2 ∑ j=1 (pjΦj)},(3.16a) u2= 2Re {2 ∑ j=1 (qjΦj)},(3.16b) where pjand qjare given by pj=a11µ2 j+a12 −a16µj,(3.17a) qj=a12µj+a22 µj−a26.(3.17b) 43 Cutout geometry 3.3 Cutout geometry This section is devoted to introduce the geometrical particularities and the variables associated with the cutout that will be used throughout the whole document. 3.3.1 Geometrical aspects It has been shown at the beginning of the present chapter, in Section 3.1, that there are methodologies based on Lekhnitskii formalism that consider many topologies of the cutouts. However, the scope of the dissertation is to develop a methodology that ideally considers, at least, elliptical and circular cutouts. Thus, the extension to other hole topologies will not be dealt in this document, and can be achieved by using a customisable conformal mapping, as in [20]. A plate with an arbitrary external shape is considered (ΓE), and an elliptical shaped internal cutout (ΓC) as in Figure 3.1. The ellipse major and minor semi-axes are denoted by aand b. Note that the ellipse axes are coincident with the local coordinate system (x1, x2)and it is rotated by an angle α with respect to the global coordinate axis (x, y). Moreover, the centre of the ellipse is defined by the point Cand is located at (xC, yC)in global coordinates. x y x2 x1 x1 x2 s n s n     a b 0' 0 0' C E C Ω E Figure 3.1: Component geometry and cutout parameters definition. Plate stretching analysis 44 The points on the boundary of the ellipse are determined according to, x1(ψ) = acos ψ, (3.18a) x2(ψ) = bsin ψ. (3.18b) Here, ψis the eccentric anomaly of the ellipse. The counter-clockwise angle between the direction of sand the positive x1is denoted by the angle θ, while the polar angle χis used for the representation of results. The relation between χand ψangles is tan χ=b atan ψ. (3.19) Furthermore, the angle θ, which is tangent to the contour, and ψare related by ρcos θ=−asin ψ, (3.20a) ρsin θ=bcos ψ, (3.20b) ρ2=a2sin2ψ+b2cos2ψ. (3.20c) The relation between the global (x, y)and the local (x1, x2)axes is given by the translation to the ellipse center and the rotation of α, hence {x y}={xC yC}+[cos α−sin α sin αcos α]{ x1 x2}.(3.21a) Finally, note that Ωis the domain of the component. The contour of the cutout is denoted by ΓC, and the external by ΓE. Both contours are swept by the arc-length s, which is directed counter-clockwise at the internal and clockwise at the external boundary. Tangential (s) and normal (n) directions comprise a coordinate axis that follows the boundary, nis always directed outward the component, s= (cos θ, sin θ),(3.22a) n= (−sin θ, cos θ).(3.22b) It must not be confused the direction swith the arc-length s, being a vector and a scalar respectively. In addition, the relation between x1,x2and sis shown below, dx1 ds= cos θ,dx2 ds= sin θ. (3.23) 45 Cutout geometry 3.3.2 Conformal mapping According to the procedure developed by Lekhnitskii [1], a conformal mapping is carried out in order to get a simple expression of the cutout geometry. This is achieved by a further transformation of the complex plane zj introduced in (3.13). To this end, the variables ζj=ζj(zj)are introduced to transform the complex domains zj, zj=a−ibµj 2ζj+a+ibµj 2ζ−1 j,(3.24) whose inverse relation is given by ζj(zj) = zj±√z2 j−a2−b2µ2 j a−ibµj .(3.25) It is important to note that the solution is not unique provided that a ± sign is ahead of the square root. The selection of the proper root requires a deep investigation on the topic, in this case, the sign selection algorithm developed by Koussios [53] is used. The conformal mapping presented transforms the external region of the elliptical hole into the external region of an unitary radius circle. Or in other words, ζjat the hole contour can be defined as a function of ψangle as ζ1=ζ2=σ=eiψ = cos(ψ) + isin(ψ).(3.26) For instance, the conformal mapping of a square plate with 200 mm edge with an elliptical cutout with a= 50 mm and b= 30 mm is shown in Figure 3.2. Moreover, given that the definition of the derivatives of ζjwith respect to zjand vice versa will be useful in subsequent equations, its expressions are given by dzj dζj =a−ibµj 2−a+ibµj 2ζ−2 j,(3.27a) ζ′ j=dζj dzj = 1±zj √z2 j−a2−b2µ2 j a−ibµj .(3.27b) Plate stretching analysis 52 boundary conditions at the cutout edge can be reworked in order to reduce the number of unknowns, and at the same time, avoid the collocation of points at this ΓCboundary. A similar procedure can be found in the peerreviewed literature although no details on the intermediate steps are given, e.g. Hufenbach and Kroll [49], and Hufenbach and Zhou [50]. Following the methodology from Lekhnitskii [1], the prescribed loads at the hole boundary can be approached by making use of Fourier series (truncated to the N-th power of eiψ). Thus, the left-hand side of (3.32), referred to the internal boundary, can be expressed by ∫ψ 0 ˜ T1 ds dψdψ=−P1 2πψ+β0+ N ∑ n=1 (βneinψ +βne−inψ) ψ 0 ,(3.42a) −∫ψ 0 ˜ T2 ds dψdψ=P2 2πψ+α0+ N ∑ n=1 (αneinψ +αne−inψ) ψ 0 ,(3.42b) where αnand βnare the coefficients of the Fourier series. Therefore, making use of (3.37) and (3.42), the boundary condition (3.32) at the hole boundary can be rewritten as follows, 2 ∑ j=1 [Φj(σ) + Φj(σ)]=P2 2πψ+α0+ N ∑ n=1 (αnσn+αnσ−n),(3.43a) 2 ∑ j=1 [µjΦj(σ) + µjΦj(σ)]=−P1 2πψ+β0+ N ∑ n=1 (βnσn+βnσ−n).(3.43b) At this point, the two expressions from (3.43) can be combined together in order to find a relation between Φ1(σ)and Φ2(σ)at the cutout boundary, Φ2(σ) = δ1Φ1(σ) + δ2Φ1(σ) + δ3(−P1 2πψ+β0+ N ∑ n=1 (βnσn+βnσ−n))+ δ4(P2 2πψ+α0+ N ∑ n=1 (αnσn+αnσ−n)),(3.44) where the coefficients δkare given by, δ1=µ2−µ1 µ2−µ2 ,(3.45a) 53 General methodology for plate stretching δ2=µ2−µ1 µ2−µ2 .(3.45b) δ3=1 µ2−µ2 ,(3.45c) δ4=−µ2 µ2−µ2 ,(3.45d) The condition in (3.43) must be accomplished for each ψat the cutout boundary. Subsequently, this condition can be generalised for the whole domain Ω, which can be achieved through the substitution of σby ζ2, and Φ1by its definition of (3.28), Φ2(ζ2) = δ1C01 +δ2C01+ N ∑ n=1 [δ1Cn1ζ−n 2+δ2Cn1ζn 2+δ1C∗ n1ζn 2+δ2C∗ n1ζ−n 2]+ δ3(β0+ N ∑ n=1 (βnζn 2+βnζ−n 2))+ δ4(α0+ N ∑ n=1 (αnζn 2+αnζ−n 2))+log terms.(3.46) At first glance, (3.46) reduces the number of unknowns to half of the initial, i.e. coefficients Cn2and C∗ n2will no longer appear in the equations due to replacement of Φ2by using (3.46). Thus, grouping the terms of the same kind (i.e. ζn j,ζ−n j,ln(ζj)) it is concluded that C02 =δ1C01 +δ2C01 +δ3β0+δ4α0,(3.47a) C12 =δ1C11 +δ2C∗ 11 +δ3β1+δ4α1,(3.47b) C∗ 12 =δ1C∗ 11 +δ2C11 +δ3β1+δ4α1,(3.47c) . . . Cn2=δ1Cn1+δ2C∗ n1+δ3βn+δ4αn,(3.47d) C∗ n2=δ1C∗ n1+δ2Cn1+δ3βn+δ4αn.(3.47e) Plate stretching analysis 54 External boundary The collocation points are placed along the external contour of the domain. Two boundary conditions can be derived per each one of the collocation points. These conditions can be defined in terms of prescribed tractions (3.34), prescribed displacements (3.36) or a combination (e.g. displacement constrained in normal direction and shear traction applied). The final expressions for each condition are found by replacing Φ1from (3.28) and Φ2from (3.46). The most used conditions are shown in Table 3.1. Table 3.1: Commonly used stretching boundary conditions for external edges. Description ncondition scondition Free edge Tn= 0 Ts= 0 Retained edge un= 0 us= 0 Retained edge (in normal dir.) un= 0 ts= 0 Prescribed edge traction Tn=˜ TnTs=˜ Ts Prescribed displacement un= ˜unus= ˜us 3.4.3 Collocation points and least squares First, it should be recalled that no collocation of points is required at the cutout boundary ΓC, since an analytical expression relating Φ2and Φ1has been obtained. Secondly, it should be mentioned that the number of collocation points placed along the external boundary ΓEmust be selected to ensure that the number equations outnumber the unknowns. An assessment on the number and distribution of the points will be carried out later in this chapter. Now, the main objective is to determine all the unknowns, and the easiest way is to create a linear system of equations (LSE). However, taking a closer look to (3.34) and (3.36), it is clear that the Re function presents a conflict with the LSE, and must be removed. Hence, the coefficient’s real and imaginary parts will be handled separately, and a vector of unknowns 55 General methodology for plate stretching can be defined by ˜ C1=                  Re {C01} Im {C01} Re {C11} . . . Re {Cn1} Im {Cn1} . . . Re {CN1} Im {CN1}                  ,˜ C2=                  Re {C∗ 11} Im {C∗ 11} Re {C∗ 21} . . . Re {C∗ n1} Im {C∗ n1} . . . Re {C∗ N1} Im {C∗ N1}                  , with ˜ C=(˜ C1 ˜ C2).