Creating a Weight and Balance Fuel-Usage Line

Not everyone treats weight and balance like they should. I've seen a pilot aft-load his airplane so severely that it fell flat on its tail after he came to a sudden stop in the runup area. It took full power to right the airplane on its nose; after which, he returned to his tiedown and removed the aft-loaded baggage. Good move, I'd say.

So why am I telling you this? It seems that some pilots (not all, of course) take weight and balance for granted. These folks assume that an airplane with X number of seats can always hold X number of people along with all their baggage (emotional and physical), family heirlooms and decades worth of aviation supplies.

In a similar cavalier manner, these same pilots often assume that taking off with a c.g. that's within limits means that they'll always land with a c.g. within limits. Say it's not so? I wish I could, but I can't. Here's why.

It's possible to depart within c.g. limits and, as fuel is used, have the c.g. gradually move beyond the allowed limits. Of course, the chance of this happening varies with several factors, one of which is the type of airplane being flown. So here's a neat little method that you can use to see how fuel usage will change the position of your c.g. during flight.

The Center of Gravity/Moment Envelope

To properly describe this problem, we need to examine a typical c.g./moment envelope as shown in Figure 1A.

This is the envelope for an early model Cessna 210. (Please do not use this graph to compute the c.g. for your airplane. Use your own graph. If you don't have one, please get one.) Point A represents a bend in the forward portion of the c.g. envelope that occurs at a weight of 3,350 lbs. You'll find that the forward portions of the c.g./moment envelope on most small airplanes have a similar bend (in contrast with the aft portion of this envelope, which is perfectly straight). Here's what the bend means.

At weights below 3,350 lbs., the airplane in Figure 1A has a forward c.g. of approximately 37 inches aft datum. At weights above 3,350 lbs., the forward c.g. limit slopes a little steeper in the aft direction, changing to approximately 40 inches aft datum.

Here's one reason why the envelope bends. It just so happens that the bend in the envelope of this Cessna 210 (as well as the A36 Bonanza, both of which are six-seat airplanes) occurs at a weight that includes full fuel and two front seat occupants who weigh-in at their maximum allowable amount (see Figure 1B).

An airplane loaded this way is likely to have a c.g. located close to the forward c.g. limit. But the moment you start loading additional weight, like rear-seat passengers and their baggage, the c.g. must move aft. After all, the c.g. can't move forward any further if you're placing passengers and baggage aft (unless there's hanky-panky going on and folks are sitting on each other's lap?but we won't go there, OK?)

Therefore, as a practical matter, engineers shift the forward c.g. limit rearward (from 37 inches to 40 inches at point at point A in this instance) at weights above 3,350 lbs.. It makes no sense to show a forward c.g. limit beyond the range of what pilots could possibly experience at these higher weights.

It's important to note, however, that the above explanation is only one of the reasons the bend in the envelope may occur for your typical six-seat airplane. And this reason doesn't always explain the location of the bend in the c.g. envelope for these types of airplanes, nor for airplanes having fewer seats (like the Cessna 172 and 152).

There is, however, another reason why a bend may occur in the c.g./moment envelope and it applies to all airplanes. It has to with something known as pitch controllability.

The forward c.g. limit is normally determined by the ability to raise the airplane's nose at low airspeed. This, of course, is important, especially if you prefer landing on a runway instead of gouging an enormous hole into it when you attempt to flare. As the c.g. moves forward, the airplane becomes more stable in pitch. Eventually, if the c.g. moves too far forward (beyond the forward c.g. limit), the pilot runs out of elevator control authority. This is no muy bueno because the pilot may not be able to flare the airplane, which means he's in deep hot sauce.

Additionally, the forward c.g. limit is designed so that the airplane can meet the manufacturer's trim requirements, low speed handling qualities and general controllability specifications. These requirements and specifications are often determined during flight tests which are conducted at varying c.g. positions. Nevertheless, you can infer from Figure 1 that if you try to operate beyond the airplane's forward c.g. limit, you're likely to run out of elevator control authority. As a note, the aft portion of the c.g./moment envelope is straight for most of the smaller airplanes we fly. The aft c.g. limit is often determined by pitch stability. In particular, it's determined by the airplane's ability to recover from stalls and spins.

Now it's time to examine how to construct something I call a fuel-usage line. You'll be able to place this line anywhere on the c.g./moment envelope and visually determine how fuel use will affect the airplane's c.g.

Constructing The Fuel-Usage Line

Figure 2A is the loading graph for this particular model Cessna 210.

The horizontal axis represents the moment, calibrated in pound-inches/1000. The vertical axis represents weight. Examine the fuel-usage line (point Z). For the sake of clarity, let's use Figure 2B, a modified and clearer version of the previous figure. Using Figure 2B, let's find out how to measure the fuel-usage line's beginning and ending points.

If you were to describe this airplane's fuel-usage line, you could say that, in the horizontal direction, it spans a numerical value of 29 lb-in/1000 as shown point B. In the vertical direction, it spans a value of 690 lbs. as shown by point C. Now we're ready to transfer these two values to our c.g./moment envelope.

