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

6

7

3 List of Symbols

Symbol Description Units

�̅�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 𝑚1

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 Chart1 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

𝑛𝑝𝑜𝑠 =𝐿

𝑊=

12⁄ 𝜌𝑣2𝐶𝐿,𝑚𝑎𝑥

𝑊/𝑆 (2)

Similarly, the maximum negative load factor is given by Equation (3).

𝑛𝑛𝑒𝑔 =𝐿

𝑊=

12⁄ 𝜌𝑣2𝐶𝐿,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 = 𝐾𝑢𝑑𝑒 (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 = 12⁄ 𝜌𝑣2𝑆 ∗ (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=

12⁄ 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 𝐶𝑙𝛼 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.

𝐶𝑀,𝐿𝐸,𝑥 = ∑ 𝐶𝑝,𝑖𝑥𝑖Δ𝑥𝑖

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.

𝑥𝑐𝑝 = −𝐶𝑀,𝐿𝐸,𝑥/𝐶𝑛 (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 = 𝐶𝑛 cos(𝛼) − 𝐶𝑎sin (𝛼) (14)

𝐶𝑛 = − ∑ 𝐶𝑝Δ𝑥𝑖

𝑖

(15)

𝐶𝑎 = ∑ 𝐶𝑝Δ𝑦𝑖

𝑖

(16)

After scaling, gauge pressures across the surface were then found from Equation (17) [5].

𝑝𝑖 =1

2𝜌∞𝑣∞

2 ∗ 𝐶𝑝,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

2𝜌∞𝑣∞

2 𝐶𝐿,𝑖 ∗ (𝑐ℎ𝑜𝑟𝑑) ∗ Δ𝑧𝑖 (19)

𝐷 =1

2𝜌∞𝑣∞

2 𝐶𝐷,𝑖 ∗ (𝑐ℎ𝑜𝑟𝑑) ∗ Δ𝑧𝑖 (20)

Shear was found as the integral of normal load from tip to root according to Equation (21).

𝑉𝑦 = ∫ 𝑁(𝑧) ∗ 𝑑𝑧𝑧=0

𝑧=𝑠𝑝𝑎𝑛

2

(21)

Moment about the 𝑥-axis was found as the integral of shear, according to Equation (22)

𝑀𝑥 = ∫ 𝑉𝑦(𝑧) ∗ 𝑑𝑧𝑧=0


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) ∗ 𝑑𝑧𝑧=0


2

(24)

Moment about the 𝑦-axis was found as the integral of shear, according to Equation (25)

𝑀𝑦 = ∫ 𝑉𝑥(𝑧) ∗ 𝑑𝑧𝑧=0


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:

𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑥,𝐵𝑒𝑎𝑚

= 𝑥𝑐,𝑢𝑝𝑝𝑒𝑟𝐴𝑟𝑒𝑎𝑢𝑝𝑝𝑒𝑟 + 𝑥𝑐,𝑚𝑖𝑑𝑑𝑙𝑒𝐴𝑟𝑒𝑎𝑚𝑖𝑑𝑑𝑙𝑒 + 𝑥𝑐,𝑙𝑜𝑤𝑒𝑟𝐴𝑟𝑒𝑎𝑙𝑜𝑤𝑒𝑟

𝐴𝑟𝑒𝑎𝑢𝑝𝑝𝑒𝑟 + 𝐴𝑟𝑒𝑎𝑚𝑖𝑑𝑑𝑙𝑒 + 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):

𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑥,𝑇𝑜𝑡𝑎𝑙

= 𝑥𝐶,𝐼 𝐵𝑒𝑎𝑚𝐴𝑟𝑒𝑎𝐼 𝐵𝑒𝑎𝑚 + 𝑥𝐶,𝐶 𝐵𝑒𝑎𝑚𝐴𝑟𝑒𝑎𝐶 𝐵𝑒𝑎𝑚 + ∑ 𝑃𝑎𝑛𝑒𝑙𝑥(𝑖) ∗ 𝑃𝑎𝑛𝑒𝑙𝐴𝑟𝑒𝑎(𝑖)𝑁

1

𝐴𝑟𝑒𝑎𝐼 𝐵𝑒𝑎𝑚 + 𝐴𝑟𝑒𝑎𝐶 𝐵𝑒𝑎𝑚 + ∑ 𝑃𝑎𝑛𝑒𝑙𝐴𝑟𝑒𝑎(𝑖)𝑁1

(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):

𝐼𝑥𝑥 =1

12∑[(𝑦𝑖

2 + 𝑦𝑖𝑦𝑖+1 + 𝑦𝑖+12 )(𝑥𝑖𝑦𝑖+1 − 𝑥𝑖+1𝑦𝑖)]

𝑁−1

1

(28)

𝐼𝑦𝑦 =1

12∑[(𝑥𝑖

2 + 𝑥𝑖𝑥𝑖+1 + 𝑥𝑖+12 )(𝑥𝑖𝑦𝑖+1 − 𝑥𝑖+1𝑦𝑖)]

𝑁−1

1

(29)

𝐼𝑥𝑦 =1

24∑[(𝑥𝑖𝑦𝑖+1 + 2𝑥𝑖𝑦𝑖 + 2𝑥𝑖+1𝑦𝑖+1 + 𝑥𝑖+1𝑦𝑖)(𝑥𝑖𝑦𝑖+1 − 𝑥𝑖+1𝑦𝑖)]

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:

𝐼𝑥𝑥 = ∑ 𝑃𝑎𝑛𝑒𝑙𝐴𝑟𝑒𝑎𝑥(𝑖)

𝑁−𝑝𝑎𝑛𝑒𝑙𝑠

1

∗ (𝑃𝑎𝑛𝑒𝑙𝐶𝑜𝑜𝑟𝑑𝑥(𝑖) − 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑥)2 (31)

38

𝐼𝑦𝑦 = ∑ 𝑃𝑎𝑛𝑒𝑙𝐴𝑟𝑒𝑎𝑦(𝑖)


1

∗ (𝑃𝑎𝑛𝑒𝑙𝐶𝑜𝑜𝑟𝑑𝑦(𝑖) − 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑦)2

(32)

𝐼𝑥𝑦 = ∑ 𝑃𝑎𝑛𝑒𝑙𝐴𝑟𝑒𝑎𝑥(𝑖)


1

∗ (𝑃𝑎𝑛𝑒𝑙𝐶𝑜𝑜𝑟𝑑𝑥(𝑖) − 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑥)

∗ (𝑃𝑎𝑛𝑒𝑙𝐶𝑜𝑜𝑟𝑑𝑦(𝑖) − 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑦)

(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 = 𝐼𝑥𝑥𝑀𝑦 − 𝐼𝑥𝑦𝑀𝑥

𝐼𝑥𝑥𝐼𝑦𝑦 − 𝐼𝑥𝑦2

𝑥 + 𝐼𝑦𝑦𝑀𝑥 − 𝐼𝑥𝑦𝑀𝑦


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:

𝑞𝑠 = − {[𝑉𝑥𝐼𝑥𝑥 − 𝑉𝑦𝐼𝑥𝑦


] ∑ 𝐵𝑟𝑥𝑟

𝑛

𝑟=1

+ [𝑉𝑥𝐼𝑦𝑦 − 𝑉𝑦𝐼𝑥𝑦


] ∑ 𝐵𝑟𝑥𝑟

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:

−𝑉𝑥𝑑𝑦 + 𝑉𝑦𝑑𝑥 = ∮ 𝑝𝑞𝑜𝑝𝑒𝑛𝑑𝑠

𝑠

0

+ 2�̅�𝑞𝑐𝑙𝑜𝑠𝑒𝑑 (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:

∮ 𝑝𝑞𝑜𝑝𝑒𝑛𝑑𝑠

𝑠

0

= − ∫ 𝑞𝑜𝑝𝑒𝑛𝑦𝑑𝑥

𝑠

0

+ ∫ 𝑞𝑜𝑝𝑒𝑛𝑥𝑑𝑦

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

2�̅�𝑛𝐺 ∮ 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 =𝑘𝜋2𝐸

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 𝜎𝑦𝑖𝑒𝑙𝑑 > 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):

𝜎𝑦𝑖𝑒𝑙𝑑 > √𝜎12 + 𝜎1𝜎2 + 𝜎2

2 (42)

44

𝑑𝑎

𝑑𝑁= 𝐶∆𝐾𝑚 (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−𝑚2 − 𝑎

𝑖𝑛𝑖𝑡𝑖𝑎𝑙

2−𝑚2 )

(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

𝑎𝑐𝑟𝑖𝑡 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

Flaw

Siz

e (

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 = √𝜋2𝐺𝐽

2𝜌𝑒𝑐2𝑠2(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(

𝛿𝐶𝐿𝛿𝜉⁄ )

12⁄ 𝜌𝑆𝐶(

𝛿𝐶𝑀,0𝛿𝜉⁄ )(

𝛿𝐶𝐿𝛿𝛼⁄ )

(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

ts ts ts

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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