Wing Bending Calculation with a Single set of Equations Deflection... · Wing Bending Calculation...

Wing Bending Calculation with a Single set of Equations

Author : Steven De Lannoy

January 2014

This document and more information can be found on the website Wingbike - a Human Powered Hydrofoil.

Abstract Wing bending calculations are complicated,

especially for hollow tapered wings. If one pursues

an exact solution, the math involved is challenging. If

one uses a numerical integration, one has to perform

a series of calculations along the span of the wing to

finally end with the tip deflection. For preliminary

designs, this is too cumbersome.

This paper builds on a paper of MIT’s

OpenCourseWare as presented by Drela [1] and

presents an improved approximation to calculate

wing tip deflection. It is very well suited for

preliminary designs of wings:

With 3 different equations to calculate Iroot:

• Solid wings

• Hollow wings (non uniform skin)

€

Iroot = Isolid 1− (1− 2Ω)3{ }

• Hollow wings (uniform skin)

€

Iroot = Isolid 1− (1−2Ωλ5)3

Some examples and a quick reference guide can

be found in the back of this paper.

1 Introduction A rectangular wing can be represented by the

following model of a beam:

Fig. 1

One end of the wing is fixed (at the root). The other

end, the tip, is free to move. The lift force generated

by the wing is proportional to the area of the wing. In

this case, since the wing has a rectangular shape,

lift is evenly distributed.

The equations to calculate bending moment, shear

force and wing deflection can be found in most

engineering handbooks as this model is a classic

engineering application of a beam. In this situation, a

rectangular wing can be regarded as a very flat

beam.

When the wing is tapered, the model changes as the

area of the wing decreases towards the tip. The

lifting force follows proportionally. Even this situation

is described in many engineering books and many

standard equations are available.

Fig. 2

WingBike–developmentofahumanpoweredhydrofoil

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However, the main difference now between the

situation in the engineering handbook and a wing is

that the first assumes a beam of constant width and

thickness whereas a tapered wing gradually

decreases in chord and thickness. So the general

engineering equations are no longer valid.

Since the wing doesn’t have a constant width and

height (chord and thickness) the bending inertia (=

resistance to bending) is no longer constant along

the span of the wing (as opposed to the engineering

situation). So not only does the lift varies along the

span, but so does the wing’s resistance to bending

as well.

The exact calculation of wing deflection involves

integrating a series of equations (Bernoulli-Euler

beam model). These steps and equations are

described by Drela [1]. Since the last two integration

steps in the procedure are complex, the author of the

document presents a numerical scheme and an

approximation. They form the basis of this paper.

In his paper, Drela [1] has derived an approximation

for the calculations of wing deflection:

€

δ =WL

12EIroot1+ 2λ1+ λ

y 2 (1)

Where

δ =deflection [m]

W = total weight [N]

L = ½ span [m]

E = Young’s Modulus [N/m2]

Iroot = bending inertia root [m4]

λ = taper ratio [-]

y= co-ordinate along span [m]

Iroot is the bending inertia at the root of the wing and

consists of an equation by itself. It depends on the

chord and thickness of the wing.

For a solid wing it is:

€

Iroot = 0,0449CT 3 (2)

Where

C = chord [m]

T = thickness [m]

This equation1 is a simplification by representing

the cross-section of the wing by a corrected

rectangular box [2,3].

Equation 1 is based on the assumption that the

wing curvature along the span of the wing is

constant (which is not the case). Therefore, it is

only accurate in certain conditions.

As described previously, the bending inertia along

the span of a tapered wing decreases, but in the

equation above, for simplification reasons, only the

bending inertia at the root is taken. Drela has

pointed out that this approximation is only accurate

for highly tapered wings. Below is a graph that

compares equation 1 with the numerical solution of

a typical hydrofoil case (for λ=0,3).

Fig. 3

The match along the span of the wing is very good

for λ=0,3.

The correlation becomes less accurate for higher

taper ratio, like in the figure below where the

equation shows a discrepancy towards the tip of

approximately 200% (for λ=1).

1The factor 0,0449 is the value for a ClarkY wing. For other shapes, see [2,3]


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Fig. 4

In the next sections we will present a modified

(improved) version of equation 1 so that tip

deflection matches better along the entire range of

taper ratios (from 0,1 to 1).

