University of Pisa
Department of Civil and Industrial engineering
Aerospace Section
WAGNER: a new code for automatic parametric
structural study of PrandtlPlane fuselages
Supervisors: Candidate:
Prof. Eng. Aldo Frediani Marco Picchi Scardaoni
Dr. Eng. Vincenzo Binante Dr. Eng. Vittorio Cipolla
Il faut imaginer Sisyphe hereux (A. Camus, Le Mythe de Sisyphe)
Πάντες ἄνθροποι τοῦ εἰδέναι ὀρέγονται φύσει (Aristotle, Metaphysics)
Il faut être absolument moderne (A. Rimbaud, Adieu, Une saison En Enfer)
The present paper presents part of the activities carried out within the research project PARSIFAL (“PrandtlPlane ARchitecture for the Sustainable Improvement of
Future AirpLanes”), which has been funded by the European Union under the Horizon 2020 Research and Innovation Program (Grant Agreement n.723149).
Abstract
In the present thesis, a code for geometrical model generation and for statical and frequency FEM analyses of PrandtlPlane fuselages is presented, thought to be used in the frame of Parsifal project.
Named WAGNER, the code aims to provide a time-cheap as well as reliable tool in the conceptual design phase of a civil transport PrandtlPlane aircraft, in order to evaluate, in a preliminary way, stress and strain fields in the whole fuselage structures, so allowing a preliminary structural efficient sizing to be used as a baseline for deeper investigations.
Furthermore, WAGNER is able to predict, more accurately then classical statistic-based approaches, the fuselage structure weight, to better estimate the empty weight and, by consequence, the maximum take-off weight of a PrandtlPlane aircraft. By varying parameters, the user can obtain a realistic configuration of the final aircraft, due to the high number of degrees of freedom.
The process is completely automatized: the user can easily set up geometrical parameters, load cases and choose analyses and output of interest. The meshing module is integrated in the code, as well.
In the second part of the thesis, some considerations about weight estimation and preliminary design are described. In this part, two layouts are taken into consideration: a single-deck 2-4-2 passengers abreast and a double-deck 3-3 passengers abreast.
Thereafter, a preliminary high structural efficiency sizing is presented for both the under-analysis layouts. FEM results for three different load cases - cruise condition, ultimate pressurization (linear and geometrical non-linear solutions) and ultimate load factor - are commented in the last three chapters, showing the validity of WAGNER and raising new critical issues.
Contents
I
Code Outline
1
1 Parsifal project 2
1.1 PrandtlPlane configuration: a brief outline . . . 2
1.2 Parsifal project . . . 6
1.2.1 Objectives . . . 6
1.2.2 Partners and Advisory Board . . . 6
1.2.3 Methodology . . . 9
1.3 The present work in the frame of PARSIFAL project . . . 12
1.4 WAGNER structure . . . 13
2 Geometry, Assembly and Meshing 15 2.1 Properties of materials . . . 15 2.2 Cross-section . . . 16 2.3 Skin . . . 18 2.4 Frames . . . 20 2.5 Stringers . . . 21 2.6 Floor beams . . . 22 2.7 Bulkheads . . . 22 2.8 Struts . . . 23 2.9 Sponson . . . 23 2.10 Link beam . . . 25 2.11 Meshing . . . 25 2.12 Assembly . . . 26
3 Loads and Constraints 30 3.1 Classification of loads . . . 30 3.2 Pressurization . . . 31 3.3 Equations of motion . . . 32 3.4 Payload . . . 35 3.5 Inertia calculation . . . 35 3.5.1 Engines . . . 36 i
3.5.2 Passengers and containers . . . 36
3.5.3 Wings . . . 38
3.6 Constraints and Inertia relief . . . 42
3.7 Frequency analysis . . . 43
3.8 Output file and MTOW estimation . . . 43
II
Results
45
4 Preliminary design and results 46 4.1 Structural preliminary sizing . . . 474.1.1 Skin . . . 47 4.1.2 Floor beams . . . 48 4.1.3 Struts . . . 48 4.1.4 Link beam . . . 50 4.1.5 Stringers . . . 50 4.1.6 Frames . . . 51
4.1.7 Bending moment and shear diagrams . . . 51
4.2 Preliminary weight results . . . 52
4.2.1 Weight fraction estimation accuracy . . . 52
4.2.2 Weight composition . . . 54
5 Low weight configurations 56 6 Cruise condition analyses 58 6.1 2-4-2 layout . . . 58
6.2 3-3 layout . . . 63
7 Ultimate pressurization analyses 71 7.1 2-4-2 layout . . . 72
7.2 3-3 layout . . . 76
8 Combined ultimate loads analyses 80 8.1 2-4-2 layout . . . 80
8.2 3-3 layout . . . 84
9 Conclusion 91 A List of WAGNER input 93 A.1 Main file structure . . . 93
A.2 inputData .py . . . 95
A.3 secondaryVariables.py . . . 103
C Input and Output data files 110 C.1 First attempt (2-4-2) . . . 110 C.1.1 Input data . . . 110 C.1.2 Output file . . . 114 C.2 First attempt (3-3) . . . 116 C.2.1 Input data . . . 116 C.2.2 Output file . . . 120 C.3 Low weight (2-4-2) . . . 122 C.3.1 Input data . . . 122 C.3.2 Output file . . . 126 C.4 Low weight (3-3) . . . 128 C.4.1 Input data . . . 128 C.4.2 Output file . . . 132
List of Figures
1.1 Future aircraft concepts . . . 3
1.2 IDINTOS [Tuscan innovative seaplane]: an example of 2-seats PrP . . 5
1.3 Artistic view of a passenger PrP . . . 11
1.4 WAGNER schematic structure . . . 13
2.1 Bilinear σ- curve for two aluminium alloys . . . 16
2.2 Cross section geometry and constraints . . . 17
2.3 The four basic cross-sections . . . 18
2.4 Skin lofting . . . 19
2.5 Frame, stringer and floor beam geometry . . . 20
2.6 Particular of frames orientation . . . 21
2.7 Particular of stringers orientation . . . 22
2.8 Passengers and cargo floor struts. In this case, the interruption has been performed only in the former ones. . . 23
2.9 Sponson detail for Lockheed C-130 Hercules . . . 24
2.10 Sponson cross section . . . 24
2.11 Assembly Tie connections (particulars) . . . 28
2.12 Frame-stringer connection by means of stringer-tie . . . 29
3.1 Scheme for the evaluation of aerodynamic forces . . . 34
3.2 Transformation of inertia tensor . . . 36
3.3 Engine and container simplified form . . . 37
3.4 Sitting human body and coordinates system . . . 37
3.5 NACA SC 20410 airfoil . . . 38
3.6 Wing part approximation . . . 39
4.1 Floor beams and strut reference model . . . 49
4.2 Maximum displacements for non optimal struts position . . . 50
4.3 Shear and bending moment diagrams . . . 52
4.4 Empty weight fraction trends from Raymer ’s . . . 53
4.5 Empty weight fraction trends from Jane ’s . . . 54
4.6 Weight fractions . . . 54 iv
4.7 Fuselage weight composition . . . 55
6.1 2-4-2 layout global deformation, cruise condition . . . 59
6.2 Hoop stress in absence of stiffeners . . . 60
6.3 Cross section deformed configuration, 2-4-2 layout, cruise condition . 61 6.4 Hoop and longitudinal skin stress, 2-4-2 layout, cruise condition . . . 62
6.5 Frames and stringers von Mises stress (particular), 2-4-2 layout, cruise condition . . . 63
6.6 Link beam-frames tie stress raise (particular), 2-4-2 layout, cruise condition . . . 64
6.7 Link beams loads, 2-4-2 layout, cruise condition . . . 64
6.8 Sponson von Mises stress field, 2-4-2 layout, cruise condition . . . 65
6.9 3-3 layout deformation, cruise condition . . . 66
6.10 3-3 cross section behaviour, 3-3 layout, cruise condition . . . 67
6.11 Reference model of arch (reduced system to the left part) . . . 68
6.12 Parametric study of β . . . 68
6.13 Cross section deformation in absence of lower deck and struts . . . . 69
6.14 Cross section deformation in absence of lower and central decks and struts . . . 69
6.15 Hoop and longitudinal stresses, 3-3 layout, cruise condition . . . 70
7.1 Skin regions below 100 MPa (in black), 2-4-2 layout, ultimate pressur-ization condition . . . 73
7.2 Stiffeners regions below 450 MPa (in black), 2-4-2 layout, ultimate pressurization condition . . . 73
7.3 Cross section deformed configuration (linear on the left), 2-4-2 layout, ultimate pressurization condition . . . 74
7.4 Von Mises stress for a central fuselage barrel (linear on the left), 2-4-2 layout, ultimate pressurization condition . . . 74
7.5 Hoop stress on skin (linear on the left), 2-4-2 layout, ultimate pressur-ization condition . . . 75
7.6 Longitudinal stress on skin (linear on the left), 2-4-2 layout, ultimate pressurization condition . . . 75
7.7 Von Mises stress for stiffeners (linear on the left), 2-4-2 layout, ultimate pressurization condition . . . 76
7.8 Cross section deformed configuration (linear on the left, 3-3 layout, ultimate pressurization condition) . . . 77
7.9 Von Mises stress for a central fuselage barrel (linear on the left), 3-3 layout, ultimate pressurization condition) . . . 77
7.10 Hoop stress on skin (linear on the left), 3-3 layout, ultimate pressur-ization condition) . . . 78
7.11 Longitudinal stress on skin (linear on the left), 3-3 layout, ultimate pressurization condition) . . . 78
7.12 Von Mises stress for stiffeners (linear on the left), 3-3 layout, ultimate pressurization condition) . . . 79 8.1 Skin elastic regions (in black), 2-4-2 layout, ultimate load condition . 81 8.2 Stiffeners elastic regions (in black), 2-4-2 layout, ultimate load condition 81 8.3 Global results, 2-4-2 layout, ultimate load condition . . . 82 8.4 Cross section displacement, 2-4-2 layout, ultimate load condition . . . 83 8.5 Frames-linkBeams connection (particular), 2-4-2 layout, ultimate load
condition . . . 84 8.6 Hoop and longitudinal stresses, 2-4-2 layout, ultimate load condition . 85 8.7 Struts stress below eulerian critical buckling tension (in black), 2-4-2
layout, ultimate load condition . . . 86 8.8 Cross Section and frames stress, 3-3 layout, ultimate load condition . 87 8.9 Global results, 3-3 layout, ultimate load condition . . . 88 8.10 Hoop and longitudinal stress, 3-3 layout, ultimate load condition . . . 89 8.11 Frames shear, 3-3 layout, ultimate load condition . . . 90 8.12 Struts below eulerian buckling critical stress (in black), 3-3 layout,
List of Tables
1.1 PARSIFAL Work Packages . . . 8
1.2 Advisory Board members . . . 9
2.1 Material properties . . . 15
2.2 Meshing default properties . . . 26
4.1 Some fuselage layouts data . . . 46
4.2 First attempt configurations weight results . . . 53
5.1 Weight saving for 2-4-2 configuration . . . 56
5.2 Weight saving for 3-3 configuration . . . 57
5.3 Optimised layouts weight composition . . . 57
7.1 Geometrical non-linear analysis non-default parameters . . . 72
Part I
Code Outline
Chapter
1
Parsifal project
1.1
PrandtlPlane configuration: a brief outline
Several years have been passed since PrandtlPlane (PrP) configuration has been presented to the scientific world. Many years of study have followed, and the attention of the worldwide scientific community has been increasing all the time.
Aerodynamic features have been firstly investigated in University of Pisa, by providing the exact solution of the problem about induced drag formulated by Prandtl in 1924 ([PRANDTL 24]).
In the most recent years, a big amount of studies, both in Europe and USA, have been conducted, and a certain knowledge and a certain experience have been collected. Although results have been very promising form an academic point of view, it is not yet enough for an industrial finalization of the PrP product.
Several works may be consulted for a deeper understanding of the state of art. This is not the place to propose this scenario in an organic and developed way. The interested reader can consult, among others, [FREDIANI 09], [CAVALLARO 16], [BOTTONI 04], [RIZZO 10], [FREDIANI 17a].
Both aircraft industry and academic world have to face with new challenges which have been risen about increasing air traffic, environment pollution and safety. In effect, it is expected that air traffic will nearly double in the next two decades on a global scale, especially in medium and small airports, due to the increment of the point-to-point connections. It comes naturally that new aircraft configurations are bound for reducing noxious emissions and noise. In addiction to that, it is necessary to reduce air traffic congestion, in particular in those areas already overcrowded; this means to reduce air traffic controllers workload, as well. In a nutshell, the new requirements on future air transport, both in the USA and in UE, can be summarized as follows:
• to satisfy the increase of air traffic with more safety and more comfort of flight; 2
CHAPTER 1. PARSIFAL PROJECT 3 • to reduce emissions and noise per unit of transport;
• to reduce time for ground operations.
The previous issues will never be fulfilled by conventional airplanes, because they have reached their maximum potential and dimensions, and further significant improvements of their efficiency are no longer possible.
New aircraft configurations have been proposed in order to satisfy those require-ments: the most likely candidate configurations for future aviation are the Blended Wing Body (BWB) (Fig. 1.1a), the Truss Braced Wings (TBW) (Fig. 1.1b) and the PrP (Fig. 1.1c) concepts.
(a) BWB (b) TBW (c) PrP
Figure 1.1: Future aircraft concepts
The BWB, in addition to some possible benefits, presents some important draw-backs. In particular, a BWB solution is possible only if a large span-length is adopted. Generally speaking, the aerodynamic efficiency of a BWB is not the best possible for a given span and total lift; the challenges of more safety and comfort in flight are critical aspects as, for example, safety during emergency evacuation, flight comfort during roll manoeuvres, lateral control and flight stability. The time for ground operations is a further critical aspect and airports infrastructures are not conceived for this aircraft configuration.
The Truss Braced Wings concept aims at reducing the induced drag by improv-ing the overall span of a conventional monoplane: the consequent structural and aero-elastic disadvantages are resolved by connecting the wings to the fuselage by means of two struts. Even if TBW configuration is able to reduce the induced drag, aerodynamic interference leads to a reduction of maximum efficiency speed. From structural design point of view, aeroelastic effects may produce a weight penalty. Fi-nally, the TBW configuration is incompatible with ICAO Aerodrome Reference Code C standard [ICAO 09], hence it cannot be considered an alternative to conventional aircraft of the class of Airbus A320 or Boeing 737.
The PrP configuration presents the minimum induced drag among all the solutions available for a given span and total lift, as proved in [FREDIANI 09]. Among the most relevant advantages, it is worth remembering that:
• compared to traditional aircraft, a PrP even with a lower wingspan of a conventional one, can provide the same aerodynamic efficiency and improve the payload capability;
CHAPTER 1. PARSIFAL PROJECT 4 • due to the higher aerodynamic efficiency, emissions can be cut during cruise and, mainly, during the take-off and landing phases, where the induced drag is maximum in percentage of the total drag; noise is minimized correspondingly; • a PrP has positive lift on both wings and is stable in any flight condition; • the safety of flight is enhanced due to the following factors:
– smooth stall and easy recovery from stall
– pitch control is actuated with a pure couple without modifying the total lift during manoeuvres;
– the emergency evacuation procedures are the same (well proven) of the conventional aircraft;
• thanks to the single and continuous cargo deck, the time for loading and unloading cargo can be reduced, also by multiple doors on both front
• many results confirm that a typical feature of a PrP is the comfort during flight, due mainly to the very high pitch damping.
• The box wing, over-constrained to the fuselage, is a natural multi-path-load damage tolerant structure.
• The PrP configuration allows adopting different propulsion system solutions (including, for example: liquid hydrogen or methane as fuels, electric propellers
distributed along both the wing span).
• freight capability is higher than that of today airplanes owing to an innovative cargo bay design;
• The PrP configuration can be adopted for aircraft of any type and category, from 2 seats, as previously investigated in [FREDIANI 15a], (Fig. 1.2), to ultra large airliners, from low to very high transonic speeds, from passenger to freighter aircraft [FREDIANI 15b].
