Week6 Wing Load Distribution W2014

2/12/2014

Aircraft Structural Design

2

Schrenk’s approximation Schrenk’s approximation •

•

•

Classical wing theory: for an elliptical wing, spanwise air load (lift) distribution is of elliptical shape. Schrenk’s approximation Schrenk’s approximation for a non-elliptical wing: assumes that the load distribution on untwisted wing or tail has a shape that is the average of the actual planform shape and an elliptical shape of the same span and area. The total area under the lift load curve must equate to the required total lift.

2/12/2014


3

Schrenk’s approximation

Schrenk’s method essentially states that the resultant load distribution is an arithmetic mean of:  A load distribution representing the actual planform shape  An elliptical distribution of the same span and area •

2/12/2014


4

Schrenk’s approximation

Here the semi-span wing area = area of an elliptical quadrant = S/2. 2/12/2014


5

Schrenk’s approximation •

Semi Elliptical Area:

 1  4 = 2 ⟹  = 2 4 4  •

But for an ellipse:





 4 2     = 1 ⟹  =  1 −   2

2/12/2014




6


Schrenks’s approximation is then to put wy (N/m) in place of cy and put L (N) in place of S, yielding the following expression for load distribution over the wing as a function of spanwise distance y (m):

4 2  = 1−   2/12/2014




7


For a tapered wing with taper ratio 

=

 : 

      = = 2 2 2    =    4  2 = 1      = 4 1  2/12/2014


8

Schrenk’s approximation Now:

 =  −

  

 −  =

2  1   − 1   2 2  = 1 −1 1   Then: replacing  with  and S with L, we obtain the expression for the load distribution over the span:

2 2  = 1 −1 1   2/12/2014


9


Schrenks’s approximation for load distribution over a tapered wing is therefore the average of the following two distributions:

4 2  = 1−   •



And:

2 2  = 1 −1 1   2/12/2014


10

Anderson’s approximation •

•

•

•

Based on results obtained for base (L b) and additional (La) lift on a tapered/twisted wing. Base lift: lift generated by the twist of a wing. We will focus here on untwisted wings whereby only the additional lift distribution is of interest. For detailed discussions see reference below:

Theory of Wing Sections by Ira H. Abbott And Albert E. Von Doenhoff, Dover Publications, Inc. NY, 1959. 2/12/2014


11


Consider an arbitrary wing with a specified surface area S, total span b, a taper ratio λ = ct/cr and an aspect ratio A = b 2/S.

cr

yi

y

Station i

c(y)

Station i+1

ct

yi+1

b/2 •

Station (i) = Station (yi/(b/2)) = Station (0, 0.2, 0.4, etc.)

2/12/2014


12


As seen before:

2  =  1  −1  •

And:

2/12/2014

  = 1  2


13


According to this method, the local lift coefficient cL at a given Station (i) can be determined using:

 =  (/)

where ci is the chord length at Station (i) and L a is a coefficient determined from Tables. •

•

Once cL is determined, calculate local lift force L i using: Where:

2/12/2014

 = ½(    )  = (+ –  )(+   )/2 Aircraft Structural Design

14

2/12/2014


15


•

Li acts midspan between station i and i+1 . Shear forces and bending/twisting moments at each station can then be determined from a balance of forces and moments. Vi

yi

Vi+1

Li

yi+1 Mi+1

Mi Inboard (wing root)

Station i

Outboard (wing tip)

Station i+1 i+1

2/12/2014


16

Anderson’s approximation

 = +    = +  (+ – )/2  + (+ –  )

2/12/2014


17


General case of a wing at an Angle of Attack α L N

α

P

z

D

x

 = ()  ()  = − ()  () 2/12/2014


18

Anderson’s approximation    

= = = =

 (/) ½(  )  (/ ) ½(  )

 = ()  ()  = − ()  ()  = +    = +  (+ – )/2  + (+ – )  = +    = +   (+ –  )/2  + (+ – ) 2/12/2014


19

Anderson’s approximation Wing torque: •

The value of the wing torque (torsion moment)  is related to the magnitude and direction of the pitching moment of the wing  plus the moment caused by the normal lift (N) acting about the shear centre of the wing box (neglecting the contribution of P).

