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) I root = I solid 1 − (1 − 2Ω) 3 { } • Hollow wings (uniform skin) I root = I solid 1 − (1 − 2Ω λ 5 ) 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
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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
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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: