FDR Report

Final Design Report:
Piper Cherokee Wing Structure
Group 8:
Edward Barber
Cullen McAlpine
Elmer Wu
MAE 154B – Design of Aerospace Structures
Spring 2015
HENRY SAMUELI SCHOOL OF ENGINEERING AND APPIED SCIENCE
Department of Mechanical and Aerospace Engineering

1
Table of Contents
TABLE OF CONTENTS................................................................................................................ 1
1 LIST OF FIGURES................................................................................................................. 3
2 LIST OF TABLES................................................................................................................... 5
3 LIST OF SYMBOLS............................................................................................................... 7
4 ABSTRACT ............................................................................................................................ 8
5 GANTT CHART ..................................................................................................................... 8
6 AIRCRAFT SELECTION..................................................................................................... 10
7 AIRCRAFT SPECIFICATIONS........................................................................................... 11
8 LOAD DIAGRAMS.............................................................................................................. 13
8.1 MANEUVER ENVELOPE ................................................................................................... 13
8.2 GUST ENVELOPE ............................................................................................................. 16
8.3 COMBINED FLIGHT ENVELOPE ........................................................................................ 19
9 AERODYNAMIC ANALYSIS ............................................................................................ 21
10 AERODYNAMIC LOADS ............................................................................................... 24
10.1 CHORDWISE PRESSURE DISTRIBUTIONS .......................................................................... 24
10.2 SURFACE PRESSURE DISTRIBUTIONS............................................................................... 25
10.3 NORMAL DISTRIBUTIONS ................................................................................................ 27
10.4 AXIAL DISTRIBUTIONS .................................................................................................... 29
10.5 3D-LOAD DISTRIBUTIONS ............................................................................................... 31
11 MATERIAL SELECTION................................................................................................ 32
12 STRUCTURAL ANALYSIS............................................................................................. 33
12.1 APPROXIMATIONS AND ASSUMPTIONS ............................................................................ 33
12.2 MATLAB CODE ............................................................................................................. 33
12.3 PANEL METHOD .............................................................................................................. 34
12.4 INITIAL GEOMETRY ......................................................................................................... 35
12.5 CENTROID CALCULATIONS.............................................................................................. 36
12.6 MOMENT AND BENDING.................................................................................................. 37
12.7 SHEAR FLOW................................................................................................................... 39
12.8 SHEAR CENTER ............................................................................................................... 40
12.9 BUCKLING....................................................................................................................... 41
12.10 FACTOR OF SAFETY...................................................................................................... 42
13 FAILURE CRITERIA ....................................................................................................... 43
13.1 VON MISES...................................................................................................................... 43
13.2 PARIS’ LAW..................................................................................................................... 43
13.3 STRUCTURAL ANALYSIS REMARKS................................................................................. 46
14 AEROELASTIC CONSIDERATIONS............................................................................. 47
14.1 DIVERGENCE ................................................................................................................... 47
FOR EXAMPLE, FOR THE DESIGNED WING AT CEILING NHAA, THE DIVERGENCE SPEED IS 692 M/S
OR APPROXIMATELY MACH 2. SINCE THE SINGLE TURBOPROP ENGINE OF THE PIPER CHEROKEE

2
COULD NEVER REACH SUCH EXTREME SPEEDS, THE DESIGNED WING STRUCTURE IS ACCEPTABLE.
................................................................................................................................................... 47
14.2 AILERON REVERSAL........................................................................................................ 47
14.3 FLUTTER ......................................................................................................................... 48
15 OPTIMIZATION............................................................................................................... 49
15.1 METHODOLOGY .............................................................................................................. 49
15.2 CALCULATIONS ............................................................................................................... 50
15.3 FINAL RESULTS ............................................................................................................... 52
15.4 REMARKS ........................................................................................................................ 54
16 FINITE ELEMENT ANALYSIS ...................................................................................... 55
17 CONCLUSIONS................................................................................................................ 59
18 REFERENCES .................................................................................................................. 60
APPENDIX A: AIR LOADS.......................................................................................................... 1
APPENDIX A.1: CHORDWISE PRESSURE DISTRIBUTIONS .............................................................. 1
APPENDIX A.2: 3D PRESSURE DISTRIBUTIONS............................................................................. 5
APPENDIX B: STRESSES FOR OPTIMAL GEOMETRY ......................................................... 9
APPENDIX B: CODE VALIDATION ........................................................................................ 13
APPENDIX C: FEA RESULTS................................................................................................... 16
Cover page photo credit: Wikipedia Commons

3
1 List of Figures
Figure 1. Piper PA-28 Cherokee 140 3-view [1].......................................................................... 12
Figure 2. Maneuvering envelope at sea level................................................................................ 15
Figure 3. Maneuvering envelope at service ceiling (4400 m). ..................................................... 15
Figure 4. Gust envelope at sea level. ............................................................................................ 18
Figure 5. Gust envelope at service ceiling (4400 m). ................................................................... 18
Figure 6. Combined flight envelope at sea level........................................................................... 20
Figure 7. Combined flight envelope at service ceiling (4400 m). ................................................ 20
Figure 8. Lift curve data for NACA 65-415 ................................................................................. 21
Figure 9. Cp vs. Chord for AoA = 9 degrees................................................................................ 22
Figure 10. Cd vs. Span at AoA of 9 degrees................................................................................. 23
Figure 11. Cl vs. Span at AoA of 9 degrees.................................................................................. 23
Figure 12. Pressure coefficient vs. chord for the PHAA case at sea level.................................... 25
Figure 13. Surface gauge pressures for the PHAA case at sea level. ........................................... 26
Figure 14. Absolute normal load and shear distributions for the PHAA case at sea level. .......... 28
Figure 15. Absolute moment distribution about the x-axis for the PHAA case at sea level. ....... 28
Figure 16. Axial load and shear distributions for the PHAA case at sea level............................. 30
Figure 17. Absolute moment about the Y-axis for the PHAA case at sea level........................... 30
Figure 18. 3D-load distribution for the PHAA case at sea level. ................................................. 31
Figure 19. Panel Method............................................................................................................... 34
Figure 20. Initial geometry for preliminary analysis. ................................................................... 35
Figure 21. k-values for plate buckling of simply-supported edges [9]......................................... 42
Figure 22: Fatigue crack growth curve to failure.......................................................................... 45
Figure 23. Optimized geometry. ................................................................................................... 52
Figure 24. Optimal design root axial stresses under PHAA at sea level. ..................................... 53
Figure 25. Optimal design root shear stresses under PHAA at sea level...................................... 53
Figure 26: Mesh for Piper Cherokee wing.................................................................................... 56
Figure 27: Representation of pressure map. ................................................................................. 57
Figure 28. Example displacement loading scenario visual results for PLAA at sea level............ 58
Figure 29. Example stress loading scenario visual results for PLAA at sea level........................ 58
Figure 31. Pressure coefficient vs. chord for the NHAA case at sea level..................................... 1
Figure 32. Pressure coefficient vs. chord for the NLAA case at sea level. .................................... 1
Figure 33. Pressure coefficient vs. chord for the PHAA case at sea level...................................... 2
Figure 34. Pressure coefficient vs. chord for the PLAA case at sea level. ..................................... 2
Figure 35. Pressure coefficient vs. chord for the NHAA case at 4400m........................................ 3
Figure 36. Pressure coefficient vs. chord for the NLAA case at 4400m. ....................................... 3
Figure 37. Pressure coefficient vs. chord for the PHAA case at 4400m. ....................................... 4
Figure 38. Pressure coefficient vs. chord for the PLAA case at 4400m......................................... 4
Figure 39. Surface gauge pressures for the NHAA case at sea level.............................................. 5
Figure 40. Surface gauge pressures for the NLAA case at sea level. ............................................. 5
Figure 41. Surface gauge pressures for the PHAA case at sea level. ............................................. 6
Figure 42. Surface gauge pressures for the PLAA case at sea level............................................... 6
Figure 43. Surface gauge pressures for the NHAA case at 4400m. ............................................... 7
Figure 44. Surface gauge pressures for the NLAA case at 4400m................................................. 7
Figure 45. Surface gauge pressures for the PHAA case at 4400m. ................................................ 8

4
Figure 46. Surface gauge pressures for the PLAA case at 4400m.................................................. 8
Figure 47. Axial stresses for NHAA at sea level............................................................................ 9
Figure 48. Shear stresses for NLAA at sea level. ........................................................................... 9
Figure 49. Axial stresses for NLAA at sea level. ......................................................................... 10
Figure 50. Shear stresses for NLAA at sea level. ......................................................................... 10
Figure 51. Axial stresses for PHAA at sea level........................................................................... 11
Figure 52. Shear stresses for PHAA at sea level. ......................................................................... 11
Figure 53. Axial stresses for PLAA at sea level. .......................................................................... 12
Figure 54. Shear stresses for PLAA at sea level........................................................................... 12
Figure 55. Test Case Geometry with Dimensions and Loads....................................................... 13
Figure 58: Stress for NGAC at sea level....................................................................................... 16
Figure 59: Displacement for NGAC at sea level. ......................................................................... 16
Figure 60: Stress for NGAC at ceiling.......................................................................................... 17
Figure 61: Displacement for NGAC at ceiling. ............................................................................ 17
Figure 62: Stress for NHAA at sea level. ..................................................................................... 18
Figure 63: Displacement for NHAA at sea level.......................................................................... 18
Figure 64: Stress for NHAA at ceiling. ........................................................................................ 19
Figure 65: Displacement for NHAA at ceiling............................................................................. 19
Figure 66: Stress for PGAC at sea level. ...................................................................................... 20
Figure 67: Displacement for PGAC at sea level........................................................................... 20
Figure 68: Stress for PGAC at ceiling. ......................................................................................... 21
Figure 69: Displacement for PGAC at ceiling.............................................................................. 21
Figure 70: Stress for PLAA at sea level........................................................................................ 22
Figure 71: Displacement for PLAA at sea level. .......................................................................... 22

5
2 List of Tables
Table 1. Aircraft comparison summary. ....................................................................................... 11
Table 2. Piper PA-28-140 Cherokee key specifications [1]. ........................................................ 11
Table 3. Critical maneuver limits.................................................................................................. 13
Table 4. Critical load conditions at sea level. ............................................................................... 19
Table 5. Critical load conditions at service ceiling....................................................................... 19
Table 6: SeaLevel Gust at AoA 9 ................................................................................................. 38
Table 7. First-pass Monte Carlo design parameters...................................................................... 50
Table 8. Second-pass Monte-Carlo design parameters................................................................. 51
Table 9. Third-pass Monte Carlo design parameters.................................................................... 51
Table 10. Fourth-pass Monte Carlo design parameters. ............................................................... 51
Table 11. Final weight and minimum factors of safety across all critical load cases.................. 52
Table 12 Comparison of Some Variables. .................................................................................... 14
Table 13. Bending Stress Comparison for Test Case Booms. ...................................................... 14
Table 14. Shear flows. .................................................................................................................. 15

7
3 List of Symbols
Symbol Description Units
𝐴̅ 𝑛 Cell section area 𝑚2
𝐴𝑜𝐴 Angle of attack 𝑑𝑒𝑔
𝐴𝑅 Aspect Ratio −
A Crack length mm
b Stringer spacing m
𝑐 Chord length 𝑚
𝐶𝐿 Lift coefficient −
𝐶𝐿,𝛼 Lift-curve slope −
E Young’s Modulus N/m
𝑒 Oswald efficiency −
G Shear modulus N/m
I Second moment of area (Inertia) 𝑚4
𝐾 Gust alleviation factor −
K Fracture toughness MPa 𝑚
1
2⁄
𝐿 Lift 𝑁
My , Mx Moment Nm
N Number of cycles -
𝑛 Load factor −
qs Open shear flow N/m
q Closed shear flow N/m
𝑆 Wing area 𝑚2
t Skin thickness m
𝑢 Gust velocity 𝑚/𝑠
𝑣 Aircraft velocity 𝑚/𝑠
Vx , Vy Shear force N
𝑊 Aircraft weight 𝑁
Y Geometric parameter -
𝛼 Angle of attack 𝑑𝑒𝑔
σ stress N/𝑚2
𝜇 Aircraft mass ratio −
ν Poisson’s ratio -
𝜌 Air density 𝑘𝑔/𝑚3
Subscript Description
𝑑𝑒 Value derived experimentally
𝑚𝑎𝑥 Maximum value
𝑛𝑒𝑔 Negative load factor
𝑝𝑜𝑠 Positive load factor

8
4 Abstract
This report discusses the design and analysis of an aircraft wing that was selected based on
interest, ease of replication, and applicability to real world scenarios. It first discusses the
research performed on various aircraft that could be used to model a wing after. Once the aircraft
was selected, the expected loads on the airframe were identified based on the aerodynamics of
the wing at various conditions. These conditions were determined from V-n diagrams that were
developed from the FAR 23 aviation regulations. By analyzing the selected wing as a multi-cell
structure with the distributed loads, a stress and buckling analysis was performed with two spars
and multiple stringers at varying positions in the airfoil. This analysis was followed by shear
calculations and refined adjustments on the wing geometry. A Monte Carlo simulation was used
to determine the best spar and stringer placement and optimized the size of components. Finite
element analysis was performed on the wing for comparison purposes. From the FEA, the
devised structural analysis provided estimates that were on the same order of magnitude but not
as accurate as expected.
5 Gantt Chart
The project Gantt chart is included on the following page. Preliminary Design Review (PDR)
tasks are divided into preliminary loads calculations and bending stresses. Critical Design
Review (CDR) tasks are focused on an in-depth analysis of stresses and structural optimization,
with confirmation through finite element modelling. Final Design Review tasks are aimed at
comparing the production PA-28-140 aircraft with the optimized model. Tasks were assigned
based on team member experience and technical ability. Unfortunately, not all tasks were
completed due to unexpected coding delays.

ID Task Name
0 Gantt Chart
1 PDR
2 Background research
3 Finalize project choice - structure type and aircraft
4 V-n Diagram - Identify critical conditions
5 Baseline geometry
6 Spanwise aerodynamic loads - XFOIL
7 Code area moment of inertia calculator for spar geometry
8 Bending stresses with simplified beam
9 PDR Presentation
10 PDR Report - compile and submit
11 CDR
12 Full pressure distribution over wing - XFOIL
13 Compute bending stress, shear flow, shear center,
deflections, buckling - MATLAB code14 Begin CAD modelling
15 Begin static FEA
16 Research material shapes available and choose material(s)
17 Analyze rivet connections
18 Discuss Paris' law and fatigue crack growth in aluminum
19 Design spar caps, stringers, rivet connections
20 Show fracture calculations and determine critical crack size
21 Refine CAD for optimized structure
22 Run refined FEA
23 Show fatigue life calculations
24 CDR Presentation
25 CDR Report - Compile
26 FDR
27 Show that no elements will fail (including rivets)
28 Divergence (address aeroelastic coupling)
29 CAD production PA-28-140 Wing
30 FEA on production model
31 Compare production model for weight, bending, and torsion
32 FDR Presentation
33 FDR Report - Compile
CM
EB
EB
CM
CM
EW
EW
4/13
EB
CM
EW
EB
CM
EB
EB
CM
EW
CM
EB
CM
CM
5/18
EB
EW
EW
EB
CM
EW
6/1
CM
3/15 3/22 3/29 4/5 4/12 4/19 4/26 5/3 5/10 5/17 5/24 5/31 6/7
March 21 April 11 May 1 May 21 June 11
MAE 154B - Design of Aerospace Structures - Spring 2015 Project Gantt Chart Edward Barber, Cullen McAlpine, Elmer Wu - Group 8
Task owner indicated by initials at right Compiled by Edward Barber

10
6 Aircraft Selection
An aircraft wing structure was chosen due to the variety of loading conditions required for
analysis, including dual-axis bending, torsion, skin buckling and fatigue. In order to size the
wing and begin analysis, a specific aircraft was required. To ensure the aircraft chosen was
applicable, several key criteria were identified:
 Data for the aircraft must be easily accessible, including: airfoil, cruise speed, stall speed,
max takeoff weight, and standard empty weight.
 The wing airfoil should have data points available for XFOIL analysis.
 The analysis required must be realistically accomplishable within ten weeks with
minimal simplification.
With these criteria in mind, several straight-wing aircraft were examined, including a stunt
aircraft (the Extra EA-300), a WWII fighter aircraft (the North American P-51 Mustang), and a
straight-winged utility aircraft (the Piper PA-28 Cherokee). A stunt aircraft was initially chosen
due to the interesting load conditions present during aerobatic maneuvers. However, due to the
added complexity of the composite structures used in most modern stunt aircraft, this option was
quickly disregarded. In order to avoid composites, older aircraft were then examined, including
several fighter aircraft of WWII and utility aircraft of the 1960’s. The P-51 and PA-28 were
chosen for comparison due to their use of readily available NACA airfoils. Of the two, the P-51
would require more simplification due to the use of taper and different airfoils in the inboard and
outboard span regions. The aircraft compared are summarized in Table 1.