2/12/2014


20

Anderson’s approximation Ni

Mo

SC

l

Myi

Station i

AC

Station i+1

Myi+1

 = +      = ½(2 )  =  (/) 2/12/2014


21


Example 1:

A tapered wing has a half span of 6 m, a root chord of 2.6 m and a tip chord of 1.6 m. Assuming the a/c is at an AOA α = 20o calculate all applicable loads using the Anderson Tables for air load approximations. Assume CL = 1.9, CD = 0.2, Cmo = -0.05 and that the a/c is in flight at an airspeed V = 100 m/s and ρ = 1.225 kg/m3.

2/12/2014


22

Anderson’s approximation Solution: λ = ct/cr = 1.6/2.6 = 0.62 ≈ 0.6 (for Anderson’s tables) S = (b/2)(1+ λ) cr = 6(1+0.62)(2.6) = 25.2 m 2 A = b2/S = (12)2/25.27 = 5.7 ≈ 6 (for Anderson’s tables) We now use the Anderson tables to obtain L a at the various stations for c t/cr = 0.6 and A = 6. For example at Station 0, L a = 1.267 while at Station 0.975, La = 0.340. From those values of L a we proceed to determine all applicable loads. •

•

•

•

2/12/2014


23

Anderson’s approximation STATION

y i

(m)

C i (m)

Si

L

2

(m )

a

c L

c D

Li

(N)

Di

(N)

c mo

Mo

0.0000

0.0000

2.6000

3.0000

1.2670

1.9444

0.2047

35728

3761

-0.0512

-2445

0.2000

1.2000

2.4000

2.7600

1.2180

2.0249

0.2132

34231

3603

-0.0533

-2162

0.4000

2.4000

2.2000

2.5200

1.1320

2.0530

0.2161

31689

3336

-0.0540

-1835

0.6000

3.6000

2.0000

2.2800

1.0020

1.9990

0.2104

27916

2939

-0.0526

-1469

0.8000

4.8000

1.8000

1.0500

0.8000

1.7733

0.1867

11405

1201

-0.0467

-540

0.9000

5.4000

1.7000

0.5025

0.6150

1.4434

0.1519

4443

468

-0.0380

-199

0.9500

5.7000

1.6500

0.2456

0.4660

1.1269

0.1186

1695

178

-0.0297

-74

0.9750

5.8500

1.6250

0.2419

0.3400

0.8348

0.0879

1237

130

-0.0220

-53

1.0000

6.0000

1.6000

0.0000

0.0000

0.0000

0

0

0.0000

0

2/12/2014

N

P

V z

M z (N.m)

M y (N.m)

34859

-8686

144737

-36063

385697

-96101

72568

33399

-8322

109878

-27377

232928

-58037

52354

30918

-7704

76479

-19056

121113

-30177

34477

27237

-6787

45560

-11352

47890

-11932

19306

11128

-2773

18323

-4565

9560

-2382

7157

4335

-1080

7196

-1793

1904

-474

2690

1654

-412

2861

-713

396

-99

1046

1207

-301

1207

-301

91

-23

437

0

0

0

0

0

0

0

(N)

V x

(N)

M x

(N.m)


24

Boeing 707 Example 2: •

Consider the wing loading of a Boeing 707 in flight at a point where the total lift on the wing L = 750 KN. Determine wing load distribution using an airload elliptical approximation (assume level flight at 0 AOA)

2/12/2014


25

Boeing 707

2/12/2014


26

Boeing 707 •

Airload elliptical approximation:

2/12/2014


27

Boeing 707 •

The wing final load distribution is established once the net effect of all relevant loads is accounted for. The plot below illustrates the net resultant of all distributed loads, i.e., before accounting for engine weights. 30 25

wy (KN/m)

20 Structures load (KN/m)

15 10

Fuel load (KN/m)

5 0 -5

Net load (KN/m) 0

10

20

30

-10

2/12/2014


28

Boeing 707 y(m) 20 19 18 17 16 15 14.5 14.5 14 13 12 11 10 9.5 9.5 9 8 6 4 2 0

wy (KN/m) 0 7.458198246 10.41138283 12.58238904 14.33121019 15.79867423 16.451 16.45100794 17.05755198 18.15129477 19.10828025 19.94820034 20.68532015 21.018 21.018769 21.33029988 21.89128517 22.78517201 23.40276824 23.7656235 23.88535032

2/12/2014

Air load delta Structures load/m 0 -1.5 3.729099123 -1.625 8.934790537 -1.75 11.49688593 -1.875 13.45679961 -2 15.06494221 -2.125 8.062418558 -2.1875 -2.1875 8.377137995 -2.25 17.60442337 -2.375 18.62978751 -2.5 19.5282403 -2.625 20.31676025 -2.75 10.42583004 -2.8125 10.58707497 21.61079252 44.67645718 46.18794025 47.16839174 47.65097382

-2.875 -3 -3.25 -3.5 -3.75 -4

Structures load delta 0 -1.5625 -1.6875 -1.8125 -1.9375 -2.0625 -1.078125

Fuel load/m Fuel load delta 0 0 0 0 -3 0 -3.277777778 -3.138888889 -3.555555556 -3.416666667 -3.833333333 -3.694444444 -3.972222222 -1.951388889

-1.109375 -2.3125 -2.4375 -2.5625 -2.6875 -1.390625

-4.111111111 -4.388888889 -4.666666667 -4.944444444 -5.222222222 -5.361111111

-2.020833333 -4.25 -4.527777778 -4.805555556 -5.083333333 -2.645833333

-1.421875 -2.9375 -6.25 -6.75 -7.25 -7.75

-5.5 -5.777777778 -6.333333333 -6.888888889 -7.444444444 -8

-2.715277778 -5.638888889 -12.11111111 -13.22222222 -14.33333333 -15.44444444


Engine load delta 0 0 0 0 0 0 0 -18 0 0 0 0 0 0 -18 0 0 0 0 0 0

29

Boeing 707 Spanwise distance (m) 20 19 18 17 16 15 14.5 14.5 14 13 12 11 10 9.5 9.5 9 8 6 4 2 0

2/12/2014

Net load/m -1.5 5.833198246 5.661382827 7.429611258 8.775654636 9.840340897 10.29127778 10.291 10.69644087 11.38740588 11.94161359 12.3787559 12.71309793 12.84438889 12.844 12.95529988 13.11350739 13.20183868 13.01387935 12.57117905 11.88535032

Net load Shear Force Bending Moment 0 0 0 2.166599123 2.166599123 3.249898685 5.747290537 7.91388966 14.03743361 6.545497043 14.4593867 31.76956884 8.102632947 22.56201965 58.38290496 9.307997766 31.87001742 94.90692126 5.032904669 36.90292208 114.6166085 -18 18.90292208 114.6166085 5.246929661 24.14985175 128.0032668 11.04192337 35.19177512 168.7160036 11.66450973 46.85628485 221.4045433 12.16018474 59.0164696 286.5011053 12.54592692 71.56239651 364.3364652 6.389371705 77.95176822 404.9096923 -18 59.95176822 404.9096923 6.449922192 66.40169041 439.723018 13.03440363 79.43609405 525.6763139 26.31534606 105.7514401 763.4945402 26.21571803 131.9671581 1053.644574 25.58505841 157.5522165 1394.334066 24.45652937 182.0087459 1782.808087


30

Week6 Wing Load Distribution W2014

Recommend Documents