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The Linearly Tapered Wing Problem

*Van Warren*
*Warren Design Vision
*

Problem: A linearly tapered wing is subjected to a uniformly
distributed load along its entire planform surface.
Determine the spanwise load constants in the distributed load equation:

in terms of the wing span, root chord, rip chord and applied load.

**Solution**:

Let:

L = half span of wing from root to tip.

r = root chord

t = tip chord

Ac = region of constant load per unit length

Av = region of varying load per unit length

Aw = Ac + Av = total wing area

Wc = load in Ac

Wv = load in Av

W = total applied load

wc(x) = load as a function of x in constant region.

wv(x) = load as a function of x in varying region.

w(x) = wc(x) + wv(x) = total spanwise load

wm = slope of spanwise load function

wb = intercept of spanwise load function.

We will solve the problem in two parts. First we will find the portion
of the load that does not vary with span, this will be called wc(x) where
the subscript 'c' stands for constant. Then we will find the varying part
of the load, called wv(x) where 'v' stands for variable. The following
diagram will help us divide the problem into these two parts:

First :

Proof:

Next :

Proof:

Now we must formulate the load distribution in two parts, for the constant
region and the varying region. We'll start with the constant case. We know
that its graph looks like:

But all that we can establish from this is that:

However, we do know that this portion of the distributed load must sum
to , which gives us a basis for finding the value of the unknown k:

Having solved the problem for the constant region we can now turn our
attention to the varying one. We know that the linear taper in the wing
gives rise to a linearly varying load whose graph is as follows:

The equation of this load function looks like this:

Again we are in the position of having to deduce the value of an unknown
constant, in this case . To do this we again refer to the fact that the
total area of the load curve must equal :

To solve the original problem, that is finding the slope and intercept
of the *combined* load we add the two load distribution functions:

but we want to factor things like so:

where:

and:

which when boiled down and substituted gives:

which is fairly simple compared to the intermediate expression swell
we suffered to get it.