11
Table 1. Aircraft comparison summary.
Aircraft Known airfoil? Primary material Taper? Other notes
EA-300 Yes Composites Yes Stunt aircraft
P-51 Yes; varies Aluminum alloy Yes Fighter aircraft
PA-28 Yes Aluminum alloy No Utility aircraft; Chosen for analysis
Ultimately, the PA-28 was chosen for analysis since it could be readily analyzed as-designed,
with very minimal simplification needed. This has the added benefit that the optimized wing
structure produced at the culmination of the project can be directly compared to the aircraft as-
produced.
7 Aircraft Specifications
Key specifications for the PA-28 are reproduced below in Table 2. A 3-view drawing of the
aircraft is shown in Figure 1.
Table 2. Piper PA-28-140 Cherokee key specifications [1].
Parameter Value
Wingspan 9.2 m
Wing Area 15.14 m2
Airfoil NACA 652-415
Standard Empty Weight 544 kg
Maximum Takeoff Weight 975 kg
Cruise Speed 200 km/h
Service Ceiling 4400 m
Using the relation between wing area and span, the approximate chord length was determined as
follows:
𝑐 =
𝑆
(𝑠𝑝𝑎𝑛)
= 1.65 𝑚
(1)

12
Figure 1. Piper PA-28 Cherokee 140 3-view [1].

13
8 Load Diagrams
In order to determine the critical load conditions for the aircraft, V-n diagrams were constructed
in accordance with the FAR 23 regulations for utility aircraft. These are comprised of a
maneuver envelope, a gust envelope, and a combined loading flight envelope. All three
diagrams were constructed at sea level and service ceiling in order to capture the effects of air
density variation between the two altitude extremes encountered.
8.1 Maneuver Envelope
The maneuver envelope shows the required rated load factors at speeds up to dive velocity.
Design dive velocity is defined as 1.5 times cruise velocity. According to FAR 23.337 [2],
utility aircraft must be rated up to a positive load factor of 4.4 and negative load factor of -1.76.
Although the positive limit must be maintained at all speeds where possible, the negative load
limit may be reduced linearly from -1.76 at cruise to -1.0 at dive speed. Combined, these
conditions produce four critical maneuver limits, as listed in Table 3.
Table 3. Critical maneuver limits.
Maneuver Limit Speed Load Factor Description
PHAA Min. required for load factor* 4.4 Positive High AoA
PLAA Dive velocity (300 km/h) 4.4 Positive Low AoA
NHAA Min. required for load factor* -1.76 Negative High AoA
NLAA Cruise velocity (200 km/h) -1.76 Negative Low AoA
*See Equations (2) and (3).
Below certain speeds the aircraft will stall before reaching the rated positive and negative
maneuver load limits. At these velocities, the maximum positive load factor is given by
Equation (2), in which density is determined by the flight altitude and weight is given as the max
takeoff weight. Using max takeoff weight results in greater lift for a given load factor and thus
was chosen in order to represent a worst-case scenario.

14
𝑛 𝑝𝑜𝑠 =
𝐿
𝑊
=
1
2⁄ 𝜌𝑣2
𝐶𝐿,𝑚𝑎𝑥
𝑊/𝑆
(2)
Similarly, the maximum negative load factor is given by Equation (3).
𝑛 𝑛𝑒𝑔 =
𝐿
𝑊
=
1
2⁄ 𝜌𝑣2
𝐶𝐿,𝑚𝑖𝑛
𝑊/𝑆
(3)
The intersection of these curves with the positive and negative maneuver load limits define the
PHAA and NHAA maneuver limits, as referenced in Table 3. The complete maneuver
envelopes for sea level and at service ceiling are shown in Figure 2 and Figure 3 respectively.
Note that the maneuver limits are labeled.

15
Figure 2. Maneuvering envelope at sea level.
Figure 3. Maneuvering envelope at service ceiling (4400 m).

16
8.2 Gust Envelope
The gust envelope shows the required rated loads encountered under gust conditions during level
flight. From FAR 23.333 [2] upward/downward gusts of 15 m/s must be accounted for at cruise
velocity below altitudes of 6100 m. These gusts linearly decrease to 7.6 m/s at dive velocity.
Although the gust magnitudes change at altitudes above 6100 m, this lies beyond the service
ceiling for the PA-28 and are thus disregarded.
Furthermore, as gust strength typically increases gradually, a so-called gust alleviation factor is
employed to more accurately describe the load experienced [2]. This factor reduces the
magnitude of the gust encountered according to Equation (4).
𝑢 = 𝐾𝑢 𝑑𝑒 (4)
Where the gust alleviation factor is given by Equation (5).
𝐾 =
0.88𝜇
5.3 + 𝜇
(5)
The gust alleviation factor varies with the mass ratio given by Equation (6).
𝜇 =
2 𝑊 𝑆⁄
𝜌𝑔𝑐𝐶𝐿,𝛼
(6)
The gust encountered creates a change in angle of attack according to Equation (7), as discussed
by Raymer [3].
Δ𝛼 = tan−1
(
𝑢
𝑣
) ≅
𝑢
𝑣
(7)
This leads to a change in lift according to Equation (8).
Δ𝐿 = 1
2⁄ 𝜌𝑣2
𝑆 ∗ (𝐶𝐿,𝛼Δ𝛼) (8)
Where the 3D lift-curve slope is approximated by Equation (9).

17
𝐶𝐿,𝛼 =
𝐶𝑙,𝛼
1 +
𝐶𝑙,𝛼
𝜋 ∗ 𝐴𝑅 ∗ 𝑒
(9)
Thus, the change in load factor can be described by Equation (10). Note, this result mirrors
analysis presented by Raymer [3] and Megson [4] and is simply an alternate form of the equation
presented in FAR 23.341 [2]. Equation (10) is used because it lends itself to the use of metric
units, unlike the FAR equations which incorporates imperial conversion factors directly.
Δ𝑛 =
Δ𝐿
𝑊
=
1
2⁄ 𝜌𝑢𝑣𝐶𝐿,𝛼
𝑊 𝑆⁄
(10)
As the FAR regulations assume gusts are encountered during steady, level flight, the final load
factor due to upward/downward gusts are given by Equation (11). The resulting gust envelopes
at sea level and service ceiling are shown in Figure 4 and Figure 5, respectively.
𝑛 = 1 ± Δ𝑛 (11)

18
Figure 4. Gust envelope at sea level.
Figure 5. Gust envelope at service ceiling (4400 m).

19
8.3 Combined Flight Envelope
By overlaying the maneuvering and gust envelopes, the combined flight envelope was found for
each altitude. The maximum positive and negative load factors were extracted and plotted in
MATLAB. Additionally, the positive and negative stall speeds were calculated as the velocities
at which the load factor was equal to 1.0 and -1.0, respectively. These speeds are marked as the
leftmost velocity boundary as the aircraft cannot fly at lower speeds. The combined flight
envelopes at sea level and service ceiling are shown in Figure 6 and Figure 7.
Critical load conditions were determined as the maximum of the maneuver limits and
upward/downward gust conditions. These are marked on the combined flight envelopes below.
Additionally, these are listed in Table 4, including lift coefficient, velocity, and load factor for
each condition at sea level. Similarly, the critical load conditions at service ceiling are listed in
Table 5. Notably, upward and downward gust loads at both altitudes did not exceed maneuver
loads and were thus discarded.
Table 4. Critical load conditions at sea level.
Load Condition PHAA PLAA NHAA NLAA Upward Gust Downward Gust
n 4.40 4.40 -1.76 -1.76 N/A N/A
CL 1.62 0.65 -1.22 -0.59 N/A N/A
V [km/h] 190 300 139 200 N/A N/A
AoA [deg] 21.2 6.0 -20.8 -11.3 N/A N/A
Table 5. Critical load conditions at service ceiling.
Load Condition PHAA PLAA NHAA NLAA Upward Gust Downward Gust
n 4.40 4.40 -1.76 -1.76 N/A N/A
CL 1.62 1.02 -1.22 -0.92 N/A N/A
V [km/h] 238 300 173 200 N/A N/A
AoA [deg] 21.3 11.4 -20.8 -16.2 N/A N/A

20
Figure 6. Combined flight envelope at sea level.
Figure 7. Combined flight envelope at service ceiling (4400 m).

21
9 Aerodynamic Analysis
Using the specs for the Piper Cherokee initial 2D calculations based on the NACA 65-425 airfoil
were performed in XFLR5 to determine the lift curve. From this lift curve the min and max 𝐶𝑙
were determined to be -1.27 and 1.63 respectively. The lift curve slope 𝐶𝑙 𝛼
was also found to be
0.114.
Figure 8. Lift curve data for NACA 65-415
In addition, the pressure coefficient was found as a function of percentage of chord at various
angles of attack. This allowed analysis of the primary forces on the wing due to the basic
principles of fluid mechanics and the pressure differences on the upper and lower surfaces of the
wing. Figure 9 shows a sample plot at an angle of attack of 9 degrees as well as a visual
representation of the pressure forces on the wing.
-1.5
-1
-0.5
0
0.5
1
1.5
2
-25 -20 -15 -10 -5 0 5 10 15 20 25
Cl
Alpha

22
Figure 9. Cp vs. Chord for AoA = 9 degrees
By taking advantage of XLFR5’s 3D Wing/Plane design features, the Piper Cherokee’s wing was
recreated and preliminary aerodynamic analysis was performed on it for a few of the major
loading conditions in the V-n diagram, specifically sea level gust conditions. Using the
associated lift coefficients, the wing was iterated through angles of attack between 25 and -25
degrees to find the 𝐶𝑙 values that matched the required lift as specified in the V-n diagram. For
example, for PLAA and NLAA at sea level, the respective 𝐶𝑙 values are 0.65 and -0.59. By
iterating through the angles of attack until a value close to the required 𝐶𝑙 was found, graphs for
drag coefficient and lift coefficient as a function of span were produced at this corresponding
angle of attack. Figure 10 and Figure 11 show two examples of this analysis.

