Study and design of a wing monocoque structure with composite materials

Study and design of a monocoque wing structure with composite materials BACHELOR’S DEGREE THESIS Bachelor’s degree in Aerospace Vehicles Engineering Miguel Alejandro Pareja Mu˜noz Director: Llu´ıs Gil Espert June 2016 DOCUMENT 1.- REPORT Universitat Polit`ecnica de Catalunya Escola Superior d’Enginyeries Industrial, Aeroespacial i Audiovisual de Terrassa ”Inspiration unlocks the future.” The Wind Rises (2013) i Acknowledgements Firstly, I would like to express my sincere gratitude to my advisor Prof. Llu´ıs Gil for the continuous support during the making of my Bachelor’s degree thesis, for his patience, motivation, and immense knowledge. His guidance has helped me in all the time of research and design, and also in the writing of this thesis. My most sincere thanks also goes to my team mate and friend, Roger Serra. He has selflessly dedicated his precious time to help me so many times, not only allowing me to reach the solution of my problems, but also making me have a great time with him. I also want to acknowledge my team mate Oriol Chandre for his great company at the Trencal`os Team workshop, in which we both have spent several hours working on our thesis. He always being in a great mood and his energy have always motivated me and influenced me very positively. Lastly, my highest appreciation goes to my family and friends, without whom I would not be who I am today. Special thanks goes to my mom, as she has supported me wholeheartedly throughout the making of this thesis - both in the good and bad moments I have experienced. ii Abstract Trencal`os Team - an aeromodelling research team, is looking to enlarge its know-how about composite-made structures. For their own experience, the best option to win the competitions they attend is with a monocoque wing structure made with composite materials. Me, as a member of Trencalos Team, I have taken this oportunity to help in the study of that matter. The thesis covers the study of the main the knowledge necessary to approach a design of such a structre in depth. This type of structure is more complex to characterize than a classical truss structure - its structural behaviour is influenced by several parameters, i.e. is multi-parametric. This parameters/features have been studied, as well as the governing laws that describe the structural beeahviour of tgis structures. Additionally, the design of a structure of this kind has been performed following a design methodology that is based in optimisation of two objectives. The design of such a structure represents a problem with multiple possible objectives, and for the sake of simplicity, just two of them have been addressed. The design methodology has been implemented in the following way: making a set of initial cases, analyse them and then apply an optimisation procedure to discard cases and find the best ones. After doing that, more cases have been made from the best cases of the last iteration. The modelling of the geometry in this thesis has been done with CATIA V5R20. The structural analysis has been performed with AbaqusTM. From the methodology premises that have been set in order to delimit such a multi-parametric study, results have been obtained. These mainly have been useful to observe the trends/effects of each of these design parameters in the final structure behaviour. Besides, an optimal solution has been found following an optimisation procedure, which is interesting from the perpspective of familiarising with this kind of tool, as any complex engineering problem depends on a lot of parameters and they are required to solve it. With all this accomplished, conclusions have been extracted about which are the best parameters combinations to include in a monocoque wing structure made with composite materials. iii Summary The thesis is divided in 8 chapters: Introduction, State of the Art, Methodology, Results, Conclusions, Future Work, Environmental Impact, Budget and Annxes. In the Introdcution chapter, a brief and concise summary of the aim, scope and objectives of the thesis is delivered. Also, the justification and a planning of the same is attached. In the State of the Art chapter, all the bibliographical research is done. All the infomation necessary to build a base on how monocoque wing structures made of composite materials behave is given there. Also, basic information about the design tools used for the design process is given here. In the Methodology chapter, the premises on which the design is based on are detailed. Then, the design procedure - set of steps - followed to obtain an optimal solution is explained. And finally, the process to prepare, execute and post-process an structural analysis is explained. In the Results chapter, the results obtained are thoroughly explained with all kinds of details and different supports - i.e. tables, plots, figures, etc. In the Conclusions chapter, all the information collected in the thesis is analysed from a general point of view and some conclusions are extracted. In the Future Work chapter, a suggestion of how the work perfomed in this thesis could be continued¸c, and in what direction, is explained. In the Environmental Impact, an insight of the impact of what has been done to accomplish this thesis on the environment is explained. In the Budget chapter, a budget of how much a project like this one would cost if it was undertaknen professionally is given. The budget is done separately in a different document. And for last, Annexes are attached providing complementary information that is of interest. The annexes are done separately in a different document. iv Contents Acknowledgements ii Abstract iii Summary iv List of Figures vii List of Tables xi List of Abbreviations and Symbols xiii 1 Introduction 1 1.1 Aim ........................................ 1 1.2 Scope ....................................... 1 1.3 Requirements................................... 2 1.4 Objectives..................................... 3 1.5 Justification.................................... 3 1.5.1 Identification of the Need . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5.2 Usefulness of the Study . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Study’sApproach................................. 6 1.7 Planning...................................... 6 1.7.1 Brief Description of Tasks . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7.2 Detailed Relation Between Tasks . . . . . . . . . . . . . . . . . . . . 8 1.7.3 Scheduling ................................ 8 2 State of the Art 10 2.1 HistoricalPreamble................................ 10 2.2 StructuralConcepts ............................... 15 2.3 CompositeMaterials............................... 18 2.3.1 Main Properties of Composites . . . . . . . . . . . . . . . . . . . . . 20 2.3.2 Classification of Composites . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.3 Laminae and Laminates . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.3.1 Laminae ............................ 26 2.3.3.2 Laminates ........................... 28 2.4 Laminae Mechanical and Physical Properties . . . . . . . . . . . . . . . . . 31 2.5 Classical Laminated Plate Theory . . . . . . . . . . . . . . . . . . . . . . . 35 2.5.1 Kinematics ................................ 37 2.5.2 TheMaterialLaw ............................ 38 2.5.3 Resultant Forces and Moments . . . . . . . . . . . . . . . . . . . . . 38 2.6 Buckling of Laminated Plates . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.7 Finite Element Analysis of Laminates . . . . . . . . . . . . . . . . . . . . . 43 v 2.7.1 Basic FEM Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.7.2 General FEM Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7.3 AbaqusTM ................................. 48 2.7.4 ShellElements .............................. 49 2.8 FailureCriteria.................................. 50 2.8.1 HashinCriterion ............................. 50 2.8.2 Tsai-Hill Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.9 ParetoOptimality ................................ 51 3 Methodology 54 3.1 PremisesoftheDesign.............................. 54 3.1.1 Type of Structural Testing . . . . . . . . . . . . . . . . . . . . . . . 54 3.1.2 Structural Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1.3 Geometrical Features of the Wing . . . . . . . . . . . . . . . . . . . 55 3.1.4 Load-bearing Elements Employed . . . . . . . . . . . . . . . . . . . . 56 3.1.5 Materials Employed and its Properties . . . . . . . . . . . . . . . . . 57 3.1.6 Maximum Weight of the Structure . . . . . . . . . . . . . . . . . . . 59 3.1.7 Minimum Buckling Constant . . . . . . . . . . . . . . . . . . . . . . 59 3.1.8 Variation of Parameters Criterion . . . . . . . . . . . . . . . . . . . . 60 3.1.9 Parameters of the Design . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Design Process Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2.1 InitialTests................................ 62 3.2.1.1 Mesh Convergence . . . . . . . . . . . . . . . . . . . . . . . 62 3.2.1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 63 3.2.1.3 Load Definition . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2.1.4 Buckling Modes Check . . . . . . . . . . . . . . . . . . . . 67 3.2.1.5 First Tendencies Observed . . . . . . . . . . . . . . . . . . 67 3.2.2 StartingPoint .............................. 68 3.2.2.1 Decision Making . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.3 Improvements............................... 69 3.2.4 FinalDecision .............................. 70 3.2.5 Flowchart of the Design Process . . . . . . . . . . . . . . . . . . . . 70 3.3 Analysis Process Description . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3.1 Pre-processing .............................. 72 3.3.1.1 Geometry Preparation . . . . . . . . . . . . . . . . . . . . . 72 3.3.1.2 AbaqusTM ModelMaking................... 73 3.3.2 Simulation of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . 76 3.3.3 Post-processing.............................. 77 3.3.4 Flowchart of the Analysis Process . . . . . . . . . . . . . . . . . . . 78 4 Results 80 4.1 InitialTestsResults ............................... 80 4.1.1 ConvergedMesh ............................. 80 4.1.2 FinalEncastre .............................. 82 4.1.3 Final Load Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.1.4 Buckling Modes Verification . . . . . . . . . . . . . . . . . . . . . . . 87 4.1.5 Trends and Structural Behaviour . . . . . . . . . . . . . . . . . . . . 95 4.2 First Batch of Cases - Starting Point . . . . . . . . . . . . . . . . . . . . . . 96 4.3 Second Batch of Cases - Improvement 1 . . . . . . . . . . . . . . . . . . . . 100 4.4 Third Batch of Cases - Improvement 2 . . . . . . . . . . . . . . . . . . . . . 101 4.5 Fourth Batch of Cases - Improvement 3 . . . . . . . . . . . . . . . . . . . . 102 4.6 Decision after Improvements 1, 2 and 3 . . . . . . . . . . . . . . . . . . . . 106 vi 4.7 Fifth Batch of Cases - Improvement 4 . . . . . . . . . . . . . . . . . . . . . 108 4.8 Sixth Batch of Cases - Improvement 5 . . . . . . . . . . . . . . . . . . . . . 110 4.9 FinalResults ...................................112 5 Conclusions 114 5.1 Conclusions about the Results . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2 Conclusions about the Numerical Model . . . . . . . . . . . . . . . . . . . . 116 5.3 Conclusions about the Design Process . . . . . . . . . . . . . . . . . . . . . 116 5.4 ProjectOverview.................................117 6 Future Work 118 7 Environmental Impact 119 Bibliography 121 vii List of Figures 1.1 Lancair IV. Retrieved from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The b.ONE, Trencal`os Team’s aircraft that competed in the Air Cargo Challenge2015................................... 5 1.3 Gantt chart of the organisation of tasks in time. . . . . . . . . . . . . . . . 9 2.1 Chanute’s biplane glider made of truss construction. Retrieved from [2]. . . 10 2.2 The Wright flyer. Retrieved from [3]. . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Bleriot’s airplane. The first monoplane. Retrieved from [4]. . . . . . . . . . 11 2.4 The Deperdussin, the first airplane with monocoque structure in the fuselage. Retrieved from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 The Junkers J1. Retrieved from [6]. . . . . . . . . . . . . . . . . . . . . . . 13 2.6 An American Airlines Boeing 707. Retrieved from [7]. . . . . . . . . . . . . 13 2.7 Research aircraft X-29. Retrieved from [8]. . . . . . . . . . . . . . . . . . . 14 2.8 Rate of materials employed in the Boeing 787. Retrieved from [9]. . . . . . 14 2.9 Example of a truss structure of an aircraft fuselage. Retrieved from [10]. . . 15 2.10 Monocoque vs. Semi-monocoque structure of a fuselage tail boom. Retrievedfrom[11]. ............................... 16 2.11 Fatigue curve of an alluminium alloy (a) and a unidirectional composite (b).Retrievedfrom[12].............................. 20 2.12 Comparison of different composites and an aluminium alloy specific properties. Retrieved from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.13 Stress-strain curves of a metal and a composite. Retrieved from [12]. . . . . 21 2.14 Comparison of characteristics of different materials: a) density (kg/m3) b) tenisle fracture strength (MPa) c) modulus of elasticity (MPa) d) price per unit mass. Retrieved from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.15 Composites classification depending on the reinforcement type. Retrieved from[13]. ..................................... 23 2.16 Anisotropic material. Retrieved from [14]. . . . . . . . . . . . . . . . . . . . 25 2.17 Rotation of axes. Retrieved from [14]. . . . . . . . . . . . . . . . . . . . . . 26 2.18 Laminae representation in space. Retrieved from [12]. . . . . . . . . . . . . 26 2.19 Variation of the elasticity modulus with θfrom the longitudinal El(fibre mudulus) to the transverse Et(matrix modulus). Retrieved from [12]. . . . 27 2.20 Laminate example with 0◦, -θ◦,θ◦, 90◦layer arrangement. Retrieved from [15]. ........................................ 28 2.21 Variation of properties with volume fraction modification of the different layers. Retrieved from [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.22 Deformation due to the anisotropy of composites.Retrieved from [12]. . . . . 29 2.23 Laminate representation in space. Retrieved from [12]. . . . . . . . . . . . . 30 2.24 Idealisation of monolayer of fibre and matrix with longitudinal force applied. Retrievedfrom[12]. ............................... 32 2.25 Idealisation of monolayer of fibre and matrix with transverse force applied. Retrievedfrom[12]. ............................... 33 viii Chapter 1 Introduction 1.1 Aim The purpose of this thesis is the design of a moncoque structure for a wing with composite materials. To achieve such a goal, thorough research on monocoque structures, composite materials, and Finite Element Method (FEM) software will be necessary. Some performance requirements for this structure will also have to be set. 1.2 Scope The scope of the project, divided in groups, is the following: Information research •Historical precedents: what wing monocoque structures made of composite materials have been made, are being made, and are intended to be developed. What are the main trends and characteristics of this specific type of structures. •Composite materials: get to know their main features, i.e. anisotropy, failure mechanisms, constitutive models, etc. Be able to predict which combination of their wide range of parameters will be best for the approach of the structure design, i.e. know how to overcome buckling issues or how to reinforce an under shear stress region by changing the laying sequence or the thickness of the layers they are made of. Get an insight of the theories that explain their mechanical behaviour. •Monocoque structures: get to know their pros and cons, what are the design parameters to take into account and be able to use them when approaching the structure of the wing. •FEM software: with a commercial FEM software chosen, get the most accurate result of the desired structure. Thus, a thorough examination of the solvers, models (static, dynamic, buckling, fatigue, etc), and other features will be necessary. Design process •Structural requirements: set the requirements for the wing’s structure. The maximum weight, the ultimate load (flight performance) and others, must be defined. Also how the wing is going to be sized is important to be defined: static load, dynamic load, buckling or fatigue. 1 Chapter 1. Introduction •Ease of use of the FEM software: get used to work with the software selected and be able to confront the design process of the wing monocoque structure made of composite materials. •Design results: obtain proof of a structure that fulfils the requirements settled. As for any engineering problem, there will be more than one solution possible. Therefore those multiple valid structures must be discussed and classified for its pros and cons, function that may be able to perform or other important aspects that must be kept in mind when developing an engineering project. •Optimisation of the process: finally, think of a way in which the whole process undergone could be improved in as much ways as possible, i.e. reduce the time required, obtain more accurate results, automation of the process, etc. Project overview •Conclusions: extract conclusions of the work done and examine the final point of the project and how it could be improved or developed from that point on. •Impact: do a brief study of how the work done influences the field of study, as well as the economical or environmental effect of the same. Deliverables Elaboration of all the deliverables of the project, which are: •Areport. •If necessary, annexes of the extra information/work done. •Abudget. 1.3 Requirements To accomplish the goal of the thesis the following requirements must be fulfilled: •The wing structure dimensioning will just take into account static loads and buckling phenomena. No dynamic loads like fatigue loads or aeroelastic loads will be considered. •The monocoque structure must have buckling limiting elements that reduce the buckling length. Also it must have shear transmitting elements. •The composite materials used must be fibrous based. •The matrix used must be epoxy. •The dimensions of the wing that will be designed will be: 120 cm of wingspan x 19 cm of chord (rectangular wing). The airfoil used is a CLARKY. •Computer-based: improve technical characteristics or use a well-equipped computer in order to run the simulations in the most optimal way. 2 Chapter 1. Introduction 1.4 Objectives The main objectives of this thesis are: •Obtain the best solution among all possible solutions that the set of designing variables can yield. It is important to find at least the trends of these variables towards the best solution, as the best solution can only be sought - and proved to be the best one - by means of optimisation methods. The secondary objectives of this thesis are: •Achieve good knowledge about a FEM software. •Get used to the design process of an engineering project. 1.5 Justification The motivation behind the selection of this matter of study lies on two main aspects: firstly, the subjects that have enthralled me the most throughout the degree have been Aerospace Structures and Science of Materials. I have always been passionate about how the right combination of materials and the proper geometry allow a structural system to withstand loads that may even surpass by large its own weight. And secondly, and the most decisive one for the making of this project, I have been enrolled in an aeromodelling research team - Trencal`os Team - for two years now. The experience of being part of a group of people that dedicates its precious time to explore new technologies has motivated me a lot. Also, applying the knowledge gained during the degree to compete against other universities in international competitions, has pushed me to go deeper into the fields of study that I love the most. Adding up to those rock solid reasons, this thesis will serve as a leap forward in Trencal`os’ know-how, as the kind of structure that I am going to study and design has never been developed by the team. That fact adds to my motivation and willingness to pursue the goal of this thesis in a substantial way. Next is detailed the identification of the need that this thesis will cover - from a wider point of view to a more specific one, and the usefulness of the study that I am going to perform. 1.5.1 Identification of the Need When designing an aircraft’s structure, its whole is divided in substructures of the different parts of the craft. Later, elements or other substructures are designed to join together all the parts. Usually different groups of people are in charge of developing each substructure. Likewise, each part’s structure is confronted at different times, following a sequence. Addressing the design of a wing’s structure is an essential task needed when an aircraft is being designed, for it is the main lifting device of the plane as well as the main limiting element in terms of its flight performance. Being so, the goal of the project is justified as it can be used as a guideline for anyone who is willing to design a monocoque wing structure made of composite materials. 3 Chapter 1. Introduction A perfectly fitting example of that need is Trencal`os Team, whose main purpose is designing unmanned aerial vehicles (UAV) to compete in aerospace engineering-related international competitions. For that very reason, the design of a wing must be tackled and, for major facts that will be exposed further in the text, a monocoque composite material structure seems to be the most effective one to choose. 1.5.2 Usefulness of the Study In terms of wing structures made of composite materials, big airliners do not use full-made composite materials wing structures but instead they combine these materials with historically and well-known metals that have been long used in the aerospace industry. A full-made composite material wing structure in an aircraft is more common in private/training, two/four-seat, single-engine aircrafts (e.g. the Cirrus SR20), as well as in drones or radio-controlled aircrafts that usually are scaled up or simply have smaller dimensions. Monocoque structures - structures designed so all the loads are taken by the skin and there is no internal framework to assist - are not the most used approach in the history of aviation. Other techniques like the semi-monocoque (or stressed skin) have been more employed as, for example, they tackle better the distribution of loads in the structure, avoiding thus a probable stress concentration that may lead to fatal failure. Despite that, monocoque structures do have their perks and have also been used in aircrafts (e.g., the Lancair IV). Up to a certain dimension of the aircraft, monocoque structures can be lighter than a semi-monocoque structure composed by an internal skeleton plus the outer skin. Furthermore, as it is a continuous shell structure there is plenty of space available that can be used for the possible necessity (e.g. fuel, baggage, control surfaces deployment system storage, etc). Figure 1.1: Lancair IV. Retrieved from [1]. So, aside from the important usefulness for Trencal`os Team that I have already mentioned before, the the kind of aircraft that would best fulfil the wing’s monocoque composite material structure intended to be studied and designed in this thesis, is a small airplane: single-engine, two or four seats, maximum wingspan of 10-11 m approximately. Therefore, this project can be useful for a company or a group of people that want to undertake the design of an aircraft of such features and their inclined to this kind of structure for the wing. 4 Chapter 1. Introduction Coming back to Trencal`os Team’s interests, it is important to stress how the UAVs that the team designs should look like, i.e, what pilar features must it have to maximize the score to be obtained. To date, Trencal`os has only taken part in the biannual international competition Air Cargo Challenge (ACC), but is currently seeking new horizons as the SAE Aero Design (i.e, Society of Aerospace Engineers) or the DBF (i.e, Design/Build/Fly) competitions. In these competitions, a UAV has to be designed under some regulations in order to undergone a series of tests and the lightest and most resistant structure can make the difference to win the contest. It has to be taken into account that the aircrafts in the ACC have to carry as many payload as possible and be as fast as possible in order to score the highest points. For that very reason the weight is a crucial aspect. Likewise, the manoeuvrability of the craft is also addressed meticulously. That’s why these UAVs have no bulky unnecessary parts and the main structure is the wing (3-3.5 m wingspan). Figure 1.2 depicts the essence of the airplanes of these competitions. Figure 1.2: The b.ONE, Trencal`os Team’s aircraft that competed in the Air Cargo Challenge 2015. Historically, Trencal`os Team has always used plywood and balsa wood as the main construction materials. In the last edition the team took a step up and changed to a high-performance foam reinforced with carbon fibre. But still, monocoque composite materials wing structures are the best for the ACC as the final results of 2015 edition proved: the first and second teams in the final ranking used this technique and stood out from the rest of teams. For that very reason, and with the next contest in mind - Air Cargo Challenge 2017 – the team has taken the opportunity of doing research projects about this subject. During this semester a wing with this characteristics will be built by the team, advised and powered by myself and another member who is doing his bachelor’s thesis of the construction process. To conclude, this project can take the team a step up and let it expand its know-how, making Trencal`os Team more competitive for the next contest(s). Moreover, it will serve me to gain more knowledge in the field of aerospace engineering that I am keener on. 5 Chapter 1. Introduction 1.6 Study’s Approach To address this project I am going to follow three main steps or parts, as stated in the scope section: information research or education in the field of study, design process or evolution with the designing tools towards the final solution (with the prior knowledge basis set), and a final project overview to give closure to the project. In the information research I want to study the main features of each distinctive characteristic of the wing structure: what parameters characterize a monocoque structure and a composite materials structure and what are the trends they follow to maximize the structures behaviour. Doing so I will not tackle the design process with the FEM software completely blind but I will be able to know which combination of parameters may be more promising than others. As I will need time to learn how to use the software, a good information research will be very useful in managing my time. In this part of the project, some requirements will be necessary to be settled and the degree of difficulty of the final structure will rely on them. So, when the time comes to define them, a good decision based on time left for delivery of the project will have to be made. Once the requirements are set, a model in the program will have to be made and then work on it until some results that agree with the requirements and the project goals are obtained. After that, a closer look to the process followed will be given and analysed in order to propose improvements of any kind. Finally, some conclusions will be extracted as well as the impact of the project at different levels: economical, environmental or others. The critical elements of this approach are the following: taking the exact necessary time to accomplish the first part so there’s enough time to perform the design through the FEM software; choosing adequate requirements for the structure in accordance to the time left when that point of the project is reached; and being able to obtain a good quality result with a FEM commercial software on time as this programs are very sophisticated and complicated to use. I feel this is the best way I can confront this project. The only disadvantage of the approach of my study is not having enough certainty of the exact timing that every little part is going to take me. That situation only leaves with the one possible way of action: perseverance and hard work. 1.7 Planning 1.7.1 Brief Description of Tasks The main tasks or work packages that must be accomplished in order to achieve the goal of the thesis are the following: •Project charter (DE): development of the project charter, document used to outline the project objectives and constraints, the directions concerning the solution, the in-scope and out-of-scope items, the reasons for undertaking the project and an 6 Chapter 1. Introduction approximate planning of the same. It is very useful to define and make clear how exactly the project is going to be done. •Follow-ups (DE): during the time doing the project, some follow-up reports will be necessary to be done. These will come handy to think of which point of the project you are at and will help in the whole planning of the project. •Information research (IR): prior to any designing task, whatever the type or field of it, a foundation knowledge is necessary to be achieved or consolidated. For that very reason, thorough research and study of the information more relevant to the attainment of the project will be undergone. That is: composite materials, monocoque structures, general theory of structures, FEM main theoretical basis and how the method is developed (i.e. solvers, models, etc), structural regulations and naturally the state of the art in order to grasp the trends and reasons in the history of monocoque wing structures made of composite materials. •Set the structural requirements (TD): before tackling the structures’ design, a set of requirements that the structure must fulfil must be defined. The maximum weight of the structure, the ultimate load it has to withstand or the maximum displacement of the wing tip has to be defined. Also how the wing is going to be sized is important to be defined: static load, dynamic load, buckling or fatigue. •FEM software learning (TD): in order to achieve the goal of this project, an analysis of the structure will have to be undergone in a commercial software. To do so I will have to learn how the program works and get used to it. Accordingly, I will have to read from beginning to end the program’s manual and extract the most valuable information for my purpose. •Wing structure design (TD): once I handle the program with ease I will develop a model of the structure and work on it, through the iterations that are necessary, and obtain the set of possible results, among the many combinations of parameters, that fulfil the requirements. •Design process optimization (TD): when the design process has been completed and results have been accomplished, the whole process will be submitted to a close examination seeking for ways of improving it: reduce the time needed, obtain more accurate results, automation of the process, etc. •Report writing (DE): as long as information is being read it will be summarized in documents. Properly scheduled dates will be set to have certain parts of the final report already written with the desired cohesion and style. •Annexes and budget writing (DE): at the end of the project the extra information that has been used must be written in the annexes and the budget of the project must be elaborated. •Conclusions and impact of the project (PO): at the end of the project some conclusions will have to be extracted. Likewise, the environmental impact, as well as the economical or technical impact of the project will have to be evaluated. DE: deliverable; IR: information research; TD: technical design; PO: project overview 7 Chapter 1. Introduction 1.7.2 Detailed Relation Between Tasks The relation between the tasks explained in the previous section will be depicted with the following table: Code of task Task identification Preceding task IR Information research IR-1 Composite materials IR-2 Monocoque structures IR-3 Theory of structures IR-4 FEM IR-5 Structural regulations IR-6 Precedents TD Technical Design TD-1 Structural requirements settlement IR TD-2 FEM software learning IR TD-3 Wing structure design IR, TD-1, TD-2 TD-4 Design process optimisation IR, TD-1, TD-2, TD-3 PO Project overview PO-1 Conclusions and impact of the project IR, TD DE Deliverables DE-1 Project charter DE-2 Follow-up 1 DE-3 Follow-up 2 DE-4 Follow-up 3 DE-5 Follow-up 4 DE-6 Report writing-1 IR DE-7 Report writing-2 IR, TD DE-8 Annexes and budget IR, TD, DE-6, DE-7 Table 1.1: Relation between tasks 1.7.3 Scheduling Finally, an allocation of the tasks in time has been done. To do so, a Gantt chart has been used. 8 Chapter 1. Introduction Figure 1.3: Gantt chart of the organisation of tasks in time. 9 Chapter 2 State of the Art 2.1 Historical Preamble During the early years, aircraft structural design relied more on intuition and empirical rules than theoretical principles or analysis. As time passed, aviation moved beyond the isolated, individual inventor and attracted the attention of engineers. The fields of elasticity, strength of materials, and structural mechanics were harnessed to tackle flight structures problems. The information and knowledge derived from structural theory enabled impressive improvements in airplane performance and eventually established the subject of airframe structures as an engineering specialty. Early airplane builders saw structural design and construction as a minor problem, easily solved using commonly available materials and trial-and-error procedures. Of the conventional materials available for construction of bridges and other stationary structures, only steel and a few varieties of wood were considered suitable for flight structures. Familiarity with woodworking and wood construction made wood the natural choice of aviation pioneers. Wood was readily available, inexpensive, required few specialized tools and could be easily worked by those with limited construction skills. Figure 2.1: Chanute’s biplane glider made of truss construction. Retrieved from [2]. Early aviation construction could be called the era of wood and piano wire since joining wooden trusses and other components together to form a truss-like structural framework was the easiest solution to early aircraft structural problems. Octave Chanute (1832-1910) is the father of airplane structures. Chanute was a successful civil engineer who, after his retirement from a successful career as a railway engineer, became interested in flight. Chanute used his knowledge of structural engineering – he had designed and built award winning truss railway bridges – to create the first biplane surfaces. This type of aerodynamic and structural design was adopted quickly by the Wright Brothers. Truss construction, fabricated from simple “two-force” (traction-compression) members that could be easily replaced when damaged, creates a stiff, lightweight structure. Reliable methods for analysis and design of beams and trusses had been 10 Chapter 2. State of the art the perimeters of the transversal sections. Below we have a table with some of the more relevant features discussed up to this point: Truss Monocoque Semi-monocoque Ease of design xxx x x Lightness xx xxx xx Number of elements xxx x xx Sensitivity to buckling x xxx xx Load transmission xxx x xx Table 2.1: Comparison between structural concepts of important features. Monocoque structures, the one kind that really matters for this thesis, have lightness as their most remarkable feature. A clever design, that is, the best possible arrangement of the skin - varying its composition, thickness and geometry - along the element, leads to a light and resilient configuration without the need of any kind of support, i.e, internal skeleton or external bracing with struts or wires. Despite that, under severe loading states or due to big dimensions, the skin of this type of structures needs to be thickened to overcome the issues that come with those conditions. Doing so, the lightweightness is partly lost. Adding to that is the fact that the buckling event of a portion of the skin can lead to a fatal failure of the whole structure, as the skin is the one and only structural element. Sudden loss of strength in a region due to failure effects results in a redistribution of loads in the rest of the skin. This two characteristics of monocoque structures make them less appealing in front of semi-monocoque structures. Firstly, because the increase of thickness will imply an increase of weight; and secondly because semi-monocoque structures deal better with buckling and with load distribution, as they add the extra elements in its configuration. A great solution to the main mode of failure of monocoque structures, that is, buckling, is the incorporation of elements to the structural skin. This can be seen as a contradiction, as the essence of this sort of structure is that the only load-bearing element is the skin and it doesn’t need any extra support. The addition of this elements has the purpose of reducing the buckling length at the skin and also transmit and distribute stresses from and to other parts of the skin so these do not concentrate at any specific region, which may cause a breakage of the same - i.e, failure of the skin. For the case that will be undertaken in this thesis - the design of a monocoque wing, these elements can be two: ribs or spar(s). On the one hand, ribs help in the transmission of loads between the intrados and extrados skin of the wing. They mainly transmit shear forces, avoiding thus a possible stress concentration at either the leading edge or the trailing edge of the section. They also help in giving torsional strength to the section at which they are placed, even though the skin has a good torsional resitance - as it is a closed profile. And the placing of ribs will reduce the buckling length in the longitudinal direction (along the span), helping the under-compression part of the skin, the extrados, due to the bending moment that is alaways generated in a wing caused by the lift distribution. Another way of enhancing the buckling resistance of the extrados is by enlarging the 17 Chapter 2. State of the art Iand the stiffness of the section with a stringer-like making of the skin. That is, design the skin with different layer sequences along the extrados, providing stiffened regions - i.e. by putting high stiffness materials like carbon or using a soft material as a core to increase the I, known as sandwich skins. Then, a spar or multiple spars - placed preferentially at the highest thickness region of the profile, will help mainly in the transmission of the shear force between the extrados and intrados skins. It has to be taken into account that neither the spar(s) nor the ribs have a load-bearing function. Their only purpose is the transmission of loads and the limitation of the buckling lenght. Dealing with the dissipation of the stresses generated on the wing is undertaken only by the skin. Summing up, monocoque structures may use extra elements and still be monocoque structures. The skin is the only part that will bear the loads to which is exposed the wing, and the arrangement of ribs or spars it is only done with the purpose of reducing the chances of failure - they limit buckling and avoid stress concentrations. For this very reason, and despite being persisten on this, the study of the compostion of the skin is very important for this kind of structures. This subject will be tackled in the next section, 2.3.Composite materials. 