Boeing 777X Wingtip Analysis - FEM Final Project

Mechanical Investigation of Boeing 777X Folding Wingtip
- CEE530 Final Project -
Matthew D. Hawkins
Master of Engineering Candidate
Duke University
Fall 2018

CEE530 - Fall 2018 Final Project Report Matthew Hawkins
Contents
Abstract …………………………………………………………………………2
Problem Description …………………………………………………………... 2
Objectives ……………………………………………………………………... 2
Background ……………………………………………………………………. 3
Approach …………………………………………………………………….… 4
Results and Discussion ………………………………………………………… 9
Summary and Conclusions …………………………………………………..… 10
References ………………………………………………………………………12
Tables and Figures ……………………………………………………………... 14
Appendices …………………………………………………………………….. 21
A. Evolution of the Boeing 777…………………………………………….............................
B. Global Demand for Widebody Aircraft and the Role of the 777 ………………………….
C. Special Conditions for Commercial Approval of Folding Wingtip …………………….....
D. Wing Loading Derivation (Full)...........................................................................................
1

I. Abstract
We have modeled a simplified wingtip section of the Boeing 777X, a commercial airliner
that can carry up to 400 passengers, or an equivalent payload of over 150,000 kg. Applying
Euler-Bernoulli beam theory for aerodynamic and structural loading, we calculate the strain
energy and rotational stiffness of the hinge mechanism that prevents its folded wingtip, a feature
that allows the aircraft to meet current regulatory and civil aviation restrictions while achieving
industry-leading aerodynamic efficiency, from unlocking in flight. As this aircraft is still in its
prototype phase, but has received conditional approval by the FAA, the analysis presented is
simplified using a variety of assumptions. We find that the latching mechanism of this hinge
must be able to produce a system of forces that create an equivalent stiffness of over 10 million
newton-meters per radian when the wing bends during steady, level flight.
II. Problem Description
The locking mechanism of the Boeing 777X wingtip hinge is one of many components of
the airliner for which failure is not an option. Under intense static and dynamic loads, the hinge
of this commercial aircraft slated for preliminary testing in 2019 must remain fully locked during
takeoff, landing, turbulence, and cruising flight. This study will focus on the static loading case
applied to the latch mechanism of this hinge during level cruise, where loads are both structural
and aerodynamic. For this case, the latch serves to maintain continuity of the wing, which can be
simplified as a cantilever beam undergoing a time-independent spanwise variable distributed
load. When subjected to this loading, the section of the wing made up by the hinge
(approximately 2m x 0.5m x 0.2m), undergoes intense strain that can be approximated by
Euler-Bernoulli Beam theory and also calculated to better approximation using ANSYS
software. Further, because this latch system must not fail, we seek to infer the necessary
rotational stiffness required by the system given a calculated deflection through both written and
computer-aided mathematical calculations. This equivalent rotational stiffness will represent the
system of forces that must be applied by the actual locking mechanism to prevent failure during
any stage of flight.
2

III. Objectives
A. To understand magnitude of loads sustained by the wingtip of a Boeing 777
B. To calculate the strain energy of the hinge section of a Boeing 777X wing
C. To calculate an equivalent rotational stiffness for the hinge mechanism
IV. Background
In the age of ever-increasing demand for fuel efficiency in human transportation, air
travel has found itself in a unique bind: that both passengers and cargo have very limited public
options for traveling great distances in short amounts of time. In this context, innovation in the
aerodynamic and propulsion systems of existing aircraft platforms presents a significantly more
stable financial case than creating new platforms, and this is why Boeing is betting their future in
high-capacity, long-haul air travel on a revision of its classic, twin-engine 777 (“triple-seven”)
model . This revision program, currently working under the title of “777X”, calls for an entirely1
new powerplant [1], significant structural changes to the fuselage and, as I will focus on in this
study, longer and more efficient wings that are quite literally redefining the industry standard.
Previous iterations of the 777 and other large twin-engine aircraft (of Code E
categorization ) have adhered to a strict wingspan limit of 65m, which was established by the2
International Civil Aviation Organization (ICAO) in order to set an upper bound on the width of
taxiways and terminal gate spaces that need to be created at airports [3]. This limit has
historically been a critical design factor for large twin engine aircraft since the size of the wing
largely dictates how large (and heavy) the fuselage can be. However, since this wingspan
regulation for Code E Aircraft only applies to the mission time that they spend on the ground,
Boeing is able to avert regulatory impediment to aircraft efficiency by creating the first-ever
commercially-available folding wingtip, to be locked out during flight and retracted on the
ground (see Figure 1).
By extending the 777X wingspan to 72m (235 feet), Boeing creates a wing of higher
aspect ratio than any other company model [1], resulting in a greater lift-to-drag ratio, which is a3
1
See Appendix A, Evolution of the Boeing 777
2
“Code E” is a category of aircraft sizing based on gate sizes at airport terminals [3]
3
Wingspan divided by mean chord [19]
3

key driver of aerodynamic efficiency. The company estimates that this will lead to a 5% increase
in aerodynamic efficiency and contribute to an overall 12% decrease in fuel consumption for the
777X when evaluated against its direct competitor, the Airbus A350-1000 [1]. For airlines, this
translates to millions of dollars in savings each year, while also reducing the carbon footprint of
the industry as a whole. Beyond the aerodynamic benefits as well, the 777X will also allow
airlines to carry larger volumes of passengers without taking on large operation and terminal
renovation costs that became a serious barrier to entry for airlines considering the high-capacity,
but economically-failing, long-haul Airbus A380 .4
While Boeing has found a way to operate inside ICAO regulations with an
industry-dominating wingspan, such an innovative device also has to pass a litany of approvals
by the Federal Aviation Administration (FAA) since the latch mechanism presents an
opportunity for catastrophic failure if not designed precisely. To achieve engineering
certification, Boeing has to meet ten specific conditions for the wingtip mechanism, some in
particular referring to the performance and durability of the locking latch mechanism (see
Appendix C). The purpose of my study, therefore, is to understand a small component of the
regulatory process for this hinge, namely the mechanical strain imposed on the latch system as
well as the rotational stiffness that must be created by the latch in order to ensure a safe flight
that achieves the benefits described above.
V. Approach
To understand the strain energy and rotational stiffness of the latch mechanism during
stable cruising flight, we need to create a simplified model of the wingtip hinge that encompasses
both theoretical and computational understanding. The following plan outlines the sequence that
we will follow:
1. Understand and calculate loading from aerospace Beam Theory
2. Calculate simplified theoretical strain energy
3. Calculate simplified theoretical deflection
4. Calculate simplified theoretical rotational stiffness
4
See Appendix B, Global Demand for Widebody Aircraft
4

5. Generate approximate simplified model of hinge mechanism in Ansys
6. Simulate loads from step 1 and obtain computational values equivalent to steps 2-4
In order to systematically approach these steps, we will first review the key mechanical
concepts applicable to the hinge system.
Solid Mechanics Review:
The wing of an aircraft is essentially a cantilever beam that, in static flight conditions,
undergoes variable distributed loading from both the structure of the wing itself as well as the
aerodynamic lift. After accounting for the weight of the engine, which can be considered a point
force, the weight of the fuel in the wing, another distributed load, and gravity, the net loading on
an aircraft wing results in a variable distributed load pattern as shown in Figure 2. Under this
loading, the wing exhibits elastic behavior equivalent to that of a spring as it stores mechanical
energy during loading and returns to its original shape after the loads are removed [16]. Further,
from Euler-Bernoulli beam theory, this loading creates a bending moment, M, that causes the
wing to deflect upwards in flight. Any given cross section of the wing will experience a shear
force, V, that can be calculated (for a simple beam) as the partial derivative of the bending
moment with respect to location on the wing,
(y) dM/dy.V = (1)
From this, it can also be shown that loading, q, is proportional to the first partial derivative of
shear
(y) dV /dyq = (2)
and we will use both as the first two governing differential equations for this analysis.
Strain Energy - When external work is done on an elastic body like the 777X wing, it is
transferred within the system into potential energy, which is commonly referred to as strain
energy in this context [2]. If we assume for this system that the relationship between applied
force, F, and displacement (in the z-direction) is linear, then the strain energy, U, can generally
be represented as
5

zF/2Ugeneral = (3)
Specifically, the strain energy that results from bending can be represented as
M l/2EIUbending = 2
(4)
where E is Young’s modulus and I is the second moment of inertia. Further, the strain energy
that can be calculated from shear is
CV l/2AGUshear = 2
(5)
where A is the cross sectional area, G is the shear modulus, and C is a correction factor that
accounts for transverse shear [2].
Once we have obtained the given strain energies that we desire, we will be able to
calculate the deflection in a representative beam by applying Castigliano’s second theorem.
While simple, it states that the displacement corresponding to any force is equal to the partial
derivative of the total strain energy with respect to that force and acts in the direction of the force
[17]. In the context of an applied moment, like the case of a wing section subject to bending,
Budynas and Nisbett prove that the rotational displacement, , equals the partial derivative ofθ
strain energy with respect to moment,
dU/dM.θ = (6)
Rotational Stiffness - A beam under its designed loading acts as a spring and stores the bending
moment, as opposed to a simple force. Stiffness refers to the rigidity of an object, which is
exactly what we want to understand when investigating this hinge mechanism that must remain
locked during flight. It depends on the material and the shape of the boundary conditions applied
to any solid object [11]. Further, the change in angular deflection is proportional to the applied
torque, T, on the system created by the moment, and is related by the constant of proportionality,
K, that we refer to as the rotational stiffness that we seek in our investigation [17]. In equation
form, we express the third governing equation of our analysis as,
K θ/dy.T = * d (7)
Now that we have our background context on strain energy and rotational stiffness, let’s dive
into the problem.
6

Assumptions and Simplifications:
As mentioned before, the model presented in this study is greatly simplified from the true
hinge mechanism about the 777X. This is primarily due to the proprietary nature of the design
itself, as well as the scope of the problem that we are trying to model for this project. Most
notably, instead of modeling the hinge in detail to all of its internal intricacies, we simplify this
problem greatly to say the hinge section acts as spring of given stiffness that we can work to
understand the high-level mechanics of. Recognizing this, there are other major assumptions we
can therefore make as we conduct this investigation, and discuss those specific to the geometric
modeling of the wing for the next section.
At the system level, we are assuming only the static case, where the airplane is in cruise
at Mach 0.75 (925 km/h). Further, we assume that all features of the wing system, including the
engine, fuel, flaps, and dampeners, are accounted for in the aeroelastic equations discussed
below when we mathematically model the wing bending and shear. With regards to bending, we
acknowledge here that bending in modern aircraft wings is significant (with respect to wingspan)
as the material composition of modern wings allow for greater bending in flight [8]. However, in
order to preserve the small-deflection assumption of castigliano’s theorem, we treat deflection
due to bending as if it were proportional to applied moment and also small with respect to total
wingspan. Finally, we assume that the system is statically determinate, where all forces can be
calculated using the equations of static equilibrium [2].
At the cross-sectional level, we simplify the analysis by removing all stress concentration
that would be present in the true wing cross-sectional structure. Next, for our hand calculations,
we also assume the beam to be of rectangular shape (as opposed to cylindrical), since the
rounded nature of the leading edge on real plans is partially designed to accounted for the
application of loads.
At the material level, we select Aluminum alloy 2909 (detailed in Table 4) as the material
to apply uniformly throughout our wing section since it is one of the most commonly used in
aircraft manufacturing[15]. We then assume a constant strain energy density in the section of the
wing that we wish to model, meaning that the volumetric strain in a given element of the system
is constant to the point that our theoretical calculations are simplified [19]. Next, for an
7

appropriate poisson’s ratio, we assume homogeneous, isotropic material such that the lateral
strain is proportional to the axial strain [2]. As we assumed linearity in the force-deflection
relationship described above, we additionally make the same assumption with rotational
stiffness, although both are likely non-linear due to the true internal structure of the 777X wing.
We will only study the case of vertical bending, and therefore assume curvature of bending to be
constant for the full length of the wing.
Parameters and variables used:
For a full list of parameters and variables, please refer to Table 1.
Modeling:
To appropriately model the section of the 777X wing that represents the hinge, we first model
and mesh the section in ANSYS and then seek to understand the boundary conditions (namely,
the bending moment and shear forces) that will be acting on the section in steady flight.
I. Geometry - Our goal in creating this model is to portray a realistic version of the wing section
that makes up the hinge mechanism. This means that we create a similar airfoil, taper ratio, and
wing sweep angle for a section of the wing that we best estimate would be present on the 777X.
However, due to the protection of intellectual property relevant to the 777X, we used publicly
available wing and loading data from the 777-300ER, which will be the closest relative of the
777X variants when they launch, and include this data as geometric parameters in Table 1. For
the airfoil, we obtained the variable thickness airfoil data of a 777 [13], and then followed the
method outlined in [21] to create a two-dimensional airfoil which we subsequently extruded with
the appropriate taper ratio and at the correct sweep angle. The resulting model is shown in Figure
4. In ANSYS, we established that this would be a linear-elastic and isotropic material such that
we could meet assumptions presented by simplified beam theory.
II. Meshing - Using a standard Q4 mesh, I arbitrarily established a test size of 0.1 m and ran a
convergence study based on directional displacement. The study revealed that tripling the
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number of nodes in the system only led to a 0.17% change in directional displacement values,
which met my 1% tolerance for convergence. The final mesh of 230 elements used adaptive
sizing and had an average surface area of 0.58 m2
.
III. Loading - Here, I briefly outline the process shown in Appendix D for deriving the boundary
conditions that I will apply to the beam theory simplification of our 777X hinge as well as my
computerized model. First, we need to generate the equations for moment and shear on the
whole system such that we can apply them at the section boundaries of our hinge. The first
relationship to define is the net wing loading as a function of wingspan, q(y), namely:
(y) L (y) Ngm (y)q = ′ − ′ (8)
where m’(y) is the local mass as a function of span y and L’(y) is the spanwise aerodynamic
loading from Lift, L. N is the load factor which, in steady flight, is equal to 1. The net loading, as
described above, is schematically shown in Figure 2, and shown graphically, alongside spanwise
moment and shear that we will derive equations for, in Figure 3.
As mentioned in the review section above, Equations (1), (2), and (7) represent the
governing equations of this analysis. Using MATLAB, we calculate the boundary conditions for
the full wing moment and shear calculations and publish them in Table 2. Then, through
numerical integration of this statically determinate problem for the variables defined in Table 1,
we arrive at the following equations for shear and moment, both as functions of normalized
location along the wingspan, :η
(η)V = b
NWFuse 2
1+λ 2
b
1 λ ) (1 )[ − η + ( − 1 2
1
− η2
] (9)
(η)M = b
NWFuse 2
1+λ 4
b2
1 (1 ) λ ) (1 (1 ))[ − η − 2
1
− η2
+ ( − 1 2
1
− η − 3
1
− η3
] (10)
Now having the ability to calculate moment and shear at a given length of the wing, we
can find their respective values at both ends of the wingtip section. Then, we finally will be able
to apply them in our theoretical and computational models for strain energy and, eventually,
rotational stiffness.
9

VI. Results and Discussion
As described earlier, the initial objectives of this problem sought to calculate strain
energy and rotational stiffness of the 777X wingtip hinge such that we can better understand the
loading effects on the real wing device.
We first calculated the strain energy and rotational stiffness for a wing of vastly
simplified geometry with MATLAB. After solving Equations (9) and (10) for both sides of the
wing section, we were able to calculate an approximate total strain energy of 256440 NM, which
represents the energy contributed by both bending and transverse shear. From there, we
calculated a rotational stiffness of 410930 NM/radian by applying equation (7). These loads are
significant and of the approximate order of magnitude with what we established after reviewing
the bending and shear results calculated in MATLAB.
Then, using ANSYS, we were able to apply the bending and shear from (9) and (10) as
boundary conditions to the section. We also fixed the inner (root-side) cross section such that the
model represented its own cantilever beam system. From there, we made adjustments to allow
for large deflection and solved for directional deformation (Figures 5 and 7), strain energy
(Figures 6 and 8), and equivalent elastic strain (Figure 9). Table 3 summarizes these results, but
we acknowledge the most significant values here. For strain energy, we obtained a total value of
20044 NM, one order of magnitude smaller than that predicted by simplified beam theory. The
maximum deflection in the computer model was 5.857 mm, which we converted to be 0.0255
radians with a calculated radius of curvature from our MATLAB results that was 0.22986 m. Our
final calculation for rotational stiffness as modeled in ANSYS was 7,0156,000 NM/radian, about
two orders of magnitude different than our approximated value.
We acknowledge that the hand modeling and the computer modeling produce noticeably
different results, and believe that the error is due to the vast simplifications that occur between a
rectangular, constant-width beam and the tapered, rounded wing section modeled in ANSYS
(both of which are greatly simplified from the actual device). As the deformation figures show,
however, the results of the study are visibly accurate to what we imagined, but the values
10

predicted by aeroelastic beam theory do not seem to accurately represent that which we modeled
in ANSYS.
All the same, looking to our primary objective for understanding the loading of a 777X
wingtip hinge, it is still significant to note that both the loading equations from MIT and the
calculated rotational stiffness establish tangible values to unpredictable quantities.
VII. Summary and Conclusions
This study was conducted to better understand the mechanical demands of a latch on the
wingtip section of a Boeing 777X. Because net displacement of this hinge in flight must be equal
to zero, we surmise that the system of forces in this locking hinge mechanism must be able to
withstand, during steady level flight, moment arms of over 1.5 million newton-meters. After
accounting for a factor of safety that is traditionally 1.5 for Boeing [22], we conclude that the
system must be able to provide a reactive stiffness of over 10 million newton-meters per radian
of deflection. This is an incredible feat of engineering.
Looking beyond this study, the next logical step is to model the effects of dynamic
loading on the same hinge and how it must also resist them. Since cruising flight is often the
lowest loading period of flight besides being on the ground, we believe that this hinge must be
modeled for significantly higher load resistance if it is to reliably meet the standards of the FAA
for an extended life. The relationships between hinge properties and the loading would inherently
be non-linear, as described by [23], and therefore would demand significantly more
pre-modeling attention.
In closing, as Boeing plans to launch the first 777X test platform in the second quarter of
2019 [24], we’ve provided context to an element of this aircraft that will undoubtedly spark
curiosity in its passengers. More significantly though, we’ve shown that the future of this
industry is all but established, as this innovative device is much more than a cosmetic flare. It is
an intricate and vital part of a system that will meet loads hardly fathomable to what we humans
experience on Earth. And, to the trained eye, it will undoubtedly be a masterpiece of engineering
as it takes to the skies.
11

VIII. References
1. 777X By Design. Boeing Company (2018). Obtained from Boeing website,
www.boeing.com/commercial/777x/by-design
2. Budynas,R. & Nisbett, J.K. (2015). Mechanical Engineering Design (Shigley’s). McGraw-Hill
Education. New York.
3. Special Conditions: The Boeing Company Model 777-8 and 777-9 Airplanes; Folding Wingtips.
Federal Register Rule (2018). Obtained from the Federal Register website,
https://www.federalregister.gov/documents/2018/05/18/2018-10576/
4. Wing Bending Calculations. MIT Open Course Ware (2006). Obtained from ocw website,
https://ocw.mit.edu/courses/aeronautics-and-astronautics
5. Aircraft Manufacturing: Global Markets to 2022. BCC Research (2018). Obtained from
bccresearch.com via Duke University proxy
6. Commercial Market Outlook. Boeing Company (2018). Obtained from Boeing website,
http://www.boeing.com/commercial/market/commercial-market-outlook/
7. Analysis: A Decade of A380, Success or Failure?. Airways Magazine (2017). Obtained from
Airways Magazine website,
https://airwaysmag.com/industry/analysis-decade-a380-success-failure-part-ii/
8. Boeing 777. Wikipedia (2018). Obtained from https://en.wikipedia.org/wiki/Boeing_777
9. Norris, G. & Wagner, M. (1996). Boeing 777. Motorbooks International. St.Paul, MN.
10. DUBAI AIRSHOW: Boeing Launches New 777X With Commitments. Morningstar/Alliance News
(2013). Obtained from Morningstar website,
http://www.morningstar.co.uk/uk/news/AN_1384756327384019100/AllianceNews
11. Rotational Stiffness.Equanalysis UG (2018). Obtained from FX Solverwebsite,
https://www.fxsolver.com/browse/formulas/Rotational+stiffness
12. Info on Boeing’s Wingsweep Angle. Online Forum (2006). Obtained from PPRuNE Forums
website, https://www.pprune.org/archive/index.php/t-214344.html
13. GOE777 Airfoil. Airfoil Tools (2018). Obtained from Airfoil tools website,
http://airfoiltools.com/airfoil/details?airfoil=goe777-il
12

14. Aluminum 2090-T83. Material Database (ND). Obtained from MatWeb website,
http://www.matweb.com/search/datasheet_print.aspx?matguid=a79a000ba9314c8d90fe75dc76efc
c8a
15. Aluminum-Lithium Alloys Fight Back. Aluminum Insider (2017). Obtained from Aluminum
Insider website, https://aluminiuminsider.com/aluminium-lithium-alloys-fight-back/
16. The Boeing 777X Folding Wingtip -Explained. DJ’s Aviation (2018). Obtained from YouTube
page, https://www.youtube.com/watch?v=u3cq2MT09lQ
17. Springs. Obtained from Sharcnet website,
https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/wb_sim/ds_Springs.html
18. The Theorem of Least Work. Course Presentation for CE474 at Purdue University. Obtained from
Purdue website,
https://engineering.purdue.edu/~ce474/Docs/The%20Theorem%20of%20Least%20Work_2012.p
df
19. Demitriadis, G. (ND). Aircraft Design - Lecture 2: Aerodynamics. Lecture to the Université de
Liege, obtained from http://www.ltas-cm3.ulg.ac.be/AERO0023-1
20. Lecture on Elastic Strain Energy. University of Auckland. Obtained from university website,
http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_I
21. (Tutorial). Ansys DesignModeler - How to Create 3D Wing using Airfoil. HTC (2016). Obtained
from YouTube page, https://www.youtube.com/watch?v=thrB-mKHGek
22. Boeing 777 Wing Test (2010). Obtained from YouTube page,
https://www.youtube.com/watch?v=Ai2HmvAXcU0
23. Castrichini, A. (2016). Nonlinear Folding Wing Tips for Gust Loads Alleviation. Aerospace
Research Journal, Volume 53, Issue 5. Obtained from ARC website,
https://arc.aiaa.org/doi/10.2514/1.C033474
24. Baitinger, B. (2018). Boeing 777X to have retractable wings, a first for passenger travel.
UPI. Obtained from UPI website,
https://www.upi.com/Boeing-777X-to-have-retractable-wings-a-first-for-passenger-travel
/7521527255838/
13