CHAPTER 1. PARSIFAL PROJECT 5
CHAPTER 1. PARSIFAL PROJECT 6
1.2
Parsifal project
1.2.1
Objectives
The main objective of PARSIFAL (Prandtlplane ARchitecture for the Sustainable Improvement of Future AirpLanes) is to improve the civil air transport of the future by introducing PrP configurations into service.
The project is focused on medium size commercial aircraft category. The adoption of PrP configuration can confer to aircraft with dimensions and fuel consumption of A320/B737 the payload capacity of higher category aircraft (i.e. A330/B767).
In addiction, PARSIFAL aims at developing design tools in order to investigate the application of PrP configuration to other aircraft categories (i.e. freighter, ultra-large airliners).
Unfortunately, databases, design tools and practical experience, gained in the case of conventional monoplanes since the middle of the last century, is not available. PARSIFAL aims at filling this gap and will face the implementation of the necessary scientific, technological and engineering background to design such an innovative aircraft.
Specific goals of PARSIFAL are listed here below:
• Definition of a PrP configuration to be compared with conventional aircraft; • Assessment of the baseline PrP performances;
• Development and tuning of design tools; • Scaling procedures for the PrP configuration.
1.2.2
Partners and Advisory Board
PartnersPARSIFAL includes cooperation between Universities, Research Centers and SMEs from four European countries: Italy, France, Germany and Netherlands.
The Consortium is made of 6 partners having well-complementary expertise and roles. They are able to cover all the aspects of the research project such as require-ments definition, scientific breakthroughs, design and technology, communication, dissemination and exploitation. The Consortium is made of 3 Universities, Pisa (IT) (as coordinator), ENSAM (Bordeaux, FR) and TUD (Delft, NL), 2 research institutes, ONERA (Meudon, FR) and DLR (Hamburg, DE), and an SME, Skybox Engineering (Pisa, IT), born as a Spin-Off company of Pisa University with the mission of promoting the technology transfer of PrandtlPlane research.
In particular:
UNIPI holds more than 20 years of experience on research concerning the PrP configuration and therefore it has the capacity of coordinating the scientific
CHAPTER 1. PARSIFAL PROJECT 7 part of the project as well as to maximize the impact of dissemination and exploitation activities. In addition to the overall expertise on aircraft design applied to PrP configurations, UNIPI will bring into PARSIFAL its wide experience on structural analyses, including the aero-elastic ones.
TUD will bring into PARSIFAL the research experience acquired on flight mechan-ics and propulsion systems of PrPs, adopting the collaborative design and knowledge-based engineering tools which have characterized a significant part of the recent TUD research activities.
ONERA will analyse the aerodynamic and aero-acoustic performance of the PrP configurations based on CFD RANS computations, will design specific elements of the PrP configurations to highlight the potential of the concept, and will provide expertise and recommendations to the Overall Aircraft Design process in order to improve the accuracy of Lo-Fi modules.
DLR will contribute on several aspects of PARSIFAL, putting on the table both the capability of providing market analysis related to the Air Transport System and to provide technological solutions as answers to market demand. For this reason DLR will put a significant part of its effort on fuselage studies and payload capability enhancement.
ENSAM will be in charge of the design/optimization of the structural parts consti-tuting the lifting system and the fuselage of the baseline PrP. The ENSAM team will provide its expertise in the fields of optimization/design of composites materials and structures which will be properly carried out thanks also to the ENSAM facilities (both experimental and numerical).
SKYBOX will bring into PARSIFAL its knowledge about PrP configuration design and optimization, acquired in past project on ultra-light PrP aircraft in which several experimental activities, such as wind tunnel tests and flight tests on scaled models, were performed in order to both validate the design tools and assess the PrP behaviour.
PARSIFAL is developed through technical and scientific Work Packages (WPs) plus a specific WP for Coordination of the activities, Dissemination, Communication and Exploitation of project results, Tab. 1.1. Each WP is subdivided into specific tasks, as reported in [PARSIFAL 17]
Advisory Board
From the organizational point of view, the Project Team will be supported by an external group of experts from the industry, airports and research, who will be part of
CHAPTER 1. PARSIFAL PROJECT 8
WP n° WP title Participant n° Leader Participant
1 Analysis of the Air Transport
Sys-tem of next future 4 DLR
2 Definition of PARSIFAL
require-ments 1 UNIPI
3 Analysis of PrP configuration 6 SKYBOX
4 Aerodynamic analysis of the
"ref-erence PrP" 3 ONERA
5 Structural analysis of the
"refer-ence PrP" 5 ENSAM
6 Flight mechanics and aircraft
con-trol 2 TUD
7 Analysis and design of the
propul-sion system 2 TUD
8 Project coordination and con-trol, dissemination, communica-tion and exploitacommunica-tion
1 UNIPI
Table 1.1: PARSIFAL Work Packages
an Advisory Board(AB), with the specific target of addressing the Research towards its maximum industrial interest and social effects. In particular, the AB will have an active role for the determination of the specific requirements that the PrP shall have in order to maximize the chances of definitely penetrating the market in the medium term period.
Further experts will confer to the PARSIFAL team the experience and prestige of academia. The role of the AB will be the definition of requirements from the different viewpoints as well as the indications of Regulations for the design and the certification of aircraft; the AB will also follow all the activities by ensuring a continuous design review, along the entire project duration. The contribution of the AB to the success of the project is fundamental for the following main reasons:
• All the studies on the PrP configuration have been conducted exclusively by scientific academic institutions or research centers so far.
• These academic institutions lack (usually) of the economical and organizational expertise necessary for planning the introduction of a new air transport into service.
• The design requirements of the aircraft to enter into service in the next two or three decades are needed to arrive from Industry and air transport organizations because academic institutions have a partial, even essential, view.
CHAPTER 1. PARSIFAL PROJECT 9 The PARSIFAL team, well aware of this gap, has proposed to eminent personalities of aeronautical industries and air services in Europe to join to the project in order to give solid and affordable guidelines, and to conduct a design review.
The AB composition is reported in Tab. 1.2.
Member Organization Role
Dieter Schmitt ARTS-DS (Germany) Independent Consultant Bernd Sträter Sträter Consulting
(Ger-many) Independent Consultant
Gina Giani Toscana Aeroporti S.p.A.,
(Italy) Chief Executive Officer &General Manager Cosimo Giulio De Metrio Società per Azioni
Eser-cizi Aeroportuali S.p.A. (Italy)
Chief Operating Officer & Deputy Chief Executive Officer
Thierry Druot Airbus SAS (France) Expert Engineer Overall Aircraft Design, Modeling & Methodologies
Daniel Reckzeh Airbus (Germany) Technology Integration
Platform Leader: Over-all Aircraft Design & Integration; Research & Technology Chief Engineering
Florent Laporte Airbus SAS (France) Head of R&T Overall Air-craft Design
Francesco Salvato Leonardo - Aircraft and
Aero-structures (Italy) Research Manager
Fabio De Donno Alitalia (Italy) Captain Boeing 777
Aurelio Pastorino Leonardo (Italy)
Table 1.2: Advisory Board members
1.2.3
Methodology
The Project will be subdivided into basic activities which will nearly correspond to dedicated WP, as follows.
Socio Economical Analysis: the analysis of the socio economic impact of the introduction of the PrP aircraft into the market has started in the first part of the research, and has been continuously updated according to project progress. This analysis has been carried out in collaboration with all the participants and with the contribution of the AB.
Requirements and Regulations: this activity concerns the definition of specific requirements for the PrP, under the constraints of the existing Regulations.
CHAPTER 1. PARSIFAL PROJECT 10 Such analysis aims to define the Top Level Aircraft Requirements, starting from the mission analysis and the application of the present regulations to PrP configurations.
Conceptual design and preliminary sizing: this activity concerns the develop-ment of the conceptual design and the preliminary sizing of the PrP configu-ration. The number of seats will vary in the ranges of A330 or B767 aircraft and the wing span is fixed (36m). The preliminary sizing will be devoted to the definition of the configuration for the problem at hand. Numerical tools have been already set up to generate initial aircraft configurations and to estimate their performances, starting from the aircraft requirements. These initial designs will provide the starting point to the application of the more sophisticated sizing and analysis tools.
