Weight and Balance: Method

Weight and Balance: Moments, the Four Loading Conditions, the Envelope Test, the Shift and Ballast Formulas and Weighing

The problem

An aircraft may fly only when its weight is no greater than the maximum allowed for the make and model and its centre of gravity (c.g.) lies within the allowable range for that weight. Both are found by the law of the lever. Every item on board has a weight W and an arm x, its horizontal distance from a reference plane called the datum, positive aft; its moment about the datum is M = W x. The c.g. of any collection of items is the point at which the whole weight would give the same moment:

x_cg = ΣM / ΣW = Σ(W_i x_i) / ΣW_i.

The datum may be anywhere, as long as every arm is measured from it: moving it by d changes every arm and the c.g. by the same d. This page follows the Federal Aviation Administration's Aircraft Weight and Balance Handbook (FAA-H-8083-1B), the weight and balance chapter of the Pilot's Handbook of Aeronautical Knowledge (FAA-H-8083-25B, chapter 10) and chapter 10 of Advisory Circular 43.13-1B. Weights are in pounds and arms in inches (kilograms and millimetres in SI); a moment is then in lb-in. Large moments are often quoted as a moment index, the moment divided by 100 or 1,000; the page prints M / 1,000.

The four conditions

The station table lists everything except the usable fuel: the empty aircraft (basic empty weight, which includes the unusable fuel, full oil and installed equipment), crew, passengers, baggage, cargo and ballast. Their sum is the zero-fuel condition. The fuel on board at the ramp is a weight W_f at the fuel arm; entered as a volume it is W_f = ρ V, with the unit weights AC 43.13-1B gives for weight and balance purposes, 6 lb/gal for gasoline and 6.7 lb/gal for turbine fuel, unless a measured density is used (the handbook multiplies the measured specific gravity by 8.345 lb/gal, the weight of a gallon of water). The page forms

ramp = zero fuel + W_f; take-off = ramp − taxi fuel; landing = take-off − trip fuel,

and for each the weight, the moment, x_cg and the comparison with the maximum ramp, take-off, landing and zero-fuel weights that were entered. A maximum zero-fuel weight means that every pound above it must be usable fuel. When the take-off weight exceeds the maximum landing weight, the excess is the fuel that must be burned before landing.

If the fuel arm varies with the quantity (swept or shaped tanks), a table of fuel weight against arm replaces the single arm; the arm is interpolated linearly at the fuel remaining, so the fuel moment is W_f x_f(W_f), and held at the end values outside the table.

Percent of the mean aerodynamic chord

Large aircraft, and the stability analysis of any aircraft, express the c.g. as a percentage of the mean aerodynamic chord (MAC) aft of its leading edge (LEMAC):

c.g. in % MAC = 100 (x_cg − LEMAC) / MAC, and back, x_cg = LEMAC + MAC ⋅ (% MAC) / 100.

The envelope test

The c.g. limits usually vary with weight: the type certificate data sheet gives them at a few weights with "straight line variation between points given". On a plot of weight against c.g. the limits and the maximum weight enclose a polygon, the c.g. envelope. The corner points are entered in order around the boundary, for the normal category and, if the aircraft has one, the utility category. A loading condition is within limits when its point (x_cg, W) lies inside or on the polygon. The page tests this by the crossing-number rule (a horizontal ray from the point crosses the boundary an odd number of times when the point is inside) after first testing whether the point lies on an edge, which counts as inside.

For each condition it also reports the forward and aft limits at that weight, which are the least and greatest c.g. at which the horizontal line W = constant crosses the boundary, the margins x_cg − x_fwd and x_aft − x_cg (negative when outside), and the highest weight of the envelope at that c.g. Below the lowest weight entered, the limits of that lowest weight are applied: this is the "or less" of the data sheets, so the envelope need not be drawn down to the empty weight. Above the highest weight the point is outside.

As fuel burns the point moves down the chart, along a straight line when the fuel arm is constant only if that arm equals the c.g. (otherwise the c.g. drifts toward or away from the tank arm as the weight falls). The chart draws this path from the ramp fuel to empty tanks, and the page tests forty-one points of the part between take-off and landing against the envelope, because a flight can start and end inside an envelope with a slanted limit and still leave it in between.

Lateral centre of gravity

Helicopters also limit the lateral c.g. Lateral arms are measured from butt line zero, the plane of symmetry, positive to the right; the lateral offset moment is ΣW y and the lateral c.g. is ΣW y / ΣW. The section appears when any lateral arm is entered, and the result is compared with the left and right limits if they are given.

Weight shift

Moving a weight w a distance d along the aircraft leaves the total weight W unchanged and changes the moment by w d, so

ΔCG = w d / W, or as the handbook writes it, w / W = ΔCG / d.

The same proportion gives the weight to shift for a wanted ΔCG, the distance to shift a given weight, or the total weight. The c.g. moves in the direction the weight moves.