IX. Tables and Figures
Figure 1: Folding Wingtip Mechanism (Conceptual Design, Diagram, and Prototype)
14

Figure 2: Distributed Aerodynamic Loading on Wing (from MIT)
Figure 3: Spanwise Loading, Shear and Moment (from MIT)
15

Figure 4: Ansys Model of Wing Section for Hinge Mechanism
Figure 5: Edge-On and Cross-Sectional Directional Deformation
16

Figure 6: Isometric View of Strain Energy
Figure 7: Plot of Deformation along Neutral Axis
Figure 8: Plot of Strain Energy Along Neutral Axis
17

Figure 9: Plot of Equivalent Strain along Neutral Axis
Figure 10: Diagram of the Trapezoidal Rule for Integration
18

Table 1: Parameter and Variable Values
Definition Value Units
Sweep Angle 31.6 [12] Degrees
Taper Ratio, λ 0.149 [12] (Dimensionless)
Hinge Width 0.5 Meters
Hinge Length (avg. Chord) 1.814 Meters
Airfoil Max Thickness 0.225 Meters
Load Factor, N 1 (Dimensionless)
Fuselage Weight, W(fuse) 299,153 (payload + fuel) Kilograms
Wingspan, b 71.8 Meters
Beam Correction Factor, C 1.2 (Dimensionless)
Section face (outer) y - 3.25 Meters
Section face (outer) y - 3.75 Meters
Table 2: Boundary Conditions for Full Wing (as Cantilever Beam)
Boundary Parameter Value
y = b/2 ( root = half wingspan) (shear)0V =
y = b/2 (moment)M = 0
y = 0 (wingtip) (deflection angle)θ = 0
y = 0 (deflection)w = 0
Table 3: Major Results from Ansys and Beam Theory
Phenomena Value
Strain Energy (max) 122.3 NM
Strain Energy (total) 2.0044 x 104
NM
Directional Deformation (max) 5.8573 mm
19

Equivalent Stress (max) 45 MPa
Moment (inner) - Beam Theory 1.7535x105
NM
Moment (outer) - Beam Theory 1.7877x105
NM
Shear (inner) - Beam Theory 1.3632x105
N
Shear (outer) - Beam Theory 1.2805x105
N
Table 4: Material Properties of Aluminum Alloy 2909
Property Value Units
E, Young’s Modulus of Al-2909 76 [14] GPa
G, Shear Modulus of Al-2909 28 [14] GPa
Density of Al-2909 2.59 g/cm3
Poisson’s Ratio for Al-2909 0.34 (Dimensionless)
20

X. Appendices
Appendix A: Evolution of the Boeing 777
The 777X, as two proposed versions (777-8 and 777-9), is the latest iteration of a platform that was
originally conceived in 1988. Back then, Boeing wanted to supplement the first wave of widebody5
aircraft from the 1970s, which included the original and iconic 747 [9]. After including launch carriers
throughout more of the design process than ever before, the first generation of the 777 family emerged as
the -200, -200ER and the -300 by 1990 and serviced long and short-haul flights for 230-330 passengers
per flight. Eight years later came the second generation of the platform, which includes the -300ER and,
eventually, the cargo-only 777F, which further expanded the market applications of the 777 platform [8].
As the first decade of the 2000s came to a close and Airbus promised to launch a twin-engine, efficient,
long haul aircraft (the A350) to supplement the A380, Boeing realized that pressure from the French
aerospace giant in the future widebody market would be too much for the 787 and legacy models of the
777 to maintain on their own. Adding to this pressure was a realization that the demand for high volume
aircraft had been initially miscalculated (discussed below) and Boeing further felt that it could not rely on
a revised 747.
Marginal efficiency revisions were made to second generation 777s that had not yet been built, but
Boeing decided to take larger steps in capacity and efficiency that could not be made through the revision
of old platforms. In November 2013, Boeing publicly launched the 777X program as a means to introduce
an essentially new aircraft in the market without starting from scratch [10]. In addition to the wingtip
innovation described earlier, Boeing partnered with GE to implement the largest commercial aircraft
engine (GE9x), and they also adopted a wing design similar to the 787 with a significantly higher
percentage of carbon-fiber reinforced polymer, like that utilised by the 787 [1]. These notable changes
are pioneering in the commercial aviation industry and were not on the design table for the last major
round of future projections for air travel in the late 1990s and early 2000s. The 777-8 and 777-9 thus
highlight that the introduction of the 777X program will complete an exciting cycle of aviation
development that has been marked by intense competition surrounding both correct and incorrect
projections for the future of commercial aviation.
5
The term “widebody” is a general and simplified description for commercial aircraft with two cabin aisles.
21

Appendix B: Global Demand for Widebody Aircraft and the Role of the 777
To understand the true significance of a widebody aircraft in our world of globalization, urbanization, and
exponential population growth, I briefly discuss the economic and historical factors most critical to the
demand for, and design of, the 777X.
Global Demand Forecast for Aircraft:
It has been estimated that the global air travel market will multiply by 2.5 its current volume by 2037,
pitting Boeing and its rival Airbus in a fierce competition to control majority share of an estimated $6.3
Trillion market. While the market for single-aisle aircraft for less than 150 passengers make up more than
50% of these market projections and global freight is estimated to account for an additional 4%, Boeing
projects that the commercial market for widebody aircraft will reach 8070 aircraft by 2037. They value
this sub-market at just under $2.5 trillion (39% of total market) over the next 20 years [6].
Need for High-Volume Aircraft
The widebody market described above is operated by nearly all of the recognizable brands in air travel,
each of which is looking for the most economically efficient way to transport high volumes of people
between the largest airports in their respective network. However, with so many options out there, what
mix of aircraft is best for each airline? The debate over which fleet orientation will allow airlines to
service the 21st century widebody market was initiated in the early 2000s, when Airbus announced the
creation of the ‘super-jumbo’ A380, which can theoretically carry up to 600 passengers between major
global “hubs”. This was to be the successor to Boeing’s 747 in the ‘very-large aircraft’ (VLA) market,
which was the portion of the widebody market taken by aircraft with capacity greater than 350
passengers. The first decade of service for the A380, however, has proven that operating an aircraft for
over 500 people is not financially sustainable for most airlines, since operational costs can only be offset
by an often-unattainable occupancy of paying passengers [7]. A similar line of reasoning can be applied to
the 747, but to a slightly lesser extent because the biggest modern airports are already built for the 747
(whereas the A380 requires multi-million dollar revision projects almost everywhere it services).
So, where does the 777X fit into all of this?
When it comes to Airbus’ bet on the future of the VLA market, some believe that the 777X is “the true
nail in the coffin for the [A380]” [7]. Comparing the two directly, the 777X has an equivalent unit cost
22

but lower operating cost per trip, and can also break-even during an airline’s off-season, which the A380
cannot offer. Further, as Boeing only plans to deliver a low number of its latest 747 revision, the 747-8i,
the airline trend seems to favor large capacity, twin engine aircraft (to which the 777-9 will be the
biggest). As Airbus counters its shortcomings of the A380 by putting significant funding into the
newly-launched A350 and an upcoming revision of the A330, both of which appear to be direct
competitors to every variant of both the 787 and 777, it is a reasonable conclusion to see that the next 20
years of the multi-trillion-dollar widebody market will be made by flights that carry between 200 and 400
passengers, not by the multi-level behemoths once believed to hold the future of the industry.
Appendix C: Special Conditions for Commercial Approval of Folding Wingtip
(Obtained from the Federal Register (cite) - abbreviated and in order of relevancy to this project)
1. The wingtips must have means to safeguard against unlocking from the extended, flight-deployed
position in flight, as a result of failures, including the failure of any single structural element.
2. The wingtip hinge structure must be designed for inertia loads acting parallel to the hinge line.
3. The folding wingtips and their operating mechanism must be designed for 65 knot, horizontal,
ground-gust conditions in any direction as specified in § 25.415(a).
4. (Boeing) must consider the effects of folding-wingtip freeplay when evaluating compliance to
the design load requirements of 14 CFR part 25, subpart C, and the aeroelastic stability (including
flutter, divergence, control reversal, and any undue loss of stability and control as a result of
structural deformation) requirements of § 25.629.
5. The wingtip-fold operating mechanism must have stops that positively limit the range of motion
of the wingtips.
6. (Boeing) must include design features that ensure the wingtips are properly secured during
ground operations, to protect ground personnel from bodily injury as well as to prevent damage to
the airframe, ground structure, and ground support equipment.
7. More than one means must be available to alert the flight crew that the wingtips are not properly
positioned and secured prior to takeoff.
8. In addition to a takeoff warning in accordance with § 25.703, a means must be provided to
prevent airplane takeoff if a wingtip is not properly positioned and secured for flight.
23

