report-wing structure design spring'16

MAE 154B PROJECT RYAN, AGRAJ, KEVIN

Kevin Landry

Ryan Rader

Agraj Sobti

MAE 154b: FDR 6/11/2016

*Thank you Professor Lynch, Peng & Auni. Our group

believes we deserve an A-. Our team put a significant

number of hours into this project, learning every concept

with minimal background beforehand. Despite not having

the familiarity with the material that many of the Aircraft

students had coming into the class, we feel as though we

were able to produce a comparable end product.*


Table of Contents

1. INTRODUCTION ....................................................................................................................7

1.1 Project Description ...................................................................................................................... 7

1.2 Assumptions ................................................................................................................................ 7

1.3 Parameters .................................................................................................................................. 8

1.4 Gantt Chart .................................................................................................................................. 9

2. Wing Loading ...................................................................................................................... 10

3. Maneuver and Loading during Banked Turns ....................................................................... 10

4. Wing Loading during Landing .............................................................................................. 12

5. Xfoil .................................................................................................................................... 12

5.1 X-Foil Tutorial ............................................................................................................................ 12

5.2 Generating the .pol File .............................................................................................................. 14

6. Xfoil, V-n Diagram, Loading.................................................................................................. 14

6.1 Interpreting the Xfoil Output...................................................................................................... 14

6.2 The Stall Curve ........................................................................................................................... 16

6.3 Maneuver Limits ........................................................................................................................ 17

6.4 Stall Velocities............................................................................................................................ 17

6.5 Gust Loading .............................................................................................................................. 18

6.6 Determining Critical Points......................................................................................................... 20

7. Wing Loading at Critical Points ............................................................................................ 21

7.1 Lift and Drag .............................................................................................................................. 21


7.2 Aircraft Coordinate Transformation ........................................................................................... 23

8. Centroid & Area Moments of Inertia .............................................................................. 27

8.1 Cantilever Beam Model .............................................................................................................. 27

8.2 Centroid Calculations ................................................................................................................. 28

9. Shear Force, Bending, Deflection .................................................................................... 31

10. Stresses .......................................................................................................................... 38

10.1 Airfoil Profile .............................................................................................................................. 38

10.2 Section Centroid Calculations ..................................................................................................... 39

10.3 Moment of Inertia Calculations .................................................................................................. 40

11. Shear Flow – Calculations ................................................................................................ 41

11.1 Shear Flow – Results .................................................................................................................. 44

12. Shear Flow-Code Verification ........................................................................................... 46

13. Buckling, Fatigue & Von Mises Stress ......................................................................... 47

13.1 Von Mises Stresses .................................................................................................................... 47

13.2 Column Buckling ........................................................................................................................ 48

13.3 Skin (Thin Plate) Buckling ........................................................................................................... 50

13.4 Shear Buckling of Skin ................................................................................................................ 50

13.5 Fracture & Fatigue ..................................................................................................................... 51

14. Aeroelasticity .................................................................................................................. 52

14.1 Divergence ................................................................................................................................. 52

14.2 Aileron Reversal ......................................................................................................................... 53

15. 3D Modelling and Simulation ........................................................................................... 54


15.1 3D Modelling ............................................................................................................................. 54

15.1.1 Modelling Steps .................................................................................................................. 54

16. Loading Analysis .............................................................................................................. 56

16.1 The Setup .................................................................................................................................. 56

16.1.1 Boundary Conditions .......................................................................................................... 56

16.2 Results ....................................................................................................................................... 56

17. Modal Analysis ................................................................................................................ 58

18. References ...................................................................................................................... 61

19. Appendix ......................................................................................................................... 61

Table of Figures

Figure 1. The Cessna 177B Cardinal Aircraft ............................................................................................. 7

Figure 2. Rectangular Planform of the wing ............................................................................................. 8

Figure 3. Gantt Chart showing schedule of tasks .................................................................................... 10

Figure 4. Cp contour over Airfoil at A.o.A of 5 deg on XFoil for NACA 2415 ............................................. 13

Figure 5. Cp contour over Airfoil at A.o.A of 0 to 20 degrees on XFoil for NACA 2415 .............................. 13

Figure 6: Lift Coefficient at Sea Level ..................................................................................................... 15

Figure 7: Lift Coefficient at Cruising Altitude .......................................................................................... 16

Figure 8: V-n Curve at Sea Level and Cruising Altitude............................................................................ 17

Figure 9: V-n Diagram with Gust Loading and Final Load Envelope ......................................................... 19

Figure 10: Critical Points from the V-n Diagrams .................................................................................... 20

Figure 11: Lift and Drag Force along the Length of the Wing .................................................................. 23


Figure 12: Inertial Lift and Drag Components and Aircraft X-Y Components ........................................... 24

Figure 13: Load Conditions at PHAA ....................................................................................................... 25

Figure 14: Lift Force for each Critical Point ............................................................................................. 25

Figure 15: wx for each critical point ........................................................................................................ 26

Figure 16: wy for each critical point ........................................................................................................ 26

Figure 17: Simplified Wing Cross Section ............................................................................................... 28

Figure 18: Shear Force (Sx) ..................................................................................................................... 32

Figure 19: Shear Force (Sy) ..................................................................................................................... 32

Figure 20: Mx vs Wing Span .................................................................................................................... 33

Figure 21: My vs Wing Span .................................................................................................................... 34

Figure 22: Deflection (u) ........................................................................................................................ 35

Figure 23: Deflection (v) ........................................................................................................................ 36

Figure 24: Direct Stress (sigma Z) ........................................................................................................... 37

Figure 25 NACA 2412 Arifoil Profile........................................................................................................ 39

Figure 26 Truncated Airfoil .................................................................................................................... 40

Figure 27 Solving the two cell problem ....................................................... Error! Bookmark not defined.

Figure 28 Shear Flow Idealization .......................................................................................................... 43

Figure 29 Magnitude of Stress along the Airfoil profile (Shear Stress) – PHAA at Sea Level ..................... 45

Figure 30 Magnitude of Stress along the Airfoil profile (Equivalent Stress) – PHAA at Sea Level ............. 45

Figure 31 Failure Stress vs Number of Loading repetitions ..................................................................... 51

Figure 32 Section View of the 3D wing model ........................................................................................ 54

Figure 33 Isometric view of the Entire wing ........................................................................................... 55

Figure 34 zz distribution accross different regions of the wing ................. Error! Bookmark not defined.

Figure 35 Deflection of the wing due to the applied load ....................................................................... 57


Figure 36 Mode-1 Frequency-7.3987 Hz ................................................................................................ 58


Figure 38 Mode-3 Frequency- 42.362 Hz ............................................................................................... 59


Figure 40 Mode-5 Frequency- 99.968 Hz ............................................................................................... 60

Figure 41 Modal Frequency vs Mode Number ....................................................................................... 61

Table of Tables

Table 1. Chosen Design Parameters ......................................................................................................... 8

Table 2: Critical Points ........................................................................................................................... 20

Table 3 Dimensions used in centroid & moment calculations ................................................................. 27

Table 4: Centroid Results ....................................................................................................................... 29

Table 5: Spar Data ................................................................................................................................. 30

Table 6: Bracket Data ............................................................................................................................ 30

Table 7: Wing Skin Data ......................................................................................................................... 31

Table 8: Area Moment of Inertia Results ................................................................................................ 31

Table 9 .................................................................................................................................................. 49

Table 10 ................................................................................................................................................ 49

Table 11 ................................................................................................................................................ 50

Table 12 ..................................................................................................... Error! Bookmark not defined.


1. INTRODUCTION

1.1 Project Description

The objective of the project is to design and analyze a wing structure for a single engine utility

Cessna 177 Cardinal aircraft (Figure 1). This structure will be certified strictly under Federal Aviation

Regulations Part 23 (FAR 23). The wing is straight with zero taper.

Figure 1. The Cessna 177B Cardinal Aircraft

In designing the wing, simulation software such as X-Foil, MATLAB, COMSOL and SolidWorks were

utilized. The data generated from MATLAB code was cross verified with manual calculations.

1.2 Assumptions

The wing was considered to be a rectangular wing with zero taper i.e. taper ratio of one. This

greatly simplifies the design which otherwise would demand complex CFD analysis to determine the

actual flow parameters (Figure 2).


Figure 2. Rectangular Planform of the wing

NACA 2415 airfoil was assumed to span throughout the span of the wing. The Oswald efficiency factor

was assumed to be 0.79 for the design.

1.3 Parameters

The following parameters were chosen for the wing design (Table 1).

Table 1. Chosen Design Parameters

Parameter Description Parameter

Airfoil NACA 2415

Maximum Gross Weight 1100 kg

Standard Empty Weight 680 kg

Cruise Speed 230 km/h

Maneuvering Speed (For XFOIL) 250 km/h

Wing Span 10.82 m

Chord 1.5 m

Oswald Efficiency 0.79

Service Ceiling 4450 m


For the XFoil calculations and plots, the Reynold’s number was calculated for both Sea level and the

Service Ceiling using equation (1).

𝑅𝑒 = 𝜌𝑉𝐿

𝜇

(1)

Where,

Re = Reynold’s number

ρ = Density of air (Sea level = 1.225 kg/m3, Service ceiling= 0.78 kg/m3)

V = Aircraft Velocity = 69.44 m/s

L = Chord Length = 1.5 m

µ = Dynamic Viscosity (Sea level = 1.983 x 10-5 Pa-s, Service Ceiling =1.65 x 10-5 Pa-s )

Re = 6.430 x 106 at Sea Level

Re = 4.924 x 106 at Service Ceiling

1.4 Gantt Chart

Before commencing the design project, a timeline was scheduled which was strictly followed to

arrive at the final design. As shown in the Gantt Chart (Figure 3), the tasks were divided into specific

time intervals to ensure smooth and timely completion of the report.


Figure 3. Gantt Chart showing schedule of tasks

2. Wing Loading

In Aerodynamics wing loading is the loaded weight of the aircraft divided by the area of the wing.

3. Maneuver and Loading during Banked Turns

When the aircraft wants to make a turn, it can normally do this in two ways. First, the pilot can deflect

the vertical stabilizer to cause a yaw to happen which then turns the nose of the aircraft in the desired

direction. The second way, which is more useful to know, is to bank the aircraft away from the desired

direction to create a centripetal force by the oblique component of the lift. This is achieved by actuating

30-Mar 09-Apr 19-Apr 29-Apr 09-May 19-May 29-May 08-Jun

X-Foil Tutorial

Generate Xfoil Data

Plotting V-n diagram

Plotting Load Distribution on Wing

Manual calculations of centroid

Manual calculations of MOI

MATLAB Calculations of centroid

MATLAB Calculations of MOI

Shear Force calculations

Bending Moment Calculations

Deflection calculation

Plotting actual Airfoil profile for NACA 2412 airfoil

Calculation of Centroid for Actual Wing section

Calculation of Area Moment of Inertia for Actual Wing section

Shear Flow Calculations

Plotting Shear Flow Results

Validaton of Shear Flow code

Failure and Safety

Stringer Buckling Calculations

Skin Bucking Calculations

Fracture Calculations

Fatigue Calculations

3D Modeling

Finite Element Analysis

Modal Analysis

Compiling the report

Final changes to the report based on feedback


the ailerons in opposite directions to cause a partial roll. When this turn happens, the vertical

component of the lift should still equal the weight of the aircraft for the aircraft to maintain its altitude.

Figure 4 Vector diagram showing the forces acting on a fixed-wing aircraft during a banked turn.

Because centripetal acceleration is given by Equation (1.a) , newton's second law in the horizontal

direction can be expressed mathematically by Equation (1.b).

𝑎 =𝑣2

𝑟

(1.a)

𝐿𝑠𝑖𝑛 θ =𝑚𝑣2

𝑟

(1.b)

Where,

L is the lift acting on the aircraft

θ is the angle of bank of the aircraft

m is the mass of the aircraft


v is the true airspeed of the aircraft

r is the radius of the turn

Also, the Vertical component of the lift should be equal to the weight of the aircraft, according to

Equation (1.c).

𝐿 cos 𝛳 = 𝑚𝑔

(1.c)

4. Wing Loading during Landing

During landing, the wing loading decreases. This is because while landing the aircraft, the pilot reduces

the velocity of the aircraft, thereby reducing the lift. Since wing loading is nothing but the lift divided by

the wing area, the wing loading is small while landing compared to when it is taking off or cruising.

Effectively, the wing loading is greatest during takeoff and at a minimum while landing.

5. Xfoil

5.1 X-Foil Tutorial

X-Foil was used to analyze the aerodynamic properties of the airfoil. The NACA 2415 airfoil was

chosen and different operations were performed including plotting the viscous flow profile over the

airfoil, plotting the Cp contour over the chord length (Figure 5) and generating the performance data ( CL,

CD, CM ) versus the Angle of Attack (Figure 6).


Figure 5. Cp contour over Airfoil at A.o.A of 5 deg on XFoil for NACA 2415

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


5.2 Generating the .pol File

The NACA 2415 airfoil was then analyzed using XFoil with the objective of generating the dot pol

file for the purpose of developing the V-n diagram.

The ‘ASEQ’ command was used to generate the CL, CD and CM plots verses the A.o.A. The A.o.A is by

default chosen to be in degrees. The range of the A.o.A was set as - 20 ͦto + 20 ͦ. Subsequently, the .pol

file was generated using the ‘hard’ and the ‘PACC’ commands. For this project, to determine the 𝐶𝑙𝛼 , the

units of the A.o.A was converted from degrees to radians. Thereby the CL values in the A.o.A range of -

7 ͦ to + 10 ͦ, were selected to determine the approximate 𝐶𝑙𝛼 of the plot. This 𝐶𝑙𝛼 would be used later

on, to obtain the 𝐶𝐿𝛼 , which is essential in developing the positive and negative stall curve equations.

6. Xfoil, V-n Diagram, Loading

6.1 Interpreting the Xfoil Output

The Xfoil lift coefficient curve is shown in Figure 7 as the 2D curve. It is important to note that the

plot takes the shape of an “S” curve and actually turns around at either end. This indicates that the

curve has a relative maximum and a relative minimum. A 2D slope can be determined by examining the

linear portion of the curve and finding a best-fit line (shown in Figure 7). With the 2D slope, a conversion

to a 3D slope (𝐶𝐿𝛼 ) can be generated using formula (2),

𝐶𝐿𝛼 =

𝐶𝑙𝛼

(1 +𝐶𝑙𝛼

𝜋𝐴𝑒)

(2)

Where A is the aspect ratio, e is the Oswald efficiency, and 𝐶𝑙𝛼 is the 2D slope.

Each of the CL points from the 2D curve can be adjusted vertically by the ratio of the slopes by simply

applying equation (3) to each point,


𝐶𝐿𝛼 = 𝐶𝑙𝛼

𝐶𝐿𝛼

𝐶𝑙𝛼

(3)

The result is an adjusted curve that follows a similar path to the 2D curve but at a slightly lower overall

slope (see Figure 7).

Figure 7: Lift Coefficient at Sea Level

The same relationships can be seen at cruising altitude (see Figure 8).


Figure 8: Lift Coefficient at Cruising Altitude

6.2 The Stall Curve

The maximum and minimum points of the 2D curve correlate to maximum and minimum angle of

attack before the plane stalls. Converting these values from the 2D curve to the 3D curve and inputting

them into the stall curve equation provides the maximum load factor (n) for a stall condition as a

function of velocity through Equation (4),

𝑛 =𝜌𝑆𝐶𝐿,𝑚𝑎𝑥

2𝑊𝑉2

(4)

Where ρ is the air density, S is the wing area, V is velocity, and W is the weight of the plane. The curve

can be applied in the positive direction for positive load factors as shown or in the negative direction by

substituting CL, min for CL, max.


6.3 Maneuver Limits

For the positive stall curve, stalling is the limiting factor in determining the load on the plane until

the load factor reaches the design limit load factor of 4.4 (per FAR 23). At this point, the limiting factor

becomes the 4.4 for all remaining velocities. The same holds true for the negative stall curve, except the

design limit in the negative direction is -1.76 (per FAR 23).

Figure 9 shows the load factor over the full range of velocities from zero to the dive velocity. The plot

shows the combination of the stall curve limited portion as well as the design load limited portion in

both the positive and negative directions.

Figure 9: V-n Curve at Sea Level and Cruising Altitude

6.4 Stall Velocities

In addition to the stall curves, maneuver limits, and stall velocities, loads from gusts are

considered in the design of the wing. This is the speed at which the plane will stall regardless of angle of

attack. For a positive load factor, the stall speed varies with the lift coefficient through Equation (5).


𝑉𝑆,𝑃𝑜𝑠 = √2𝑊

𝐶𝐿𝑚𝑎𝑥𝜌𝑆

(5)

Where,

W is the weight of the plane,

ρ is the air density, and

S is the wing surface area.

The equation holds true for negative load factors except CLmax is replaced by CLmin. The equation is also

applicable for sea level and at cruising altitude. Using the appropriate CLmax, CLmin, and ρ will provide a

total of four stall velocities, a positive and negative at each altitude.

6.5 Gust Loading

In addition to the stall curves, maneuver limits, and stall velocities, loads from gusts must be

considered in the design of the wing. The gust loads are calculated using the FAR 23 specifications of 50

ft/s at cruise velocity and 25 ft/s at dive velocity. The load factor is determined at each of these

velocities and assumed to be linear between the two. Determination of the load factor at each location

is achieved through Equations 6-8:

𝑛 = 1 +𝐾𝑔𝑎𝑈𝑒𝑉

498 (𝑊𝑆

)

(6)

𝐾𝑔 =0.88𝜇

5.3 + 𝜇

(7)

𝜇 =2 (

𝑊𝑆

)

𝜌𝑐𝑎𝑔

(8)


Where a is the previously determined slope of the 3D lift curve (rad-1), Ue is the FAR 23 specified gust

velocity (ft/s), V is the aircraft velocity (ft/s), W is the weight of the plane (lbf), S is the wing area (ft2), ρ

is the air density (slug/ft3), c is the chord length of the plane (ft), and g is the gravitational constant

(ft/s2).

With the positive load factors at cruise velocity and dive velocity calculated, the third and final

point used in plotting the gust load profile is based on the fact that the gust load factor is 1 when the

velocity is equivalent to zero. The triangle formed by these three points results in the positive half of the

gust profile. The negative portion is found by simply mirroring the positive gust load about the n=1 line.

The resulting gust profile is plotted on top of the maneuver profile as seen in Figure 10. The envelope of

the possible load cases is defined on the left portion of the plot by the stall velocity. The upper portion

of the envelope is controlled initially by the stall curve until the FAR requirement of 4.4 is reached. For

all successive velocities, the max loading is determined by the larger of the 4.4 requirement and the gust

loading. The result is shown by the dashed green line in Figure 10.

Figure 10: V-n Diagram with Gust Loading and Final Load Envelope


6.6 Determining Critical Points

From the combined loading, a few critical points can be determined from the extreme points

along the curve. Figure 11 shows the 5 points for both the sea level and cruising altitude critical points. It

is significant to note that there may be a wide range of load cases depending on the aircraft, however

our aircraft has a total of 5 critical points with 2 of them in the negative region being defined by the gust

loading.

Figure 11: Critical Points from the V-n Diagrams

Table 2 shows the details of each of the critical points.

Table 2: Critical Points


7. Wing Loading at Critical Points

7.1 Lift and Drag

With the critical points identified, it is possible to calculate the lift and drag forces along the

length of the wing. First the total lift is calculated using the load factor (n) and the weight (W) by,

𝐿 = 𝑛𝑊

(9)

The total lift value is used to calculate both a rectangular and elliptical distribution for lift along the

length of the wing. The rectangular distribution is simply the total lift divided by the length or half span

(b) as follows,

𝐿(𝑧) =𝐿

𝑏

(10)

The elliptical distribution varies with the length such that the lift is at its peak at the half span and is zero

at either of the wing tips. This relationship is defined by equation 11,

𝐿(𝑧) =4𝐿

𝜋𝑏√1 − (

2𝑧

𝑏)

2

(11)

The rectangular and elliptical distributions are averaged point by point to get a final lift distribution as a

function of location along the wing.

Using the total lift and the standard lift equation, the coefficient of lift can be calculated as,

𝐶𝐿 =2𝐿

𝜌𝑉2𝑆

(12)

Where ρ is the air density, V is the velocity at the given critical point, and S is the wing area.


The angle of attack can be determined by interpolating the 3D lift curve shown in figure X for coefficient

of lift calculated above.

Similarly, the total drag is first calculated using the relationship,

𝐷 =𝜌𝑉2𝑆

2(𝐶𝐷,0 +

𝐶𝐿2

𝜋𝐴𝑒)

(13)

Where ρ is the air density, V is the velocity at the given critical point, CD,0 is the zero lift drag coefficient

(determined from the Xfoil output), A is the aspect ratio, and e is the Oswald efficiency.

As with the lift, the drag is initially assumed to be rectangularly distributed expressed by

equation 14,

𝐷(𝑧) =𝐷

𝑏

(14)

An exception is made to account for additional drag due to wingtip vortices. These are assumed to

increase the drag by 10% over the final 20% of the half span.

The result is the lift and drag profile shown in Figure 12. For the convention used in the

equations above, the z dimension is defined with zero being the center of the plane and increasing z

moving outward along the wing length.


Figure 12: Lift and Drag Force along the Length of the Wing

7.2 Aircraft Coordinate Transformation

The lift and drag calculations provide the distributed loads across the length of the wing, but are

centered in the inertial coordinate system. For each of the critical analysis points, the aircraft is at some

angle of attack which causes a difference between the inertial and aircraft coordinates (see Figure 13).


Figure 13: Inertial Lift and Drag Components and Aircraft X-Y Components

The result is that the lift and drag must be rotated into the aircraft coordinate system. This is

accomplished by the standard equations,

𝑤𝑥(𝑧) = − L(z)sin 𝛼 + D(z)cos 𝛼

(15)

𝑤𝑦(𝑧) = 𝐿(𝑧) cos 𝛼 + 𝐷(𝑧) sin 𝛼

(16)

These equations provide the load conditions shown in Figure 14 at PHAA and sea level.


Figure 14: Load Conditions at PHAA

This process is repeated for each of the critical points providing the total Lift Curves shown in Figure 15

and the wing loads shown in Figure 16 and Figure 17.

Figure 15: Lift Force for each Critical Point


Figure 16: wx for each critical point

Figure 17: wy for each critical point


8. Centroid & Area Moments of Inertia

8.1 Cantilever Beam Model

The calculations for centroids and moments of inertia in relation to our wing design are all based

on the underlying assumption for the shape of the wing. We began our analysis by taking a simplified

cross section of the wing (Figure 18). The design assumes a uniform cross section from root to tip. All

parts of the wing are also assumed to be composed of straight segments. The cross section is

approximated using three spars, eight brackets and four skin panels.

Table 3 Dimensions used in centroid & moment calculations

VARIABLE DIMENSIONS

Chord Length 1.5 m

Height 0.2235 m

Spar Thickness 0.0025 m

Skin Thickness 0.001016 m

Spar Height 0.999797 m

Bracket Height 0.012 m

Bracket Thickness 0.0025 m

Theta 30 deg

Alpha 10 deg

Half Chord Length [m] 0.75

Half Height [m] 0.11175

Half Spar Thickness [m] 0.00125

Half Spar Height [m] 0.4998985

Half Bracket Thickness [m] 0.00125

Theta [rad] 0.523598333


8.2 Centroid Calculations

Part of our project involved carrying out hand calculations to corroborate the results of our

MATLAB code. For the centroid calculation, many of the specifications were assumed to be the same as

NACA 2412. Among them were: spar thickness (.0025 meters), skin thickness (.001016 meters) and the

area of the bracket (.0003 square meters). Other data was updated where applicable, such as chord

length (1.5 meters). We were given freedom over the location of the spars, and chose initial locations of

0 [m], 0.75 [m] and 1.5 [m]. Another assumption that we made was to assume that the brackets are

point masses located at the joint of the spar and the skin. In other words, the bracket height is

neglected. The bracket height is only used to calculate the area of the bracket in the cross section.

Figure 18: Simplified Wing Cross Section

In general, the centroid is calculated from an area-weighted average of the centroid of each of the

individual components. In our case this consists of 15 individual components. The MATLAB code

provided additional formulas to work with, some of which provided helpful assumptions in the

calculations. The x-coordinate of the centroid for the spars is set at their position coordinate on the

wing. The y-coordinate of the centroid for the spars is set to zero for root spar and middle spar, while

the spar out at the wing tip is calculated using trigonometry. This equation is given by:


−(𝑥3 − 𝑥2) ∗ tan (𝜃)

(17)

where x3 and x2 are the x-coordinate locations of the second and third spar, and theta is the angle of

the trailing edge in radians (in degrees theta is 30, in radians 0.524).