Figure 3A shows how I reconstructed the fuel-usage line (point A) on the c.g./moment envelope.

I created this line by starting in the lower left-hand corner of the graph and moving horizontally to the right an amount equal to 29 lb-in/1000 and vertically an amount equal to 690 lbs. Then I drew a line (the blue line) from this point to the bottom left-hand corner of the graph. This is now my fuel-usage line.

Interpreting the Fuel-Usage Line

Our main concern with creating the fuel-usage line is ensuring that the airplane remains within its allowable c.g. limits as fuel is consumed. Common sense suggests that as fuel is burned, the airplane's weight will decrease. But we see on the table that there will also be a change in moment. Will this change in weight and moment move us beyond the allowable c.g. limits? All you need to do is compare the slope of the fuel-usage line with that of the boundaries of the c.g/moment envelope.

Referring to Figure 3B, notice that the fuel-usage line (point A) appears to have a slope similar to the lower forward part of the c.g./moment envelope (point B). Let's shift the fuel-usage line to the right as shown in the figure (this is like drawing an exact replica of the line but with the top at point C). If the airplane's present c.g. position is located at point C (the green dot), the c.g. will move down along the slope of the fuel-usage line as fuel is burned. Therefore, the airplane remains within allowable c.g. limits as fuel is consumed.

Of course, I'm assuming that nothing else will change that might affect the airplane's c.g. In other words, baggage or passengers shouldn't leave the airplane (at least not without your permission, anyway). I'm also assuming that no one will call a huddle at the far ends of the cockpit, causing a radical shift in c.g. (and a possible unauthorized touchdown short of a runway).

The fact that fuel burn doesn't radically shift the c.g. in most small airplanes shouldn't come as much of a surprise to you. After all, engineers try to place the fuel tanks at a point close to where they expect to find the airplane's center of gravity. This is simply good design. Additionally, there aren't too many other convenient places to put fuel tanks in an airplane. Nevertheless, it's still possible to move out of the allowable c.g. limits with fuel burn, especially if the airplane is operated close to these limits to begin with.

In Figure 4A I've placed the fuel-usage lines at several strategic locations on the c.g./moment envelope (Note: you can conveniently move the fuel-usage line by precisely copying it onto a rectangular piece of plastic and keeping the borders of the plastic parallel to the horizontal and vertical graph lines. Simply move the top of this fuel-usage line to the airplane's present c.g. point. Then examine the fuel-usage line in relation to the c.g/moment envelope as demonstrated below).

If the green dot on line A represents the airplane's present c.g. location, then a fuel burn will move the c.g. deeper within the envelope. No problem here.

If the green dot at the top of line B represents the airplane's present c.g. location, then a fuel burn of 690 lbs. (represented by the full length of the fuel-usage line) will move the c.g. to the very bottom of the c.g./moment envelope, but not out of the envelope. No problem here.

But what happens if the starting c.g. is represented by the green dot at the top of line C? Do you see how the fuel-usage line slopes below the envelope's aft borders? This means that as the airplane consumes fuel, the c.g. moves aft of the allowable limit. Now that's a problem. Of course, to have this problem, you'd have to load the airplane full aft to begin with. And this isn't necessarily difficult, especially if the weight limitations of the baggage compartments are exceeded.

But what happens if this airplane is loaded as shown by line E? Clearly, as soon as you burn fuel, the airplane's c.g. moves beyond its forward limits. Granted, it may be difficult, if not impossible, to load this airplane to the forward limit. Yet, some airplanes are easier to load this way, especially if they have forward luggage compartments.

Finally, let's load the airplane as shown by line D in Figure 4B.

After consuming 240 lbs. (40 gallons) of fuel , the c.g. would shift downward, from the green dot to point F, the brown dot. How do I know the c.g. shifts downward an amount equal to 240 lbs. of fuel usage? Just measure the vertical distance between the green dot and the brown dot. At point F, as more fuel is consumed, the c.g. will move aft of its allowable limits. At 12 gallons per hour, this airplane can fly for a little less than 3.3 hours before having a c.g. problem. This is one reason why you want to perform a weight and balance calculation for your departure weight as well as your expected landing weight.

What It All Means

Creating a fuel-usage line is helpful when you fly larger general aviation airplanes. Additionally, because the fuel-usage line has a slope fairly similar to that of the c.g. envelope, you're not likely to have c.g. trouble unless you're near one end of the envelope to begin with. With most of the general aviation airplanes we fly, it's very difficult to exceed the forward c.g. limit, especially if the airplane doesn't have a forward baggage compartment. Exceeding the aft c.g. limit is another story, and a more common one. It's much easier to exceed this limit, thus the reason for constructing the fuel-usage line. Nevertheless, creating this line and comparing it to the c.g. envelope is an interesting exercise even if fuel-weight variations aren't likely to affect your airplane.

Next month we'll consider the weight and balance limitations of a few other airplanes. In particular, we'll examine how to construct a fuel-usage line for airplanes that use the weight vs. c.g. location graph instead of the weight vs. moment graph I've used here. So stay tuned.

For more information on this subject, see "Hot and Heavy Performance: Computing Weight and Balance and Density Altitude."