This discussion topic came up in the thread on beams and how they work. I think that we need to talk about how wings work as beams.

I ran out of time, as the wife and I are leaving on a trip. If the writing is not up to my usual standards, I apologize. Have at, and I will be back on when the trip is finished...

For the moment, let's stick with positive AOA, positive g type of lift, and establish the two frames of reference. One frame of reference is based on direction of airflow, the other the wing’s chordline. The airplane is flying horizontally; lift is in the vertical direction, and the wing AOA can vary from zero to about 18 degrees. Wings carry loads in the vertical direction varying from zero to four, six, or even more times the weight of the airplane. Eighteen degrees is about as much AOA as the wing chord line can make before stall, some a little more, others a little less. In some airplanes, wings can be flown to much higher AOA, but that is post-stall - lift decreases and drag goes up by an order of magnitude when AOA gets higher than about 18 degrees... Getting high CsubL and high speed simultaneously is outside of the envelope we design for in little airplanes.

To frame the problem, lift and drag are described using the same equation:

dynamic pressure * coefficient * projected area

Coefficient of lift on airfoil sections can go to around 1.8, while Cd is not plotted beyond 0.03. Max drag at AOA less than stall is on the order of 1/60 of max lift. That is for sections. Real 2D wings do about 78% that well on lift and drag of real wing is bigger, but there are still about 30 times as many units of lift as there is drag possible in most wings. The forces aft on the wing due to drag are pretty small compared to the lift that can be obtained.

There is also pitching moment, and which is significant and it is the genesis of torque in the wing – its equation is similar to the above:

dynamic pressure * coefficient * chord * projected area

The coefficient is usually between 0 and -0.10, chord in feet is around 3 to 6 feet, so pitxhing moment in ft-lbs is bigger than drag in lbs.

The lift occurring at high AOA is still wing frame of reference UP, but the wing chord line can be tilted up as much as about 18 degrees. So let’s think about positive g AOA near stall. The sine of 18 degrees is about 0.31, while cosine is about 0.95. The lift the wing is making vertically is distributed 31% forward along the chordline and 95% up perpendicular to the chordline. This 31% of wing lift forward accounts for some early wing failures with the wing wracked forward and not up. Let's follow through how these loads are handled in our wings, which are beams in both vertical and horizontal directions.

The lift components forward and up on the wing get distributed along the wing approximately elliptically. Since the lift is spread out chordwise along the wing, and there is pitching moment from the lift, there is also torsion applied to the wing, usually pitching the wing nose down. These loads do get distributed. Then there is some drag which gets subtracted from the forward component of lift. All of these loads are zero at the wingtip and accumulate towards the root end.

Ok, we have lift, pitching moment and drag, and then they load up the wing’s structure. Lift perpendicular to the chord line can get huge, forward component of lift plus drag can get to about 30% of total lift, and moment is smaller...

Beams carry shear load, bending and, when needed, torque. So what is a beam? What makes it carry torque? A little on those questions first.

A beam is intended primarily to carry shear and bending loads. Usually you are looking to carry those loads in some kind of efficient manner, whether the criteria are cost or weight or space or resonance avoidance. Sometimes you have functions you want to perform besides just carrying loads. Like lifting and controlling an airplane. You can build up several elements into a beam, even if some of those elements look like beams. And the total beam can look like other things, like airplane wings.

Beams become stressed when carrying shear, bending and torsion. In a wing where the elements are well attached to each other, all of the parts move together in carrying that shear, bending and torsion. Much of how the loads are distributed has to do with what fraction of the total stiffness against bending and torsion each element has.

Imagine two channels, one really beefy, another somewhat thinner. Bolt them together web to web. The bending stiffness of the two are combined to make a larger bending stiffness. And the fraction of bending moment carried by each will be the ratio of each piece’s bending stiffness divided by the whole bending stiffness. Bending load is spread among the pieces according to their relative stiffness.

This combination can be extended to the whole wing. Imagine a big beam, a smaller one, and the wing skin, all nicely tied together so that they deform together under bending and torsion… The bending stiffness in the fore-aft direction, bending stiffness in the up-down direction and torsional stiffness about the centroid all can be calculated. The terms are EIxx, EIyy, and GJ. Here is an interesting one, Ixx + Iyy = J.

How to calculate EI and GJ? There are equations for common shapes, other stuff has to be done numerically. Ixx = Sum (dA*y^2), Iyy = sum(dA*x^2), and J = sum(dA*r^2).

E and G? These are characteristics of the materials. I always have to look up everything except steel.

You can carry bending moments a bunch of ways. The lightest structures for this usually look like the shapes we are accustomed to for beams. And you will find them inside wings…

Now let’s imagine a metal skinned cantilever wing. There is one very identifiable spar near or on the thickest part of the wing with upper and lower caps and a web or two tying them together. There ribs in front of and behind the main spar giving the wings shape and extending back to where a much lighter spar is installed. The drag spar is usually just a light formed sheet metal structure. Then there are skins attached to the spars and ribs. Take a cross section through the wing and you cut the spars and skin – the ribs are few enough that they hardly contribute to the wing, and are usually ignored for bending stiffness. Properly done these parts all move together under bending and torsion. We can thus figure out the cross section’s ability to carry load and deflections while doing so.

We can calculate the area of vertical elements – they carry the vast majority of shear, and usually you only include the vertical elements of the main spar. Timoshenko and others talk about this phenomenon in their books on mechanics of materials.

We can calculate the centroid of the wing and then the I’s and J for the whole cross section. There are simple formulae for the normal beam looking parts, but you have to do some piece wise calcs to do the skin. Once you have I’s and positions for all the pieces you can calculate stresses everywhere and figure out if you made something too heavy or if something needs a little more beef.

Something to remember here, lift perpendicular to the chord line can get big, but the wing is a wide hollow beam only a few inches thick resisting that moment. Since depth of beams is so important, we need a significant beam to do this job.

Next we have somewhat lower force pulling forward on the same wing, but the beam is several feet deep in that direction. Because x^2 is big, dA does not need to be very big to be adequate to carrying these moments.

Usually if the main spar is beefy enough to carry lift perpendicular to the chord line, you have a drag spar, and some kind of bracing tying the spars together (skins and ribs or ribs and drag/anti drag wires), your wing can be made to carry this fore/aft load pretty lightly.

The spar is tailored by doing this operation at the root, and then at intervals out the wing until you get to spar and skin dimensions that you will not go below. Have fun.

Sorry, this is as far as I got...

See you guys later..

Billski

I ran out of time, as the wife and I are leaving on a trip. If the writing is not up to my usual standards, I apologize. Have at, and I will be back on when the trip is finished...

**Air Loads on Wings**For the moment, let's stick with positive AOA, positive g type of lift, and establish the two frames of reference. One frame of reference is based on direction of airflow, the other the wing’s chordline. The airplane is flying horizontally; lift is in the vertical direction, and the wing AOA can vary from zero to about 18 degrees. Wings carry loads in the vertical direction varying from zero to four, six, or even more times the weight of the airplane. Eighteen degrees is about as much AOA as the wing chord line can make before stall, some a little more, others a little less. In some airplanes, wings can be flown to much higher AOA, but that is post-stall - lift decreases and drag goes up by an order of magnitude when AOA gets higher than about 18 degrees... Getting high CsubL and high speed simultaneously is outside of the envelope we design for in little airplanes.

To frame the problem, lift and drag are described using the same equation:

dynamic pressure * coefficient * projected area

Coefficient of lift on airfoil sections can go to around 1.8, while Cd is not plotted beyond 0.03. Max drag at AOA less than stall is on the order of 1/60 of max lift. That is for sections. Real 2D wings do about 78% that well on lift and drag of real wing is bigger, but there are still about 30 times as many units of lift as there is drag possible in most wings. The forces aft on the wing due to drag are pretty small compared to the lift that can be obtained.

There is also pitching moment, and which is significant and it is the genesis of torque in the wing – its equation is similar to the above:

dynamic pressure * coefficient * chord * projected area

The coefficient is usually between 0 and -0.10, chord in feet is around 3 to 6 feet, so pitxhing moment in ft-lbs is bigger than drag in lbs.

The lift occurring at high AOA is still wing frame of reference UP, but the wing chord line can be tilted up as much as about 18 degrees. So let’s think about positive g AOA near stall. The sine of 18 degrees is about 0.31, while cosine is about 0.95. The lift the wing is making vertically is distributed 31% forward along the chordline and 95% up perpendicular to the chordline. This 31% of wing lift forward accounts for some early wing failures with the wing wracked forward and not up. Let's follow through how these loads are handled in our wings, which are beams in both vertical and horizontal directions.

The lift components forward and up on the wing get distributed along the wing approximately elliptically. Since the lift is spread out chordwise along the wing, and there is pitching moment from the lift, there is also torsion applied to the wing, usually pitching the wing nose down. These loads do get distributed. Then there is some drag which gets subtracted from the forward component of lift. All of these loads are zero at the wingtip and accumulate towards the root end.

Ok, we have lift, pitching moment and drag, and then they load up the wing’s structure. Lift perpendicular to the chord line can get huge, forward component of lift plus drag can get to about 30% of total lift, and moment is smaller...

**Wings are Beams**Beams carry shear load, bending and, when needed, torque. So what is a beam? What makes it carry torque? A little on those questions first.

A beam is intended primarily to carry shear and bending loads. Usually you are looking to carry those loads in some kind of efficient manner, whether the criteria are cost or weight or space or resonance avoidance. Sometimes you have functions you want to perform besides just carrying loads. Like lifting and controlling an airplane. You can build up several elements into a beam, even if some of those elements look like beams. And the total beam can look like other things, like airplane wings.

Beams become stressed when carrying shear, bending and torsion. In a wing where the elements are well attached to each other, all of the parts move together in carrying that shear, bending and torsion. Much of how the loads are distributed has to do with what fraction of the total stiffness against bending and torsion each element has.

Imagine two channels, one really beefy, another somewhat thinner. Bolt them together web to web. The bending stiffness of the two are combined to make a larger bending stiffness. And the fraction of bending moment carried by each will be the ratio of each piece’s bending stiffness divided by the whole bending stiffness. Bending load is spread among the pieces according to their relative stiffness.

This combination can be extended to the whole wing. Imagine a big beam, a smaller one, and the wing skin, all nicely tied together so that they deform together under bending and torsion… The bending stiffness in the fore-aft direction, bending stiffness in the up-down direction and torsional stiffness about the centroid all can be calculated. The terms are EIxx, EIyy, and GJ. Here is an interesting one, Ixx + Iyy = J.

How to calculate EI and GJ? There are equations for common shapes, other stuff has to be done numerically. Ixx = Sum (dA*y^2), Iyy = sum(dA*x^2), and J = sum(dA*r^2).

E and G? These are characteristics of the materials. I always have to look up everything except steel.

You can carry bending moments a bunch of ways. The lightest structures for this usually look like the shapes we are accustomed to for beams. And you will find them inside wings…

Now let’s imagine a metal skinned cantilever wing. There is one very identifiable spar near or on the thickest part of the wing with upper and lower caps and a web or two tying them together. There ribs in front of and behind the main spar giving the wings shape and extending back to where a much lighter spar is installed. The drag spar is usually just a light formed sheet metal structure. Then there are skins attached to the spars and ribs. Take a cross section through the wing and you cut the spars and skin – the ribs are few enough that they hardly contribute to the wing, and are usually ignored for bending stiffness. Properly done these parts all move together under bending and torsion. We can thus figure out the cross section’s ability to carry load and deflections while doing so.

We can calculate the area of vertical elements – they carry the vast majority of shear, and usually you only include the vertical elements of the main spar. Timoshenko and others talk about this phenomenon in their books on mechanics of materials.

We can calculate the centroid of the wing and then the I’s and J for the whole cross section. There are simple formulae for the normal beam looking parts, but you have to do some piece wise calcs to do the skin. Once you have I’s and positions for all the pieces you can calculate stresses everywhere and figure out if you made something too heavy or if something needs a little more beef.

Something to remember here, lift perpendicular to the chord line can get big, but the wing is a wide hollow beam only a few inches thick resisting that moment. Since depth of beams is so important, we need a significant beam to do this job.

Next we have somewhat lower force pulling forward on the same wing, but the beam is several feet deep in that direction. Because x^2 is big, dA does not need to be very big to be adequate to carrying these moments.

Usually if the main spar is beefy enough to carry lift perpendicular to the chord line, you have a drag spar, and some kind of bracing tying the spars together (skins and ribs or ribs and drag/anti drag wires), your wing can be made to carry this fore/aft load pretty lightly.

The spar is tailored by doing this operation at the root, and then at intervals out the wing until you get to spar and skin dimensions that you will not go below. Have fun.

Sorry, this is as far as I got...

See you guys later..

Billski

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