23
Figure 10. Cd vs. Span at AoA of 9 degrees.
Figure 11. Cl vs. Span at AoA of 9 degrees.
0.035
0.037
0.039
0.041
0.043
0.045
0.047
0.049
-20 -15 -10 -5 0 5 10 15 20
Cd
Span
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
-20 -15 -10 -5 0 5 10 15 20
Cl
Span

24
10 Aerodynamic Loads
For each critical load case, the corresponding XFLR5 case data was reinterpreted to extract
pressure data across the wing, as well as normal and axial force, shear, and moment distributions
along the span length. The data manipulation was performed in MATLAB, with the intention
that the data extracted can then be used for structural analysis in subsequent scripts and Abaqus
FEA. The following sections discuss the procedures used to create each dataset. Furthermore,
each section includes a sample plot from the results for the PHAA case at sea level. The
complete plots can be found in Appendix A: Air Loads.
10.1 Chordwise Pressure Distributions
From XFLR5 pressure coefficient vs. chord data, the chordwise pressure distributions were
simply found by scaling the airfoil surface normal vectors by their associated pressure
coefficient. The center of pressure was found by applying Equation 1.17 in Anderson [5] to find
the moment coefficient due to pressures acting along 𝑥. It is reproduced as Equation (12) below.
𝐶 𝑀,𝐿𝐸,𝑥 = ∑ 𝐶 𝑝,𝑖 𝑥𝑖Δ𝑥𝑖
𝑖
(12)
From the moment coefficient about the leading edge, the center of pressure 𝑥-coordinate was
found from Equation 1.18 in Anderson [5], reproduced as Equation (13). The center of pressure
𝑦-coordinate was found using the same method.
𝑥 𝑐𝑝 = −𝐶 𝑀,𝐿𝐸,𝑥/𝐶 𝑛 (13)
The pressure coefficient distribution and center of pressure for the PHAA case at sea level is
shown in Figure 12. The results are sufficiently similar to the plot produced within XFLR5 to
validate the method – in general results are accurate to within 5% error.

25
Figure 12. Pressure coefficient vs. chord for the PHAA case at sea level.
10.2 Surface Pressure Distributions
Surface pressures across the wing were found by scaling the chordwise pressure distribution at
each spanwise location by the lift coefficient produced by XFLR5 at that location. This method
is based on the assumption that the 2D pressure profile generated using the same velocity and
angle of attack as the 3D lift coefficient profile will not vary significantly across the wing.
Although this may neglect some 3D effects, the results seem reasonable. The 2D “baseline”
coefficient of lift used for scaling was found by rotating the normal and axial force coefficients
according to Equation (14). These were found from Equations 1.15 and 1.16 in Anderson,
reproduced as Equation (15) and Equation (16), respectively.

26
𝐶𝑙 = 𝐶 𝑛 cos(𝛼) − 𝐶 𝑎sin(𝛼) (14)
𝐶 𝑛 = − ∑ 𝐶 𝑝Δ𝑥𝑖
𝑖
(15)
𝐶 𝑎 = ∑ 𝐶 𝑝Δ𝑦𝑖
𝑖
(16)
After scaling, gauge pressures across the surface were then found from Equation (17) [5].
𝑝𝑖 =
1
2
𝜌∞ 𝑣∞
2
∗ 𝐶 𝑝,𝑖 (17)
The distributions were then exported as CSV files for use with Abaqus FEA. Gauge pressures
across the wing for the PHAA case at sea level are shown in Figure 13. Surface gauge pressures
for the PHAA case at sea level.Figure 13.
Figure 13. Surface gauge pressures for the PHAA case at sea level.

27
10.3 Normal Distributions
Normal load distributions were produced by rotating the lift and drag distributions according to
Equation (18).
𝑁 = 𝐿𝑐𝑜𝑠(𝛼) + 𝐷𝑠𝑖𝑛(𝛼) (18)
Lift and drag distributions were found from the spanwise lift and drag coefficient data using
Equation (19) and Equation (20), respectively.
𝐿 =
1
2
𝜌∞ 𝑣∞
2
𝐶𝐿,𝑖 ∗ (𝑐ℎ𝑜𝑟𝑑) ∗ Δ𝑧𝑖 (19)
𝐷 =
1
2
𝜌∞ 𝑣∞
2
𝐶 𝐷,𝑖 ∗ (𝑐ℎ𝑜𝑟𝑑) ∗ Δ𝑧𝑖 (20)
Shear was found as the integral of normal load from tip to root according to Equation (21).
𝑉𝑦 = ∫ 𝑁(𝑧) ∗ 𝑑𝑧
𝑧=0
𝑧=
𝑠𝑝𝑎𝑛
2
(21)
Moment about the 𝑥-axis was found as the integral of shear, according to Equation (22)
𝑀𝑥 = ∫ 𝑉𝑦(𝑧) ∗ 𝑑𝑧
𝑧=0
𝑧=
𝑠𝑝𝑎𝑛
2
(22)
The final normal load and shear distributions for the PHAA case at sea level are shown in Figure
14. The moment distribution about the 𝑥-axis for the PHAA case at sea level are shown in
Figure 15. Note, that in both figures the absolute value of the distributions are plotted for clarity.
The results were verified by matching the total halfspan load to the shear load at the root.

28
Figure 14. Absolute normal load and shear distributions for the PHAA case at sea level.
Figure 15. Absolute moment distribution about the x-axis for the PHAA case at sea level.

29
10.4 Axial Distributions
Normal load distributions were produced by rotating the lift and drag distributions according to
Equation (18).
𝐴 = −𝐿𝑠𝑖𝑛(𝛼) + 𝐷𝑐𝑜𝑠(𝛼) (23)
Lift and drag distributions were found from the spanwise lift and drag coefficient data using
Equation (19) (19) and Equation (20)(20), respectively.
Shear was found as the integral of normal load from tip to root according to Equation (24).
𝑉𝑥 = ∫ 𝐴(𝑧) ∗ 𝑑𝑧
𝑧=0
𝑧=
𝑠𝑝𝑎𝑛
2
(24)
Moment about the 𝑦-axis was found as the integral of shear, according to Equation (25)
𝑀 𝑦 = ∫ 𝑉𝑥(𝑧) ∗ 𝑑𝑧
𝑧=0
𝑧=
𝑠𝑝𝑎𝑛
2
(25)
The final axial load and shear distributions for the PHAA case at sea level are shown in Figure
16. The moment distribution about the 𝑦-axis for the PHAA case at sea level are shown in
Figure 17. Note, that in both figures the absolute value of the distributions are plotted for clarity.
The results were verified by matching the total halfspan load to the shear load at the root.

30
Figure 16. Axial load and shear distributions for the PHAA case at sea level.
Figure 17. Absolute moment about the Y-axis for the PHAA case at sea level.

31
10.5 3D-Load Distributions
A set of load distributions were generated with MATLAB 3D graphics to visualize the
application of normal and axial load distributions of the wing. For plotted purposes, these loads
were normalized and centered at the center of pressure. Interestingly, the axial forces point
toward the airfoil leading edge, rather than the trailing edge as initially expected. This can be
rationalized by the dominance of lift force over drag: at the angles of attack examined, the lift
vector is always directed towards the leading edge. An example distribution is shown in Figure
18.
Figure 18. 3D-load distribution for the PHAA case at sea level.

32
11 Material Selection
For simplicity of calculation and realism for a utility aircraft, material selection was limited to
aluminum alloys only. A range of possible alloys were identified from aluminum manufacturers,
such as ALCOA [6], and aerospace extrusions suppliers, such as MS Aerospace Materials [7].
Ultimately, material properties were sourced from Aerospace Specification Metals (ASM) [8], as
they included more details than most other suppliers. All properties were sourced through them
to maintain consistency. Alloys include: 2024-T3, 6061-T6, 7050-T7, 7075-T6, and 7178-T6.
The choice of alloy was analyzed during optimization, however in general terms the 2XXX
series was primarily examined for skins due to high fatigue strength and the 7XXX series for
stringers and spar caps due to high strength [6]. The 6XXX series will likely be used for
fasteners only due to unremarkable properties overall.

33
12 Structural Analysis
12.1 Approximations and Assumptions
A structural idealization involving point mass booms and webs as outlined in the Megson
textbook was used in the structural analysis of the wing structure. This structural idealization was
implemented in a Monte Carlo optimization code via MATLAB. The following is a list of the
assumptions and simplifications made for the realization of such an analysis. In each sub-section,
additional assumptions and idealizations may also be listed in order to clarify certain processes
and methodologies.
 Wing is analyzed as a cantilever beam in the spanwise direction.
 Lift and drag data are constant distributed loads across cross sectional area in both the x
and y directions.
 Bending stresses are calculated at each boom location along the wing.
 Max shear and moment is at the root of the wing. Counter clockwise moment is negative.
 Shear flows and resulting analyses utilize Megson’s structural idealization.
12.2 MATLAB Code
A MATLAB code was written in order to implement a Monte Carlo optimization script. The
Monte Carlo process utilizes a variety of pre-defined parameters such as flight conditions (air
density ρ and airspeed v), shear web thickness (t1, t2, t3,…), sparcap dimensions, and spar and
stringer placement to output factor of safety values corresponding to bending stresses, shear
stresses, and buckling. This process is explained in detail under the section discussing
optimization.

34
12.3 Panel Method
The geometry of the airfoil represents boundary conditions for the shaping and placing of
irregularly shaped spars, stringers, and other internal structures. To create a standardized baseline
upon which structural analysis can be used for later sections, a panel method was utilized. The
airfoil coordinates set (plotted form trailing edge to leading edge) was converted into a set that
contained information on the panel length, its corresponding midpoint coordinate, and the
tangential vector defining that panel. Both the tangential and normal vectors were normalized
into unit vectors for ease of calculations in future steps. Figure 19 shows an illustration of how
this was done.
Figure 19. Panel Method.
Skin Thickness
Tangent (i) Normal
y
X
Airfoil Coordinate
Panel Coordinate
Airfoil Coordinate dx (i)
dy (i)

35
12.4 Initial Geometry
The MATLAB code allows the user to pre-define specifications, namely: spar cap area, web
thickness, and the percentile chord position of placement. Panel information deduced from
Figure 19 was used in the placement and geometry of the spars. The preliminary first iteration
analysis of the wingbox involves two spars, an I-beam and a C-beam. These were initially placed
at the 40% and 70% chord respectively. The rear spar is a C-beam, represent the farthest rear
edge of the wing box, due to the attached hinged control surfaces The I-beam is a typical
bending-resistant cross section with flanges shaped parallel to the airfoil boundary. These were
simplified by treating the spar caps simply as point areas. The initial geometry is shown in
Figure 20.
Figure 20. Initial geometry for preliminary analysis.

36
12.5 Centroid Calculations
To tabulate the total centroid of the cross sectional area shown in Error! Reference source not
ound. and Error! Reference source not found., a two-step method was used: first determine the
centroid of the two spar sections, then determine the centroid of the spars combined with the
wing skin. Centroid calculations for spars were done in parts: for the I-beam, the two rectangular
flanges and the middle trapezoidal cross section is found individually and then combined using
the centroid formula. The C-beam is calculated in a similar fashion, this time with 3 trapezoidal
cross sections instead. The simplified process is shown below in Equation (26) for one
coordinate:
𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑥,𝐵𝑒𝑎𝑚
=
𝑥 𝑐,𝑢𝑝𝑝𝑒𝑟 𝐴𝑟𝑒𝑎 𝑢𝑝𝑝𝑒𝑟 + 𝑥 𝑐,𝑚𝑖𝑑𝑑𝑙𝑒 𝐴𝑟𝑒𝑎 𝑚𝑖𝑑𝑑𝑙𝑒 + 𝑥 𝑐,𝑙𝑜𝑤𝑒𝑟 𝐴𝑟𝑒𝑎𝑙𝑜𝑤𝑒𝑟
𝐴𝑟𝑒𝑎 𝑢𝑝𝑝𝑒𝑟 + 𝐴𝑟𝑒𝑎 𝑚𝑖𝑑𝑑𝑙𝑒 + 𝐴𝑟𝑒𝑎𝑙𝑜𝑤𝑒𝑟
(26)
Note that, since the material is homogeneous, then the mean of the endpoints of the polygon
would be the centroid of the enclosed polygon. As such, the MATLAB function “mean(X)” was
used. Likewise, the area of the enclosed geometry was calculated using the “polyarea”
MATLAB function. Once the centroid for the spars were found, the wing skin was combined
into the formula, yielding Equation (27):
𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑥,𝑇𝑜𝑡𝑎𝑙
=
𝑥 𝐶,𝐼 𝐵𝑒𝑎𝑚 𝐴𝑟𝑒𝑎𝐼 𝐵𝑒𝑎𝑚 + 𝑥 𝐶,𝐶 𝐵𝑒𝑎𝑚 𝐴𝑟𝑒𝑎 𝐶 𝐵𝑒𝑎𝑚 + ∑ 𝑃𝑎𝑛𝑒𝑙 𝑥(𝑖) ∗ 𝑃𝑎𝑛𝑒𝑙 𝐴𝑟𝑒𝑎(𝑖)𝑁
1
𝐴𝑟𝑒𝑎𝐼 𝐵𝑒𝑎𝑚 + 𝐴𝑟𝑒𝑎 𝐶 𝐵𝑒𝑎𝑚 + ∑ 𝑃𝑎𝑛𝑒𝑙 𝐴𝑟𝑒𝑎(𝑖)𝑁
1
(27)
This yields the coordinates marked in Figure 20.

37
12.6 Moment and Bending
The second moment of area about the x and y-axes were used in determining axial stresses
exerted on the wing box structure. In determining this characteristic, a major approximation was
made in the contribution of the wing skin. Each panel contribution was assumed as a point mass-
area located at their respective coordinates (see Figure 19 for an explanation of the panel
coordinate system). Using the polygon formula for determining inertia, the spar contributions are
tabulated according to N-coordinates corresponding to endpoints, numbered in a
counterclockwise fashion. These are shown below in Equations (28) - (30):
𝐼 𝑥𝑥 =
1
12
∑[(𝑦𝑖
2
+ 𝑦𝑖 𝑦𝑖+1 + 𝑦𝑖+1
2
)(𝑥𝑖 𝑦𝑖+1 − 𝑥𝑖+1 𝑦𝑖)]
𝑁−1
1
(28)
𝐼 𝑦𝑦 =
1
12
∑[(𝑥𝑖
2
+ 𝑥𝑖 𝑥𝑖+1 + 𝑥𝑖+1
2
)(𝑥𝑖 𝑦𝑖+1 − 𝑥𝑖+1 𝑦𝑖)]
𝑁−1
1
(29)
𝐼 𝑥𝑦 =
1
24
∑[(𝑥𝑖 𝑦𝑖+1 + 2𝑥𝑖 𝑦𝑖 + 2𝑥𝑖+1 𝑦𝑖+1 + 𝑥𝑖+1 𝑦𝑖)(𝑥𝑖 𝑦𝑖+1 − 𝑥𝑖+1 𝑦𝑖)]
𝑁−1
1
(30)
Since a point mass has no inertia, the parallel axis theorem is all that is needed, thus Equations
(31) - (33) below define the contributions from the skin:
𝐼 𝑥𝑥 = ∑ 𝑃𝑎𝑛𝑒𝑙𝐴𝑟𝑒𝑎 𝑥(𝑖)
𝑁−𝑝𝑎𝑛𝑒𝑙𝑠
1
∗ (𝑃𝑎𝑛𝑒𝑙𝐶𝑜𝑜𝑟𝑑 𝑥(𝑖) − 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑥)2
(31)

38
𝐼 𝑦𝑦 = ∑ 𝑃𝑎𝑛𝑒𝑙𝐴𝑟𝑒𝑎 𝑦(𝑖)
1
∗ (𝑃𝑎𝑛𝑒𝑙𝐶𝑜𝑜𝑟𝑑 𝑦(𝑖) − 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑦)2
(32)
𝐼 𝑥𝑦 = ∑ 𝑃𝑎𝑛𝑒𝑙𝐴𝑟𝑒𝑎 𝑥(𝑖)
1
∗ (𝑃𝑎𝑛𝑒𝑙𝐶𝑜𝑜𝑟𝑑 𝑥(𝑖) − 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑥)
∗ (𝑃𝑎𝑛𝑒𝑙𝐶𝑜𝑜𝑟𝑑 𝑦(𝑖) − 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑦)
(33)
The total inertia of the entire cross section is the sum of the contributions of each of the above
parts. These values are substituted into the bi-directional bending equation shown below as
Equation (34), where the x and y coordinates are boom positions in relation to the centroid. Note
that the moments are tabulated from the root of the wing and that drag creates a negative y-
moment while lift creates a negative x-moment.
𝜎𝑧 =
𝐼 𝑥𝑥 𝑀 𝑦 − 𝐼 𝑥𝑦 𝑀 𝑥
𝐼 𝑥𝑥 𝐼 𝑦𝑦 − 𝐼 𝑥𝑦
2
𝑥 +
𝐼 𝑦𝑦 𝑀𝑥 − 𝐼 𝑥𝑦 𝑀 𝑦
2
𝑦
(34)
A sample result of bending stresses under PHAA load cases are shown below for the optimized
geometry:
Table 6: SeaLevel Gust at AoA 9
X Position (m) Y-Position (m) Axial Stress (MPa)
0.743 -0.086 216
0.658 0.159 -231

39
12.7 Shear Flow
Shear forces were considered in the design of the wingbox. The values tabulated are used in
determining factor of safety and in buckling considerations. In order to calculate the shear flow
of the given geometry, the following assumptions were made:
 Cuts are made left of the spar placement
 Panel walls along skin between boom placements have no effective stress carrying
capabilities
 Pass the resultant shear due to lift and drag through the center of pressure
 Utilize the cross sectional centroid as the reference origin
 Omit the control surfaces in the last 25% of the chord from analysis
 Counterclockwise positive convention for flow
 All cell sections must share the same rate of twist
Shear flow for a closed multicell section is determined from making cuts and finding the open
shear flow; and then combining that result with the closed shear flow numbers of the cut. The
open shear flow Equation (35) is listed below:
𝑞 𝑠 = − {[
𝑉𝑥 𝐼 𝑥𝑥 − 𝑉𝑦 𝐼 𝑥𝑦
2
] ∑ 𝐵𝑟 𝑥 𝑟
𝑛
𝑟=1
+ [
𝑉𝑥 𝐼 𝑦𝑦 − 𝑉𝑦 𝐼 𝑥𝑦
2
] ∑ 𝐵𝑟 𝑥 𝑟
𝑛
𝑟=1
}
(35)
Equation (35), combined with the panel method is executed as follows: the coordinate of the
boom is the x and y coordinates and the inertias and shears were calculated through beam
analyses in previous sections. These values correspond to an open cross section and are assumed
to be constant along the walls connecting adjacent booms.
The next step of the shear flow calculations involves a summing of the moment contribution
from internal shear flow and equating it to the moments due to the external shear. In order to