Optimization of Aircraft Configuration: the studies on the aerodynamic opti-mization of the aircraft is one of the main items of the Project. This optimiza-tion process is will be conducted by means of a code which was previously successfully tested in the design of the IDINTOS aircraft. The optimization of the PrP shape is a constrained optimization problem: the goal is the definition of the aerodynamic parameters minimizing the fuel consumption in the presence of the constraints on the static stability of flight with a given interval of the margins. A significant contribution of Mathematics (especially in the field of calculus of variation and the so called Operative Research) and Informatics is needed in this phase.
Aerodynamic Analysis: the aerodynamics of the optimal configurations provided by the previous phase will be analysed by means of CFD tools, to determine aerodynamic forces and moments as well as all the aerodynamic derivatives (without and with high lift devices and control surfaces deflection) at both cruise speed with transonic effects, and low speed regimes. Moreover, this analysis aims at eliminating possible local shock waves (some effects are already known) by local shape modifications.
Flight Mechanics and Controls: in a PrP all the horizontal wings produce a positive lift in any flight condition, and the Mechanics of Flight becomes totally different when compared to a conventional aircraft. The stability of flight within given intervals is guaranteed by the fulfilment of the optimization constraints. Experience teaches that the margin of stability can be reduced when compared to a conventional aircraft, because both the pitch damping and the inertia of a PrP are higher.
Fuselage structures: fuselage design aims to investigate which cross-section best meets the requirement of improving transport capacity without weight penalties. Oval cross-sections will be object of study, focusing on both horizontally enlarged and vertically enlarged fuselages, in order to:
CHAPTER 1. PARSIFAL PROJECT 11
Figure 1.3: Artistic view of a passenger PrP
• enhance passenger plus luggage and/or cargo capacity;
• connect the rear wing to the fuselage avoiding empty weight increase and aerodynamic problems such as flutter;
• optimize the main undercarriage design;
• optimize the wetted fuselage surface per passenger; • reduce the aircraft length.
In PARSIFAL project, the structural solutions of the fuselage are conducted in the presence of a few loading conditions and the contributions of plants, main undercarriage, lateral sponsons, hydraulic plants, etc. to the total weight, will be obtained from statistical data.
Lifting system structures: an initial beam model of the wings is analysed; the beam section are made by a front and a rear spar and by lower and upper skins between the spars. The structural design and optimization of the lifting system is based on an optimization procedure already tested in previous analyses; the optimization relies on a sequence of steps:
• design towards an uniform maximum stress level (typically 0.75 of the yielding stress) at a vertical load factor equal to n1 (= 2.5);
• design against the compression loads;
• design to take into account static aero-elasticity requirements and, in particular, aileron efficiency
• design against flutter.
A further interesting aspect of the structural design, to be analysed more deeply in PARSIFAL, is the utilization of composites solutions. As a matter of fact, the thickness of a single PrP wing is nearly the half of a monoplane and the resulting skin thickness is much higher; thus, in the case of composites, the skin thickness is sufficiently high to possibly avoid the presence of glued stiffeners. These aspects will be investigated in PARSIFAL. The loading conditions will be limited to the Limit Loads of 2.5 under linear elasticity. Both metallic and composite solutions will be analysed in PARSIFAL.
CHAPTER 1. PARSIFAL PROJECT 12 Propulsion system: engines are the major source of environmental impact, i.e.
emissions and noise. This topic will be deeply investigated in order to maximize the potential offered by the particular configuration of the aircraft. It is expected that the PrP will have both pollution and noise considerably reduced with respect to conventional aircraft, and this will happen in particular during take-off and landing, with consequent major benefits for the area surrounding the airport. The possibility of reducing the engine power installed on a PrP, compared with an equivalent conventional aircraft, will be studied deeply in PARSIFAL. Different positions of the turbofan engines will be considered for the PrP configuration, as well as the adoption of Very Large bypass ratio turbofans.
Aero-elasticity: the aero-elastic studies will be focused on static phenomena and flutter. Analyses will be carried out to fulfil flutter speed requirements and to determine the weight penalty due to flutter.
Aircraft performances evaluation: starting from the conceptual phase, the air-craft performance of PrP will be evaluated and compared both with existing aircraft (based on data from literature and manufactures) and conceptual de-signs of conventional aircraft with same top level requirements, generated using the same conceptual design tools and approach used for the PrP. In the final part of the project, this activity will get more connected to the socio-economic analysis and will include all the aspects related to the air transport of the future.
1.3
The present work in the frame of PARSIFAL
project
This work takes place in the activities concerning the aforementioned fuselage structure definition and analyses .
In the preliminary phases of the project, it is necessary to evaluate a large amount of structural configurations (in this case, attention has been focused on fuselages) in order to compare benefits and drawbacks of different layouts, verify effects of eventual modifications in the global model or in geometrical as well as material properties.
It has been requested a code aiming at fulfilling those aspects in a reasonable way, i.e. in order to explore different configurations behaviour at a global scale. Neverthe-less, the need for realistic and real physical phenomena adequately representative outputs has been defined of primary importance. Last, but not the least, the time to generate the FEM model, to elaborate and to give back a solution has been requested to be as short as possible.
These are the principal aspects on which the code, called WAGNER, is based and has been developed. WAGNER is the natural answer to the need of a parametric
CHAPTER 1. PARSIFAL PROJECT 13 and automatic code for preliminary global PrP fuselages analysis and sizing.
WAGNER is drawn on two previous Master’s Degree thesis developed at the University of Pisa, (i.e. [PACCHIARINI 17] and [FELICE 17]), and represents the generalization as well as the improvement of those works: its final goal, in fact, is to provide a preliminary sizing to be used as a baseline for future and deeper studies.
WAGNER can perform both linear and non-linear analyses, both static and quasi-static analyses and eigenfrequencies extraction.
WAGNER is written in Python (release 2.7) language and is thought to be run in Abaqus FEA software, in particular in the Abaqus/Standard environment. Aerospace section of University of Pisa has licences facilities of Abaqus FEA, release 6.14, installed on the workstations in the Aerospace Department Computing Center.
The very following chapters of this work will presents the structure of WAGNER, the parameters the user can set and the facilities both for the user and the analysis.
1.4
WAGNER structure
WAGNER has been structured in a main file, that calls for the others generating the FEM model, the mesh and the required solution. A conceptual scheme is depicted in Fig. 1.4, while the main file structure in reported in AppendixA.
CHAPTER 1. PARSIFAL PROJECT 14 In particular:
main .py is WAGNER main file, which has to be run by Abaqus CAE or Abaqus Command. It sets the Working directory, where results will be saved. For each run, a new folder is created: its name is in the format yyyy − mm − dd_hh − mm − ss, to avoid any confusion. Furthermore, in each folder a inputData .txt file is saved, containing the input data set for that particular analysis. At the end of the run process, a output.txt file will be also saved therein.
The main file contains also the preamble, calling for Abaqus internal methods and WAGNER’ s own functions. In addiction to that, at the beginning of the main file, the user must insert the WAGNER folder path.
inputData .py contains all the variables and parameters settable by the user. From this file, the user can set also some flags allowing or retaining some specific modelling and analysis features. A complete list is reported in Appendix A. secondaryVariables.py contains variables somehow defined by the author. The
user can re-define them, but it is not suggested for normal purposes. A complete list is reported in Appendix A.
In addition to this, a folder named classes contains all the functions called during WAGNER execution. In fact, it has been chosen to introduce some functions to improve the maintainability, the modularity and the readability of the code.
Chapter
2
Geometry, Assembly and Meshing
In this chapter, the most important features about model geometric realization, parts assembly and meshing module are presented.
2.1
Properties of materials
By default, WAGNER implements four different materials: • aluminium alloy of AA 2XXX family;
• aluminium alloy of AA 7XXX family;
• an equivalent isotropic sandwich panel material;
• a composite material (for wires, for instance a generic Carbon/Epoxy one). For each of them, elastic properties of density, Young ’s modulus and Poisson ’s ratio are defined. For metallic materials, also plastic properties, according to the so-called bilinear stress-strain model (Fig. 2.1) are defined .
The user can choose whether elastic or elasto-plastic behaviour will be considered during the analysis he/she is up to run.
