Report-Wing Structure Design Spring'16

MAE 154B PROJECT RYAN, AGRAJ, KEVIN
Kevin Landry
Ryan Rader
Agraj Sobti
MAE 154b: FDR
6/11/2016
*Thank you Professor Lynch, Peng & Auni. Our group
believes we deserve an A-. Our team put a significant
number of hours into this project, learning every concept
with minimal background beforehand. Despite not having
the familiarity with the material that many of the Aircraft
students had coming into the class, we feel as though we
were able to produce a comparable end product.*

Table of Contents
1. INTRODUCTION ....................................................................................................................7
1.1 Project Description ......................................................................................................................7
1.2 Assumptions................................................................................................................................7
1.3 Parameters..................................................................................................................................8
1.4 Gantt Chart..................................................................................................................................9
2. Wing Loading......................................................................................................................10
3. Maneuver and Loading during Banked Turns .......................................................................10
4. Wing Loading during Landing ..............................................................................................12
5. Xfoil....................................................................................................................................12
5.1 X-Foil Tutorial ............................................................................................................................12
5.2 Generating the .pol File..............................................................................................................14
6. Xfoil, V-n Diagram, Loading..................................................................................................14
6.1 Interpreting the Xfoil Output......................................................................................................14
6.2 The Stall Curve...........................................................................................................................16
6.3 Maneuver Limits........................................................................................................................17
6.4 Stall Velocities............................................................................................................................17
6.5 Gust Loading..............................................................................................................................18
6.6 Determining Critical Points.........................................................................................................20
7. Wing Loading at Critical Points ............................................................................................21
7.1 Lift and Drag ..............................................................................................................................21

7.2 Aircraft Coordinate Transformation ...........................................................................................23
8. Centroid & Area Moments of Inertia..............................................................................27
8.1 Cantilever Beam Model..............................................................................................................27
8.2 Centroid Calculations.................................................................................................................28
9. Shear Force, Bending, Deflection ....................................................................................31
10. Stresses ..........................................................................................................................38
10.1 Airfoil Profile..............................................................................................................................38
10.2 Section Centroid Calculations.....................................................................................................39
10.3 Moment of Inertia Calculations..................................................................................................40
11. Shear Flow – Calculations ................................................................................................41
11.1 Shear Flow – Results ..................................................................................................................44
12. Shear Flow-Code Verification...........................................................................................46
13. Buckling, Fatigue & Von Mises Stress.........................................................................47
13.1 Von Mises Stresses ....................................................................................................................47
13.2 Column Buckling ........................................................................................................................48
13.3 Skin (Thin Plate) Buckling...........................................................................................................50
13.4 Shear Buckling of Skin................................................................................................................50
13.5 Fracture & Fatigue .....................................................................................................................51
14. Aeroelasticity ..................................................................................................................52
14.1 Divergence.................................................................................................................................52
14.2 Aileron Reversal.........................................................................................................................53
15. 3D Modelling and Simulation...........................................................................................54

15.1 3D Modelling .............................................................................................................................54
15.1.1 Modelling Steps..................................................................................................................54
16. Loading Analysis..............................................................................................................56
16.1 The Setup ..................................................................................................................................56
16.1.1 Boundary Conditions ..........................................................................................................56
16.2 Results.......................................................................................................................................56
17. Modal Analysis................................................................................................................58
18. References......................................................................................................................61
19. Appendix.........................................................................................................................61
Table of Figures
Figure 1. The Cessna 177B Cardinal Aircraft.............................................................................................7
Figure 2. Rectangular Planform of the wing .............................................................................................8
Figure 3. Gantt Chart showing schedule of tasks....................................................................................10
Figure 4. Cp contour over Airfoil at A.o.A of 5 deg on XFoil for NACA 2415 .............................................13
Figure 5. Cp contour over Airfoil at A.o.A of 0 to 20 degrees on XFoil for NACA 2415..............................13
Figure 6: Lift Coefficient at Sea Level .....................................................................................................15
Figure 7: Lift Coefficient at Cruising Altitude..........................................................................................16
Figure 8: V-n Curve at Sea Level and Cruising Altitude............................................................................17
Figure 9: V-n Diagram with Gust Loading and Final Load Envelope.........................................................19
Figure 10: Critical Points from the V-n Diagrams....................................................................................20
Figure 11: Lift and Drag Force along the Length of the Wing..................................................................23

Figure 12: Inertial Lift and Drag Components and Aircraft X-Y Components ...........................................24
Figure 13: Load Conditions at PHAA.......................................................................................................25
Figure 14: Lift Force for each Critical Point.............................................................................................25
Figure 15: wx for each critical point........................................................................................................26
Figure 16: wy for each critical point........................................................................................................26
Figure 17: Simplified Wing Cross Section ...............................................................................................28
Figure 18: Shear Force (Sx).....................................................................................................................32
Figure 19: Shear Force (Sy).....................................................................................................................32
Figure 20: Mx vs Wing Span....................................................................................................................33
Figure 21: My vs Wing Span....................................................................................................................34
Figure 22: Deflection (u) ........................................................................................................................35
Figure 23: Deflection (v) ........................................................................................................................36
Figure 24: Direct Stress (sigma Z)...........................................................................................................37
Figure 25 NACA 2412 Arifoil Profile........................................................................................................39
Figure 26 Truncated Airfoil ....................................................................................................................40
Figure 27 Solving the two cell problem ....................................................... Error! Bookmark not defined.
Figure 28 Shear Flow Idealization ..........................................................................................................43
Figure 29 Magnitude of Stress along the Airfoil profile (Shear Stress) – PHAA at Sea Level.....................45
Figure 30 Magnitude of Stress along the Airfoil profile (Equivalent Stress) – PHAA at Sea Level .............45
Figure 31 Failure Stress vs Number of Loading repetitions .....................................................................51
Figure 32 Section View of the 3D wing model........................................................................................54
Figure 33 Isometric view of the Entire wing ...........................................................................................55
Figure 34 zz distribution accross different regions of the wing ................. Error! Bookmark not defined.
Figure 35 Deflection of the wing due to the applied load.......................................................................57

Figure 36 Mode-1 Frequency-7.3987 Hz ................................................................................................58
Figure 38 Mode-3 Frequency- 42.362 Hz ...............................................................................................59
Figure 40 Mode-5 Frequency- 99.968 Hz ...............................................................................................60
Figure 41 Modal Frequency vs Mode Number .......................................................................................61
Table of Tables
Table 1. Chosen Design Parameters.........................................................................................................8
Table 2: Critical Points ...........................................................................................................................20
Table 3 Dimensions used in centroid & moment calculations.................................................................27
Table 4: Centroid Results.......................................................................................................................29
Table 5: Spar Data .................................................................................................................................30
Table 6: Bracket Data ............................................................................................................................30
Table 7: Wing Skin Data.........................................................................................................................31
Table 8: Area Moment of Inertia Results................................................................................................31
Table 9 ..................................................................................................................................................49
Table 10 ................................................................................................................................................49
Table 11 ................................................................................................................................................50
Table 12 ..................................................................................................... Error! Bookmark not defined.

1. INTRODUCTION
1.1 Project Description
The objective of the project is to design and analyze a wing structure for a single engine utility
Cessna 177 Cardinal aircraft (Figure 1). This structure will be certified strictly under Federal Aviation
Regulations Part 23 (FAR 23). The wing is straight with zero taper.
Figure 1. The Cessna 177B Cardinal Aircraft
In designing the wing, simulation software such as X-Foil, MATLAB, COMSOL and SolidWorks were
utilized. The data generated from MATLAB code was cross verified with manual calculations.
1.2 Assumptions
The wing was considered to be a rectangular wing with zero taper i.e. taper ratio of one. This
greatly simplifies the design which otherwise would demand complex CFD analysis to determine the
actual flow parameters (Figure 2).

Figure 2. Rectangular Planform of the wing
NACA 2415 airfoil was assumed to span throughout the span of the wing. The Oswald efficiency factor
was assumed to be 0.79 for the design.
1.3 Parameters
The following parameters were chosen for the wing design (Table 1).
Table 1. Chosen Design Parameters
Parameter Description Parameter
Airfoil NACA 2415
Maximum Gross Weight 1100 kg
Standard Empty Weight 680 kg
Cruise Speed 230 km/h
Maneuvering Speed (For XFOIL) 250 km/h
Wing Span 10.82 m
Chord 1.5 m
Oswald Efficiency 0.79
Service Ceiling 4450 m

For the XFoil calculations and plots, the Reynold’s number was calculated for both Sea level and the
Service Ceiling using equation (1).
𝑅𝑒 =
𝜌𝑉𝐿
𝜇
(1)
Where,
 Re = Reynold’s number
 ρ = Density of air (Sea level = 1.225 kg/m3
, Service ceiling= 0.78 kg/m3
)
 V = Aircraft Velocity = 69.44 m/s
 L = Chord Length = 1.5 m
 µ = Dynamic Viscosity (Sea level = 1.983 x 10-5
Pa-s, Service Ceiling =1.65 x 10-5
Pa-s )
 Re = 6.430 x 106
at Sea Level
 Re = 4.924 x 106
at Service Ceiling
1.4 Gantt Chart
Before commencing the design project, a timeline was scheduled which was strictly followed to
arrive at the final design. As shown in the Gantt Chart (Figure 3), the tasks were divided into specific
time intervals to ensure smooth and timely completion of the report.

Figure 3. Gantt Chart showing schedule of tasks
2. Wing Loading
In Aerodynamics wing loading is the loaded weight of the aircraft divided by the area of the wing.
3. Maneuver and Loading during Banked Turns
When the aircraft wants to make a turn, it can normally do this in two ways. First, the pilot can deflect
the vertical stabilizer to cause a yaw to happen which then turns the nose of the aircraft in the desired
direction. The second way, which is more useful to know, is to bank the aircraft away from the desired
direction to create a centripetal force by the oblique component of the lift. This is achieved by actuating
30-Mar 09-Apr 19-Apr 29-Apr 09-May 19-May 29-May 08-Jun
X-Foil Tutorial
Generate Xfoil Data
Plotting V-n diagram
Plotting Load Distribution on Wing
Manual calculations of centroid
Manual calculations of MOI
MATLAB Calculations of centroid
MATLAB Calculations of MOI
Shear Force calculations
Bending Moment Calculations
Deflection calculation
Plotting actual Airfoil profile for NACA 2412 airfoil
Calculation of Centroid for Actual Wing section
Calculation of Area Moment of Inertia for Actual Wing section
Shear Flow Calculations
Plotting Shear Flow Results
Validaton of Shear Flow code
Failure and Safety
Stringer Buckling Calculations
Skin Bucking Calculations
Fracture Calculations
Fatigue Calculations
3D Modeling
Finite Element Analysis
Modal Analysis
Compiling the report
Final changes to the report based on feedback

the ailerons in opposite directions to cause a partial roll. When this turn happens, the vertical
component of the lift should still equal the weight of the aircraft for the aircraft to maintain its altitude.
Figure 4 Vector diagram showing the forces acting on a fixed-wing aircraft during a banked turn.
Because centripetal acceleration is given by Equation (1.a) , newton's second law in the horizontal
direction can be expressed mathematically by Equation (1.b).
𝑎 =
𝑣2
𝑟
(1.a)
𝐿𝑠𝑖𝑛 θ =
𝑚𝑣2
𝑟
(1.b)
Where,
L is the lift acting on the aircraft
θ is the angle of bank of the aircraft
m is the mass of the aircraft

v is the true airspeed of the aircraft
r is the radius of the turn
Also, the Vertical component of the lift should be equal to the weight of the aircraft, according to
Equation (1.c).
𝐿 cos 𝛳 = 𝑚𝑔
(1.c)
4. Wing Loading during Landing
During landing, the wing loading decreases. This is because while landing the aircraft, the pilot reduces
the velocity of the aircraft, thereby reducing the lift. Since wing loading is nothing but the lift divided by
the wing area, the wing loading is small while landing compared to when it is taking off or cruising.
Effectively, the wing loading is greatest during takeoff and at a minimum while landing.
5. Xfoil
5.1 X-Foil Tutorial
X-Foil was used to analyze the aerodynamic properties of the airfoil. The NACA 2415 airfoil was
chosen and different operations were performed including plotting the viscous flow profile over the
airfoil, plotting the Cp contour over the chord length (Figure 5) and generating the performance data ( CL,
CD, CM ) versus the Angle of Attack (Figure 6).