2.3 Composite Materials The term composite2could mean almost anything if taken at face value, since all materials are composed of dissimilar subunits if examined at close enough detail. But in modern materials engineering, the term usually refers to a matrix material that is reinforced with particles or fibres (reinforcement). The definition of Zagainov [15] of this type of materials is the following: ”volumetrically formed special combinations of two or more components dissimilar in form and properties, exhibiting clear boundaries between components, using the advantages of each component. Whatever the material, a composite is formed by a matrix (binder) or low-modulus material and reinforcing elements with strength and stiffness properties 10 to 1000 times higher than those of the matrix.” Many composites used today are at the leading edge of materials technology - fibre-reinforced ones being the most widely used, with performance and costs appropriate to ultrademanding applications such as aircrafts. But heterogeneous materials combining the best aspects of dissimilar constituents have been used by nature for millions of years. Ancient society, imitating nature, used this approach as well: straw was employed to reinforce mud in brickmaking, without which the bricks would have almost no strength. The fibres used in modern composites have strengths and stiffnesses far above those of traditional bulk materials (e.g, wood, aluminium, and even some steels). For instance, the high strengths of the glass fibres are due to processing that avoids the internal or surface flaws which normally weaken glass; and the strength and stiffness of the polymeric aramid fibre is a consequence of the nearly perfect alignment of the molecular chains with the fibre axis. 2The word composite is used as a synonym of composite materials. 18 Chapter 2. State of the art Of course, these materials are not generally usable as fibres alone, and typically they are impregnated by a matrix material that acts to transfer loads to the fibres, and also to protect the fibres from abrasion and environmental attack. The matrix dilutes the properties to some degree, but even so very high specific (weight-adjusted) properties are available from these materials. The fibres may be oriented randomly within the material, but it is also possible to arrange for them to be oriented preferentially in the direction expected to have the highest stresses. Such a material is said to be anisotropic - different properties in different directions, and control of the anisotropy is an important means of optimizing the material for specific applications. Composite materials have a wide range of other great properties besides anisotropy, despite the latter being the most important. In the following sections, a discussion of the features of composites and also some of the different classifications of them will be delivered. But prior to that, it is worth mentioning how composite materials influence the structure design, and specially an aircraft’s structue. A great feature associated with the development of composite structures is the possibility to design concurrently both the structure and the material for the latter. As a rule, composite materials do not exist separately from the structure and specific fabrication practice. Manufacturing processes possibilities are broad but limited. Conversely, the restrictions of those processes are the ones that play a much more important role in the design of composite structures than in that of metal structures. Designers that use isotropic materials must select an existing material and once taken that decision, they can jump into the sizing of the part or element. On the other hand, when composite materials are employed, the designer builds the material according to the needs defined by the functional element. Experience of composite application shows that, provided the design and fabrication problems are adequately taken into account, the composite structure usually has much lower number of parts, units and, especially, connecting elements. A high material utilization factor, high potentialities for automation of the production process and robotization, decrease the labour expenditures and the cost of production. Their application in flying vehicles structures has led to an increase of weight efficiency and a noticeable improvement of their flight performance. This increase directly affects its fuel efficiency. Hence, in assessing the total expenses associated with aircraft service time, the use of the composite structures may be more economically profitable as compared to an aircraft made of conventional metallic alloys despite the existing high cost of composites. Employing composites causes the so-called “cascade effect”: smaller weight →smaller lift →smaller wing →drag reduction →required thrust reduction →smaller engine weight →fuel reserve reduction →ultimate load reduction. “Investigations indicate that 1 kg of weight saved during design results in reduction of the take-off weight by 4-5 kg, being able to develop an aircraft with take-off weight of about 14 tons instead of 18 tons for an all-metal aircraft.” [15, p. 6] 19 Chapter 2. State of the art 2.3.1 Main Properties of Composites The most outstanding and worth mentioning properties of composites are the following: •Weight efficient: as it has been emphasised previously, composites are very lightweight (see Figure 2.14 a)) - low densities - and their application always entails less elements to build the same structure as it would take to with traditional materials like metals or wood - thus, reducing the overall weigth. •Less subelemetns required: composites have the advantage of needing less joints and fixations systems, thus reducing the number of parts (simplicity of the final product) and less machining time (reduction of manufacturing costs). •Complex geometries: any type of geometry can be achieved by moulding (formation process). •Anisotropy: physical properties are different depending on the direction of the applied force. •Tailor-made mechanical properties: thanks to the control of the anisotropy, enhancement of the mechanical properties along a specific direction can be achieved by means of proper fibre orientation. •Good fatigue resistance: composites withstand high strengths under fatigue loads and up to higher number of cylces than metals. This enhanced life that entails savings in the long-term costs of the product. Their specific fatigue strength (σ/ρ) is three times higher than that of an aluminium alloy and two times higher than that of a high-strength steel or titanium alloy. Besides, up to a great number of cycles (∼106), composite materials fatigue strength is equal to 90% of their static tensile strength while for aluminium alloys and steel it is just equal to 35% and 50%, respectively. See Figure 2.11. Figure 2.11: Fatigue curve of an alluminium alloy (a) and a unidirectional composite (b). Retrieved from [12]. •Mechanical properties: high tenisle fracture strength and modulus of elasticity. Even though their mechanical properties are of the same order of magnitude as some metals, their specific mechancial properties are way better that that of metals thanks to their lightweightness. See Figure 2.12 and Figure 2.14 b) and c). 20 Chapter 2. State of the art Figure 2.12: Comparison of different composites and an aluminium alloy specific properties. Retrieved from [12]. •Do not yield: unlike metals, the elastic limit corresponds to the rupture limit. See Figure 2.13. Figure 2.13: Stress-strain curves of a metal and a composite. Retrieved from [12]. •Good corrosion resistance: less inspections required that leads to savings on maintenance costs. It is fair to say that composite materials do not corrode, except in the case of aluminium with carbon fibre in which galvanic phenomenon creates rapid corrosion. •Ageing: the main originators are heat and moisture. •Sensitivity to chemicals: no sensitivity to chemicals used in engines like grease, oils, hydraulic liquids, paints and solvents, petroleum, etc. However, they are sensible to cleaners for paint, which attack epoxy resins. •Impact resistance: they have medium-to low-level impact resistance (inferior to that of metallic materials). Aramid fibres are an exception to that as they exhibit great impact resistance. For that reason they are widely used to protect regions on the aircraft that are susceptible to receive an impact - e.g. leading edge of the wings, cockpit, etc. 21 Chapter 2. State of the art •Fire resistance: excellent fire resistance as compared with light alloys of identical thicknesses. However, the smokes emitted from the combustion of certain matrices can be toxic. •Price: comoposites are very expensive materials. See Figure 2.14 d). Figure 2.14: Comparison of characteristics of different materials: a) density (kg/m3) b) tenisle fracture strength (MPa) c) modulus of elasticity (MPa) d) price per unit mass. Retrieved from [12]. 2.3.2 Classification of Composites Composite materials are commonly classified at two distinct levels: one first level of classification is made with respect to the matrix constituent, and the second level is made with respect to the reinforcement form. In the classfication by matrix we find three main types of composites: •Metal matrix composites (MMCs): materials consisting of a metallic matrix combined with a ceramic (oxides, carbides) or metallic (lead, tungsten, molybdenum) reinforcement. •Ceramic matrix composites (CMCs): materials consisting of a ceramic matrix combined with a ceramic (oxides, carbides) reinforcement. •Polimer matrix composites (PMCs): materials consisting of a polymer (resin) matrix combined with a polymeric (aramid fibre, glass fibre), ceramic (oxides, carbides) or metallic (lead, tungsten, molybdenum) reinforcement. It must be noted the following: as ceramics are materials with a higher elasticity modulus, or Young’s modulus (E) - less deformation by an increase in the stress in the elastic region - than metals and polymers, the latter cannot be used as reinforcement for a CMC. The same happens with MMCs and polymers. And lastly, PMCs, having the 22 Chapter 2. State of the art lowest E of the three families, can be reinforced with any type of material. Typical value of E for a ceramic like silicon carbide (SiC) is E = 300 GPa; for a metal like an steel alloy is E = 207 GPa; and for a polymer like epoxy is E = 2.41 GPa. Then, in the classification by reinforcement we find three main types of composites as well: •Fibre-reinforced composites: are composed of fibres embedded in matrix material. Such a composite is considered to be a discontinuous fibre or short fibre composite if its properties vary with fibre length. This type or fibre reinforced composites can have the fibres aligned or randomly oriented. On the other hand, when the length of the fibre is such that any further increase in length does not increase the elastic modulus of the composite, the latter is considered to be continuous fibre reinforced. Fibres are small in diameter and when pushed axially, they bend easily although they have very good tensile properties. These fibres must be supported to keep individual fibres from bending and buckling. •Structural composites: materials comprised of layers bonded together - also known as laminatively or in a determined order for many times as required. Sandwich structures fall under this category as well. •Particulate-reinforced composites: are composed of particles distributed or embedded in a matrix body. The particles may be flakes or in powder form. Figure 2.15: Composites classification depending on the reinforcement type. Retrieved from [13]. The most commonly used composite materials in the aerospace industry are the ones with polymeric matrix and fibrous reinforcements. They are almost always in the form of laminates, that is, monolayers with the fibres oriented in different directions, that are then stacked up following a sequence . Polymers are ideal materials as they can be processed easily, are very lightweight, and have desirable mechanical properties - obviously, metalic and ceramic materilas have higher mechanical properties but they are not as lightweight as polymers. Two main kinds of polymers are thermosets and thermoplastics. Thermosets are the most popular among the fibre composite matrices without which, a lot of research and development in structural engineering field could have not been possible. Aerospace components, automobile parts, defense systems etc., use a great deal of this type of fibre composites. Epoxy matrix, among other thermosets like phenolic polyamide resins or polyester, is the most used and well-known resin as its properties stand oud from the rest 23 Chapter 2. State of the art of thermoset resins. And among the wide variety of reinforcements that a polymeric matrix can use, the ones that are more used in modern aircrafts and have the highest mechanical properties, are continuous polymeric fibres - e.g. carbon,glass or aramid fibres being the most employed. A set of long continuous fibres embedded in a matrix in one direction constitute the basic structural element of composite construction: the unidirectional layers or monolayers. This basic structural element can be formed with fabrics instead of unidirectional layers. Fabrics are two unidirectional sets of fibres weaved together perpendicular to each other - i.e. are bidirectional elements. Being so, the main structural element in composite designing is more known as laminae. When a laminae is formed from a fabric embedded in a matrix material, it can be considered like two laminae formed with unidirectional layers of half the thickness each and stacked up at 0/90◦. A laminae by itself can hardly form a functional material that has to fit some strength and stifness requirements. That is why, a combination of several monolayers - with different thicknesses and properties - is employed, making up laminates. If you add a soft material in between a laminate - to gain second moment of inertia of the section or other purposes, you obtain a so-called sandwich panel. Layers of different materials may be used, resulting in a hybrid laminate. The individual layers generally are orthotropic - that is, with principal properties in orthogonal directions, or transversely isotropic - with isotropic properties in the transverse plane. The laminate then will exhibit anisotropic, orthotropic, or quasi-isotropic properties (depending on the sequence of stacking and orientation of the monolayers). Quasi-isotropic laminates exhibit isotropic - that is, independent of direction - inplane response but are not restricted to isotropic out-of-plane (bending) response. 2.3.3 Laminae and Laminates As it has been mentioned in the previous section, laminates of polymeric matrixes with fibrous reinforcements are the composites that are mostly employed in the aerospace sector - and thus, are the ones that must be studied in this thesis. As a laminate is made up by several laminae, in this section a discussion of laminates and laminae main characteristics will be delivered. Recall that in Section 2.3.2. Classification of Composites we have explained that laminaes can be either made by unidirectional fibres or fabric, but in either way the laminae can be treated as a composition of laminae. Next to this section, in 2.4. Laminae Mechanical and Physical Properties, the deduction of the calculation of the properties of laminae will be delivered. Prior to the discussion on laminae and laminates fundamental knowledge, it is important to know which elastic properties are needed to be known to characterize an anisotropic material. The relation between strains and stresses for an isotropic material under plane stress - plane stress state is very important for the development of the Classical Laminated Plate Theory (see Section 2.5. Classical Laminated Plate Theory) - only counts with two independent variables. 24 Chapter 2. State of the art    εx εy γxy   =  1/E −ν/E 0 −ν/E 1/E 0 0 0 1/G    σx σy τxy   (2.1) This two independent variables are E and the Poisson coefficient (ν), as it is important to recall that the shear modulus (G) is related with those two variables in the following way: G=E 2(1+ν)(2.2). In plane stress there is also a strain in the z direction due to the Poisson effect. This strain component will be ignored in the sections to follow. Figure 2.16: Anisotropic material. Retrieved from [14]. When E16=E26=E3, the material is orthotropic. It is common, however, for the properties in the plane transverse to the fiber direction to be isotropic to a good approximation (E2=E3); such a material is transversely isotropic - laminae, as it will be discussed later on, will be considered transversely isotropic. The elastic constitutive laws must be modified to account for this anisotropy, and the following form is an extension of the usual equations of isotropic elasticity to transversely isotropic materials:    ε1 ε2 γ12   =  1/E1−ν21/E10 −ν12/E21/E20 0 0 1/G12    σ1 σ2 τ12   (2.3) The parameter ν12 is the principal Poisson’s ratio; it is the ratio of the strain induced in the 2-direction by a strain applied in the 1-direction. This parameter is not limited to values less than 0.5 as in isotropic materials. Conversely, ν21 gives the strain induced in the 1-direction by a strain applied in the 2-direction. Since the 2-direction (transverse to the fibres) usually has much less stiffness than the 1-direction, a given strain in the 1-direction will usually develop a much larger strain in the 2-direction than will the same strain in the 2-direction induce a strain in the 1-direction. Hence we will usually have ν12>ν21. There are five constants in equation (2.3) (E1,E2, ν12,ν21 and G12). However, only four of them are independent; since the matrix, named S, is symmetric, we have ν21/E2=ν12/E1. ε=S·σ(2.4) σ=D·ε(2.5) where D=S−1is the stiffness matrix. The simple form of equation (2.3), with zeroes in the terms representing coupling between normal and shearing components, is obtained only when the axes are aligned along the principal material directions; i.e. along and transverse to the fibre axes. If the axes are oriented along some other direction, all terms of the compliance matrix will be populated, and the symmetry of the 25 Chapter 2. State of the art material will not be evident. If for instance the fibre direction is off-axis from the loading direction, the material will develop shear strain as the fibres try to orient along the loading direction. There will therefore be a coupling between a normal stress and a shearing strain, which does not occur in an isotropic material. Figure 2.17: Rotation of axes. Retrieved from [14]. The transformation when the fibre direction is off-axis from the loading direction is the follwing [14]:    σ1=σxcos2θ+σysin2θ+ 2τxy sin θcos θ σ2=σxsin2θ+σycos2θ−2τxy sin θcos θ τ12 = (σy−σx) sin θcos θ+τxy(cos2θ−sin2θ) (2.6) where θis the angle from the x axis to the 1 (fibre) axis. These relations can be written in matrix form as    σ1 σ2 τ12   =  c2s22sc s2c2−2sc −sc sc c2−s2    σx σy τxy   (2.7) where c= cos θand s= sin θ. This can be abbreviated as σ0=Tσ, where Tis the transformation matrix in brackets above. 2.3.3.1 Laminae Figure 2.18: Laminae representation in space. Retrieved from [12]. They are composed by the reinforcing fibres, the binder (matrix) and the surface of contact between the two. Each part has clearly differentiated functions: •The stiff fibres bear the major stresses in loading, providing strength and stiffness in the direction of fibre orientation. The total strength and stiffness of the composite material are determined mainly by the properties of the fibres, their dimensions, orientation and content by volume. 