Material default properties are shown in Tab. 2.1, where ρ is the density, E the
Material ρ E ν σy y σu u
Mg m−3 MPa - MPa - MPa
-AA 2XXX 2780 × 10−12 72400 0.33 300 4.10 × 10−3 400 0.2 AA 7XXX 2830 × 10−12 71000 0.33 450 6.27 × 10−3 520 0.11
Honeycomb 1700 × 10−12 70000 0.3 - - -
-Composite 1600 × 10−12 130000 0.37 600 - -
-Table 2.1: Material properties
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 16 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 50 100 150 200 250 300 350 400 450 500 ǫ σ [M P a ] AA2XXX AA7XXX
Figure 2.1: Bilinear σ- curve for two aluminium alloys
Young ’s modulus, ν the Poisson’s ratio, σy and y the yield stress and strain, σu and
u the ultimate stress and strain. Data references are [FANTERIA 14] and [NIU 92].
Note that composite materials do not take into account for plasticity: that is to simulate their typical fragile attitude.
Furthermore, wire-simulating composite material reacts only to tension, in order to simulate a more realistic cables behaviour.
2.2
Cross-section
PrP cross-section is realized by means of circular tangent arcs, so that the contour results in a C1 piecewise-continuous function, with continuous first derivative, as
well. WAGNER implements a 12-arc-formed cross-section that, taking into account the longitudinal symmetry plane, it results in 6 independent arcs.
Without loss of generality, with reference to Fig. 2.2a, the following relations have been found for given radii r0, r1, r2, r3, r4, arc lengths a0, a1, a2, a3, a4 and centre
C0:
A = (0, C0y+ r0)
B = (r0sin(a0), r0cos(a0) + C0y)
C1 = (Bx− r1sin(a0), By− r1cos(a0))
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 17 C2 = (Cx− r2cos(π/2 − a0 − a1), Cy− r2sin(π/2 − a0− a1)) D = (C2x+ r2cos(a2− (π/2 − a0− a1)), C2y− r2sin(a2− (π/2 − a0− a1))) C3 = (Dx− r3cos(a2− (π/2 − a0− a1)), Dy + r3sin(a2− (π/2 − a0− a1))) E = (C3x+ r3 cos(a2− (π/2 − a0− a1) + a3), C3y− r3sin(a2− (π/2 − a0− a1) + a3) C4 = (Ex− r4cos(a2− (π/2 − a0− a1) + a3), Ey+ r4sin(a2− (π/2 − a0− a1) + a3)) F = (C4x+ r4sin(π − a0− a1− a2− a3− a4), C4y− r4cos(π − a0− a1− a2− a3− a4)) a5 = π − a0− a1− a2− a3− a4 C5 = (0, C4y− C4y− Fy C4x− Fx C4x) r5 = q (C5x− Fx)2+ (C5y− Fy)2 G = (0, C5y− r5)
where points A, B, C, D, E, F , G delimit the six arcs, Ci and ri, i = 0, 1, . . . , 5, are
the centre and the radius of the ith arc, respectively. Subscripts x and y denote the
horizontal and vertical component of points coordinates in the xy plane.
(a) PrP general cross-section (one half)
(b) Abreast seat configuration for A340-500/-600
Figure 2.2: Cross section geometry and constraints
The required arc lengths and angles can be imported in WAGNER from a CAD software, in which the user has previously defined the fuselage layout with all the required constraints about passengers comfort and space (in Fig. 2.2b they are shown for a conventional airliner), that can hardly be implemented in an organic as well as parametric way.
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 18
2.3
Skin
Skin generation is based on extrusion and multi-section lofting of some reference cross-section profiles. In particular, WAGNER requires the definition of four cross-cross-sections (Fig. 2.3):
Figure 2.3: The four basic cross-sections
• the so-called nose-section; • two main cross-sections; • the so-called tail-section
which the user can completely set from a geometrical point of view. Central cross-sections are partitioned along the whole curve path with a pitch equal to the stringers pitch, defined by the user. WAGNER has the authority to modify this value if arch length
stringer pitch is not an integer number.
By using the Shell-loft and Shell-extrusion methods, WAGNER generates the skin surface (Fig. 2.4).
As concern thickness attribution, the skin has been divided into six regions, numbered in red circles in Fig. 2.4 (from here onwards, they will be referenced as zone or region 1, 2 etc.). For each of them, independently, the user can define the thickness value.
To make it clearer, the skin is composed by shell elements. It is not a strong assumption since, as stated in Chapter 1, we are not interested in transversal stresses and strain. Therefore, the choice of 2D shell elements appears as justified.
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 19
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 20
2.4
Frames
Frames are generated having a Z-shaped profile (Fig. 2.5a).
(a) Frame Z-shaped profile (b) Stringer W-shaped profile
(c) Floor beam profile
Figure 2.5: Frame, stringer and floor beam geometry
Web height is equal to a scale factor the user can define for each of the six aforementioned regions. As for the flanges length, it is equal to the web height multiplied by the f lange length
web height ratio, which can be defined for each of the six regions,
as well. The default value is 0.3, as common in aircraft design and minimum weight structures theory [FARRAR 49].
Once the profile has been sketched, Sweep Abaqus internal method provides a Z-shaped frame following the skin profile contour.
WAGNER tries to maintain the web direction parallel to the skin local normal, to simulate a reliable physical situation. In region 1, 2, 6, owing to the variant curvature of the skin in both circumferential and longitudinal directions, frames orientation follows an averaged reference normal computed by WAGNER (Fig. 2.6).
Frames pitch is given as an input by the user. WAGNER maintains it constant if the ratio region length
f rame pitch gives an integer number. If not, WAGNER modifies the pitch
for that sector.
As for thickness, it can be defined for each of the six regions. Thickness is constant for web and flanges and does not vary circumferentially.
WAGNER takes into account the fact that there exist some frames (nominally, bulk frames) stiffer than others, used to introduce loads from wings, empennage or
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 21
Figure 2.6: Particular of frames orientation
landing gears in the fuselage structure. For those bulk frames, the scale factor is increasable by a factor chosen by the user and thickness can be assigned independently.
By default, frames material is an aluminium alloy of the AA 7XXX family. It is well worth comparing shell-element frames with beam-elements ones, as previewed in the primitive phases of WAGNER development. It has been shown that shell elements model in a more realistic way the interaction between plate and stiffeners under transversal pressure load. Switching to shell elements has drastically reduced computational times both for model generation and solution. Furthermore, stress and strain fields are definitely more accurate than the ones provided from beam elements; stiffeners final deformed configuration can be visualized, as well.
2.5
Stringers
Stringers are designed having a Hat-shaped profile (Fig. 2.5b).
Stringer height is equal to a scale factor the user can define for each of the six arcs composing fuselage cross-sections. As for flanges lengths, they are equal to the web height multiplied by a f lange length
web height ratio, different from primary and secondary
flanges, but constant along fuselage circumferential path. At points B, C, D, E, F W-shaped profiles are substituted by symmetric closed box-shaped ones, in order to preserve local curvature continuity. Consequently, membrane stresses integrity is not jeopardized by less efficient bending stresses that would have existed if abrupt changes had occurred in curvature sign.
WAGNER maintains the stringers direction parallel to the skin local normal to simulate a reliable physical situation (Fig. 2.7).
As for thickness, it can be defined for each of the six fuselage regions. It is constant for both web and flanges.
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 22
Figure 2.7: Particular of stringers orientation
2.6
Floor beams
Floor beams are generated as I-profiles. Analogously to frames and stringers, web height is equal to a scale factor, constant for all beams. Flanges length is obtained multiplying the scale factor by the f lange length
web height ratio (Fig. 2.5c).
The same is done for the cargo floor and, if exists, for the central floor, which have similar parameters to be set.
The user can specify the relative position between fuselage and the decks beams. Once the profiles have been sketched, extrusion is performed, until the proper width (previously computed) is reached. That allows a better connection between frames and floor beams.
Thickness is constant for both passengers and cargo floor beams and does not vary from webs to flanges. This extreme simplicity is due to the fact that floor beams are not primary elements for sizing. Therefore, a set of parameters larger than the one implemented in WAGNER has been judged as overwhelming.