Figure 5. Cp contour over Airfoil at A.o.A of 5 deg on XFoil for NACA 2415
Figure 6. Cp contour over Airfoil at A.o.A of 0 to 20 degrees on XFoil for NACA 2415

5.2 Generating the .pol File
The NACA 2415 airfoil was then analyzed using XFoil with the objective of generating the dot pol
file for the purpose of developing the V-n diagram.
The ‘ASEQ’ command was used to generate the CL, CD and CM plots verses the A.o.A. The A.o.A is by
default chosen to be in degrees. The range of the A.o.A was set as - 20 ͦto + 20 ͦ. Subsequently, the .pol
file was generated using the ‘hard’ and the ‘PACC’ commands. For this project, to determine the 𝐶𝑙 𝛼
, the
units of the A.o.A was converted from degrees to radians. Thereby the CL values in the A.o.A range of -
7 ͦ to + 10 ͦ, were selected to determine the approximate 𝐶𝑙 𝛼
of the plot. This 𝐶𝑙 𝛼
would be used later
on, to obtain the 𝐶𝐿 𝛼
, which is essential in developing the positive and negative stall curve equations.
6. Xfoil, V-n Diagram, Loading
6.1 Interpreting the Xfoil Output
The Xfoil lift coefficient curve is shown in Figure 7 as the 2D curve. It is important to note that the
plot takes the shape of an “S” curve and actually turns around at either end. This indicates that the
curve has a relative maximum and a relative minimum. A 2D slope can be determined by examining the
linear portion of the curve and finding a best-fit line (shown in Figure 7). With the 2D slope, a conversion
to a 3D slope (𝐶𝐿 𝛼
) can be generated using formula (2),
𝐶𝐿 𝛼
=
𝐶𝑙 𝛼
(1 +
𝐶𝑙 𝛼
𝜋𝐴𝑒
)
(2)
Where A is the aspect ratio, e is the Oswald efficiency, and 𝐶𝑙 𝛼
is the 2D slope.
Each of the CL points from the 2D curve can be adjusted vertically by the ratio of the slopes by simply
applying equation (3) to each point,

𝐶𝐿 𝛼
= 𝐶𝑙 𝛼
𝐶𝐿 𝛼
𝐶𝑙 𝛼
(3)
The result is an adjusted curve that follows a similar path to the 2D curve but at a slightly lower overall
slope (see Figure 7).
Figure 7: Lift Coefficient at Sea Level
The same relationships can be seen at cruising altitude (see Figure 8).

Figure 8: Lift Coefficient at Cruising Altitude
6.2 The Stall Curve
The maximum and minimum points of the 2D curve correlate to maximum and minimum angle of
attack before the plane stalls. Converting these values from the 2D curve to the 3D curve and inputting
them into the stall curve equation provides the maximum load factor (n) for a stall condition as a
function of velocity through Equation (4),
𝑛 =
𝜌𝑆𝐶𝐿,𝑚𝑎𝑥
2𝑊
𝑉2
(4)
Where ρ is the air density, S is the wing area, V is velocity, and W is the weight of the plane. The curve
can be applied in the positive direction for positive load factors as shown or in the negative direction by
substituting CL, min for CL, max.

6.3 Maneuver Limits
For the positive stall curve, stalling is the limiting factor in determining the load on the plane until
the load factor reaches the design limit load factor of 4.4 (per FAR 23). At this point, the limiting factor
becomes the 4.4 for all remaining velocities. The same holds true for the negative stall curve, except the
design limit in the negative direction is -1.76 (per FAR 23).
Figure 9 shows the load factor over the full range of velocities from zero to the dive velocity. The plot
shows the combination of the stall curve limited portion as well as the design load limited portion in
both the positive and negative directions.
Figure 9: V-n Curve at Sea Level and Cruising Altitude
6.4 Stall Velocities
In addition to the stall curves, maneuver limits, and stall velocities, loads from gusts are
considered in the design of the wing. This is the speed at which the plane will stall regardless of angle of
attack. For a positive load factor, the stall speed varies with the lift coefficient through Equation (5).

𝑉𝑆,𝑃𝑜𝑠 = √
2𝑊
𝐶𝐿𝑚𝑎𝑥 𝜌𝑆
(5)
Where,
 W is the weight of the plane,
 ρ is the air density, and
 S is the wing surface area.
The equation holds true for negative load factors except CLmax is replaced by CLmin. The equation is also
applicable for sea level and at cruising altitude. Using the appropriate CLmax, CLmin, and ρ will provide a
total of four stall velocities, a positive and negative at each altitude.
6.5 Gust Loading
In addition to the stall curves, maneuver limits, and stall velocities, loads from gusts must be
considered in the design of the wing. The gust loads are calculated using the FAR 23 specifications of 50
ft/s at cruise velocity and 25 ft/s at dive velocity. The load factor is determined at each of these
velocities and assumed to be linear between the two. Determination of the load factor at each location
is achieved through Equations 6-8:
𝑛 = 1 +
𝐾𝑔 𝑎𝑈𝑒 𝑉
498 (
𝑊
𝑆
)
(6)
𝐾𝑔 =
0.88𝜇
5.3 + 𝜇
(7)
𝜇 =
2 (
𝑊
𝑆
)
𝜌𝑐𝑎𝑔
(8)

Where a is the previously determined slope of the 3D lift curve (rad-1
), Ue is the FAR 23 specified gust
velocity (ft/s), V is the aircraft velocity (ft/s), W is the weight of the plane (lbf), S is the wing area (ft2
), ρ
is the air density (slug/ft3
), c is the chord length of the plane (ft), and g is the gravitational constant
(ft/s2
).
With the positive load factors at cruise velocity and dive velocity calculated, the third and final
point used in plotting the gust load profile is based on the fact that the gust load factor is 1 when the
velocity is equivalent to zero. The triangle formed by these three points results in the positive half of the
gust profile. The negative portion is found by simply mirroring the positive gust load about the n=1 line.
The resulting gust profile is plotted on top of the maneuver profile as seen in Figure 10. The envelope of
the possible load cases is defined on the left portion of the plot by the stall velocity. The upper portion
of the envelope is controlled initially by the stall curve until the FAR requirement of 4.4 is reached. For
all successive velocities, the max loading is determined by the larger of the 4.4 requirement and the gust
loading. The result is shown by the dashed green line in Figure 10.
Figure 10: V-n Diagram with Gust Loading and Final Load Envelope

6.6 Determining Critical Points
From the combined loading, a few critical points can be determined from the extreme points
along the curve. Figure 11 shows the 5 points for both the sea level and cruising altitude critical points. It
is significant to note that there may be a wide range of load cases depending on the aircraft, however
our aircraft has a total of 5 critical points with 2 of them in the negative region being defined by the gust
loading.
Figure 11: Critical Points from the V-n Diagrams
Table 2 shows the details of each of the critical points.
Table 2: Critical Points

7. Wing Loading at Critical Points
7.1 Lift and Drag
With the critical points identified, it is possible to calculate the lift and drag forces along the
length of the wing. First the total lift is calculated using the load factor (n) and the weight (W) by,
𝐿 = 𝑛𝑊
(9)
The total lift value is used to calculate both a rectangular and elliptical distribution for lift along the
length of the wing. The rectangular distribution is simply the total lift divided by the length or half span
(b) as follows,
𝐿( 𝑧) =
𝐿
𝑏
(10)
The elliptical distribution varies with the length such that the lift is at its peak at the half span and is zero
at either of the wing tips. This relationship is defined by equation 11,
𝐿( 𝑧) =
4𝐿
𝜋𝑏
√1 − (
2𝑧
𝑏
)
2
(11)
The rectangular and elliptical distributions are averaged point by point to get a final lift distribution as a
function of location along the wing.
Using the total lift and the standard lift equation, the coefficient of lift can be calculated as,
𝐶𝐿 =
2𝐿
𝜌𝑉2 𝑆
(12)
Where ρ is the air density, V is the velocity at the given critical point, and S is the wing area.

The angle of attack can be determined by interpolating the 3D lift curve shown in figure X for coefficient
of lift calculated above.
Similarly, the total drag is first calculated using the relationship,
𝐷 =
𝜌𝑉2
𝑆
2
(𝐶 𝐷,0 +
𝐶𝐿
2
𝜋𝐴𝑒
)
(13)
Where ρ is the air density, V is the velocity at the given critical point, CD,0 is the zero lift drag coefficient
(determined from the Xfoil output), A is the aspect ratio, and e is the Oswald efficiency.
As with the lift, the drag is initially assumed to be rectangularly distributed expressed by
equation 14,
𝐷( 𝑧) =
𝐷
𝑏
(14)
An exception is made to account for additional drag due to wingtip vortices. These are assumed to
increase the drag by 10% over the final 20% of the half span.
The result is the lift and drag profile shown in Figure 12. For the convention used in the
equations above, the z dimension is defined with zero being the center of the plane and increasing z
moving outward along the wing length.

Figure 12: Lift and Drag Force along the Length of the Wing
7.2 Aircraft Coordinate Transformation
The lift and drag calculations provide the distributed loads across the length of the wing, but are
centered in the inertial coordinate system. For each of the critical analysis points, the aircraft is at some
angle of attack which causes a difference between the inertial and aircraft coordinates (see Figure 13).

Figure 13: Inertial Lift and Drag Components and Aircraft X-Y Components
The result is that the lift and drag must be rotated into the aircraft coordinate system. This is
accomplished by the standard equations,
𝑤𝑥( 𝑧) = − L(z)sin 𝛼 + D(z)cos 𝛼
(15)
𝑤 𝑦( 𝑧) = 𝐿( 𝑧) cos 𝛼 + 𝐷(𝑧) sin 𝛼
(16)
These equations provide the load conditions shown in Figure 14 at PHAA and sea level.

Figure 14: Load Conditions at PHAA
This process is repeated for each of the critical points providing the total Lift Curves shown in Figure 15
and the wing loads shown in Figure 16 and Figure 17.
Figure 15: Lift Force for each Critical Point

Figure 16: wx for each critical point
Figure 17: wy for each critical point

8. Centroid & Area Moments of Inertia
8.1 Cantilever Beam Model
The calculations for centroids and moments of inertia in relation to our wing design are all based
on the underlying assumption for the shape of the wing. We began our analysis by taking a simplified
cross section of the wing (Figure 18). The design assumes a uniform cross section from root to tip. All
parts of the wing are also assumed to be composed of straight segments. The cross section is
approximated using three spars, eight brackets and four skin panels.
Table 3 Dimensions used in centroid & moment calculations
VARIABLE DIMENSIONS
Chord Length 1.5 m
Height 0.2235 m
SparThickness 0.0025 m
SkinThickness 0.001016 m
Spar Height 0.999797 m
Bracket Height 0.012 m
BracketThickness 0.0025 m
Theta 30 deg
Alpha 10 deg
Half Chord Length [m] 0.75
Half Height [m] 0.11175
Half SparThickness [m] 0.00125
Half Spar Height [m] 0.4998985
Half BracketThickness [m] 0.00125
Theta [rad] 0.523598333

8.2 Centroid Calculations
Part of our project involved carrying out hand calculations to corroborate the results of our
MATLAB code. For the centroid calculation, many of the specifications were assumed to be the same as
NACA 2412. Among them were: spar thickness (.0025 meters), skin thickness (.001016 meters) and the
area of the bracket (.0003 square meters). Other data was updated where applicable, such as chord
length (1.5 meters). We were given freedom over the location of the spars, and chose initial locations of
0 [m], 0.75 [m] and 1.5 [m]. Another assumption that we made was to assume that the brackets are
point masses located at the joint of the spar and the skin. In other words, the bracket height is
neglected. The bracket height is only used to calculate the area of the bracket in the cross section.
Figure 18: Simplified Wing Cross Section
In general, the centroid is calculated from an area-weighted average of the centroid of each of the
individual components. In our case this consists of 15 individual components. The MATLAB code
provided additional formulas to work with, some of which provided helpful assumptions in the
calculations. The x-coordinate of the centroid for the spars is set at their position coordinate on the
wing. The y-coordinate of the centroid for the spars is set to zero for root spar and middle spar, while
the spar out at the wing tip is calculated using trigonometry. This equation is given by:

−( 𝑥3 − 𝑥2) ∗ tan(𝜃)
(17)
where x3 and x2 are the x-coordinate locations of the second and third spar, and theta is the angle of
the trailing edge in radians (in degrees theta is 30, in radians 0.524).
To calculate the centroid of the entire wing, we need to take an area-weighted average of the
locations of the centroid of all 15 components. The formula can be expressed as,
𝐶 𝑦 =
∑ 𝐴 𝑛 𝐶 𝑦 𝑛𝑛
∑ 𝐴 𝑛
(18)
The calculated values for the centroid are shown in Table 4. These numbers were verified using
MATLAB, Microsoft Excel, as well as hand calculations. As one can see in Figure 18, the centroid lies
outside of the wing in the negative y direction.
Table 4: Centroid Results
Resultant Centroid of Wing Section Location (m)
Cx 0.7670
Cy -0.1248
The area moment of inertia calculation makes use of the centroid data and the parallel axis theorem,
shown in equations 19-21:
I 𝑥𝑥 = ∑ 𝐴𝑖 ∗ (
𝑛
𝑖=1
𝑦𝑖 − 𝑦̅)2
(19)
I 𝑦𝑦 = ∑ 𝐴𝑖 ∗ (
𝑛
𝑖=1
𝑥𝑖 − 𝑥̅)2
(20)

I 𝑥𝑦 = ∑ 𝐴𝑖 ∗ (
𝑛
𝑖=1
𝑥𝑖 − 𝑥̅) ∗ (𝑦𝑖 − 𝑦̅)
(21)
We used the formulas provided to calculate the area moment of inertia for the leading edge of the wing.
The values of Ixx, Iyy and Ixy for the entire wing are found by summing the values for the individual
components. The data used in the calculations for the three spars is shown in Table 5, the data for the
brackets in
Table 6 and the data for the wing’s skin in
. The net values for the leading edge of the wing are displayed in
Table 8.
Table 5: Spar Data
SPARS (3)
_
x
[m]
_
y
[m]
Area [m
2
]
Ixx
[m
4
]
Iyy
[m
4
]
Ixy
[m
4
]
1 0 0 0.00055875 0.00001103 0.00032871 -0.00005348
2 0.75 0 0.00055875 0.00001103 0.00000016 -0.00000119
3 1.5 -0.433 0.00055875 0.00005540 0.00030021 -0.00012623
Table 6: Bracket Data
BRACKETS (8)
_
x
[m]
_
y
[m]
Area [m
2
]
Ixx
[m
4
]
Iyy
[m
4
]
Ixy
[m
4
]
SKIN (4)
_
x
[m]
_
y
[m]
Area [m
2
]
Ixx
[m
4
]
Iyy
[m
4
]
Ixy
[m
4
]
1 top 0.375 0.1118 0.000762 4.26E-05 0.00015 -7.066E-05
1 bottom 0.375 -0.1118 0.000762 1.3E-07 0.00015 -3.896E-06
2 top 1.125 -0.1048 0.00087988 1.41E-05 0.000154 -1.75E-05
2 bottom 1.125 -0.3283 0.00087988 5.02E-05 0.000154 -8.79E-05

1 top 0.0013 0.1118 0.00003 1.679E-06 1.76E-05 -5.4E-06
1 bottom 0.0013 -0.1118 0.00003 5.103E-09 1.76E-05 -3E-07
2 top 0.74875 0.1118 0.00003 1.679E-06 9.99E-09 -1.3E-07
2 bottom 0.74875 -0.1118 0.00003 5.103E-09 9.99E-09 -7.1E-09
3 top 0.75125 0.1118 0.00003 1.679E-06 7.44E-09 -1.1E-07
3 bottom 0.75125 -0.1118 0.00003 5.103E-09 7.44E-09 -6.2E-09
4 top 1.49875 -0.3213 0.00003 1.158E-06 1.61E-05 -4.3E-06
4 bottom 1.49875 -0.5448 0.00003 5.291E-06 1.61E-05 -9.2E-06
Table 7: Wing Skin Data
Table 8: Area Moment of Inertia Results
Net Area Moment of Inertia [m
4
]
Ixx 0.00019600
Iyy 0.00131007
Ixy -0.00038037
9. Shear Force, Bending, Deflection
This section of the report will go through the steps from load to deflection. Using the load intensity
equations calculated for wx and wy. The equation for shear force in the y-direction is found through a
SKIN (4)
_
x
[m]
_
y
[m]
Area [m
2
]
Ixx
[m
4
]
Iyy
[m
4
]
Ixy
[m
4
]
1 top 0.375 0.1118 0.000762 4.26E-05 0.00015 -7.066E-05
1 bottom 0.375 -0.1118 0.000762 1.3E-07 0.00015 -3.896E-06
2 top 1.125 -0.1048 0.00087988 1.41E-05 0.000154 -1.75E-05
2 bottom 1.125 -0.3283 0.00087988 5.02E-05 0.000154 -8.79E-05

force balance equation. The result of the force balance equation is the shear force equation, which is the
negative integral of the load intensity functions. The equations for the shear forces are given by:
𝑆 𝑦 = −∫ 𝑤 𝑦 𝑑𝑧
(22)
𝑆 𝑥 = −∫ 𝑤𝑥 𝑑𝑧
(23)
The plots of the shear forces Sx and Sy are shown in Figure 19 and Figure 20 (below).
Figure 19: Shear Force (Sx)

Figure 20: Shear Force (Sy)

The moments Mx and My were calculated by taking the integral of the shear force equations.
𝑀𝑥 = ∫ 𝑆 𝑦 𝑑𝑧 (24)
𝑀 𝑦 = ∫ 𝑆 𝑥 𝑑𝑧 (25)
The proof of this result is shown by a moment equilibrium. Plots of the moments at sea level and
service ceiling are shown in Figure 21 and Figure 22.
Figure 21: Mx vs Wing Span

Figure 22: My vs Wing Span
The equations for u’’ and v’’ are given by equations 26 and 27:
𝑑2 𝑢
𝑑𝑧2 = −𝐾 ∗ [−𝑀𝑥 ∗ 𝐼𝑥𝑦 + 𝑀 𝑦 ∗ 𝐼𝑥𝑥] (26)
𝑑2 𝑣
𝑑𝑧2 = −𝐾 ∗ [𝑀𝑥 ∗ 𝐼 𝑦𝑦 − 𝑀 𝑦 ∗ 𝐼𝑥𝑦] (27)
𝐾 =
1
𝐸∗(𝐼 𝑥𝑥∗𝐼 𝑦𝑦−𝐼 𝑥𝑦
2 )
(28)
The deflections due to bending are found by integrating u’’ and v’’ twice. The constants of integration go
to zero because we integrate from the LHS. Ultimately, we compare the deflections at the tip by
executing numerical integration in MATLAB with equations 29 and 30:
𝑢( 𝑧) = ∑
𝑑𝑢
𝑑𝑧
∗ (𝑧𝑖+1 − 𝑧𝑖) (29)
𝑣( 𝑧) = ∑
𝑑𝑣
𝑑𝑧
∗ (𝑧𝑖+1 − 𝑧𝑖) (30)

The plots for deflection are shown below in Figure 23 and Figure 24.
Figure 23: Deflection (u)

Figure 24: Deflection (v)

The direct stress is also plotted versus Z. This is accomplished by taking the given equation for direct
stress and populating the direct stress matrix point by point in MATLAB. The resulting plot for direct
stress is shown in Figure 25.
Figure 25: Direct Stress (sigma Z)

10. Stresses
The following coordinate system was used for the airfoil:
x – Along the chord of the airfoil
y – The span-wise coordinate from the tip to the root
z – Axis perpendicular to the Airfoil Chord
10.1 Airfoil Profile
To do this part of the calculation, the airfoil was split into approximately 801 skin segments, that is,
there are skin elements for every 0.03m along the chord of the airfoil.
To plot the profile of the airfoil, the Equation for a camber 4-digit NACA Airfoil was obtained from online
sources and implemented in the MATLAB code. The following formula was used:
𝑦𝑐 = {
𝑚
𝑥
𝑝2
(2𝑝 −
𝑥
𝑐
) , 0 ≤ 𝑥 ≤ 𝑝𝑐
𝑚
(𝑐−𝑥)
(1−𝑝)2
(1 +
𝑥
𝑐
− 2𝑝) , 𝑝𝑐 ≤ 𝑥 < 𝑐
(31)
where:
 m is the maximum camber (100 m is the first of the four digits),
 p is the location of maximum camber (10 p is the second digit in the NACA xxxx description).
For this cambered airfoil, because the thickness needs to be applied perpendicular to the camber line,
the coordinates (𝑥 𝑈, 𝑦 𝑈) and (𝑥 𝐿, 𝑦 𝐿), of respectively the upper and lower airfoil surface, become:
𝑥 𝑈 = 𝑥 − 𝑦𝑡 𝑠𝑖𝑛𝜃, 𝑦 𝑈 = 𝑦𝑐 + 𝑦𝑡 𝑐𝑜𝑠𝜃 (32)
𝑥 𝐿 = 𝑥 + 𝑦𝑡 𝑠𝑖𝑛𝜃, 𝑦 𝐿 = 𝑦𝑐 − 𝑦𝑡 𝑐𝑜𝑠𝜃 (33)
Where,
𝜃 = arctan (
𝑑𝑦 𝑐
𝑑𝑥
) (34)
𝑑𝑦 𝑐
𝑑𝑥
= {
2𝑚
𝑝2
(𝑝 −
𝑥
𝑐
) , 0 ≤ 𝑥 ≤ 𝑝𝑐
2𝑚
(1−𝑝)2
(𝑝 −
𝑥
𝑐
), 𝑝𝑐 ≤ 𝑥 ≤ 𝑐
(35)

In our case, we had the NACA 2412 Airfoil which had the following specifications :
m = 2 % (1st digit - maximum camber (m) in percentage of the chord )
p = 40 % (Second digit - position of the maximum camber (p) in tenths of chord )
t = 0.15 (last 2 digits - maximum thickness (t) of the airfoil in percentage of chord)
Figure 26 NACA 2412 Arifoil Profile
While plotting the airfoil profile on MATLAB, the rear 20 % of the chord length was truncated to make
space for control surfaces. A spar was placed at the rear end which came out to be 1.2 m from the nose
of the airfoil given the chord length of 1.5 m.
10.2 Section Centroid Calculations
For a more accurate sizing of the wing the section centroid was computed for the actual wing
section. This is performed by splitting the skin into smaller rectangular sections. The area of these
elements is calculated and then attributed to point areas whose distances from the axes origin is the
same as that of centroid of the rectangles. Sum of these distances weighted on the areas yields the
centroid location.
The centroid calculations were done and all the cross section components were included in the
calculation, as shown in Figure 27:

Figure 27 Truncated Airfoil
The following formula was used for centroid calculations:
𝐶 𝑥 =
∑ 𝐴 𝑖 𝑥 𝑖
𝑛
1
∑ 𝐴 𝑖
𝑛
1
𝑎𝑛𝑑 𝐶 𝑦 =
∑ 𝐴 𝑖 𝑦 𝑖
𝑛
1
∑ 𝐴 𝑖
𝑛
1
(36)
Where,
(𝐶𝑥 , 𝐶 𝑦) is the centroid location of the airfoil
𝐴𝑖 is the area of the ith
component
𝑥𝑖 is the x-coordinate of the ith
component
𝑦𝑖 is the y-coordinate of the ith
component
10.3 Moment of Inertia Calculations
Similar to the Centroid calculations the moment of inertia was also recalculated for the actual
wing section. Since each small section of area was considered to be a point area, the actual moment of
inertia of that section is negligible. However, there is a contribution to the overall moment of inertia
from each of these sections when the axis is moved from the one passing through the centroid of the
elements to the centroid of the actual wing section. Parallel axis theorem is used to calculate this
contribution, which is given by Equations (37) and (38).
𝐼𝑥 = ∑ 𝐼 𝑥′,𝑖 + 𝐴𝑖(𝐶 𝑦 − 𝑦𝑖)
2𝑛
1 (37)
𝐼 𝑦 = ∑ 𝐼 𝑦′,𝑖 + 𝐴𝑖( 𝐶𝑥 − 𝑥𝑖 )2𝑛
1 (38)
Where,

𝐼 𝑥′,𝑖 is the Area MOI of the ith
component about its own centroid about the x-axis (Negligible)
𝐼 𝑦′,𝑖 is the Area MOI of the ith
component about its own centroid about the y-axis (Negligible)
11. Shear Flow – Calculations
Using the calculated x-y coordinates to define the profile of the wing and the associated areas at each
point for spar caps and stringers, shear flow throughout the structure can be determined. Ideally, the
coordinate system would be centered at the centroid of the cross section, but this can be corrected for
through a simple coordinate transfer. Each point will be treated as a boom where the area of the boom
corresponds to the effective area of the skin, stringers, or spar caps adjacent to that point. This is
calculated such that axial stress is conserved between the actual cross section and the simplified boom
cross section. The final equation is shown in Equation (39).
𝐴 𝐵𝑜𝑜𝑚 = 𝐴 𝑠𝑝𝑎𝑟 𝑐𝑎𝑝 𝑜𝑟 𝑠𝑡𝑟𝑖𝑛𝑔𝑒𝑟 + ∑ [
𝑡𝑏
6
(2 +
𝜎 𝑧𝑧(𝑛+1)
𝜎 𝑧𝑧(𝑛)
)]𝑛
𝑖=1 (39)
Where the area of the spar cap or stringer are only for that specific point, n is the number of adjacent
panels, t is the thickness of that adjacent panel, b is the length of the adjacent panel, σzz(n+1) is the stress
at the point on the other end of the adjacent panel, and σzz is the stress at the current point. Typically n
is equal to 2, one for each of the adjacent skin panels. However at the locations where the spar meets
the skin, n is equivalent to 3 to include the contributions due to the spar. σzz in this equation is calculated
using the moment in the X and Y directions and implemented into Equation (40).
𝜎𝑧𝑧,𝑖 =
𝑀 𝑥(𝐼 𝑦𝑦(𝑦 𝑖−𝑦 𝑐)−𝐼 𝑥𝑦(𝑥 𝑖−𝑥 𝑐))
𝐼 𝑥𝑥 𝐼 𝑦𝑦−𝐼 𝑥𝑦
2 +
𝑀 𝑦(𝐼 𝑥𝑥(𝑥 𝑖−𝑥 𝑐)−𝐼 𝑥𝑦(𝑦 𝑖−𝑦 𝑐))
2 (40)
Where the moments of inertia are the previously calculated inertia terms for the wing cross section, x
and y are the coordinates of the point, and xc and yc are the x-y coordinates of the centroid if the points
are not already normalized.