26 Chapter 2. State of the art El=EfVf+EmVm(2.22) •Poisson coefficient νlt Considering again the load applied to deduce the longitudinal modulus El, the transverse strain for the composite can be written as εtm+f=−νlt El σlm+f=νltεl(2.23) The transverse length variations can be written as εtm+f=∆(εm+εf) εm+εf =εtmVm+εtfVf(2.24) And applying equation (2.23), and doing the same for both the transverse strain of the fibre and matrix, we get to −νltεl=−νmεlVm−νfεlVf νlt =νmVm+νfVf(2.25) •Transverse modulus Et(perpendicular to the fibre direction) Figure 2.25: Idealisation of monolayer of fibre and matrix with transverse force applied. Retrieved from [12]. A force in the transverse direction is applied. At the interface between the two materials, the following hypothesis are taken: (a) Freedom of movement in the ldirection εlm6=εlf. (b) Freedom of movement in the zdirection εzm6=εzf. (c) Both materials are under the same state of stress σt. 33 Chapter 2. State of the art As stated previously: εtm+f=σt Et =∆(εm+εf) εm+εf =εtmVm+εtfVf where Vfand Vmare the volume fraction of the fibre and matrix, respectively. Now, considering that each material – matrix and fibre – are isotropic materials, and taken into account their behaviour law (the Hooke’s law), we can state the following: σt Et =σt Em Vm+σt Ef Vf(2.26) then, 1 Et =Vm Em +Vf Ef (2.27) or Et=Em"1 (1 −Vf) + Em EfVf#(2.28) •Shear modulus Glt Figure 2.26: Idealisation of monolayer of fibre and matrix with shear force applied. Retrieved from [12]. A shear stress τlt is applied. The following hypothesis are taken: (a) The angular strains of the matrix and fibre are different γltm6=γltf. (b) Both materials are under the same state of stress τlt. Keeping in mind that γlt =τlt Glt (2.29), we can then say that: γltm+f(em+ef) = γltmem+γltfef(2.30) 34 Chapter 2. State of the art where emand efare the thickness of the matrix and fibre respectively. Equation (2.30) can be rewritten as γltm+f=γltmVm+γltfVf(2.31) where Vfand Vmare the volume fraction of the fibre and matrix, respectively. Now, considering that each material – matrix and fibre – are isotropic materials, and taken into account the previous stated relation between the angular strain and the shear modulus, we can state the following: τlt Glt =τlt Gm Vm+τlt Gf Vf(2.32) then, 1 Glt =Vm Gm +Vf Gf (2.33) or Glt =Gm"1 (1 −Vf) + Gm GfVf#(2.34) For last, it is important to remark that all this formulas apply to a laminae of a certain thickness made from unidirectional fibres. If the laminae is made from fabric, the resultant laminae is a two monolayer laminae stacked up at 0/90◦with half the thickness each. 2.5 Classical Laminated Plate Theory Analysis of composite plates is based on the following approaches: equivalent single-layer theories (2-D) or three dimensional elasticity theory (3-D). For the goal of this thesis, focusing on 2-D theories to explain how composite plates behave is enough. The two theories that lay inside the 2-D theories are the Classical Laminated Plate Theory (CLPT) and the First-order Shear Deformation laminated plate Theory (FSDT). The simplest theory is the CLPT and requires that the Kirchhoff hypothesis holds, which assumes that plane cross sections remain plane and normal to the middle-plane during deformations. This implies that the transverse shear strains vanish. The FSDT is a bit more complicated and is build on the Reissner-Mindlin hypothesis, where plane cross sections remain plane after deformation, but not necessarily normal to the reference plane. This results inclusion of out-of-plane shear deformation. As the study performed in this thesis will be executed with a commercial software, the usefulness of this section is to be brief and concise providing a general understanding of composite plates structural behaviour: distribution of stresses and deformations. Being so, the CLPT is the only theory that is going to be explained. ([19] and [32]) The CLPT is a direct extension of the classical plate theory for isotropic and homogeneous material as proposed by Kirchhoff – Love [33]. However, the extension of 35 Chapter 2. State of the art this theory to laminates requires some modifications to take into account the inhomogeneity in thickness direction. In the following, the assumptions made in this theory along with the assumptions made for classical plate theory are given. Assumptions of Classical Laminated Plate Theory 1. The laminate consists of perfectly bonded layers. There is no slip between the adjacent layers. In other words, it is equivalent to saying that the displacement components are continuous through the thickness. 2. Each lamina is considered to be a homogeneous layer such that its effective properties are known. 3. Each lamina is in a state of plane stress. 4. The strain in the direction perpendicular to the layer due to the Poisson effect is ignored. 5. The individual lamina can be isotropic, orthotropic or transversely isotropic. 6. The laminate deforms according to the Kirchhoff - Love assumptions for bending and stretching of thin plates (as assumed in classical plate theory). The assumptions are: (a) The normals to the midplane remain straight and normal to the midplane even after deformation. (b) The normals to the midplane do not change their lengths. CLPT is a theory for bending of homogeneous plates, but accounting for in-plane tractions in addition to bending moments. Also, the stiffness variation through layers (or plies) is taken into consideration. In general cases, the determination of the tractions and moments at a given location will require a solution of the general equations for equilibrium and displacement compatibility of plates. This theory is treated in a number of standard texts, and will not be discussed here. We begin by assuming a knowledge of the tractions Nand moments Mapplied to a plate at a position x,y, as shown in Figure 2.27. It will be convenient to normalize these tractions and moments by the width of the plate, so they have units of N/m and N-m/m, or simply N, respectively. Coordinates xand yare the directions in the plane of the plate, and zis customarily taken as positive downward. Figure 2.27: Applied tractions and moments in plate. Retrieved from [16]. 36 Chapter 2. State of the art N=   Nx Ny Nxy   M=   Mx My Mxy   (2.35) 2.5.1 Kinematics When this tractions and moments are applied, displacements occur in the plate. The plate bends, as shown in Figure 2.28 and the different in-plane displacements can be described as follows: Figure 2.28: Undeformed and deformed geometries of an edge of a plate under the CLPT assumptions. Retrieved from [17]. u(x, y, z) = u0−z∂w ∂x , v(x, y, z) = v0−z∂w ∂y , w(x, y, z) = w0(x, y) (2.36) So uis displacement in xdirection, vis displacement in ydirection and wis displacement in zdirection, while u0,v0and w0are displacements of the midplane in x, yand zdirections, respectively. It is important to recall that wis constant as it was stated on assumption 4: the strain in zdirection due to Poissson effects is neglected. Being so, equation (2.36) can be reqritten as u(x, y, z) = u0−z∂w0 ∂x , v(x, y, z) = v0−z∂w0 ∂y (2.37) Based on the displacement field above, we can find the strains as follows: 37 Chapter 2. State of the art ε=   εx εy γxy   =   ∂ ∂x 0 0∂ ∂y ∂ ∂y ∂ ∂x   u v=     ∂u0 ∂x ∂v0 ∂y ∂u0 ∂y +∂v0 ∂x      +z     −∂2w0 ∂x2 −∂2w0 ∂y2 −2∂2w0 ∂x∂y      = =   εx0 εy0 γxy0   +z   κx κy κxy   =ε0+zκ (2.38) where ε0is the midplane strain and kis the vector of second derivatives of the displacement, called the curvature. The component kxy is a twisting curvature, stating how the x-direction midplane slope changes with y(or equivalently how the y-direction slope changes with x). 2.5.2 The Material Law The stresses relative to the x-y axes are now determined from the strains, and this must take consideration that each ply will in general have a different stiffness, depending on its own properties and also its orientation with respect to the x-y axes. This is accounted for by computing the transformed stiffness matrix D, which is the same matrix D as in equation (2.5) but multiplied by the trasnformation matrix T. Recall that the ply stiffnesses as given by equation (2.3), are those along the fiber and transverse directions of that particular ply. The properties of each ply must be transformed to a common x-y axes, chosen arbitrarily for the entire laminate. The stresses at any vertical position are then: σ=Dε =Dε0+zDκ (2.39) where here Dis the transfomed stiffness of the ply at which the stresses are being computed. 2.5.3 Resultant Forces and Moments Each of these ply stresses must add to balance the traction per unit width N: N=Z+h 2 −h 2 σdz = N X k=1 Zzk+1 zk σkdz (2.40) where σkis the stress in the kth ply and zkis the distance from the laminate midplane to the bottom of the kth ply. Using equation (2.39) to write the stresses in terms of the midplane strains and curvatures: N= N X k=1 Zzk+1 zk Dkε0dz +Zzk+1 zk Dkκzdz(2.41) The curvature κand midplane strain ε0are constant throughout z, and the transformed stiffness Dkdoes not change within a given ply. Removing these quantities from within the integrals: 38 Chapter 2. State of the art N= N X k=1 Dkε0Zzk+1 zk dz +DkκZzk+1 zk zdz(2.42) After evaluating the integrals, this expression can be written in the compact form: N=Aε0+Bκ(2.43) where Ais an ”extensional stiffness matrix” defined as: A= N X k=1 Dk(zk+1 −zk) (2.44) and Bis a ”coupling stiffness matrix” defined as: B=1 2 N X k=1 Dk(zk+12−zk2) (2.45) The Amatrix gives the influence of an extensional midplane strain (ε0) on the in-plane traction N, and the Bmatrix gives the contribution of a curvature (κ) to the traction (coupling). ”It may not be obvious why bending the plate will require an in-plane traction, or conversely why pulling the plate in its plane will cause it to bend. But visualize the plate containing plies all of the same stiffness, except for some very low-modulus plies somewhere above its midplane. When the plate is pulled, the more compliant plies above the midplane will tend to stretch more than the stiffer plies below the midplane. The top half of the laminate stretches more than the bottom half, so it takes on a concave-downward curvature.”[14, p. 9] Similarly, the moment resultants per unit width must be balanced by the moments contributed by the internal stresses. Following the same procedure as with the tractions: M=Z+h 2 −h 2 σzdz = N X k=1 Zzk+1 zk σkzdz (2.46) M= N X k=1 Zzk+1 zk Dkε0zdz +Zzk+1 zk Dkκz2dz= N X k=1 Dkε0Zzk+1 zk zdz +DkκZzk+1 zk z2dz (2.47) M=Bε0+Dκ(2.48) where Dis a ”bending stiffness matrix” defined as: D=1 3 N X k=1 Dk(zk+13−zk3) (2.49) The complete set of relations between applied forces and moments, and the resulting midplane strains and curvatures, can be summarized as a single matrix equation: 39 Chapter 2. State of the art N M=A B B Dε0 κ(2.50) The A/B/B/Dmatrix in brackets is the laminate stiffness matrix, and its inverse will be the laminate compliance matrix. The presence of nonzero elements in the coupling matrix Bindicates that the application of an in-plane traction will lead to a curvature or warping of the plate, or that an applied bending moment will also generate an extensional strain. These effects are usually undesirable. However, they can be avoided by making the laminate symmetric about the midplane, as examination of equation (2.45) can reveal. The above relations provide a straightforward - although tedious, unless a computer is used (see Section 2.7. Finite Element Analysis of Laminates) - means of determining stresses and displacements in laminated composites subjected to in-plane traction or bending loads. 2.6 Buckling of Laminated Plates From Annex A.1. Governing Equations for Plates, the equations A.1, A.2 and A.7 are the equilibrium equations for a laminated thin plate: ∂Nx ∂x +∂Nxy ∂y = 0 ∂Ny ∂y +∂Nxy ∂x = 0 ∂2Mx ∂x2+ 2∂2Mxy ∂x∂y +∂2My ∂y2+p∗= 0 where p∗=p+Nx∂2w ∂x2+Ny∂2w ∂y2+ 2Nxy ∂2w ∂x∂y +ρ∗∂2w ∂t2. The last equation solves a buckling problem [19]. Buckling is the loss of stability due to geometric effects rather than material failure. But it can lead to material failure and collapse if the ensuing deformations are not restrained. It is characterized by a sudden sideways failure of a structural member subjected to high compressive stress - and/or in-plane shear stress for composites, where the stress at the point of failure is less than the ultimate stress that the material is capable of withstanding. Buckling is commonly referred as a bifurcation problem. That is, there is a bifurcation point where the primary (without lateral or out-of-plane deformations w= 0) and secondary (when the instability occurs) loading paths intersect - point A in Figure 2.29. At this point, more than one equilibrium position is possible. The primary path is not usually followed after loading exceeds this point and the structure is in the post-buckling state. The slope of the secondary path at the bifurcation point determines the nature of the post-buckling. A positive slope indicates that the structure will have post buckling strength whilst a negative slope means that the structure will snap through or simply collapse. 40 Chapter 2. State of the art Figure 2.29: Bifurcation problem scheme. Retrieved from [18]. Let’s keep developing equation ??.7. Insert equation (2.48), we obtain: −D11 ∂4w ∂x4−4D16 ∂4w ∂x3∂y −(2D12 + 4D66)∂4w ∂x2∂y2−4D26 ∂4w ∂x∂y3 −D22 ∂4w ∂y4+B11 ∂3u0 ∂x3+ 3B16 ∂3u0 ∂x2∂y + (B12 + 2B66)∂3u0 ∂x∂y2+B26 ∂3u0 ∂y3 +B16 ∂3v0 ∂x3+ (B12 + 2B66)∂3u0 ∂x2∂y + 3B26 ∂3u0 ∂x∂y2+B22 ∂3v0 ∂y3+p∗= 0 (2.51) If we particularise equation (2.51) for an orthotropic laminate - with principal properties in orthogonal directions, its constitutive equations satisfy the following conditions [26]: A12 =A26 = 0 Bij = 0 D12 =D26 = 0 Incorporation of conditions above into equation (2.51) simplifies the equilibrium equation (buckling problem) as follows: D11 ∂4w ∂x4+ (2D12 + 4D66)∂4w ∂x2∂y2+D22 ∂4w ∂y4=p∗(2.52) The solution of equation (2.52) depends on the boundary conditions to which the laminated plate is subjected to and the force p∗that is applied to it. Here are some examples of typical forces that can make a plate buckle: Uniaxial compressive load p∗=Nx∂2w ∂x2 Biaxial compressive load p∗=Nx∂2w ∂x2+Ny∂2w ∂y2 In −plane shear load p∗=Nxy∂2w ∂x∂y There also can be combined load states - e.g. uniaxial compressive load plus in-plane shear load. And here are some examples of boundary conditions that can be applied to a plate like the one in Figure 2.30: 41 Chapter 2. State of the art Figure 2.30: Plate of dimensions a x b. Retrieved from [19]. Simply supported plate x=0: w(0, y)=0 Mx(0, y)=0 x=a:w(a, y) = 0 Mx(a, y)=0 y=0: w(x, 0) = 0 My(x, 0) = 0 y=b:w(x, b) = 0 My(x, b)=0 Clamped plate x=0: w(0, y)=0 ∂w(0,y) ∂x = 0 x=a:w(a, y)=0 ,∂w(a,y) ∂x = 0 y= 0 : w(x, 0) = 0 ∂w(x,0) ∂y = 0 y=b:w(x, b) = 0 ∂w(x,b) ∂y = 0 Each case - a set of boundary conditions and forces applied - has a particular solution, which also depends on the theory adopted for the problem (not the same solution for CLPT than for FSDT). Each particular solution yields a set of critical loads, each of them causing failure by buckling to the plate when applied. The number of possible solutions (critical loads) to a particular case are called buckling modes. Every critical load is a product of the original load - being this a distribution of an out-of-plane load like aerodynamic lift or a concentrated force, with the possibility of being lower or higher. If it is lower it means that the plate fails with the original load multiplied by a factor less than 1. Conversely, if the critical load is higher than the original load, it means that buckling occurs when the plate is subjected to the original load multiplied by a factor greater than 1. As recalled prior in this section, the study performed in this thesis is going to be tackled with a commercial software. This kind of software solve this type of analysis as an eigenvalue analysis because the critical loads are the eigenvalues λiof the discretised system of equations 42 Chapter 2. State of the art •Analysis type: the type of analysis must be defined depending on the study that the user is performing. The types are many, and the ones of interest for this thesis are the static and buckle types. •Output requests: the output data to be generated must be selected. For last, just to say that AbaqusTM doesn’t use units. The units you use to model the geometry are the ones that determine the rest of the units of the output variables - see Table 2.4. Table 2.4: AbaqusTM units logic. 2.7.4 Shell Elements Shell elements are used to model structures in which one dimension (the thickness) is significantly smaller than the other dimensions and in which the stresses in the thickness direction are negligible. Plate theories like the one explained in Section 2.5. Classical Laminated Plate Theory are implemented and differentiated by element types. AbaqusTM has two types of shell elements, which are described in the user’s manual [21] as: ”Two types of shell elements are available in AbaqusTM: conventional shell elements and continuum shell elements. Conventional shell elements discretise a reference surface by defining the element’s planar dimensions, its surface normal, and its initial curvature. The nodes of a conventional shell element, however, do not define the shell thickness; the thickness is defined through section properties. Continuum shell elements, on the other hand, resemble three-dimensional solid elements in that they discretise an entire three-dimensional body yet are formulated so that their kinematic and constitutive behaviour is similar to conventional shell elements. Continuum shell elements are more accurate in contact modelling than conventional shell elements, since they employ two-sided contact taking into account changes in thickness. For thin shell applications, however, conventional shell elements provide superior performance.” Study of composites can be done at different level of detail: michromechanics, lamina level or laminate level. For the purposes of the thesis the lamina level or meso-scale level, is going to be used. Shell elements have the ability of computing the laminate properties with the laminate stacking sequence and the unidirectional layers (known as laminae) properties. Shell elements, unlike continuum elements, use material directions local to each element. Anisotropic material data, such as the one of laminates, and element output 49 Chapter 2. State of the art variables, such as stress and strain, are defined in terms of these local material directions. In large-displacement analyses the local material axes on a shell surface rotate with the average motion of the material at each integration point. 2.8 Failure Criteria Failure criteria [35] are curve fits of experimental data that attempt to predict failure under multiaxial stress based on experimental data obtained under uniaxial stress. All failure criteria described in this section predict the first occurrence of failure in one of the laminae but are unable to track failure propagation until complete laminate failure. Continuum damage mechanics that track damage evolution up to laminate failure are out of the scope of this thesis. Failure criteria are presented using the notion of failure index, which is defined as IF=stress strength (2.80) Failure is predcited when IF≥1. 2.8.1 Hashin Criterion Hashin criterion [36] is implemented for elements with a plane stress formulation, thus being a 2-D failure criterion for unidirectional laminae. Failure indices for Hashin criteria are related to fibre and matrix failures and involve four failure modes: tensile fibre failure, compressive fibre failure, tensile matrix failure, and compressive matrix failure. Prior to announce the criterion, let’s define the failure properties: S+ 11 value of σ11 at longitudinal tensile failure S− 11 value of σ11 at longitudinal compressive failure S+ 22 value of σ22 at transverse tensile failure S− 22 value of σ22 at transverse compressive failure S12 absolute value of σ12 at longitudinal shear failure S23 absolute value of σ23 at transverse shear failure If σ11 ≥0 the tensile fibre failure criterion is: I+ F,f ≡σ11 S+ 11 2 +ασ12 S12 2 ≥1 (2.81) If σ11 <0 the compressive fibre failure criterion is: I− F,f ≡σ11 S− 11 2 ≥1 (2.82) If σ22 ≥0 the tensile matrix failure criterion is: I+ F,m ≡σ22 S+ 22 2 +σ12 S12 2 ≥1 (2.83) 50 Chapter 2. State of the art If σ22 <0 the compressive matrix failure criterion is: I− F,m ≡σ22 2S23 2 +"S− 22 2S23 2 −1#σ22 S− 22 +σ12 S12 2 ≥1 (2.84) Note that subscript fstands for fibre, and mfor matrix. Then, there is the parameter α, which is a coefficient that determines the contribution of the longitudinal shear stress to fibre tensile failure. Allowable range is 0 ≤α≤1. AbaqusTM only has this criterion implemented and the default value of αis α= 0. 