Banking On Airspeed

If there is any one aspect of aeronautical knowledge that has great potential to either expand one’s understanding of flight, or take us out of the game altogether, it is the relationship between bank angle and stall speed. Unfortunately, most student pilots resign themselves to accepting the information on blind faith, without much understanding of the underlying insights that can seem hopelessly unattainable.

WYSI (is not) WYG
First, stalling is not a function of airspeed. Yes, you can stall from a lack of airspeed, but you can also stall at cruise speed. The relationship has more to do with your wing’s design and the angle of attack at which you fly. Note that angle of bank can increase angle of attack. As Paul Craig has already pointed out in “When Does the Airspeed Indicator Lie?” the stall speeds at the bottom of the green and white arcs on your airspeed indicator are only good at one-g, and this can bite you. It can nibble at a sailplane, lazily circling in a thermal; it can chomp down suddenly on a race pilot wracking around a pylon in a steep turn at Reno (in September, by the way); it can even ambush an overly-ambitious bank in a Skyhawk.

It’s what happens when a wing exceeds its critical angle of attack. Okay … then, what’s a critical angle of attack? Basically, it occurs when the wing can provide no further increase in lift — when the difference between the direction of the relative wind and the wing’s upper surface becomes so great that air can no longer follow the wing’s upper surface. At that point, the distribution of pressure above the wing is such that significant airflow from the trailing edge actually moves forward against the relative wind to the center of low pressure. (That spot is normally just forward of the aircraft’s spar, and roughly a third of the way back from the leading edge). Of course, air is still moving back over the wing from the leading edge. (In fact, some is being sucked from just under the leading edge, too.) As the flows meet, air burbles up away from the airfoil — that’s the buffet you feel.

Lift depends on several things: the density of the air, the area of the wings, the speed at which you fly (actually, the square of that airspeed), and the properties of the wing lumped into what is called a coefficient of lift. (This in turn is a function of the angle of attack and the shape of the airfoil.) We’ll get back to that in a minute — first, some groundwork.

Here we are in a cartoon world, flying in trail behind our buddy’s 172. Buddy (that’s his name) just cranked into a 55-degree left turn. He’ll have to trim a bit nose-up and add some power, but at this instant, his wings now use part of their lifting potential to cause the turn (horizontal) in addition to the lift they must provide to oppose gravity (vertical). So, even though he probably doesn’t know it, he’s now effectively flying with little more than one wing — the horizontal component of the surface area of his two wings just got whacked down almost to the equivalent of one. (Five more degrees, and it would be just one.) You can see it there represented by that thin horizontal line just beneath his airplane.

HIGHER LEARNING (This won’t hurt … I promise)
The reduction in both the vertical component of lift as well as the wing area can be represented by the trigonometric cosine function, which is a relationship between the side adjacent to a given angle and the longest side (the hypotenuse) in a right triangle. Here, the cosine of that angle equals the ratios of both the reduced lift and wing area with respect to their original values (the equations in the boxes, above). This relationship holds true whether the bank is just fifteen degrees, fifty… or eighty-nine.

As Buddy enters his turn, the wings (both of them) are still providing lift, but the total lift vector is displaced from the vertical. Its the other component — horizontal — that turns (centripetally accelerates) the aircraft, while the lift nearly equivalent to “one wing” remains in the vertical. To maintain altitude, Buddy must increase lift by increasing the angle of attack so that the vertical component of lift once again equals the airplane’s weight. Translation: He pulls back on the yoke. He also finds that he must add power. But this only works up to a point. In a 172, he’ll run out of power somewhere around 60 degrees … at that point, if he tries to maintain altitude, he’ll risk stalling the aircraft. We’ll see why, later.

There are some useful “reference points” to keep in mind, using bank angles of 30°, 45°, and 60°, whenever you enter a turn. One easy way to remember this trig stuff is the “9-7-5” method.

  • The cosine of 30° is about 0.866, or very roughly, 0.9. At 30° bank, the aircraft is producing only about 90% (actually about 86.6%) of the vertical lift it produced with wings level — for any given airspeed.
  • The cosine of 45° is about 0.707. Round that to 0.7. and at 45° the aircraft is producing only 70% of its wings-level vertical lift. — this will get your attention.
  • The cosine of 60° is exactly one-half, or 0.5 — leaving you with half of your original vertical lift. Go past 60° bank and things start changing rapidly in ways that are not friendly to most generic light single-engine aircraft.