To calculate the centroid of the entire wing, we need to take an area-weighted average of the

locations of the centroid of all 15 components. The formula can be expressed as,

𝐶𝑦 =∑ 𝐴𝑛𝐶𝑦𝑛𝑛

∑ 𝐴𝑛

(18)

The calculated values for the centroid are shown in Table 4. These numbers were verified using

MATLAB, Microsoft Excel, as well as hand calculations. As one can see in Figure 18, the centroid lies

outside of the wing in the negative y direction.

Table 4: Centroid Results

Resultant Centroid of Wing Section Location (m)

Cx 0.7670 Cy -0.1248

The area moment of inertia calculation makes use of the centroid data and the parallel axis theorem,

shown in equations 19-21:

I𝑥𝑥 = ∑ 𝐴𝑖 ∗ (

𝑛

𝑖=1

𝑦𝑖 − �̅�)2

(19)

I𝑦𝑦 = ∑ 𝐴𝑖 ∗ (

𝑛

𝑖=1

𝑥𝑖 − �̅�)2

(20)


I𝑥𝑦 = ∑ 𝐴𝑖 ∗ (

𝑛

𝑖=1

𝑥𝑖 − �̅�) ∗ (𝑦𝑖 − �̅�)

(21)

We used the formulas provided to calculate the area moment of inertia for the leading edge of the wing.

The values of Ixx, Iyy and Ixy for the entire wing are found by summing the values for the individual

components. The data used in the calculations for the three spars is shown in Table 5, the data for the

brackets in

Table 6 and the data for the wing’s skin in

. The net values for the leading edge of the wing are displayed in

Table 8.

Table 5: Spar Data

SPARS (3) _ x

[m]

_ y

[m] Area [m

2]

Ixx

[m4]

Iyy

[m4]

Ixy

[m4]

1 0 0 0.00055875 0.00001103 0.00032871 -0.00005348

2 0.75 0 0.00055875 0.00001103 0.00000016 -0.00000119

3 1.5 -0.433 0.00055875 0.00005540 0.00030021 -0.00012623

Table 6: Bracket Data

BRACKETS (8) _ x

[m]

_ y

[m] Area [m

2]

Ixx

[m4]

Iyy

[m4]

Ixy

[m4]

SKIN (4) _ x

[m]

_ y

[m] Area [m

2]

Ixx [m

4]

Iyy [m

4]

Ixy [m

4]

1 top 0.375 0.1118 0.000762 4.26E-05 0.00015 -7.066E-05 1 bottom 0.375 -0.1118 0.000762 1.3E-07 0.00015 -3.896E-06

2 top 1.125 -0.1048 0.00087988 1.41E-05 0.000154 -1.75E-05 2 bottom 1.125 -0.3283 0.00087988 5.02E-05 0.000154 -8.79E-05


1 top 0.0013 0.1118 0.00003 1.679E-06 1.76E-05 -5.4E-06

1 bottom 0.0013 -0.1118 0.00003 5.103E-09 1.76E-05 -3E-07

2 top 0.74875 0.1118 0.00003 1.679E-06 9.99E-09 -1.3E-07

2 bottom 0.74875 -0.1118 0.00003 5.103E-09 9.99E-09 -7.1E-09

3 top 0.75125 0.1118 0.00003 1.679E-06 7.44E-09 -1.1E-07

3 bottom 0.75125 -0.1118 0.00003 5.103E-09 7.44E-09 -6.2E-09

4 top 1.49875 -0.3213 0.00003 1.158E-06 1.61E-05 -4.3E-06

4 bottom 1.49875 -0.5448 0.00003 5.291E-06 1.61E-05 -9.2E-06

Table 7: Wing Skin Data

Table 8: Area Moment of Inertia Results

Net Area Moment of Inertia [m4]

Ixx 0.00019600

Iyy 0.00131007

Ixy -0.00038037

9. Shear Force, Bending, Deflection

This section of the report will go through the steps from load to deflection. Using the load intensity

equations calculated for wx and wy. The equation for shear force in the y-direction is found through a

SKIN (4) _ x

[m]

_ y

[m] Area [m

2]

Ixx [m

4]

Iyy [m

4]

Ixy [m

4]

1 top 0.375 0.1118 0.000762 4.26E-05 0.00015 -7.066E-05 1 bottom 0.375 -0.1118 0.000762 1.3E-07 0.00015 -3.896E-06

2 top 1.125 -0.1048 0.00087988 1.41E-05 0.000154 -1.75E-05 2 bottom 1.125 -0.3283 0.00087988 5.02E-05 0.000154 -8.79E-05


force balance equation. The result of the force balance equation is the shear force equation, which is the

negative integral of the load intensity functions. The equations for the shear forces are given by:

𝑆𝑦 = − ∫ 𝑤𝑦𝑑𝑧

(22)

𝑆𝑥 = − ∫ 𝑤𝑥𝑑𝑧

(23)

The plots of the shear forces Sx and Sy are shown in Figure 19 and Figure 20 (below).

Figure 19: Shear Force (Sx)


Figure 20: Shear Force (Sy)


The moments Mx and My were calculated by taking the integral of the shear force equations.

𝑀𝑥 = ∫ 𝑆𝑦𝑑𝑧 (24)

𝑀𝑦 = ∫ 𝑆𝑥𝑑𝑧 (25)

The proof of this result is shown by a moment equilibrium. Plots of the moments at sea level and

service ceiling are shown in Figure 21 and Figure 22.

Figure 21: Mx vs Wing Span


Figure 22: My vs Wing Span

The equations for u’’ and v’’ are given by equations 26 and 27:

𝑑2𝑢

𝑑𝑧2 = −𝐾 ∗ [−𝑀𝑥 ∗ 𝐼𝑥𝑦 + 𝑀𝑦 ∗ 𝐼𝑥𝑥] (26)

𝑑2𝑣

𝑑𝑧2 = −𝐾 ∗ [𝑀𝑥 ∗ 𝐼𝑦𝑦 − 𝑀𝑦 ∗ 𝐼𝑥𝑦] (27)

𝐾 =1

𝐸∗(𝐼𝑥𝑥∗𝐼𝑦𝑦−𝐼𝑥𝑦2 )

(28)

The deflections due to bending are found by integrating u’’ and v’’ twice. The constants of integration go

to zero because we integrate from the LHS. Ultimately, we compare the deflections at the tip by

executing numerical integration in MATLAB with equations 29 and 30:

𝑢(𝑧) = ∑𝑑𝑢

𝑑𝑧∗ (𝑧𝑖+1 − 𝑧𝑖) (29)

𝑣(𝑧) = ∑𝑑𝑣

𝑑𝑧∗ (𝑧𝑖+1 − 𝑧𝑖) (30)


The plots for deflection are shown below in Figure 23 and Figure 24.

Figure 23: Deflection (u)


Figure 24: Deflection (v)


The direct stress is also plotted versus Z. This is accomplished by taking the given equation for direct

stress and populating the direct stress matrix point by point in MATLAB. The resulting plot for direct

stress is shown in Figure 25.

Figure 25: Direct Stress (sigma Z)


10. Stresses

The following coordinate system was used for the airfoil:

x – Along the chord of the airfoil

y – The span-wise coordinate from the tip to the root

z – Axis perpendicular to the Airfoil Chord

10.1 Airfoil Profile

To do this part of the calculation, the airfoil was split into approximately 801 skin segments, that is,

there are skin elements for every 0.03m along the chord of the airfoil.

To plot the profile of the airfoil, the Equation for a camber 4-digit NACA Airfoil was obtained from online

sources and implemented in the MATLAB code. The following formula was used:

𝑦𝑐 = {𝑚

𝑥

𝑝2(2𝑝 −

𝑥

𝑐) , 0 ≤ 𝑥 ≤ 𝑝𝑐

𝑚(𝑐−𝑥)

(1−𝑝)2(1 +

𝑥

𝑐− 2𝑝) , 𝑝𝑐 ≤ 𝑥 < 𝑐

(31)

where:

m is the maximum camber (100 m is the first of the four digits),

p is the location of maximum camber (10 p is the second digit in the NACA xxxx description).

For this cambered airfoil, because the thickness needs to be applied perpendicular to the camber line,

the coordinates (𝑥𝑈 , 𝑦𝑈) and (𝑥𝐿 , 𝑦𝐿), of respectively the upper and lower airfoil surface, become:

𝑥𝑈 = 𝑥 − 𝑦𝑡𝑠𝑖𝑛𝜃, 𝑦𝑈 = 𝑦𝑐 + 𝑦𝑡𝑐𝑜𝑠𝜃 (32)

𝑥𝐿 = 𝑥 + 𝑦𝑡𝑠𝑖𝑛𝜃, 𝑦𝐿 = 𝑦𝑐 − 𝑦𝑡𝑐𝑜𝑠𝜃 (33)

Where,

𝜃 = arctan (𝑑𝑦𝑐

𝑑𝑥) (34)

𝑑𝑦𝑐

𝑑𝑥= {

2𝑚

𝑝2(𝑝 −

𝑥

𝑐) , 0 ≤ 𝑥 ≤ 𝑝𝑐

2𝑚

(1−𝑝)2(𝑝 −

𝑥

𝑐) , 𝑝𝑐 ≤ 𝑥 ≤ 𝑐

(35)


In our case, we had the NACA 2412 Airfoil which had the following specifications :

m = 2 % (1st digit - maximum camber (m) in percentage of the chord )

p = 40 % (Second digit - position of the maximum camber (p) in tenths of chord )

t = 0.15 (last 2 digits - maximum thickness (t) of the airfoil in percentage of chord)

Figure 26 NACA 2412 Arifoil Profile

While plotting the airfoil profile on MATLAB, the rear 20 % of the chord length was truncated to make

space for control surfaces. A spar was placed at the rear end which came out to be 1.2 m from the nose

of the airfoil given the chord length of 1.5 m.

10.2 Section Centroid Calculations

For a more accurate sizing of the wing the section centroid was computed for the actual wing

section. This is performed by splitting the skin into smaller rectangular sections. The area of these

elements is calculated and then attributed to point areas whose distances from the axes origin is the

same as that of centroid of the rectangles. Sum of these distances weighted on the areas yields the

centroid location.

The centroid calculations were done and all the cross section components were included in the

calculation, as shown in Figure 27:


Figure 27 Truncated Airfoil

The following formula was used for centroid calculations:

𝐶𝑥 =∑ 𝐴𝑖𝑥𝑖

𝑛1

∑ 𝐴𝑖𝑛1

𝑎𝑛𝑑 𝐶𝑦 =∑ 𝐴𝑖𝑦𝑖

𝑛1

∑ 𝐴𝑖𝑛1

(36)

Where,

(𝐶𝑥 , 𝐶𝑦) is the centroid location of the airfoil

𝐴𝑖 is the area of the ith component

𝑥𝑖 is the x-coordinate of the ith component

𝑦𝑖 is the y-coordinate of the ith component

10.3 Moment of Inertia Calculations

Similar to the Centroid calculations the moment of inertia was also recalculated for the actual

wing section. Since each small section of area was considered to be a point area, the actual moment of

inertia of that section is negligible. However, there is a contribution to the overall moment of inertia

from each of these sections when the axis is moved from the one passing through the centroid of the

elements to the centroid of the actual wing section. Parallel axis theorem is used to calculate this

contribution, which is given by Equations (37) and (38).

𝐼𝑥 = ∑ 𝐼𝑥′,𝑖 + 𝐴𝑖(𝐶𝑦 − 𝑦𝑖)2

𝑛1 (37)

𝐼𝑦 = ∑ 𝐼𝑦′,𝑖 + 𝐴𝑖(𝐶𝑥 − 𝑥𝑖 )2 𝑛1 (38)

Where,


𝐼𝑥′,𝑖 is the Area MOI of the ith component about its own centroid about the x-axis (Negligible)

𝐼𝑦′ ,𝑖 is the Area MOI of the ith component about its own centroid about the y-axis (Negligible)

11. Shear Flow – Calculations

Using the calculated x-y coordinates to define the profile of the wing and the associated areas at each

point for spar caps and stringers, shear flow throughout the structure can be determined. Ideally, the

coordinate system would be centered at the centroid of the cross section, but this can be corrected for

through a simple coordinate transfer. Each point will be treated as a boom where the area of the boom

corresponds to the effective area of the skin, stringers, or spar caps adjacent to that point. This is

calculated such that axial stress is conserved between the actual cross section and the simplified boom

cross section. The final equation is shown in Equation (39).

𝐴𝐵𝑜𝑜𝑚 = 𝐴𝑠𝑝𝑎𝑟 𝑐𝑎𝑝 𝑜𝑟 𝑠𝑡𝑟𝑖𝑛𝑔𝑒𝑟 + ∑ [𝑡𝑏

6(2 +

𝜎𝑧𝑧(𝑛+1)

𝜎𝑧𝑧(𝑛))]𝑛

𝑖=1 (39)

Where the area of the spar cap or stringer are only for that specific point, n is the number of adjacent

panels, t is the thickness of that adjacent panel, b is the length of the adjacent panel, σzz(n+1) is the stress

at the point on the other end of the adjacent panel, and σzz is the stress at the current point. Typically n

is equal to 2, one for each of the adjacent skin panels. However at the locations where the spar meets

the skin, n is equivalent to 3 to include the contributions due to the spar. σzz in this equation is calculated

using the moment in the X and Y directions and implemented into Equation (40).

𝜎𝑧𝑧,𝑖 =𝑀𝑥(𝐼𝑦𝑦(𝑦𝑖−𝑦𝑐)−𝐼𝑥𝑦(𝑥𝑖−𝑥𝑐))

𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦2 +

𝑀𝑦(𝐼𝑥𝑥(𝑥𝑖−𝑥𝑐)−𝐼𝑥𝑦(𝑦𝑖−𝑦𝑐))

𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦2 (40)

Where the moments of inertia are the previously calculated inertia terms for the wing cross section, x

and y are the coordinates of the point, and xc and yc are the x-y coordinates of the centroid if the points

are not already normalized.


To solve for shear flow, q, the cross section of the wing must be divided into two sections separated by

the spar cap. Figure 28 shows a sample airfoil cross section divided into two cells. The area for each of

these cells can be found by taking the cross product of consecutive points and dividing by 2 as shown in

Equation (40.a).

Figure 28 Solving the two cell problem

2𝐴 = (𝑥1, 𝑦1, 𝑧1) × (𝑥2, 𝑦2, 𝑧2)

(40.a)

To solve for the total shear flow, q, at each point the wing is first cut at the spar and the back plate to

make a singular open section as shown in Figure 28. By creating an open section, the qb at the open end,

between points 41 and 1 in Figure 28 is known to be 0. Having previously simplified the wing section

into boom areas, the assumption is made that the skin or spar panels are of infinitely small skin

thicknesses. The result is that the shear stress varies from one side of the boom to the other, but

remains constant throughout the length of skin. This concept is depicted by Figure 29.


Figure 29 Shear Flow Idealization

Starting with point 1, the Δqb, or change in sheer flow from one side of the boom to the other is

calculated at each point around the profile using Equation (41).

Δ𝑞𝑏,𝑖 =𝑆𝑦𝐼𝑥𝑦−𝑆𝑥𝐼𝑥𝑥

𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦2 𝐴𝐵𝑜𝑜𝑚,𝑖(𝑥𝑖 − 𝑥𝑐) +

𝑆𝑥𝐼𝑥𝑦−𝑆𝑦𝐼𝑦𝑦

𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦2 𝐴𝐵𝑜𝑜𝑚,𝑖(𝑦𝑖 − 𝑦𝑐) (41)

Where Sx and Sy are the shear forces, ABoom,I is the effective boom area at that point, xi and yi are the x-y

components of the point, and xc and yc are the centroid coordinates if not normalized. From this the

total qb at a given point is the sum of the Δqb’s for all previous points as shown in equation X. The values

of qb are shown in the Appendix.

𝑞𝑏,𝑘 = ∑ Δ𝑞𝑏,𝑖𝑘𝑖=1 (42)

To calculate the total q, the relationship between cell 1 and cell 2 must be accounted for. This is

considered by examining the total moment on the system in Equation (43) and the twist rate of each cell

using Equations (44) and (45).

𝑀0 + 𝑆𝑦𝜉0 − 𝑆𝑥𝜂0 = 2𝐴1𝑞0,1 + 2𝐴2𝑞0,2 + ∑ 2𝑞𝑏,𝑖Δ𝐴𝑖𝑛𝑖=1 (43)

𝑑𝜃

𝑑𝑧=

1

2𝐴1𝐺[𝑞0,1 (∑

((𝑥𝑖+1−𝑥𝑖)2+(𝑦𝑖+1−𝑦𝑖)2)12

𝑡𝑠𝑘𝑖𝑛

𝑛𝑖=1 ) + (𝑞0,1 − 𝑞0,2)

(𝑦𝑏𝑜𝑡−𝑦𝑡𝑜𝑝)

𝑡𝑠𝑝𝑎𝑟+

(𝑆𝑦𝐼𝑥𝑦−𝑆𝑥𝐼𝑥𝑥)

𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦2 ∑ 𝐴𝑏𝑜𝑜𝑚,𝑖𝑥𝑖

((𝑥𝑖+1−𝑥𝑖)2+(𝑦𝑖+1−𝑦𝑖)2)12


𝑛𝑖=1 +

(𝑆𝑥𝐼𝑥𝑦−𝑆𝑦𝐼𝑦𝑦)

𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦2 ∑ 𝐴𝑏𝑜𝑜𝑚,𝑖𝑦𝑖

((𝑥𝑖+1−𝑥𝑖)2+(𝑦𝑖+1−𝑦𝑖)2)12


𝑛𝑖=1 ]

(44)


𝑑𝜃

𝑑𝑧=

1

2𝐴2𝐺[𝑞0,2 (∑

((𝑥𝑖+1−𝑥𝑖)2+(𝑦𝑖+1−𝑦𝑖)2)12


𝑛𝑖=1 ) + (𝑞0,2 − 𝑞0,1)

(𝑦𝑡𝑜𝑝−𝑦𝑏𝑜𝑡)

𝑡𝑠𝑝𝑎𝑟+ 𝑞0,2

(𝑦𝑡𝑜𝑝−𝑦𝑏𝑜𝑡)

𝑡𝑠𝑝𝑎𝑟+

(𝑆𝑦𝐼𝑥𝑦−𝑆𝑥𝐼𝑥𝑥)

𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦2 ∑ 𝐴𝑏𝑜𝑜𝑚,𝑖𝑥𝑖

((𝑥𝑖+1−𝑥𝑖)2+(𝑦𝑖+1−𝑦𝑖)2)12


𝑛𝑖=1 +

(𝑆𝑥𝐼𝑥𝑦−𝑆𝑦𝐼𝑦𝑦)

𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦2 ∑ 𝐴𝑏𝑜𝑜𝑚,𝑖𝑦𝑖

((𝑥𝑖+1−𝑥𝑖)2+(𝑦𝑖+1−𝑦𝑖)2)12


𝑛𝑖=1 ]

(45)

It can be assumed that the wing cross section behaves as a rigid body and thus the twist rates are

equivalent. This leaves three equations with three unknowns; dθ/dz, q0,1, and q0,2. Solving the system of

equations gives the q0,1 and q0,2 to complete Equation (46).

𝑞𝑡𝑜𝑡𝑎𝑙,𝑖 = 𝑞𝑏,𝑖 + 𝑞0,1 𝑜𝑟 2 (46)

Where the q0 is the appropriate term for the cell containing the point in question. This resulting shear

flow (qtotal) is in terms of the shear force per unit length. To get this into shear stress, the shear flow

must be divided by the thickness, resulting in units of shear force per unit area (Equation (47)).

𝜎𝑧𝑠,𝑖 =𝑞𝑡𝑜𝑡𝑎𝑙,𝑖

𝑡𝑠𝑘𝑖𝑛 (47)

Combining the shear stress and the axial stress from Equation (47) at each point, the equivalent

principle stress can be calculated. Equation (48) defines this relationship.

𝜎𝑒𝑞,𝑖 = √2𝜎𝑧𝑧,𝑖2 + 6𝜎𝑧𝑠,𝑖

2 (48)

11.1 Shear Flow – Results

The results of these calculations for the first of the 10 critical points being analyzed (PHAA at Sea Level)

can be seen in Figure 30, shear stress, and Figure 31, equivalent stress. As the legend on the right

indicates, the colors correspond to the magnitude of the stress at that location.


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

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


12. Shear Flow-Code Verification

Variations in the frequency of points and method of calculating shear flow may lead to slight variations

in the results shown, however a few main principles should still hold true.

First, the shear flow should return to zero for the open section (qb). Figure 32 shows that the shear flow

in fact does return to zero for each of the ten load cases considered. It is somewhat misleading to

examine the values of open section shear flow directly as they are on the order of ±1,000, but with

respect to the magnitude of the maximum values of qb this is relatively zero.

Figure 32 Open Section Shear Flow

A second check is to verify that the sum of the shear stresses times the boom areas also returns to zero.

Figure 33 shows that this principle also holds true.


Figure 33 Product of Shear Flow and Boom Area

13. Buckling, Fatigue & Von Mises Stress

13.1 Von Mises Stresses

An important test in the design of our wing was to check the yield stress against the Von Mises

yield criterion. The tensile yield strength of AL 2024 T3 is 345 MPa and the ultimate tensile

strength is 483 MPa. The Von Mises equation is given in terms of principal stresses by equation

49 below.

𝜎𝑒𝑞 = √[(𝜎11 − 𝜎22)2 + (𝜎22 − 𝜎33)2 + (𝜎33 − 𝜎11)2 + 6(𝜎12

2 + 𝜎232 + 𝜎31

2 )]

2

(49)


In our analysis, equation 49 reduces to equation 50 because we only have one longitudinal

component and one shear component.

𝜎𝑒𝑞 = √[2(𝜎𝑧𝑧)2 + 6(𝜎𝑧𝑠)2]

2

(50)

Table 9: Von Mises Stress Calculation

The calculation verifies that our wing design meets the Von Mises Yield Criterion with a factor of

safety of 1.4 for yield and 2.0 for fracture.

13.2 Column Buckling

We considered column buckling in our wing design to avoid stringer buckling, which is achieved

by adjusting the spacing of the ribs. The theory for the buckling analysis was worked out by Euler in

the eighteenth century. Euler’s equations for a load P applied to the ends of a beam can be seen in

the equations and Figure 34 below:

Figure 34: Column Buckling


We use two equations to solve for the rib spacing:

𝑃𝐶𝑅 = (𝜋2𝐸𝐼)

𝑙𝑒2

(51)

The variables, their values, and the units for each term are outlined in tables 10 & 11 below.

𝑃𝐶𝑅 = 1.5𝜎𝑧𝑧𝐴𝑠𝑡𝑟𝑖𝑛𝑔𝑒𝑟

(52)

There is an n-squared factor in equation 51 that disappears when we set n equal to one (setting n equal

to one gives the smallest value for P where the column remains in equilibrium [Megson 271]). Next, by

setting equations (51) & (52) equal to each other and solving for effective length, we obtain the results

in Table 10.

Table 10: Rib Spacing

Table 11: Stringer Area

We use the following equation (53) from Megson [Table 8.1, pg. 272] to solve for the rib spacing given

the effective length:

𝑙𝑒

𝑙= 2 (53)

Setting Equation 53 equal to 2 gives us the most conservative value for rib spacing, as a larger effective

length decreases the value for P critical that yields failure in the column. Given a half-span of 5.41

meters, this would suggest 10 ribs evenly spaced along the wing.


13.3 Skin (Thin Plate) Buckling

We consider thin plate buckling theory to determine the design for the skin plates that are

strengthened by the ribs and stringers. This analysis will compare the critical buckling load of the

individual plates with the total compressive plate buckling stress. The critical buckling load is given by

equation 54:

𝑁𝑋,𝐶𝑅 =𝑘𝜋2𝐸𝑡3

12𝑏2(1 − 𝑣2)

(54)

We set the buckling coefficient k equal to 8.5, set the Elastic Modulus for AL 2024 T3 to 73.1

[GPa], set Poisson’s ratio, v, equal to 0.33, while t and b are the skin thickness and plate length,

respectively. Next, we summed the critical buckling load for each section and divided by the skin

thickness to find the total compressive plate buckling stress.

𝜎𝐶𝑅 =𝑛 ∗ 𝑁𝑋,𝐶𝑅

𝑡

(55)

Here n is the number of plates, and t is the skin thickness. The calculations are shown in Table 12 below.

Table 12

13.4 Shear Buckling of Skin

In addition to column buckling and skin (thin plate) buckling, plate buckling from shear loading

of the wing’s skin was considered. The coefficient, k, used for plate buckling from shear loading is

given by Figure 35. Using the appropriate value for k, the buckling stress is calculated using

equation 54. With the calculated values for shear stress in the skin below the critical buckling

stress, the geometry of the wing design passed this test.


Figure 35: Shear Buckling Coefficients for Flat Plates (Megson 315)

13.5 Fracture & Fatigue

Fatigue is defined as the progressive deterioration of the strength of a material or structural

component during service such that failure can occur at much lower stress levels than the

ultimate stress level [Megson 455]. Fracture can also occur at stresses below the yield stress if an

initial crack is present. A small crack, if undetected, can manifest into catastrophic failure. Figure

36 below from Megson (pg. 423) illustrates the reduction in failure stress as the number of

repetitions of this stress increases.

Figure 36 Failure Stress vs Number of Loading repetitions

Here we analyze the number of flight cycles our wing can endure before a crack grows by 1.8

centimeters, given an initial crack size of 0.2 centimeters.


The equation for the number of loading/flight cycles (N) until failure is given by Equation 56:

𝑁 = 𝑎𝐶𝑅

1−𝑚2 − 𝑎

𝑖

1−𝑚2

𝐶(1 −𝑚2

)𝜋2𝜎∞2

(56)

To solve for aCR, we plug in a value for KIC (26 [MPa*m^.5]) into Equation (57)

𝑎𝐶𝑅 =1

𝜋(𝐾𝐼𝐶

𝜎∞)2

(57)

The results are given in Error! Reference source not found.. The critical crack length is calculated as

approximately 5.4mm.

Table 13: Flight Cycles to Failure

14. Aeroelasticity

14.1 Divergence

Wing divergence is the process of an applied load increasing due to aerodynamic loads (i.e. a

positive torsional pitch moment) that eventually build and cause the wing to reach a divergence

point. The deflection due to the increasing load causes the angle of attack to increase, and the

increasing wing deflection and twist this creates can lead to failure. The torsional divergence

speed is calculated from equation 58, where U is the torsional divergence speed:

𝑀 = 𝐶𝑈2(𝜃 + 𝛼0)

(58)


C is a coefficient, M is the moment per unit length, theta is the elastic twist of the beam, and

alpha not is the initial angle of attack. These values are related to the torsional stiffness of the

beam (GJ) by equation 59:

𝐺𝐽𝑑2𝜃

𝑑𝑦2= −𝑀

(59)

When designing the wing, it is necessary that the torsional divergence speed, U, does not

exceed the maximum flight velocity.

14.2 Aileron Reversal

Aileron Reversal is defined as the process where an aircraft rolls in the opposite direction as the

aileron input, often caused by twisting of the wing. As speed increases, the wing twist caused by the

aileron reversal will also increase. The rigidity of the wing is another important factor when

considering aileron reversal, as aileron reversal will occur more easily if the wing is more susceptible

to torsion. In the design of the wing, increasing wing rigidity decreases the likelihood of aileron

reversal. We can define an aileron reversal speed, which is the speed at which the aileron deflection

fails to cause any moment about the wing – at this point the aircraft rolls in the opposite direction

and experiences aileron reversal.


15. 3D Modelling and Simulation

15.1 3D Modelling

Figure 37 Section View of the 3D wing model

A CAD model of the wing design has been developed using SolidWorks, as shown in Figure 37. This

shows an example wing configuration which is not necessarily the most optimized, but gives an idea on

how all the structural components are mounted together. The wing is a three-cell box beam and its

structural portion extends from the airfoils leading edge to 80 % of the chord length.

In the drawings appear all the structural elements: the wing skin, the spars with spar caps, the stringers

and the ribs.

15.1.1 Modelling Steps

To start with the .dat file for NACA 2415 was imported using the Insert Curve option in Solidworks. This

generated the airfoil profile. The cross section of the airfoil was then extruded with the desired features

to generate the 3D wing model.


Figure 38 Isometric view of the Entire wing

The spars are connected to the airfoil skin through L-shaped spar caps that also hold a skin panel, which

closes the third cell at the back end of the structure. The ribs are shaped to fit the internal cavity of the

wing and some holes are cut through them in order to lighten the structure. Top-hat stringers run

parallel to the spars and are connected uniquely to the skin.


16. Loading Analysis

16.1 The Setup

The testing of the wing under a set of experimental loading conditions was performed on

Solidworks. For this type of analysis, the Solidworks Simulation package was used. The Static analysis

option was selected while setting up the loading conditions. Aluminum 2024 T3 with a yield stress of

3.45 x 108 N/m2 .

The complexity of the complete CAD model was too high to execute an accurate FEA on it.

Particularly, the very small skin thickness would require an excessively fine mesh strongly increasing the

computational time. For this reason, a simpler model was designed so that reasonably accurate results

can be obtained at the cost of an acceptable level of approximation.

16.1.1 Boundary Conditions

The root of the wing was specified to be fixed in the inertial frame.

The tip of the wing was left free.

A load of 200 N was uniformly applied on the bottom surface of the wing in the upward

direction.

16.2 Results

On applying an experimental load of 200 N uniformly on the bottom surface of the wing, it was observed

that the maximum stress on the wing was 4.89 x 105 N/m2 which was much lower than the yield stress

of Aluminum 2024 T3 of 3.45 x 108 N/m2 as can be seen from Error! Reference source not found..


Figure 39: Sigma_zz Distribution

With such a small value of loading, a corresponding maximum deflection of 42.66 mm was observed at

the wing-tips as can be seen from Figure 40.

Figure 40 Deflection of the wing due to the applied load


17. Modal Analysis

The modal analysis was conducted on the simplified wing model with one end fixed and assuming no lift

conditions (i.e. no wing loading). An iterative solver was used in the Frequency Study conducted on

Solidworks 2016 to generate the modal vibration data. The simulation generated the Modal Vibration

Frequency and Shape for the first five modes. This has been shown here from Figure 41 - Figure 45. The

plot of the Modal frequencies verses the Mode Number is shown in Figure 46.

Figure 41 Mode-1 Frequency-7.3987 Hz



Figure 43 Mode-3 Frequency- 42.362 Hz



Figure 45 Mode-5 Frequency- 99.968 Hz


Figure 46 Modal Frequency vs Mode Number

18. References

1) Megson, T. H. G. Aircraft Structures for Engineering Students. 5th ed. Oxford: Butterworth-

Heinemann, 2013. Print.

2) Lynch, C. S. Design of Aerospace Structures Lecture Series. UCLA Engineering: Mechanical and

Aerospace Engineering, 2016. Print.

19. Appendix

PLAA Sea Level

7.3987

32.997

42.362

55.554

99.968

0

20

40

60

80

100

120

0 1 2 3 4 5 6

Res

on

ant

Freq

uen

cy (H

z)

Mode #

Modal Analysis


Gust 1 Sea Level


Gust 2 Sea Level


NHAA Sea Level


PHAA Cruise Altitude


PLAA Cruise Altitude


Gust 1 Cruise Altitude


Gust 2 Cruise Altitude


NHAA Cruise Altitude

Appendix B

%% MAE 154B

% Ryan Rader

% Kevin Landry

% Agraj Sobti

clear all

close all

clc

% Establish basic parameters for plane

b=10.82; % meters

c=1.5; % meters

e=.79;

S=b*c; % meters^2

A= b^2/S;


W=1100*9.8; % Newtons

rho0=1.225; % Kg/m^3

rhoC=.7809; % Kg/m^3

VC=230*1000/3600; % m/s

VD=VC*1.5; % m/s

V=[0:0.5:VD]; % m/s

mu=1.983e-5; % pa*sec

g=32.17; % ft/s^2

A_cap=.0003; % m^2

A_str=.0001; % m^2

t_spar=.0025; % m

t_skin= .001; % m

x_spar=c/3; % m

x_strU=[0:5:95]*1.2/100; % m

x_strL=x_strU; % m

G=27579029160; % pa for 2024 all tempers

% Calculate a Reynolds Number

Re0=rho0*VC*c/mu;

ReC=rhoC*VC*c/mu;

[alpha1,CL1,CD1,CDp1,CM1,Top_Xtr1,Bot_Xtr1]=textread('naca24150.

pol','%f %f %f %f %f %f %f','headerlines', 12);

[alpha2,CL2,CD2,CDp2,CM2,Top_Xtr2,Bot_Xtr2]=textread('naca2415C.

pol','%f %f %f %f %f %f %f','headerlines', 12);

alpha={alpha1,alpha2};

CL={CL1,CL2};

CD={CD1,CD2};

CDp={CDp1,CDp2};

CM={CM1,CM2};

Top_Xtr={Top_Xtr1,Top_Xtr2};

Bot_Xtr={Bot_Xtr1,Bot_Xtr2};

alphar{1}=deg2rad(alpha{1});

alphar{2}=deg2rad(alpha{2});

ind=find(CL{1}>0,1);

CD0(1)=CD{1}(ind);

clear ind

ind=find(CL{2}>0,1);

CD0(2)=CD{2}(ind);

clear ind


[CLmax(1),mind(1)]=max(CL{1});

[CLmin(1),mind(2)]=min(CL{1});

[CLmax(2),mind(3)]=max(CL{2});

[CLmin(2),mind(4)]=min(CL{2});

Vspos(1)=sqrt(2*W/(CLmax(1)*rho0*S));

Vsneg(1)=sqrt(2*W/(abs((1))*rho0*S));

Vspos(2)=sqrt(2*W/(CLmax(2)*rhoC*S));

Vsneg(2)=sqrt(2*W/(abs(CLmin(2))*rhoC*S));

ind1(1)=find(alpha{1}==-7);

ind2(1)=find(alpha{1}==10);

ind1(2)=find(alpha{2}==-7);

ind2(2)=find(alpha{2}==10);

linefit(1,:)=polyfit(alphar{1}(ind1(1):ind2(1)),CL{1}(ind1(1):in

d2(1)),1);

linefit(2,:)=polyfit(alphar{2}(ind1(2):ind2(2)),CL{2}(ind1(2):in

d2(2)),1);

x=linspace(-.3,.3)';

y(:,1)=linefit(1,1).*x+linefit(1,2);

xinter(:,1)=-linefit(1,2)/linefit(1,1);

y(:,2)=linefit(2,1).*x+linefit(2,2);

xinter(:,2)=-linefit(2,2)/linefit(2,1);

CL3Dslope(1,1)=linefit(1,1)/(1+linefit(1,1)/(pi*A*e));

CL3Dline(1,1)=CL3Dslope(1,1);

CL3Dline(1,2)=-CL3Dslope(1,1)*xinter(:,1);

CL3Dslope(2,1)=linefit(2,1)/(1+linefit(2,1)/(pi*A*e));

CL3Dline(2,1)=CL3Dslope(2,1);

CL3Dline(2,2)=-CL3Dslope(2,1)*xinter(:,2);

CLmax(1)=CLmax(1)*CL3Dline(1,1)/linefit(1,1);

CLmin(1)=CLmin(1)*CL3Dline(1,1)/linefit(1,1);

CLmax(2)=CLmax(2)*CL3Dline(2,1)/linefit(2,1);

CLmin(2)=CLmin(2)*CL3Dline(2,1)/linefit(2,1);

% CLmax(1)=CL3Dline(1,1)*alphar{1}(mind(1))+CL3Dline(1,2);

% CLmin(1)=CL3Dline(1,1)*alphar{1}(mind(2))+CL3Dline(1,2);

% CLmax(2)=CL3Dline(2,1)*alphar{2}(mind(3))+CL3Dline(2,2);

% CLmin(2)=CL3Dline(2,1)*alphar{2}(mind(4))+CL3Dline(2,2);

% y3D(:,1)=CL3Dline(1,1).*x+CL3Dline(1,2);

% y3D(:,2)=CL3Dline(2,1).*x+CL3Dline(2,2);

y3Dline1{1}=(CL3Dline(1,1)/linefit(1,1)).*CL{1};


y3Dline1{2}=(CL3Dline(2,1)/linefit(2,1)).*CL{2};

p1=plot(alphar{1},CL{1});

hold on

plot(x,y(:,1))

%plot(x,y3D(:,1))

plot(alphar{1},y3Dline1{1})

grid on

legend('2D CL Curve','Best Fit Line','3D CL

Curve','location','best')

xlabel('\alpha (radians)')

ylabel('C_L (Coefficient of Lift)')

figure

p2=plot(alphar{2},CL{2});

hold on

plot(x,y(:,2))

% plot(x,y3D(:,1))

plot(alphar{1},y3Dline1{1})

grid on

legend('2D CL Curve','Best Fit Line','3D CL

Curve','location','best')

xlabel('\alpha (radians)')

ylabel('C_L (Coefficient of Lift)')

% Calculate n curves. Column 1 is sea level and column 2 is

cruise alt.

for n=1:length(V)

npos(n,1)=.5*rho0*CLmax(1)*V(n)^2*S/W;

nneg(n,1)=.5*rho0*CLmin(1)*V(n)^2*S/W;

npos(n,2)=.5*rhoC*CLmax(2)*V(n)^2*S/W;

nneg(n,2)=.5*rhoC*CLmin(2)*V(n)^2*S/W;

end

% take indicies less than the design load of 4.4 and -1.76

nposi{1}=find(npos(:,1)<4.4);

nnegi{1}=find(nneg(:,1)>-1.76);

nposi{2}=find(npos(:,2)<4.4);

nnegi{2}=find(nneg(:,2)>-1.76);

nnegi{3}=find(V>=VC,1);

% Calculate the slope and intercept for the maneuver limit using

the points

% (VC,-1.76) and (VD,-1)

manslope=(-1-(-1.76))/(VD-VC);

manint=-(VD*manslope+1);


for j=1:2

maxpos(j)=max(nposi{j});

maxneg(j)=max(nnegi{j});

npos(maxpos(j)+1:end,j)=4.4;

nneg(maxneg(j)+1:nnegi{3},j)=-1.76;

for n=nnegi{3}:length(V)

nneg(n,j)=manslope*V(n)+manint;

end

end

npos(end+1,:)=zeros(1,2);

nneg(end+1,:)=zeros(1,2);

V(end+1)=V(end);

%% Calculate the Gust Load Factor

UeC=50; % ft/s

UeD=25; % ft/s

VCk=VC*1.944; % knots

VDk=VD*1.944; % knots

cft=c*3.28084; % ft

Wlbf=W*0.224809; % lbf

Sft2=S*10.7639; % ft^2

rho0s=rho0*0.00194032; % slug/ft^3

rhoCs=rhoC*0.00194032; % slug/ft^3

u0=2*Wlbf/Sft2/(rho0s*cft*CL3Dslope(1)*g);

uC=2*Wlbf/Sft2/(rhoCs*cft*CL3Dslope(2)*g);

Kg0=.88*u0/(5.3+u0);

KgC=.88*uC/(5.3+uC);

nC(1)=1+Kg0*CL3Dslope(1)*UeC*VCk/(498*(Wlbf/Sft2));

nC(2)=1+KgC*CL3Dslope(2)*UeC*VCk/(498*(Wlbf/Sft2));

nD(1)=1+Kg0*CL3Dslope(1)*UeD*VDk/(498*(Wlbf/Sft2));

nD(2)=1+KgC*CL3Dslope(2)*UeD*VDk/(498*(Wlbf/Sft2));

ngustposS(:,1)=[1,nC(1),nD(1),1];

ngustnegS(:,1)=[1,-(nC(1)-2),-(nD(1)-2),1];

ngustposS(:,2)=[1,nC(2),nD(2),1];

ngustnegS(:,2)=[1,-(nC(2)-2),-(nD(2)-2),1];

vgust=[0,VC,VD,0];

ngustpos(:,1)=interp1(vgust(1:3),ngustposS(1:3,1),V);

ngustneg(:,1)=interp1(vgust(1:3),ngustnegS(1:3,1),V);


ngustpos(:,2)=interp1(vgust(1:3),ngustposS(1:3,2),V);

ngustneg(:,2)=interp1(vgust(1:3),ngustnegS(1:3,2),V);

count=0;

countn=0;

for j=1:2

for n=1:length(V)

if V(n)<Vspos(j)

else

count=count+1;

Vcomp{j}(count)=V(n);

end

if V(n)>Vspos(j)&& n<=maxpos(j)

nComp{j}(count)=npos(n,j);

elseif n>maxpos(j)

nComp{j}(count)=max(npos(n,j),ngustpos(n,j));

end

if V(n)<Vsneg(j)

else

countn=countn+1;

Vcomn{j}(countn)=V(n);

end

if V(n)>Vsneg(j)&&n<=maxneg(j)

nComn{j}(countn)=nneg(n,j);

elseif n>maxneg(j)

nComn{j}(countn)=min(nneg(n,j),ngustneg(n,j));

end

end

count=0;

countn=0;

end

nComp{1}=[0,nComp{1},0];

Vcomp{1}=[Vspos(1),Vcomp{1},VD];

nComp{2}=[0,nComp{2},0];

Vcomp{2}=[Vspos(2),Vcomp{2},VD];

nComn{1}=[0,0,nComn{1},0];

Vcomn{1}=[Vspos(1),Vsneg(1),Vcomn{1},VD];

nComn{2}=[0,0,nComn{2},0];

Vcomn{2}=[Vspos(2),Vsneg(2),Vcomn{2},VD];

figure('Position',[50,50,1200,500])

subplot(1,2,1)

plot(V,npos(:,1),'b')

hold on

plot(vgust,ngustposS(:,1),'r')


plot(Vcomp{1},nComp{1},'g--','linewidth',2)

plot(V,nneg(:,1),'b')

plot(vgust,ngustnegS(:,1),'r')

plot(Vcomn{1},nComn{1},'g--','linewidth',2)

legend('Maneuver Limit','Gust Loading','Combined

Loading','Location','northwest')

title('V-n Diagram at Sea Level')

xlabel('Velocity (m/s)')

ylabel('Load Factor (n)')

subplot(1,2,2)

plot(V,npos(:,2),'b')

hold on

plot(vgust,ngustposS(:,2),'r')

plot(Vcomp{2},nComp{2},'g--','linewidth',2)

plot(V,nneg(:,2),'b')

plot(vgust,ngustnegS(:,2),'r')

plot(Vcomn{2},nComn{2},'g--','linewidth',2)

legend('Maneuver Limit','Gust Loading','Combined

Loading','Location','northwest')

title('V-n Diagram at 4450 m')

xlabel('Velocity (m/s)')

ylabel('Load Factor (n)')

%% Calculating Wx and Wy

LCs=[59.5 4.4

95.5 4.4

95.5 -1.1

64 -1.78

39.5 -1.76

75.5 4.4

95.5 4.4

95.5 -1.298

64 -2.051

50.5 -1.76];

nz=100;

for n=1:length(LCs)

if n>length(LCs)/2

rho=rhoC;

CD0T=CD0(2);

m=2;

else

rho=rho0;

CD0T=CD0(1);


m=1;

end

L(n) = LCs(n,2)*W; % N

lift

CLCP(n) = 2*L(n)/rho/LCs(n,1)^2/S; %

lift coefficient

% Alpha(n)=(CLCP(n)-CL3Dline(2))/CL3Dline(1);

Alpha(n)=interp1(y3Dline1{m},alphar{m},CLCP(n));

D(n) = 0.5*rho*LCs(n,1)^2*S*(CD0(m) + CLCP(n)^2/pi/A/e); %

N drag

z = 0:b/2/nz:b/2; % root to tip (half

span)

% lift distribution

l_rect(:,n) = L(n)/b.*ones(1,nz+1);

l_ellip (:,n)= (4*L(n)/pi/b).*sqrt(1-(2.*z./b).^2);

l (:,n)= (l_rect(:,n) + l_ellip(:,n))./2; %

N/m

d (:,n)=D(n)/b.*ones(1,nz+1);

for m=1:length(d)

if m>.8*length(d)

d(m,n)=d(m,n)*1.1;

end

end


subplot(1,2,1)

p=plotyy([z',z',z'],[l_ellip(:,n),l_rect(:,n),l(:,n)],z,d(:,n));

ylabel(p(1),'Lift Force (N/m)')

xlabel('z (m)')

ylabel(p(2),'Drag Force (N/m)')

legend('lift elliptic distribution','lift rectangular

distribution','combined lift distribution','Drag

Distribution','location','best')

% rotate into x-y coordinate

wy(:,n) = cos(Alpha(n)).*l(:,n) + sin(Alpha(n)).*d(:,n);

wx(:,n) = -sin(Alpha(n)).*l(:,n) + cos(Alpha(n)).*d(:,n);

% Note: wx and wy are defined from root to tip

subplot(1,2,2)

plot(z,wy,z,wx,'linewidth',2)


xlabel('z (m)')

ylabel('Distributed Load (N/m)')

legend('w_y','w_x','location','best')

end


plot(z,l(:,1),'-',z,l(:,2),'-',z,l(:,3),'-',z,l(:,4),'-

',z,l(:,5),'-',...

z,l(:,6),'--',z,l(:,7),'--',z,l(:,8),'--',z,l(:,9),'--

',z,l(:,10),'--','linewidth',2)

ylabel('Lift Force (N/m)')

xlabel('z (m)')

legend('PHAA SL','PLAA SL','Gust 1 SL','Gust 2 SL','NHAA

SL','PHAA CA','PLAA CA','Gust 1 CA','Gust 2 CA','NHAA

CA','location','west')

Rho(1:5)=rho0;

Rho(6:10)=rhoC;

Cd(1:5)=CD(1);

Cd(6:10)=CD(2);

for n=1:length(LCs)

[z1t,Mx0t,My0t,Sx0t,Sy0t,sig_z0t,u0t,v0t] =

DeflectionLoadscode(wx(:,n),wy(:,n),LCs(n,2),Rho(n),LCs(n,1),Alp

ha(n),Cd(n),nz);

z1(:,n)=z1t;

Mx0(:,n)=Mx0t;

My0(:,n)=My0t;

Sx0(:,n)=Sx0t;

Sy0(:,n)=Sy0t;

sig_z0(:,n)=sig_z0t;

u0a(:,n)=u0t;

v0a(:,n)=v0t;

clear z1t Mx0t My0t Sx0t Sy0t sig_z0t u0t v0t

end

%% X Moments

figure

subplot(2,1,1)

plot(z,Mx0(:,1),z,Mx0(:,2),z,Mx0(:,3),z,Mx0(:,4),z,Mx0(:,5))

legend('PHAA','PLAA','GL 1','GL 2','NHAA','location','best')

ylabel('Mx')

xlabel('Half Span Distance')

title('At Sea Level')

subplot(2,1,2)

plot(z,Mx0(:,6),z,Mx0(:,7),z,Mx0(:,8),z,Mx0(:,9),z,Mx0(:,10))



ylabel('Mx')


title('At Service Ceiling')

%% Y Moment

figure

subplot(2,1,1)

plot(z,My0(:,1),z,My0(:,2),z,My0(:,3),z,My0(:,4),z,My0(:,5))


ylabel('My')



subplot(2,1,2)

plot(z,My0(:,6),z,My0(:,7),z,My0(:,8),z,My0(:,9),z,My0(:,10))


ylabel('My')



%% X Shear

figure

subplot(2,1,1)

plot(z,Sx0(:,1),z,Sx0(:,2),z,Sx0(:,3),z,Sx0(:,4),z,Sx0(:,5))


ylabel('Sx')



subplot(2,1,2)

plot(z,Sx0(:,6),z,Sx0(:,7),z,Sx0(:,8),z,Sx0(:,9),z,Sx0(:,10))


ylabel('Sx')



%% Y Shear

figure

subplot(2,1,1)

plot(z,Sy0(:,1),z,Sy0(:,2),z,Sy0(:,3),z,Sy0(:,4),z,Sy0(:,5))


ylabel('Sy')



subplot(2,1,2)

plot(z,Sy0(:,6),z,Sy0(:,7),z,Sy0(:,8),z,Sy0(:,9),z,Sy0(:,10))



ylabel('Sy')



%% Direct Stress

figure

subplot(2,1,1)

plot(z,sig_z0(:,1),z,sig_z0(:,2),z,sig_z0(:,3),z,sig_z0(:,4),z,s

ig_z0(:,5))


ylabel('Direct Stress')



subplot(2,1,2)

plot(z,sig_z0(:,6),z,sig_z0(:,7),z,sig_z0(:,8),z,sig_z0(:,9),z,s

ig_z0(:,10))


ylabel('Direct Stress')