2.8.2 Tsai-Hill Criterion The Tsai-Hill criterion [37] is a quadratic, interactive stress-based criterion that identifies failure, but does not distinguish between different modes of failure. Failure occurs whenever the following condition is satisfied: IF≡σ2 11 S2 11 +σ2 22 S2 22 −σ11σ22 S2 11 +σ2 12 S2 12 ≥1 (2.85) where the Sij coefficients are the same as the ones in the Hashin criterion and if σ11 ≥0, then S11 =S+ 11; if σ11 <0, then S11 =S− 11; if σ22 ≥0, then S22 =S+ 22 and if σ22 <0, then S22 =S− 22. 2.9 Pareto Optimality For a nontrivial multi-objective optimization problem, there does not exist a single solution that simultaneously optimizes each objective. In that case, the objective functions are said to be conflicting, and there exists a (possibly infinite) number of Pareto optimal solutions [38]. A solution is called nondominated, Pareto optimal, Pareto efficient or noninferior, if none of the objective functions can be improved in value without degrading some of the other objective values. The term is named after Vilfredo Pareto (1848–1923), an Italian engineer and economist who used the concept in his studies of economic efficiency and income distribution. Figure 2.36 shows four geometric examples of Pareto optimality. In these figures, the circles represent objectives that are satisfied best when the area of the circle is maximized. The constraints are that the circles may not overlap and must fit within the triangle. 51 Chapter 2. State of the art Figure 2.36: Examples illustrating the Pareto optimality. Retrieved from [22]. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto front is the set of choices that are Pareto efficient. The Pareto frontier, P(Y) is described as follows. Consider a system with function f:Rn→Rm, where X is a compact set of feasible decisions in the metric space Rn, and Y is the feasible set of criterion vectors in Rm, such that Y={y∈Rm:y=f(x), x ∈X}. We assume that the preferred directions of criteria values are known. A point y00 ∈Rm is preferred to (strictly dominates) another point y0∈Rm, written as y00 y0. The Pareto frontier is thus written as: P(Y) = {y0∈Y:{y00 ∈Y:y00 y0, y00 6=y0}= Ø}. An example of a Pareto front is depicted in Figure 2.37, where we can see that point C does not belong to the front as it is dominated by points A and B. The latter points are non-dominated points or Pareto efficient points. 52 Chapter 2. State of the art Figure 2.37: Example of a Pareto front. Retrieved from [23]. 53 Chapter 3 Methodology In this chapter, the process undergone to obtain the design of a moncoque wing structure with composite materials is explained. The first aspect of the methodology that is detailed are the premises from which the design process has been addressed. Following to that, a thorough description of the whole design process - steps and decision making along the procedure - is given - supported by a flowchart. And for last, a brief and schematic description, also supported with a flowchart, of the process of the making of an AbaqusTM simulation is delivered. 3.1 Premises of the Design In the design of a monocoque composite structure there are several parameters to take into account, as each of them vary the overall behaviour of the structure. As for that, the iteration over all of those parameters has been discarded for being too complicated to be tackled - highly time-consuming and expert programming required. For that reason, it has been mandatory to select the possible ranges of the parameters of the study - some of them have been fixed. In other words, the design of the wing’s structure has been delimited. The main parameters that have been delimited are geometrical and material-related. The criterion of varying parameters has been delimited as well. Then, the structural requirements that the structure must withstand, the structure’s maximum weight, the structure’s minimum buckling constant value, and the type of structural testing to which the structure is going to be subjected to in order to be sized, have been set. Following, each of these premises are individually explained. After that, a list of all the parameters used for the design is going to be presented. 3.1.1 Type of Structural Testing The type of structural testing performed to the structure is the following: symmetrical testing of a cantilevered semi-wing with a clamped end (restricted displacements and rotations), and a distributed force applied over the semi-wing (the lift generated by the semi-wing) - see Figure 3.1. 54 Chapter 3. Methodology Figure 3.1: Schematic representation of the structural testing that is going to be performed on the wing. Retrieved from [24]. The clamped end is assumed to emulate the union between the wing and the fuselage of the aircraft, which is ideally placed at the right center of the wing, allowing the maximum wing surface unaltered by the union. 3.1.2 Structural Requirements As it has been explained in Section 1.5. Justification, this thesis has the goal to be useful for Trencal`os Team in developing a know-how about monocoque composite structures. The type of competitions in which the team participates require an aircraft capable of turning fast and carrying the highest amount of pay-load. Accordingly, the structural requirements have been set keeping those two factors in mind. The maximum take-off weight(MTOW) has been set to 10 kg and the load factor n, which is related with making a steep turn and going very fast, is set to 4. Summing up: •MTOW = 10 kg. •n= 4. •Lift per semi-wing = 200 N (approximating ~g to 10 m/s2). 3.1.3 Geometrical Features of the Wing The geometrical characteristics of the wing chosen to develop this design are the following: •Wingspan = 1.2 m. •Planform: rectangular. •Chord = 0.19 m. •Airfoil: CLARKY. •Airfoil’s maximum thickness and position = 0.0222 m (11.71% of the chord) at 0.0532 m (28% of the chord). 55 Chapter 3. Methodology As I have mentioned in Section 1.5. Justification, another member of the team has studied the construction technique of monocoque composite structures for his bachelor’s thesis. He has picked a 0.6 m wingspan wing, with CLARKY airfoil, to make his tests. For that reason, I have chosen this geometrical characteristics so I can be of any help with the numerical results that I have yield - mainly, the tendencies that I have observed. However, I have decided to change the wingspan. I have doubled the wingspan for two reasons: first, the sizing of a wing lies mainly on the bending moment that is created along its wingspan. Taking into account that, I have made the design more challenging by adding more lever arm for the lift, which will generate a higher bending moment. And secondly, the aircrafts employed in the competitions in which Trencal`os Team takes part in have wingspans greater than 3 metres. Being so, doubling the wingspan up to 1.2 m makes the wing more alike to the one that would be used in such a competition. For last, I want to mention that the airfoil has been cut at the 0.17 m point of the chord - see Figure 3.2. That is because the ailerons are always built separately and do not have to withstand high loads. Thus, I have not included them in the wing structure. Figure 3.2: Clarky airfoil with a cut at 0.17 m of the chord line. 3.1.4 Load-bearing Elements Employed As explained in Section 2.2. Structural Concepts, monocoque structures can be complemented with auxiliary elements that help in the transmission of loads and increase the buckling resistance of the skin - e.g. by enlarging the Iof a certain region or by limiting the buckling length. The main load-bearing element though, which takes the greatest part of the loads, is the skin. The auxiliary members that are going to be tested in the wing’s structure are: spar or spars (transmission of the shear stress and torsional resistance) and ribs (limit the buckling length in the main buckling direction , i.e. the span-wise direction). The main variable in relation with this elements is its position along the chord line. For a spar of a certain width, there are infinite possible positions along the 0.17 m of chord available. For that reason, the positions that will be tested are going to be few but representative. Intuitively, the most appropriate position to put the spar is at the maximum thickness point of the profile, as it has the highest Iand thus helps more with the bearing of shear loads. So, the spar will be placed at the maximum thickness point for one of the study cases, and placed before and after that point for some other cases. Putting a second spar will be based on the optimal position of the first spar. Regarding ribs, there is the same problem as with the spar: there are infinite possibilities. Being so, a finite number of combinations will be tested, which are: 3, 4 and 5 ribs per semi-wing. 56 Chapter 3. Methodology Lastly, in relation with the skin, there is the possibility of enhancing the Iof the extrados by using different layer sequences - e.g. using different materials and/or different core thickness. Doing so, buckling resistance is enlarged. This stiffened regions - known as stringers - are going to be used. The main variables regarding this ”elements” (as they actually belong to the skin) are: materials applied, core thickness and region of appliance - position and width. Summing up, we have: Figure 3.3: Structural elements employed. 3.1.5 Materials Employed and its Properties The reinforcing materials employed for the making of the skin have been decided to be fibre-based. Unidirectional fibres and fabrics will both be considered. Glass fibre and carbon fibre will be the two types of fibres used for being the most commonly used for this type of structures. And the matrix material for the making of the skin will be a polymeric resin, specifically, epoxy. A set of fibres plus the matrix form a laminae. The skin will be made up with laminae stacked up in different orientations, forming a laminate or sandwich panel, if a soft core is added. If a sandwich panel is used, the core employed will be Rohacell R  110 wf, a high performance foam. Secondary elements added to the structure, like ribs or spar(s), will be made out of balsa wood. Summing up: Materials employed Skin fibre Glass fibre / Carbon fibre matrix Epoxy core (if employed) RohacellR 110 wf Auxiliary elements Balsa wood Table 3.1: Summary of the materials employed Based on the discussion on laminates in Section 2.3.3.2. Laminates, the orientations that are going to be used for the skin are 0,90 and ±45◦. The possible combinations for the laminate with this 4 different orientations is very large. For that reason, the possible skins considered are the following: 57 Chapter 3. Methodology Skins employed Type of fibre Skin code Region Layer stacking sequence Fabric ALow load-bearing region (+45/−45/core)sor (+45/−45)s High load-bearing region (+45/−45/0/core)sor (+45/−45/0)s BLow load-bearing region (0/90/core)sor (0/90)s High load-bearing region (0/90/0/core)sor (0/90/0)s Unidirectional C Low load-bearing region (+45/core/ −45) or (+45/−45) High load-bearing region (+45/0/core/0/−45) or (+45/02/−45) Table 3.2: Different types of skin considered for the study. where the plies that are in bold are laminae made of carbon fibre/epoxy and the ones that are not are made of glass fibre/epoxy. Each skin is considered with and without core. There are two made with fabrics and one made with unidirectional layers - that refers to the glass fibre/epoxy laminaes, the carbon fibre/epoxy ones are unidirectional in the 3 skins. Note that each skin is mainly differentiated by this. Furthermore, the skin is not continuous along the wing, making a distinction between high and low load-bearing regions - see Figure 3.4, where these regions are defined (where carbon fibre/epoxy is applied and where is not) and the skin layer sequence defined in Table 3.2 is depicted. Figure 3.4: Skin layer sequence depiction and differentiation between high and low loadbearing regions. For last, I will present the mechanical and physical properties of the 4 materials employed for the study. First of all, I am going to expose balsa wood and Rohacell R  110 wf, which have been considered isotropic materials. Balsa and Rohacell properties Material ρ(kg/m3) E (MPa) νSource Balsa 160 3000 0.38 [39] Rohacell 110 180 0.33 [40] Table 3.3: Balsa and Rohacell R 110 wf properties. Regarding carbon fibre/epoxy(CE) and glass fibre/epoxy(GE) composites, there are a wide range of them. Only by changing the grammage - i.e. weight per unit of area - of the fibre, the properties of the resulting composite can vary a lot. Additionally, as seen in Section 2.4. Laminae Mechanical and Physical Properties, the volume fraction of each individual that makes up the composite, also influences a lot the final properties of the same. Besides those facts, composite manufacturers do not provide the whole range of mechanical properties necessary to characterize a composite material. For that reason, the properties of this two composite monolayers have been extracted both from research articles, which also provided the failure properties necessary to implement the Hashin an Tsai-Hill failure criteria - see Section 2.8. Failure Criteria. 58 Chapter 3. Methodology in Figure 3.11 the effect of it its observed with the tip vortex generated which adds to the total drag force generated by the wing. Figure 3.10: Induced angle along the span. Figure 3.11: Lift distribution and tip vortex effect downstream. Modelling such a load distribution in AbaqusTM is not an easy task. For that reason, some simplifications of this load distribution which yield the equivalent lift per semi-wing defined in Section 3.1.9. Parameters of the Design,Lsw = 200N, have been considered. I have discretised the airfoil (see Figure 3.12) in elements and I have treated it with an ExcelR file - loaddefinition.xlsx (see Annex B.4. Loaddefinition.xlsx). In that file I have applied different pressure distributions over the extrados and the intrados such that the Lsw is accomplished. 65 Chapter 3. Methodology Figure 3.12: Discretisation in elements of the airfoil. To do so I have used Excel R ’s solver, which allows you to compute the value of a cell that depend on other cells by changing this dependent cells values aiming for an objective - i.e. in my case reducing the difference with the goal lift (Lsw) and the calculated lift with the input pressure (pressure cells are the one that are changed). Using this file I have modelled the force needed to be generated in three ways. I have considered that the percentage of the force generated by extrados and intrados is a 75 and 25%, respectively - see Figure 3.13. Figure 3.13: Pressure distribution along extrados and intrados. Retrieved from [25]. The three load distributions tested in the initial tests have been the following: •Distribution 1: uniformly distributed pressure over the extrados and the intrados in order to generate 150 N and 50 N in each, respectively. Figure 3.14: First load distribution. •Distribution 2: uniformly distributed pressure over a band comprised between the 25 and 40% of the chord - where the resulting force is commonly found - at the extrados and the intrados in order to generate 150 N and 50 N in each, respectively. 66 Chapter 3. Methodology Figure 3.15: Second load distribution. •Distribution 3: non-uniformly distributed pressure over the extrados and the intrados - such that the suction peak can be modelled (Figure 3.13)- in order to generate 150 N and 50 N in each, respectively. There have been 5 regions defined - 3 in the extrados and 2 in the intrados - to which 4 different uniformly distributed pressure values have been applied. Figure 3.16: Third load distribution. 3.2.1.4 Buckling Modes Check The bifurcation problem is a non-linear problem and its solution can be mathematically possible but without physical meaning. Adding to that, as it has been remarked in Section 2.6 Buckling of Laminated Plates, boundary conditions determine the buckling mode of failure - shape and critical load. To ensure that a composite-made skin structure (higher complexity in the matrix Kg) does not yield a non-logical solution in a FEA model, I have made some tests to check it out. The procedure taken has been the following: analyse the buckling modes of the same structure but made with an isotropic equivalent material. To do so, I have preserved the structural behaviour of the composite-made structure under static loads. I have varied the thickness of the skin of the new structure and the Eand νuntil I have obtained the same deformed shape under the same static loads. Once done that, I have run in both composite and equivalent isotropic material structures a buckling analysis and compared them. 3.2.1.5 First Tendencies Observed For last, from all the initials tests performed to specify the mesh, the boundary conditions, the load definition and some extra tests, some trends have been observed. This trends will be noted and taken into account to define the starting point of the design - see Section 3.2.2. Starting Point. 67 Chapter 3. Methodology 3.2.2 Starting Point Having performed the initial tests, a starting point of the design must be defined - i.e. an initial family of points (m, 1/λ) from which an optimum is going to be sought. Taking into account all the design parameters seen in Section 3.1.9. Parameters of the Design, the following set of cases have been decided to be analysed: •3types of skin. The 3 skins defined in Section 3.1.5. Materials Employed and its Properties, with its differentiation of region within the skin - high and low load-bearing regions. •The high load-bearing region position has been placed at the maximum thickness of the airfoil, taking profit of this geometric condition that allows the highest I(x) - where x axis is pointed chord-wise towards the leading edge - of the wing’s section. The width of this region that will be studied is 20 or 30 mm. •6types of core: from 0 to 5 mm, with 1 mm increments. •5positions of the spar. One at the maximum thickness of the airfoil (53.2 mm from the leading edge), two ahead of that position (25 and 35 mm from the leading edge), and two behind that position (65 and 75 mm from the leading edge). •The spar width is going to be fixed with 10 mm. That adds to 3 x 2 x 6 x 5 = 180 different cases, which traduces to 360 AbaqusTM simulations, as each ase requires 2 simulations: one for the static condition in which static failure indices are obtained, and the other to analyse the resistance to failure by buckling. Summing up this step of the process, the cases run are: Starting Point Parameter Value(s) Skin 3 types Width of reinforced region 20 / 30 mm Core thickness 0 / 1 / 2 / 3 / 4 / 5 mm Spar position 25.5 / 35.5 / 53.2 / 65.5 / 75.5 mm Spar width 10 mm Total = 180 cases Table 3.5: Set of parameters analysed in the first family of points. 3.2.2.1 Decision Making Once this family of points is obtained, the points that do not fulfil the conditions specified for the Hashin and Tsai-Hill criteria (see Section 2.8. Failure Criteria) will be discarded. Also, the points which have a mass m > 0.450 kg or a buckling constant λ < 1will also be discarded. With only the points that do not fail statically or by buckling remaining, the Pareto optimality is applied - see Section 2.9. Pareto Optimality. The Pareto front is obtained, and the next iterations (variation of parameters) will be done only from this non-dominated points. The rest of valid points, from the failure point of view, are now discarded too. 68 Chapter 3. Methodology Figure 3.17: Decision making for each point of the set of points analysed after each step. This procedure has been implemented with a Matlab code (Decisionmaking.m) - see Annex B.1. Decisionmaking.m. 3.2.3 Improvements Once the Pareto optimal points are achieved, the step that follows is: select the best points of the Pareto front and introduce improvements to those points. The best points of the Pareto front will be the ones closer to the optimal region, as Figure 3.5 (a) depicts. The improvements that are going to be tested on the non-dominated points are the following. First, to each Pareto optimal point each of these improvements are going to be tested separately: •Improvement 1: vary the spar width to 5,15,20 and 25 mm. •Improvement 2: add a second spar of the same width (10 mm) placed at 110, 120,130 and 140 mm from the leading edge. •Improvement 3: add stringers with 3 different widths. The core combinations for the stringers region (stiffened region), and the no-stringer region (non-stiffened region) are shown in Table 3.6. Core combinations No-stringer region thickness (mm) Stringer region thickness (mm) 0 1 / 2 / 3 / 4 / 5 1 2 / 3 / 4 / 5 2 3 / 4 / 5 3 4 / 5 4 5 Table 3.6: Core combinations for the stringers improvement. That adds to (Z x 4) + (Z x 4) + (Z x 45) = 53Z possible new different cases, where Z is the number of non-dominated points selected in the previous step. After obtaining these new points, the same process of decision making explain in the previous Section 3.2.2.1. Decision Making is followed. See Figure 3.17 to recall the procedure to which all the points are subjected to in order to determine their optimality for the study of this thesis. 69 Chapter 3. Methodology Once applied this decision making, next step is apply some of the new points’ improvements at the same time. Then, we have: •Improvement 4:combination of a spar position, spar width, set of stringers or second spar, based on the best points (Pareto optimal points) selected from the set of points obtained from the 3 prior improvements. For instance, if there is a very good point with a certain spar width, and another very good point with a certain set of stringers, it will be interesting to try a combination of those two improvements and see what is the outcome of that. The total amount of different cases up to this point is very complicated to estimate. The process to follow after this new cases are run is the same as the one explained before. The decision making is applied and new non-dominated points are obtained. Finally, with the best remaining points, a fifth and final improvement is implemented in such points. •Improvement 5: add ribs to the non-dominated points. The tests that will be performed will be with 3,4, and 5 ribs per semi-wing. Decision making procedure is applied again and the final Pareto front is obtained. Summing up, the sets of steps explained in this section are the following: Steps followed after the first family of points are obtained Improvement Important parameter Possible values 1 Spar width 5 / 15 / 20 / 25 mm 2 Second spar position 110 / 120 / 130 / 140 mm 3 Stringers 3 different widths Decision making 4 Combination Combination or improvements Decision making 5 Ribs 3 / 4 / 5 per semi-wing Table 3.7: Steps followed after the first family of points are obtained. 3.2.4 Final Decision With the final Pareto front, there are a certain number of points - i.e. materially-constituted and geometrically different moncoque composite structures, which are optimal points. These points are each of them better than the rest of the Pareto front points in one of the two objectives of my interest: mor 1/λ. With this cases on the table, a proper consideration of all of them will be done, and by thorough and detailed reasoning some of the will be chosen as the most optimal for the goal of this thesis. 3.2.5 Flowchart of the Design Process Before ending this section, a flowchart of the whole design process is attached. The Analysis procedure flowchart and the Decision making procedure flowchart are shown in Figure 3.25 and 3.17, respectively. 70 Chapter 3. Methodology Figure 3.18: Flowchart of the design process. 71 Chapter 3. Methodology 3.3 Analysis Process Description In this section, the process followed to perform an analysis - in order to obtain the information necessary for every case - is explained. The process is divided in 4 sub-processes: geometry preparation,AbaqusTM model making,simulation of the analysis and post-processing. In relation with Section 2.7.3. AbaqusTM, the pre-processing part is the geometry preparation and the AbaqusTM model making, the simulation part is the simulation of the analysis and the post-processing part is the equally-defined post-processing. 3.3.1 Pre-processing 3.3.1.1 Geometry Preparation Each case has its own geometrical parameters. Being so, the first thing to do in order to analyse the case in AbaqusTM is to adjust the geometry to the case’s geometric features. Let’s recall which are the main geometrical parts that comprise this study: wing skin,main spar,secondary spar and ribs. In order to quicken the process, the geometrical parts have been modelled (with CATIA V5R20) using parametric formulas. These geometric parameters have been stored in an ExcelR file - parameters.xlsx (See Annex B.6. Parameters.xlsx), from which the update of any of the values has a modification effect of the part. The wing skin has been modelled as a surface - allowing me to define different material sequences at different regions of it in AbaqusTM; and the main spar,secondary spar and ribs as solid parts. The parameters that have been defined to model the geometry are the following: •spar w and spar o, which represent the main spar width and offset from the leading edge of the wing, respectively. With these two parameters the main spar has been modelled. Also, the skin surface has been partitioned with these two parameters - creating edges on it. The latter has been done in order to avoid wrong imported geometry in AbaqusTM. If the skin is partitioned as mentioned, as both the main spar and the surface share the extrados and intrados edges, perfect geometric coupling between the two parts is achieved - see Figure 3.19. Figure 3.19: (a) Wrong geometry due to lack of partitioning (b) Good geometry with skin partitioning. •spar2 w and spar2 o, which represent the secondary spar width and offset from the leading edge of the wing, respectively. With these two parameters the secondary spar has been modelled. Also, for the same reasons as I have already explained, the skin surface has been partitioned with these two parameters - creating the reference edges on it. 72 Chapter 3. Methodology •S1 w,S1 o,S2 w,S2 o,S3 w and S3 o, which are the width and offset from the leading edge of each stringer region on the skin. With these parameters, the skin extrados has been partitioned to define the stringer regions. The partitioning process can be also done in AbaqusTM, where you can define datum planes to partition the imported geometry. But doing it with CATIA formulas assignation feature has allowed me to do it in a more time-efficient way. •rib4 o and rib5 o, which are the offset of ribs 4 and 5 from the plane of symmetry of the semi-wing. As detailed in Section 3.1.9. Parameters of the Design, three combinations of ribs are going to be tested: 3, 4 and 5 ribs per semi-wing. The 5 ribs have been modelled as follows: rib 1 and rib 3 are the wing symmetry plane and tip rib, respectively. Rib 2 is the rib fixed at the symmetry plane of the semi-wing, for 3 and 5 ribs combinations. And the last two ribs, rib 4 and 5, can be placed either at an offset of ±100 mm - 4 ribs combination - or ±150 mm - 5 ribs combination - from the symmetry plane of the semi-wing. For last, as it has been detailed in Section 3.2.1.3. Load Definition, there are 3 different ways of defining the sizing load of the structure. If either load distribution 2 or 3 detailed in that section is the one that results the most appropriate for the study, then partitioning of the wing skin - to define the different uniformly distributed pressure values - must be done. To end this section, a summary of the geometry preparation is delivered: Case geometry −→ parameters.xlsx modification −→ open CATIA V5R20 −→ update each part modified −→ save modified parts −→ Geometry is ready 3.3.1.2 AbaqusTM Model Making With the geometry ready, the AbaqusTM model can be built. The set of steps that takes to do so are the following: 1. Import the geometry. AbaqusTM has compatibility with CATIA V5R20. 2. Assign surfaces and sets. For the sake of quickening the process, I have seen that it is convenient to assign surfaces and sets in each part if they are going to be used to define a load, boundary condition or an interaction. So, skin surfaces must be defined - recall that different pressure values may be assigned, as long as surfaces in skin and spar/ribs which are going to have a perfect contact interaction. Then, the sets of points to define a boundary condition must be selected too. 3. Assign materials. For the non-composite material part, I have created a solid homogeneous section with balsa wood. This section is assigned to the spar and ribs. For the skin, a composite layup is created, in which the following has to be defined: local coordinate system (1 direction is the fibre direction, 2 direction is the matrix direction, and 3 direction is the matrix and out-of plane direction), number of plies, orientation and thickness of each ply, the number of integration points per ply and the region of the part for each ply - see Figure 3.20. When stringers are defined, two composite layups must be created - as different core thickness are to be assigned. 73 Chapter 3. Methodology Figure 3.20: Definition of a composite layup in AbaqusTM. The great thing about the composite layup feature of AbaqusTM is that you define 7 layers, as in Figure 3.20, but defining the regions of each layer the skin can have different layer sequences - high and low load-bearing regions (see Section 3.1.5. Material Employed and its Properties. The next thre figures, Figure 3.21, Figure 3.22 and Figure 3.23 depict the composite layup defined in Figure 3.20. Figure 3.21: Stack-up plot of a 5 layer region, where no CE is employed. 74 Chapter 4. Results Model settings Parameter Value Skin A Spar width 10 mm Spar position 53.2 mm Core thickness 2 mm CE reinforcement width 10 mm Table 4.1: Parameters used to perform the mesh convergence tests. Regarding the load, the load distribution that has been applied is the first one proposed in Section 3.2.1.3. Load Definition. And regarding the boundary condition application, the encastre used has been the first one proposed in Section 3.2.1.2. Boundary Conditions. As explained in Section 3.2.1.1. Mesh Convergence, a static analysis has been performed with different meshes. For each analysis the curve displacement (δ) vs. semi-span (b/2) has been obtained. The final mesh has been picked when the difference between the δtipnew and δtipold represents less than a 1% of the δtipold - see equation 3.1. Next, the results are presented - recall that seeds are the points that AbaqusTM places at the edges to generate the mesh: Mesh Skin elements Spar elements Seeds (seed/mm) δtip (mm) Difference (%) 1 362 20 30 25.541 - 2 1122 40 15 25.244 1.16 3 4082 480 7.5 22.674 10.18 4 17444 3078 3.5 22.125 2.42 5 52802 16500 2 22.013 0.51 Table 4.2: Results of the convergence mesh tests. Figure 4.1: Displacement vs. semi-span for the 5 meshes. 81 Chapter 4. Results Figure 4.2: Close-up of the Displacement vs. semi-span for the 5 meshes. The mesh that has been used for the rest of simulations of the thesis is mesh 5, which has the following characteristics: 52802 elements for the skin and 16500 elements for the main spar. For other parts that may be included in the model (e.g. second spar, ribs), a seeding of 2 seeds/mm will be used. 4.1.2 Final Encastre The tests performed to determine which encastre (U1=U2=U3=UR1=UR2=UR3= 0) is the best to minimise as much as possible stress concentrators have been done with the settings and load distributions as in the previous Section 4.1.1. Converged Mesh, except for the encastre which has been varied. The results obtained for the three encastres proposed in Section 3.2.1.2. Boundary Conditions are presented next. The main output variable that is going to be compared between the three options is the stress σ11, as it is the highest stress generated in the structure - due to the bending moment generated by the lift distribution. 82 Chapter 4. Results •Encastre 1. Maximum |σ11|is 490.5 MPa (compression stress). Figure 4.3: Stress distribution with the encastre 1. •Encastre 2. Maximum |σ11|is 484.0 MPa (compression stress). Figure 4.4: Stress distribution with the encastre 2. 83 Chapter 4. Results •Encastre 3. Maximum |σ11|is 479.3 MPa (compression stress). Figure 4.5: Stress distribution with the encastre 3. Note that in the last three Figures, the spar has been hidden in the .odb fie in order to visualize better the stress distribution at the skin. Also, the stress field shown is the envelope one (all plies together) with the option Max absolute value of the display options of AbaqusTM, which shows the max absolute value at each element. As the stress distributions with the three encastres are very similar, the final encastre selected is the third one (encastre 3) it has the less stress σ11 at the symmetry plane of the wing. 4.1.3 Final Load Distribution The tests performed to determine which load distribution is the best to resemble the real lift distribution over the wing have been done with the settings as in Section 4.1.1. Converged Mesh, but with the final encastre already incorporated and varying the load distribution. The criterion to select one load distribution over another is difficult to set as I don’t know how the structure should precisely behave - i.e. how much should it displace at the tip or how much the stress values should rise. So, firstly, I am going to compare both the stress and displacement field of the three configurations. The stress field obtained for the three load distributions is the same as in Figure 4.5. That Figure corresponds to the first proposed load distribution - the three possible distributions are explained in Section 3.2.1.3. Load Definition. That stress field seems reasonable, being the region where carbon fibre/epoxy is placed the high load-bearing region and leaving the glass fibre/region with less stress concentration. As I said, the three distributions yield the same stress field with minor differences. The maximum stress σ11 yielded in each case, from distribution 1 to 3, are 479.3 MPa, 466.7 MPa and 473.6 MPa, respectively. Regarding the displacement field, the three also 84 Chapter 4. Results exhibit the the same displacement distribution (see Figure 4.6) with the only difference only laying on the maximum displacement at the tip. In each case, from distribution 1 to 3, the latter is 22.99 mm, 22.32 mm and 22.67 mm, respectively. Figure 4.6: Displacement field exhibit by the three load distributions, this one belonging to the first one. As the static analysis doesn’t help to decide which of the distributions is better, I have performed a buckling analysis. I have only analysed the three first buckling modes for each distribution, and the deformed shape of the latter is the same for the three distributions (see Figures 4.7, 4.8 and 4.9). Furthermore, the bucking constants (λ) do not vary much from one load distribution to the other. In each case, from distribution 1 to 3, the first mode λis 3.35, 3.45 and 3.37, respectively. 85 Chapter 4. Results Figure 4.7: First buckling mode. Figure 4.8: Second buckling mode. Figure 4.9: Third buckling mode. The first and second modes happen in the extrados - due to the compression stress caused by the bending moment. The third mode happens in the intrados, which seems a bit odd as that is the tension region of the wing. But due to the net force produced by the wing (the lift), there is a shear stress distribution along the wing, which causes a compression force in the intrados - this phenomena is known as shear buckling. The latter will be explained more in detail in next Section 4.1.4. Buckling Modes Verification. Lastly, the laminae at which the section fails is the CE laminae, as CE regions are the ones that bear the highest stresses. 86 Chapter 4. Results Coming back to the matter of this section, that is choosing which is the best load distribution to mimic the lift distribution, I have seen so far: stress and displacement distribution, as well as the buckling modes, are the same for the three possibilities - so are the characteristic values of each field. Being so, the decision I have taken is not based on numerical results but on a theoretical basis. The load distribution selected is the load distribution 3 - see Figure 3.16 - for being the one that best resembles the characteristic suction peak of an airfoil, and thus the lift distribution along the wing. 4.1.4 Buckling Modes Verification In this section is intended to ensure that a composite-made skin structure (higher complexity in the matrix Kg) does not yield a non-logical buckling solution in a FEA model. To do so I have followed the following settings: load distribution 3, encastre 3 and the material configuration shown in Table 4.3. Model settings Parameter Value Skin A Core thickness 0 mm CE reinforcement width 0 mm Table 4.3: Parameters used to perform the buckling modes verification. Note that no spar nor CE has been used. But prior to jumping to this tests, I want to explain the buckling phenomena depicted in Figure 4.9, which happened in the intrados - as it is going to be one of the modes obtained in this section. That buckling phenomena is mainly caused by the shear stress at the wing. let me show the only three non-zero stresses distribution at the intrados (σ11,σ22 and τ12) - model settings of Section 4.1.1. Converged Mesh, encastre 3 and load distribution 3. Figure 4.10: Intrados σ11 distribution. 87 Chapter 4. Results Figure 4.11: Intrados σ22 distribution. Figure 4.12: Intrados τ12 distribution. It is important to recall that shell elements are employed in this thesis - in the modelling of the skin, and that this elements use a local coordinate system as a reference (see Section 2.7.4. Shell Elements). Being so, the results depicted in Figures 4.10, 4.11 and 4.12 are referenced to that local coordinate system. The latter is different for every shell element, as it follows the normal direction of each element. The 1 direction is in the span-wise direction, the 3 direction is the normal of the element, and the 2 direction is the one that forms an orthogonal coordinate system with the 1 and 3 directions. Some local coordinate systems of the intrados are depicted in Figure 4.13: 88 Chapter 4. Results Figure 4.13: Local coordinate systems at the intrados. Following that local coordinate systems, and the stress fields depicted in Figures 4.10, 4.11 and 4.12, the stresses (at the intrados) per element are: Figure 4.14: Stresses in an intrados element. Rotate now the element in Figure 4.14 ±45◦and in you have compression stress in the intrados - some in the span-wise derection and other in the chord-wise direction. Having understood that, I made a prediction which was the following: the buckling modes of the Figures 4.7, 4.8 and 4.9 with a fabric skin oriented at ±45◦, and the buckling modes of a fabric skin oriented at 0/90◦should be way different. Figure 4.15 depicts what I have explained in this paragraph: Figure 4.15: Stress at the intrados and lamina equivalence of a pair of laminae at ±45 and 0/90◦. The last Figure shows the equivalence of the two laminae to a laminae of equivalent longitudinal E0 land transverse E0 tmodulus, rotated a certain θ. The 0/90◦is equivalent to ±45◦laminae of equivalent modulus, and vice versa with the two laminae at ±45◦. 89 Chapter 4. Results That fact makes the 0/90◦set of laminae weaker in the directions where compression stresses appear at the intrados. Being so, this configuration will be much more susceptible to have buckling modes at the intrados than the ±45◦set. I have extracted the first three modes of the same structure as for Figures 4.7, 4.8 and 4.9, just changing the skin to 0/90◦. The results are the following: Figure 4.16: First buckling mode of the 0/90◦GE skin configuration. Figure 4.17: Second buckling mode of the 0/90◦GE skin configuration. 90 Chapter 4. Results 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 mass (kg) 1/λ Batch 1 points m = 0.45 kg 1/λ = 1 Figure 4.32: Batch 1 with limiting lines m= 0.45 and 1/λ=1. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mass (kg) 1/λ Batch 1 points Figure 4.33: Batch 1 with the first step of the decision making procedure applied. After this step, 19 points have been rejected. Hence, points that fail by buckling or exceed the maximum mass set are discarded. The second step of the decision making procedure is reject any point that has at least one failure index which satisfies IF≥1. By inspection of every point failure indices, invalid points are discarded, leaving just the ones that satisfy the failure (statical and buckling) and mass premises. 97 Chapter 4. Results 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mass (kg) 1/λ Batch 1 points Figure 4.34: Batch 1 with the second step of the decision making procedure applied. After this step, 102 points have been rejected, with only 29 points remaining. All this points failed statically. As seen in Figures 4.10, 4.11 and 4.12, the stress perturbations of the encastre take a great portion of the semi-wing. That concentration - that is for sure increased by the numerical simulation - has caused stress peaks that make the structure suffer statically. It must be recalled that the load factor ntested is 4, so it is a very demanding flight performance. Despite that, failure indices IFare very influenced by this stress concentration, resulting in this large amount of points rejected in this step. The last step of the decision making procedure, which recall is implemented with a Matlab code named Decisionmaking.m (see Annex B.1. Decisionmaking.m.), is find the Pareto optimal points or the also-known Pareto front. The points belonging to this front are the non-dominated points, and thus are the ones from which the design process is continued - see Section 2.9. Pareto Optimality. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mass (kg) 1/λ Batch 1 points Pareto points Pareto front Axes bisector Figure 4.35: Batch 1 valid points and the Pareto front. As all the cases I have worked with in this thesis have a mass >0.25 kg, from now on 98 Chapter 4. Results the plots that are going to be presented will have m = 0.25 as the initial value. Also, it will help depicting the points on the plots, as there will be a lot of them. 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mass (kg) 1/λ Batch 1 points Pareto points Pareto front Axes bisector Figure 4.36: Close-up of the Batch 1 valid points and the Pareto front. With the Pareto front achieved - made of 17 points out of the 29 - it is time to consider which are the most optimal within the Pareto optimal. As depicted in Figure 4.35, the axes bisector is indicated with an arrow. The arrow points towards the optimal region where I want all the points to be. Points near the bisector are the most optimal points I can get, as they are points which are both strong in the two objective functions - minimizing the mass and minimizing the inverse of the first buckling mode constant (i.e. maximizing the first buckling mode constant). Following that reasoning, I have focused on the two columns of non-dominated points near the bisector - see Figure 4.35. From those points, the most optimal ones are the following two: 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 88 117 mass (kg) 1/λ Batch 1 valid points Pareto points Pareto front Axes bisector Optimal points Figure 4.37: Batch 1 valid points, the Pareto front, and the two selected points to continue the design process. The points/cases 88 and 117 are the best options to continue with the design process. The criterion to choose both points is the following: besides being close to the bisector, 99 Chapter 4. Results both belong to a column of points, which all have almost the same mass, but they two are the ones with the best 1/λ in their respective groups. The trends that have been observed with the Pareto front points are the following: •7 points have skin A, 8 points have skin B and 2 points have skin C. Both selected points, 88 and 117, have skin B. Conclusion: it is way better to use a symmetric skin layer sequence - see the skins in Section 3.1.5. Materials Employed and its Properties. •4 points have 1 mm core thickness, 4 points have 2 mm core thickness, 4 points have 3 mm core thickness and 5 points have 4 mm core thickness. None Pareto optimal point has 5 mm core thickness. Also, the two columns of points mentioned before (8 points, exactly), are the 2 and 3 mm core thickness points. The selected points, thus, have a core thickness of 2 mm, one, and 3 mm, the other. •2 points have the spar positioned at 35 mm from the leading edge, 3 at 53.2 mm, 10 at 65 mm and 2 at 75 mm. This shows a clear trend towards having the spar positioned towards the 34.21% of the chord (that’s the percentage that 65 mm represents). This is logical, as the suction peak modelled in Section 3.2.1.3. Load Definition for option 3 was centred in between the 14% and the 54% of the chord - being the resultant of the force approximately centred towards the 33% of the chord. The spar position of the points selected is 53.2 mm (maximum thickness) and 65 mm. •The region where CE is applied (reinforced region placed at the maximum thickness point) has had a width of 30 mm in 16 of the 17 points, whereas only 1 point has a 20 mm width. Therefore, for the weight that add 10 mm more of reinforced section, it is clearly optimal to choose this option. A summary of the two selected points is detailed in the next table: Features 88 117 Mass (kg) 0.362 0.3157 1/λ 0.2304 0.3401 Skin B B Core thickness (mm) 3 2 Spar position (mm) 53.2 65 CE reinforcement width (mm) 30 30 Table 4.6: Main features of the two selected points of Batch 1. From now one, the improvements explained in Section 3.2.3. Improvements have been applied to this points first, and then two some of their children - i.e. optimal points which have been obtained from points 88 and 117. 4.3 Second Batch of Cases - Improvement 1 The improvement 1 detailed in Section 3.2.3. Improvements is now applied to points 88 and 117. Recall that this improvement consists on changing the spar width and maintain the rest of the parameters unaltered. The spar widths that have been tested are 5,15,20 and 25 mm. Being so, 8 new cases have been tested: 4 with point 88 and 4 with point 117. Applying the decision making procedure, of this 8 cases, 7 are valid - see Annex C. 100 Chapter 4. Results Numerical Results to consult the parameters of this points. Next, a plot with the Batch 1 valid points and the valid points of this new Batch 2 are represented: 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 88 117 mass (kg) 1/λ Batch 1 valid points Pareto points Batch 2 valid points Old Pareto front New Pareto front Axes bisector Optimal points Figure 4.38: Batch 1 valid points, old and new Pareto front and Batch 2 valid points. As it can be seen in Figure 4.38, the bacth 2 adds 4 points to the new Pareto front, which has been reduced from 17 to 14 points. In other words, 7 old non-dominated points are now dominated points - i.e. are no longer optimal points. Finally, the trends that have been observed with the Batch 2 points are the following: •Increasing the spar width leads to more total mass of the structure. But increasing the section of the spar allows it to transmit more stresses unloading the skin, and thus statical failure indices are lower and the first buckling mode constant increases. •Decreasing the spar width leads to less total mass of the structure. But decreasing the section of the spar reduces the ability of the spar to unload the skin, and thus statical failure indices are higher and the first buckling mode constant decreases. As explained in Section 3.2.3. Improvements, a decision of the next points to optimise from the new Pareto front will not be taken until the 3 improvements and tested and all children points are compared all together. 4.4 Third Batch of Cases - Improvement 2 The improvement 2 detailed in Section 3.2.3. Improvements is now applied to points 88 and 117. Recall that this improvement consists on adding a second spar, of the same width as the main spar, a certain distance away and maintain the rest of the parameters unaltered. The second spar positions that have been tested are 105,115, 125 and 135 mm from the leading edge. Being so, 8 new cases have been tested: 4 with point 88 and 4 with point 117. Applying the decision making procedure, of this 8 cases, 8 are valid - see Annex C. Numerical Results to consult the parameters of this points. 101 Chapter 4. Results Next, a plot with the Batch 1 valid points and the valid points of this new Batch 3 are represented: 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 88 117 mass (kg) 1/λ Batch 1 valid points Pareto points Batch 3 valid points Old Pareto front New Pareto front Axes bisector Optimal points Figure 4.39: Batch 1 valid points, old and new Pareto front and Batch 3 valid points. As it can be seen in Figure 4.39, the batch 3 adds 8 points to the new Pareto front, which has been increased from 17 to 24 points. As the 8 points of the Batch 3 belong to the new Pareto front, 1 old non-dominated point is now a dominated point - i.e. is no longer an optimal point. To conclude, the trends that have been observed with the Batch 3 points are the following: •Adding a second spar increases the total mass of the structure. It also increases the section of stress transmission between extrados and intrados, which unloads the skin. Thus, statical failure indices are lower and the first buckling mode constant increases. •Decreasing or increasing the distance of the second spar will result in more or less mass addition. As the distance is increased, the thickness of the airfoil decreases, and thus the second spar has less volume -i.e. less mass. Accordingly, the Idecreases and so does the unloading ability of the spar, resulting in higher failure indices and lower first buckling mode constant than a structure with a second spar closer to the main spar. As explained in Section 3.2.3. Improvements, a decision of the next points to optimise from the new Pareto front will not be taken until the 3 improvements and tested and all children points are compared all together. 4.5 Fourth Batch of Cases - Improvement 3 The improvement 3 detailed in Section 3.2.3. Improvements is now applied to points 88 and 117. Recall that this improvement consists on adding stringers of different widths and maintain the rest of the parameters unaltered. Points 88 and 177 have 3 and 2 mm of core thickness, respectively. Thus, the core combinations that are going to be tested on each are: 102 Chapter 4. Results Features 88 117 Original core thickness (mm) 3 2 Core thickness no-stringer region (mm) 2 3 1 2 Core thickness stringer region (mm) 3 / 4 / 5 4 / 5 2 / 3 / 4 / 5 3 / 4 / 5 Table 4.7: Core combinations for the stringers of improvement 3. Core thickness thicker than the original core thickness have not been considered as the increase of weight would be very substantial. Besides, the stringer regions are going to have the same skin composition as the CE reinforced region already applied up to now - and that adds weight too. Figure 4.40: Stringers improvement implemented in Bacth 4. As shown by Figure 4.40, one stringer has been placed before the spar and two behind. Stringer one is centred at 16.6 mm from he leading edge, stringer two is centred at 96.25 mm from the leading edge, and stringer three is centred at 143.75 mm from the leading edge. Their width has been varied equally for the two optimal points (88 and 117) - see Table 4.8. Stringer configuration S1 w (mm) S2 w (mm) S3 w (mm) 1 10 20 20 2 15 30 30 3 24.2 42.5 42.5 Table 4.8: Stringer configurations for improvement 3 points. Being so, 15 new cases have been tested for point 88, and 21 new cases for point 117. Applying the decision making procedure, of this 36 cases, 26 are valid - see Annex C. Numerical Results to consult the parameters of this points. Next, a plot with the Batch 1 valid points and the valid points of the Batch 4 are represented: 103 Chapter 4. Results 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 88 117 mass (kg) 1/λ Batch 1 valid points Pareto points Batch 4 valid points Old Pareto front New Pareto front Axes bisector Optimal points Figure 4.41: Batch 1 valid points, old and new Pareto front and Batch 4 valid points. Figure 4.41 shows the new Pareto front, which has 9 points added by the batch 4 - the Pareto front has been increased from 17 to 25 points. Subtracting the 9 front points of the Batch 4, there are 16 points leftthat means that 1 old non-dominated point is now a dominated point. After carrying out this improvement, I thought that it would be smart to test another skin sequence at the stringers. As the skin already has the CE reinforced region, I have removed one laminae of CE in the skin that is applied at the stringers regions. Besides, I have moved this laminae to gain I. I have done the following: (0/90/0/core)s→ (0/90/core/90/0/0), changing from a symmetric layer sequence to a unsymmetrical one - note that the bold 0 denotes the CE laminae. This removal of the laminae has lowered the total weight and as I am going to show you next, given Ito the section - as the λhave increased. Again, 15 new cases have been tested for point 88, and 21 new cases for point 117. Applying the decision making procedure, of this 36 cases, 15 are valid - see Annex C. Numerical Results to consult the parameters of this points. This is logical, the reduction in CE - a high stiffness material - has increased the maximum stresses and thus more specimens have failed. Next, a plot with the Batch 1 valid points and the valid points of this extra points for Batch 4 are represented: 104 Chapter 4. Results 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 88 117 mass (kg) 1/λ Batch 1 valid points Pareto points Batch 4 extra valid points Old Pareto front New Pareto front Axes bisector Optimal points Figure 4.42: Batch 1 valid points, old and new Pareto front and extra (valid) points for Batch 4. Figure 4.42 shows the new Pareto front, which has 10 points added by the batch 4 - the Pareto front has been increased from 17 to 26 points. Subtracting the 10 front points of the Batch 4, there are 16 points leftthat means that 1 old non-dominated point is now a dominated point. Now, it is convenient to compare the 26 valid points of the first stringer skin configuration and the 15 valid points of the second. To do that, a plot with the Batch 1 valid points and the 41 stringer valid points of Batch 4 is represented next: 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 88 117 mass (kg) 1/λ Batch 1 valid points Pareto points Stringer configuration 1 Stringer configuration 2 Old Pareto front New Pareto front Axes bisector Optimal points Figure 4.43: Batch 1 valid points, old and new Pareto front and final Batch 4 points. Figure 4.43 shows the new Pareto front, which has 11 points added by the batch 4 - the Pareto front has been increased from 17 to 27 points. Subtracting the 11 front points of the Batch 4, there are 16 points leftthat means that 1 old non-dominated point is now a dominated point. What is important to remark is the following: 10 of the 11 points added to the Pareto front belong to the second stringer configuration. Even though the valid points of this 105 Chapter 4. Results second configuration were less than the first - 15 versus 26, they have proven to be more optimal. See Figure 4.44 to visualise with clarity that almost all new front points belong to the second stringer configuration. 0.3 0.35 0.4 0.1 0.2 0.3 0.4 0.5 88 117 mass (kg) 1/λ Batch 1 valid points Pareto points Stringer configuration 1 Stringer configuration 2 Old Pareto front New Pareto front Axes bisector Optimal points Figure 4.44: Close-up of Batch 1 valid points, old and new Pareto front and final Batch 4 points. Now, the trends observed with Batch 4 points are the following: •The mass trend is difficult to predict because there are two main influencing variables: the core combination and the stringer width combination. A general trend can be described as: for low thickness core combinations and low stringer widths, the mass gets reduced; for low thickness core combinations and mid-high stringer widths, the mass can either be reduced or increased; and for high thickness core combinations and whatever the stringer width, the mass gets increased. •Following the same reasoning for the first buckling mode constant: or low thickness core combinations and low stringer widths, λgets reduced; for low thickness core combinations and mid-high stringer widths, λcan either be reduced or increased; and for high thickness core combinations and whatever the stringer width, λgets increased. Now that the 3 improvements have been tested, it is time to compare the first batch of points, Batch 1, and the Batches 2 to 4 - created from the improvements - all together. After that, new optimal points will be extracted and the design will be followed from those points on. 4.6 Decision after Improvements 1, 2 and 3 To continue with the design process, new points must be selected. Let’s see if new Pareto optimal points are obtained from the Batches 2 to 4. Next is the representation of the valid points of Batches 1 to 4, and the new Pareto front: 106 Chapter 4. Results Point mass (kg) 1/λ 131 0.2691 0.7077 106 0.2697 0.6897 111 0.2697 0.6667 116 0.2697 0.5952 198 0.3026 0.5732 221 0.3067 0.5668 132 0.3150 0.4705 107 0.3157 0.3846 112 0.3157 0.3717 117 0.3157 0.3401 246 0.3229 0.3215 222 0.3321 0.2955 156 0.3371 0.2809 255 0.3393 0.2793 151 0.3408 0.2447 249 0.3483 0.2272 243 0.3692 0.2186 185 0.3784 0.2071 158 0.3790 0.1580 154 0.4256 0.1553 Table 4.12: Final Pareto front points. Of those 3 points, the best one in optimisation terms is point 246, as it almost lays in the bisector. It is the point that best minimises both objective functions. Besides, point 246 and 117 have another key feature that I think it is a great advantage over point 222: simplicity of construction. Recall from Table 4.11 that this two points do not have stringers. Thus, the building process of such a structure is easier as stringers incorporate geometrical features that are difficult to tackle. Taking a closer look at the building point of view, point 117 is really the most desirable one, as unlike 246, doesn’t have ribs. Thus, it eliminates a tedious building part, as the ribs have to fit the extrados and intrados shape. The latter are not going to have the theoretical shape after the build of the skin, and the ribs will have to adapt to its shape by a manual process - e.g. progressive sanding until it fits. In conclusion, point 246 is the best point from an optimisation point of view. And from a building point of view, point 117 is the best one as it is the simplest one to construct. And leaving aside those points of view, points 246,117 and 222 are the best points of the total mount of 260 points tested for this design process. Tabel 4.13 sums up the features of this 3 points. Features 117 222 246 mass (kg) 0.3157 0.3321 0.3228 1/λ 0.3401 0.2954 0.3215 Skin B B B Core thickness (mm) 2 2 2 Core thickness stringer region (mm) - 3 - Stringer skin configuration - 2 - Main spar position (mm) 65 65 65 Main spar width (mm) 10 10 10 CE reinforcement width (mm) 30 30 30 Ribs per semi-wing - - 3 Table 4.13: Main features of the points 117, 222 and 246. 113 Chapter 5 Conclusions In this chapter, some conclusions are extracted to put an end to this thesis. The conclusions that have been obtained are of different nature. These are: •Conclusions about the results obtained. •Conclusions about the numerical model employed in AbaqusTM. •Conclusions about the design procedure followed. •Aproject overview about the organisation, goals achieved, and self-valuation of the thesis. Next, each of this sections are going to be covered. 