Weight added or removed

Adding a weight w at arm x changes both sums: the new c.g. is (W x_cg + w x) / (W + w). Subtracting the old c.g. gives the proportion of the Pilot's Handbook,

ΔCG = w (x − x_cg) / (W + w),

the added weight over the new total weight equals the c.g. change over the distance between the weight and the old c.g. A removal is the same formula with w negative.

Ballast

When the c.g. is outside a limit and nothing can be moved, ballast is carried (temporary) or installed (permanent). To bring the c.g. of an aircraft of weight W from x_cg to a wanted value x_t with ballast at station x_b, set (W x_cg + B x_b) / (W + B) = x_t:

B = W (x_t − x_cg) / (x_b − x_t),

the handbook's "aircraft weight times distance out of limits, divided by the distance between the ballast and the desired c.g.". This is exact for a fixed target. When the limit itself varies with weight, the ballast also moves the point up the chart to a weight where the limit is different, so the page instead searches for the least ballast that puts the point on the boundary of the envelope (a scan followed by bisection) and quotes the fixed-limit formula beside it. The ballast is limited to what the maximum weight of the condition allows. A negative result of the formula means the station is on the wrong side of the target.

Adverse-loading check

After an alteration, unless the data sheet gives an empty-weight c.g. range, the mechanic must show that no permitted loading can take the c.g. outside the limits. Each station is given the least and the greatest weight it can carry in flight (a pilot for the pilot's seat, nothing for a passenger seat, the placarded maximum for baggage, the minimum fuel and full fuel for the tanks). For a reciprocating engine the minimum fuel for balance purposes is 1/12 gallon per maximum-except-take-off horsepower, half the METO horsepower in pounds of gasoline.

The rule of the handbook and of AC 43.13-1B. Most forward c.g.: stations ahead of the forward limit at their greatest weight, stations behind it at their least. Most aft c.g.: stations behind the aft limit at their greatest weight, stations ahead of it at their least.

The exact extreme. A weight ahead of the current c.g. moves it forward, whatever its position relative to the limit. Starting from the rule's loading, the page reassigns every station about the resulting c.g. and repeats until nothing changes. Each pass can only move the c.g. further in the wanted direction, and the final loading is the true least (or greatest) value of ΣW x / ΣW over the weight ranges. It differs from the rule only when a station lies between the limit and the resulting c.g.; both are printed. Each result is tested against the envelope at its own weight, and flagged if it exceeds the maximum take-off weight (the handbook's own examples stay below it by the choice of loads).

Weighing

The empty weight and its c.g. come from weighing the level aircraft on scales under the wheels or jack points. The net weight at each point is the scale reading less the tare (chocks, blocks); the moment of each point is its net weight times the arm of the point; the c.g. is ΣM / ΣW as before. With the main wheels at arm D and a nose or tail wheel a distance L from them carrying the net weight P, this is the handbook's set of four formulas, all of them

x_cg = x_main + P (x_P − x_main) / W,

that is D − P L / W for a nose wheel and D + P L / W for a tail wheel when the datum is ahead of the main wheels, with D negative when the datum is behind them. Fluids on board that are not part of the empty weight (fuel, or oil for aircraft certificated without it) are subtracted with their moments, and unusable fuel or undrainable oil that was drained is added; these go in the adjustments box.

Equipment changes

Installing or removing equipment changes the empty weight by the item's weight and the empty moment by its weight times its arm. The record is a running sum: new weight = old weight ± items, new moment = old moment ± item moments, new empty-weight c.g. = new moment / new weight, and the new useful load is the maximum weight less the new empty weight. The loading on the page then uses the new empty aircraft.

Floor loading

A cargo floor has a structural load limit in weight per unit area. The load a pallet imposes is its loaded weight (cargo, pallet and tie-downs) divided by its area, with the area in square feet as length times width in inches over 144 (square metres in SI). The limit times the area, less the pallet and tie-downs, is the cargo the pallet may carry.

What the weight and the c.g. do to the aircraft

Weight raises the stall speed in proportion to its square root and lengthens the take-off and landing runs; those are computed by Aircraft Performance. The c.g. sets the longitudinal stability: the static margin is the distance of the c.g. ahead of the neutral point as a fraction of the MAC, so an aft c.g. reduces stability and stick force gradient, and a forward c.g. increases the elevator deflection and tail down-load needed to trim and to flare; those are computed by Longitudinal Static Stability, which takes the c.g. in fractions of the MAC.

Limits of the page

The page does arithmetic on the numbers entered. The arms, limits and maximum weights of a particular aircraft are those of its type certificate data sheet, approved flight manual and current weight and balance record, and operators under an approved weight and balance programme follow that programme (AC 120-27F), including its passenger and baggage weights and curtailments of the envelope, none of which are built in here. Moment-limit tables and loading graphs of a flight manual are replaced by the direct products W x. The envelope is tested as drawn, with no allowance for in-flight movement of passengers, landing-gear retraction or fuel slosh. Vertical c.g. is not computed.