To solve for shear flow, q, the cross section of the wing must be divided into two sections separated by
the spar cap. Figure 28 shows a sample airfoil cross section divided into two cells. The area for each of
these cells can be found by taking the cross product of consecutive points and dividing by 2 as shown in
Equation (40.a).
Figure 28 Solving the two cell problem
2𝐴 = ( 𝑥1, 𝑦1, 𝑧1) × ( 𝑥2, 𝑦2, 𝑧2)
(40.a)
To solve for the total shear flow, q, at each point the wing is first cut at the spar and the back plate to
make a singular open section as shown in Figure 28. By creating an open section, the qb at the open end,
between points 41 and 1 in Figure 28 is known to be 0. Having previously simplified the wing section
into boom areas, the assumption is made that the skin or spar panels are of infinitely small skin
thicknesses. The result is that the shear stress varies from one side of the boom to the other, but
remains constant throughout the length of skin. This concept is depicted by Figure 29.

Figure 29 Shear Flow Idealization
Starting with point 1, the Δqb, or change in sheer flow from one side of the boom to the other is
calculated at each point around the profile using Equation (41).
Δ𝑞 𝑏,𝑖 =
𝑆 𝑦 𝐼 𝑥𝑦−𝑆 𝑥 𝐼 𝑥𝑥
2 𝐴 𝐵𝑜𝑜𝑚,𝑖( 𝑥𝑖 − 𝑥 𝑐) +
𝑆 𝑥 𝐼 𝑥𝑦−𝑆 𝑦 𝐼 𝑦𝑦
2 𝐴 𝐵𝑜𝑜𝑚,𝑖( 𝑦𝑖 − 𝑦𝑐) (41)
Where Sx and Sy are the shear forces, ABoom,I is the effective boom area at that point, xi and yi are the x-y
components of the point, and xc and yc are the centroid coordinates if not normalized. From this the
total qb at a given point is the sum of the Δqb’s for all previous points as shown in equation X. The values
of qb are shown in the Appendix.
𝑞 𝑏,𝑘 = ∑ Δ𝑞 𝑏,𝑖
𝑘
𝑖=1 (42)
To calculate the total q, the relationship between cell 1 and cell 2 must be accounted for. This is
considered by examining the total moment on the system in Equation (43) and the twist rate of each cell
using Equations (44) and (45).
𝑀0 + 𝑆 𝑦 𝜉0 − 𝑆 𝑥 𝜂0 = 2𝐴1 𝑞0,1 + 2𝐴2 𝑞0,2 + ∑ 2𝑞 𝑏,𝑖Δ𝐴𝑖
𝑛
𝑖=1 (43)
𝑑𝜃
𝑑𝑧
=
1
2𝐴1 𝐺
[𝑞0,1 (∑
((𝑥 𝑖+1−𝑥 𝑖)2+(𝑦 𝑖+1−𝑦 𝑖)2)
1
2
𝑡 𝑠𝑘𝑖𝑛
𝑛
𝑖=1 ) + (𝑞0,1 − 𝑞0,2)
(𝑦 𝑏𝑜𝑡−𝑦𝑡𝑜𝑝)
𝑡 𝑠𝑝𝑎𝑟
+
(𝑆 𝑦 𝐼 𝑥𝑦−𝑆 𝑥 𝐼 𝑥𝑥)
2 ∑ 𝐴 𝑏𝑜𝑜𝑚,𝑖 𝑥𝑖
((𝑥 𝑖+1−𝑥 𝑖)2+(𝑦 𝑖+1−𝑦 𝑖)2)
1
2
𝑛
𝑖=1 +
(𝑆 𝑥 𝐼 𝑥𝑦−𝑆 𝑦 𝐼 𝑦𝑦)
2 ∑ 𝐴 𝑏𝑜𝑜𝑚,𝑖 𝑦𝑖
((𝑥 𝑖+1−𝑥 𝑖)2+(𝑦 𝑖+1−𝑦 𝑖)2)
1
2
𝑛
𝑖=1 ]
(44)

𝑑𝜃
𝑑𝑧
=
1
2𝐴2 𝐺
[𝑞0,2 (∑
((𝑥 𝑖+1−𝑥 𝑖)2+(𝑦 𝑖+1−𝑦 𝑖)2)
1
2
𝑛
𝑖=1 ) + (𝑞0,2 − 𝑞0,1)
(𝑦𝑡𝑜𝑝−𝑦 𝑏𝑜𝑡)
+ 𝑞0,2
(𝑦𝑡𝑜𝑝−𝑦 𝑏𝑜𝑡)
+
(𝑆 𝑦 𝐼 𝑥𝑦−𝑆 𝑥 𝐼 𝑥𝑥)
2 ∑ 𝐴 𝑏𝑜𝑜𝑚,𝑖 𝑥𝑖
((𝑥 𝑖+1−𝑥 𝑖)2+(𝑦 𝑖+1−𝑦 𝑖)2)
1
2
𝑛
𝑖=1 +
(𝑆 𝑥 𝐼 𝑥𝑦−𝑆 𝑦 𝐼 𝑦𝑦)
2 ∑ 𝐴 𝑏𝑜𝑜𝑚,𝑖 𝑦𝑖
((𝑥 𝑖+1−𝑥 𝑖)2+(𝑦 𝑖+1−𝑦 𝑖)2)
1
2
𝑛
𝑖=1 ]
(45)
It can be assumed that the wing cross section behaves as a rigid body and thus the twist rates are
equivalent. This leaves three equations with three unknowns; dθ/dz, q0,1, and q0,2. Solving the system of
equations gives the q0,1 and q0,2 to complete Equation (46).
𝑞𝑡𝑜𝑡𝑎𝑙,𝑖 = 𝑞 𝑏,𝑖 + 𝑞0,1 𝑜𝑟 2 (46)
Where the q0 is the appropriate term for the cell containing the point in question. This resulting shear
flow (qtotal) is in terms of the shear force per unit length. To get this into shear stress, the shear flow
must be divided by the thickness, resulting in units of shear force per unit area (Equation (47)).
𝜎𝑧𝑠,𝑖 =
𝑞 𝑡𝑜𝑡𝑎𝑙,𝑖
(47)
Combining the shear stress and the axial stress from Equation (47) at each point, the equivalent
principle stress can be calculated. Equation (48) defines this relationship.
𝜎𝑒𝑞,𝑖 = √2𝜎𝑧𝑧,𝑖
2 + 6𝜎𝑧𝑠,𝑖
2 (48)
11.1 Shear Flow – Results
The results of these calculations for the first of the 10 critical points being analyzed (PHAA at Sea Level)
can be seen in Figure 30, shear stress, and Figure 31, equivalent stress. As the legend on the right
indicates, the colors correspond to the magnitude of the stress at that location.

Figure 30 Magnitude of Stress along the Airfoil profile (Shear Stress) – PHAA at Sea Level
Figure 31 Magnitude of Stress along the Airfoil profile (Equivalent Stress) – PHAA at Sea Level

12. Shear Flow-Code Verification
Variations in the frequency of points and method of calculating shear flow may lead to slight variations
in the results shown, however a few main principles should still hold true.
First, the shear flow should return to zero for the open section (qb). Figure 32 shows that the shear flow
in fact does return to zero for each of the ten load cases considered. It is somewhat misleading to
examine the values of open section shear flow directly as they are on the order of ±1,000, but with
respect to the magnitude of the maximum values of qb this is relatively zero.
Figure 32 Open Section Shear Flow
A second check is to verify that the sum of the shear stresses times the boom areas also returns to zero.
Figure 33 shows that this principle also holds true.

Figure 33 Product of Shear Flow and Boom Area
13.Buckling, Fatigue & Von Mises Stress
13.1 Von Mises Stresses
An important test in the design of our wing was to check the yield stress against the Von Mises
yield criterion. The tensile yield strength of AL 2024 T3 is 345 MPa and the ultimate tensile
strength is 483 MPa. The Von Mises equation is given in terms of principal stresses by equation
49 below.
𝜎𝑒𝑞 = √
[( 𝜎11 − 𝜎22)2 + ( 𝜎22 − 𝜎33)2 + ( 𝜎33 − 𝜎11)2 + 6( 𝜎12
2
+ 𝜎23
2
+ 𝜎31
2 )]
2
(49)

In our analysis, equation 49 reduces to equation 50 because we only have one longitudinal
component and one shear component.
𝜎𝑒𝑞 = √
[2( 𝜎𝑧𝑧)2 + 6( 𝜎𝑧𝑠)2]
2
(50)
Table 9: Von Mises Stress Calculation
The calculation verifies that our wing design meets the Von Mises Yield Criterion with a factor of
safety of 1.4 for yield and 2.0 for fracture.
13.2 Column Buckling
We considered column buckling in our wing design to avoid stringer buckling, which is achieved
by adjusting the spacing of the ribs. The theory for the buckling analysis was worked out by Euler in
the eighteenth century. Euler’s equations for a load P applied to the ends of a beam can be seen in
the equations and Figure 34 below:
Figure 34: Column Buckling

We use two equations to solve for the rib spacing:
𝑃𝐶𝑅 =
(𝜋2
𝐸𝐼)
𝑙 𝑒
2
(51)
The variables, their values, and the units for each term are outlined in tables 10 & 11 below.
𝑃𝐶𝑅 = 1.5𝜎𝑧𝑧 𝐴 𝑠𝑡𝑟𝑖𝑛𝑔𝑒𝑟
(52)
There is an n-squared factor in equation 51 that disappears when we set n equal to one (setting n equal
to one gives the smallest value for P where the column remains in equilibrium [Megson 271]). Next, by
setting equations (51) & (52) equal to each other and solving for effective length, we obtain the results
in Table 10.
Table 10: Rib Spacing
Table 11: Stringer Area
We use the following equation (53) from Megson [Table 8.1, pg. 272] to solve for the rib spacing given
the effective length:
𝑙 𝑒
𝑙
= 2 (53)
Setting Equation 53 equal to 2 gives us the most conservative value for rib spacing, as a larger effective
length decreases the value for P critical that yields failure in the column. Given a half-span of 5.41
meters, this would suggest 10 ribs evenly spaced along the wing.

13.3 Skin (Thin Plate) Buckling
We consider thin plate buckling theory to determine the design for the skin plates that are
strengthened by the ribs and stringers. This analysis will compare the critical buckling load of the
individual plates with the total compressive plate buckling stress. The critical buckling load is given by
equation 54:
𝑁𝑋,𝐶𝑅 =
𝑘𝜋2
𝐸𝑡3
12𝑏2(1 − 𝑣2)
(54)
We set the buckling coefficient k equal to 8.5, set the Elastic Modulus for AL 2024 T3 to 73.1
[GPa], set Poisson’s ratio, v, equal to 0.33, while t and b are the skin thickness and plate length,
respectively. Next, we summed the critical buckling load for each section and divided by the skin
thickness to find the total compressive plate buckling stress.
𝜎 𝐶𝑅 =
𝑛 ∗ 𝑁𝑋,𝐶𝑅
𝑡
(55)
Here n is the number of plates, and t is the skin thickness. The calculations are shown in Table 12 below.
Table 12
13.4 Shear Buckling of Skin
In addition to column buckling and skin (thin plate) buckling, plate buckling from shear loading
of the wing’s skin was considered. The coefficient, k, used for plate buckling from shear loading is
given by Figure 35. Using the appropriate value for k, the buckling stress is calculated using
equation 54. With the calculated values for shear stress in the skin below the critical buckling
stress, the geometry of the wing design passed this test.

Figure 35: Shear Buckling Coefficients for Flat Plates (Megson 315)
13.5 Fracture & Fatigue
Fatigue is defined as the progressive deterioration of the strength of a material or structural
component during service such that failure can occur at much lower stress levels than the
ultimate stress level [Megson 455]. Fracture can also occur at stresses below the yield stress if an
initial crack is present. A small crack, if undetected, can manifest into catastrophic failure. Figure
36 below from Megson (pg. 423) illustrates the reduction in failure stress as the number of
repetitions of this stress increases.
Figure 36 Failure Stress vs Number of Loading repetitions
Here we analyze the number of flight cycles our wing can endure before a crack grows by 1.8
centimeters, given an initial crack size of 0.2 centimeters.

The equation for the number of loading/flight cycles (N) until failure is given by Equation 56:
𝑁 =
𝑎 𝐶𝑅
1−
𝑚
2
− 𝑎𝑖
1−
𝑚
2
𝐶(1 −
𝑚
2 )𝜋2 𝜎∞
2
(56)
To solve for aCR, we plug in a value for KIC (26 [MPa*m^.5]) into Equation (57)
𝑎 𝐶𝑅 =
1
𝜋
(
𝐾𝐼𝐶
𝜎∞
)2
(57)
The results are given in Error! Reference source not found.. The critical crack length is calculated as
approximately 5.4mm.
Table 13: Flight Cycles to Failure
14. Aeroelasticity
14.1 Divergence
Wing divergence is the process of an applied load increasing due to aerodynamic loads (i.e. a
positive torsional pitch moment) that eventually build and cause the wing to reach a divergence
point. The deflection due to the increasing load causes the angle of attack to increase, and the
increasing wing deflection and twist this creates can lead to failure. The torsional divergence
speed is calculated from equation 58, where U is the torsional divergence speed:
𝑀 = 𝐶𝑈2
(𝜃 + 𝛼0)
(58)

C is a coefficient, M is the moment per unit length, theta is the elastic twist of the beam, and
alpha not is the initial angle of attack. These values are related to the torsional stiffness of the
beam (GJ) by equation 59:
𝐺𝐽
𝑑2
𝜃
𝑑𝑦2
= −𝑀
(59)
When designing the wing, it is necessary that the torsional divergence speed, U, does not
exceed the maximum flight velocity.
14.2 Aileron Reversal
Aileron Reversal is defined as the process where an aircraft rolls in the opposite direction as the
aileron input, often caused by twisting of the wing. As speed increases, the wing twist caused by the
aileron reversal will also increase. The rigidity of the wing is another important factor when
considering aileron reversal, as aileron reversal will occur more easily if the wing is more susceptible
to torsion. In the design of the wing, increasing wing rigidity decreases the likelihood of aileron
reversal. We can define an aileron reversal speed, which is the speed at which the aileron deflection
fails to cause any moment about the wing – at this point the aircraft rolls in the opposite direction
and experiences aileron reversal.

15. 3D Modelling and Simulation
15.1 3D Modelling
Figure 37 Section View of the 3D wing model
A CAD model of the wing design has been developed using SolidWorks, as shown in Figure 37. This
shows an example wing configuration which is not necessarily the most optimized, but gives an idea on
how all the structural components are mounted together. The wing is a three-cell box beam and its
structural portion extends from the airfoils leading edge to 80 % of the chord length.
In the drawings appear all the structural elements: the wing skin, the spars with spar caps, the stringers
and the ribs.
15.1.1 Modelling Steps
To start with the .dat file for NACA 2415 was imported using the Insert Curve option in Solidworks. This
generated the airfoil profile. The cross section of the airfoil was then extruded with the desired features
to generate the 3D wing model.

Figure 38 Isometric view of the Entire wing
The spars are connected to the airfoil skin through L-shaped spar caps that also hold a skin panel, which
closes the third cell at the back end of the structure. The ribs are shaped to fit the internal cavity of the
wing and some holes are cut through them in order to lighten the structure. Top-hat stringers run
parallel to the spars and are connected uniquely to the skin.

16. Loading Analysis
16.1 The Setup
The testing of the wing under a set of experimental loading conditions was performed on
Solidworks. For this type of analysis, the Solidworks Simulation package was used. The Static analysis
option was selected while setting up the loading conditions. Aluminum 2024 T3 with a yield stress of
3.45 x 108
N/m2
.
The complexity of the complete CAD model was too high to execute an accurate FEA on it.
Particularly, the very small skin thickness would require an excessively fine mesh strongly increasing the
computational time. For this reason, a simpler model was designed so that reasonably accurate results
can be obtained at the cost of an acceptable level of approximation.
16.1.1 Boundary Conditions
 The root of the wing was specified to be fixed in the inertial frame.
 The tip of the wing was left free.
 A load of 200 N was uniformly applied on the bottom surface of the wing in the upward
direction.
16.2 Results
On applying an experimental load of 200 N uniformly on the bottom surface of the wing, it was observed
that the maximum stress on the wing was 4.89 x 105
N/m2
which was much lower than the yield stress
of Aluminum 2024 T3 of 3.45 x 108
N/m2
as can be seen from Error! Reference source not found..

Figure 39: Sigma_zz Distribution
With such a small value of loading, a corresponding maximum deflection of 42.66 mm was observed at
the wing-tips as can be seen from Figure 40.
Figure 40 Deflection of the wing due to the applied load

17. Modal Analysis
The modal analysis was conducted on the simplified wing model with one end fixed and assuming no lift
conditions (i.e. no wing loading). An iterative solver was used in the Frequency Study conducted on
Solidworks 2016 to generate the modal vibration data. The simulation generated the Modal Vibration
Frequency and Shape for the first five modes. This has been shown here from Figure 41 - Figure 45. The
plot of the Modal frequencies verses the Mode Number is shown in Figure 46.
Figure 41 Mode-1 Frequency-7.3987 Hz

Figure 43 Mode-3 Frequency- 42.362 Hz

Figure 45 Mode-5 Frequency- 99.968 Hz

Figure 46 Modal Frequency vs Mode Number
18. References
1) Megson, T. H. G. Aircraft Structures for Engineering Students. 5th ed. Oxford: Butterworth-
Heinemann, 2013. Print.
2) Lynch, C. S. Design of Aerospace Structures Lecture Series. UCLA Engineering: Mechanical and
Aerospace Engineering, 2016. Print.
19. Appendix
PLAA Sea Level
7.3987
32.997
42.362
55.554
99.968
0
20
40
60
80
100
120
0 1 2 3 4 5 6
ResonantFrequency(Hz)
Mode #
Modal Analysis

Gust 1 Sea Level

Gust 2 Sea Level

NHAA Sea Level

PHAA Cruise Altitude

PLAA Cruise Altitude

Gust 1 Cruise Altitude

Gust 2 Cruise Altitude

NHAA Cruise Altitude
Appendix B
%% MAE 154B
% Ryan Rader
% Kevin Landry
% Agraj Sobti
clear all
close all
clc
% Establish basic parameters for plane
b=10.82; % meters
c=1.5; % meters
e=.79;
S=b*c; % meters^2
A= b^2/S;

W=1100*9.8; % Newtons
rho0=1.225; % Kg/m^3
rhoC=.7809; % Kg/m^3
VC=230*1000/3600; % m/s
VD=VC*1.5; % m/s
V=[0:0.5:VD]; % m/s
mu=1.983e-5; % pa*sec
g=32.17; % ft/s^2
A_cap=.0003; % m^2
A_str=.0001; % m^2
t_spar=.0025; % m
t_skin= .001; % m
x_spar=c/3; % m
x_strU=[0:5:95]*1.2/100; % m
x_strL=x_strU; % m
G=27579029160; % pa for 2024 all tempers
% Calculate a Reynolds Number
Re0=rho0*VC*c/mu;
ReC=rhoC*VC*c/mu;
[alpha1,CL1,CD1,CDp1,CM1,Top_Xtr1,Bot_Xtr1]=textread('naca24150.
pol','%f %f %f %f %f %f %f','headerlines', 12);
[alpha2,CL2,CD2,CDp2,CM2,Top_Xtr2,Bot_Xtr2]=textread('naca2415C.
pol','%f %f %f %f %f %f %f','headerlines', 12);
alpha={alpha1,alpha2};
CL={CL1,CL2};
CD={CD1,CD2};
CDp={CDp1,CDp2};
CM={CM1,CM2};
Top_Xtr={Top_Xtr1,Top_Xtr2};
Bot_Xtr={Bot_Xtr1,Bot_Xtr2};
alphar{1}=deg2rad(alpha{1});
alphar{2}=deg2rad(alpha{2});
ind=find(CL{1}>0,1);
CD0(1)=CD{1}(ind);
clear ind
ind=find(CL{2}>0,1);
CD0(2)=CD{2}(ind);
clear ind

[CLmax(1),mind(1)]=max(CL{1});
[CLmin(1),mind(2)]=min(CL{1});
[CLmax(2),mind(3)]=max(CL{2});
[CLmin(2),mind(4)]=min(CL{2});
Vspos(1)=sqrt(2*W/(CLmax(1)*rho0*S));
Vsneg(1)=sqrt(2*W/(abs((1))*rho0*S));
Vspos(2)=sqrt(2*W/(CLmax(2)*rhoC*S));
Vsneg(2)=sqrt(2*W/(abs(CLmin(2))*rhoC*S));
ind1(1)=find(alpha{1}==-7);
ind2(1)=find(alpha{1}==10);
ind1(2)=find(alpha{2}==-7);
ind2(2)=find(alpha{2}==10);
linefit(1,:)=polyfit(alphar{1}(ind1(1):ind2(1)),CL{1}(ind1(1):in
d2(1)),1);
linefit(2,:)=polyfit(alphar{2}(ind1(2):ind2(2)),CL{2}(ind1(2):in
d2(2)),1);
x=linspace(-.3,.3)';
y(:,1)=linefit(1,1).*x+linefit(1,2);
xinter(:,1)=-linefit(1,2)/linefit(1,1);
y(:,2)=linefit(2,1).*x+linefit(2,2);
xinter(:,2)=-linefit(2,2)/linefit(2,1);
CL3Dslope(1,1)=linefit(1,1)/(1+linefit(1,1)/(pi*A*e));
CL3Dline(1,1)=CL3Dslope(1,1);
CL3Dline(1,2)=-CL3Dslope(1,1)*xinter(:,1);
CL3Dslope(2,1)=linefit(2,1)/(1+linefit(2,1)/(pi*A*e));
CL3Dline(2,1)=CL3Dslope(2,1);
CL3Dline(2,2)=-CL3Dslope(2,1)*xinter(:,2);
CLmax(1)=CLmax(1)*CL3Dline(1,1)/linefit(1,1);
CLmin(1)=CLmin(1)*CL3Dline(1,1)/linefit(1,1);
CLmax(2)=CLmax(2)*CL3Dline(2,1)/linefit(2,1);
CLmin(2)=CLmin(2)*CL3Dline(2,1)/linefit(2,1);
% CLmax(1)=CL3Dline(1,1)*alphar{1}(mind(1))+CL3Dline(1,2);
% CLmin(1)=CL3Dline(1,1)*alphar{1}(mind(2))+CL3Dline(1,2);
% CLmax(2)=CL3Dline(2,1)*alphar{2}(mind(3))+CL3Dline(2,2);
% CLmin(2)=CL3Dline(2,1)*alphar{2}(mind(4))+CL3Dline(2,2);
% y3D(:,1)=CL3Dline(1,1).*x+CL3Dline(1,2);
% y3D(:,2)=CL3Dline(2,1).*x+CL3Dline(2,2);
y3Dline1{1}=(CL3Dline(1,1)/linefit(1,1)).*CL{1};

y3Dline1{2}=(CL3Dline(2,1)/linefit(2,1)).*CL{2};
p1=plot(alphar{1},CL{1});
hold on
plot(x,y(:,1))
%plot(x,y3D(:,1))
plot(alphar{1},y3Dline1{1})
grid on
legend('2D CL Curve','Best Fit Line','3D CL
Curve','location','best')
xlabel('alpha (radians)')
ylabel('C_L (Coefficient of Lift)')
figure
p2=plot(alphar{2},CL{2});
hold on
plot(x,y(:,2))
% plot(x,y3D(:,1))
plot(alphar{1},y3Dline1{1})
grid on
legend('2D CL Curve','Best Fit Line','3D CL
Curve','location','best')
xlabel('alpha (radians)')
ylabel('C_L (Coefficient of Lift)')
% Calculate n curves. Column 1 is sea level and column 2 is
cruise alt.
for n=1:length(V)
npos(n,1)=.5*rho0*CLmax(1)*V(n)^2*S/W;
nneg(n,1)=.5*rho0*CLmin(1)*V(n)^2*S/W;
npos(n,2)=.5*rhoC*CLmax(2)*V(n)^2*S/W;
nneg(n,2)=.5*rhoC*CLmin(2)*V(n)^2*S/W;
end
% take indicies less than the design load of 4.4 and -1.76
nposi{1}=find(npos(:,1)<4.4);
nnegi{1}=find(nneg(:,1)>-1.76);
nposi{2}=find(npos(:,2)<4.4);
nnegi{2}=find(nneg(:,2)>-1.76);
nnegi{3}=find(V>=VC,1);
% Calculate the slope and intercept for the maneuver limit using
the points
% (VC,-1.76) and (VD,-1)
manslope=(-1-(-1.76))/(VD-VC);
manint=-(VD*manslope+1);

for j=1:2
maxpos(j)=max(nposi{j});
maxneg(j)=max(nnegi{j});
npos(maxpos(j)+1:end,j)=4.4;
nneg(maxneg(j)+1:nnegi{3},j)=-1.76;
for n=nnegi{3}:length(V)
nneg(n,j)=manslope*V(n)+manint;
end
end
npos(end+1,:)=zeros(1,2);
nneg(end+1,:)=zeros(1,2);
V(end+1)=V(end);
%% Calculate the Gust Load Factor
UeC=50; % ft/s
UeD=25; % ft/s
VCk=VC*1.944; % knots
VDk=VD*1.944; % knots
cft=c*3.28084; % ft
Wlbf=W*0.224809; % lbf
Sft2=S*10.7639; % ft^2
rho0s=rho0*0.00194032; % slug/ft^3
rhoCs=rhoC*0.00194032; % slug/ft^3
u0=2*Wlbf/Sft2/(rho0s*cft*CL3Dslope(1)*g);
uC=2*Wlbf/Sft2/(rhoCs*cft*CL3Dslope(2)*g);
Kg0=.88*u0/(5.3+u0);
KgC=.88*uC/(5.3+uC);
nC(1)=1+Kg0*CL3Dslope(1)*UeC*VCk/(498*(Wlbf/Sft2));
nC(2)=1+KgC*CL3Dslope(2)*UeC*VCk/(498*(Wlbf/Sft2));
nD(1)=1+Kg0*CL3Dslope(1)*UeD*VDk/(498*(Wlbf/Sft2));
nD(2)=1+KgC*CL3Dslope(2)*UeD*VDk/(498*(Wlbf/Sft2));
ngustposS(:,1)=[1,nC(1),nD(1),1];
ngustnegS(:,1)=[1,-(nC(1)-2),-(nD(1)-2),1];
ngustposS(:,2)=[1,nC(2),nD(2),1];
ngustnegS(:,2)=[1,-(nC(2)-2),-(nD(2)-2),1];
vgust=[0,VC,VD,0];
ngustpos(:,1)=interp1(vgust(1:3),ngustposS(1:3,1),V);
ngustneg(:,1)=interp1(vgust(1:3),ngustnegS(1:3,1),V);

ngustpos(:,2)=interp1(vgust(1:3),ngustposS(1:3,2),V);
ngustneg(:,2)=interp1(vgust(1:3),ngustnegS(1:3,2),V);
count=0;
countn=0;
for j=1:2
for n=1:length(V)
if V(n)<Vspos(j)
else
count=count+1;
Vcomp{j}(count)=V(n);
end
if V(n)>Vspos(j)&& n<=maxpos(j)
nComp{j}(count)=npos(n,j);
elseif n>maxpos(j)
nComp{j}(count)=max(npos(n,j),ngustpos(n,j));
end
if V(n)<Vsneg(j)
else
countn=countn+1;
Vcomn{j}(countn)=V(n);
end
if V(n)>Vsneg(j)&&n<=maxneg(j)
nComn{j}(countn)=nneg(n,j);
elseif n>maxneg(j)
nComn{j}(countn)=min(nneg(n,j),ngustneg(n,j));
end
end
count=0;
countn=0;
end
nComp{1}=[0,nComp{1},0];
Vcomp{1}=[Vspos(1),Vcomp{1},VD];
nComp{2}=[0,nComp{2},0];
Vcomp{2}=[Vspos(2),Vcomp{2},VD];
nComn{1}=[0,0,nComn{1},0];
Vcomn{1}=[Vspos(1),Vsneg(1),Vcomn{1},VD];
nComn{2}=[0,0,nComn{2},0];
Vcomn{2}=[Vspos(2),Vsneg(2),Vcomn{2},VD];
figure('Position',[50,50,1200,500])
subplot(1,2,1)
plot(V,npos(:,1),'b')
hold on
plot(vgust,ngustposS(:,1),'r')

plot(Vcomp{1},nComp{1},'g--','linewidth',2)
plot(V,nneg(:,1),'b')
plot(vgust,ngustnegS(:,1),'r')
plot(Vcomn{1},nComn{1},'g--','linewidth',2)
legend('Maneuver Limit','Gust Loading','Combined
Loading','Location','northwest')
title('V-n Diagram at Sea Level')
xlabel('Velocity (m/s)')
ylabel('Load Factor (n)')
subplot(1,2,2)
plot(V,npos(:,2),'b')
hold on
plot(vgust,ngustposS(:,2),'r')
plot(Vcomp{2},nComp{2},'g--','linewidth',2)
plot(V,nneg(:,2),'b')
plot(vgust,ngustnegS(:,2),'r')
plot(Vcomn{2},nComn{2},'g--','linewidth',2)
legend('Maneuver Limit','Gust Loading','Combined
Loading','Location','northwest')
title('V-n Diagram at 4450 m')
xlabel('Velocity (m/s)')
ylabel('Load Factor (n)')
%% Calculating Wx and Wy
LCs=[59.5 4.4
95.5 4.4
95.5 -1.1
64 -1.78
39.5 -1.76
75.5 4.4
95.5 4.4
95.5 -1.298
64 -2.051
50.5 -1.76];
nz=100;
for n=1:length(LCs)
if n>length(LCs)/2
rho=rhoC;
CD0T=CD0(2);
m=2;
else
rho=rho0;
CD0T=CD0(1);

m=1;
end
L(n) = LCs(n,2)*W; % N
lift
CLCP(n) = 2*L(n)/rho/LCs(n,1)^2/S; %
lift coefficient
% Alpha(n)=(CLCP(n)-CL3Dline(2))/CL3Dline(1);
Alpha(n)=interp1(y3Dline1{m},alphar{m},CLCP(n));
D(n) = 0.5*rho*LCs(n,1)^2*S*(CD0(m) + CLCP(n)^2/pi/A/e); %
N drag
z = 0:b/2/nz:b/2; % root to tip (half
span)
% lift distribution
l_rect(:,n) = L(n)/b.*ones(1,nz+1);
l_ellip (:,n)= (4*L(n)/pi/b).*sqrt(1-(2.*z./b).^2);
l (:,n)= (l_rect(:,n) + l_ellip(:,n))./2; %
N/m
d (:,n)=D(n)/b.*ones(1,nz+1);
for m=1:length(d)
if m>.8*length(d)
d(m,n)=d(m,n)*1.1;
end
end
subplot(1,2,1)
p=plotyy([z',z',z'],[l_ellip(:,n),l_rect(:,n),l(:,n)],z,d(:,n));
ylabel(p(1),'Lift Force (N/m)')
xlabel('z (m)')
ylabel(p(2),'Drag Force (N/m)')
legend('lift elliptic distribution','lift rectangular
distribution','combined lift distribution','Drag
Distribution','location','best')
% rotate into x-y coordinate
wy(:,n) = cos(Alpha(n)).*l(:,n) + sin(Alpha(n)).*d(:,n);
wx(:,n) = -sin(Alpha(n)).*l(:,n) + cos(Alpha(n)).*d(:,n);
% Note: wx and wy are defined from root to tip
subplot(1,2,2)
plot(z,wy,z,wx,'linewidth',2)

xlabel('z (m)')
ylabel('Distributed Load (N/m)')
legend('w_y','w_x','location','best')
end
plot(z,l(:,1),'-',z,l(:,2),'-',z,l(:,3),'-',z,l(:,4),'-
',z,l(:,5),'-',...
z,l(:,6),'--',z,l(:,7),'--',z,l(:,8),'--',z,l(:,9),'--
',z,l(:,10),'--','linewidth',2)
ylabel('Lift Force (N/m)')
xlabel('z (m)')
legend('PHAA SL','PLAA SL','Gust 1 SL','Gust 2 SL','NHAA
SL','PHAA CA','PLAA CA','Gust 1 CA','Gust 2 CA','NHAA
CA','location','west')
Rho(1:5)=rho0;
Rho(6:10)=rhoC;
Cd(1:5)=CD(1);
Cd(6:10)=CD(2);
for n=1:length(LCs)
[z1t,Mx0t,My0t,Sx0t,Sy0t,sig_z0t,u0t,v0t] =
DeflectionLoadscode(wx(:,n),wy(:,n),LCs(n,2),Rho(n),LCs(n,1),Alp
ha(n),Cd(n),nz);
z1(:,n)=z1t;
Mx0(:,n)=Mx0t;
My0(:,n)=My0t;
Sx0(:,n)=Sx0t;
Sy0(:,n)=Sy0t;
sig_z0(:,n)=sig_z0t;
u0a(:,n)=u0t;
v0a(:,n)=v0t;
clear z1t Mx0t My0t Sx0t Sy0t sig_z0t u0t v0t
end
%% X Moments
figure
subplot(2,1,1)
plot(z,Mx0(:,1),z,Mx0(:,2),z,Mx0(:,3),z,Mx0(:,4),z,Mx0(:,5))
legend('PHAA','PLAA','GL 1','GL 2','NHAA','location','best')
ylabel('Mx')
xlabel('Half Span Distance')
title('At Sea Level')
subplot(2,1,2)
plot(z,Mx0(:,6),z,Mx0(:,7),z,Mx0(:,8),z,Mx0(:,9),z,Mx0(:,10))

ylabel('Mx')
title('At Service Ceiling')
%% Y Moment
figure
subplot(2,1,1)
plot(z,My0(:,1),z,My0(:,2),z,My0(:,3),z,My0(:,4),z,My0(:,5))
ylabel('My')
subplot(2,1,2)
plot(z,My0(:,6),z,My0(:,7),z,My0(:,8),z,My0(:,9),z,My0(:,10))
ylabel('My')
%% X Shear
figure
subplot(2,1,1)
plot(z,Sx0(:,1),z,Sx0(:,2),z,Sx0(:,3),z,Sx0(:,4),z,Sx0(:,5))
ylabel('Sx')
subplot(2,1,2)
plot(z,Sx0(:,6),z,Sx0(:,7),z,Sx0(:,8),z,Sx0(:,9),z,Sx0(:,10))
ylabel('Sx')
%% Y Shear
figure
subplot(2,1,1)
plot(z,Sy0(:,1),z,Sy0(:,2),z,Sy0(:,3),z,Sy0(:,4),z,Sy0(:,5))
ylabel('Sy')
subplot(2,1,2)
plot(z,Sy0(:,6),z,Sy0(:,7),z,Sy0(:,8),z,Sy0(:,9),z,Sy0(:,10))