40
execute this procedure, the moment was taken about the centroid of the cross section. With the
positive x pointing from leading edge to trailing edge, and the positive y pointing upwards from
the centerline of the airfoil, the moment Equation (36) taken about the centroid is denoted as:
−𝑉𝑥 𝑑𝑦 + 𝑉𝑦 𝑑𝑥 = ∮ 𝑝𝑞 𝑜𝑝𝑒𝑛 𝑑𝑠
𝑠
0
+ 2𝐴̅ 𝑞 𝑐𝑙𝑜𝑠𝑒𝑑
(36)
The fine details of the expanded formula will not be shown due to the large number of terms;
however, the first integral can be expanded to show that:
∮ 𝑝𝑞 𝑜𝑝𝑒𝑛 𝑑𝑠
𝑠
0
= − ∫ 𝑞 𝑜𝑝𝑒𝑛 𝑦𝑑𝑥
𝑠
0
+ ∫ 𝑞 𝑜𝑝𝑒𝑛 𝑥𝑑𝑦
𝑠
0
(37)
The second term of Equation (36) (37)is dependent upon the number of cell sections, one q value
per cell, used in the wingbox. In order to solve the equation, however, another set of equations
must be used. The angle of twist equation is used to describe the torsion of a section and shown
below as Equation (38), where n represents the cell section in question:
𝑑𝜃
𝑑𝑧
=
1
2𝐴̅ 𝑛 𝐺
∮ 𝑞
𝑑𝑠
𝑡 (38)
By assuming that each cell section twists the same amount, n-1 equations of the above form can
be created to show the equivalence of twists.
𝑑𝜃
𝑑𝑧
)
1
=
𝑑𝜃
𝑑𝑧
)
2 (39)
12.8 Shear Center
To calculate shear center, the internal moments was equated to the external shear assuming that it
passed through the shear center. Summing the moments about the centroid and dropping the y-

41
term, Equation (36) would then yield the x-distance from the centroid that the shear center would
be. Then, since shear center is defined as the location where if a shear force were passed through
it no torsion would occur, the angle of twist equations should all be equal to zero. With these
three equations, the unknowns dx, q1, and q2 can be solved. The y-coordinate of shear center
would be found similarly by passing the y-component of shear force through the coordinate
found before and solving the three equations. Thus, the shear center of the optimized wing is:
(.616, .03) m.
12.9 Buckling
A major issue with high compressive stresses exerted upon thin, long plate elements is in the
buckling of the plate. The formula used for calculating the critical buckling stress is dependent
upon the dimensions of the plate. In this case, the dimensions a and b correspond to the rib
spacing and the stringer spacing, respectively. Although only dummy stringers have been placed
in the structure, the basic code is already implemented utilizing the below relation in Equation
(40).
𝜎𝑐𝑟 =
𝑘𝜋2
𝐸
12(1 − 𝜈2)
(
𝑡
𝑏
)2
(40)
Where: 𝑘 = (
𝑚𝑏
𝑎
+
𝑚𝑎
𝑏
)
2
(41)
Each m corresponds to a mode of buckling. These modes of buckling will switch depending on
the ratio of a to b, governed by the equation in Sun’s book:
𝑎
𝑏
= √𝑚(𝑚 + 1). Due to the low
variance of k values for
𝑎
𝑏
ratios corresponding to mode numbers higher than 5, any ratio beyond
the fifth mode was treated with a k = 4 value. A graph of how these modes are related is included
in Figure 21. The output value of critical stress is compared to the axial stresses and shear

42
stresses that are exerted upon the panel member of that section (defined by the placement of the
stringers).
Figure 21. k-values for plate buckling of simply-supported edges [9].
12.10 Factor of safety
Factor of safety was the chosen metric to aid in optimization of the wing structure. For each
analysis done, bending, shears, and buckling, the critical results and the calculated results were
compared in a factor of safety ratio. This ratio is found by dividing the critical values by the
values yielded by the analysis. An example would be dividing the yield stress of the chosen
material by the bending stresses tabulated in Section 12.6. If the ratio is below 1, then the
structure fails at that point since the calculated numbers exceed that of the critical values. A
factor of safety corresponding to above 1 would be desired; however, it should not be above 1.5,
since excessively high factors of safety are indicative of overdesign.

43
13 Failure Criteria
13.1 Von Mises
In using the factor of safety metric, the von Mises yield criterion was also considered. A 2D
plane stress von Mises criteria was explored, which is defined by the below equation:
For the analysis of the skin of the wing, these 2-dimensional plates are considered to have no
stresses in the x-direction and only exhibit axial stress components in the z-direction and shear
along xy-directions. Given that the axial z-direction stresses are greater than the shear stresses by
a hundred times, it can be shown that 𝜎1 ≫ 𝜎2 and therefore the von Mises yield criterion is
approximated simply as 𝜎 𝑦𝑖𝑒𝑙𝑑 > 𝜎𝑥.
13.2 Paris’ Law
Fatigue is defined as the potential for a structure to fail due to cyclic loading, and Paris’ Law was
used to examine its effects on the designed wing. Fatigue is considered to be a three-part
process: crack initiation (stage I), crack growth (stage II), and eventually accelerated growth to
fracture (stage III). Generally fatigue prediction is based on experimental data, but with the use
of fracture mechanics predictions can be made (Pugno, 2006). One of the most popular methods
of predicting fatigue crack growth involves the use of Paris’ Law. Put simply, Paris’ Law can be
represented by Equation (43):
𝜎 𝑦𝑖𝑒𝑙𝑑 > √𝜎1
2
+ 𝜎1 𝜎2 + 𝜎2
2 (42)

44
𝑑𝑎
𝑑𝑁
= 𝐶∆𝐾 𝑚
(43)
which describes the crack growth rate as a function of the material properties and the variance in
the stress intensity factor. The stress intensity factor K can be represented in terms of the tensile
stress, crack length, and geometric parameters in Equation (44):
𝐾 = 𝜎𝑌√ 𝜋𝑎 (44)
Rearranging, the critical crack length can be found by substituting the fracture toughness for the
desired material and solving for the corresponding crack length. Using the values for Aluminum
6061, this critical crack length was determined to be 43.5 mm. This law is usually applied to
crack growth classified as Stage II, which means that the intensity alternates in a stable manner
while still remaining above a specified threshold value while the crack propagates. For Stage II
crack growth, the general relationship between C and m is of the form of Equation (45).
ln(𝐶) = 𝑎 + 𝑏𝑚
(45)
This is determined based on the linear relationship between C and m when plotted on a
logarithmic graph, similar to the general stress-strain correlation (Cortie and Garrett, 1988).
Using these definitions along with the relevant material properties, the number of cycles that will
result in fracture of the material were determined for the designed wing. This was accomplished
by separating variables and integrating both sides to produce Equation (46).
𝑁𝑓𝑟𝑎𝑐𝑡𝑢𝑟𝑒 =
2(𝑎 𝑐𝑟𝑖𝑡
2−𝑚
2
− 𝑎𝑖𝑛𝑖𝑡𝑖𝑎𝑙
2−𝑚
2
)
(2 − 𝑚)𝐶(∆𝜎𝑌√ 𝜋) 𝑚 (46)
Unfortunately, it can be difficult to determine both the initial crack length and the dimensionless
parameter Y. For this reason, it is generally assumed that initial crack length in an aircraft spar is
0.25 mm and the geometry suggests a Y value of approximately 1.12. In addition, Paris’ Law is

45
only supposed to be valid during low intensity load cycling or large values of a, so it doesn’t
apply to all the possible scenarios under which an aircraft wing could fail due to fatigue.
Regardless, a preliminary calculation was undertaken to estimate the number of cycles at which
the Piper Cherokee wing would fail. The following table contains the values for each parameter
used in this calculation.
Variable Value
𝑎 𝑐𝑟𝑖𝑡 2.22 mm
𝑎𝑖𝑛𝑖𝑡𝑖𝑎𝑙 0.25 mm
M 4.19
C 3.7E-12 mm/cycle
Y 1.12
σ 310 MPa
Using these values and equations, the number of cycles to failure is 175173. This is more than an
acceptable number of cycles, especially considering that it is highly unlikely that small aircraft
like the Piper Cherokee would frequently experience the maximum stress cycle at the critical
flight conditions or that there would be routine ultrasonic checks of the wing structure integrity.
Figure 22 shows the fatigue crack growth curve to failure for the designed wing structure
composed of Aluminum 6061.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
-20000 30000 80000 130000 180000
FlawSize(m)
Number of Loading Cycles
Figure 22: Fatigue crack growth curve to failure.

46
13.3 Structural Analysis Remarks
A number of bugs in the structural code need to be fixed before validating the numerical output
of the code. Although a number of the values outputted are on the right order of magnitude of
expected answers, a validation of the code must be also applied to a simpler model from perhaps
a textbook or other reference material. By the submission of the final report, such example
conditions will be explored before running the optimization code. Since the code is already in
place, the remaining work will involve debugging and correcting known errors. In reference to
the shear flow, however, a new method of integration will be attempted as outlined in the
Megson textbook An Introduction to Aircraft Structural Analysis under the “structural
idealization” section as opposed to the panel method described above. In addition, work will
need to be done on the buckling functions to encompass a wider range of buckling modes.

47
14 Aeroelastic Considerations
14.1 Divergence
During periods of flight when there are extremes in wing loading, there can be substantial
changes in the lift distribution due to the structural distortion of the wing. The structure of the
wing therefore must be designed in such a way that these extreme conditions result in a
balancing force to prevent a phenomenon known as wing divergence. Wing divergence occurs
when the lift vector creates a moment about the shear center that in turn increases the incidence
of the wing. This can create a positive feedback loop that can lead to the destruction of the wing
structure. In order to determine the critical divergence speed for the chosen wing design,
Equation (47) was used.
𝑉𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 = √
𝜋2 𝐺𝐽
2𝜌𝑒𝑐2 𝑠2(
𝛿𝑐1
𝛿𝛼
⁄ ) (47)
For example, for the designed wing at ceiling NHAA, the divergence speed is
692 m/s or approximately Mach 2. Since the single turboprop engine of the
Piper Cherokee could never reach such extreme speeds, the designed wing
structure is acceptable.
14.2 Aileron Reversal
When wings flex and are distorted, it can severely limit the effectiveness of the control surfaces.
At high speeds the forces on the aileron can cause the wing to twist significantly. When this
happens, the aileron that is meant to change the incidence of the wing will have markedly less of
an effect due to it’s own decreasing incidence. This phenomenon is known as aileron reversal.
Aileron reversal can be avoided by increasing the torsional rigidity of the wing structure. To
calculate the velocity at which aileron reversal occurs, Equation (48) is used.

48
𝑉𝑟𝑒𝑣𝑒𝑟𝑠𝑎𝑙 = √
−𝐾(
𝛿𝐶𝐿
𝛿𝜉⁄ )
1
2⁄ 𝜌𝑆𝐶(
𝛿𝐶 𝑀,0
𝛿𝜉⁄ )(
𝛿𝐶𝐿
𝛿𝛼
⁄ )
(48)
Using comparisons between similar small aluminum aircraft, the aileron reversal speed is
estimated at 244 m/s, or approximately Mach 0.71, which is far greater than the determined dive
speed of 83.3 m/s.
14.3 Flutter
Due to the ability of a wing structure to experience bending and torsion simultaneously, it can
experience a phenomena called flutter. Flutter occurs when an aerodynamic structure reaches a
harmonic frequency at a specific airspeed and then begins to oscillate. If the speed is increased,
this oscillations increase in magnitude and rather than stabilize, they begin to diverge. This
occurs due to positive feedback between the lifting force and the deflection of the wing, similar
to divergence. The best method of determining whether or not flutter will occur is generally
through extensive testing of the wing since it is generally a very complex structure. Since the
designed wing was not physically built, flutter analysis was not conducted on the finalized wing
structure.

49
15 Optimization
Although certain aspects of the aircraft are fixed, such as its chord length, airfoil choice, and
aerodynamic properties, its internal wing structure is generally unrestricted. Spar and stringer
placement, as well as material selection and component thicknesses are variable. An optimization
MATLAB script was devised in order to quickly evaluate multiple designs, using a Monte Carlo-
style method of analysis.
15.1 Methodology
In order to begin optimization, spar, stringer, rib, and skin properties were chosen to vary,
including: number, placement, and thickness. A minimum, maximum, and increment value were
then assigned to each property. For each property, the chosen increment determined how much
its value increased by for each iteration, up to the maximum value allowed. Each design case
was defined by compiling a single value from each property. Within MATLAB, a series of
design cases were developed from all the possible combinations of values.
From the set of design cases, the wing cross-sectional geometric properties were calculated for
each case, including centroid, area, moments of inertia, and weight per unit length. These
properties were fed into the structural analysis MATLAB code discussed in Section 12.2 to
check for skin buckling and to calculate stresses due to bending and shear at each critical load
condition. From these values, the minimum factors of safety for axial stress, shear stress, axial
buckling, and shear buckling were returned, along with the half-wing weight for each design
case. Designs were considered “successful” if all minimum factors of safety were greater than
1.5, as required by the FAR 23 regulations [2].
An “optimal” design was selected from the successful test cases as the design with minimum
weight. Ideally, the maximum factor should not be much greater than the minimum factor of

50
safety, however this metric was quickly abandoned: the variance in factor of safety was simply
too great to provide any meaningful information. Finally, fatigue life, divergence, aileron
reversal, and aeroelastic effects must be considered, although these are not currently incorporated
into a structural analysis script – these were calculated after an optimal design was chosen in
order to verify its failure modes.
15.2 Calculations
A total of four design passes were conducted. Initially, a broad range was set for each design
parameter with generally course increments. Each subsequent study was a refinement of the
previous study, by narrowing the focus of each parameter and decreasing the increment size,
based upon the optimal result of the previous case. Table 7 through Table 10 show the
parameters modified in each pass, as well as the minimum, maximum, and increments used. The
final column in each shows the optimal results for the given pass.
Table 7. First-pass Monte Carlo design parameters.
Parameter Units Minimum Maximum Increment Optimal
Fore spar location 𝑥/𝑐 0.20 0.60 0.20 0.20
Spar cap area 𝑚𝑚2 100 300 100 100
Spar web thickness 𝑚𝑚 1.0 3.0 1.0 1.0
Num. upper stringers N/A 10 40 15 25
Num. lower stringers N/A 10 40 15 25
Stringer areas 𝑚𝑚2 50 250 100 150
Skin thickness 𝑚𝑚 0.5 2.5 1.0 2.5
Rib spacing 𝑚 0.25 1.0 0.25 1.0

51
Table 8. Second-pass Monte-Carlo design parameters.
Spar cap area 𝑚𝑚2 150 250 50 150
Rib spacing 𝑚 0.5 1.0 0.5 1.0
Table 9. Third-pass Monte Carlo design parameters.
Rib spacing 𝑚 1.0 1.0 0.0 1.0
Table 10. Fourth-pass Monte Carlo design parameters.
Rib spacing 𝑚 1 1 0 1

52
15.3 Final Results
The optimal result from the forth-pass Monte Carlo was deemed acceptable. The final input
parameters are shown in Table 10. Material choice was varied in a number of subsequent tests
for the specific geometry, however the slight difference in material properties between
Aluminum 2024-T3 and the 7XXX series of alloys were found to have minimal impact. In the
end, 2024-T3 was chosen for all components for simplicity and fatigue strength. The resulting
half-span weight and factors of safety are shown in Table 11. The final geometry, including spar
and stringer locations, is shown in Figure 23. Axial stresses at the root for the PHAA load case
are shown in Figure 24, whilst shear stresses at the root are shown in Figure 25. The final
stresses at additional load cases can be found in Appendix 0.
Table 11. Final weight and minimum factors of safety across all critical load cases.
Half-span
weight [kg]
Min. axial
stress FoS
Min. shear
stress FoS
Min. axial
buckling FoS
Min. shear
buckling FoS
144 1.54 157 1.67 783
Figure 23. Optimized geometry.

53
Figure 24. Optimal design root axial stresses under PHAA at sea level.
Figure 25. Optimal design root shear stresses under PHAA at sea level.

54
15.4 Remarks
In all optimization tests, axial stresses and buckling under axial load were clearly the limiting
factors. Although axial stresses were expected, axial buckling was surprising. After
examination, the initial evenly-placed stringer distribution appeared adequate for most buckling
cases, but inadequate under certain conditions. The code was modified to increase stringer
density towards the wing center, from 0.20c to 0.50c, which caused the greatest increase in factor
of safety from testing. Although this reduced the number of stringers required, the “optimal”
skin thickness is still twice what was initially expected. A future iteration would ideally vary
stringer placement automatically, at the expense of many more test cases required. Alternatively,
the stringers could be placed by hand after examining local stresses, however this would not lend
itself to a Monte Carlo-style analysis.
On the whole, the Monte Carlo method was an efficient way to evaluate a vast multitude of
designs with relative ease, however the results are somewhat unexpected. Although as-built
PA-28-140 specifications were not available, from drawings the forward spar appears much
further back. Additionally, the full wing weight for the optimized wing is 288kg, which seems
unreasonable for an aircraft with an empty weight of 544kg. Further testing and refinements of
the method would be required if it were to be used for the detailed design of an actual aircraft.

55
16 Finite Element Analysis
During the study of our Piper Cherokee wing, it was important to utilize finite element analysis
(FEA). Although COMSOL was suggested, Abaqus was chosen to perform this analysis.
To begin the FEA on the wing, a CAD assembly model was created to accurately represent the
physical wing, with two ribs, spars, stringers, and control surfaces, as shown in XX. Once the
CAD model was determined to fit specifications, it was saved as a STEP file and imported into
Abaqus for FEA. The assembly was imported maintaining part independence; so all components
could be defined using the appropriate mesh to assure accuracy. Upon completing these steps,
the upper and lower surfaces of the wing were defined as Aluminum 6061 “shell” cells, and the
spar caps, ribs, and stringers were designated Aluminum 6061 “solid” cells due to their relative
thickness.
Once all elements were accurately represented, boundary conditions were placed on all faces that
were coincident with the fuselage of the aircraft to prevent displacement and/or rotation in any
direction.
After applying all necessary boundary conditions and assuring the correct definition of all
components, a mesh was created using free hex elements at a medium resolution. The created
mesh is shown in Figure 27. After creating the mesh and applying all necessary boundary
conditions, the pressure profile created using the data from XLFR5 at the critical load conditions
was imported and applied to the upper and lower surfaces of the wing. This pressure map is
shown in Figure 28.

56
Figure 26. Wing CAD model.
Figure 27: Mesh for Piper Cherokee wing.

57
Figure 28: Representation of pressure map.
Once all conditions were applied, the job was submitted for structural analysis. The results for
one loading scenario are shown in Figure 29 and Figure 30Error! Reference source not found..
Based on the loading scenarios run in Abaqus, the FEA results are on the same order of
magnitude and but do not provide similar values to the code. Based on cases run without
stringers the skin was observed to bulge outward like a pressure vessel. Therefore, it is possible
that by applying a gage pressure in our calculations to account for the internal wing pressure, the
way Abaqus defined the pressure profiles was incorrect. Due to time constraints the analysis
could not be recomputed. The results for all load cases run are included in the appendix; however
due to memory allocation issue on the SEAS computers, 3 cases could not be run to completion.

58
Figure 29. Example displacement loading scenario visual results for PLAA at sea level.
Figure 30. Example stress loading scenario visual results for PLAA at sea level.
.

59
17 Conclusions
Based on the structural calculations and optimization, there are some questionable results for the
designed wing. One of the most suspect outcomes is the optimized skin thickness of 3.2 mm.
This seems excessively thick when compared to similar aircraft, and it increases the weight
significantly. The proposed total wing weight comes to approximately 282 kg, which seems to be
far greater than expected for a 544 kg empty weight aircraft. In addition, the optimized wing has
more stringers than anticipated, despite lack of a direct comparison to the actual Piper Cherokee.
Finally, the spar webs seem too thin and the forward spar was predicted to be closer to the
maximum thickness of the selected airfoil rather than approaching the leading edge. Although
the FEA also does not agree well with the derived results, this is assumed to be due to a default
pressure definition in Abaqus that was discovered too late. For future work, the results of the
optimization should include a revised analysis on stringer placement. Currently, the limiting
factor for the wing structure is the axial buckling of the plates due to compressive stresses.

60
18 References
[1] Wikipedia, "Piper PA-28 Cherokee," 13 April 2015. [Online]. Available:
http://en.wikipedia.org/wiki/Piper_PA-28_Cherokee. [Accessed 16 April 2015].
[2] FAA Federal Aviation Regulations (FARS, 14 CFR), "FAR Part 23: Airworthiness
Standards: Normal, Utility, Aerobatic, Commuter Category Airplanes," 30 March 1967.
[Online]. [Accessed 16 April 2015].
[3] D. P. Raymer, "Structures and Loads," in Aircraft Design: A Conceptual Approach, 4th
Edition, Washington D.C., American Institute for Aeronautics and Astronautics, 1992, pp.
333-345.
[4] T. Megson, An Introduction to Aircraft Structural Analysis, Burlington: Elsevier Ltd., 2012.
[5] J. D. Anderson, Fundamentals of Aerodynamics, New York: McGraw-Hill, 2010.
[6] ALCOA, [Online]. Available: https://www.alcoa.com/global/en/home.asp.
[7] MS Aerospace Materials, [Online]. Available: http://www.msaerospacematerials.com/.
[8] Aerospace Specification Metals, Inc., [Online]. Available:
http://www.aerospacemetals.com/index.html.
[9] C. Sun, Mechanics of Aircraft Structures, 2nd Ed., New Jersey: Wiley & Sons, 2006.

1
Appendix A: Air Loads
Appendix A.1: Chordwise Pressure Distributions
Figure 31. Pressure coefficient vs. chord for the NHAA case at sea level.
Figure 32. Pressure coefficient vs. chord for the NLAA case at sea level.

2
Figure 33. Pressure coefficient vs. chord for the PHAA case at sea level.
Figure 34. Pressure coefficient vs. chord for the PLAA case at sea level.

3
Figure 35. Pressure coefficient vs. chord for the NHAA case at 4400m.
Figure 36. Pressure coefficient vs. chord for the NLAA case at 4400m.

4
Figure 37. Pressure coefficient vs. chord for the PHAA case at 4400m.
Figure 38. Pressure coefficient vs. chord for the PLAA case at 4400m.

5
Appendix A.2: 3D Pressure Distributions
Figure 39. Surface gauge pressures for the NHAA case at sea level.
Figure 40. Surface gauge pressures for the NLAA case at sea level.

6
Figure 41. Surface gauge pressures for the PHAA case at sea level.
Figure 42. Surface gauge pressures for the PLAA case at sea level.

7
Figure 43. Surface gauge pressures for the NHAA case at 4400m.
Figure 44. Surface gauge pressures for the NLAA case at 4400m.

8
Figure 45. Surface gauge pressures for the PHAA case at 4400m.
Figure 46. Surface gauge pressures for the PLAA case at 4400m.

9
Appendix B: Stresses for Optimal Geometry
Figure 47. Axial stresses for NHAA at sea level.
Figure 48. Shear stresses for NLAA at sea level.

10
Figure 49. Axial stresses for NLAA at sea level.
Figure 50. Shear stresses for NLAA at sea level.

11
Figure 51. Axial stresses for PHAA at sea level.
Figure 52. Shear stresses for PHAA at sea level.

12
Figure 53. Axial stresses for PLAA at sea level.
Figure 54. Shear stresses for PLAA at sea level.

13
Appendix B: Code Validation
This section utilizes a simplified example of a wing cross section loosely based upon Example
5.9 from the Second Edition of CT Sun’s Mechanics of Aircraft Structures. Modifications were
made to this example in order to explore fringe test cases and validate the MATLAB code used
in this report’s analysis. Figure 55, shows the geometry and values used for this test case. Table
12, Table 13, and Table 14 below also show the comparison of MATLAB outputs to the hand
calculations.
Figure 55. Test Case Geometry with Dimensions and Loads
Vy = 4800 N My = 1000 N
Mx = 48000 N
Vx = 100 N
A1 = A8 = .0002 m2
A2 = A7 = .0003 m2
= A4 = A5
A3 = A6 = .0004 m2
ts ts ts
tststs
ts
tC
tI
ts = 2 mm
tI = 4 mm
tC = 3 mm

14
Table 12 Comparison of Some Variables.
MATLAB Code Outputs Hand Calculations
Centroid (.5207, 0) m (.535, 0) m
Ixx 2.454E-04 m4
2.561E-04 m4
Iyy .0013 m4
7.977E-04 m4
Ixy -1.694E-21 m4
0.000E+00 m4
Shear Center (.479, 8.88E-16) m (.488, 2.655E-15) m
Table 13. Bending Stress Comparison for Test Case Booms.
Boom Number x y
Hand Calculations
(Pascals)
MATLAB Output
(Pascals)
1 1 0.2 3.807E+07 3.875E+07
2 0.6 0.2 3.757E+07 3.906E+07
3 0.4 0.2 3.732E+07 3.922E+07
4 0 0.2 3.682E+07 3.954E+07
5 0 -0.2 -3.816E+07 -3.872E+07
6 0.4 -0.2 -3.766E+07 -3.903E+07
7 0.6 -0.2 -3.741E+07 -3.919E+07
8 1 -0.2 -3.691E+07 -3.950E+07

15
Table 14. Shear flows.
Shear Panel MATLAB Output Hand Calculations
q12 -1781.75 -1939.98
q23 -5305.52 -5326.52
q34 -2870.40 -2835.65
q45 -6103.33 -5892.82
q56 -2815.59 -2701.53
q67 -5243.08 -5121.56
q78 -1724.25 -1759.76
q81 1386.32 1176.20
q36 -7387.18 -7204.27

16
Appendix C: FEA Results
Figure 56: Stress for NGAC at sea level.
Figure 57: Displacement for NGAC at sea level.

17
Figure 58: Stress for NGAC at ceiling.
Figure 59: Displacement for NGAC at ceiling.

18
Figure 60: Stress for NHAA at sea level.
Figure 61: Displacement for NHAA at sea level.

19
Figure 62: Stress for NHAA at ceiling.
Figure 63: Displacement for NHAA at ceiling.

20
Figure 64: Stress for PGAC at sea level.
Figure 65: Displacement for PGAC at sea level.

21
Figure 66: Stress for PGAC at ceiling.
Figure 67: Displacement for PGAC at ceiling.

22
Figure 68: Stress for PLAA at sea level.
Figure 69: Displacement for PLAA at sea level.

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