%% u-Deflections

figure

subplot(2,1,1)

plot(z,u0a(:,1),z,u0a(:,2),z,u0a(:,3),z,u0a(:,4),z,u0a(:,5))


ylabel('u-Deflection')



subplot(2,1,2)

plot(z,u0a(:,6),z,u0a(:,7),z,u0a(:,8),z,u0a(:,9),z,u0a(:,10))


ylabel('u-Deflection')



%% v-Deflection

figure

subplot(2,1,1)

plot(z,v0a(:,1),z,v0a(:,2),z,v0a(:,3),z,v0a(:,4),z,v0a(:,5))

legend('PHAA','PLAA','GL 1','GL

2','NHAA','location','northeast')

ylabel('v-Deflection')

xlabel('Half Span Distance (m)')



subplot(2,1,2)

plot(z,v0a(:,6),z,v0a(:,7),z,v0a(:,8),z,v0a(:,9),z,v0a(:,10))

legend('PHAA','PLAA','GL 1','GL

2','NHAA','location','northeast')

ylabel('v-Deflection')

xlabel('Half Span Distance (m)')


%% Recalculating the Airfoil Section

[cq,Ixx,Iyy,Ixy,x,yU,yL,Bigmat,i_strU,dx,i_spar,i_strL] =

airfoil_section(c,A_cap,A_str,t_spar,t_skin,x_spar,x_strU,x_strL

);

Ivals=[Ixx,Iyy,Ixy];

for n=1:length(LCs)

SMvals=[Mx0(1,n),My0(1,n),Sx0(1,n),Sy0(1,n)]; % at the root

[sigeq(:,n),sigzs(:,n),sigzz(:,n),qtot(:,n),xyc,xycent,qb(:,n),s

igcheck(:,n)]=shearflow(Bigmat,cq,SMvals,Ivals,t_skin,t_spar,G);

figure

scatter(xycent(:,1),xycent(:,2),10,sigzs(:,n),'filled')

c=colorbar;

c.Label.String='\sigma_z_s (pa)';

ylim ([-.3 .3])

xlabel('X Coordinate (m)')

ylabel('Y Coordinate (m)')

figure

scatter(xycent(:,1),xycent(:,2),10,sigeq(:,n),'filled')

c=colorbar;

c.Label.String='\sigma_e_q (pa)';

ylim ([-.3 .3])



end

figure

scatter(xycent(:,1),xycent(:,2),10,sigeq(:,1),'filled')

c=colorbar;

c.Label.String='\sigma_e_q';

ylim ([-.3 .3])




figure

plot(x,yU,'k','Linewidth',2);

ylim([-0.3 0.3])

hold on

plot(i_strU*dx,yU(i_strU),'or','markersize',5);

plot([x_spar(1),x_spar(1)],[yU(i_spar(1)),yL(i_spar(1))],'b','Li

newidth',3);

scatter(xyc(1,1),xyc(1,2),'m*')

plot(x,yL,'k','Linewidth',2);

plot(i_strL*dx,yL(i_strL),'or','markersize',5);

plot([x(end),x(end)],[yU(end),yL(end)],'b','Linewidth',3);

ylabel('y (m)')

xlabel('x (m)')

grid on

legend('Airfoil','Stringers','Spars','Centriod')

figure

plot(qb)

ylabel('q_b (N/m)')

xlabel('Point Number')



CA','location','best')

figure

plot(sigcheck)

ylabel('Sigma_z_s*Boom Area (lbf)')

xlabel('Point Number')



CA','location','best')

function[sigeq,sigzs,sigzz,qtot,xyc,xycent,q,sigcheck]=shearflow

(Bigmat,xs,SMvals,Ivals,tskin,tspar,G)

xy=Bigmat(:,1:2);

if xy(1,2)~=xy(end,2)

xy(end+1,:)=xy(1,:);

end

b=ones(length(xy),1);


xy=[xy,b*0];

% calculate area of wing cross section

for n=1:length(xy)-1

dA2(n)=norm(cross(xy(n,:),xy(n+1,:)));

A2(n)=sum(dA2);

ldist(n,1)=sqrt((xy(n,1)-xy(n+1,1))^2+(xy(n,2)-

xy(n+1,2))^2);

askin(n,1)=Bigmat(n,6);

xyA(n,:)=[xy(n,1)*askin(n),xy(n,2)*askin(n)];

atot(n,1)=sum(Bigmat(n,3:6));

end

xyc(1,:)=[sum(xyA(:,1))/sum(askin),sum(xyA(:,2))/sum(askin)];

%normalize x and y vectors to centriod

xycent=xy-[b*xyc(1),b*xyc(2),b*0];

% calculate area of wing cross section

for n=1:length(xy)-1

dA2(n)=norm(cross(xycent(n,:),xycent(n+1,:)));

A2(n)=sum(dA2);

ldist(n,1)=sqrt((xycent(n,1)-xycent(n+1,1))^2+(xycent(n,2)-

xycent(n+1,2))^2);

askin(n,1)=Bigmat(n,6);

xyA(n,:)=[xycent(n,1)*askin(n),xycent(n,2)*askin(n)];

atot(n,1)=sum([Bigmat(n,3),Bigmat(n,5:6)]);

end

xyc(2,:)=[sum(xyA(:,1))/sum(askin),sum(xyA(:,2))/sum(askin)];

for n=1:length(xycent)-1

Ivec(n,:)=[askin(n)*(xycent(n,2)-

xyc(2))^2,askin(n)*(xycent(n,1)-xyc(1))^2,askin(n)*(xycent(n,1)-

xyc(1))*(xycent(n,2)-xyc(2))];

end

IvecT=[sum(Ivec(:,1)),sum(Ivec(:,2)),sum(Ivec(:,3))];

den1=IvecT(1)*IvecT(2)-IvecT(3)^2;

Cts=[(SMvals(2)*IvecT(1)-

SMvals(1)*IvecT(3))/den1,(SMvals(1)*IvecT(2)-

SMvals(2)*IvecT(3))/den1,...

(SMvals(4)*IvecT(3)-

SMvals(3)*IvecT(1))/den1,(SMvals(3)*IvecT(3)-

SMvals(4)*IvecT(2))/den1];



sigzz(n,1)=Cts(1)*xycent(n,1)+Cts(2)*xycent(n,2);

end

ind=find(Bigmat(:,4)>0);

Lspar=sqrt((xycent(ind(2),1)-

xycent(ind(3),1))^2+(xycent(ind(2),2)-xycent(ind(3),2))^2);


if n==1

BoomA(n,1)=(atot(n)-

askin(n))+tskin*ldist(n)/6*(2+sigzz(n+1)/sigzz(n))+tspar*ldist(e

nd)/6*(2+sigzz(end)/sigzz(n));

elseif n==length(xycent)-1

BoomA(n,1)=(atot(n)-askin(n))+tskin*ldist(n-

1)/6*(2+sigzz(n-

1)/sigzz(n))+tspar*ldist(n)/6*(2+sigzz(1)/sigzz(n));

elseif n==ind(2)


1)/6*(2+sigzz(n-1)/sigzz(n))...

+tskin*ldist(n)/6*(2+sigzz(n+1)/sigzz(n))+tspar*Lspar/6*(2+sigzz

(ind(3))/sigzz(n));

elseif n==ind(3)


1)/6*(2+sigzz(n-1)/sigzz(n))...

+tskin*ldist(n)/6*(2+sigzz(n+1)/sigzz(n))+tspar*Lspar/6*(2+sigzz

(ind(2))/sigzz(n));

else

BoomA(n,1)=(atot(n)-

askin(n))+tskin*ldist(n)/6*(2+sigzz(n+1)/sigzz(n))...

+tskin*ldist(n-1)/6*(2+sigzz(n-1)/sigzz(n));

end

dq(n,1)=Cts(3)*BoomA(n)*(xycent(n,1))+Cts(4)*BoomA(n)*(xycent(n,

2));

q(n,1)=sum(dq);

q2dA(n,1)=dq(n,1)*dA2(n);

end

At(1)=sum(dA2(ind(2):ind(3)))+norm(cross(xycent(ind(3),:),xycent

(ind(2),:)));

At(2)=sum(dA2(ind(1):ind(2)))+norm(cross(xycent(ind(4),:),xycent

(ind(1),:)))...


+norm(cross(xycent(ind(2),:),xycent(ind(3),:)))+sum(dA2(ind(3):i

nd(4)));

Bx(1)=sum(BoomA(ind(2):ind(3)).*xycent(ind(2):ind(3),1).*ldist(i

nd(2):ind(3))/tskin);

Bx(2)=sum(BoomA(ind(1):ind(2)).*xycent(ind(1):ind(2),1).*ldist(i

nd(1):ind(2))/tskin)+...

sum(BoomA(ind(3):ind(4)).*xycent(ind(3):ind(4),1).*ldist(ind(3):

ind(4))/tskin);

By(1)=sum(BoomA(ind(2):ind(3)).*xycent(ind(2):ind(3),2).*ldist(i

nd(2):ind(3))/tskin);

By(2)=sum(BoomA(ind(1):ind(2)).*xycent(ind(1):ind(2),2).*ldist(i

nd(1):ind(2))/tskin)+...

sum(BoomA(ind(3):ind(4)).*xycent(ind(3):ind(4),2).*ldist(ind(3):

ind(4))/tskin);

dytspar=(xycent(ind(2),2)-xycent(ind(3),2))/tspar;

dytback=(xycent(ind(1),2)-xycent(ind(4),2))/tspar;

ldtsk(1)=sum(ldist(ind(2):ind(3))/tskin);

ldtsk(2)=(sum(ldist(ind(1):ind(2)))+sum(ldist(ind(3):ind(4))))/t

skin;

b=[-Cts(3)*Bx(1)-Cts(4)*By(1);-Cts(3)*Bx(2)-

Cts(4)*By(2);SMvals(3)+SMvals(2)*xs-sum(q2dA)];

A=[ldtsk(1)+(-dytspar),-(-dytspar),-At(1)*G;-

dytspar,ldtsk(2)+dytspar+dytback,-At(2)*G;...

sum(q2dA(ind(2):ind(3))),sum(q2dA(ind(1):ind(2)))+sum(q2dA(ind(3

):ind(4))),0];

x=A^-1*b;

q01=x(1);

q02=x(2);

dthdz=x(3);

qtot=[q(ind(1):ind(2))+q02;q(ind(2)+1:ind(3))+q01;q(ind(3)+1:ind

(4))+q02];

sigzs=qtot/tskin;

for n=1:length(sigzs)

sigeq(n,1)=sqrt(2*sigzz(n)^2+6*sigzs(n)^2);


end

sigcheck=BoomA.*sigzs;

xycent=xycent(1:end-1,:);

function [cq,Ixx,Iyy,Ixy,x,yU,yL,Bigmat,i_strU,dx,i_spar,i_strL]

=

airfoil_section(c,A_cap,A_str,t_spar,t_skin,x_spar,x_strU,x_strL

)

%% airfoil section profile

% NACA 2415

m = 0.02; % 1st digit maximum camber (m) in percentage of the

chord

p = 0.4; % 2nd digit position of the maximum camber (p) in

tenths of chord

t = 0.15; % last 2 digits(Maximum thickness (t) of the airfoil

in percentage of chord)

cq = c/4;

nx = 500; % number of increments

dx = c/nx;

x = 0:dx:c; % even spacing

yc = zeros(1,nx+1);

yt = zeros(1,nx+1);

yU0 = zeros(1,nx+1);

yL0 = zeros(1,nx+1);

theta = zeros(1,nx+1);

xb = p*c;

i_xb = xb/dx + 1;

for i = 1:nx+1

yt(i) = 5*t*c*(0.2969*sqrt(x(i)/c) - 0.1260*(x(i)/c) -

0.3516*(x(i)/c)^2 + 0.2843*(x(i)/c)^3 - 0.1015*(x(i)/c)^4);

if i <= i_xb

yc(i) = m*x(i)/p^2*(2*p - x(i)/c);

theta(i) = atan(2*m/p^2*(p - x(i)/c));

else

yc(i) = m*(c - x(i))/(1-p)^2*(1 + x(i)/c - 2*p);

theta(i) = atan(2*m/(1-p)^2*(p-x(i)/c));

end

yU0(i) = yc(i) + yt(i)*cos(theta(i));

yL0(i) = yc(i) - yt(i)*cos(theta(i));

end


% The last 20% of the chord length of the airfoil was neglected

under the assumption that

% this section contained flaps and ailerons, and would therefore

not support aerodynamic loads.

i_xend = round(0.8*c/dx)+1;

x = x(1:i_xend);

yU = yU0(1:i_xend);

yL = yL0(1:i_xend);

% Here the airfoil profile is approximated by assuming xU0=x &

xL0=x.

%% adding stringers, spar caps and spars

% spars only

x_spar = [x_spar,x(end)]; % x_spar: location of

spars and spar caps

n_spar = length(x_spar); % Number of spars

i_spar = round(x_spar./dx)+1; % number of divisions to

get to x_spar

h_spar = yU(i_spar) - yL(i_spar); % height of spar

Cy_spar = (yU(i_spar) + yL(i_spar))/2; % y coord of centroid of

spar

A_spar = t_spar.*h_spar; % Area of spar = t*h

array_spar = zeros(1,length(x));

j=1;

for i = 1:length(x)

if i == i_spar(j)

array_spar(i)=A_spar(j);

j=j+1;

else

array_spar(i)=0;

end

end

array_cap = zeros(1,length(x));

array_cap(i_spar) = A_cap;

%% stringers

i_strU = round(x_strU./dx)+1; % index in x array

corresponding to the Upper stringer locations

i_strL = round(x_strL./dx)+1; % index in x array

corresponding to the Lower stringer locations

% Remove Stringer where there are spars for Upper Part

commonind = [];

for i=1:length(i_spar)


for j=1:length(i_strU)

if i_spar(i) == i_strU(j)

commonind = [commonind , j];

end

end

end

i_strU (commonind) = [];

% Remove Stringer where there are spars for Lower Part

commonind = [];

for i=1:length(i_spar)

for j=1:length(i_strL)

if i_spar(i) == i_strL(j);

commonind = [commonind , j];

end

end

end

i_strL(commonind) = [];

n_strU = length(i_strU); % Number of stringers in upper

half

n_strL = length(i_strL); % Number of stringers in lower

half

% Creating aray of Stringer areas for putting in Bigmatrix

array_strU = zeros(1,length(x));

array_strU(i_strU) = A_str;

array_strL = zeros(1,length(x));

array_strL(i_strL) = A_str;

%% skins

% nodes include spar caps and stringers

x_nodeU = [x_spar,x_strU]; % x-coord of (stringers,spar

caps) combined

x_nodeU = sort(x_nodeU); % x-coord of nodes (Sorted)

n_nodeU = length(x_nodeU);

i_nodeU = round(x_nodeU./dx)+1; % index of nodes

% MODIFIED SKIN CODE

n_skinU = length(x)-1;

for i = 1:n_skinU

L_skinU(i) = sqrt((x(i)-x(i+1))^2 + (yU(i)-yU(i+1))^2);

A_skinU(i) = t_skin*L_skinU(i);

Cx_skinU(i) = (x(i)+x(i+1))/2;

Cy_skinU(i) = (yU(i)+yU(i+1))/2;

end

n_skinL = length(x)-1;

for i = 1:n_skinL


L_skinL(i) = sqrt((x(i)-x(i+1))^2 + (yL(i)-yL(i+1))^2);

A_skinL(i) = t_skin*L_skinL(i);

Cx_skinL(i) = (x(i)+x(i+1))/2;

Cy_skinL(i) = (yU(i)+yU(i+1))/2;

end

A_skinU = [0 A_skinU]; % adding zero to the front of the array

A_skinL = [A_skinL 0]; % same as above (Now 401 elements)

% skin should be broken into smaller elements for higher

accuracy of calculation

% break one skin element into two by adding one more node in

between

%%

% x_skinU = zeros(1,2*length(x_nodeU)-1);

% for i = 1:length(x_nodeU)-1

% x_skinU(2*i-1) = x_nodeU(i);

% x_skinU(2*i) = (x_nodeU(i) + x_nodeU(i+1))/2;

% end

% x_skinU(end) = x_nodeU(end);

%

% i_skinU = round(x_skinU/dx)+1;

% n_skinU = length(x_skinU)-1;

% L_skinU = zeros(1,n_skinU);

% A_skinU = zeros(1,n_skinU);

% Cx_skinU = zeros(1,n_skinU);

% Cy_skinU = zeros(1,n_skinU);

% for i = 1:n_skinU

% L_skinU(i) = sqrt((yU(i_skinU(i+1)) - yU(i_skinU(i)))^2 +

(x_skinU(i+1) - x_skinU(i))^2);

% A_skinU(i) = t_skin*L_skinU(i);

% Cx_skinU(i) = (x_skinU(i+1) + x_skinU(i))/2;

% Cy_skinU(i) = (yU(i_skinU(i+1)) + yU(i_skinU(i)))/2;

% end

%

% % lower part

% x_nodeL = [x_spar,x_strL]; % x-coord of (stringers,spar

caps) combined

% n_nodeL = length(x_nodeL);

% x_nodeL = sort(x_nodeL); % x-coord of nodes (Sorted)

% i_nodeL = round(x_nodeL./dx)+1; % index of nodes

%

% x_skinL = zeros(1,2*length(x_nodeL)-1);

% for i = 1:length(x_nodeL)-1

% x_skinL(2*i-1) = x_nodeL(i);

% x_skinL(2*i) = (x_nodeL(i) + x_nodeL(i+1))/2;

% end


% x_skinL(end) = x_nodeL(end);

%

% i_skinL = round(x_skinL/dx)+1;

% n_skinL = length(x_skinL)-1;

% L_skinL = zeros(1,n_skinL);

% A_skinL = zeros(1,n_skinL);

% Cx_skinL = zeros(1,n_skinL);

% Cy_skinL = zeros(1,n_skinL);

% for i = 1:n_skinL

% L_skinL(i) = sqrt((yL(i_skinL(i+1)) - yL(i_skinL(i)))^2 +

(x_skinL(i+1) - x_skinL(i))^2);

% A_skinL(i) = t_skin*L_skinL(i);

% Cx_skinL(i) = (x_skinL(i+1) + x_skinL(i))/2;

% Cy_skinL(i) = (yL(i_skinL(i+1)) + yL(i_skinL(i)))/2;

% end

%% centroid of the wing section

% initial value

Cx_sum = 0;

Cy_sum = 0;

A_sum = 0;

% spars

for i = 1:n_spar

Cx_sum = Cx_sum + x_spar(i)*A_spar(i);

Cy_sum = Cy_sum + Cy_spar(i)*A_spar(i);

A_sum = A_sum + A_spar(i);

end

%Upper Skin

for i = 1:n_skinU

Cx_sum = Cx_sum + Cx_skinU(i)*A_skinU(i);

Cy_sum = Cy_sum + Cy_skinU(i)*A_skinU(i);

A_sum = A_sum + A_skinU(i);

end

%Lower skin

for i = 1:n_skinL

Cx_sum = Cx_sum + Cx_skinL(i)*A_skinL(i);

Cy_sum = Cy_sum + Cx_skinL(i)*A_skinL(i);

A_sum = A_sum + A_skinL(i);

end

%Upper Stringers

for i = 1:n_strU

Cx_sum = Cx_sum + x_strU(i)*A_str;

Cy_sum = Cy_sum + yU(i_strU(i))*A_str;

A_sum = A_sum + A_str;

end


%Lower Stringers

for i = 1:n_strL

Cx_sum = Cx_sum + x_strL(i)*A_str;

Cy_sum = Cy_sum + yL(i_strL(i))*A_str;

A_sum = A_sum + A_str;

end

%All spar caps

for i = 1:n_spar

Cx_sum = Cx_sum + 2*A_cap*x_spar(i);

Cy_sum = Cy_sum + A_cap*Cy_spar(i);

A_sum = A_sum + A_cap;

end

Cx = Cx_sum/A_sum;

Cy = Cy_sum/A_sum;

% figure

% plot(x,yU,'k',x,yL,'k','Linewidth',2);

% ylim([-0.3 0.3])

% hold on

%

plot(i_strU*dx,yU(i_strU),'or',i_strL*dx,yL(i_strL),'or','marker

size',5);

%

plot([x_spar(1),x_spar(1)],[yU(i_spar(1)),yL(i_spar(1))],'b',[x(

end),x(end)],[yU(end),yL(end)],'b','Linewidth',3);

%

plot(x_spar,yU(i_spar),'sg',x_spar,yL(i_spar),'sg','markersize',

7);

% scatter(Cx,Cy,'m*')

% ylabel('y (m)')

% xlabel('x (m)')

% grid on

%% Area moments of inertia

% initial value

Ixx = 0;

Iyy = 0;

Ixy = 0;

% Spars MOI

for i = 1:n_spar

Ixx = Ixx + t_spar*h_spar(i)^3/12 + A_spar(i)*(Cy_spar(i)-

Cy)^2;

Iyy = Iyy + t_spar^3*h_spar(i)/12 + A_spar(i)*(x_spar(i)-

Cx)^2;


Ixy = Ixy + A_spar(i)*(Cy_spar(i)-Cy)*(x_spar(i)-Cx);

end

% Spar caps MOI

for i = 1:n_spar

Ixx = Ixx + A_cap*(yU(i_spar(i))-Cy)^2 +

A_cap*(yL(i_spar(i))-Cy)^2;

Iyy = Iyy + 2*A_cap*(x_spar(i)-Cx)^2; %

2*A since both upper and lower caps

Ixy = Ixy + A_cap*(yU(i_spar(i))-Cy)*(x_spar(i)-Cx)+... %

upper spar caps

A_cap*(yL(i_spar(i))-Cy)*(x_spar(i)-Cx); %

lower spar caps

end

% Upper Skin MOI

for i = 1:n_skinU

Ixx = Ixx + A_skinU(i)*(Cx_skinU(i)-Cx)^2;

Iyy = Iyy + A_skinU(i)*(Cy_skinU(i)-Cy)^2;

Ixy = Ixy + A_skinU(i)*(Cx_skinU(i)-Cx)*(Cy_skinU(i)-Cy);

end

% Lower Skin MOI

for i = 1:n_skinL

Ixx = Ixx + A_skinL(i)*(Cx_skinL(i)-Cx)^2;

Iyy = Iyy + A_skinL(i)*(Cy_skinL(i)-Cy)^2;

Ixy = Ixy + A_skinL(i)*(Cx_skinL(i)-Cx)*(Cy_skinU(i)-Cy);

end

% Upper Stringers MOI

for i = 1:n_strU

Ixx = Ixx + A_str*(yU(i_strU(i))-Cy)^2;

Iyy = Iyy + A_str*(x_strU(i)-Cx)^2;

Ixy = Ixy + A_str*(x_strU(i)-Cx)*(yU(i_strU(i))-Cy);

end

%Lower Stringers MOI

for i = 1:n_strL

Ixx = Ixx + A_str*(yL(i_strL(i))-Cy)^2;

Iyy = Iyy + A_str*(x_strL(i)-Cx)^2;

Ixy = Ixy + A_str*(x_strL(i)-Cx)*(yU(i_strL(i))-Cy);

end

%% Shifting the origin to the centroid:

% x = x-Cx;

% yU = yU-Cy;

% yL = yL-Cy;

% x_skinU = Cx_skinU-Cx;

% x_skinL = Cx_skinL-Cx;

% x_spar = x_spar-Cx;

% x_strU = x_strU-Cx;


% x_strL = x_strL-Cx;

%% Big Matrix containing all the values

for i = 1:length(x) %inverting the coordinates

x2(i) = x(end-(i-1));

yUtemp(i) = yU(end-(i-1));

array_captemp(i) = array_cap(end-(i-1));

array_spartemp(i) = array_spar(end-(i-1));

array_strUtemp(i) = array_strU(end-(i-1));

A_skinUtemp(i) = A_skinU(end-(i-1));

end

% yU = yUtemp;

% array_cap = array_captemp;

% array_spar = array_spartemp;

% array_strU = array_strUtemp;

% A_skinU = A_skinUtemp;

Bigmat = [x2(1:end-1)' yUtemp(1:end-1)' array_captemp(1:end-1)'

array_spartemp(1:end-1)' array_strUtemp(1:end-1)'

A_skinUtemp(1:end-1)' ; ...

x',yL',array_cap',array_spar',array_strL',A_skinL'] ;

%Bigmat = [x2' yU' array_cap' array_spar' array_strU' A_skinU' ;

...

%x',yL',array_cap',array_spar',array_strL',A_skinL'] ;

end

report-wing structure design spring'16

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