(3.48) Note that every Cnj with j= 2 have been dropped on ˜ Cdue to the relation between Cn2and Cn1exposed in (3.47). From (3.48) the total number of unknowns will be 2 (2N+ 1), where Nis the truncated order of the series (3.28). Thus, the number of collocation points must be of at least 2N+1. For a proper performance of the methodology, the equations must outnumber the unknowns, and hence the least squares method (LSQ) can be used to solve the LSE. Finally, the LSE is assembled into a matrix form. Each term multiplying the unknowns will be included into the ˜ Amatrix. The unknowns have been already considered into the ˜ Cvector. And lastly, the free terms, that proceed from the boundary conditions, will be included into the ˜ Bvector. Each one of the collocation points will define two equations (or two rows of ˜ Aand ˜ B). With the present formulation, the LSE is written as ˜ A˜ C−˜ B= 0,(3.49) where ˜ A=          ˜ A(1,1) r0˜ A(1,1) i0˜ A(1,1) r1. . . ˜ A(1,1) r(2N+1) ˜ A(1,1) i(2N+1) ˜ A(1,2) r0˜ A(1,2) i0˜ A(1,2) r1. . . ˜ A(1,2) r(2N+1) ˜ A(1,2) i(2N+1) ˜ A(2,1) r0˜ A(2,1) i0˜ A(2,1) r1. . . ˜ A(2,1) r(2N+1) ˜ A(2,1) i(2N+1) . . .. . .. . ..... . .. . . ˜ A(NCP ,2) r0˜ A(NCP ,2) i0˜ A(NCP ,2) r1. . . ˜ A(NCP ,2) r(2N+1) ˜ A(NCP ,2) i(2N+1)          ,(3.50) Plate stretching analysis 56 and ˜ B=         ˜ B(1,1) ˜ B(1,2) ˜ B(2,1) . . . ˜ B(NCP ,2)         .(3.51) ˜ A(k,i) mn coefficients proceed from (3.34) and/or (3.36) by using the definition of Re and Im functions, Cnj =Re {Cnj}+iIm {Cnj},Cnj =Re {Cnj}−iIm {Cnj },(3.52) and in the same way for coefficients C∗ nj. Moreover, the free terms ˜ B(k)are the terms non-dependent on Cnj,C∗ nj. These terms are given by the boundary conditions by applying (3.34) and/or (3.36). Subsequently, (3.49) is solved by using the LSQ method. In this way, the vector of unknowns ˜ Cis calculated and substituted into expressions (3.15) and (3.16) in order to determine the resultant forces and displacements at any point of the domain (x1, x2)∈Ω. Unless otherwise indicated, the points at the external boundaries will be evenly collocated, including points at the corners of the component. In this case, the uniform distribution of points is the most efficient solution because of the smooth evolution of the stresses and displacements along the edges. On the contrary, in a case in which the stress tends to concentrate around a point or a segment, a different distribution of points would be preferable. This fact will be discussed in Chapter 6, since the stiffened edges work better under non-uniform distribution of points. 3.5 Extended methodology for plate stretching A closed form methodology dealing with the structural performance of finite composite plates with cutouts have been presented in last section. Although the formulation provided is useful for a great variety of geometries and load conditions, it is susceptible to be improved. The enhancements of the formulation presented during this section are original of this dissertation. In this regard, one challenging aspect would be to consider a component consisting of many regions, each one with different laminate mechanical 57 Extended methodology for plate stretching properties. In this way, ply drop-offs and local reinforcements can be simulated. This added feature have been already published as a part of the present dissertation by Pastorino et al. [24]. To this end, let the geometry under study to be comprised by Mregions, each one of them with a different extensional stiffness matrix (A). An arbitrary region will be denoted as m, assuming that m= 1,2...M. 3.5.1 Boundary conditions on a shared boundary In the first place, it is assumed an arbitrary domain comprised by two adjacent regions Ω(m)and Ω(m′). Both regions share a contour, which is denoted by ΓS. In addition, the external or non-shared boundaries are denoted by ΓE. See Figure 3.3 for a better understanding. External boundary Interface boundary u(m) u(m) s Ts Tn T ～ E S u ～ (m) (m) n s u(m') 2 u(m') 1T2 (m') T1 (m') s n  E x2 x1  (m')  (m) S Ts ～ Tn ～ u ～ s s s 0 u(m) 2 u(m') 1 T2 (m') T1 (m') x2x1 s s Figure 3.3: Boundary condition definition based on the location of the collocation points. Stretching analysis. Boundary conditions for external contours ΓEare given by (3.34) (integrated between 0and s), and/or (3.36), in terms of loads and displacements respectively. Plate stretching analysis 58 However, a different approach must be followed for the shared boundaries ΓS. In this case, it is required that: (1) Equilibrium of resultant forces and (2) continuity of displacement are accomplished at each collocation point at the interface between (m)and (m′). Thus, according to the diagram of Figure 3.3, T(m) 1+T(m′) 1= 0,T(m) 2+T(m′) 2= 0,(3.53a) u(m) 1=u(m′) 1,u(m) 2=u(m′) 2.(3.53b) Note that (3.53b) can be used directly by applying (3.35). However, (3.53a) shall be integrated to allow a straightforward formulation. To this end, let 0be a common origin, and the arc-length s(see Figure 3.3) be shared for both mand m′regions at ΓS. Thus, (3.53a) becomes, ∫s 0(T(m) 1+T(m′) 1)ds= 0,∫s 0(T(m) 2+T(m′) 2)ds= 0.(3.54) Note that, in order to use a common arc-length sat the shared boundary, the direction of sin the region m′is defined as counterclockwise. Hence, the sign of the left-hand side of (3.32) must be the opposite for this region. Finally, substituting (3.32) and (3.35) into (3.54) and (3.53b), a set of four equations that defines the boundary conditions at an arbitrary shared boundary ΓSis obtained, Re {2 ∑ j=1 (Φ(m) j−Φ(m′) j)}= 0,(3.55a) Re {2 ∑ j=1 (µjmΦ(m) j−µjm′Φ(m′) j)}= 0,(3.55b) Re {2 ∑ j=1 (pjmΦ(m) j−pjm′Φ(m′) j)}= 0,(3.55c) Re {2 ∑ j=1 (qjmΦ(m) j−qjm′Φ(m′) j)}= 0.(3.55d) In summary, each collocation point must be distinguished according to whether it belongs to a shared boundary ΓS(i.e. equations (3.55)), or to an external boundary ΓE(i.e. equations (3.34) or (3.36)). 59 Extended methodology for plate stretching 3.5.2 Reinforced cutout A reinforcement around the cutout can be considered as a particular case of the methodology presented right before, with some particularities. In this case, the geometry is comprised by an arbitrary plate with a cutout, which is is surrounded by a region with different mechanical properties than the rest of the plate, as in Figure 3.4. Reinforcement boundary R u(m') 2 u(m') 1 T2 (m') T1 (m') 0 E x2 x1  (m')  (m) R s s 0' u(m) 2 u(m') 1 T2 (m') T1 (m') x2 x1 0'' s s C s Figure 3.4: Boundary condition definition on a reinforcement shared boundary. Stretching analysis. It is important to distinguish among the internal (m′) and the external or surrounding (m) regions as in Figure 3.4, both of them share a common boundary referred as the reinforcement boundary ΓR. The boundary equations to consider equilibrium of forces and continuity of displacements at a reinforced boundary are exactly the same as in expressions (3.55). Nevertheless, some particularities must be accounted in this case, • The outer boundary of the internal region (m′) is given by the geometry of the reinforcement. The equations in (3.47) must be used to reduce the number of unknowns to half in (3.48). • The external region (m) is treated as a region in which the cutout is defined by the outer contour of the reinforcement. In this particular case is not possible to reduce the number of unknowns, this is because Plate stretching analysis 60 the boundary conditions at the cutout are unknown, hence αnand βn cannot be substituted into the expression (3.47). Thus, the full set of unknowns must be included into the matrix of unknowns ˜ C. Or in other words, the components Cn2and C∗ n2must be incorporated to (3.48). • A set of collocation points must be placed along the reinforcement boundary ΓR, in order to enforce (3.55) in each one of them. In this way, the equilibrium of forces and the continuity of displacements are accomplished at each collocation point. 