ylabel('Sy')
%% Direct Stress
figure
subplot(2,1,1)
plot(z,sig_z0(:,1),z,sig_z0(:,2),z,sig_z0(:,3),z,sig_z0(:,4),z,s
ig_z0(:,5))
ylabel('Direct Stress')
subplot(2,1,2)
plot(z,sig_z0(:,6),z,sig_z0(:,7),z,sig_z0(:,8),z,sig_z0(:,9),z,s
ig_z0(:,10))
ylabel('Direct Stress')
%% u-Deflections
figure
subplot(2,1,1)
plot(z,u0a(:,1),z,u0a(:,2),z,u0a(:,3),z,u0a(:,4),z,u0a(:,5))
ylabel('u-Deflection')
subplot(2,1,2)
plot(z,u0a(:,6),z,u0a(:,7),z,u0a(:,8),z,u0a(:,9),z,u0a(:,10))
ylabel('u-Deflection')
%% v-Deflection
figure
subplot(2,1,1)
plot(z,v0a(:,1),z,v0a(:,2),z,v0a(:,3),z,v0a(:,4),z,v0a(:,5))
legend('PHAA','PLAA','GL 1','GL
2','NHAA','location','northeast')
ylabel('v-Deflection')
xlabel('Half Span Distance (m)')

subplot(2,1,2)
plot(z,v0a(:,6),z,v0a(:,7),z,v0a(:,8),z,v0a(:,9),z,v0a(:,10))
legend('PHAA','PLAA','GL 1','GL
2','NHAA','location','northeast')
ylabel('v-Deflection')
xlabel('Half Span Distance (m)')
%% Recalculating the Airfoil Section
[cq,Ixx,Iyy,Ixy,x,yU,yL,Bigmat,i_strU,dx,i_spar,i_strL] =
airfoil_section(c,A_cap,A_str,t_spar,t_skin,x_spar,x_strU,x_strL
);
Ivals=[Ixx,Iyy,Ixy];
for n=1:length(LCs)
SMvals=[Mx0(1,n),My0(1,n),Sx0(1,n),Sy0(1,n)]; % at the root
[sigeq(:,n),sigzs(:,n),sigzz(:,n),qtot(:,n),xyc,xycent,qb(:,n),s
igcheck(:,n)]=shearflow(Bigmat,cq,SMvals,Ivals,t_skin,t_spar,G);
figure
scatter(xycent(:,1),xycent(:,2),10,sigzs(:,n),'filled')
c=colorbar;
c.Label.String='sigma_z_s (pa)';
ylim ([-.3 .3])
xlabel('X Coordinate (m)')
ylabel('Y Coordinate (m)')
figure
scatter(xycent(:,1),xycent(:,2),10,sigeq(:,n),'filled')
c=colorbar;
c.Label.String='sigma_e_q (pa)';
ylim ([-.3 .3])
end
figure
scatter(xycent(:,1),xycent(:,2),10,sigeq(:,1),'filled')
c=colorbar;
c.Label.String='sigma_e_q';
ylim ([-.3 .3])

figure
plot(x,yU,'k','Linewidth',2);
ylim([-0.3 0.3])
hold on
plot(i_strU*dx,yU(i_strU),'or','markersize',5);
plot([x_spar(1),x_spar(1)],[yU(i_spar(1)),yL(i_spar(1))],'b','Li
newidth',3);
scatter(xyc(1,1),xyc(1,2),'m*')
plot(x,yL,'k','Linewidth',2);
plot(i_strL*dx,yL(i_strL),'or','markersize',5);
plot([x(end),x(end)],[yU(end),yL(end)],'b','Linewidth',3);
ylabel('y (m)')
xlabel('x (m)')
grid on
legend('Airfoil','Stringers','Spars','Centriod')
figure
plot(qb)
ylabel('q_b (N/m)')
xlabel('Point Number')
CA','location','best')
figure
plot(sigcheck)
ylabel('Sigma_z_s*Boom Area (lbf)')
xlabel('Point Number')
CA','location','best')
function[sigeq,sigzs,sigzz,qtot,xyc,xycent,q,sigcheck]=shearflow
(Bigmat,xs,SMvals,Ivals,tskin,tspar,G)
xy=Bigmat(:,1:2);
if xy(1,2)~=xy(end,2)
xy(end+1,:)=xy(1,:);
end
b=ones(length(xy),1);

xy=[xy,b*0];
% calculate area of wing cross section
for n=1:length(xy)-1
dA2(n)=norm(cross(xy(n,:),xy(n+1,:)));
A2(n)=sum(dA2);
ldist(n,1)=sqrt((xy(n,1)-xy(n+1,1))^2+(xy(n,2)-
xy(n+1,2))^2);
askin(n,1)=Bigmat(n,6);
xyA(n,:)=[xy(n,1)*askin(n),xy(n,2)*askin(n)];
atot(n,1)=sum(Bigmat(n,3:6));
end
xyc(1,:)=[sum(xyA(:,1))/sum(askin),sum(xyA(:,2))/sum(askin)];
%normalize x and y vectors to centriod
xycent=xy-[b*xyc(1),b*xyc(2),b*0];
% calculate area of wing cross section
for n=1:length(xy)-1
dA2(n)=norm(cross(xycent(n,:),xycent(n+1,:)));
A2(n)=sum(dA2);
ldist(n,1)=sqrt((xycent(n,1)-xycent(n+1,1))^2+(xycent(n,2)-
xycent(n+1,2))^2);
askin(n,1)=Bigmat(n,6);
xyA(n,:)=[xycent(n,1)*askin(n),xycent(n,2)*askin(n)];
atot(n,1)=sum([Bigmat(n,3),Bigmat(n,5:6)]);
end
xyc(2,:)=[sum(xyA(:,1))/sum(askin),sum(xyA(:,2))/sum(askin)];
for n=1:length(xycent)-1
Ivec(n,:)=[askin(n)*(xycent(n,2)-
xyc(2))^2,askin(n)*(xycent(n,1)-xyc(1))^2,askin(n)*(xycent(n,1)-
xyc(1))*(xycent(n,2)-xyc(2))];
end
IvecT=[sum(Ivec(:,1)),sum(Ivec(:,2)),sum(Ivec(:,3))];
den1=IvecT(1)*IvecT(2)-IvecT(3)^2;
Cts=[(SMvals(2)*IvecT(1)-
SMvals(1)*IvecT(3))/den1,(SMvals(1)*IvecT(2)-
SMvals(2)*IvecT(3))/den1,...
(SMvals(4)*IvecT(3)-
SMvals(3)*IvecT(1))/den1,(SMvals(3)*IvecT(3)-
SMvals(4)*IvecT(2))/den1];

sigzz(n,1)=Cts(1)*xycent(n,1)+Cts(2)*xycent(n,2);
end
ind=find(Bigmat(:,4)>0);
Lspar=sqrt((xycent(ind(2),1)-
xycent(ind(3),1))^2+(xycent(ind(2),2)-xycent(ind(3),2))^2);
if n==1
BoomA(n,1)=(atot(n)-
askin(n))+tskin*ldist(n)/6*(2+sigzz(n+1)/sigzz(n))+tspar*ldist(e
nd)/6*(2+sigzz(end)/sigzz(n));
elseif n==length(xycent)-1
BoomA(n,1)=(atot(n)-askin(n))+tskin*ldist(n-
1)/6*(2+sigzz(n-
1)/sigzz(n))+tspar*ldist(n)/6*(2+sigzz(1)/sigzz(n));
elseif n==ind(2)
1)/6*(2+sigzz(n-1)/sigzz(n))...
+tskin*ldist(n)/6*(2+sigzz(n+1)/sigzz(n))+tspar*Lspar/6*(2+sigzz
(ind(3))/sigzz(n));
elseif n==ind(3)
1)/6*(2+sigzz(n-1)/sigzz(n))...
+tskin*ldist(n)/6*(2+sigzz(n+1)/sigzz(n))+tspar*Lspar/6*(2+sigzz
(ind(2))/sigzz(n));
else
BoomA(n,1)=(atot(n)-
askin(n))+tskin*ldist(n)/6*(2+sigzz(n+1)/sigzz(n))...
+tskin*ldist(n-1)/6*(2+sigzz(n-1)/sigzz(n));
end
dq(n,1)=Cts(3)*BoomA(n)*(xycent(n,1))+Cts(4)*BoomA(n)*(xycent(n,
2));
q(n,1)=sum(dq);
q2dA(n,1)=dq(n,1)*dA2(n);
end
At(1)=sum(dA2(ind(2):ind(3)))+norm(cross(xycent(ind(3),:),xycent
(ind(2),:)));
At(2)=sum(dA2(ind(1):ind(2)))+norm(cross(xycent(ind(4),:),xycent
(ind(1),:)))...

+norm(cross(xycent(ind(2),:),xycent(ind(3),:)))+sum(dA2(ind(3):i
nd(4)));
Bx(1)=sum(BoomA(ind(2):ind(3)).*xycent(ind(2):ind(3),1).*ldist(i
nd(2):ind(3))/tskin);
Bx(2)=sum(BoomA(ind(1):ind(2)).*xycent(ind(1):ind(2),1).*ldist(i
nd(1):ind(2))/tskin)+...
sum(BoomA(ind(3):ind(4)).*xycent(ind(3):ind(4),1).*ldist(ind(3):
ind(4))/tskin);
By(1)=sum(BoomA(ind(2):ind(3)).*xycent(ind(2):ind(3),2).*ldist(i
nd(2):ind(3))/tskin);
By(2)=sum(BoomA(ind(1):ind(2)).*xycent(ind(1):ind(2),2).*ldist(i
nd(1):ind(2))/tskin)+...
sum(BoomA(ind(3):ind(4)).*xycent(ind(3):ind(4),2).*ldist(ind(3):
ind(4))/tskin);
dytspar=(xycent(ind(2),2)-xycent(ind(3),2))/tspar;
dytback=(xycent(ind(1),2)-xycent(ind(4),2))/tspar;
ldtsk(1)=sum(ldist(ind(2):ind(3))/tskin);
ldtsk(2)=(sum(ldist(ind(1):ind(2)))+sum(ldist(ind(3):ind(4))))/t
skin;
b=[-Cts(3)*Bx(1)-Cts(4)*By(1);-Cts(3)*Bx(2)-
Cts(4)*By(2);SMvals(3)+SMvals(2)*xs-sum(q2dA)];
A=[ldtsk(1)+(-dytspar),-(-dytspar),-At(1)*G;-
dytspar,ldtsk(2)+dytspar+dytback,-At(2)*G;...
sum(q2dA(ind(2):ind(3))),sum(q2dA(ind(1):ind(2)))+sum(q2dA(ind(3
):ind(4))),0];
x=A^-1*b;
q01=x(1);
q02=x(2);
dthdz=x(3);
qtot=[q(ind(1):ind(2))+q02;q(ind(2)+1:ind(3))+q01;q(ind(3)+1:ind
(4))+q02];
sigzs=qtot/tskin;
for n=1:length(sigzs)
sigeq(n,1)=sqrt(2*sigzz(n)^2+6*sigzs(n)^2);

end
sigcheck=BoomA.*sigzs;
xycent=xycent(1:end-1,:);
function [cq,Ixx,Iyy,Ixy,x,yU,yL,Bigmat,i_strU,dx,i_spar,i_strL]
=
airfoil_section(c,A_cap,A_str,t_spar,t_skin,x_spar,x_strU,x_strL
)
%% airfoil section profile
% NACA 2415
m = 0.02; % 1st digit maximum camber (m) in percentage of the
chord
p = 0.4; % 2nd digit position of the maximum camber (p) in
tenths of chord
t = 0.15; % last 2 digits(Maximum thickness (t) of the airfoil
in percentage of chord)
cq = c/4;
nx = 500; % number of increments
dx = c/nx;
x = 0:dx:c; % even spacing
yc = zeros(1,nx+1);
yt = zeros(1,nx+1);
yU0 = zeros(1,nx+1);
yL0 = zeros(1,nx+1);
theta = zeros(1,nx+1);
xb = p*c;
i_xb = xb/dx + 1;
for i = 1:nx+1
yt(i) = 5*t*c*(0.2969*sqrt(x(i)/c) - 0.1260*(x(i)/c) -
0.3516*(x(i)/c)^2 + 0.2843*(x(i)/c)^3 - 0.1015*(x(i)/c)^4);
if i <= i_xb
yc(i) = m*x(i)/p^2*(2*p - x(i)/c);
theta(i) = atan(2*m/p^2*(p - x(i)/c));
else
yc(i) = m*(c - x(i))/(1-p)^2*(1 + x(i)/c - 2*p);
theta(i) = atan(2*m/(1-p)^2*(p-x(i)/c));
end
yU0(i) = yc(i) + yt(i)*cos(theta(i));
yL0(i) = yc(i) - yt(i)*cos(theta(i));
end