Equation 1 can be used for solid wings and some

types of hollow wings provided the correct Iroot is

used. Before we can proceed and derive an

improved equation we need to explain skin

distribution first as it influences the bending inertia

along the span of the wing.

2 Skin distribution When using a single equation to calculate wing

bending, one has to pay extra attention to the skin

distribution along the span of the wing. The

thickness has a large influence on the total bending

inertia of the wing.

Equation 1 calculates wing bending by using the

bending inertia of the root. It does not account for

how the skin is distributed along the span of the

wing.

Two types of skin distribution can be distinguished

for tapered hollow wings: uniform skin or non-

uniform skin.

The first means that the skin thickness is constant

along the span of the wing. The latter means that

the skin thickness decreases towards the tip of the

wing, proportional to the dimension of the chord

and thickness of the wing. It is illustrated in the

figure below.

Non-uniform skin (skin = 0,4 mm)

Uniform skin (skin = 1 mm)

Fig. 5

Three hollow cross-sections are presented. The

largest one, represents the cross-section at the

root of a wing and has a skin thickness of 1mm.

The two smaller cross-sections represent the tip of

the wing (for λ=0,4). One for non-uniform skin

distribution, one for uniform skin distribution.

The non-uniform skin is scaled proportional to the

chord and decreases to 0,4mm. The uniform skin is

constant and remains 1 mm.

It’s not hard to conclude that a uniform skin

distribution will lead to less tip deflection as the skin

is relatively thick towards the tip of the wing (higher

bending inertia).

When using equation 1 to calculate tip deflection it

does not account for this skin distribution and the

bending inertia as the root of the wing is identical in

both cases.

Let’s use an example to illustrate this, but first the

equations of bending inertia are presented. How

these equations were derived can be found in [2,3].

Bending inertia for a hollow wing (non-uniform

skin)2:

€

I = 0,0449C T 3 −T 3(1− 2Ω)3{ } (3)

Where

C = chord [m]

T = thickness [m]

Ω = skin fraction [-]

2 The factor 0,0449 is the value for a ClarkY wing. For other shapes, see [2,3]


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Skin fraction (Ω) is a value between 0 and 1 and

relates the skin to the thickness of the wing.

(4)

So if along the span, T decreases (for a tapered

wing), so will the skin thickness.

The equation for bending inertia for uniform skin

distribution is:

€

I = 0,0449C T 3 − (T − 2Sk)3{ } (5)

Where Sk is the actual skin thickness (and

constant along the span).

We will illustrate the differences with an example:

Chord 150 mm Thickness 20 mm Skin 2 mm Ω 0,1

Chord Wing

Thickn. Uniform Non-

uniform 150 20,0 22.487 22.487 130 17,3 14.164 12.686 110 14,7 8.200 6.503 90 12,0 4.202 2.914 70 9,3 1.778 1.066 50 6,7 532 278

The bending inertia at the root is identical (22.487

mm4) as the skin is equal in both cases. Towards

the tip of the wing, values diverge (almost a factor

2).

We can illustrate that the calculation of the bending

inertia of a solid wing (equation 2) is very similar to

the calculation of the bending inertia of a hollow

wing with non-uniform skin distribution.

We simplify equation 3 by isolating T:

€

I = 0,0449CT 3 1− (1− 2Ω)3{ }

This equals

€

I = Isolid 1− (1− 2Ω)3{ } (6)

So to obtain the bending inertia of a hollow wing

(non-uniform skin) one can take the bending of the

solid wing multiplied by a constant.

In the next sections we will show that equation 1

can be improved for

• Solid wings

• Hollow (non-uniform) wings

• Hollow (uniform) wings

provided that the bending inertia at the root is

corrected accordingly in some cases.

3 Solid and hollow wings (non-uniform)

Finally, before we can derive the improved

equation, we need to discuss the Bernoulli-Euler

beam model in very general terms first as it forms

the basis of the calculations and justifies the

approximations.

