6. REFINED WEIGHT AND BALANCE ESTIMATE
6.1 Process for Refining the Weight Estimate The methods for making a refined weight
estimate of the design aircraft are described in some detail in Chapter 8 of Torenbeek
(Ref. 6-1). The notes shown here are offered in aid of pursuing the weight estimation
process. Generally the initial weight estimate Wto is used as the scale factor in applying
the methods of Ref. 6-1. When the revised take-off weight is found it may differ from
the initially chosen value. This is so because the weight of the individual components is
taken as a function of the take-off weight. It is possible to consider the value of Wto to be
the independent variable such that it is computed at the end of the component weight
estimation process. On the other hand, it is generally found that any difference between
the initial Wto and the revised Wto is small enough to permit a second iteration with a
reasonable expectation of convergence. Thus, the initial take-off weight estimate,
denoted by Wto,1 leads to a revised take-off weight, Wto,2. This value is now used as the
starting point in the component weight estimation process to arrive at a third estimate,
Wto,3. If the difference is such that (Wto,3 - Wto,2)/Wto,2 is less than, say 0.005 (i.e. 0.5%),
then the iteration process may be halted and the last value of will be considered the
revised take-off weight. Of course, all the component weights will be those
corresponding to the final choice of the take-off weight.
6.2 Limit Load Factor The load factor is defined as n = L/W with n>0 denoting wing
pulls up and n<0 denoting wings pulled down. A load factor n=1 denotes steady level
flight with L=W. Load factors different from n=1 are caused by maneuvers such as turns,
dives, climbs, etc. as shown in Figs. 6-1and 6-2.
trajectory
L=nW
W
Figure 6-1 Load factor during pull-up maneuver
The structural strength required of the airplane components is determined by the design
maximum load factor specified for the airplane and will vary with the function of the
airplane, with fighters having load limits set not by structural strength achievable, but
rather by the ability of pilots to withstand the accelerations causing the load factor
(generally n<9). In any maneuver the maximum lift that can be generated is Lmax=nmaxW
which means that
C
q C

nmax  L ,max  L ,max s.l . VE2
(6-1)
W / S  2 W / S 
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Lcos
L=nW

Lsin
W
Figure 6-2 Load factor during a level coordinated turn
We may use Eq. 6-1 to illustrate the safe operating regime for an aircraft on a so-called
V-n diagram as shown in Fig. 6-3. The limit normal load factor shown in the figure
depends upon the take-off weight as given in FAR Part 25 paragraph 25.337 according to
the following equation:
24, 000
(6-2)
nlim it  2.1 
Wto  10, 000 
This is for 2.5 < nlimit < 3.8. For aircraft weighing more than 50,000 lbs or less than 4,118
pounds, nlimit is constant and equal to 2.5 or 3.8, respectively.
limit load
factor, n
design limit
nmax
CL,max limit
speed limit
0
VE
design limit, n<0
Figure 6-3 V-n diagram for an airplane showing limits of operation
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6.3 Wing Group Weight The wing weight fraction, Ww /Wzf, depends upon the design
limit normal maneuvering load factor through nult =1.5nlimit. Since the wing weight is
approximately 8% of the aircraft's weight it is suggested that for aircraft weights in the
range where nlimit is variable the wing weight fraction be varied with the limit normal load
factor within the iteration process described previously. Torenbeek (Ref. 6-1) offers the
following equation for initially estimating the weight of the wing group
1.05
0.3
Wwg
 
6.25cos c / 2   Wzf 
b
0.55 0.3 
(6-3)
 0.0017nult tr ,max 
 
 1 

Wzf
b
S


 cos c / 2  

This equation is written for lengths in feet and weights in pounds; the quantities Wzf and
tr,max denote aircraft zero-fuel weight and wing root maximum thickness, respectively. A
schematic diagram of the wing group and the associated notation is shown in Fig. 6-4.
This wing weight expression includes high lift devices and ailerons, but not spoilers or
wing-mounted engines. These may be accounted for by increasing the wing weight given
by this equation by 2% for spoilers. To account for 2 or 4 wing mounted engines reduce
the wing weight by 5% or 10%, respectively. The actual weight of the propulsion group,
that is, the weight of the engines and associated equipment is calculated separately.
cr/2
cr
c/2
b/2
Figure 6-4 Schematic diagram of the wing group and its notation
6.4 Fuselage Group Weight The fuselage weight is proportional to the square root of the
design dive equivalent airspeed VD,E. FAR Part 25 paragraph 25.335 specifies that this
speed be equal to or greater than 125% of the design cruise equivalent airspeed VE. For
turboprops and other low to moderate speed aircraft a value of 130% to 140% may be
used at this stage in the design process. For high subsonic speed aircraft the pertinent
factor is the Mach number when determining the design dive speed. FAR Part 25
paragraph 25.335 specifies a prescribed dive maneuver for calculating VD,E, but this is
beyond the scope of the design process considered here. For present purposes it may be
69
assumed that MD = M +0.10 where M is the cruise Mach number. This factor is
reasonable, but arbitrary, and may be reduced somewhat if weight problems begin to
accumulate for the design.
It should be noted that the speeds discussed above are taken as equivalent air speeds
(EAS) rather than true airspeed, where VE = 1/2V. The equivalent airspeed is a measure
of the dynamic pressure experienced by the aircraft and VD,E is therefore a measure of the
maximum dynamic pressure experienced in flight. It is surprising that the design normal
load factor does not appear in the fuselage weight equation. In Appendix D of Torenbeek
(Ref. 6-1) a more detailed weight estimation method is presented and it does include nlimit.
It is suggested that pressure forces acting on the fuselage shell are more significant than
the fore and aft bending moments acting at the wing-fuselage juncture. A summary table
of all component weights and weight fractions must be presented at the end of the chapter
on weight estimation.
The fuselage weight is difficult to estimate because it is a complex structure with many
openings, support attachments, floors, etc., but it is strongly dependent on the gross shell
area, Sg. This is the surface area of the complete fuselage treated as an ideal surface, that
is, with no cutouts for windows or wing and tail attachments. Methods for approximating
the gross shell area are given in Appendix B in Torenbeek (Ref. 6-1). For cylindrical
cabin sections of fuselages with high fineness ratio, L/d >5, the gross area may be
estimated with the following equation:
2/3

2  
1 
S g   dL 1 
(6-4)
 1 
2
  L / d     L / d  
The fuselage weight may then be approximated by
W f  0.021S g1.2 VD,E
lt
d
(6-5)
In this equation the lengths are in feet, the weight is in pounds, and the design dive speed,
VD,E, is in knots. The length lt is the distance between the root quarter-chord points of the
tail and the wing, and, for a first approximation, it may be taken to be the estimated value
for lh found in the previous chapter. To this basic weight, 8% should be added to account
for a pressurized cabin and 7% added if the engines are mounted on the aft fuselage.
6.5 Landing Gear Group Weight The landing gear weight is not linearly related to the
take-off weight, as can be seen in Eq. 8-17 on p. 282 of Torenbeek (Ref. 6-1). This is not
a problem because the weight fraction of the landing gear is nearly constant at about
3.5% to 4.5% of the take-off weight for aircraft whose weight exceeds 10,000 lbs. The
approximation given by Torenbeek is presented below for airliner type aircraft; the
subscript mg refers to the main gear while the subscript ng refers to the nose gear, and the
weights are all in pounds.
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Wmg  40  0.16Wto3/ 4  0.019Wto  1.5 105Wto3/ 2
Wng  20  0.10Wto3/ 4  2 106 Wto3/ 2
(6-6)
6.6 Tail Group Weight This group also represents a small fraction of the take-off weight,
about 2% to 3%, but that weight does have an effect on center of gravity location because
of the long moment arms. For airliner type aircraft the weight of the tail surfaces is
mainly dependent on the design dive speed and Torenbeek (Ref. 6-1) suggests the
following functional relationships:
 Sh0.2VD , E 
Wh
 f

 cos  
kh S h
h 

(6-7)
0.2


S
V
Wv
 g  v D,E 
 cos  
kv S v
v 

The coefficients kh and kv account for different tail configurations. For example, current
practice for airliners is to have variable incidence tails, and kh=1.1, while a fixed
horizontal stabilizer would have kh=1.0, reflecting the lighter structure typical of fixed
Sh
equipment. For fuselage-mounted vertical tails =1.0 while for T-tails kv  1  0.15 h h . In
Sv bv
this last equation the quantities hh and bv correspond to the height of the horizontal tail
above the fuselage centerline and the height of the tip of the vertical tail above the
fuselage centerline, respectively. In Fig. 8-5 of Ref. 6-1 a curve is presented illustrating
the functional relationships given in Eq. (6-7) and a fit to this curve yields the following
approximations:

 S h0.2VD , E

Wh
 kh  2  4.15erf  3
 0.65  
 10 cos 

Sh

h


(6-8)
0.2




S V
Wv
 kv  2  4.15erf  3 v D , E  0.65  
 10 cos 

Sv

v


The quantity erf(x) is the error function and is tabulated in various mathematics reference
books. The definition of the error function is
x
2
2
erf ( x) 
e  d
(6-9)

 0
6.7 Propulsion Group Weight In addition to a detailed method for estimating the weight
of the propulsion group, Torenbeek (Ref. 6-1, chapter 8.4.2), offers the following simpler
approximation for podded jet engines equipped with thrust reversers and water injection
systems:
W pg  1.42 N eWe
(6-10)
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In this equation Ne is the number of engines and We is the weight of one engine in
pounds.
6.8 Wing Group Center of Gravity The center of gravity (CG) of the wing group may be
estimated according to the suggestions provided in Table 8-15 of Torenbeek (Ref. 6-1).
We may examine this case more closely by consulting the schematic diagram of the wing
shown in Fig. 6-5.
fuselage skin
centerline
wing group CG at 0.7(Xrs-Xfs)
mean aerodynamic chord (MAC)
front spar at 0.25C
rear spar at 0.55C to 0.6C
0.35b/2
YMAC
b/2
Figure 6-5 Schematic diagram of wing layout for estimating the
location of the wing CG
6.9 Fuselage Group Center of Gravity The fuselage center of gravity (CG) may be taken
from the estimates given by Torenbeek (Ref. 6-1, Table 8-15) and illustrated in Figs. 6-6
and 6-7.
0.42 to 0.45 L
L
Figure 6-6 Approximate location of CG of fuselage group alone
for wing-mounted engines
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0.47L
L
Figure 6-7Approximate location of CG of fuselage group alone
for fuselage-mounted engines
6.10 Landing Gear Group Center of Gravity The nose gear is placed near the nose of the
aircraft and the main landing gear must be placed aft of the overall CG of the complete
aircraft. A first approximation would place the nose and main landing gear at the
approximate locations shown in Fig. 6-8, depending upon the engine mounting
configuration.
0.17L
0.55L
0.14L
0.6L
Figure 6-8 Approximate locations of landing gear components as
functions of fuselage length for different engine mounting configurations
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Using the estimated weights of the nose and main landing gear and the fuselage length
one may approximate the location of the CG of the complete landing gear system
6.11 Tail Group Center of Gravity The CG of the tail group is dependent on the nature of
the tail configuration. Torenbeek (Ref. 6-1, Table 8-15) provides some estimates of the
CG location for conventional and T-tail arrangements, as shown in Figs. 6-9 and 6-10.
0.42c
0.42c
0.38bh/2
0.38bh/2
(a)
(b)
Figure 6-9 Approximate location of the CG location of the horizontal tail
for (a) wing-mounted engines and (b) fuselage-mounted engines
0.42c
hv
0.42c
0.38hv
hv
0.55hv
Fig 6-10 Approximate location of the CG location of the vertical tail
for (a) wing-mounted engines and (b) fuselage-mounted engines
These approximate locations may be used along with the estimated weights of the tail
surfaces to develop the location of the CG opf the entire aircraft.
6.12 Propulsion Group Center of Gravity The engine CG should be obtained from the
engine manufacturer, or from and estimate based upon the general configuration of the
engine using actual dimensions. The nacelle housing the engines may be assumed to have