3.6 Stretching benchmark exercises The methodology explained during this chapter has been implemented and tested using the mathematical software Matlab, three benchmark exercises will be solved in order to demonstrate the capabilities of the methodology. For comparison purposes, the benchmark exercises will be also solved using Abaqus 6.14 FE software with quadratic (S8R) and linear (S4R) shell elements, see Abaqus documentation [55]. The best procedure to verify the agreement between both models (present and FEM) is to plot the tangential resultant forces Nsalong the cutout boundary, as a function of the χangle, which is defined according to (3.19). In this section, the solutions obtained will be plotted and analysed. In regard of the material, the lamina properties to be used for all the exercises are shown in Table 3.2, and corresponds to a standard UD CFRP lamina. Table 3.2: Mechanical properties of a lamina. E11 (GPa) E22 (GPa) G12 (GPa) ν12 Thickness (mm) 140 9 4.65 0.3 0.184 Similarly, the stacking sequences to be used in the exercises are shown in Table 3.3. 61 Stretching benchmark exercises Table 3.3: Lay-ups used in the benchmark exercises. Designation Lay-up Thickness (mm) LU1 [45,−45,0,90,0]2s3.680 LU2 [45,−45,0,90,0,0,−45,45]2s5.888 LU3 [45,−45,0,90,0,0,−45,45,0]2s6.624 LU4 [45,−45,0,90,0,0,−45, 45,45,−45,0,−45,45]2s 9.568 3.6.1 Benchmark exercise 1 A geometry composed by two regions with different laminate stacking sequences is proposed in Figure 3.5. An off-centred hole is placed in one of the regions. The boundary conditions are all defined in terms of loads, with a parabolic normal distribution and a uniform shear load. 100 N/mm 300 300 400 100 160 50 N/mm 160 220 LU1 LU2 (All dimensions in mm) Figure 3.5: Geometry and loads of benchmark exercise 1. In the first instance, it is interesting to perform an assessment on the parameters of the methodology. To this end, the effect of the number of collocation points and the truncation order of the Laurent series Ncan be investigated through a parametric analysis. The solution should stabilise for a certain selection of parameters, indicating that the convergence has Plate stretching analysis 68 0 45 90 135 180 225 270 315 360 Angle (deg) -1500 -1000 -500 0 500 1000 1500 2000 2500 Resultant forces NS (N/mm) Present NS FEM NS Figure 3.12: Resultant forces around the cutout boundary. Present method & FEM. N= 20 and Ru= 2. Benchmark exercise 2. (a) Present Nx(N/mm)(b) FEM Nx(N/mm) (c) Present Ny(N/mm)(d) FEM Ny(N/mm) (e) Present Nxy (N/mm)(f) FEM Nxy (N/mm) Figure 3.13: Resultant forces in global coordinates. Benchmark exercise 2. 69 Stretching benchmark exercises 3.6.3 Benchmark exercise 3 The geometry of the third benchmark exercise is comprised by four regions and two cutouts, as shown in Figure 3.14. The boundary conditions are defined in terms of loads (in blue) and displacements (in red). 0.25 mm 0.45 mm 200 150 80 75 N/mm 120 75 90 150 150 LU1 LU2 LU3 LU4 0.45 mm 0.25 mm 100 N/mm Cutout 2 Cutout 1 55 30 (All dimensions in mm) Figure 3.14: Geometry and loads of benchmark exercise 3. The resultant forces around the cutouts (1) and (2) are shown in Figure 3.15 for both the present methodology and FEM. The agreement between both models is again outstanding. As before, the present model has been launched with N= 20 and Ru= 2, and the FEM is prepared using Abaqus S8R elements, with a total of 3.02·105DOFs. The in-plane resultant forces over the component are plotted in Figure 3.16, the comparison with respect to the FE model reveals that the results are also satisfactory at the whole domain. Plate stretching analysis 70 0 45 90 135 180 225 270 315 360 Angle (deg) -2000 -1000 0 1000 2000 Resultant forces N S (N/mm) Present (1) FEM (1) Present (2) FEM (2) Figure 3.15: Resultant forces around the boundaries of cutouts (1) and (2). Present method & FEM. N= 20 and Ru= 2. Benchmark exercise 3. 71 Stretching benchmark exercises (a) Present Nx(N/mm)(b) FEM Nx(N/mm) (c) Present Ny(N/mm)(d) FEM Ny(N/mm) (e) Present Nxy (N/mm)(f) FEM Nxy (N/mm) Figure 3.16: Resultant forces in global coordinates. Benchmark exercise 3. Plate stretching analysis 72 3.7 Discussion A methodology for the structural analysis of composite components, weakened by the presence of elliptical cutouts has been presented and verified by comparing with respect to FE results. Only the stretching performance of the plate has been considered in this chapter. An innovative approach has been presented in order to consider that the geometry is comprised by more than one region with different laminate mechanical properties. In addition to that, the present methodology allows the introduction of customizable boundary conditions, in terms of loads and/or displacements, as shown in the exercises solved. This original formulation increases the range of application of the closed-form methodologies based on the Lekhnitskii formalism with respect to the current SoA. The results from the analysis are proven to be efficient and accurate when using a constant order of approximation N= 20, and a constant equationunknown ratio Ru= 2. As a summary, Table 3.4 collects the results obtained from the three benchmark exercises presented in this article for different orders of approximation N. The computational time considers that the models run individually (not in parallel). Additionally, Table 3.4 includes the results and the computational time of two FE models per benchmark exercise. First, the exercise has been modelled and simulated with a coarse mesh of S4R linear shell elements. Here, an analogous computational time to that of the closed-form formulation with an order of approximation N= 20 was pursued for comparison purposes. Such a time reduction was reached at the expense of the accuracy of the analysis. Second, the results of the FE models used as references in Section 3.6 are shown. These models were solved with a high level of accuracy using a fine optimised mesh of S8R quadratic shell elements. It should be noted also that the programmed script is written in Matlab, which is an interpreted language. The closed-form methodology could be further optimised by using compiled languages. As shown in Table 3.4, the present methodology delivers highly accurate results, if compared with respect to computationally heavy FE models. The convergence of the solution is clearly appreciated from the solutions for truncation orders of 10,20 and 30, that have been added to the table for all the exercises. In addition, the present formulation demonstrates that is highly competitive in terms of analysis computational time, requiring just a few seconds to solve problems with several regions. Apart from the 73 Discussion analysis time, great time savings can be achieved in preparation, launching of the model and representation of results, but it will be discussed in detail at the last chapter of this dissertation. Plate stretching analysis 74 Table 3.4: Results of the benchmark exercises. Stretching analysis. Benchmark Formulation N/DOFs (1) min NsMax Nst(2) (N/mm) (N/mm) (s) 1 Closed-form 10 −288.89 483.45 0.57 20 −318.15 500.96 1.77 30 −318.30 500.94 3.68 FEM 3.74 ·104(S4R) −287.75 479.11 3 3.11 ·105(S8R) −317.82 500.77 21 2 Closed-form 10 −1087.87 2331.91 3.13 20 −1093.09 2346.61 10.07 30 −1093.33 2346.92 21.78 FEM 3.8·104(S4R) −1001.92 2238.33 7 3.02 ·105(S8R) −1091.74 2346.80 30 3(3) Closed-form 10 −1608.06 1126.00 1.21 20 −1638.85 1143.84 3.95 30 −1639.39 1138.61 8.29 FEM 4.49 ·104(S4R) −1546.94 1056.76 6 5.25 ·105(S8R) −1639.66 1139.87 57 (1) Order of approximation (closed-form) / Degrees of freedom (FEM). (2) Computational time (Intel Core i7-7700 CPU @ 3.60 GHz and 16 GB DDR3 RAM) (3) Only the cutout (1) results are collected in this table. Chapter 4 Plate bending analysis The closed-form formulation for anisotropic plates under stretching boundary conditions has been exposed in Chapter 3. A methodology considering plate bending is analogously developed in the present chapter. For the time being, the formulations are developed under the assumption that the stretching and the bending performances are completely uncoupled and independent, i.e. in-plane deformation of the plate does not induce out-of- plane deformations (or vice versa). It is important to note that the same thought on the closed-form and the FE methodologies conducted at the introduction of Chapter 3 is valid for this case; the closed-form formulations are optimal for the resolution of parametric studies, and can be an excellent complement to the FE formulations for the structural analysis of composite components. The bending analytical methodology for anisotropic plates with cutouts is under investigation, again, since Savin [4] and Lekhnitskii [1] built the methodology known as Lekhnitskii formalism. It is based on the Kirchhoff- Love hypotheses [35], that were presented and analysed in Section 2.3. These hypotheses are also known as the CPT for isotropic or the CLT for composite plates. Likewise, the laminate behaviour described in Chapter 2 were developed under the same hypotheses. Finally, the structure of the chapter is provided. In the first place, a state of the art on the topic is explored in Section 4.1. The Lekhnitskii formalism for plate bending, which constitutes the basis of the present development, is described in Section 4.2. Subsequently, Section 4.3 collects the bending formulation for finite plates. Followed by the extended formulation, to account for different regions in Section 4.4. In order to close the chapter, Plate bending analysis 76 some benchmark exercises are solved and analysed in Section 4.5, while the discussion is conducted in Section 4.6. 4.1 State of the art for bending analysis 4.1.1 Analysis of infinite plates Although most of the articles in this field address the resolution of composite laminated plates under stretching loads only, many investigations in the peer-reviewed literature are also focused on the production of a formulation for the bending of anisotropic plates. In particular, Lekhnitskii [1] provided the so-called Lekhnitskii formalism for plate bending, while Lu and Mahrenholtz [56], Hsie and Hwu [57], and Hwu [58] developed a Stroh-like bending formalism, based on the studies of Ting [14]. It should be noted here, that both Lekhnitskii and Hwu raised the bending complex variable formulations, equivalent to Lekhnitskii and Stroh formulations for stretching that were exposed in Chapter 3. Both the Lekhnitskii and the Stroh formalisms are based on the CLT [35]. Or in other words, both of them assume that the out-of-plane normal and shear strains are negligible. The advantage of the CLT is that it is simple and straightforward to implement, also it is accurate enough for thin plates, which are usual in composite components. Additionally, there are many other well known plate theories that model the behaviour of the twodimensional plate bending problem, see the review from Khandan et al. [59]. In a general basis, these other theories involve greater accuracy at the expense of complexity e.g. the First Shear order Deformation Theory (FSDT). On the basis of the Lekhnitskii formalism for plate bending, Prasad and Shuart [60] conducted a study on the bending of anisotropic plates with elliptical cutouts. The results for UD, QI, and cross-ply laminates are given for a bi-axial moment applied at infinity. Another work was presented by Chen and Nie [61], who derived the Green’s functions for thin anisotropic plates with an elliptical hole. These functions can be used for the boundary element method. Ko and Lin [62] extracted the solution for the out-of-plane stresses at the cutout boundary, when a remote out-of-plane bending moment is applied. The solution to several exercises is given, and excellent agreement is 77 State of the art for bending analysis achieved when comparing them with respect to the FEM results. However, the solution provided is only valid for cross-ply laminates. Ukadgaonker and Rao [63] derived the solution for infinite anisotropic plates with holes different than elliptical. For instance, triangular, squared and star-shaped cutouts are studied and solved. As in their previous work [20], the modifiable cutout shape is obtained by adjusting the conformal mapping, that were introduced in Section 3.3. A similar investigation was conducted by Patel and Sharma [64]. Here, the authors performed many parametric analysis on infinite plates with square-shaped holes under bending loads. In contrast to the latter works presented, which are all based on the Lekhnitskii formalism, Wu and Hsiao [65] derived the exact solution for infinite anisotropic plates with an elliptical hole under a remote uniform out-of- plane bending on the basis of the Stroh-like bending formalism. The article focuses on the resolution of a single-valued deflection function, which is accomplished by means of a correction constant. Explicit expressions for the deflection and moments at the hole edge are given. 4.1.2 Analysis of finite plates On the basis of the work from Ogonowski [36] for the stretching analysis, the plate bending formulation is susceptible to allow finite boundaries by using the BCM. In this regard, Hufenbach et al. [5] proposed the application of the BCM for the bending of finite anisotropic plates. An important improvement is included in this study, the out-of-plane shear strains are considered, which is achieved by using the Reissner-Mindlin or FSDT formulation, instead of the CLT. This formulation is particularly convenient for the estimation of stresses at the cutout vicinity of thick-walled composites. Results from experimental tests are measured and compared with respect to the semi-analytical formulation developed. Additionally, an assessment on the effect of the coupon thickness is conducted by comparing the results by using the FSDT and the CLT models. On the basis of the Lekhnitskii formalism and the BCM, Mao and Xu [66] provided a methodology for finite anisotropic plates with multiple elliptical cutouts under bending loads. The structural analysis of the component is performed by making use of the Faber polynomial series in order to approach the potential function. Several academic exercises with multiple cutouts are successfully solved at the end of the article, although no refer- Plate bending analysis 84 moments (either normal or tangential to the edge) and/or transversal shear forces applied at the edge. On the other hand, as argued before, the Dirichlet boundary conditions are given in terms of the deflection’s derivatives at the edge, with respect to both normal (n) and tangential (s) directions of the edge. The moment boundary conditions can be derived by means of integration of the bending moments’ projection over the principal directions x1and x2. Thus, ∓∫s 0 (Mndx2−V∗ ndx1)−cx1= 2Re {p1 µ1 w′ 1+p2 µ2 w′ 2} s 0 ,(4.20a) ±∫s 0 (−Mndx1−V∗ ndx2) + cx2= 2Re {q1w′ 1+q2w′ 2}|s 0.(4.20b) Using the same nomenclature than Hwu [7], the integral of the effective transverse shear is denoted as V∗ n, V∗ n=∫Vnds. (4.21) Mnand Vnare respectively the bending moments and forces per unit length in normal direction. The upper sign of equations (4.20) must be used for the external contours, and the lower for the internal ones. cis a real constant determined through enforcing the deflection function wto be single-valued (see Wu and Hsiao [65]). The understanding of the boundary conditions as exposed in (4.20) is more complicated than the conditions for the stretching problem, as in (3.32). In the case of bending, Lekhnitskii [1] does not explain the origin of such boundary conditions, however Hwu [7] provides a successful clarification by using a stress function and the surface moments. According to (4.20), the boundary conditions are introduced as prescribed bending moments ˜ Mn(s)and effective transverse shear forces ˜ Vn(s), which are given as functions of the arc length sof the plate edges (see Figure 4.1). Note that the tilde ˜ •stands for enforced boundary conditions. The conditions (4.20) are given in x1and x2local directions, a rotation matrix could be used to transform the conditions into normal and tangential directions of the edge (nand srespectively). ∓∫s 0 Mnds= 2Re {2 ∑ j=1 [(qjcos θ+pj µj sin θ)w′ j]} s 0 ,(4.22a) 85 General methodology for plate bending x2 Vn x3 x1 sMn ～ ～ s nx2 x1  E Ω Figure 4.1: Diagram of normal moments and shear forces on an arbitrary finite plate. ±∫s 0 V∗ nds−cs = 2Re {2 ∑ j=1 [(−qjsin θ+pj µj cos θ)w′ j]} s 0 .