By default, floor beams material is an aluminium alloy of the AA 7XXX family.
2.7
Bulkheads
Pressurization bulkheads are generated as planar shells. During WAGNER devel-opment, it has been tried to implement more realistic revolution-shells bulkheads, but this feature has seemed to be dramatically dependant on region 6 shape. In the most of cases, thou, Abaqus could not warrant the geometrical continuity of the part. Hence, in order to protect the robustness of the code, it has been decided to generate bulkheads as planar shells. Bulkheads shape follows exactly the skin one, at the longitudinal point their are situated. The user can define thickness independently for each bulkhead.
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 23
2.8
Struts
Floor struts are generated as circular cylinders. For simplicity,the generation modal-ities being completely similar, only passengers floor struts creation procedure in described.
The user can set diameter and thickness, which parameters will be kept constant during the procedure. For regions 3, 4, and 5, the user can also choose whether to interrupt struts generation in the wing box zone or not. In region 1, 2 and 6 struts disposition follows a straight line, as depicted in Fig. 2.8.
Figure 2.8: Passengers and cargo floor struts. In this case, the interruption has been performed only in the former ones.
This allows to generate struts all at once. In any case, extrusion depth is properly computed.
By default, struts material is an aluminium alloy of the AA 7XXX family.
2.9
Sponson
Sponson is a new feature introduced by PrP configuration in civil airliners. In effect, they are most used in military aircraft (as C130, C27) or, more generally speaking, in high wing aircraft (Fig. 2.9).
The sponson has not a primary structural function: it is a fairing for main landing gears, which cannot be hosted in the wing. Due to its mainly aerodynamic function, its exact shape should be given by means of an aerodynamic optimization process.
In WAGNER, drawing on aforementioned military aircraft, a simple profile has been implemented which, by using lofting Abaqus methods, is joined up to the fuselage main cross section. With reference to Fig. 2.10, the arc [RGQ is obtained as a spline, the arc \P CCQ is a circular one, the segment DP is a straight line. Points D and G belong to the ones defining the fuselage main cross-section.
The user can define its dimension both in longitudinal and transversal directions and can choose its relative position with reference to the fuselage. If the first and
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 24
Figure 2.9: Sponson detail for Lockheed C-130 Hercules
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 25 last profiles do not match with frames positions, they are shifted, in longitudinal direction, to match the closest frame site.
The geometric tangency problem has been analytically studied as a problem between the circular arc of center CC = (x0, y0)and radius r (both parameters can
be set) and two straight lines: the first one, representing the segment DP , of equation y − Py = m1(x − Px) and the second one, approximating the arc GQd, of equation y − Qy = m2(x − Qx). (Note that P and Q coordinates and angular coefficients m1
and m2 are unknown).
The following formula apply:
m1 = (Dy− y0)(x0− Dx) − rp(Dx− x0)2+ (Dy− y0)2− r2 (Dx+ r − x0)(r − Dx+ x0) (2.1) m2 = (Gy− y0)(x0− Gx) + rp(Gx− x0)2+ (Gy− y0)2− r2 (Gx+ r − x0)(r − Gx+ x0) (2.2) Px = x0+ m1(y0− Dy+ m1Dx) m2 1+ 1 (2.3) Py = x0+ m1(x0− Dx+ m1y0) m2 1+ 1 (2.4) Qx = x0+ m2(y0− Gy + m2Gx) m2 2+ 1 (2.5) Qy = x0+ m2(x0− Gx+ m2y0) m2 2+ 1 (2.6)
2.10
Link beam
Vertical link beams are the only 1D part present in the model and are generated as Truss elements, reacting only to tension. Default material is a composite one. The user can set the traction reagent area, material elastic properties and the pitch above fuselage frames. Their function is to reduce the vertical relative displacements of the fuselage, under pressurization load.
2.11
Meshing
After geometry generation, WAGNER performs the meshing of parts. As said before, all parts but link beams are formed by shell elements. For each link beam, only one element is present, owing to the fact that the traction normal stress is physically constant along the beam axis. For each of the remaining parts, WAGNER assigns a
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 26 seed size, an element shape and a meshing technique (all settable by the user). The default situation is presented in Tab. 2.2.
Part Element Seed size Technique Shape N° elements
skin S4R 100 Sweep Quad dominated 7 × 104
frames S4R 100 Sweep Quad 4 × 104
stringers S4R 100 Free Quad dominated 2 × 105
floorUp S4R 100 Structured Quad 2 × 104
floorDown S4R 100 Structured Quad 1 × 104
strutsUp S4R 100 Sweep Quad 5 × 104
strutsDown S4R 100 Sweep Quad 6 × 103
sponson S4R 50 Sweep Quad dominated 3 × 104
bulkheads S4R 100 Free Quad dominated 1 × 103
linkBeam T3D2 - - - 1 × 101
Table 2.2: Meshing default properties
Because of no particular reasons, standard elements have been chosen from Abaqus internal library.
By default, the size of the problem is about half million elements.
2.12
Assembly
WAGNER’ s following step is parts assembly. Parts need to communicate each other during solution computation in a realistic and physical-likely fashion. In Abaqus Interaction module, the Tie method has been chosen to this purpose. In effect, Tie command is able to connect regions having different mesh grids at the boundary interface. Attention must be paid to the fact that master nodes suppress degrees of freedom (DOF) of slave ones. Therefore, the latter cannot be used again either as master or slave for a new Tie connection definition. On the contrary, master nodes can have multiple sets of slave ones.
By keeping this in mind, WAGNER implements the following connections (the former is the master, the latter the slave):
Skin-Stringers: Stringers flanges which must be in contact with the skin, are picked up and gathered in a set (Fig. 2.11a);
Skin-Sponson: Sponson boundary edges are connected with skin faces (Fig. 2.11b); Frames webs-Link beams: Frames webs faces are tied with link beams boundary vertices. This fact will produce an unnatural as well as meaningless stress rise in the neighbourhood of the connection points, if a purely elastic material behaviour is applied. In post-processing phase, the user must take into account this phenomenon (Fig. 2.11c);
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 27 Frames webs-Bulkheads: Bulkheads vertices are tied to the matching frame webs
(Fig. 2.11d);
Frames webs-Floor beams: External edges of each floor beams (of both the decks) are tied to the matching frame web faces (Fig. 2.11e).
Floor beams-Struts: upper struts edges are tied to the matching lower floor beam flange face (Fig. 2.11f);
Frames webs-Struts: lower struts edges are tied to the matching frame web faces (Fig. 2.11g);
Frames flanges-Skin: The webs flanges faces which must be in contact with Skin are picked up and tied to the Skin faces (Fig. 2.11h).
It is worth presenting a brief explanation of the absence of Frames-Stringers tie. In actual aircraft, stringer ties connect frames webs to stringers webs, as depicted in Fig. 2.12, ([BOEING 90]). This connection implies that a minimum of two elements must be present both in frames and stringer webs. Consequently, it would result in a number of elements more than necessary for both parts or in a dramatically complicated meshing algorithm, attempting at containing the final element quantity. More importantly, by connecting the central nodes, the situation would be similar to the one previously described in the Frames webs-Link beams tie section. For each frame, the output would be invalidated by the presence of several spots, along the circumferential direction, in which neighbourhood stresses would be consistently risen and thus they would be non representative of the actual situation. For these reasons, WAGNER does not contemplate any simulation of stringer ties.
Moreover, in [FELICE 17], a comparison between the two cases has been explicitly performed, showing similar stress fields.
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 28
(a) Skin-Stringers Tie
(b) Skin-Sponson Tie
(c) Frames-LinkBeams Tie
(d) Frames-Bulkhead Tie
(e) Frames-FloorBeams Tie
(f ) FloorBeams-Struts Tie
(g) Frames-Struts Tie (h) Frames-Skin Tie
CHAPTER 2. GEOMETRY, ASSEMBLY AND MESHING 29
Chapter
3
Loads and Constraints
In the frame of aircraft structural project, attention must be paid to the fuselage design. In fact, fuselage is a quite complex structure and the simultaneous fulfilment of structural, aerodynamic and commercial requirements is mandatory.
In this chapter, the main hypotheses on loads modelling and their introduction in fuselage structure are discussed.