a CG located 40% of the length of the nacelle, as measured from the lip of the nacelle.
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fuselage skin
centerline
main landing gear CG at rear spar
and Y=0.22(b/2)
wing group CG at 0.7(Xrs-Xfs)
fuel tank
front spar at 0.25C
rear spar at 0.55C to 0.6C
0.35b/2
0.45b/22
b/2
Figure 6-11 Composite sketch of wing group, fuel tank, wing-mounted engine,
and landing gear from which a collective CG may be determined
6.13 Aircraft Center of Gravity The center of gravity (CG) of the aircraft is of great
importance with respect to stability and control. This aspect of the design process
follows directly after the weight estimation process and is described in some detail in
Section 8.5 of Torenbeek (Ref. 6-1). Table 8-16 of Ref. 6-1 gives CG limits for a number
of different aircraft and Section 8.5.4 outlines a design procedure to obtain a balanced
aircraft. As pointed out in previous sections of this chapter, Table 8-15 gives the CG
locations of various aircraft components. In addition, information on nacelle placement
is given on p. 211 and on wing spar locations on p. 261.
One method for proceeding with the determination of the center of gravity of the
complete airplane involves dividing the airplane into two groups: the fuselage group that
includes the fuselage and the tail surfaces, and the wing group that includes the wing
engines, and landing gear. Side and plan views of these two groups with appropriate
dimensions are shown in Figs. 6-11 and 6-12. Taking moments about the nose of the
aircraft yields
WOEXOE = WFGXFG + WWG(XLEMAC + XWG)
Setting X*=XOE – XLEMAC and solving for XLEMAC leads to the following result:
XLEMAC = XFG + (WWG /WFG)XWG – (1 + WWG /WFG)X*
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XFG
XOE
cMAC
XW
XLEMAC
G
Figure 6-11 Schematic diagram of the two mass groups used in determining the
center of gravity of the complete airplane
XFG
XOE
cMAC
XWG
XLEMAC
Figure 6-12 Plan view of the two mass groups for determining the center of
gravity of the complete airplane
76
The displacement of the center of gravity of the airplane ahead of its aerodynamic center
determines the degree of the airplane’s longitudinal static stability. If the two points
coincide the stability is neutral, while if the center of gravity falls aft of the aerodynamic
center the airplane will be unstable. It is desirable in a commercial passenger transport to
have sufficient static stability for comfort and robustness of safety margins while
maintaining a level of maneuvering agility suitable to its mission. A reasonable location
of the center of gravity to meet these conditions is to have the center of gravity location
ahead of the aerodynamic center by an amount equal to 5% to 10% of cMAC, the mean
aerodynamic chord of the wing. That means that the distance X* lies in the following
range: 0.15 < X*/cMAC <0.20. Choosing a value in this range makes the equation for
XLEMAC determinate, and the wing can be finally placed. With the successful conclusion
of the CG evaluation the position of the wing with respect to the fuselage is determined
and the travel of the CG for different operating conditions can be determined. The entire
aircraft may then be drawn as a complete unit in a three-view representation.
6.5 Presentation of Weight and Balance Results The results of this chapter are to be
presented in a table of group weights as suggested by Table 6-1, the diagram of CG
locations and travel, and the three-view of the design aircraft showing pertinent
dimensions.
Table 6-1 Table of aircraft weight breakdown by groups
Aircraft Designation: ___________
References
6-1, p.280
6-1, p. 281
6-1, p. 282
6-1, p. 282
6-1, p. 283
6-1, p. 283
6-1, p. 285
6-1, p. 287
Group
Wing group
Tail group
Body group
Landing gear group
Surface controls group
Nacelle group
Propulsion group
Airframe services and
equipment
Empty weight (WE)
6-1, p.292
Operational items
Operational empty
weight (WOE)
Chapter 2, your
report
Chapter 2, your
report
Payload weight (WPL)
Fuel Weight (WF)
Take-off Weight
77
Weight (lbs)
XCG(in.)
Reference
6-1. Torenbeek, E.: Synthesis of Subsonic Airplane Design, Kluwer Academic Publishers,
Dordrecht, The Netherlands, 1982
78