(4.22b) Similarly, the slope boundary conditions are given in terms of the deflection’s derivatives (see Lekhnitskii [1], Hwu [7]), on the basis of the expressions in (4.13) as follows, ∂w ∂x1 =∂w ∂s cos θ−∂w ∂n sin θ= 2Re {w′ 1+w′ 2},(4.23a) ∂w ∂x2 =∂w ∂s sin θ+∂w ∂n cos θ= 2Re {µ1w′ 1+µ2w′ 2}.(4.23b) The edge’s slope conditions of (4.23) are also derived in sand ndirections by using the rotation matrix, ∂w ∂s = 2Re {2 ∑ j=1 [(cos θ+µjsin θ)w′ j]},(4.24a) ∂w ∂n = 2Re {2 ∑ j=1 [(−sin θ+µjcos θ)w′ j]}.(4.24b) At this point, the discussion about the conventional boundary conditions can be resumed. It should be noted that the conditions defined in (4.22) and (4.24) are given exclusively in terms of the variable w′ j(without deriving or integrating), as in the stretching problem with Φj. Plate bending analysis 86 In this way, the Lekhnitskii formalism allows a straightforward substitution of the boundary conditions. Naturally, other boundary conditions could be considered (as the conventional deflection w) at the expense of a lengthy and tedious formulation. The present dissertation is built up on the basis of the existing literature, and the boundary conditions presented in (4.22) and (4.24), that were initially introduced by Lekhnitskii [1], and further used by Hwu [58] and Mao and Xu [66] will be considered for the application of bending boundary conditions hereinafter. Cutout boundary Now, the expressions for the boundary conditions at the hole edge must be developed. In accordance to the research from Lekhnitskii [1], the noncompensated moments and the shear forces around the cutout will define the coefficients of the logarithmic terms. Hence, making use of (4.23) and assuming that the deflections are uni-evaluated from ψ= 0 to ψ= 2π, Im {2 ∑ j=1 (C′ 0j)}= 0,(4.25a) Im {2 ∑ j=1 (µjC′ 0j)}= 0,(4.25b) Im {2 ∑ j=1 (C′′ 0j)}= 0,(4.25c) Im {2 ∑ j=1 (µjC′′ 0j)}= 0,(4.25d) Proceeding similarly with (4.20) and considering again the uni-evaluated deflection at ψ= 0 and ψ= 2π, Im {2 ∑ j=1 (qjC′ 0j)}=m1 4π,(4.26a) Im {2 ∑ j=1 (pj µj C′ 0j)}=m2 4π,(4.26b) Im {2 ∑ j=1 (qjC′′ 0j)}= 0,(4.26c) 87 General methodology for plate bending Im {2 ∑ j=1 (pj µj C′′ 0j)}=P3 4π.(4.26d) Here m1and m2are the resultant of moments in x1and x2directions along the hole edge. Similarly, P3is the resultant of forces in x3direction. Using (4.25) and (4.26), the complex coefficients C′ 0jand C′′ 0jcan be calculated. If the moments and loads acting on the cutout edge are equilibrated, all the aforementioned terms will be equal to zero. The terms of (4.18) depending on the coefficients Cnj and C∗ nj are also investigated. As for the stretching problem, the boundary conditions at the cutout edge can be used to reduce the number of unknowns, without requiring the use of collocation points at the cutout edge ΓC. A similar procedure has been carried out by Hufenbach and Kroll [49], and Hufenbach and Zhou [50] for the stretching analysis. But no investigations deal with this procedure for bending, to the knowledge of the author this is original of the present investigation. On the basis of the Lekhnitskii formalism [1], the bending moments and the transverse shear forces applied at the hole boundary can be approached by making use of a Fourier series development as a function of αnand βn coefficients. Hence, taking the right-hand side of (4.20), referred to the internal boundary, ∫ψ 0(˜ Mn dx2 dψ−˜ V∗ n dx1 dψ)dψ= P3a 2πψcos ψ−m2 2πψ+α0+ N ∑ n=1 (αneinψ +αne−inψ) ψ 0 ,(4.27a) −∫s 0(−˜ Mn dx1 dψ−˜ V∗ n dx2 dψ)dψ= −P3b 2πψsin ψ−m1 2πψ+β0+ N ∑ n=1 (βneinψ +βne−inψ) ψ 0 .(4.27b) Subsequently, it should be recalled the definition of ζjat the hole boundary as in (3.37), ζj=σ=eiψ = cos ψ+isin ψ ζ−1 j=σ=e−iψ = cos ψ−isin ψ}with ζj∈ΓC.(4.28) Plate bending analysis 88 Which is substituted into the right-hand side of (4.20), 2 ∑ j=1 [pj µj w′ j(σ) + pj µj w′j(σ)]= P3a 2πψcos ψ−m2 2πψ+α0+ N ∑ n=1 (αnσn+αnσ−n),(4.29a) 2 ∑ j=1 [qjw′ j(σ) + qjw′j(σ)]= −P3b 2πψsin ψ−m1 2πψ+β0+ N ∑ n=1 (βnσn+βnσ−n).(4.29b) Note that the definition of real part of complex numbers has been used in (4.29), with 2Re {η}=η+η, valid ∀η∈C. A relation between w′ 1(σ)and w′ 2(σ)can be obtained from (4.29). Proceeding as in Section 3.4.2, by extrapolation of this relation to the whole domain as σ→ζ2, and using (4.18), the following relation is found: w′ 2(ζ2) = δ1C01 +δ2C01+ N ∑ n=1 [δ1Cn1ζ−n 2+δ2Cn1ζn 2+δ1C∗ n1ζn 2+δ2C∗ n1ζ−n 2]+ δ3[β0−ibc 2(ζ2−ζ−1 2)+ N ∑ n=1 (βnζn 2+βnζ−n 2)]+ δ4[α0−ac 2(ζ2+ζ−1 2)+ N ∑ n=1 (αnζn 2+αnζ−n 2)]+logarithmic terms, (4.30) where the coefficients δkare given by, δ1=iµ2(p1q2µ2−p2q1µ1) µ12Im (p2q2µ2),(4.31a) δ2=iµ2(p1q2µ2−p2q1µ1) µ12Im (p2q2µ2).(4.31b) δ3=iµ2p2 2Im (p2q2µ2),(4.31c) 89 General methodology for plate bending δ4=−iµ2µ2q2 2Im (p2q2µ2),(4.31d) Ignoring the logarithmic terms that are known from (4.26), and grouping together all the terms of the same kind, the relations for each Cn2and C∗ n2 with Cn1and C∗ n1are found, C02 =δ1C01 +δ2C01 +δ3β0+δ4α0,(4.32a) C12 =δ1C11 +δ2C∗ 11 +δ3(−ibc 2+β1)+δ4(−ac 2+α1),(4.32b) C∗ 12 =δ1C∗ 11 +δ2C11 +δ3(ibc 2+β1)+δ4(−ac 2+α1),(4.32c) . . . Cn2=δ1Cn1+δ2C∗ n1+δ3βn+δ4αn,(4.32d) C∗ n2=δ1C∗ n1+δ2Cn1+δ3βn+δ4αn.(4.32e) It should be noted that (4.32) follows the same structure than (3.47), but with different δkparameters as defined in (4.31). Similarly, (4.32) reduces the number of unknowns to half, since the Cn2and C∗ n2coefficients can be expressed as a function of the other coefficients. External boundary The boundary conditions at the external edges must be individually defined in each collocation point, which are placed along the external contour ΓE. The conventional boundary conditions for plate bending can be defined by a deflection and a slope, or a shear force and a moment, or a combination of them. See for instance Hwu [58], or Bauchau and Craig [67]. These conditions are given by, ∂w ∂n =∂˜w ∂n , or Mn=˜ Mn,(4.33) and, w= ˜w, or Vn=˜ Vn.(4.34) Plate bending analysis 90 Note that the slope of the deflection (∂w/∂s) can be enforced by using (4.24b), while the moment (Mn) and the shear force (Vn) conditions can be fulfilled in the integral sense according to (4.22). However, as argued before, the deflection condition cannot be completely fulfilled with (4.24a). In this regard, the condition w= ˜wrequires that ∂w/∂s =∂˜w/∂s, which is the condition used in the existing literature (see [1, 58, 66]). A set of standard boundary conditions are defined in Table 4.1. Note that the pseudo prefix indicates that the clamped and the simply supported conditions are not strictly fulfilled. Table 4.1: Commonly used bending boundary conditions for external edges. Description ncondition scondition Pseudo simply supported Mn= 0 ∂w ∂s = 0(1) Free edge Mn= 0 Vn= 0 Pseudo-clamped ∂w ∂n = 0 ∂w ∂s = 0(1) Enforced moment Mn=˜ MnVn=˜ Vn Enforced slope ∂w ∂n =∂˜w ∂n ∂w ∂s =∂˜w ∂s (1) (1) The condition is given in terms of ∂w ∂s instead of w. With the current notation the only certainty is that wwill be constant if ∂w ∂s = 0. 4.3.3 Collocation points and least squares No changes on the resolution method are introduced with respect to the methodology explained in Section 3.4.2. Thus, a vector of unknowns ˜ C with all the Cnj and C∗ nj parameters is created as in (3.48). In addition, a coefficient matrix ˜ A, and a free-term vector ˜ Bare built according to (3.50) and (3.51) respectively. Finally, the unknowns are calculated by solving the LSE obtained with the LSQ method, as in (3.49), ˜ A˜ C−˜ B= 0.(4.35) 91 Extended methodology for plate bending 4.4 Extended methodology for plate bending It is interesting to extend the formulation to account for different regions in the model. To this end, an analogous formulation to the one presented in Section 3.5 is developed, but for the case of bending. The contents of this section are original of this dissertation. Furthermore, the particular solution of (4.4) has not been investigated by the time being. This topic will be also explored. 4.4.1 Boundary conditions on a shared boundary The boundary conditions defined on a collocation point depend entirely on its position. According to Figure 4.2, if the point is placed at the external edge ΓE, two boundary conditions must be given, as defined in (4.22) and (4.24). Mn2 (m) Mn1 (m) Mn1 (m') Mn2 (m') w ,1 (m) w ,2 (m) w ,1 (m') w ,2 (m') External boundary Interface boundary E S x2 Vn x3 x1 s Mn ～ ～ E  (m)  (m') E S s Mn ～ Vn ～ w ,s ～ w ,n ～ w ,s (m') Mnw ,n (m') Vn (m') *(m') * x3 x1 x2 s s s x1 x2 0 s Figure 4.2: Boundary condition definition based on the location of the collocation points. Bending analysis. Otherwise, if the point is placed at a shared boundary ΓS, the conditions must be selected in order to accomplish the continuity of the deflection’s derivative, and the equilibrium of forces and moments. According to Figure Plate bending analysis 92 4.2, the projection of the resultant moments in x1and x2directions have been defined as Mn1and Mn2, which are equivalent to T1and T2of (3.31). Their definition is given by, Mn1=−Mn dx2 ds+V∗ n dx1 ds,(4.36a) Mn2=−Mn dx1 ds−V∗ n dx2 ds.(4.36b) Thus, the equilibrium of forces and moments at the shared boundary, as well as the continuity of slopes, will be given by, M(m) n1+M(m′) n1= 0, M(m) n2+M(m′) n2= 0,(4.37a) ∂w(m) ∂n =∂w(m′) ∂n ,∂w(m) ∂s =∂w(m′) ∂s .(4.37b) A set of four equations must be accomplished at each collocation point in a shared boundary. In order to facilitates the definition of the conditions, it is advisable to integrate (4.37a) with respect to sas depicted in Figure 4.2. In this way, the equations will be written in the same terms than (4.20). ∫s 0(M(m) n1+M(m′) n1)ds = 0,∫s 0(M(m) n2+M(m′) n2)ds = 0.(4.38) Note that the arc-length parameter sused is the same for both of the regions, and subsequently the region (m′)will be swept in couterclockwise direction (instead of the natural clockwise). It is important to consider this fact since it will cause the right-hand side of (4.20) to change its sign. Now, (4.38) and (4.37b) can be expressed in terms of w′ j, by using (4.20) and (4.23) respectively, the boundary conditions become, 2Re {2 ∑ j=1 (pjm µjm w′(m) j−pjm′ µjm′ w′(m′) j)}= 0,(4.39a) 2Re {2 ∑ j=1 (qjmw′(m) j−qjm′w′(m′) j)}= 0,(4.39b) 2Re {2 ∑ j=1 (w′(m) j−w′(m′) j)}= 0,(4.39c) 93 Extended methodology for plate bending 2Re {2 ∑ j=1 (µjmw′(m) j−µjm′w′(m′) j)}= 0.(4.39d) Therefore, (4.39) constitutes a set of four equations, two of them guarantee the equilibrium of moments at the shared boundary. While the other two simulate the continuity of the edge rotations at the interface, although the rigid body motions are not limited with this condition. It should be remarked that, although the deflection at the both sides of the shared boundary might be different since only the slope continuity is enforced, the rigid body motions do not affect to the resultant moments at any of the regions. 4.4.2 Reinforced cutout The reinforcement of the hole has been dealt previously in Section 3.5.2, some particularities of the reinforcements were highlighted, and must been taken into account in this case as well, i.e. a set of collocation points must be placed alongside the reinforcement boundary ΓR, and none is required at the cutout boundary ΓC. Figure 4.3 depicts the conditions at the reinforcement boundary, which is denoted as ΓR. In this case, it is clear that (4.39) must be fulfilled in order to ensure the continuity of rotations and the equilibrium of moments at the interface. The equations must be fulfilled at the collocation points on the reinforcement boundary ΓR. Reinforcement boundary R x2 x3 x1 C R s s  (m')  (m) E s Mn2 (m') Mn1 (m') Mn1 (m) Mn2 (m) w ,1 (m') w ,2 (m') w ,1 (m) w ,2 (m) x1 x2 s s Figure 4.3: Boundary condition definition on a reinforcement shared boundary. Bending analysis. Plate bending analysis 100 Present FEM (CPT) 0 45 90 135 180 225 270 315 360 Angle (deg) Resultant moments Ms (Nmm/mm) 0 10 50 60 70 80 90 100 110 120 Present FEM (CLT) Figure 4.9: Resultant moments around cutout boundary. Present method & FEM (CLT). N= 20 and Ru= 2. Benchmark exercise 4. 101 Bending benchmark exercises (a) Present Mx(Nmm/mm)(b) FEM (CLT)Mx(Nmm/mm) (c) Present My(Nmm/mm)(d) FEM (CLT)My(Nmm/mm) (e) Present Mxy (Nmm/mm)(f) FEM (CLT)Mxy (Nmm/mm) Figure 4.10: Resultant moments in global coordinates. Benchmark exercise 4. Plate bending analysis 102 Kirchhoff-Love hypotheses influence Previously in this chapter, it has been remarked that the present formulation is based on the CLT. However, the use of thick-plate theories (e.g. FSDT) is usually contemplated when solving two-dimensional plates under bending loads. In fact, all the predefined shell elements in the Abaqus 6.14 catalogue (except for the STRI3) assume FSDT hypotheses when the element is subjected to bending stresses. Hence, in order to provide a better picture of the CLT assumptions made, Figure 4.11 plots the results around the cutout with the present formulation, against the results obtained from FEM by assuming FSDT hypotheses. The FEM model has 5.32 ·105DOFs and S8R elements. Present FEM (FSDT) 0 45 90 135 180 225 270 315 360 Angle (deg) Resultant moments Ms (Nmm/mm) 0 10 50 60 70 80 90 100 110 120 130 Present FEM (FSDT) Figure 4.11: Resultant moments around the cutout boundary. Present method & FEM (FSDT). N= 20 and Ru= 2. Benchmark exercise 4. The concentration of out-of-plane shear strains is specially noticeable at the free edges of the component. Hence, the region of the cutout edge is particularly affected by the shear deformations. Actually, from Figure 4.11, the maximum relative difference between the two models measured with the Rdparameter is around 5%. Note the differences with respect to the Figure 4.9, in which the FEM and the present formulation results are completely overlapped. 103 Bending benchmark exercises 4.5.2 Benchmark exercise 5 The second benchmark exercise is based on the geometry presented in Section 3.6.2, with boundary conditions in terms of edge rotations (in red) and moments (in blue). In this case, the boundary conditions consist on two edges pseudo simply supported (i.e. wdeflection is constant along the edge), while a bi-moment is applied to the component according to Figure 4.12, where Mx= 100 Nmm/mm and My= 50 Nmm/mm. As mentioned before, the pseudo simply supported condition shown in Figure 4.12 is not w= 0 but ∂w/∂s = 0, that only warrants a constant w. Actually, it is defined this way because of the limitations on the boundary conditions defined in (4.23). This only applies for the model solved with the present methodology. A FE model is used for validation, and it is modelled under the assumption of a strict simply supported condition w= 0. If both conditions (w= 0 and ∂w/∂s = 0) are equivalent except for the rigid body motions, the solution obtained from both the FE and the present methods should be equal in terms of the resultant moments. 50 Nmm/mm dw/ds = 0 dw/ds = 0 100 Nmm/mm (All dimensions in mm) 100 200 150 80 120 104 156 LU2 LU1 LU4 Figure 4.12: Geometry and loads of benchmark exercise 5. Plate bending. The results obtained for N= 20 and Ru= 2 with the present formulation are compared with respect to the ones from FEM in Figure 4.13. Excellent agreement is shown between both models, with a relative difference below 0.5%. It should be mentioned that the FE model is launched with 9.38·104 DOFs and STRI3 elements. Plate bending analysis 104 Finally, the resultant moments in Cartesian coordinates, acting on the full surface of the component, are shown in Figure 4.14. The results of the present methodology are of great quality if compared with respect to the reference. It should be noted that, despite the pseudo simply supported boundary condition have been applied to the closed-form model, the results reveal that this condition is equivalent to the strict simply supported condition used to generate the FE reference on these problems. Present FEM (CPT) 0 45 90 135 180 225 270 315 360 Angle (deg) Resultant moments Ms (Nmm/mm) 0 150 200 250 300 350 400 450 Present FEM (CLT) Figure 4.13: Resultant moments around the cutout boundary. Present method & FEM (CLT). N= 20 and Ru= 2. Benchmark exercise 5. 105 Bending benchmark exercises (a) Present Mx(Nmm/mm)(b) FEM (CLT)Mx(Nmm/mm) (c) Present My(Nmm/mm)(d) FEM (CLT)My(Nmm/mm) (e) Present Mxy (Nmm/mm)(f) FEM (CLT)Mxy (Nmm/mm) Figure 4.14: Resultant moments in global coordinates. Benchmark exercise 5. Plate bending analysis 106 4.5.3 Benchmark exercise 6 The geometry of this benchmark exercise is identical than the one shown in Section 3.6.3. In this case, the component is pseudo-clamped at its left side, while a constant resultant moment of 100 Nmm/mm is applied at its right side, see Figure 4.15. As in the previous exercise, let it be noted that the pseudo-clamped edge condition is simulated under the assumption: ∂w ∂s =∂w ∂n = 0, although the strictly clamped condition would be: w=∂w ∂n = 0. Once again, the FE model has been performed by assuming a strictly clamped condition, since the simplification performed in the closed-form method is only expected to affect to the rigid body motions. 200 150 80 120 75 90 150 150 LU1 LU2 LU3 LU4 Cutout 2 Cutout 1 55 30 (All dimensions in mm) dw/dn = dw/ds = 0 100 Nmm/mm Figure 4.15: Geometry and loads of benchmark exercise 6. Plate bending. The resultant moments obtained with the present formulation are shown in Figure 4.16, in which an excellent agreement with respect to the FEM model is appreciated around both cutouts (less than 1% of maximum difference between both models). The parameters of the closed-form model are N= 20 and Ru= 2, while the FEM is launched with 1.95·105DOFs and STRI3 elements. 107 Bending benchmark exercises The resultant moments in the whole domain are represented in Figure 4.17 for both the present formulation and the reference. The quality of the results is again outstanding. 0 45 90 135 180 225 270 315 360 Angle (deg) 0 200 400 600 800 1000 Resultant moments Ms (Nmm/mm) Present (1) FEM (1) (CLT) Present (2) FEM (2) (CLT) Figure 4.16: Resultant moments around the boundaries of cutouts (1) and (2). Present method & FEM (CLT). N= 20 and Ru= 2. Benchmark exercise 6. Plate bending analysis 108 (a) Present Mx(Nmm/mm)(b) FEM (CLT)Mx(Nmm/mm) (c) Present My(Nmm/mm)(d) FEM (CLT)My(Nmm/mm) (e) Present Mxy (Nmm/mm)(f) FEM (CLT)Mxy (Nmm/mm) Figure 4.17: Resultant moments in global coordinates. Benchmark exercise 6. 109 Discussion 4.6 Discussion Some benchmark exercises have been solved by using the closed-form formulation developed throughout the present chapter. The accuracy of the method using constant parameters, with a truncation order of N= 20 and a equation-unknown ratio of Ru= 2, have been demonstrated. Moreover, all the exercises include some of the special features developed. The innovative approach presented allows to efficiently calculate plates with several regions, which increases the range of application of the closed-form methodologies based on the Lekhnitskii formalism. In order to verify the formulation, the exercises have been compared to FEM results. In Table 4.2, the results of both formulations are compared. Here, the maximum and minimum resultant moments around the hole for different orders of approximation Nare collected. In regard of the FEM, three different results are given per exercise. The first one corresponds with a coarse mesh making use of the CLT hypotheses, the second analysis is performed under the same basis but with a refined mesh, this is the result used for comparison purposes in the Figures 4.9, 4.10, 4.13, 4.14, 4.16 and 4.17. Finally, the last result is obtained by means of a finer mesh under the FSDT hypotheses, this model is used in the Figure 4.11. In addition, the analysis computational time is given. It should be noticed that the present formulation is quite competitive in terms of computing time. The accuracy of the present formulation has been also demonstrated by comparing with respect to FEM under the same CLT hypotheses. Finally, the comparison by using a higher order shear methodology (i.e. FSDT), reveals that a slight error is induced with the CLT hypotheses, at least at the vicinity of the hole edge. Coupled stretching-bending analysis 116 u0 2= 2Re {4 ∑ j=1 (qj dφj dzj)}= 2Re {4 ∑ j=1 (qjΦj)},(5.5c) where Φjis the derivative of φjwith respect to zj. The roots µj, and the complex constants pj,qj, can be calculated by substitution of (5.5) into (5.3) by assuming arbitrary complex functions Φj, [A11 + 2A16µj+A66µ2 j]pj+[A16 + (A12 +A66)µj+A26µ2 j]qj −[B11 + 3B16µj+ (B12 + 2B66)µ2 j+B26µ3 j]= 0,(5.6a) [A16 + (A12 +A66)µj+A26µ2 j]pj+[A66 + 2A26µj+A22µ2 j]qj −[B16 + (B12 + 2B66)µj+ 3B26µ2 j+B22µ3 j]= 0,(5.6b) [D11 + 4D16µj+ 2 (D12 + 2D66)µ2 j+ 4D26µ3 j+D22µ4 j] −[B11 + 3B16µj+ (B12 + 2B66)µ2 j+B26µ3 j]pj −[B16 + (B12 + 2B66)µj+ 3B26µ2 j+B22µ3 j]qj= 0.(5.6c) Note that the solution of µj,pjand qjis not immediate due to the crossed terms involving the three unknowns. Hence, operating with (5.6) in order to eliminate the unknown parameters pjand qj, an eighth-order polynomial expression dependent only on µjis obtained [19], P0+P1µj+P2µ2 j+P3µ3 j+P4µ4 j+P5µ5 j+P6µ6 j+P7µ7 j+P8µ8 j= 0.(5.7) From (5.7), eight complex µjroots conjugated two-by-two are obtained. Furthermore, the real parameters P0,P1, ... P8are given by, P0=B11R0+B16S0−D11T0,(5.8a) P1=B11R1+ 3B16R0+B16S1 + (B12 + 2B66)S0−D11T1−4D16T0,(5.8b) P2=B11R2+ 3B16R1+ (B12 + 2B66)R0+B16S2+ (B12 + 2B66)S1 + 3B26S0−D11T2−4D16T1−(2D12 + 4D66)T0,(5.8c) 117 Lekhnitskii formalism for stretching-bending coupling P3=B11R3+ 3B16R2+ (B12 + 2B66)R1+B26R0+B16S3 + (B12 + 2B66)S2+ 3B26S1+B22S0−D11T3−4D16T2 −(2D12 + 4D66)T1−4D26T0,(5.8d) P4=B11R4+ 3B16R3+ (B12 + 2B66)R2+B26R1+B16S4 + (B12 + 2B66)S3+ 3B26S2+B22S1−D11T4−4D16T3 −(2D12 + 4D66)T2−4D26T1−D22T0,(5.8e) P5=B11R5+ 3B16R4+ (B12 + 2B66)R3+B26R2+B16S5 + (B12 + 2B66)S4+ 3B26S3+B22S2−4D16T4 −(2D12 + 4D66)T3−4D26T2−D22T1,(5.8f) P6= 3B16R5+ (B12 + 2B66)R4+B26R3 + (B12 + 2B66)S5+ 3B26S4+B22S3 −(2D12 + 4D66)T4−4D26T3−D22T2,(5.8g) P7= (B12 + 2B66)R5+B26R4+ 3B26S5+B22S4 −4D26T4−D22T3,(5.8h) P8=B26R5+B22S5−D22T4,(5.8i) where Si,Riand Tiare, R0=A66B11 −A16B16,(5.9a) R1= 2A66B16 + 2A26B11 −A16B12 −2A16B66 −A12B16,(5.9b) R2=A22B11 + 5A26B16 −3A16B26 −A12B12 −2A12B66,(5.9c) R3=−2A66B26 +A26B12 + 2A26B66 + 3A22B16 −A16B22 −3A12B26,(5.9d) R4=−A26B26 +A22B12 + 2A22B66 −A12B22 −A66B22,(5.9e) R5=A22B26 −A26B22,(5.9f) S0=A11B16 −A16B11,(5.10a) Coupled stretching-bending analysis 118 S1=A11B12 + 2A11B66 −A16B16 −A12B11 −A66B11,(5.10b) S2= 3A11B26 +A16B12 + 2A16B66 −2A66B16 −3A12B16 −A26B11,(5.10c) S3=A11B22 + 5A16B26 −A12B12 −2A12B66 −3A26B16,(5.10d) S4= 2A16B22 + 2A66B26 −A12B26 −A26B12 −2A26B66,(5.10e) S5=A66B22 −A26B26,(5.10f) T0=A11A66 −A2 16,(5.11a) T1= 2A11A26 −2A12A16,(5.11b) T2= 2A16A26 +A11A22 −A2 12 −2A12A66,(5.11c) T3= 2A16A22 −2A12A26,(5.11d) T4=A22A66 −A2 26.(5.11e) Finally, the coefficients pjand qjcan be estimated by substituting the µj roots into (5.6a) and (5.6b). A total of eight complex coefficients conjugated two-by-two is obtained as a solution for µj,pjand qj. 5.2.2 Resultant forces and moments from Φj The expressions for the resultant forces and moments by unit length, Nij and Mij, are obtained by substituting (5.5) into the compatibility equations in (2.35), and subsequently into (5.1). See [19]. N1= 2Re {4 ∑ j=1 (d1jΦ′ j)},N2= 2Re {4 ∑ j=1 (d2jΦ′ j)}, N12 = 2Re {4 ∑ j=1 (d3jΦ′ j)},M1= 2Re {4 ∑ j=1 (e1jΦ′ j)}, M2= 2Re {4 ∑ j=1 (e2jΦ′ j)},M12 = 2Re {4 ∑ j=1 (e3jΦ′ j)}. (5.12) Note that the summation only considers the first four functions Φ′ j. This is because the eight original functions Φ′ j(zj)have been set to be conjugated two-by-two, in order to produce a real solution for Nij and Mij (the same procedure was followed in Chapters 3 and 4). 119 General methodology for stretching-bending coupling Φ′ jfrom (5.12) is given by, Φ′ j=dΦj dzj .(5.13) And the coefficients dij and eij,                  d1j d2j d3j e1j e2j e3j                  =          A11 A12 A16 B11 B12 B16 A12 A22 A26 B12 B22 B26 A16 A26 A66 B16 B26 B66 B11 B12 B16 D11 D12 D16 B12 B22 B26 D12 D22 D26 B16 B26 B66 D16 D26 D66                           pj qjµj pjµj+qj −1 −µ2 j −2µj                  .(5.14) Furthermore, from the equilibrium equations in (5.2a) and (5.2b), the dij coefficients are related as follows, d1j=µ2 jd2j=−µjd3j.(5.15) And from the equilibrium of moments in (5.2c), e1j=−µj(2e2j+µje3j).(5.16) 5.3 General methodology for stretching-bending coupling Note that the complex functions Φjintroduced in this chapter are analogous to the functions Φjof Chapter 3 and w′ jof Chapter 4. Hence, the methodology to be used for the resolution is analogous to the ones previously developed. 5.3.1 Approximation function for Φj The Laurent series used in (3.28) and (4.18), can be also applied in this case. Assuming a plate with a cutout, the Φjfunction can be defined as, Φj(ζj) = (C′ 0j+C′′ 0jzj)ln ζj+C0j+ N ∑ n=1 (Cnjζ−n j+C∗ njζn j).(5.17) On the contrary, if there is no cutout the same series used in (3.29) and (4.19) can be applied, Φj(ζj) = C0j+ N ∑ n=1 (C∗ njzn j).(5.18) Coupled stretching-bending analysis 120 5.3.2 Boundary conditions The boundary conditions will be used to estimate the C0j,C′ 0j,C′′ 0j,Cnj and C∗ 0jcoefficients. Once that the µjroots have been obtained, the present method is perfectly analogous to the non-coupled methodologies presented in Chapters 3 and 4. However, there is a slight difference, in the coupled methodology there are a total of eight Φjfunctions, which can be reduced to four by using the Re function. On the contrary, the uncoupled methodologies require just four Φjfunctions (or two after the reduction). The expressions to define the boundary conditions were introduced in Sections 3.4.2 and 4.3.2 for stretching and bending conditions respectively. However, the expressions must be rewritten to account for some minor modifications, e.g. the change of jsubindex that is now swept from 1to 4. First, the stretching boundary conditions in local n−sdirections are given by (3.34) and (3.36). Using the expressions of Nij in (5.12) and the relation between the dij coefficients in (5.15), the boundary conditions in terms of the stretching loads become ±∫s 0 Tsds= 2Re {4 ∑ j=1 [(−µjcos θ+ sin θ)d2jΦj]} s 0 ,(5.19a) ±∫s 0 Tnds= 2Re {4 ∑ j=1 [(µjsin θ+ cos θ)d2jΦj]} s 0 ,(5.19b) and in terms of the in-plane displacements, us= 2Re {4 ∑ j=1 [(pjcosθ +qjsinθ) Φj]},(5.20a) un= 2Re {4 ∑ j=1 [(pjsin θ−qjcos θ) Φj]}.(5.20b) Secondly, the bending boundary conditions can be rewritten from (4.22) and (4.24), by making use of (5.12) and (5.16). In terms of the bending moments, ∓∫s 0 Mnds= 2Re {4 ∑ j=1 [(e2jcos θ+e1j µj sin θ)Φj]},(5.21a) 121 General methodology for stretching-bending coupling ±∫s 0 V∗ nds−cs = 2Re {4 ∑ j=1 [(−e2jsin θ+e1j µj cos θ)Φj]}.(5.21b) It should be recalled that V∗ nis the integral of the effective transverse shear loads as defined previously in (4.21), V∗ n=∫Vnds. (5.21c) The rotation boundary conditions in terms of the deflection’s derivatives are ∂w ∂s = 2Re {4 ∑ j=1 [(cos θ+µjsin θ) Φj]},(5.22a) ∂w ∂n = 2Re {4 ∑ j=1 [(−sin θ+µjcos θ) Φj]}.(5.22b) It should be noted that the upper and lower signs of (5.19) and (5.21) are given for the external (ΓE) and the internal (ΓC) boundaries of the plate respectively. The external edges (ΓE) must be swept in clockwise direction and the internal (ΓC) in counterclockwise direction as depicted initially in Figure 3.1. Otherwise the signs of the left-hand side of (5.19) and (5.21) must be the opposite. Cutout boundary The cutout boundary can be used to reduce to half the number of unknowns without placing collocation points on it, as it has been done previously in Sections 3.4.2 and 4.3.2. This procedure is original from the present dissertation, although it is a simple application of the work done in previous chapters. Once again, it is convenient to rewrite the expressions in order to account for some minor changes in the nomenclature and to consider j= 1,2,3,4. It should be recalled that at the cutout boundary ζ1=ζ2=ζ3=ζ4= σ=eiψ. The first coefficients to solve are those associated with the noncompensated loads at the cutout, which are C′ 0jand C′′ 0j. As in (3.39), the uni-evaluated displacement field at ψ= 0 and ψ= 2πrequires that Im {4 ∑ j=1 (pjC′ 0j)}= 0,(5.23a) Coupled stretching-bending analysis 122 Im {4 ∑ j=1 (qjC′ 0j)}= 0.(5.23b) And doing the same with the forces as in (3.40), Im {4 ∑ j=1 (µjd2jC′ 0j)}=P1 4π,(5.24a) Im {4 ∑ j=1 (µjd2jC′ 0j)}=−P2 4π,(5.24b) Similarly, as in (4.25), the deflection uni-evaluated field requires that Im {4 ∑ j=1 (C′ 0j)}= 0,(5.25a) Im {4 ∑ j=1 (µjC′ 0j)}= 0,(5.25b) Im {4 ∑ j=1 (C′′ 0j)}= 0,(5.25c) Im {4 ∑ j=1 (µjC′′ 0j)}= 0.(5.25d) And finally, by using the moments as in (4.26), Im {4 ∑ j=1 (e2jC′ 0j)}=m1 4π,(5.26a) Im {4 ∑ j=1 (e1j µj C′ 0j)}=m2 4π,(5.26b) Im {4 ∑ j=1 (e2jC′′ 0j)}= 0,(5.26c) Im {4 ∑ j=1 (e1j µj C′′ 0j)}=P3 4π.(5.26d) 123 General methodology for stretching-bending coupling The set of expressions conformed by (5.23), (5.24), (5.25) and (5.26) allows the estimation of the complex coefficients C′ 0jand C′′ 0j. If the loads and moments applied at the cutout edge are self-compensated, all these terms will be strictly zero. In addition, the stretching loads and bending moments at the hole boundary can be approximated by making use of a Fourier series. To this end, the previous expressions (3.43) and (4.29) are rewritten by making use of the superscripts (S)and (B)to indicate stretching and bending respectively. 4 ∑ j=1 [d2jΦj(σ) + d2jΦj(σ)]= λ1=P2 2πψ+α(S) 0+ N ∑ n=1 (α(S) nσn+α(S) nσ−n),(5.27a) 4 ∑ j=1 [µjd2jΦj(σ) + µjΦjd2j(σ)]= λ2=−P1 2πψ+β(S) 0+ N ∑ n=1 (β(S) nσn+β(S) nσ−n).(5.27b) 4 ∑ j=1 [e1j µj Φj(σ) + e1j µj Φj(σ)]= λ3=P3a 2πψcos ψ−m2 2πψ+α(B) 0+ N ∑ n=1 (α(B) nσn+α(B) nσ−n),(5.27c) 4 ∑ j=1 [e2jΦj(σ) + e2jΦj(σ)]= λ4=−P3b 2πψsin ψ−m1 2πψ+β(B) 0+ N ∑ n=1 (β(B) nσn+β(B) nσ−n).(5.27d) Note that the right-hand side of the expressions in (5.27) have been denoted by λj. Furthermore, as in previous chapters, the number of unknowns can Coupled stretching-bending analysis 124 be reduced to half by using these expressions. In fact, operating with (5.27) results in a relation that links Φ3and Φ4with Φ1and Φ2, i.e. Φ3=f(Φ1,Φ2),Φ4=f(Φ1,Φ2).(5.28) Or in matrix notation, by using a similar nomenclature than in Chapters 3 and 4 with the δkl coefficients, {Φ3 Φ4}=[δ31 δ32 δ′ 31 δ′ 32 δ41 δ42 δ′ 41 δ′ 42 ]       Φ1 Φ2 Φ1 Φ2        +[ˆ δ31 ˆ δ32 ˆ δ33 ˆ δ34 ˆ δ41 ˆ δ42 ˆ δ43 ˆ δ44 ]       λ1 λ2 λ3 λ4        .(5.29) No explicit expression of the coefficients δkl will be given, since it would be quite complex to be useful. On the contrary, the linear system of equations (5.27) can be written in matrix form as [Λ]{ΦD ΦU}= 0, with ΦD=       Φ3 Φ4 Φ3 Φ4        , and ΦU=                          Φ1 Φ2 Φ1 Φ2 λ1 λ2 λ3 λ4                          .(5.30) Using the Matlab function rref in order to produce the reduced row echelon form of Λ, the first four columns of Λwill be reduced to the identity, while the rest of the columns will define the coefficients δkl and ˆ δkl. Finally, the coefficients Cn3and C∗ n3can be substituted into the formulation by, C03 =δ31C01 +δ32C02 +δ′ 31C01 +δ′ 32C02 +ˆ δ31α(S) 0+ˆ δ32β(S) 0+ˆ δ33α(B) 0+ˆ δ34β(B) 0,(5.31a) . . . 125 General methodology for stretching-bending coupling Cn3=δ31Cn1+δ32Cn2+δ′ 31C∗ n1+δ′ 32C∗ n2 +ˆ δ31α(S) n+ˆ δ32β(S) n+ˆ δ33α(B) n+ˆ δ34β(B) n,(5.31b) C∗ n3=δ31C∗ n1+δ32C∗ n2+δ′ 31Cn1+δ′ 32Cn2 +ˆ δ31α(S) n+ˆ δ32β(S) n+ˆ δ33α(B) n+ˆ δ34β(B) n.(5.31c) Similarly, the coefficients Cn4and C∗ n4are given by, C04 =δ41C01 +δ42C02 +δ′ 41C01 +δ′ 42C02 +ˆ δ41α(S) 0+ˆ δ42β(S) 0+ˆ δ43α(B) 0+ˆ δ44β(B) 0,(5.32a) . . . Cn4=δ41Cn1+δ42Cn2+δ′ 41C∗ n1+δ′ 42C∗ n2 +ˆ δ41α(S) n+ˆ δ42β(S) n+ˆ δ43α(B) n+ˆ δ44β(B) n,(5.32b) C∗ n4=δ41C∗ n1+δ42C∗ n2+δ′ 41Cn1+δ′ 42Cn2 +ˆ δ41α(S) n+ˆ δ42β(S) n+ˆ δ43α(B) n+ˆ δ44β(B) n.(5.32c) External boundary First, the stretching boundary conditions in any external boundary ΓEmust be defined by making use of (5.19) and (5.20), for the stretching conditions defined in Table 3.1. 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