5.1 Conclusions about the Results First of all, the results obtained mainly help me see the trends and effects that every parameter of design has on the structural behaviour. These results are not a definite optimised solution to the monocoque structure made with composite materials. They just represent a possible iteration from a starting point of parameter values. To obtain such final solution, a full iteration loop over a wider range of parameters and with different starting points should be done. Being so, the conclusions I want to subtract from the results are the trends that each parameter has shown in modifying the structural behaviour. The trends are: •Skin: of the 3 types of skin proposed for this thesis - which are actually 6 depending on the thickness of the core - I have concluded the following: first, sandwich panels (with non-zero core thickness) are better than laminates (zero core thickness) for the expected performance of the wing; secondly, skins with layers stacked symmetrically are way better than if the sequence is unsymmetrical. This was predicted in Section 2.3.3.2. Laminates and proved in Section 4.2. First Batch of Cases - Starting Point. •Core thickness: as commented with the skin, sandwich panels have proven to be the best option for the wing and for that reason, zero core thickness is not an option - a high section second moment of inertia Iis necessary to have a good behaviour against the bending moment. Then, regarding the resting 5 options I have tested, the optimum is found with 2/3 mm core thickness, feature that have had the most promising points tested throughout the whole design process. Despite that, there are also Pareto optimal points from the final Pareto front with the 5 core thicknesses. 114 Chapter 5. Conclusions So, it just depends on how much you want to strengthen one objective function or the other. •Spar width: different spar widths have been tested. Increasing the spar width leads to an increment of the total mass but an unloading of the skin stresses. The latter results in higher resistance to statical and buckling failure. In the other hand, reducing the spar width leads to a lower total mass but has the contrary effect with statical and buckling failure. The final selected points have all 10 mm spar width, but within the final Pareto front there are also cases with 5 and 15 mm spar widths. •Spar position: 5 positions have been tested in the design process. Of this 5, two of them have proven to be the best of them: spar placed at the maximum thickness point of the airfoil (53.2 mm) and placed at 65 mm from the leading edge. The front-chord positions tested have had a lack of I, as there the thickness is reduced, and thus the skin has suffered more both statically and by buckling. Although 53.2 and 65 mm options have add more mass, as the spar has more height in those positions, the Igained has compensated with the good structural behaviour shown. Plus, as the airfoil suction peak has been modelled between the 14% and 54% of the chord - to simulate the resultant lift force at about a 33% of the chord, those two configurations are more optimal to transmit the stresses more directly without having to deal with a noticeable torsion moment. •Second spar: adding a second spar is an option that adds mass but improves the statical and buckling behaviour of the structure - it adds more surface of stress transmission, thus unloading the skin. Then, the position of this second spar adds more or less mass depending on how far from the main spar it is placed: the further it is placed, the less mass it adds, and vice versa. Accordingly, the further it is placed, the less it improves the structural behaviour of the structure, and vice versa. I have nothing more to say about the second spar, as it has not thrived in the optimisation process. •Stringers: 3 widths of stringers and different core combinations have been tested in the design process - see Section 4.5. Fourth Batch of Cases - Improvement 3. Also two stringers skin combinations have been tested. The mass trend is difficult to predict because there are two main influencing variables: the core combination and the stringer width combination. A general trend can be described as: for low thickness core combinations and low stringer widths, the mass gets reduced; for low thickness core combinations and mid-high stringer widths, the mass can either be reduced or increased; and for high thickness core combinations and whatever the stringer width, the mass gets increased. If the skin with more CE is used, it will add more mass than the other. Following the same reasoning for the structural behaviour of the structure: for low thickness core combinations and low stringer widths, the structural behaviour worsens; for low thickness core combinations and mid-high stringer widths, the structural behaviour can either worsen or get better; and for high thickness core combinations and whatever the stringer width, the structural behaviour get better. The optimal combinations, though, have been the ones in which the stringer has had the following combination of parameters: –Original core thickness in the no-stringer region. –Stringer core thickness 1 mm higher than the original. –Width configuration 1: the one with the less width (10, 20 and 20 mm). –Skin configuration 2 at the stringers: less CE but similar I. 115 Chapter 5. Conclusions •Ribs: the use of ribs have proven that the structural behaviour against buckling improves. That is due to the reduction of the buckling length. Despite that, adding ribs also adds to the total structure mass. 3 possible combinations have been tested - see Section 4.8. Sixth Batch of Cases - Improvement 5. From those 3, the one that has turned to be the most optimal one has been the 3 ribs per semi-wing configuration - it has add 4 non-dominated points to the final Pareto front. Finally, I want to extract some conclusions about the failure criteria employed in this thesis. Regardless of the values obtained for each criterion indices (as numerical parameters influence the latter), I want to state the following: Hashin indices have always been lower than Tsai-Hill ones, which is completely logical: the latter criterion takes into account the influence of different type of stresses - σ11,σ22 and τ12 - at the same time. The Hashin criterion, conversely, doesn’t consider the influence of different type of stresses. 5.2 Conclusions about the Numerical Model Regarding the numerical model made with AbaqusTM, I feel that the way I have approached it is a good one - see Section 3.3.1. Pre-processing - but I want to state the following: FEA is a complicated method of solving an structural problem such as mine, and for that reason AbaqusTM has a very long set of possibilities (different solvers, mesh types, material types, etc) that I haven’t had time to master. Being so, I cannot be sure that there is not a better way of tackling the thesis problem - i.e. modelling the geometry differently, modelling and assignation of materials, meshing of parts, interaction properties between parts, etc. I wanted to state the latter as I feel that, as remarked in previous Section 5.1. Conclusions about the Results, the numerical results obtained cannot be trusted with a 100% sureness. For instance, with the ribs models I have noticed a strange behaviour in some points, of which I don’t understand its reasons. Despite re-doing the model, it has kept happening. That behaviour is the following: even though the case to which ribs were added was statically correct (IF<1), when adding the ribs, this case now failed statically. As perfect bonding between rib and skin and rib and spar is simulated, I don’t understand why the ribs can create the stresses to rise, when they should unload the skin. Accordingly, other rare event is the fact that out 150 starting cases, 102 failed statically. That is mainly due to the boundary conditions effect - singularities (see Section 3.2.1.2. Boundary Conditions), which cannot be completely solved as it is a price that must be paid when using FEA software. Maybe, the most important thing about this, is not how to prevent it from happening, but to know how to treat it - see next Section 5.3. Conclusions about the Design Process. To conclude, I’m satisfied with the numerical model but I am sure it can be improved as some flaws have shown to exist. 5.3 Conclusions about the Design Process Regarding the design process, I have learned a precious lesson. When addressing multi-objective and multi-parametric problem as the one in this thesis, the preparation of the design process so it is as automatised as possible is crucial. If the latter is achieved, then more tests can be run - more parameters can be tested and the range of the design 116 Chapter 5. Conclusions gets wider - and less time (and patience/energy) is lost getting models ready to run. One of the main obstacles has been the modelling of the new geometry for every case to pass it to AbaqusTM. Not for the complexity of that process - which was quasi-automatised in CATIA V5R20 - but for the time lost in the making of every model. As I haven’t been able to parametrize the geometry in AbaqusTM, the latter has had to come from outside the program. AbaqusTM models can be entirely made by a script: the geometry, materials definition and assignation, mesh, loads and boundary conditions definition, etc. But I haven’t had enough time to learn how to do so, as learning how to use the program itself has already been complex. If I had been able to achieve such a modelling (geometry and numerical model) technique, the number of iterations over the parameters could have been way larger than it has been. Throughout the design process I have picked the most promising points after each step. The reasoning employed to do so is completely correct but if I had had this means implemented, I could have followed the design process after each step with all the points in the Pareto front instead of just a few, as I have done. Also, more parameters could have been tested. Instead of limiting the spar positions, or the types of skin, or even the material properties, a wider range of values could have been tested for every parameter. But to do so the process has to be automatised or else you will need three times (or more) the time I have employed to accomplish this thesis. In other words, I am satisfied with the design process I have followed: do some initial tests to be able to define a not-so-imprecise starting point; define a set of simple cases as the starting point, and from those cases select the best points and keep adding improvements to see how they change. But the process has had to be delimited due to the time expense of every model-making and running of the latter with the tools I have been able to develop. Lastly, I want to comment something about the post-processing of the output variables. As occurred to me when runnign the last cases for the thesis, the failure indices could have been collected at a certain distance from the encastre to avoid the singularities that are always generated in a FEA software. Therefore, following the St. Venant’s Principle the results obtained regarding statical failure would have been more logical - composite materials are high stiffness materials which are hard to break statically and low thickness composite-made structures almost always fail by buckling, not statically. 5.4 Project Overview Overall, I am satisfied with the output of the thesis. The scope and objectives set at the beginning of the project have been fully achieved. The main issue I have had is time. Getting to know what I needed to know to be able to jump onto the next step has always been slower than I expected. For that reason, I have needed to employ more hours than the 600 that I first thought I would need. Finally, just say that this thesis has encouraged me to keep up with my education in the aerospace engineering field, as it has been very useful to make me see what I already know and that I have still a lot new things to know. 117 Chapter 6 Future Work The future work for this thesis is the automation of the whole design process. Doing so, a wide range of parameters could be tested and there would be no need for delimitation of the study. AbaqusTM has the option of creating a parametric study with a script. In the AbaqusTM Scripting User’s Guide there are some examples and some guidelines on how to create such a script. In order to develop such a designing tool, it is necessary to know Python language of programming, the basic commands to communicate with AbaqusTM, and as the problem of this thesis is a multi-objective one, also knowledge of optimisation algorithms. A parametric study allows you to create just one time the model and then in the script you assign the variables that are going to have different parameters. With a loop iteration, each parameter is assigned and the case is run each time a change is performed - for each case the geometry, assembly, mesh, loads, and others, are actualised. Being able to automatise that would save a lot of time as the main loss of time I have suffered in this thesis has been in the making of the models and jobs definition. But in addition to that, an optimisation algorithm is necessary to be employed. Otherwise, the only part that would be automatised of the process would be the analysis process - from the making of the model to the execution of the job. With an optimisation algorithm, the decision making procedure - that has been performed after each set of new cases has been obtained - would be included in the script. By doing so, once the script is ready, the whole design process can be run without interruptions. To be able to do so, some optimisation algorithms - that can make the appropriate decisions based on the objective functions defined - are necessary. The most well-known optimisation algorithms are the genetic algorithms, which generate solutions using techniques inspired by natural evolution, such as inheritance, mutation, selection and crossover. 118 Chapter 7 Environmental Impact As a part of every project, the environmental impact that this thesis has must be covered. Next I will explain the advantages and disadvantages that the making of this thesis has in the environment. •Advantages –Achievement of a multi-objective and multi-parametric design faster. This can eventually increase the development of more composite-made structures, as the biggest brake for this structures against conventional-material-made ones is the complexity of design and thus, time required. Finally, if more compositemade structures are build for aircraft wings, or any other moving vehicle, this will result in more fuel efficient and less polluting vehicles - as this type of structure has proven to be more lightweight than conventional structures. – Saving of raw materials. Thanks to the optimisation process carried out in the thesis, the results obtained reduce a lot the possibilities of such a multi-parametric problem. This allows less need of prototyping and, consequently, less tests required - with all it entails: test-benches, special tooling and qualified personnel. All this means a great money saving. And the environmental advantage is: ∗Reduction of wastes: as less materials are required, the production of the latter will be lower as well. That implies that the wastes generated throughout the process are also cut down. [27] has a very interesting table that sums up which are the wastes (and the effects of those) generated in composite materials production, which now can be reduced - see Table 7.1. – Savings of paper: as the computer is the main working tool for this thesis - and in nowadays society, paper has not barely used. •Disadvantages – Energy consumption: ∗Electricity: the main tool of design is the computer. Throughout the time spent to do this thesis, the latter has needed power to be functioning. Also, it must be taken into account that the working place to do the thesis needs eletricity to be suitable: air-conditioning, lights or other office/indoor devices also need electricity. This electricity must be generated by either hydroelectric or nuclear power station or others means like fossil fuels. The point is that the production of that electricity pollutes the environment by emitting hazardous emissions to the air, contaminating the water of rivers or adulterating the food that we eat as a result of the first two. 119 Chapter 7. Environmental Impact ∗Natural gas: in regard with the needs of the place of development of the thesis, some heating infrastructure is needed in any housing, and most of this heating systems work on natural gas. As explained with electricity, natural gas production also harms the environment. Table 7.1: Typical composites fabrication processes, products used and wastes. Retrieved from [27]. 120 Bibliography [1] “Lancair IV.” [Online]. Available: http://www.aerodynamicsllc.com/planes we build.html [2] “Chanute’s biplane glider.” [Online]. Available: http://media-2.web.britannica.com/ eb-media/13/99713-004-3D59FB9D.jpg [3] “The Wright flyer.” [Online]. Available: http://robustdesignconcepts.com/files/ spaceshipone/www.nasm.si.edu/exhibitions/gal100/wright flight.jpg [4] “Bleriot’s airplane.” [Online]. Available: http://i.dailymail.co.uk/i/pix/2009/07/17/ article-1200178-03125B93000005DC-448 634x286.jpg [5] “The Deperdussin.” [Online]. Available: http://www.ctie.monash.edu.au/hargrave/ images/dep gb provost 041013 500.jpg [6] “The Junkers J1.” [Online]. Available: http://www.junkers.de/sites/default/files/ media/J1 THF 024 1175 2.jpg [7] “The Boeing 707.” [Online]. Available: http://cdn-www.airliners.net/ aviation-photos/photos/1/7/4/1948471.jpg [8] “X-29.” [Online]. Available: https://www.nasa.gov/sites/default/files/images/ 299396main EC87-0182-14 full.jpg [9] “Materials employed in the Boeing 787.” [Online]. Available: http://www.1001crash. com/dossier/composite/B787 composite.jpg [10] “Truss structure.” [Online]. Available: hhttp://legendaryaircraft.hellobit.hu/gallery/ yak-3-2/fuselage frame 1.jpg [11] “Monocoque vs. Semi-monocoque.” [Online]. Available: http://source-http//www. aero-mechanic.com/wp-content/uploads/2011/03/2-14.gif [12] D. Gray, Composite Materials: Design and Applications. CRC Press, 2014. [13] W. Callister and D. Rethwisch, Materials science and engineering: an introduction. Wiley, 2007. [14] D. Roylance, “Laminated composite plates,” 2000. [15] G. Zagainov and G. Lozino-Lozinski, Composite Materials in Aerospace Design. Chapman & Hall, 1996. [16] “Tractions and moments in a plate.” [Online]. Available: http://www.mathworks.com/matlabcentral/mlc-downloads/downloads/ submissions/44717/versions/6/screenshot.jpg 121 Chapter 7. Environmental Impact [17] J. N. Reddy, Mechanics of Laminated Composite Plates - Theory and Analysis. CRC Press, 1997. [18] “Force-displacement bifurcation diagram.” [Online]. Available: http://www.strand7. com/images/equations/image19.gif [19] Q. J. Yang, “Simplified Approaches to Buckling of Composite Plates,” Master’s thesis, 2009. [20] E. J. Barbero, Finite Element Analysis of Composite Materials. CRC Press, 2007. [21] “Abaqus 6.14 user manual,” DS Simulia, 2014. [22] “Pareto optimality example.” [Online]. Available: http://www-cdr.stanford.edu/ NextLink/images/pareto.gif [23] “Pareto front.” [Online]. Available: https://upload.wikimedia.org/wikipedia/ commons/thumb/b/b7/Front pareto.svg/640px-Front pareto.svg.png [24] “Cantilever beam.” [Online]. Available: http://www.codecogs.com/users/23287/ Cantilever-Beams-101.png [25] “Pressure distribution over an airofil.” [Online]. Available: http://www.efm.leeds.ac. uk/CIVE/CIVE1400/Section4/img00048.gif [26] B. D. Agarwal, L. J. Broutman, and K. Chandrashekara, Analysis and Performance of Fiber Composites. Wiley, 2006. [27] “A Guide for Composites Fabrication,” Washington State Department of Ecology, 2004. [28] T. A. Weisshaar, Aerospace Structures - an Introduction to Fundamental Problems, 2011. [29] S. Krenk and J. Høgsberg, Statics and Mechanics of Structures. Springer Science+Business Media, 2013. [30] M. C.-Y. Niu, Airframe Structural Design: Practical Design Information and Data on Aircraft Structures. Conmilit Press, 1988. [31] T. H. G. Megson, Aircraft Structures for engineering students, 2007. [32] “Introduction to Classical Laminated Plate Theory,” MHRD, Govt. of India. [33] C. R. Steele and C. D. Balch, Introduction to the Theory of Plates. [34] J. N. Reddy, Energy and Variational Methods in Applied Mechanics. Wiley, 1984. [35] S. S´adaba, “High fidelity simulations of failure in fiber-reinforced composites,” Ph.D. dissertation, Universidad Polit´ecnica de Madrid, 2014. [36] “Hashin Criterion,” AUTODESK. [Online]. Available: http://help.autodesk.com/view/ACMPAN/2016/ENU/?guid= GUID-A2597BFC-606B-4C21-ABCA-8F34D37FBC41 [37] “Tsai-Hill Criterion,” AUTODESK. [Online]. Available: http://help.autodesk.com/view/ACMPAN/2016/ENU/?guid= GUID-01A4F20B-09D0-4DED-91F7-967277BEDBD6 122