% The last 20% of the chord length of the airfoil was neglected
under the assumption that
% this section contained flaps and ailerons, and would therefore
not support aerodynamic loads.
i_xend = round(0.8*c/dx)+1;
x = x(1:i_xend);
yU = yU0(1:i_xend);
yL = yL0(1:i_xend);
% Here the airfoil profile is approximated by assuming xU0=x &
xL0=x.
%% adding stringers, spar caps and spars
% spars only
x_spar = [x_spar,x(end)]; % x_spar: location of
spars and spar caps
n_spar = length(x_spar); % Number of spars
i_spar = round(x_spar./dx)+1; % number of divisions to
get to x_spar
h_spar = yU(i_spar) - yL(i_spar); % height of spar
Cy_spar = (yU(i_spar) + yL(i_spar))/2; % y coord of centroid of
spar
A_spar = t_spar.*h_spar; % Area of spar = t*h
array_spar = zeros(1,length(x));
j=1;
for i = 1:length(x)
if i == i_spar(j)
array_spar(i)=A_spar(j);
j=j+1;
else
array_spar(i)=0;
end
end
array_cap = zeros(1,length(x));
array_cap(i_spar) = A_cap;
%% stringers
i_strU = round(x_strU./dx)+1; % index in x array
corresponding to the Upper stringer locations
i_strL = round(x_strL./dx)+1; % index in x array
corresponding to the Lower stringer locations
% Remove Stringer where there are spars for Upper Part
commonind = [];
for i=1:length(i_spar)

for j=1:length(i_strU)
if i_spar(i) == i_strU(j)
commonind = [commonind , j];
end
end
end
i_strU (commonind) = [];
% Remove Stringer where there are spars for Lower Part
commonind = [];
for i=1:length(i_spar)
for j=1:length(i_strL)
if i_spar(i) == i_strL(j);
commonind = [commonind , j];
end
end
end
i_strL(commonind) = [];
n_strU = length(i_strU); % Number of stringers in upper
half
n_strL = length(i_strL); % Number of stringers in lower
half
% Creating aray of Stringer areas for putting in Bigmatrix
array_strU = zeros(1,length(x));
array_strU(i_strU) = A_str;
array_strL = zeros(1,length(x));
array_strL(i_strL) = A_str;
%% skins
% nodes include spar caps and stringers
x_nodeU = [x_spar,x_strU]; % x-coord of (stringers,spar
caps) combined
x_nodeU = sort(x_nodeU); % x-coord of nodes (Sorted)
n_nodeU = length(x_nodeU);
i_nodeU = round(x_nodeU./dx)+1; % index of nodes
% MODIFIED SKIN CODE
n_skinU = length(x)-1;
for i = 1:n_skinU
L_skinU(i) = sqrt((x(i)-x(i+1))^2 + (yU(i)-yU(i+1))^2);
A_skinU(i) = t_skin*L_skinU(i);
Cx_skinU(i) = (x(i)+x(i+1))/2;
Cy_skinU(i) = (yU(i)+yU(i+1))/2;
end
n_skinL = length(x)-1;
for i = 1:n_skinL

L_skinL(i) = sqrt((x(i)-x(i+1))^2 + (yL(i)-yL(i+1))^2);
A_skinL(i) = t_skin*L_skinL(i);
Cx_skinL(i) = (x(i)+x(i+1))/2;
Cy_skinL(i) = (yU(i)+yU(i+1))/2;
end
A_skinU = [0 A_skinU]; % adding zero to the front of the array
A_skinL = [A_skinL 0]; % same as above (Now 401 elements)
% skin should be broken into smaller elements for higher
accuracy of calculation
% break one skin element into two by adding one more node in
between
%%
% x_skinU = zeros(1,2*length(x_nodeU)-1);
% for i = 1:length(x_nodeU)-1
% x_skinU(2*i-1) = x_nodeU(i);
% x_skinU(2*i) = (x_nodeU(i) + x_nodeU(i+1))/2;
% end
% x_skinU(end) = x_nodeU(end);
%
% i_skinU = round(x_skinU/dx)+1;
% n_skinU = length(x_skinU)-1;
% L_skinU = zeros(1,n_skinU);
% A_skinU = zeros(1,n_skinU);
% Cx_skinU = zeros(1,n_skinU);
% Cy_skinU = zeros(1,n_skinU);
% for i = 1:n_skinU
% L_skinU(i) = sqrt((yU(i_skinU(i+1)) - yU(i_skinU(i)))^2 +
(x_skinU(i+1) - x_skinU(i))^2);
% A_skinU(i) = t_skin*L_skinU(i);
% Cx_skinU(i) = (x_skinU(i+1) + x_skinU(i))/2;
% Cy_skinU(i) = (yU(i_skinU(i+1)) + yU(i_skinU(i)))/2;
% end
%
% % lower part
% x_nodeL = [x_spar,x_strL]; % x-coord of (stringers,spar
caps) combined
% n_nodeL = length(x_nodeL);
% x_nodeL = sort(x_nodeL); % x-coord of nodes (Sorted)
% i_nodeL = round(x_nodeL./dx)+1; % index of nodes
%
% x_skinL = zeros(1,2*length(x_nodeL)-1);
% for i = 1:length(x_nodeL)-1
% x_skinL(2*i-1) = x_nodeL(i);
% x_skinL(2*i) = (x_nodeL(i) + x_nodeL(i+1))/2;
% end

% x_skinL(end) = x_nodeL(end);
%
% i_skinL = round(x_skinL/dx)+1;
% n_skinL = length(x_skinL)-1;
% L_skinL = zeros(1,n_skinL);
% A_skinL = zeros(1,n_skinL);
% Cx_skinL = zeros(1,n_skinL);
% Cy_skinL = zeros(1,n_skinL);
% for i = 1:n_skinL
% L_skinL(i) = sqrt((yL(i_skinL(i+1)) - yL(i_skinL(i)))^2 +
(x_skinL(i+1) - x_skinL(i))^2);
% A_skinL(i) = t_skin*L_skinL(i);
% Cx_skinL(i) = (x_skinL(i+1) + x_skinL(i))/2;
% Cy_skinL(i) = (yL(i_skinL(i+1)) + yL(i_skinL(i)))/2;
% end
%% centroid of the wing section
% initial value
Cx_sum = 0;
Cy_sum = 0;
A_sum = 0;
% spars
for i = 1:n_spar
Cx_sum = Cx_sum + x_spar(i)*A_spar(i);
Cy_sum = Cy_sum + Cy_spar(i)*A_spar(i);
A_sum = A_sum + A_spar(i);
end
%Upper Skin
for i = 1:n_skinU
Cx_sum = Cx_sum + Cx_skinU(i)*A_skinU(i);
Cy_sum = Cy_sum + Cy_skinU(i)*A_skinU(i);
A_sum = A_sum + A_skinU(i);
end
%Lower skin
for i = 1:n_skinL
Cx_sum = Cx_sum + Cx_skinL(i)*A_skinL(i);
Cy_sum = Cy_sum + Cx_skinL(i)*A_skinL(i);
A_sum = A_sum + A_skinL(i);
end
%Upper Stringers
for i = 1:n_strU
Cx_sum = Cx_sum + x_strU(i)*A_str;
Cy_sum = Cy_sum + yU(i_strU(i))*A_str;
A_sum = A_sum + A_str;
end

%Lower Stringers
for i = 1:n_strL
Cx_sum = Cx_sum + x_strL(i)*A_str;
Cy_sum = Cy_sum + yL(i_strL(i))*A_str;
A_sum = A_sum + A_str;
end
%All spar caps
for i = 1:n_spar
Cx_sum = Cx_sum + 2*A_cap*x_spar(i);
Cy_sum = Cy_sum + A_cap*Cy_spar(i);
A_sum = A_sum + A_cap;
end
Cx = Cx_sum/A_sum;
Cy = Cy_sum/A_sum;
% figure
% plot(x,yU,'k',x,yL,'k','Linewidth',2);
% ylim([-0.3 0.3])
% hold on
%
plot(i_strU*dx,yU(i_strU),'or',i_strL*dx,yL(i_strL),'or','marker
size',5);
%
plot([x_spar(1),x_spar(1)],[yU(i_spar(1)),yL(i_spar(1))],'b',[x(
end),x(end)],[yU(end),yL(end)],'b','Linewidth',3);
%
plot(x_spar,yU(i_spar),'sg',x_spar,yL(i_spar),'sg','markersize',
7);
% scatter(Cx,Cy,'m*')
% ylabel('y (m)')
% xlabel('x (m)')
% grid on
%% Area moments of inertia
% initial value
Ixx = 0;
Iyy = 0;
Ixy = 0;
% Spars MOI
for i = 1:n_spar
Ixx = Ixx + t_spar*h_spar(i)^3/12 + A_spar(i)*(Cy_spar(i)-
Cy)^2;
Iyy = Iyy + t_spar^3*h_spar(i)/12 + A_spar(i)*(x_spar(i)-
Cx)^2;

Ixy = Ixy + A_spar(i)*(Cy_spar(i)-Cy)*(x_spar(i)-Cx);
end
% Spar caps MOI
for i = 1:n_spar
Ixx = Ixx + A_cap*(yU(i_spar(i))-Cy)^2 +
A_cap*(yL(i_spar(i))-Cy)^2;
Iyy = Iyy + 2*A_cap*(x_spar(i)-Cx)^2; %
2*A since both upper and lower caps
Ixy = Ixy + A_cap*(yU(i_spar(i))-Cy)*(x_spar(i)-Cx)+... %
upper spar caps
A_cap*(yL(i_spar(i))-Cy)*(x_spar(i)-Cx); %
lower spar caps
end
% Upper Skin MOI
for i = 1:n_skinU
Ixx = Ixx + A_skinU(i)*(Cx_skinU(i)-Cx)^2;
Iyy = Iyy + A_skinU(i)*(Cy_skinU(i)-Cy)^2;
Ixy = Ixy + A_skinU(i)*(Cx_skinU(i)-Cx)*(Cy_skinU(i)-Cy);
end
% Lower Skin MOI
for i = 1:n_skinL
Ixx = Ixx + A_skinL(i)*(Cx_skinL(i)-Cx)^2;
Iyy = Iyy + A_skinL(i)*(Cy_skinL(i)-Cy)^2;
Ixy = Ixy + A_skinL(i)*(Cx_skinL(i)-Cx)*(Cy_skinU(i)-Cy);
end
% Upper Stringers MOI
for i = 1:n_strU
Ixx = Ixx + A_str*(yU(i_strU(i))-Cy)^2;
Iyy = Iyy + A_str*(x_strU(i)-Cx)^2;
Ixy = Ixy + A_str*(x_strU(i)-Cx)*(yU(i_strU(i))-Cy);
end
%Lower Stringers MOI
for i = 1:n_strL
Ixx = Ixx + A_str*(yL(i_strL(i))-Cy)^2;
Iyy = Iyy + A_str*(x_strL(i)-Cx)^2;
Ixy = Ixy + A_str*(x_strL(i)-Cx)*(yU(i_strL(i))-Cy);
end
%% Shifting the origin to the centroid:
% x = x-Cx;
% yU = yU-Cy;
% yL = yL-Cy;
% x_skinU = Cx_skinU-Cx;
% x_skinL = Cx_skinL-Cx;
% x_spar = x_spar-Cx;
% x_strU = x_strU-Cx;

% x_strL = x_strL-Cx;
%% Big Matrix containing all the values
for i = 1:length(x) %inverting the coordinates
x2(i) = x(end-(i-1));
yUtemp(i) = yU(end-(i-1));
array_captemp(i) = array_cap(end-(i-1));
array_spartemp(i) = array_spar(end-(i-1));
array_strUtemp(i) = array_strU(end-(i-1));
A_skinUtemp(i) = A_skinU(end-(i-1));
end
% yU = yUtemp;
% array_cap = array_captemp;
% array_spar = array_spartemp;
% array_strU = array_strUtemp;
% A_skinU = A_skinUtemp;
Bigmat = [x2(1:end-1)' yUtemp(1:end-1)' array_captemp(1:end-1)'
array_spartemp(1:end-1)' array_strUtemp(1:end-1)'
A_skinUtemp(1:end-1)' ; ...
x',yL',array_cap',array_spar',array_strL',A_skinL'] ;
%Bigmat = [x2' yU' array_cap' array_spar' array_strU' A_skinU' ;
...
%x',yL',array_cap',array_spar',array_strL',A_skinL'] ;
end

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