9. The airplane must demonstrate acceptable handling qualities during rollout in a crosswind
environment, as wingtips transition from the flight-deployed to folded position, as well as during
the unlikely event of asymmetric wingtip folding.
10. The forward position lights must be installed such that they consist of a red and a green light
spaced laterally as far apart as practicable, and installed forward on the airplane.
Appendix D: Full Derivation of Wing Loading Equations
The following derivation is obtained from an open source course in aeronautics at the Massachusetts
Institute of Technology [4]. Please see Table 1 for all variables.
Loading Relations
The net load distribution along the span of a wing (simulated beam) can be expressed as:
(y) L (y) Ngm (y)q = ′ − ′ (8)
For mass distribution m’ and lift distribution L’ along the span. The resulting distributions are
shown in Tables 2 and 3. As reminder, Euler-Bernoulli beam theory gives the the standard
differential equations:
(y) dM/dyV = (1)
(y) dV /dyq = (2)
θ w/dy= d (11)
θ/dy T/K /EId = = T (7)
The boundary conditions we now use for integration are given in Table 2.
Load Distribution:
We simplify true aerodynamic effects by assuming that the net aerodynamic and weight loading
(Equation 8) is proportional to local chord of the wing,
(y) K c(y)q = q (12)
24

which allows us to assume a constant local coefficient of lift, meaning that the local wing mass
distribution scales with the chord. Eventually, one obtains the relation:
=Kq = Swing
L−NWwing
Swing
NWfuse
(13)
where W(fuse) is the difference between the total weight of the aircraft and the weight of the
wings.
Numerical Integration:
The above equations can be numerically integrated for any given nonuniform q(y) and EI(y)
distributions. All spanwise variables are defined at a suitable number of discrete spanwise
locations y0, y1 . . . yi . . . yn−1, yn. The differential equations above can then be approximated over
the yi . . . yi+1 interval via averages and finite differences. This is equivalent to integration via the
Trapezoidal Rule, shown in Figure 11.
(y )V i+1 − V i = 2
q +qi+1 i
i+1 − yi (14)
(y )Mi+1 − Mi = 2
V +Vi+1 i
i+1 − yi (15)
y )θi+1 − θi = 2
1
(Mi+1
EIi+1
+
Mi
EIi
)( i+1 − yi (16)
(y )wi+1 − wi = 2
θ −θi+1 i
i+1 − yi (17)
For this statically-determinate problem, the beam equations can be discretely integrated in the
order shown above. The summation of each equation starts at the end where its boundary
condition is applied. This means that (14) and (15) are summed inward from the tip and (16) and
(17) are summed outward from the root.
Simplified Deflection Calculations:
For our estimation purposes, it’s a simple approximation to assume that beam curvature is
constant and taken from a representative location (such as the root, where y = 0). For a straight
taper wing, the chord distribution is
(y)c = b
Swing 2
1+λ
1 λ )[ + ( − 1 b
2y
] (18)
25

We can assume this relationship holds for the 777X since its wing is essentially a straight
taper for the section of interest to us.
The corresponding approximate loading given by (12) and (13) is therefore,
(y) K c(y)q = q = b
NWFuse 2
1+λ 2
b
1 λ )η[ + ( − 1 ] (19)
Where , the normalized span location which runs between 0 and 1 (root to tip). We canη = b
2y
now calculate Shear and Bending Moment as a function of span:
Shear:
V (η) (η)dη= 2
b
∫
1
η
q (20)
V (η) η= b
NWFuse 2
1+λ 2
b
∫
1
η
1[ + (λ η)− 1 ] d (21)
(η)V = b
NWFuse 2
1+λ 2
b
1 λ ) (1 )[ − η + ( − 1 2
1
− η2
] (9)
Bending Moment:
M(η) (η)dη= 2
b
∫
1
η
V (22)
M(η) η= b
NWFuse 2
1+λ 4
b2
∫
1
η
1 λ ) (1 )[ − η + ( − 1 2
1
− η2
]d (23)
(η)M = b
NWFuse 2
1+λ 4
b2
1 (1 ) λ ) (1 (1 ))[ − η − 2
1
− η2
+ ( − 1 2
1
− η − 3
1
− η3
] (10)
26

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