Week6 Wing Load Distribution W2014

2/12/2014 Aircraft Structural Design 2

Schrenk’s approximation

• Classical wing theory: for an elliptical wing, spanwise air load (lift) distribution is of elliptical shape.

• 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.



• 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



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




• Semi Elliptical Area: 𝑆

2=1

4

𝜋

42𝑎𝑏 ⟹ 𝑎 =

4𝑆

𝜋𝑏

• But for an ellipse:

𝑦2

𝑏2

2 +𝑐𝑦

2

𝑎2= 1 ⟹ 𝑐𝑦=

4𝑆

𝜋𝑏1 −

2𝑦

𝑏

2



• 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):

𝑤𝑦=4𝐿

𝜋𝑏1 −

2𝑦

𝑏

2



• For a tapered wing with taper ratio 𝜆 =𝑐𝑡

𝑐𝑟:

𝐴𝑟𝑒𝑎 =𝑆

2=

𝑐𝑡 + 𝑐𝑟2

𝑏

2

=𝑐𝑟𝑐𝑟

+𝑐𝑡𝑐𝑟

𝑏𝑐𝑟4

= 1 + 𝜆 𝑏𝑐𝑟4 𝑜𝑟 𝑐𝑟 =

2𝑆

1 + 𝜆 𝑏



Now: 𝑐𝑦 = 𝑐𝑟 −𝑦𝑏

2

𝑐𝑟 − 𝑐𝑡 =

𝑐𝑟 1 +2𝑦

𝑏𝜆 − 1 𝑜𝑟

𝑐𝑦=2𝑆

1 + 𝜆 𝑏1 +

2𝑦

𝑏𝜆 − 1

Then: replacing 𝑐𝑦 with 𝑤𝑦and S with L, we obtain the

expression for the load distribution over the span:

𝑤𝑦=2𝐿

1 + 𝜆 𝑏1 +

2𝑦

𝑏𝜆 − 1



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

𝑤𝑦=4𝐿

𝜋𝑏1 −

2𝑦

𝑏

2

• And:

𝑤𝑦=2𝐿

1 + 𝜆 𝑏1 +

2𝑦

𝑏𝜆 − 1

Anderson’s approximation


• Based on results obtained for base (Lb) 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.



• Consider an arbitrary wing with a specified surface area S, total span b, a taper ratio λ = ct/cr and an aspect ratio A = b2/S.

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

cr

yi

yi+1

ct

b/2

Station i Station i+1 y

c(y)



• As seen before:

𝑐𝑦 = 𝑐𝑟 1 +2𝑦

𝑏𝜆 − 1

• And:

𝑆 = 1 + 𝜆 𝑏𝑐𝑟2



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

𝑐𝐿 = 𝐶𝐿(𝑆/𝑏𝑐𝑖)𝐿𝑎

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

• Once cL is determined, calculate local lift force Li using:

𝐿𝑖 = ½(𝜌𝑉2𝑐𝐿 𝑆𝑖)

• Where: 𝑆𝑖 = (𝑦𝑖+1 – 𝑦𝑖)(𝑐𝑖+1 + 𝑐𝑖)/2




• 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.

yi+1 yi Li

Station i+1

i+1

Station i

Vi+1 Vi

Inboard (wing root) Outboard (wing tip)

Mi+1 Mi



𝑉𝑖 = 𝑉𝑖+1 + 𝐿𝑖

𝑀𝑖 = 𝑀𝑖+1

+ 𝐿𝑖(𝑦𝑖+1– 𝑦𝑖)/2 + 𝑉𝑖+1(𝑦𝑖+1– 𝑦𝑖)



• General case of a wing at an Angle of Attack α

𝑁 = 𝐿𝑐𝑜𝑠(𝛼) + 𝐷𝑠𝑖𝑛(𝛼) 𝑃 = − 𝐿𝑠𝑖𝑛(𝛼) + 𝐷𝑐𝑜𝑠(𝛼)

α

L

D

N

P x

z


𝑐𝐷 = 𝐶𝐷(𝑆/𝑏𝑐𝑖)𝐿𝑎 𝑫𝒊 = ½(𝝆𝑽𝟐𝒄𝑫 𝑺𝒊) 𝑐𝐿 = 𝐶𝐿(𝑆/𝑏𝑐𝑖)𝐿𝑎 𝑳𝒊 = ½(𝝆𝑽𝟐𝒄𝑳𝑺𝒊)

𝑁𝑖 = 𝐿𝑖𝑐𝑜𝑠(𝛼) + 𝐷𝑖𝑠𝑖𝑛(𝛼) 𝑃𝑖 = − 𝐿𝑖𝑠𝑖𝑛(𝛼) + 𝐷𝑖𝑐𝑜𝑠(𝛼)

𝑉𝑧𝑖 = 𝑉𝑧𝑖+1 + 𝑁𝑖

𝑀𝑥𝑖 = 𝑀𝑥𝑖+1 + 𝑁𝑖(𝑦𝑖+1– 𝑦𝑖)/2 + 𝑉𝑧𝑖+1(𝑦𝑖+1– 𝑦𝑖) 𝑉𝑥𝑖 = 𝑉𝑥𝑖+1 + 𝑃𝑖

𝑀𝑧𝑖 = 𝑀𝑧𝑖+1 + 𝑃𝑖(𝑦𝑖+1– 𝑦𝑖)/2 + 𝑉𝑥𝑖+1(𝑦𝑖+1– 𝑦𝑖)


Anderson’s approximation Wing torque:

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



𝑀𝑦𝑖 = 𝑀𝑦𝑖+1 +𝑀𝑜 + 𝑁𝑖𝑙𝑖

𝑀𝑜 = ½(𝜌𝑉2𝑐𝑚𝑜 𝑐𝑖𝑆𝑖) 𝑐𝑚𝑜 = 𝐶𝑚𝑜 (𝑆/𝑏𝑐𝑖)𝐿𝑎


AC

SC

Ni

l

Mo

Myi+1 Myi Station i+1 Station i

Anderson’s approximation • 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.


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 m2

• A = b2/S = (12)2/25.27 = 5.7 ≈ 6 (for Anderson’s tables)

• We now use the Anderson tables to obtain La at the various stations for ct/cr = 0.6 and A = 6. For example at Station 0, La = 1.267 while at Station 0.975, La = 0.340. From those values of La we proceed to determine all applicable loads.




STATION y i (m) C i (m) S i (m2) L a c L c D L i (N) D i (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

N P V z (N) V x (N) M x (N.m) 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

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)


Boeing 707


Boeing 707 • Airload elliptical approximation:


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.


-10

-5

0

5

10

15

20

25

30

0 10 20 30

wy (KN/m)

Structures load(KN/m)

Fuel load (KN/m)

Net load (KN/m)

Boeing 707


y(m) wy (KN/m) Air load delta Structures load/m Structures load delta Fuel load/m Fuel load delta Engine load delta

20 0 0 -1.5 0 0 0 0

19 7.458198246 3.729099123 -1.625 -1.5625 0 0 0

18 10.41138283 8.934790537 -1.75 -1.6875 -3 0 0

17 12.58238904 11.49688593 -1.875 -1.8125 -3.277777778 -3.138888889 0

16 14.33121019 13.45679961 -2 -1.9375 -3.555555556 -3.416666667 0

15 15.79867423 15.06494221 -2.125 -2.0625 -3.833333333 -3.694444444 0

14.5 16.451 8.062418558 -2.1875 -1.078125 -3.972222222 -1.951388889 0

14.5 16.45100794 -2.1875 -18

14 17.05755198 8.377137995 -2.25 -1.109375 -4.111111111 -2.020833333 0

13 18.15129477 17.60442337 -2.375 -2.3125 -4.388888889 -4.25 0

12 19.10828025 18.62978751 -2.5 -2.4375 -4.666666667 -4.527777778 0

11 19.94820034 19.5282403 -2.625 -2.5625 -4.944444444 -4.805555556 0

10 20.68532015 20.31676025 -2.75 -2.6875 -5.222222222 -5.083333333 0

9.5 21.018 10.42583004 -2.8125 -1.390625 -5.361111111 -2.645833333 0

9.5 21.018769 -18

9 21.33029988 10.58707497 -2.875 -1.421875 -5.5 -2.715277778 0

8 21.89128517 21.61079252 -3 -2.9375 -5.777777778 -5.638888889 0

6 22.78517201 44.67645718 -3.25 -6.25 -6.333333333 -12.11111111 0

4 23.40276824 46.18794025 -3.5 -6.75 -6.888888889 -13.22222222 0

2 23.7656235 47.16839174 -3.75 -7.25 -7.444444444 -14.33333333 0

0 23.88535032 47.65097382 -4 -7.75 -8 -15.44444444 0

Boeing 707


Spanwise distance (m) Net load/m Net load Shear Force Bending Moment

20 -1.5 0 0 0

19 5.833198246 2.166599123 2.166599123 3.249898685

18 5.661382827 5.747290537 7.91388966 14.03743361

17 7.429611258 6.545497043 14.4593867 31.76956884

16 8.775654636 8.102632947 22.56201965 58.38290496

15 9.840340897 9.307997766 31.87001742 94.90692126

14.5 10.29127778 5.032904669 36.90292208 114.6166085

14.5 10.291 -18 18.90292208 114.6166085

14 10.69644087 5.246929661 24.14985175 128.0032668

13 11.38740588 11.04192337 35.19177512 168.7160036

12 11.94161359 11.66450973 46.85628485 221.4045433

11 12.3787559 12.16018474 59.0164696 286.5011053

10 12.71309793 12.54592692 71.56239651 364.3364652

9.5 12.84438889 6.389371705 77.95176822 404.9096923

9.5 12.844 -18 59.95176822 404.9096923

9 12.95529988 6.449922192 66.40169041 439.723018

8 13.11350739 13.03440363 79.43609405 525.6763139

6 13.20183868 26.31534606 105.7514401 763.4945402

4 13.01387935 26.21571803 131.9671581 1053.644574

2 12.57117905 25.58505841 157.5522165 1394.334066

0 11.88535032 24.45652937 182.0087459 1782.808087

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Week6 Wing Load Distribution W2014

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