Remarkable results are presented in analytical closed form.
3.1
Classification of loads
A generic fuselage undergoes loads of different nature: aerodynamic, inertial and pressurization-induced. For a final design, several are the load combinations required for airworthiness certification.
In particular, regulations require [EASA 17]: • analyses with reference at:
– flight loads;
– flight loads and cabin pressurization at maximum design differential pressure;
– cabin pressurization at ultimate design pressure; – landing loads;
– ground loads. • fatigue analysis; • fail safe analysis; • peculiar cases such as:
– de-pressurization;
CHAPTER 3. LOADS AND CONSTRAINTS 31 – birds impact;
– concentrated loads applied to passengers deck; – accident loads.
Among the flight loads, one can define: • gust loads: – symmetric – non-symmetric • manoeuvre loads: – symmetric – non-symmetric.
Among pressurization loads, [EASA 17] specifies: • pressurization with landing loads;
• pressure differential loads corresponding to the maximum relief valve setting multiplied by a factor of 1.33, omitting other loads;
• flight loads combined with pressure differential loads from zero up to the maximum relief valve setting.
WAGNER being a preliminary design tool, it has been chosen not to implement all the various aforementioned scenarios. Attention has been focused primarily on load of inertial and aerodynamic nature and pressurization.
3.2
Pressurization
As reported in [LOMAX 96], design limit pressure (corresponding to the maximum relief valve setting) is
pdl = 9.1 psi,
or, in SI,:
pdl = 0.0627 MPa. (3.1)
Therefore, design ultimate pressure ( by definition 1.33 pdl) is
pul = 0.0834 MPa. (3.2)
Obviously, pressure is applied to the two bulkheads and to the skin portion between them.
CHAPTER 3. LOADS AND CONSTRAINTS 32
3.3
Equations of motion
It is necessary to report some important result of classic Flight Mechanics about Euler ’s dynamics equations.
Reference for this section is [McRUER 73]; used symbols are almost the same of the reference.
Under classical hypotheses: • aircraft as a perfect rigid body; • Earth assumed fixed in space; • time-invariant mass distribution;
• xbzb plane as a symmetry one for the aircraft;
• steady state conditions,
dynamics equations of forces and moments can be written, in the body reference system frame (CG, xb, yb, zb)as:
Xb = m(QW − RV ) − T (3.3) Yb = m(RU − P W ) (3.4) Zb = m(P V − QU ) (3.5) Lb = QR(Izz− Iyy) − P QIxz (3.6) Mb = P R(Ixx− Izz) + (P2− R2)Ixz − T beng (3.7) Nb = P Q(Iyy− Ixx) + QRIxz (3.8)
where m is the total aircraft mass, {U, V, W } the velocity vector, {P, Q, R} the angular velocity vector, T the installed thrust, Xb, Yb, Zb respectively the resultant
of aerodynamic and weight forces along xb, yb, zb directions, Lb, Mb, Nb respectively
the resultant of aerodynamic moments around xb, yb, zb directions, Iij the second
order moment of inertia with reference to ib and jb axes, beng the engines distance
from CG (along zb direction).
Note that weight moments are null, by definition, the centroid being the pole. Xb, Yb and Zb can be expressed as follows:
Xb Yb Zb = mg − sin Θ cos Θ sin Φ cos Θ cos Φ −
D cos α cos β − L sin α − S cos α sin β S cos β + D sin β
L cos α + D cos β sin α − S sin α sin β
(3.9)
CHAPTER 3. LOADS AND CONSTRAINTS 33 while P Q R = ˙ Φ − ˙Ψ sin Θ ˙
Θ cos Φ − ˙Ψ sin Φ cos Θ − ˙Θ sin Φ + ˙Ψ cos Φ cos Θ
(3.11) and U V W = V0 cos α cos β sin β sin α cos β (3.12) where V0 is the asymptotic aircraft speed module, α is the aerodynamic angle of
attack (AOA), β the side-slip angle, Φ the roll angle, Θ the pitch angle, Ψ the yaw angle, D the drag force, S the side force, L the lift force. Therefore, by providing as an input either
{V0, α, β, Φ, Θ, ˙Φ, ˙Θ, ˙Ψ} (3.13)
or
{V0, α, β, Φ, Θ, P, Q, R}, (3.14)
force and moment resultants are completely known.
Unfortunately, this knowledge in quite useless for our purpose: one should know how the components are shared between the two wings to sketch stress diagrams and to perform FEM stress analyses.
Indeed, for a given reduced system of generalised forces (one resultant force and one resultant moment), infinite equivalent systems having the same force and moment resultants exist. We are interested in finding out the one representing the actual flight condition.
The approach used in WAGNER, in order to try to close the problem, is the following.
Let CG = {CGx, CGy, CGz}be the aircraft centroid coordinates, P1 = {P1x, P1y, P1z}
and P2 = {P2x, P2y, P2z} the wings aerodynamic centres coordinates (for simplicity,
positioned at 1
4 of the root airfoil chord from the leading edge), with reference to the
body frame.
By imposing (3.13), pure aerodynamic generalized forces resultants are known. For equilibrium, it is necessary that (Fig. 3.1):
X1(1 + αX) Y1(1 + αY) Z1(1 + αZ) = Faerodynamic (3.15) χ1(1 + αχ) η1(1 + αη) ζ1(1 + αζ) + (P1− CG) ∧ X1 Y1 Z1 + (P2− CG) ∧ X1αX Y1αY Z1αZ = Maerodynamic (3.16) .
CHAPTER 3. LOADS AND CONSTRAINTS 34
Figure 3.1: Scheme for the evaluation of aerodynamic forces
By imposing the various αi and by solving the system, it is possible to know wing
loads and then to apply, to each wing, its correspondent forces. The αi coefficients
can be estimated: for example, from CFD simulations, it turns out that front wing lift is about 3
2 of the rear wing one ([FREDIANI 17a]). Furthermore, drag force can
be shared proportionally to wings own area. For simply load cases, such as pitch equilibrium in the longitudinal plane, most of these parameters become zero.
In the longitudinal plane, by taking β = 0 and Φ = 0, one finds the following expression of the vertical load factor increment
∆ nz =
˙ Θ V0
g cos α, (3.17)
useful for simulating different points in the flight envelope at nz = 1 + ∆ nz. That
assumes a physical meaning by considering an angular velocity Q at the beginning of pull up manoeuvre.
For more complicated combinations, it is needed CFD help or estimations on statistical bases.
Anyway, WAGNER is able to manage any load combination as long as αi parameters
are estimated, even if in more complex cases uncertainties on these values may be non negligible.
CHAPTER 3. LOADS AND CONSTRAINTS 35
3.4
Payload
As stated in [AEA 87], passengers have been considered: • having a body mass of 75 kg
• carrying a luggage of 20 kg • sitting on 11.5 kg mass seat 1
while containers have been considered having a density of
ρcontainer = 176 kg m−3, (3.18)
as prescribed in [AEA 87]. Containers empty weight has been taken into account, as well.
WAGNER computes how many passengers and how many containers (geometric data must be given in input) fill the available space. After that, the total weight (separately for passengers and containers) is distributed as a pressure load to the
relative floor beam upper flanges.
Further pressures have been implemented in WAGNER to simulate weight force components along xb and yb directions, if they exist.
3.5
Inertia calculation
WAGNER implements some computations to evaluate, in an preliminary way, the inertia of some relevant elements in the aircraft. In this section main results are presented. Let first report some well-known aspects.
Let Jxx := Z C ρ(y2+ z2) dC Jyy := Z C ρ(x2+ z2) dC Jzz := Z C ρ(x2+ y2) dC Jyz:= Z C ρyz dC Jxz := Z C ρxz dC Jxy := Z C ρxy dC
be the second order moments of inertia (integration is intended over the considered body 3D domain) and let
I := Jxx −Jxy −Jxz Jyy −Jyz symm Jzz
be the moment of inertia tensor. With reference to Fig. 3.2, for the generic point P of mass ρdC, it is valid that (P − G) + (G − O) = (P − O). By substituting (P − O)
CHAPTER 3. LOADS AND CONSTRAINTS 36
Figure 3.2: Transformation of inertia tensor
coordinates in the afore-defined moment of inertia and by noticing that, if G is the centroid of the body, the first order moments of inertia vanish, one obtain the rules of transformation of inertia tensor between two translated coordinate systems frames:
c Jxx = Jxx+ m((G − O)2y+ (G − O) 2 z) (3.19a) c Jyy = Jyy+ m((G − O)2z+ (G − O) 2 x) (3.19b) c Jzz = Jzz + m((G − O)2x+ (G − O) 2 y) (3.19c) c Jyz= Jyz+ m(G − O)y(G − O)z (3.19d) c Jxz = Jxz+ m(G − O)x(G − O)z (3.19e) c Jxy = Jxy + m(G − O)x(G − O)y (3.19f)
3.5.1
Engines
Engines are modelled as circular cylinder, having characteristic reference radius r, length L and mass m. With reference to Fig. 3.3a:
Jxx = m(3r2+ L2) 12 Jyy = Jxx Jzz = mr2 2 Jyz = 0 Jxz = 0 Jxy = 0
3.5.2
Passengers and containers
For passengers and containers, the procedure is conceptually the same.
In [SANTSCHI 63], first estimations of sitting human body moments of inertia (Fig. 3.4) are reported, as a function of body mass and height. By default, it has been assumed an human height equal to 175 mm and a mass equal to the sum of
CHAPTER 3. LOADS AND CONSTRAINTS 37
(a) Engine-cylinder model (b) Container-parallelepiped model
Figure 3.3: Engine and container simplified form
CHAPTER 3. LOADS AND CONSTRAINTS 38 body, seat and luggage ones, previously defined. In effect, seats follow the shape of a sitting human body; therefore, considering an equivalent mass formed by the actual body and a seat is like considering an oversize person body. On the other hand, the adding of luggage mass can be questionable. The writer has chosen to account for this mass by simply summing to the others, without being boaster. The statistical formula are (h in in and W in lb):
Jxx = −91.6 + 1.43h + 0.322W (3.20)
Jyy = −135.0 + 2.268h + 0.268W (3.21)
Jzz = −52.8 + 0.768h + 0.201W (3.22)
while the position of the center of mass can be deducted from Fig. 3.4.
As for containers, they have been considered as parallelepipeds. The well known moments of inertia are, then (Fig. 3.3b):
Jxx = m(h2+ L2) 12 Jyy = m(w2+ L2) 12 Jzz = m(w2+ h2) 12
Once the single inertia tensor has been defined, by means of (3.19) it has been computed the resultant inertia (and mass, as well) of both passengers and containers. Those resultants have been applied to the centroid of total passenger system or container system, respectively.
3.5.3
Wings
As reported in [FREDIANI 17a], the supercritical airfoil NACA-SC-20410 has been chosen as baseline airfoil for aerodynamic preliminary evaluations (Fig. 3.5). By
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 x c t c NACA SC 24101
Figure 3.5: NACA SC 20410 airfoil
means of simple computations, it can be shown that all airfoil geometric quantities such area, moments of inertia etc. can be expressed as a non-dimensional coefficient multiplied by a certain power of the chord c, (to get the proper dimension of the
CHAPTER 3. LOADS AND CONSTRAINTS 39 considered quantity). For example, the airfoil area can be expressed as A = 0.0675 c2
and the airfoil centroid is positioned (from the leading edge) at xCG= 0.4155 c.
WAGNER assumes a wingbox between 0.1 c and 0.7 c. By doing so, the centroid position of the wingbox (0.3904 c) is very close to the aforementioned whole airfoil centroid. Keeping in mind that the wingbox will be filled with fuel, it results that the final centroid position will be dominated by the wingbox mass, the wingbox sector being remarkably heavier than "empty" leading and trailing edges ones.
For those considerations, it has been assumed to consider a rectangular shaped wingbox in wing inertia computation. As a-posteriori confirm, the rectangular-wingbox area is 0.06 c2 while the actual wingbox area would be 0.054c2. Centroid
position is far 0.4 c from leading edge.
Every section of each wing (root-kink, kink-tip) has been approximated as reported in Fig. 3.6. Chord is assumed to vary according to this law:
Figure 3.6: Wing part approximation
c(x) = cR 1 − 1 − λ L x , (3.23) having defined λ := cT cR , αΛ:= − arctan cR (λ − 1) 2 L .
CHAPTER 3. LOADS AND CONSTRAINTS 40 In order to find geometrical properties, integrations must be performed between:
−cR 2 + x tan(Λ + αΛ) ≤ z(x) ≤ − cR 2 + x tan(Λ + αΛ) + c(x) − t c c(x) 2 ≤ y(x) ≤ t c c(x) 2 0 ≤ x ≤ L. One finds: V = L 3 t c c2R(λ2+ λ + 1) (3.24) xCG = L (3 λ2+ 2 λ + 1) 4 (λ2+ λ + 1) (3.25) yCG= 0 (3.26) zCG= cRλ − cR+ 2 L tan Λ − arctancR(λ−1) 2 L (3 λ2+ 2 λ + 1) 8 (λ2+ λ + 1) (3.27)
For centroid-referred moments of inertia, it is necessary change variable: x = ˆx + xCG −cR 2 + x tan(Λ + αΛ) − zCG≤ z(x) ≤ − cR 2 + x tan(Λ + αΛ) − zCG+ c(x) −xCG≤ ˆx ≤ L − xCG.
Central moments of inertia are: Jxx ρ = λ4 60+ λ3 60 + λ2 60 + λ 60+ 1 60 cR4L tc3+ −13 λ 480 − 3 (λ2+ λ + 1) 64 + 19 λ2 960 + 19 λ3 960 + 19 λ4 960 + 1 15 cR4L tc+ 3 λ 80 + 4 λ + 8 (λ2+ λ + 1) 64 + λ2 40+ λ3 80− 11 80 cR3L2tc tan Λ − arctan cR (λ − 1) 2 L + 3 λ 80 − 4 λ + 4 (λ2+ λ + 1) 64+ λ2 80 + 3 40 cR2L3tctan Λ − arctan cR (λ − 1) 2 L 2 (3.28)
CHAPTER 3. LOADS AND CONSTRAINTS 41 Jyy ρ = −13 λ 480 − 3 (λ2+ λ + 1) 64 + 19 λ2 960 + 19 λ3 960 + 19 λ4 960 + 1 15 cR4L tc+ 3 λ 80 + 4 λ + 8 (λ2+ λ + 1) 64 + λ2 40+ λ3 80− 11 80 cR3L2tc tan Λ − arctan cR (λ − 1) 2 L + 3 λ 80 − 4 λ + 4 (λ2+ λ + 1) 64 + λ2 80+ 3 40 cR2L3tctan Λ − arctan cR (λ − 1) 2 L 2 + 3 λ 80 − 4 λ + 4 (λ2+ λ + 1) 64+ λ2 80+ 3 40 cR2L3tc (3.29) Jzz ρ = λ4 60 + λ3 60+ λ2 60+ λ 60+ 1 60 cR4L tc3+ 3 λ 80 − λ + 1 (λ2+ λ + 1) 16+ λ2 80+ 3 40 cR2L3tc (3.30) Jyz = 0 (3.31) Jxz ρ = ((λ − 1) (λ4+ 4 λ3+ 10 λ2+ 4 λ + 1)) ((λ2+ λ + 1) 160) cR 3L2t c+ λ4+ 4 λ3+ 10 λ2+ 4 λ + 1 ((λ2+ λ + 1) 80) cR2L3tc tan Λ − arctan cR (λ − 1) 2 L (3.32) Jxy = 0 (3.33)
Expressions are known if a right estimation of ρ is made.
In [COSTA 11] a structural preliminary study of wings structure and weight has been conducted. By dividing the cumulative wing weight presented in that work by the total volume of their wing, volume calculated applying (3.24), an averaged density of structural parts is found to be about
ρempty = 440 kg m−3, (3.34)
while a typical aeronautic fuel has a density of about ρf uel = 800 kg m−3.
As stated in [COSTA 11], only the 80 % of the wingbox volume will filled by fuel, because of ribs, stringers, spares and other aspect which reduce the available volume. Therefore, by averaging, a final density for the full wingbox is obtained:
