Calculating chordwise lift distribution
If you’re building an aircraft from scratch, there are few who argue in favor of not testing the strength of the design before flying. There are many extreme forces acting on an aircraft during flight, especially on the wings. Lift is mainly applied to the wings, and it tends to be quite powerful, as it has to produce enough force to lift the entire weight of the aircraft. Therefore, it’s important to make sure that your wings can withstand this force without breaking or deforming before flight. Otherwise, you’ll quickly be faced with an aircraft that is in several pieces, and a pilot that is very much dead.
Before building our entire wing, we wanted to test two main structural components of the wings: the spars (which go across the length of the wing, the span) and the ribs (which go across the width of the wing, the chord). Lift is not evenly distributed along the chord of the wing. Instead, the force tends to be much stronger closer to the leading edge of the wing, and much weaker towards the trailing edge. In order to accurately test the strength of the ribs, we had to distribute weight along it in a similar way lift would be distributed. In order to do this, one of our lovely designers, Matt Gibson, modeled the chordwise lift distribution by plotting one hundred data points, each with a value showing the proportional lifting force corresponding to a distance from the leading edge of the wing. The model can be found on this spreadsheet.
Using this model, we split the six-foot long chord into six sections, each roughly a foot long. I then had to calculate two properties for each section. The first was the centroid of each section, or the area at which the total force within that section can be placed (analogous to how the center of gravity of an object is the average location of the gravitational force acting on said object). This point can be described by its distance from the leading edge of the wing. The second value is a percentage of force that each of the six sections carry. Let’s start with the centroid.
Calculating this point is fairly simple. All you have to do is the following: For each data point in a section multiply the lift force at that point by its distance from the wing’s leading edge. Then, sum up those values and divide it by the total lift experienced by that section. This will give you the centroid as a distance from the leading edge of the wing. These values are recorded in the table mentioned above, but they are roughly as follows: section 1’s center of pressure is about 7.4 inches from the leading edge, section 2’s is about 18.5 inches away, section 3’s is around 30.4 inches, 4’s is about 42.1 inches, 5’s is about 54 inches, and 6’s is around 64.6 inches away.
Calculating the percentage of the total force felt by each section is even easier. All you have to do is add up the total amount of force experienced by the whole wing, and then, divide the total force experienced by each section. The value for total force in this model is about 0.73 lbf (this number is very low as chordwise lift distribution was estimated at very low speed), but please note that this number is not the actual amount of force experienced by the wing. Instead, all of the data points are accurate representations in proportion to the total lift value, as this is all that is needed for rib testing. These percentages are recorded in our spreadsheet, but as a quick summary, the first section experiences about 22% of the total force, the second experiences 26.6%, the third gets 21.1%, the fourth feels 17.1%, the fifth 10.25%, and the sixth gets about 3%. These are rough numbers (the more accurate ones are in the spreadsheet), but you get the idea. The front of the wing experiences a majority of the lift, with the force tapering off as you approach the trailing edge of the wing.
Once we knew how to distribute our weight along with the six points, we loaded each of those points with a proportional amount of weight, slowly increasing it while maintaining our ratios until the rib broke. The results from the rib test landed fairly conclusively in the “not strong enough” category. The two best-designed ribs broke at around 100 pounds, and the third broke before we loaded it to 100 pounds. That means, using this design and assuming that the ribs are all built perfectly, they can safely hold 66 pounds (1.5 factor of safety). The total force the wing experiences is about 2,800 pounds, accounting for factors of safety. This means we’d need 42 ribs for the wing. This is possible, but it’s not ideal. Instead, we’re going to modify the design in two main ways. First off, the original design’s weak point was the spar on the leading edge, since it was not completely surrounded by the foam; there was a semicircular indentation cut out of the end of the rib where the spar would fit. This meant that almost half of the spar’s cross-section wasn’t touching the rib, and indeed, this is where all three ribs failed. For the new design, we will be moving that spar back, such that it is completely inside the rib. The second edition is that there will be a metal plate affixed to the rib in the location of both spars, which will act to reinforce the foam and make those points of high pressure much stronger.