3.1 The theory behind the assumptions. To calculate wing bending, one uses 4 differential

equations that have to be solved in 4 subsequent

steps:

Step1: (7)

Step 2: (8)

Step 3: (9)

Step 4: (10)

Step 1 involves integrating q (function of wing

loading) along the span of the wing (dy). This yields

the distribution of the shear force (S) along the

span. Integrating this (step 2) yields the bending

moment along the span M. Step 3 involves

integrating the quotient of M, E (Young’s modulus)

and I (bending inertia) to obtain the deflection angle

(θ) along the span. Finally, the deflection (ω) is

obtained in step 4 by integrating the deflection

angle equation.


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The integration of step 3 is especially difficult as M

and I are by themselves fairly complicated

equations and both dependant of y. The math to

solve this analytically is very challenging. Therefore

a numerical scheme was presented [1] and an

approximation that leads to equation 1 of this

paper.

The quotient of M, E, I in step 3 is called the beam

curvature and it depends on y.

(11)

See figure 6 below for a typical case.

Fig. 6

The wing curvature changes along the span and is

different for different λ. Figure 6 is only presented

for λ=0,3, λ=0,5 and λ=1,0.

The assumption that leads to equation 1 is that the

wing curvature is constant along the span and that

it equals the value at the root (dotted lines).

(12)

Now one can see why equation 1 provides a

relative good match for λ=0,3 (fig. 3) compared to

λ=1,0 (fig. 4). The line is nearer most of the time.

The assumption that the wing curvature is constant

(and equals the value at the root) is more or less

arbitrarily chosen to simplify the last integrations.

One is free to choose other values to obtain a

better match just as long as they are

mathematically correct.

Based on figure 6, it seems logical to correct

equation 1 for different taper ratios although

equation 1 is already dependent on λ (through M

and I). This correction is the basis of the improved

equation that will be presented in the next section

(3.2), but to demonstrate the validity of this choice,

first a final remark on the wing curvature graphs.

Figure 6 was calculated for a typical case of a

human powered hydrofoil. If one normalizes the

wing curvature for the normalized span, one

obtains the following graph.

Fig. 7

This graph is now independent of W, L, E, I, C, T

and Ω. It is valid for solid wings AND hollow wings

(only for non-uniform skin).

This observation is certainly not the case for

uniform skin distribution as will be showed in

section 4.

3.2 Improved equation When improving equation 1, we will focus on

matching the tip deflection. In doing so, we might

lose some accuracy along the span. Since we are

mainly interested in tip deflection, that is

acceptable for now.

We have taken the same values that were used in

fig. 3 and fig. 4. We have set y=L in equation 1


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and kept all other factors constant. Now varying λ

through the entire range (from λ=0,1 to λ=1) has

resulted in a range of tip deflections. Then we have

calculated the same range using the numerical

scheme.

We have normalized the outcome: dividing the

numerical tip deflection by itself and dividing the

outcome of equation 1 (&2) by the outcome of the

numerical method. In this way, the relative

deflection is obtained. Now the result is case

independent as well (meaning independent of W, L,

E, Iroot). The result can be seen in the figure 8.

Fig. 8

Now we have another way to demonstrate that tip

deflection is indeed very accurately calculated for

λ=0,3 and very poorly for λ=1 (up to 200% over-

estimation). At λ=0,1, the equation even under-

predicts the tip deflection by 30%.

As equation 1 is based on an assumption

previously described, it is by no means an exact

solution. This justifies applying an additional

mathematical correction to improve its accuracy.

From fig. 8, one can estimate that the correction to

improve equation 1, has to be dependent on λ as

with increasing λ, the correction needs to be larger.

In appendix 1, a detailed description on how we

have matched the data is presented. For now, we

will describe the method in general terms.

Since equation 1 results in a straight line in the

relative deflection graph (fig. 8) we have used a

curve fitting technique to fit a straight line through

it. From this, we have derived the required

correction factor so that the sloped line (equation

1&2) will match the flat line (numerical solution) .