When Buddy honked back on the controls, he increased the angle of attack of the wings. In a 60° bank (just five more degrees from his angle), he must double the total lift so the vertical component once again equals the aircraft’s weight. Remember: the lift is cut in half, so he doubles it, and the houses don’t get any bigger (yet). There’s a name for this extra oomph, and it’s called the load factor, which is simply the ratio of the lift the aircraft is producing, to its normal weight. You feel it as “g-force“. In coordinated flight, the load factor equals the reciprocal of that cosine of the bank angle.

Example 1: At a bank angle of 60° the load factor is 2, because the cosine of 60 is 0.5 (one-half).

Example 2: At about 75° bank, the load factor is nowhere near 2 — it’s 4. That’s because the cosine of 75 is 0.25 (one-fourth). Your wings are producing only 25% of their wings-level vertical lift and are supporting four times the normal load (4 g’s) … as are you. Those 15-degrees bought you 2 more g’s. Danger: Your average 172 might be able to maintain altitude with a bank that steep, but does not have the power to maintain it — or the load limits to sustain it. The positive 4 g’s you’d impose on the aircraft is over the positive load limit of 3.8 g’s for normal category aircraft.

IMPORTANT: Roll it over just three and a half degrees more (from 75° to 78.5°) and your load factor has jumped another full integer to 5. Those three-plus extra degrees just bought you another clone to sit on your lap. What’s more, at 78.5°, the cosine is about 0.2 (one-fifth) — your wings are now producing only 20% of their wings-level vertical lift and your airframe is extremely unhappy with you … if it’s still intact.

Now let’s say Buddy overshoots his base-to-final turn. He knows all about the stuff I just said (because I told him. And you probably know where I’m going with this.) It’s a windy day. He cranks in over 30 degrees of bank. (Maybe it’s 40; it’s hard to be sure.) He’s already going fairly slowly. He pulls back a bit to shorten the turn, applies some opposite rudder to straighten out the nose, and the last thing he sees through his windshield is someone’s swimming pool. (Or, to make a happy ending, he senses the onset of a stall, lowers the nose, adds full power, reduces bank angle, and goes around for another try.)

BOTTOM LINE: In a turn, the airspeed safety zone gets smaller and smaller, the steeper the bank. And if you think you can cheat a steep turn by descending, think again. Even in a gliding turn, the effect of bank angle on stall speed is exactly the same as in level flight. (On the other hand if you dive, or accelerate downward, you might bank with impunity-until the ground intervened.) But if your bottom is getting pulled into your seat, you’ll have a need for speed-or else.

CONTINUING-EDUCATION (More math for the purists)
First, here’s that lift equation, showing how lift is a function of the coefficient of lift, air density (the Greek letter rho), velocity squared, and wing area, respectively:

Figure 2

If you solved for velocity, it would equal the square root of all those other terms: 2L divided by the product of C1 x rho x A:

Figure 3

But for now, just look at that L in the numerator and the A in the denominator. What that says is that velocity is proportional to the square root of lift divided by area:

Figure 4

If you think of effective weight instead of lift, that weight per unit area is nothing more than wing loading.

Figure 5


There are some important lessons in this. For one, this V can also represent stall speed (at the maximum coefficient of lift), and the W in the numerator under that square root can also be thought of as the load factor: the stall speed increases with the square root of the load factor, as well as weight.

  • At 30°, the load factor rises to only 1.15, and stall speed goes up about seven and a half percent — no biggie.
  • At 45°, the load factor rises to 1.41 (actually the square root of two), and stall speed increases by about 19%.
  • At 60°, load factor doubles, and stall speed jumps by 41% (or what the load factor was at 45 degrees).
  • At just over a 75° bank, the load factor is four, and stall speed doubles — at any speed less than twice the normal stall speed, you’ll stall. But above it, structural damage could occur because a load factor greater than 3.8 has already been invoked. For many modern airplanes, maneuvering speed is about two times stall speed. For older airplanes though, it can be as low as about 1.7, so be careful!

Reference: FAA Advisory Circular 61-67b: “STALL AND SPIN AWARENESS TRAINING”