The question is best illustrated with a concrete example. Let’s use a hypothetical Cherokee-like 2,500-pound airplane of 35-foot wingspan. For the sake of simplicity, let’s look at relative speeds at points 20 feet apart along the wing. This is a bit beyond the midpoint of each wing panel, but lift is a function of the square of speed and so whatever speed-related effects arise in turning flight are going to be biased toward the outer part of the wings. We’ll try bank angles of 10, 30, 60 and 80 degrees. Let’s say that our true airspeed is 119 knots — a convenient 200 feet per second — which yields turn radii of 7,043, 2,157, 718 and 220 feet respectively and G-loadings of 1.015, 1.155, 2.0 and 5.76. As the wings tilt increasingly, the 20 feet measured horizontally between our reference points shrinks, becoming 19.7 feet at 10 degrees, 17.3 at 30, 10 at 60 and 3.5 at 80.
Taking the 60-degree bank as an example, since it is the steepest coordinated turn that most pilots are likely to encounter, the outer wing is five feet farther from the center and the inner one five feet closer. If the radius of the turn is 718 feet, their radii are 723 and 713 feet respectively. The ratio of those two numbers, 1.014, is the ratio of the speeds of those points on the wing. In other words, the outer wing travels 1.4 percent farther than the inner in the same period of time, and so must be going 1.4 percent faster.
If we do the same calculation for each bank angle, we find that the ratio increases gradually, but not very rapidly, as the bank angle increases. At 89 degrees, the point 10 feet out along the outer wing is moving 2 percent faster than the corresponding point on the inner.
Now, the overbanking tendency, if it exists, is what is called a “couple” — a force applied at a distance from an object’s pivot, such as to make the object rotate. The pivot in this case is the rolling axis of the airplane — roughly, a line running from nose to tail down the middle of its fuselage — and the effect of the couple, if any, is to make the airplane roll into a steeper bank. The total force consists of an excess of lift on the outer wing and a deficit on the inner, and the lever arm, which is the same regardless of the bank angle, is 10 feet for each wing.
To get the size of this couple, or “rolling moment,” at 60 degrees of bank, we use the square of the speed ratio, because lift is proportional to the square of speed. The square of 1.014 is 1.028. The outer wing therefore has 2.8 percent more lift than the inner. Now, the total lift is 5,000 pounds, because this is a 2-G turn, and so it turns out that the inner wing’s lift is 2,465 pounds (5,000 / 2.028), the outer one’s 2,535, and the rolling moment is 700 pound-feet (35 x 10 x 2).
I assume that by now your eyes have glazed over or you’re experiencing uncomfortable flashbacks to high school, but that’s all right. This is all just a back-of-an-envelope exercise to get a sense of the magnitude of the overbanking force, if such there be. It’s not exact, because we’re using a single point on the wing to represent the entire wing, and all these distances and forces really vary along the wing. They’re smaller closer to the fuselage and larger out toward the wingtip.
But we have a rough idea now of the overbanking force in a 60-degree bank, and it seems quite large. The corresponding figures for the other bank angles are 140 pound-feet at 10 degrees, 230 at 30 and 2,270 at 80 degrees.