Since the curve fitting technique is a statistical

technique, it results in a correction factor with

decimal factors that complicate equation 1. We

have looked at multiples of these factors to see if

we could find a correction factor with (near) integer

number (see appendix 1). This has resulted in the

following correction factor:

(13)

Multiplying with equation 1 results into:

(14)

With for bending inertia at the root (solid wing):

(15)

And for a hollow wing (non-uniform skin):

€

I = 0,0449C T 3 −T 3(1− 2Ω)3{ }

€

= Isolid 1− (1− 2Ω)3{ } (16)

Or simplified into a practical equation:

€

δ =2WLEIroot

Ay 2 (17)

λ A 0,1 0,0661 0,2 0,0583 0,3 0,0524 0,4 0,0476 0,5 0,0437 0,6 0,0404 0,7 0,0376 0,8 0,0352 0,9 0,0331 1 0,0313


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With this new equation (14&2), we have again

calculated the normalized deflection curves,

leading to the following figure:

Fig. 9

The correlation has become near perfect.

Since we have focused on a way to match the tip

deflection, let’s check if this correction has had a

negative effect on the deflection along the span of

the wing.

Fig. 10

Both equation 1 and 14 provide a good match for

λ=0,3. So the correction factor hasn’t changed the

original correlation.

For λ=1 (figure 11), there is a slight mismatch

between the numerical calculations and the result

of equation 14. It seems that halfway the wing,

equation 14 (&2) under-predicts the wing

deflection.

Fig. 11

One could apply an additional correction to

equation 14, but this would complicate this

equation since y has now to be matched. Since

prediction of the tip deflection is more important

than the deflection distribution along the span of

the wing, we will not go further into this.

The correction factor we have applied to convert

equation 1 to equation 14, can be illustrated with

the wing curvature graph we used earlier.

Fig. 12

Equation 1 used the values at the root of the wing

(figure 6). For equation 14, these values were

corrected in such a way that the tip deflection

provided a better match. Compare figure 6 with

figure 12 to see the difference.

We have stated earlier that the normalized wing

curvature graph is equal for solid wings and hollow

wings (figure 7). Therefore equation 14 would be


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valid for hollow wings as well (non-uniform skin).

The reason is that the equation to calculate the

bending inertia of a solid wing, only differs from the

equation to calculate the bending inertia of a hollow

wing by a factor (see equation 6).

To illustrate the validity, we have performed a new

set of tip calculations (numerically and with

equations 1&3 and 14&3 respectively for a hollow

wing). The result can be found in the figure below.

Fig. 13

The line that is determined by equation 1&3 is

identical to the line that was determined by

equation 1&2 in figure 8. Hence, the same

correction factor is valid. In appendix 1, we have

used a curve fitting technique to prove this.

4 Hollow wing (uniform skin) In the previous section, we could simply convert

equation 1 into equation 14 by correcting for λ. The

basis for this was that the normalized wing

curvature graph was independent for all cases

(independent of W, E, L, I, C, T). We used,

depending on the situation, equation 2 (solid) or

equation 3 (hollow) to calculate the required

bending inertia at the root of the wing.

It is logical to use equation 14 in combination with

equation 5 to calculate tip deflection for the uniform

skin scenario.

However, this will not lead to correct values. For

uniform skin distribution, the normalized wing

curvature is not only dependent on λ, but on Sk as

well. This can be seen in the figure below.

Fig. 14

We have taken the wing curvature line from figure

6 (solid wing and λ=0,3). We have then expanded

the graph for three different hollow wings (uniform

skin thickness 1, 2 and 3mm). The normalized

graph is clearly dependent on Sk as well. The

combination of equation 14 with equation 5 is

clearly not universal.

Below we have illustrated this with a deflection

curve for λ=0,3 (using equation 14&5).

Fig. 15


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For λ=1 (below) the match at the tip is perfect,

along the span there is a discrepancy.

Fig. 16

To illustrate the discrepancies, again, we have

performed a range of calculations for a hollow wing

in for the range λ=0,1 to λ=1,0. Then we have

normalized the outcome.

Fig. 17

Equation 1 (in combination with equation 5)

provides an even worse match than previously.

Equation 14 in combination with equation 5

provides at least a better match. Especially for

values close to λ=1. The reason why the match

improves towards higher taper values is that at

λ=1, there is no difference between a wing with

uniform or non-uniform skin. Since the chord does

not decreases toward the tip, the skin remains

constant (equation 3 and equation 5 are equal

now). This is why figure 16 showed a good match

as it is not different from the solution using

equation 14 and 3.

Note that the discrepancy between the numerical

solution and equation 14&5 is no longer linear.

To rewrite equation 14 in combination with

equation 5 to represent uniform skin distribution,

one would have to dive deep into the Bernoulli-

Euler equations. This would make the math even

more challenging. It is more convenient to rewrite

equation 5 so that it can be used in combination

with 14. Even if it means applying a mathematical

trick.

A wing with a uniform skin is stiffer compared to the

same wing with non-uniform skin as the skin is

relatively thicker towards the tip of the wing.

For a wing with non-uniform skin to have the same

tip deflection as a wing with uniform skin, it would

require a thicker skin at the root to begin with

(provided all other factors are equal).

We could choose this thickness, which is not the

real thickness, in such a way that the tip deflection

matches the tip deflection of the same wing with

uniform skin.

We know that the bending inertia for hollow wing

(non-uniform skin) is:

€

Iroot = Isolid 1− (1− 2Ω)3{ } (18)

Now we will correct (increase) Ω with factor K in

such a way that the tip deflection corresponds with

the uniform skin thickness case.

€

Iroot = Isolid 1− (1− 2KΩ)3{ } (19)

So, by starting with a higher value of Ω, the skin at

the root is thicker and we will end with the same tip

deflection as it were a uniform skin. In this way

equation 14 can still be used. This again is a

mathematical trick rather than based on exact

physics as it will under predict the deflection closer

to the root of the wing.

There are two reasons why this is acceptable.

Firstly, as stated earlier, our main focus is to


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predict tip deflection. Secondly, the assumption

that lead to equation 1 was in fact comparable:

a mathematical trick applied to the root of the

wing.

We have performed a large set of numerical

simulations (varying W, L, E, C, T, λ, Sk and Ω) in

which we matched the tip deflection of the two

methods.

We have found that K is only dependant on λ (see

figure below)

Fig. 18

Figure 18 shows how much the skin fraction (Ω) of

a hollow wing (non-uniform skin) much be

increased to match the tip deflection of the same

wing with uniform skin distribution (all other factors

equal).

The slope (=K) is constant for each specific taper

ratio. Plotting this slope versus taper ratio results in

figure 19.

Fig. 19

We have found that K depends on λ in the

following way (correlation R2=0,999):

€

K =1λ5

(20)

Now the bending inertia equation for uniform skin

is:

€


(21)

Or simplified:

€

Iroot = Isolid 1− (1− 2KΩ)3{ } (22)

λ K 0,1 1,585 0,2 1,380 0,3 1,272 0,4 1,201 0,5 1,149 0,6 1,108 0,7 1,074 0,8 1,046 0,9 1,021 1 1,000

We have re-calculated the example from above

(figure 15) but instead of using the combination of

equations 14 and 5, we have used the combination

of equations 14 and 21.

Fig. 20

The conclusions are similar to the previous

conclusions: a perfect match at the tip of the wing

in combination with under prediction of the


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deflection along the span. This is logical as the

wing has been artificially made stiffer near the root.

To check the entire range, we have again

performed normalized tip deflection calculations

from λ=0,1 to λ=1,0.

Fig. 21

There is still a small discrepancy for λ=0,1, but

very acceptable.

5 Examples We will illustrate all the equations we have derived

by an example. We have chosen a fairly large span

and weight so that differences are clear.

Span 2,5 [m] Weight 1200 [N] Chord root 0,15 [m] Taper 0,3 [-] th/c 13,33% [-] Youngs Mod 2,0E+11 [N/m2] Skin 0,0015 [m]

Calculate Ω = 0,0015 / 0,02 = 0,075

We will consider 3 scenarios:

• Solid wing

• Hollow wing (non-uniform skin)

• Hollow wing (uniform skin)

We will compare the equations with the numerical

scheme we have composed earlier and we will

discuss the results.

Solid wing

First calculate the bending inertia at the root of the

wing using equation 2:

Now wing bending is calculated with equation 14:

€

δ =2WLEIroot

1+ 2λ(1+ λ)(13+ 35λ)

y 2

€

=2(1200)(1,25)

(2,0∗1011)(5,4 ∗10−8)0,0524{ }y 2 = 0,0146y 2

We have divided the span into 5 sections and

calculated the wing deflection and compared it with

the outcome of the numerical scheme:

y num eq14&2 0,00 0,000 0,000 0,25 0,091 0,091 0,50 0,373 0,365 1,00 1,508 1,458 1,25 2,281 2,278

The deflection at the tip is: 2,3 cm.

Fig. 22


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As stated before, the corrections we have applied,

focus on matching the tip deflections and accepting

a deviation in wing deflection at other stations

along the wing span. However, if we calculate the

correlation, the series yield R2=0,9998

Hollow wing (non-uniform skin)



€


€

= (5,4 ∗10−8)(0,3859) = 2,1∗10−8

Then calculate the deflection with equation 14:

€

δ =2WLEIroot

1+ 2λ(1+ λ)(13+ 35λ)

y 2

€

=2(1200)(1,25)

(2,0∗1011)(2,1∗10−8)0,0524{ }y 2 = 0,0378y 2

Again, for 5 sections we have calculated the wing

deflection and compared it with the outcome of the

numerical scheme:

Y num eq14&3 0,00 0,000 0,000 0,25 0,237 0,236 0,50 0,966 0,945 1,00 3,907 3,779 1,25 5,910 5,904

The deflection at the tip is 5,9 cm.

The correlation between both series is R2=0,9998

Fig. 23

Hollow wing (uniform skin)



€


€

= (5,4 ∗10−8) 1− (1− 2(0,075)0,35

)3

= (5,4 ∗10−8)(0,470)

Then calculate the deflection using equation 14:

€

=2(1200)(1,25)

(2,0∗1011)(2,5∗10−8)0,0524{ }y 2 = 0,0310y 2

Again, for 5 sections we have calculated the wing

deflection and compared it with the outcome of the

numerical scheme:

y num eq14 &5 eq14&21 0,00 0,000 0,000 0,000 0,25 0,227 0,236 0,194 0,50 0,888 0,945 0,775 1,00 3,294 3,779 3,101 1,25 4,817 5,904 4,845

The deflection at the tip is: 4,8 cm.

The correlation between both series is R2=0,9998


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Fig. 24

Finally we have combined the wing deflection

graphs for the 3 cases to illustrate the effect:

Fig. 25

Conclusion For preliminary design purposes, the single

equations to calculate tip deflection are surprisingly

accurate. One has to pay attention to the type of

skin distribution.

Deflection calculations can never be made without

checking the shear force and stress in the wing.

IMPORTANT NOTE

It should be emphasized that when performing deflection calculations one should always

simultaneously calculate the shear force (at

least at the root of the wing) and stress as the

wing might fail before the calculated deflection is reached !!!!

References [1] Drela, M, “Wing Bending Calculations”.

MIT OpenCourseWare, Unified

Engineering Course Notes, MIT

Department of Aeronautics and

Astronautics,

http://ocw.mit.edu/courses/aeronautics-

and-astronautics/16-01-unified-

engineering-i-ii-iii-iv-fall-2005-spring-

2006/systems-labs-06/spl10.pdf

[2] Lannoy, De, S., “Section Modulus and

Wing Bending Inertia of Wings”. Website:

Wingbike - a Human Powered Hydrofoil

[3] Lannoy, De, S., “Method to determine

correction factors for Section Modulus and

Bending Inertia Equations of wings.

Website: Wingbike - a Human Powered

Hydrofoil


StevenDeLannoy

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Appendix 1 Solid tapered wing and hollow wing with non-uniform skin distribution.

A set of wing tip deflection calculations was performed using the numerical scheme explained in [1]. The

range was set from λ=0,1 to λ=1. This outcome was then normalized to itself, meaning that the outcome (tip

deflection) was simply divided by itself. This results in a flat line. Next equations 1 and 2 were used to

calculate the tip deflection for the same interval by setting y=L in the equation. The outcome of this

calculation was then normalized to the outcome of the numerical scheme (divided by the outcome of the

numerical scheme). This results in the following table and figure:

taper num eq1&2

0,1 100% 68% 0,2 100% 83% 0,3 100% 98%

0,4 100% 112%

0,5 100% 127%

0,6 100% 142% 0,7 100% 156%

0,8 100% 171%

0,9 100% 185%

1 100% 200%

Normalizing the outcome of both methods is a way to present relative differences between the two methods

and is an indication on how well both cases match. One can see that both lines intersect at λ=0,3, meaning

that the calculated wing tip deflection is identical for λ=0,3 for both methods. As described in the

introduction, the match becomes worse with increasing λ . Equation 1 under-predicts the tip deflection for

λ<0,3.

Since normalizing presents relative differences, the discrepancy between the normalized numerical

calculation and the normalized approximation of equation 1 is independent of W, L, E, and I.

Using a linear curve fitting technique, one can obtain an accurate fit (R2 = 1) of the normalized line of

equation 1.

The correction factor to match equation 1 to the numerical scheme now is:

Multiplying equation 1 with this factor, will result in a match between the two methods, but this will result in a

new equation with “odd” looking decimal factors. It’s better to try and find a more elegant factor by

multiplying the numerator and the denominator of the correction factor by the same number:


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15

Linear = 1,4649 λ+ 0,5368 Numerator Denominator

1 1,465 λ+ 0,54 13 19,044 λ+ 6,98 2 2,930 λ+ 1,07 14 20,509 λ+ 7,52 3 4,395 λ+ 1,61 15 21,974 λ+ 8,05 4 5,860 λ+ 2,15 16 23,438 λ+ 8,59 5 7,325 λ+ 2,68 17 24,903 λ+ 9,13 6 8,789 λ+ 3,22 18 26,368 λ+ 9,66 7 10,254 λ+ 3,76 19 27,833 λ+ 10,20 8 11,719 λ+ 4,29 20 29,298 λ+ 10,74 9 13,184 λ+ 4,83 21 30,763 λ+ 11,27

10 14,649 λ+ 5,37 22 32,228 λ+ 11,81 11 16,114 λ+ 5,90 23 33,693 λ+ 12,35 12 17,579 λ+ 6,44 24 35,158 λ+ 12,88

There are several options that yield values close to an integer number: 11, 13, 15, 17, 24. Since equation 1

has a factor 12 in the denominator, it’s best to choose a combination that will simplify the new equation. The

following correction factor meets these criteria:

Now the new equation 1 becomes:

€

δ =2WLEIroot

1+ 2λ(1+ λ)(13+ 35λ)

y 2

Performing the simulation again and comparing it with equation 14, yields the following values:

taper num eq1&2 Eq14&2

0,1 100% 68% 99% 0,2 100% 83% 100% 0,3 100% 98% 100%

0,4 100% 112% 100%

0,5 100% 127% 100%

0,6 100% 142% 100% 0,7 100% 156% 100%

0,8 100% 171% 100%

0,9 100% 185% 100%

1 100% 200% 100%

With the new equation 14, a near perfect match is obtained over the entire range.


StevenDeLannoy

16

We have stated earlier that the normalized wing curvature graph is equal for solid wings and hollow wings

(non-uniform). Therefore equation 14 would be valid for hollow wings as well. To illustrate this, we have

performed a new set of tip calculations (numerically and with equations. The results can be found in the

figure below.

Equation 14 (in combination with equation 3) gives the same match over the entire range.


StevenDeLannoy

17

Appendix 2 Quick Reference Guide

Decide on type of cross-section

Hollow Solid Non-uniform skin distribution Uniform skin distribution

€

Iroot = 0,0449CT 3

€


€


or simplified

€

Iroot = Isolid 1− (1− 2KΩ)3{ }

€

δ =2WLEIroot

1+ 2λ(1+ λ)(13+ 35λ)

y 2

or simplified

€

δ =2WLEIroot

Ay 2

λ A K 0,1 0,0661 1,585 0,2 0,0583 1,380 0,3 0,0524 1,272 0,4 0,0476 1,201 0,5 0,0437 1,149 0,6 0,0404 1,108 0,7 0,0376 1,074 0,8 0,0352 1,046 0,9 0,0331 1,021 1 0,0313 1,000

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