STATISTICAL ANALYSIS OF MISSION PROFILE PARAMETERS

OF CIVIL TRANSPORT AIRPLANES

By Otto Buxbaum

Laboratorium f_/r Betriebsfestigkeit

Darmstadt-Eberstadt, Germany

SUMMARY

To evaluate fatigue life, manufacturers must define and use typical mission pro-

files. The probability with which a mission profile (or one of its parameters) occurs

can be used to quantitatively describe the term "typical." The airplane weight at any

point in the mission is, of course, a very important parameter; the present paper pre-

sents some weight data and analyses from several types of airplanes. Several long-,

medium-, and short-range airplanes, flown either in passenger or in cargo service

of Lufthansa German Airlines, were observed between January and April 1969.

The statistical analysis of flight times as well as airplane gross weights and

fuel weights of jet-powered civil transport airplanes has shown that the distributions

of their frequency of occurrence per flight can be presented approximately in general

form. Before, however, these results may be used during the project stage of an air-

plane for defining a typical mission profile (the parameters of which are assumed to

occur, for example, with a probability of 50 percent), the following points have to be

taken into account.

Because the individual airplanes were rotated during service, the scatter

between the distributions of mission profile parameters for airplanes of the same

type, which were flown with similar payload, has proven to be very small. Signifi-

cant deviations from the generalized distributions may occur if an operator uses one

airplane preferably on one or two specific routes.

Another reason for larger deviations could be that the maintenance services of

the operators of the observed airplanes are not representative of other airlines.

Although there are indications that this is unlikely, similar information should be

obtained from other operators. Such information would improve the reliability of

the data of the present report.

INTRODUCTION

The airworthiness standards for transport category airplanes require that the

fatigue strength evaluation include the typical loading spectrum expected in service.

(See ref. 1.) The loading spectrum, however, depends on the mission profile, which has

541

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been chosen in agreement with the requirements of the customers. Since the operational

conditions will vary from flight to flight and probably different customers may operate

the same type of airplane differently, several mission profiles will always be discussed

for an airplane that is in the project stage. (See ref. 2.) The manufacturer then has to

combine the various mission profiles into one or two so-called representative or typical

ones on which to base the fatigue-life evaluation.

A quantitative description of the term "typical" may be obtained by defining the

probability with which a mission profile or one of its parameters will occur. This defi-

nition can be achieved possibly for parameters like flight altitude and airspeed by means

of results from measurements which have been carried out on airplanes of similar design

features, for example, from VGtt recordings. There is, however, still a lack of informa-

tion insofar as parameters such as flight-time airplane weight and weight distribution are

concerned.

In order to investigate the variation of mission-profile parameters and to gather

information which could be used for the design of similar airplanes, the following analy-

sis has been performed.

The author is indebted to the German Government, Ministry of Defense, for finan-

cial support and to the departments of structural engineering and performance and opera-

tion engineering of Lufthansa for their assistance and collaboration.

AIRPLANE TYPES, ANALYZED PARAMETERS, AND

PERIODS OF OBSERVATION

Several long-, medium-, and short-range airplanes flown either in passenger or in

cargo service of Lufthansa German Airlines have been observed during a period lasting

from January to April 1969. As far as it was possible, the following parameters have

been taken for each flight from flight and fuel logs as well as from the so-called "load

sheets:" airborne time, take-off gross weight, landing gross weight, fuel take-off weight,

and fuel landing weight. Information about the individual airplanes and their characteris-

tics is presented in table 1. In addition to the analysis performed for the airplanes and

the period of observation as mentioned, results from earlier similar investigations have

been included for information.

PRESENTATION AND DISCUSSION OF RESULTS

The results are presented for each of the mission-profile parameters in form of

cumulative frequency distributions, from which the number of occurrences per flight and

the respective magnitude can be read, and in form of cross plots for any two parameters,

542

which have occurred during the same flight. In order to achieve the intended generaliza-

tion, the airplane and fuel weights have been related to the maximum allowable weights

as specified in table 1.

Airborne Time

The cumulative frequency distributions of airborne times for the different types of

airplanes are presented in figure 1. The longest flight time has been observed ior a

707C airplane in cargo service; its flight time was 9.75 hours. Also a difference in flight

times between cargo and passenger airplanes of the same type can be noted.

If the cumulative frequency distributions of airborne times are plotted on Gaussian

probability paper with a logarithmic grid for the variate, then the distributions for the

individual airplanes may be approximated by one or by a combination of several straight

lines (fig. 2); that is, they correspond to logarithmic normal distributions, as it was

demonstrated in reference 2. Only those data have been included in figure 2, which were

obtained during the same period of observation. The following conclusions can be drawn:

(a) The scatter between airplanes of the same type which were flown with similar

payload is very low.

(b) The distributions can be separated into three groups which actually correspond

to short-range, medium-range, and long-range airplanes.

(c) The difference between passenger and cargo airplanes increases with the range.

(d) All long-range airplanes show the same asymptotic behaviour, which has been

observed in a previous investigation. That behaviour could be caused either by the

specific station-to-station distances as flown in service by the operator concerned, or by

the limitation of fuel capacity, or - and that seems to be very likely - by a combination

of the two reasons.

If it is assumed that other airlines operate similarly and the scatter for very short

flights (which occur with probabilities above 99.5 percent) is neglected, then the following

generalized information may be derived for the airborne times of jet-powered civil trans-

port airplanes. The logarithmic mean value of the airborne time amounts for short-range

airplanes to 37 minutes and for medium-range airplanes to 60 minutes. (See fig. 3.) The

corresponding standard deviations, by which the slope in the probability paper is defined,

are 0.155 and 0.215. The two logarithmic normal distributions intersect at a flight time

of 11 minutes and are assumed to occur with a probability of 99.5 percent. At the same

point also, the distributions for the long-range airplanes are assumed to have their origin.

As has been mentioned before, the long-range airplanes show an asymptotic behaviour,

which may be expressed by a mean value of 440 minutes and a standard deviation of 0.040;

they do not, however, follow this distribution completely but only to a certain percentage,

543

which is about 25 for the passenger and 70 for the cargo airplanes. As the distribution

for the long-range passenger airplanes leaves the asymptote already at a probability of

25 percent, its mean value is about 245 minutes instead of 440 minutes for the cargo

version.

Two distributions for short- and medium-range airplanes can be used directly for

an estimation of an airborne time belonging to a typical mission profile; in the case of the

long-range airplanes a distinction has to be made between cargo and passenger service,

and prior to the estimation, an assumption has to be made about the percentage of flights

which will follow the asymptote, that is, which actually can be called long-range flights.

Take-Off Weight

A similar analysis has been made for the take-off gross weights. As it has been

said before, the results are presented in relation to the corresponding maximum allowable

take-off weight. (See the cumulative frequency distributions in fig. 4.) It has to be noted

here that for the 737 type airplanes only those take-off weights which have occurred at

flights departing from and arriving at Frankfurt airport could be obtained. The data for

the other airplanes resulted from succeeding flights in the periods of observation as given

in table 1.

The scatter between the cumulative frequency distributions for the individual air-

planes of the same type, which flew with the same payload, was very small. (See, as an

example, that of passenger and cargo long-range airplanes in fig. 5.) This graph shows

also that the cargo airplanes are generally flown with a much higher take-off weight than

the passenger airplanes. An indication that this happens not only with the long-range air-

planes as investigated for one operator but also with the whole fleet of all airplanes from

all operators may be derived from the fact that a certain type of fatigue failure in the

wing structure has occurred at a significantly shorter service life for cargo airplanes

than for passenger airplanes. A careful fatigue-life evaluation has demonstrated that the

reason why cargo airplanes have the shorter life must result from generally higher air-

plane gross weights. The data as presented in figure 5 confirm that prediction.

In order to obtain the intended generalization, the data as observed during the same

period of time for jet-powered short-, medium-, and long-range airplanes have been

plotted on probability paper. (See fig. 6.) The distributions for the short- and

medium-range airplanes can be approximated by a rather small scatter band of two

straight lines with a standard deviation of 0.03. It says that 99.95 percent of all flights

were made with a take-off weight exceeding 70 to 75 percent of the maximum allowable

one, and that in about 5 percent of all flights, 100 percent of the maximum take-off weight

was reached. The variation of the relative take-off weight of long-range airplanes is

larger than that of short- and medium-range types. But also the difference between

544

passenger andcargo service is larger for the long-range airplanes, becauseonly 0.5 per-cent of all flights of passengerairplanes took place with the maximum allowable take-offweight, whereas in the caseof cargo airplanes it was almost every secondflight.

As supplementary information, a cross plot of the variation of take-off weight withairborne time as observedon three long-range passenger airplanes is shown. (Seefig. 7.)This examplehas beenselectedbecauseit was the best correlation which has beenobtained. For the other types of airplanes, the trend wasnot as clear. More detailsaboutthis subject are given in refererice 3.

LandingWeight

If the cumulative frequency distributions of relative airplane landing gross weight(fig. 8) are comparedwith thoseof the take-off weight as shownin figure 4, it is evidentthat the curves for the landing weight of the individual types of airplanes are much moreconsistent and conformable. Whenplotting thesedistributions on logarithmic probabilitypaper andapproximating them by straight lines (fig. 9), it becomesapparent that for alltypes of airplanes, between2 and 15percent of all landings occurred with the maximumallowable landingweight. The distributions have almost the same slopewith one excep-tion, which is again the long-range cargo-type airplane. It has to be mentionedfurtherthat the scatter betweenthe distributions for the individual airplanes of the same type wassimilar to that of the take-off weight and wasvery small. Unfortunately, for the short-range airplanes, only the landing weights for flights from andto Frankfurt airport couldbe obtainedbecauseof matters of organisation. This fact seems, however, to beofsecondary importance with regard to the result.

In order to investigate the relation betweenairborne time and the respective landingweight, cross plots havebeenmadewhich showedthat the landingweight is more or lessindependentof the flight time. An exampleof this type of plotting is shownfor three long-rangepassenger airplanes in figure 10.

Take-Off Fuel Weight

The definition of a mission profile to beused for fatigue analysis has to include notonly the airplane gross weight but also the appropriate weight distribution. Sincetheweight of the fuel, which the airplane is carrying, allows information to be derived aboutthe weight distribution, ananalysis similar to that for the airplane weights has beenper-formed also for the fuel weights.

Figure 11showsthe cumulative frequency distributions of take-off fuel weights forthe different types of airplanes in relation to the respective maximum fuel weights. Thisform of presentation is not very suitable for deducinga general trend, becausethe indi-vidual curves intersect at several points. Therefore anattempt wasmadeto plot the ratio

545

between take-off fuel weights and the respective allowable airplane take-off weights on

logarithmic probability paper. (See fig. 12.) The distributions appear as a family of

curves with increasing standard deviation for increasing airplane size. They are clipped

at the respective value of the ratio of maximum fuel to maximum allowable airplane take-

off weight. Only the distributions for the long-range cargo airplanes behave as excep-

tions because they consist of two parts, each of which can be described by a logarithmic

normal distribution. It has been demonstrated that almost every second flight of long-

range cargo airplanes is made with the maximum allowable take-off weight. (See fig. 6.)

If the maximum allowable payload was reached, the fuel weight had to be restricted in

order not to exceed the maximum allowable airplane gross weight. That may have led to

this combination of two logarithmic normal distributions. Furthermore, it can be seen in

figure 12 that the variation between the cumulative frequency distributions as observed for

airplanes of the same type which flew with similar payload is very small.

A generalized presentation and a good approximation to the results is obtained when

the scatter as occurring in the range of probabilities between 90 and 99.5 percent is

ignored and is replaced by a fictitious point at 95 percent, where all distributions are

assumed to intersect at a weight ratio of 13 percent. (See fig. 13.)

Landing Fuel Weight

In opposition to the fuel weights as observed during take-off, it is not necessary to

relate those occurring during landing to the respective airplane gross weight, it is suffi-

cient for obtaining general information to relate them to the maximum fuel capacity of the

airplane type. The results of the analysis are presented again in form of cumulative fre-

quency distributions for the different types of airplanes. (See fig. 14.) From this graph,

a further confirmation can be derived for the assumption which was made when explaining

the fuel take-off weights of long-range cargo airplanes because it shows that these air-

planes have generally the lowest percentage of maximum fuel weight during landing.

From the presentation of the distributions in a probability paper (fig. 15), the percentages

of maximum fuel weight as occurring during every second landing can be defined as 14.5

for the cargo and 20 for the passenger long-range airplanes. The corresponding figures

for medium- and short-range airplanes are 38 and 49 percent, respectively. The latter

value seems to be very high; it can, however, be explained by the fact that in short-range

service, up to three flights were flown without refueling. It is interesting to note that the

distributions for the individual airplane types are almost parallel to each other, a tendency

which already has been observed for the airplane landing weights. (See fig. 9.)

546

CONC LUDING REMARKS

The statistical analysis of flight times as well as airplane gross weights and fuel

weights of jet-powered civil transport airplanes has shown that the distributions of their

frequency of occurrence per flight can be presented approximatively in general form.

Before, however, these results may be used during the project stage of an airplane for

defining a typical mission profile (the parameters of which are assumed to occur, for

example, with a probability of 50 percent), the following points have to be taken into

account.

Because the individualairplanes were rotated during service, the scatter between

the distributionsof mission profileparameters for airplanes of the same type, which were

flown with similar payload, has proven to be very small. Significantdeviations from the

generalized distributionsmay occur ifan operator uses one airplane preferably on one or

two specific routes.

Another reason for larger deviations could be thatthe maintenance services of the

operators of the observed airplanes are not representative of other airlines. Although

there are indicationsthat thisis unlikely,similar information should be obtained from

other operators. Such information would improve the reliabilityof the data of the present

report.

RE FERENCES

1. Anon.: Airworthiness Standards: Transport Category Airplanes. Federal Aviation

Regulations, Pts. 25 and 25.571, FAA, May 8, 1970.

2. Buxbaum, O.; and Gassner, E.: HS.ufigkeitsverteilungen als Bestandteil der

Lastannahmen f[ir Verkehrsflugzeuge Zeitschr. Luftfahrttechnik-Raumfahrttechnik,

Bd. 13, Nr. 4, 1967, pp. 78-84. (Also available as Library Translation No. 1303,

Brit. R.A.E., June 1968.)

3. Buxbaum, O.; and Reinhold, P.: Statistische Auswertung der das Einsatzprofil von

Transportflugzeugen kennzeichnenden Parameter. Tech. Rep. TB-88, Laboratorium

f[ir Betriebsfestigkeit, 1971. (To be distributed as ICAF-Document.)

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