Operating Flight Strength

A surprising amount of the science of maneuvering flight still applies to us in corporate and airline aviation. Using a V-g diagram, for example, tells me a G450 cannot really pull 2 G's at most landing weights on approach, it will stall first. While my earliest Air Force text says "The chart usually is included in the aircraft flight handbook in the section dealing with operating limitations," I've never seen a V-g diagram published in any airplane without an ejection seat. Of course if you fly your airplane with a minimum level of finesse you may never need such a chart. But if you would like a chart, you have what you need to build one. I will show you how.

Whether or not your aircraft has a V-g diagram or you build your own, you must still understand that V_A maneuvering speed is not what you've been taught. See: V_A - Maneuvering Speed for more about this.

1 — V-g diagrams, theory

2 — V-g diagrams and V_A

3 — Constructing V-g diagrams, theory

4 — Constructing V-g diagrams, example

Load factor in many classic texts is labeled "n" but have become "G" in keeping with pilot phraseology.

When I learned this in 1979 there was a well known caveat that was articulated this way: "Maneuver speed protection is not available for a rolling pull up." In other words, if you combine roll with pull, all bets are off. After the crash of American Airlines 587 it has been articulated in another, more useful way: Maneuvering speed is valid only when considering pitching moments with no aileron or rudder input. The old school house wisdom that "you cannot stall or overstress an airplane at V_A is wrong. The formula for V_A deals only with pitch and stall speed varies with g (you can stall an airplane above V_A if you increase the load factor.

It is vitally important to realize this number is valid only for the computed grossweight, altitude, and configuration.

Notice that nothing in the text or the diagram references altitude or weight. The aircraft obviously stalls at a different speed for varying altitudes and weights, and the available G changes as well. The V_A published in your flight manual has more to do with the FAA's certification rules than it does about actually maneuvering your aircraft. For now, just realize that the published maneuvering speed in your aircraft flight manual is not what you think it is. More about this below when we construct a V-g diagram for an example aircraft.

The V-g Diagram, a plot of the aircraft's available load factor (G) versus velocity (V), is a fundamental tool in determining aircraft performance, most notably its cornering speed. To an air combat pilot, cornering speed is the point where the pilot can pull in the pitch axis without fear of stalling or over-stressing the aircraft, producing the tightest possible turn. (An airplane pulls its way around a turn, the roll only serves to point the pitch vector.) To a corporate or airline pilot, this speed is known as Maneuvering Speed, V_A and provides a topic for discussion but little more. More on this at V_A.

To construct a V-g diagram, you must first plot the stalling speed of the aircraft at various load factors. Note that this speed changes with weight an altitude, so you will need a different chart for every combination of weight and altitude that interests you.

How do you compute stall speed for other than 1-G flight when that is all the aircraft flight manual gives you? With mathematical slight of hand. We've already seen, at Lift, the following:

$$L=C_L½\rho V^2$$

and,

$$G=\frac{L}{W}$$

therefore,

$$\mathrm{GW}=C_L½\rho V^2$$

If you set V_S to be the stall speed at the G-Limit, V_S(1G) to be the stall speed for 1-G flight, and assume C_L, W, and ρ do not change:

$$V_S=\sqrt{\frac{GW}{CL½\rho }}$$ $$V_{\mathrm{S(1G)}}=\sqrt{\frac{1W}{CL½\rho }}$$

Then with some division:

$$V_S=V_{\mathrm{S(1G)}}\sqrt{\mathrm{|G|}}$$

I took the absolute value of G because we will be dealing with negative G's and you can't very well be dealing with imaginary numbers when talking about maneuvering speed. There is a mathematical justification for this, but for the life of me I can't remember it. But it works.

The second step in constructing a V-g diagram is to plot the positive and negative load factor limits. This tends to be fairly straight forward.

The third step in constructing a V-g diagram is to add the limiting speed.

The fourth step is to add maneuvering speed, which occurs at the intersection of the positive load factor limit and the aerodynamic stall line.

The V_A maneuvering speed in your flight manual is just a number picked by your manufacturer to satisfy the requirements of 14 CFR 25.335 which only requires the speed cannot be less than the stalling speed of the aircraft with the flaps retracted and does not specify a weight or altitude. If you would like the maneuvering speed for any other configuration, weight, or altitude, you will need to construct a V-g diagram for those conditions.

I did precisely that when called upon to conduct combat operations in an Air Force Gulfstream III into Sarajevo back when there was a shooting war going on. (See At Least They Aren't Shooting at Us.) The AFM published V_A of 206 knots would have been the wrong speed to "yank and bank" under enemy fire on takeoff where the correct speed would have been 196 knots, or on arrival where the correct speed was 151 knots. A 55 knot error seemed significant enough, but how many people are using V_A in the peacetime world?

There is an increased emphasis on maneuvering aircraft in an Unusual Attitude situation and some are advocating that flying at V_A gives you a license to maneuver to your heart's content. It does not but studying a V-g diagram helps you understand your aircraft's operating envelope.

I shall construct three V-g diagrams and derive V_A for a Gulfstream G450 for these conditions as a demonstration for others wanting to do the same, and to provide the numbers for myself:

• Typical Landing at 1,000' Pressure Altitude (Gear Down, Flaps 39°, 50,000 lbs)
• Heavy Weight Takeoff at 1,000' Pressure Altitude (Gear Up, Flaps 20°, 74,600 lbs)
• Heavy Weight Climb at 3,000' Pressure Altitude (Gear Up, Flaps Up, 70,000 lbs)

I'll go through the steps in detail for the first example and then more quickly for those that follow.

Landing - stall charts

Using Gulfstream G450 performance charts, we see the flaps 39° stall speed at 50,000 lbs and 1,000 feet pressure altitude is 98 KCAS.

Landing - Derive Range of Stall Speeds for G-Envelope

Recall that

$$V_S=V_{\mathrm{S(1G)}}\sqrt{\mathrm{|G|}}$$

We can use that formula to derive a range of stall speeds for varying G loads:

G	VS
0	0
0.5	69
1.0	98
1.5	120
2.0	139

We plot that on a graph of load factors on the vertical axis and speeds on the horizontal axis.

We can fit a curve to replace the dots. Because of the stall chart value was rounded to the nearest 1 knot, errors are magnified on this chart so the curve may not fit exactly. As will be clear a few steps later, the critical bit is the KCAS for the load factor limit.

Landing - add G-limits

We know from G450 limitations the flight load acceleration limits with Flaps 39° up to landing weight are 0 to 2.0 G. We can plot these as well.

Because the V-g Diagram should show the aircraft's envelope, we can add the limiting airspeed to the chart.

Landing - determine corner speed (V_A)

V_A maneuvering speed is where the maximum stall speed intersects with the maximum load factor. Graphically, it occurs in the top left corner, the reason some call this "corner speed.'

The G450 published V_A of 206 KCAS is much higher than the true V_A for a G450 in landing configuration at 50,000 lbs gross weight and 1,000 feet pressure altitude. Full control deflection at 206 knots in this situation could very well overstress the aircraft.

Takeoff - stall charts

Using Gulfstream performance charts, we see the flaps 20° stall speed at 74,600 lbs and 1,000 feet pressure altitude is 132 KCAS.

Takeoff - derive range of stall speeds for G-envelope

Recall that:

$$V_S=V_{\mathrm{S(1G)}}\sqrt{\mathrm{|G|}}$$

We can use that formula to derive a range of stall speeds for varying G loads:

G	VS
0	0
0.5	93
1.0	132
1.5	162
2.0	187

Takeoff - add G-limits

We know from G450 limitations the flight load acceleration limits with Flaps 20° are 0 to 2.0 G.

Takeoff - Determine Corner Speed (V_A)

Just as we did for the landing example, we can plot the G-Limits, airspeed limits, and find the corner speed. In this case, V_A = 187 KCAS, still much lower than the published number of 206 KCAS.

Climb - stall charts

Using Gulfstream performance charts, we see the flaps 0° stall speed at 70,000 lbs and 3,000' pressure altitude is 130 KCAS.

Recall that

$$V_S=V_{\mathrm{S(1G)}}\sqrt{\mathrm{|G|}}$$

G	VS
-1.0	130
0	0
1.0	130
2.0	184
2.5	206

Because the load factor limit now includes negative G-loads, we must also plot the speeds at -1.0 G.

Climb - Add G-limits and determine corner speed

We know from G450 limitations the flight load acceleration limits with Flaps 0° are -1.0 to 2.5 G.

Now we finally see a V_A that equals the published number of 206 KCAS. This isn't the only one, there are many combinations of weight and pressure altitudes that will yield this number.

G450 V_A

Gulfstream G450 Maneuvering Speed (Sea Level)
Gear	Flaps	Weight	VA
Down	39	50,000 lbs	139 knots
Up	20	74,600 lbs	187 knots
Up	Up	70,000 lbs	206 knots

As we can see, the published V_A is rarely right. Do you need to memorize an continuum of V_A's? No, not at all. Maneuvering speed is a certification issue. When controlling your aircraft, do so with a gentle hand and you can avoid overstressing or stalling without the benefit of this made up number. The next time a simulator instructor recommends you fly this speed and the maneuver as required, nod politely and ignore the suggestion.

Air Training Command Manual 51-3, Aerodynamics for Pilots, 15 November 1963

Dole, Charles E., Flight Theory and Aerodynamics, 1981, John Wiley & Sons, Inc, New York, NY, 1981.

Please note: I've used some Gulfstream performance charts for the purposes of education. Gulfstream Aerospace Corporation has no affiliation or connection whatsoever with this website, and Gulfstream does not review, endorse, or approve any of the content included on the site. As a result, Gulfstream is not responsible or liable for your use of any materials or information obtained from this site.