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.
2/12/2014 Aircraft Structural Design 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 Aircraft Structural Design 4
Schrenk’s approximation
Here the semi-span wing area = area of an elliptical quadrant = S/2.
2/12/2014 Aircraft Structural Design 5
Schrenk’s approximation
2/12/2014 Aircraft Structural Design 6
• 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
Schrenk’s approximation
2/12/2014 Aircraft Structural Design 7
• 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
Schrenk’s approximation
2/12/2014 Aircraft Structural Design 8
• 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
Schrenk’s approximation
2/12/2014 Aircraft Structural Design 9
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
Schrenk’s approximation
2/12/2014 Aircraft Structural Design 10
• 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
2/12/2014 Aircraft Structural Design 11
• 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.
Anderson’s approximation
2/12/2014 Aircraft Structural Design 12
• 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)
Anderson’s approximation
2/12/2014 Aircraft Structural Design 13
• As seen before:
𝑐𝑦 = 𝑐𝑟 1 +2𝑦
𝑏𝜆 − 1
• And:
𝑆 = 1 + 𝜆 𝑏𝑐𝑟2
Anderson’s approximation
2/12/2014 Aircraft Structural Design 14
• 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
2/12/2014 Aircraft Structural Design 15
Anderson’s approximation
2/12/2014 Aircraft Structural Design 16
• 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
Anderson’s approximation
2/12/2014 Aircraft Structural Design 17
𝑉𝑖 = 𝑉𝑖+1 + 𝐿𝑖
𝑀𝑖 = 𝑀𝑖+1
+ 𝐿𝑖(𝑦𝑖+1– 𝑦𝑖)/2 + 𝑉𝑖+1(𝑦𝑖+1– 𝑦𝑖)
Anderson’s approximation
2/12/2014 Aircraft Structural Design 18
• 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
Anderson’s approximation
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)
2/12/2014 Aircraft Structural Design 19
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).
2/12/2014 Aircraft Structural Design 20
Anderson’s approximation
𝑀𝑦𝑖 = 𝑀𝑦𝑖+1 +𝑀𝑜 + 𝑁𝑖𝑙𝑖
𝑀𝑜 = ½(𝜌𝑉2𝑐𝑚𝑜 𝑐𝑖𝑆𝑖) 𝑐𝑚𝑜 = 𝐶𝑚𝑜 (𝑆/𝑏𝑐𝑖)𝐿𝑎
2/12/2014 Aircraft Structural Design 21
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.
2/12/2014 Aircraft Structural Design 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 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.
2/12/2014 Aircraft Structural Design 23
Anderson’s approximation
2/12/2014 Aircraft Structural Design 24
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)
2/12/2014 Aircraft Structural Design 25
Boeing 707
2/12/2014 Aircraft Structural Design 26
Boeing 707 • Airload elliptical approximation:
2/12/2014 Aircraft Structural Design 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.
2/12/2014 Aircraft Structural Design 28
-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
2/12/2014 Aircraft Structural Design 29
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
2/12/2014 Aircraft Structural Design 30
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
Recommended