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3573
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MINISTRY OF TECHNOLOGY

AERONAUTICAL RESEARCH COUNCIL

REPORTS A N D M E M O R A N D A

The Formulation of an Influence-Coefficient Method

for Determining Static Aeroelasfic Effects, and its
Application to a Slender Aircraft in Symmetric
Flight at M = 2°2

By A. S. TAYLOR, M.Sc., A.F.R.Ae.S. and D. J. ECKFORD, B.Sc., A.F.I.M.A.

StructuresDept., R.A.E., Farnborough

L O N D O N : H E R MAJESTY'S S T A T I O N E R Y O F F I C E
1969

PRICE £1 2S. 6d. NET

The Formulation of an Influence-Coefficient Method
for Determining Static Aeroelastic Effects, and its
Application to a Slender Aircraft in Symmetric
Flight at M 2"2
By 'A. S. TAYLOR, M.Sc., A.F.R.Ae.S. and D. J. ECKFORD, B.Sc., A.F.I.M.A.
Structures Dept., R.A.E., Farnborough

Reports and Memoranda No. 3573*

September, 1967

Summary.
This Report presents a matrix formulation of a method which employs influence coefficients to solve
static aeroelastic problems. Linear structural characteristics are assumed and the aerodynamic loading,
including that due to elastic deformation, is assumed to be compounded from a linear combination of
'elementary' distributions. Details of the method are here considered only for symmetric flight cases,
though in essence it is universally applicable.
The method has been applied to a 'Concorde'-like supersonic transport aircraft in symmetric flight
at M = 2.2. Linearised supersonic theory has been used to derive the load due to elastic deformation.
Results are presented for the incidences and elevator angles to trim and to sustain quasi-steady
manoeuvres, for the longitudinal distributions of shear force and bending moment, and for the elastic
deformation acquired.
The potentialities of the method are considered sufficient to justify the expenditure of the further
effort which is necessary to develop it into a fully automated design procedure of wide applicability.

CONTENTS
i° Introduction

2. Theoretical Basis
2.1. Some introductory remarks
2.2. General outline of the method
2.3. The theory in a practical form
2.4. Some concepts in the analysis of aeroelastic effects
2.5. Method of calculation for an aircraft with built-in compensatory warp

. An Example of the Application of the Method

3.1. The matrices employed
3.1.1. 6e, the structural flexibility influence matrix

*Replaces R.A.E. Tech. Report No. 67 237 (A.R.C. 30 059).

 3.1.2. 8, the load vector transformation matrix between Z 1 and Y~2
3.1.3. N, the aerodynamic influence matrix
3.1.4. The matrices ff and o~
3.2. The component loadings
3.2.1. Inertial loading for the aircraft with zero fuel
3.2.2. Inertial loading due to fuel
3.2.3. Aerodynamic loading for the aircraft with undeflected controls and zero rate of
pitch (components {~?i}cand {C?i}])
Y2.4. Aerodynamic load per radian elevator deflection
3.2.5. Aerodynamic load due to rate of pitch
3.3. The conditions of overall equilibrium
3.4. The resultant load on the aircraft
3.5. Definition of the design point and other 'standard' conditions
3.6. Results
3.6.1. General scope of calculations performed and results presented
3.6.2. Incidences to trim and incremental incidences per g, et and de/dn
3.6.3. Elevator deflections to trim, rh
3.6.4. Incremental elevator deflections per 9, drl/dn
3.6.5. Longitudinal distributions of shear force and bending moment
3.6.6. Elastic warp
3.6.7. Comparison with results of other investigations

4. The State of Development of the Method

5. Concluding Discussion
List of Symbols
References
Appendix A On choosing the sets of points E,
Tables 1 to 2
Illustrations--Figs. 1 to 36
Detachable Abstract Cards

1. Introduction.
In Part I of R. & M. 34261 one of the present authors, A. S. Taylor, and W. F. W. Urich developed
an approximate method of estimating the effects of aeroelasticity on the longitudinal trim and quasi-
steady manoeuvrability of slender aircraft; in Part II of the same Report Taylor extended the investigation
to include the estimation of the associated effects on the longitudinal distributions of shear force and
bending moment. For this work it was assumed that, structurally, the aircraft behaved as a 'free-free'
beam, subject only to longitudinal bending, and that the total chordwise aerodynamic loading (including
that due to elastic camber) varied linearly with local incidence and with elevator deflection, ancl was thus
calculable by superposition of a number of 'elementary' distributions. This two-dimensional approach
was conceived at an early stage of the exploratory investigations into the suitability of the slender near-
delta planform for supersonic transport aircraft. At that time attention was focused on the 'completely
integrated' slender-wing configuration, for which the spanwise stiffness might well have been large enough
to justify the assumption of beam-like behaviour. In the event, however, the layout adopted for the first
Anglo-French supersonic transport aircraft ('Concorde') featured a discrete fuselage, in conjunction
with a thin slender wing, and from the project studies of the British Aircraft Corporation and Sud Aviation
it soon became apparent that for such a configuration the hypothesis of purely longitudinal bending
would be untenable. It thus seemed desirable to formulate a three-dimensional counterpart of the two-
dimensional method given in R. & M. 3426, so that proper account could be taken of the spanwise de-
formability in the estimation of steady and quasi-steady aeroelastic effects. For the development of such
a method, a matrix formulation of the problem, employing structural and aerodynamic influence co-
efficients, was selected.
The two-dimensional method was, of course, inherently applicable only to aircraft of slender delta
configuration. No such restriction need be applied in the development of a (linearised) three-dimensional
method which should, in principle, be able to cope with a layout of any type, operating in any flow regime,
for which there exists an appropriate theory to provide the aerodynamic influence matrix to be associated
with the incremental incidences due to distortion. Thus, at the outset, the object has been to enunciate
the basis of the theory and to develop a practical form of considerable generality (Section 2). Later
(Section 3), in order to provide a numerical illustration of the method, the theory has been specialised
for the case of a 'Concorde'-like slender aircraft flying at M = 2.2, and the results of calculations are
presented in Section 3.6. For this application much of the data was provided by the British Aircraft
Corporation. In the derivation of the aerodynamic influence matrix to be associated with the elastic
distortion, linearised supersonic-flow theory was employed, via the method of D. E. Lees 2.
At this juncture no attempt has been made to assess the accuracy of the various parts of the me.thod,
nor its overall accuracy. The intention has been to present a generalised formulation of the matrix
approach to the problem, and to gain experience in its numerical application, while obtaining a qualitative
(and perhaps roughly quantitative) assessment of the quasi-steady aeroelastic effects on longitudinal
trim, manoeuvrability and loading of a slender supersonic transport aircraft in cruising flight.
In Section 4 some discussion of the state of development achieved by the present investigation is given,
together with indications of the further work that would be necessary to establish the method as a routine
design procedure, of general applicability. In the final discussion (Section 5) it is concluded that the
expenditure of effort on such work, which would include the automation of many of the processes involved,
would be justifiable. For, inasmuch as it provides a direct and 'exact' solution of an idealised problem,
the present method is superior to those which involve iteration or assumed modes of deformation while,
in an automated version, it should compare quite favourably with them as regards computational
economy.

2. Theoretical Basis.
2.1. Some Introductory Remarks.
Before proceeding to the development of a 'three-dimensional' theory it may be worthwhile to recall
the salient features of the 'two-dimensional' approach since some useful concepts from the latter may be
carried over to the former.
As already remarked in the Introduction, it was assumed for the 'two-dimensional' work that, structur-
ally, the aircraft behaved as a 'free-free' beam, subject only to longitudinal bending, and that 'the total
chordwise aerodynamic loading (including that due to elastic camber) varied linearly with local incidence
and with elevator deflection, and was thus calculable by superposition of a number of 'elementary'
distributions. The various distr~utions, other than that due to elastic camber, could be independently
specified in accordance with file best available experimental or theoretical data. For the specification
of loading due to elastic camber it was necessary to employ a simple (linearised) theory and for the
nunaerical work of R. & M. 3426 slender-wing and piston theories were selected. The differential equation
for the deflected beam could then be set up and solved on a digital computer in conjunction with the
equations of overall equilibrium. The distributions of elastic camber, the incidences of the centreline
chord and the elevator angles appropriate to the cases of trimmed level flight and of steady pull-up
manoeuvres were thus obtained.
In the formulation of a 'three-dimensional' method which shall be applicable over a wide range of
planforms (including those of low aspect ratio) the choice of flexibility influence coefficients as a means
of expressing the deformability characteristics of the structure is a natural one which leads on to the idea
of expressing the aerodynamic loading due to distortion also by means of influence coefficients. The general
principles underlying the solution of steady aeroelastic problems by means of influence coefficients
were discussed by Williamsa as long ago as 1954, and he sketched in some details of their application to
specific problems, including wing divergence and the determination of aileron rolling power and elevator
power. Four years earlier E. G. Broadbent, in investigating the rolling power of a swept wing, had
used rather special forms of structural influence coefficients 4. These forms were employed by Foody
and Reid, in conjunction with aerodynamic influence coefficients derived by Ktichemann's modified
lifting-line theory, in a routine computational technique for the estimation of loading (in subsonic flow)
on deformable wings of arbitrary planform. They first described this technique in 1954 in a report of
Short Brothers and Harland Ltd; an abridged version of this report was published by the Royal Aero-
nautical Society in 19555. However, the present authors are not aware of any detailed matrix formulation
of the trim and manoeuvrability problems which is generally available in published form, and it has
therefore seemed worthwhile to present such a formulation here.
2.2. General Outline of the Method.
The general outline of the theory is given below for symmetric flight conditions. The theory is not
inherently so restricted and the principles governing its extension to asymmetric conditions are obvious.
So long as any asymmetric case can be regarded as a planar problem, at least for a particular component,
the method for symmetric cases can be applied with few changes other than in notation and interpretation
of certain terms. However, in the case of completely general motion, involving significant deformations
of the whole aircraft under a loading which has components in and normal to the plane of symmetry,
the practical difficulties would be considerable.
An aircraft in a particular steady (or quasi-steady) symmetric flight condition is subject to distributed
aerodynamic, inertial and gravitational loadings, and loads due to the propulsion system. These may
all be resolved into components in and normal to a datum plane which lies near and approximately
parallel to the wing. For simplicity it will be assumed that all propulsion system loads act in this plane
and that, in common with all other load components in the plane, they cause negligible distortion of the
aircraft structure. For our purposes, therefore, we may hereafter consider the aircraft as subject to a
self-equilibrating distributed loading acting normally to the datum plane. This will have caused a de-
formation of the structure and, in particular, the streamwise slope at any point on the lifting surfaces
will have been changed by an amount which we term elastic incidence. The lifting surfaces as a whole may
be said to have acquired elastic warp*. One of the contributions to the net loading will therefore be the
aerodynamic load which results from this elastic warp. We may express the total aerodynamic load
as the sum of the load due to elastic warp and the load which would act over the aircraft if the deformation
were removed, without changing the datum incidence or control deflection**. The configuration on
which this latter load would act may be identified with that of a hypothetical rigid aircraft which can

*Elastic warp corresponds in the present three-dimensional approach to what was termed 'elastic
camber' in the two-dimensional treatment in Ref. 1.
**The terms 'datum incidence' and 'datum control deflection' refer to quantities defined for the aircraft
as a whole: 'datum' will be omitted when no confusion with local quantities is likely to arise. Their
precise meanings will be considered later (Section 2.3): their immediate intuitive meanings are sufficient
at this stage.

,4
serve as a datum for assessing aeroelastic effects. It should be noted that the rigid aircraft thus envisaged
will not in general be in steady flight for those combinations of incidence and control deflection that
produce such conditions for the actual flexible aircraft.
It is now assumed that the distributed loading may be adequately represented by a number of discrete
loads Q~ at a set Y.1 of n~ points and we write

(Q,}I = {Q,}~ + {AQ,}I (1)

where AQi is the contribution to Q, resulting from elastic warp. ( S e e list of symbols for explanatory notes
on notation).
Further, it is assumed that the structural flexibility is described by a square matrix of flexibility influence
coefficients 5a which relates the deflections w~ at a set ~Ez of n 2 points to a set of loads L~ at these same
points by

(w,}2, = 5:{L/}2. (2)

To find the deflections w~ due to the loads Q~, in the general case where Z1 and Zz do not coincide,
we transform the vector {Q~}~ to an equivalent vector {L,}2, expressing the transformation through a
matrix g :

{L~}2 = g{Q,}x • (3)

Hence

{w,}2 = (4)

We now turn to the calculation of {AQi}I corresponding to the deflections {wi}2. Assume that an aero-
dynamic influence matrix ~ can be determined of which the general element ~jk (load/radian) represents
the increment in load at a general point j of a set Ya of na points due to an increment in local incidence
at a general point k of a set E4 of n4 points. Denoting the vectors of loads in Y3 and incidences in E4
by {Pi}3 and {~i}4 respectively, we have

{P~}a = ~{a,}4. (5)

Suppose that elastic incidences ~ are determined from the deflections w~ through the relationship

{a,}4 = Cg{W,}z • (6)

Then the aerodynamic loading due to distortion is represented by a set of loads P, in E3 given by

{P,}3 = ~WSaS{Qi}I • (7)

Finally, {AQI}I is determined by replacing the loads Pi in Ea by an equivalent set of loads AQ~ in El,
the relationship between them being expressed by a matrix ~- :

(AQ~}~ = : { P , } 3 . (8)

Hence

{AQ~}~ = :t~cgAaS{Q,}~ (9)

= Aa {Q~}I, say. (10)

We now have from equation (l)

= {0,}, = (11)

or

= {C),11 (12)

If, then, we have a discrete load approximation (the loads ~)i) to the continuous loading on a hypothetical
rigid aircraft we may determine the corresponding approximation (the loads Qi) for the flexible aircraft
through the application of equation (12). For this purpose it is necessary, in the most general case, to
determine
(i) a structural flexibility influence matrix 5O associated with a set of points Z2,
(ii) a matrix ~ which transforms a vector of loads in Z1 to an equivalent vector of loads in E2,
(iii) an aerodynamic influence matrix ~ which gives the aerodynamic loads in E3 corresponding to
local elastic incidences in Z4,
(iv) a matrix cg which transforms deflections in Y2 to local elastic incidences in Z4,
(v) a matrix ~- which transforms a vector of loads in E3 to an equivalent vector of loads in El.
The above analysis applies only if [ J - c~] is not singular. Since £o depends on kinetic pressure there
may exist airspeeds for which [J-.L,¢] becomes singular corresponding to the physical phenomenon
of static aeroelastic divergence. In principle these divergence speeds could therefore be found by de-
termining the conditions for [ J - 5¢] to become singular; in practice, however, the numerical processes
involved would become unreliable as such conditions were approached. Further discussion of such
difficulties will not be attempted here and we shall assume that all cases to be investigated are sufficiently
far from cases of divergence for the inversion of [ J - £~o] to be a well-conditioned process.

2.3. The Theory in a Practical Form.

The above sketch of the theory has presented the idea behind the method. The purpose of this Section
is to consider some of the practical difficulties which must be resolved before it can be applied, while
still retaining considerable generality.
For a particular quasi-steady flight condition the loading system {~} ~ can be compounded from the
known mass distribution and the component aerodynamic loads (derived from calculations and/or
wind tunnel tests for the rigid aircraft) provided that the appropriate incidence and control deflection
have been specified. These are, however, not known in advance and have in fact to be determined from
considerations of the overall equilibrium of the flexible aircraft. Indeed, the interpretations to be placed
upon the terms 'incidence' and 'control deflection' for the flexible aircraft are not yet defined since no
particular system of axes has been assumed.
A question arises in connection with the specification of the structural influence matrix 5 °. Influence
coefficients are normally evaluated experimentally or theoretically for the aircraft subjected to a statically
determinate system of constraints and it is not immediately apparent whether or not a matrix of coefficients
so obtained can be directly applied in the determination of the deformation of a freely flying aircraft.
It will transpire that, for the method here developed, we may proceed on the assumption that 5O relates
to the aircraft constrained in any convenient statically determinate manner. The deflections so obtained
will be referred to a certain datum, the selection of which depends on the chosen constraints. For sym-
metric load systems it is most natural to apply the constraints at points along the centreline of the aircraft.
One possibility is to assume that the aircraft is 'simply supported' at two points (at each of which a force
but no moment may be applied), another to assume that the aircraft is 'built in' at a single point. In the
former situation a suitable datum from which to measure the deflections (and the elastic incidences)
is the plane, perpendicular to the plane of symmetry, which contains the two support points, while in
the latter situation a suitable datum plane is tangential to the line of material particles adjacent to the
point where the aircraft is 'built in'.
When the datum plane for measuring deflections has been decided upon, the basis for defining
'incidence' is fixed. This is defined as the angle of the datum plane to the free stream (or an angle differing
from it by a constant amount). Here we define 'control deflection' simply by identifying it with the de-
flection of the control surface when the elastic deformation is removed, that is with the control deflection
for the hypothetical rigid aircraft. This is convenient for our present purposes, whatever might be its
shortcomings as a general definition for a flexible aircraft.
We now turn to the problem of the determination of {~2i}1. In any particular case {~2i)1 will be com-
pounded from a number of loading systems; aerodynamic, inertial and gravitational. Let a typical unit
component system be {Lgi}] and its contribution to {0.i}1 be determined by factoring it by a,; i.e.

(13)
{ Q,} , = Sa,
r

and

{Q,}, (14)

where
{Q,}~ = [ J - ~ ] - ' {~,}~. (15)
Two problems remain :
(i) The validity of using the structural influence matrix 6a appropriate to the constrained aircraft has
still to be established.
(ii) A method of determining the factors ar has to be found. (For a particular flight condition all of these
are known except the two which define the datum incidence and control deflection.)
These problems may be resolved by the following method. Assume that the incidence is defined in a
way compatible with the constraints used in the determination of :T. Then the load on the free-flying
aircraft may be found by determining the aetoelastically modified loading system {Qi}q corresponding
to each system {Q.i}~,with the aircraft constrained, and then combining these modified systems according
to equation (14), for which the unknown factors ar are found by applying the conditions of overall equi-
librium of the aircraft. Since the resulting {Qi} 1 is a self-equilibrating system, the aggregate of the Con-:
straining loads corresponding to the component loadings a, {Qi}~ must also form a self-equilibrating
system and, since a statically determinate system of constraints has been assumed, it follows that each of
the resultant constraining loads must be zero. {Qi}1 is thus appropriate to the free-flying condition.

2.4. Some Concepts in the Analysis of Aeroelastic Effects.

The most usual type of investigation into the significance of aeroelasticity proceeds as follows. Suppose
that the shape of a certain aircraft, actual or projected, is known when the aircraft is unloaded. This shape
will usually be One for which extensive programmes of calculations and wind-tunnel tests have been
carried out. Then, if the structural properties of the aircraft are also known, stability and control and
loading analyses may be performed:
(a) assuming the aerodynamic shape to remain unchanged from the unloaded condition, apart from
control deflections, for all conditions of flight, and
(b) taking full account of structural deformations and the attendant changes in aerodynamic shape
and loading.
The differences between the results of (a) and (b) indicate the degree of importance to be attached to
 aeroelastic effects in the stability and control and loading analyses for this particular aircraft. We term
such differences gross aeroelastic effects.
In the aircraft design stage a rather different approach can be used. Suppose that in his attempts to
obtain the best possible performance from a design, which can be varied within limits, a designer assumes
that it is sufficient to concentrate on the performance at a particular combination of height, speed, weight,
etc. This combination we will term the desion point. When, on the basis of wind-tunnel tests (on nominally
rigid models) and calculation, a designer is satisfied with the design-point performance he may be loath
to risk any seriously adverse aeroelastic effects on that performance. These may be avoided if the de-
formations that would occur under the design-point loading are determined and equal but opposite
deformations, which we call compensatory warp, are built into the aircraft. The aircraft will then assume
the required shape, and hence loading, at the design point. It should be noted that the procedure is rather
simpler than in most aeroelastic problems in that the deflected shape of the flexible aircraft (with com-
pensatory warp) is known and hence the loading may be found directly: the compensatory warp then
follows from a purely structural analysis. Stability and control and loading analyses may now be per-
formed :
(c) for a hypothetical rigid aircraft having the shape required at the design point (identical to (a)
above), and
(d) for the actual flexible aircraft built with compensatory warp.
The differences between (c) and (d) represent the residual aeroelastic effects that remain after those
associated with the deformations which would occur under the design-point loading have been nullified
by the incorporation of compensatory warp. These we term net aeroelastic effects.
By definition the net effects on trim conditions are zero at the design point and in most cases they are
likely to be smaller than the corresponding gross effects although in certain circumstances they can be
greater. For those quantities which depend not on a particular initial condition but on the behaviour
when passing from one steady condition to a neighbouring one at constant speed and Mach number,
such as the elevator deflection per g, net and gross effects are equal. For those quantities which relate
to changes of speed and/or Mach number, such as the slope of the curve of elevator deflection to trim,
net and gross effects are in general not equal: the former are, however, not zero even when the initial
condition is the design point.
Both gross and net effects are significant to the designer. The order of magnitude of the former should
be determined at a fairly early design stage so that the importance of allowing for aeroelastic effects can
be assessed. This assessment could indicate that, while in certain flight conditions the effects of aircraft
flexibility should be taken into account, such effects would be so small at the design point as to render
the incorporation of compensatory warp unnecessary. In such circumstances the aircraft would be built
to the original design shape and the concept of net aeroelastic effects would be irrelevant. However,
when compensatory warp is incorporated it is still necessary to know what are the net effects.
It seems desirable that in any aeroelastic investigation one should consider both gross and net effects
although it will usually be unnecessary to cover both in equal detail. Gross effects should probably be
gi~,en greater emphasis since they relate to the somewhat more fundamental concept.

2.5. Method of Calculation for an Aircraft with Built-in Compensatory Warp.

Tile previous section considered the possibility of employing built-in compensatory warp to nullify
aeroelastic effects at a design condition, and introduced the concept of net aeroelastic effects. We now
consider how the method of this Report can allow us to compute such net effects.
Suppose that the loading at the design point is represented by a vector {Q~}1. Then the required
compensatory warp is given by deflections w~ where

{w~}2 = - S e e {Q~°}I. (16)

At any other flight condition, where the net load on the aircraft is {Q~}1, the deflections of the aircraft
with compensatory warp, relative to the 'design' shape, are

8
 {w,}2 = bog {Q~-Q?}~. (17)

Hence, the load due to the divergence from the design shape, which arises from the difference between the
design-point loading and the loading at any other flight condition, is given by

{AQ,}, = ~cg~9°6~ {Q,- Q?}, = £: {Q,- Q?},. (18)

Therefore

{e,},-{Qi}, = {Q,-Q?}, (19)

or

{Q,}, = {Q,},-[:- {el},. (2o)

It is seen that the above equation is identical with equation (12) save for the addition of a term on the
right hand side. The vector - ~ o {Q~}I must be calculated and premultiplied by [ J - £#]- 1 in the same
way as the load vectors {Qi}]- The resulting vector is then simply combined with the vectors {Qi}] subject
to the conditions of overall equilibrium.
Equation (20) may be looked at from two points of view. The first is that which was adopted in its
development; that the deflections are measured relative to the shape of the hypothetical rigid aircraft
(without compensatory warp) and that the additional term arises purely from the incorporation of the
compensatory warp in these deflections, in accordance with equation (17). A second point of view is that
the 'datum' shape is the unloaded shape of the flexible aircraft with compensatory warp. Then to the set
of load vectors must be added one giving the aerodynamic load due to the difference between this datum
shape and the shape of the hypothetical rigid aircraft. If this load vector is calculated through the same
aerodynamic matrix as is employed to calculate the aeroelastic deflections it becomes - ~ {Q~}I. The
additional term in equation (20) is then the aeroelastically modified form of this extra load vector.
The second of these points of view is useful in providing us with terminology: we speak of the additional
vector - ~ {Q~)}a as the 'load due to compensatory warp'. The first point of view is perhaps the better
one for analytical discussions as it makes clear the fact that the method used to calculate loads due to
aeroelastic distortion has to be sufficiently accurate only for loads due to deflections (wi- w~), which will
usually be smaller than deflections wi. Calculations of net aeroelastic effects should therefore be rather
more reliable than those of gross effects, at least for those trimmed level-flight conditions where the
overall loading resembles the design-point loading. An additional indication is that it will normally be
unprofitable to use a method to calculate the load due to compensatory warp which differs from that
applied to aeroelastic distortion, as could justifiably be done in accordance with our complete freedom
in determining individual load systems, so replacing the term - ~ {Q~}1 by some other vector. (The use
of a different method would mean that the design-point conditions would not be exactly satisfied, but
this would be of no practical significance.)

3. An Example of the Application of the Method.

The above method has been Used to determine the static aeroelastic effects on a slender aircraft in
symmetric flight at a Mach number of 2.2. The general arrangement of the aircraft, which is an early
projected version of the Anglo-French 'Concorde' airliner, is shown in Fig. 1. The weights and kinetic
pressures assumed for the start, middle and end of cruise are appropriate to the medium range version.
The structural flexibility matrix and the data from which were derived tt/e inertial and (untrimmed)
aerodynamic loadings for the rigid aircraft were provided by the British Aircraft Corporation (Operating)
Ltd., Filton Division. The sizes of the matrices which could be employed were initially dictated by limita-
tions of computer capacity, on the assumption that the calculations would be performed on the R.A.E.
'Mercury' computer. In the event, 'Mercury' failed inexplicably to perform certain operations pro-
grammed for it and some of the work had to be performed on the 'Atlas' computer at Manchester. Had
the use of this computer been envisaged at the outset, larger matrices could have been employed with
(presumably) beneficial effects on accuracy.
The British system of units is that basically employed. When certain numerical values first appear
in the text their equivalents in SI units are given in parentheses. Values in SI units are indicated on the
illustrations by means of auxiliary scales.
Since this is a symmetric problem we need consider directly only the loads acting on either the port
or the starboard half of the aircraft. We take account of the loads and elastic incidences on the other
half by modifying the matrices 5¢ and ~. (The starboard half was actually chosen.)

3.1. The Matrices Employed.

3.1.1..c/, the structural flexihility i~fluence matrix. Three such matrices were made available by
the British Aircraft Corporation. These were associated with sets of 75, 121 and 200 points respectively
(per half-wing) and had been obtained in the course of a displacement-type structural analysis. The present
investigation was initially limited in the choice of Ea to sets which contained less than 100 points and so,
following the considerations in Appendix A, the matrix associated with 75 points was chosen. The set
of structural grid points is shown in Fig. 2. The 'half-aircraft' is constrained by two simple supports.
These do not lie on the centreline; however, as the problem is symmetric, the implied complete constraint
system of four simple supports is still statically determinate.
The matrix 5o took account of the fact that for every load at a grid point on the starboard half of
the aircraft there is an equal load at the corresponding point on the port half, which causes additional
deflections on the starboard half.
It is seen that the grid points give good coverage oi" most of the half-aircraft; the main regions which
are omitted are (i) the fuselage inboard of a line near the wing root, (ii) the elevators, (iii) the portion of
the wing to the rear of (approximately) the elevator hinge line, and (iv) the portion of the fuselage to the
rear of *.he aft support point. The assumptions which were made to allow for these omissions are :
(i) the fuselage is rigid in a spanwise direction,
(ii), (iii) to the rear of the last row of grid points there is no change in elastic incidence along any chord-
wise line, and
(iv) the fuselage load to the rear of the aft support point causes no deflections.
Of these the last is probably the most questionable but the choice of Z2 and the method of deriving the
component load systems made it necessary - it is thought to lead to only small errors.

3.1.2. g, the load vector transformation matrix between El and E 2. The twin considerations of
computer capacity and the need to ensure that, over the region of action of any component loading,
Y.~ gives a coverage at least as dense as E2 gave close limits for the number of points in El. Accordingly
it was decided to choose E~ to coincide with Z z. Hence the matrix ~ becomes the unit matrix and may be
omitted from the equations.

3.1.3. ~, the aerodynamic influence matrix. A matrix of aerodynamic influence coefficients based
on steady, linearised, supersonic-flow theory was required. The method described by Lees 2 offered
a direct approach to the calculation of such influence coefficients, and this method was accordingly
adopted. For its application the wing is subdivided into diamond-shaped panels by a network of equi-
spaced Mach lines, and the velocity potential induced at each receiving panel by unit downwash at
every other panel in turn is calculated. The corresponding lift per unit downwash is obtained by integrating
the potential over the edges of the receiving panel. Only panels lying within the forward Mach cone
from the aftmost comer of a given panel can contribute to the lift at that panel and when, as in the present
example, the leading edges of the wing are subsonic, the precise area within which 'effective' panels lie
is defined by application of Evvard's area cancellation technique. The influence of the jth panel on the
ith panel depends only on parameters m, n defining the relative positions of the two panels in a system

10
of oblique axes parallel to the Mach lines*. In his paper, Lees has tabulated the relevant influence
function A(m,n) for values of m and n over the ranges 1 ~<m~<11, 1 ~<n~<21, and this provides the basis
for a straightforward, if somewhat tedious, compilation of the influence coefficient matrix for the complete
wing. The matrix thus obtained actually gives loads per unit kinetic pressure-we denote it by ~*. ~ is
obtained by multiplying ~* by Q.
In the present application the method was applied to the complete aircraft planform, excluding the
portion of the fuselage extending behind the wing trailing edge. Advantage was again taken of the sym-
metrical nature of the problem so that E3 and E4 (which in this case coincide) were defined over the
starboard half of the aircraft only. Mesh size was fixed by taking the length of the longer diagonal of a
panel to be 1/11 of the overall length of the assumed planform. The resulting mesh divided the half-
aircraft into 60 panels of diamond or partial diamond shape as shown in Fig. 2. For clarity the mesh is
shown on the port half. The points of E3 and E¢ are located at the centroids of panels. For each panel
1 on the starboard half-wing we may envisage a symmetrically disposed panel l' on the port half-wing.
The element ~Zk of ~ then represents the point load at grid point k due to unit incidence of the pair
of panels l, l'.
With this size of mesh, the influence coefficients could be calculated directly from Lees's tabulated
values of A(m,n). Although the number of points in the E3 thereby obtained is lower than the number
in El, the 'densities' of the two sets of points are comparable Over most of the aircraft. Hence there is
no serious violation of the requirement (Appendix A) that the density of E 3 be at least as high as that
of El.
As was noted earlier, assessment of accuracy was not one of the aims of the present investigation.
However, for interest's sake, the load per radian incidence of the rigid aircraft as predicted by ~ was
compared with that which was used as a basis for the loading component {L)i}] (see Section 3.2.3). The
lift-curve slope which was obtained was 2.23 whereas a value of 2.0 was assumed for {Lgi}]. The aero-
dynamic centre was given at 0"559c0 by ~ and at 0.592Co by {Qi}]. Examination of the two lengthwise
ioadings shows that most of the discrepancy in the position of the aerodynamic centre is due to the fact
that the use of ~ leads to overestimation of the loading on the forward part of the fuselage. Further
refinement was not thought worthwhile at this stage.
3.1.4. The matrices cg and ~ . It will be recalled that, in the general theory, cg is the matrix which
transforms deflections at the set E2 of structural grid points into local elastic incidences at the se~ E4
of downwash points associated with ~, while matrix ~ transforms a vector of loads in E3 to an equivalent
vector of loads in E~. As already noted, in the present application, E~ = Z2 and E3 = Z4, so that in
effect ~ transforms deflections at the structural grid points into local incidences at the aerodynamic
grid points and ~ transforms a vector of loads at the aerodynamic grid points into an equivalent vector
of loads at the structural grid points.
Basically, the methods adopted for the calculation of elements of these matrices were those suggested
by Williams 3. Some extensions were necessary in order to deal with certain regions of the planform
(notably the fuselage and elevon portions) where the ideas of Williams are not directly applicable.
3.2. The Component Loadings.
For this investigation all the component loadings were taken to act over the same set of points Z1
although, as discussed in Appendix A, this need not generally be the case. The following rigid aircraft
component loadings were considered.
{~}~ Inertial load (per g) for the aircraft with zero fuel.
{Lgi}~ Inertial load (per g) due to fuel.
{~i}c Aerodynamic load for a CL of 0'1.

*If Q is the aftmost corner of panel i, and P is the foremost corner of panel j, the oblique co-ordinates
of P relative to Q are ml and nl where I is the length of the panel side.

11
 {Qi}~ Aerodynamic load due to incidence.
{~)~}] Aerodynamic load due to elevator deflection.
{Q~}'I Aerodynamic load due to rate of pitch.

We now discuss the formation of each of the above loading vectors. For the aerodynamic components
the so-called load vectors are in fact to be interpreted as vectors of loads divided by the free stream kinetic
pressure.
In addition to these vectors, the vector - J ~ * c g S ~ {Q~} 1, giving the load due to compensatory warp,
was computed. We denote this by {Q~}w. The design point is defined in Section 3.5.
3.2.1. Inertial loading for the aircraft with zero fuel. The distribution of mass for the aircraft with
zero fuel was supplied by the British Aircraft Corporation in the form of discrete loads at the nodal
points of the 200-point structural grid. These were re-distributed to the nodes of the 75-point grid. The loads
at points of the 200-point grid lying within the wing planform aft of the elevon hinge line were transferred
forward to the rearmost grid line of the 75-point grid (approximately coincident with the elevon hinge
line) and then shared in appropriate proportions between the nearest grid points on either hand. Transfer
moments about the grid line in question were considered to have no effect on deflections at points of the
75-point grid, but were preserved for inclusion in the equations of overall equilibrium for the aircraft.
Similarly, loads at points on the rearbody aft of the rearmost grid line were neglected as regards their
effect on deflections at points of the 75-point grid, but were accounted for in the equations of overall
equilibrium.
3.2.2. Inertial loading due to fuel. As shown in Fig. 3, the aircraft has six fuel tanks per side and a
single tank in the tail cone. The mass of fuel contained in each tank when full is given below (that for
the tail tank being per side).

Tank No. 1 2 3 4 5 6 7

Mass of fuel, lb 5080 9790 7110 14133 12870 7050 6615

(kg) (2304~ (4441) (3225) (6411) (5838) (3198) (3000)

The inertial loading for each tank when full was derived in the same way as the inertial loading for
the aircraft with zero fuel. Let the load vector corresponding to the rth tank be {(2~}~. Then we express
the inertial loading due to a particular fuel loading as

)l
6

(0 ~<kr ~< 1) (21)

r=l

Note that there is no vector corresponding to the fuel in the tail tank, by virtue of assumption (iv) in
Section 3.1. l : the force and moment due to this fuel were included in the conditions of overall equilibrium.

3.2.3. Aerodynamic loading for the aircraft with undeflected controls and zero rate of pitch (com-
ponents {Ol} c and {Oi}])- Aerodynamic loading data for the rigid aircraft at M = 2.2 were made available
by British Aircraft Corporation in the form of:
(1) Longitudinal load distribution curves (L(x): load per unit length per unit kinetic pressure, versus
x: distance from nose of aircraft) for values of the untrimmed lift coefficient CL from --0"1 to 0'4 in steps
of 0"l.

12
 (2) Spanwise load distribution curves (local loading coefficient versus fraction of local semispan from
centreline) for eight chordwise stations at the same CL values as for (1).
This information was based partly on wind-tunnel tests and partly on theory. As already indicated
in Section 3.2, it was decided to express the aerodynamic loading for the aircraft with undeflected controls
and zero rate of pitch as the sum of two components; one corresponding to an arbitrarily chosen CL
of 0"1 (corresponding roughly to the design point) and the other to the incremental incidence relative to
the CL = 0.1 value. The starting points for the derivation of the loading vectors {~)i}c and {~?i}] were
therefore the L(x) versus x curve for CL = 0-1, taken directly from the B.A.C. data, and a curve of in-
cremental loading AL(x), per radian of incidence, versus x, deduced from the set of L(x) versus x curves,
dCL
under the assumption that ~ - 2.0 over the range -0.1 < CL < 0.4. The two basic loading curves

AL(x)
[L(x)]cL = o.1 and T ~ versus x are shown in Fig. 4. To convert these distributed loadings to the equiva-

lent discrete load vectors {~)~}cand {0.~}~,the total load in each case had first to be apportioned to chord-
wise stations corresponding to spanwise grid lines of the structural grid; the load at each chordwise
station had then to be distributed to the grid points at that station in a manner consistent with the spanwise
load distribution curves supplied by B.A.C. Graphical interpolation was used to derive from these curves
the spanwise distribution curves appropriate to the chordwise stations of the structural grid.
In the apportionment of load to chordwise stations the general principle adopted was to assume the
longitudinal distribution between two adjacent stations to be trapezoidal and to replace it by a pair of
concentrated loads at those stations, which gave the same total load and c.p. position. Thus, in general, the
load allocated to a particular chordwise station comprised two contributions, derived from the continuous
load distributions in the two intervals between that station and its immediate neighbours. The loading
ahead of the foremost structural grid point, G.P.75, was replaced by concentrated loads at G.P.'s 74 and
75 to give the same total load and c.p. position, while the loading aft of the elevon hinge line was assumed
to be concentrated along the rearmost structural grid line, an appropriate transfer moment being com-
puted for inclusion in the overall balance equations.
In general, for a given chordwise station, the (spanwise) distribution of the total load between the
structural grid points at that station was determined in a manner analogous to that adopted for the
chordwise allocation. In order to derive the point loads corresponding to the distributed load inboard
of the first row of grid points a row of 'dummy' points along the centreline was introduced. This done,
the procedure was perfectly straightforward for stations with a fu!l complement of grid points; where,
however, a spanwise row of points was incomplete (see Fig. 2), further 'dummy' grid points were in-
troduced to complete the set, and the load was first distributed between this complete set of actual and
'dummy' grid points. The !oad at each 'dummy' point was subsequently replaced by a statically equivalent
pair of loads at the nearest two actual grid points on the same chordwise grid line. At stations on the
forward part of the wing where there were only two or three grid points, 'dummy' points were introduced
at locations chosen so as to ensure that the continuous spanwise distribution was broken down into
approximately trapezoidal portions. In this case, the load at each 'dummy' point was replaced by a
statically equivalent pair of loads at the nearest two actual grid points on the spanwise grid line under
consideration. Loads at the aircraft centreline were transferred directly outboard to the first chordwise
row of grid points. The resultant of the continuous loading outboard of the last grid point on a spanwise
grid line was estimated separately and replaced by a pair of loads at the nearest two grid points.
A digital computer ('Mercury') program was prepared to perform the initial allocation of load to the
grid points (actual and 'dummy') of the various chordwise stations. Subsequent manipulation of the
loads was performed by hand.
3.2.4. Aerodynamic load per radian elevator deflection. The aircraft has three trailing edge control
surfaces (elevons) per side, as shown in Fig. 3. It was assumed that any of these could be used for pitch
control (i.e. as elevators) and that the ratios of the deflections of the individual surfaces could take any
non-negative values. Accordingly, the deflection of any surface, say the rth, may be expressed as grq,
where q is the largest of the three control surface deflections and is termed the 'control (elevator) deflection'.

13
Therefore if the load vector corresponding to unit deflection of the rth control surface is {~)~}~"the total
load vector per unit elevator deflection is

(22)
r=l

The loadings were calculated by linear supersonic theory 6. The load vectors were derived by transfer-
ring the load on the control surfaces to the rearmost line of grid points, while preserving the spanwise
positions of the centre s~of pressure (no load was transferred to the grid point nearest the tip). Again, the
correct balance was preserved by the addition of transfer moments in the equations of overall equilibrium.
3.2.5. Aerodynamic load due to rate of pitch. No wind-tunnel data exist on the aerodynamic load
due to rate of pitch. This loading was therefore calculated using the aerodynamic influence matrix.
(For the present example of a slender aircraft at high speed its effect will be small but it was included
for the sake of completeness.) Consider first the loading to act in Z3. Let a typical point of Z4 and the
centre of gravity of the aircraft be distances x'i and 2 ahead of an arbitrary fixed point (here taken as
the aft support point). Then the incidence induced at this typical point by a rate of pitch q about the centre
of gravity is

- (x'i - ~) q / V . (23)
Thus the load vector in Z3 due to unit rate of pitch is

= {x3, {J,},, (24)

where {0¢i}4 is a unit column.

Now q = ( n - 1) g/V, where ng is the total normal acceleration% and since V is constant over the range
of kinetic pressure (or altitude) being considered, we may express the above loading as a loading due to
incremental normal acceleration. Hence, in Zx, the loading due to rate of pitch can be expressed as

(n - 1) {g,}~ = ( n - 1) {Q,}7~+ 2 (n - 1) {~)i}~2 (25)

where

(26)

{tZ}7 2 = o/v2}4. (27)

3.3. The Conditions of Overall Equilibrium.

The conditions of overall equilibrium for the aircraft in quasi-steady flight with a normal acceleration
of n9 were satisfied by equilibrating the forces normal to the datum plane and the moments about the
rearmost line of structural grid points.
Let the force and moment corresponding to the load vector {Qi}[, for example, be Zx and M1 re-
spectively. Included in such forces and moments are any additional forces and moments not accounted
for in the load vectors (see for example Section 3.2.1). Then the equilibrium equations are

+Note that in Ref. 1 the total normal acceleration was denoted by ~g, ng being there the incremental
normal acceleration from level flight. This revision is in accordance with the usual practice when dis-
cussing structural loading.

14
 Q{Zc + o~Z=+ 11Z n+ (n- 1) Zq + Zw} + n(Z, + Ze) = 0 (28)

and

Q{Mc+e M~+rl M, +(n- 1) Ma+ Mw} +n(M~+ Me) = O, (29)

where

ZF =
>,7

i
k~ ZF~ (30.1)
r=l

Zk
7

Me = r Mer (30.2)
r=l

Zn = Z g r Zn~ (30.3)
r=l

Mn = Z g r Mn, (30.4)
r=l

(Zw, Me7 and kv relate to'the fuel in the tail tank).

These equations may be solved for c~and t/. The chosen method of presentation of results separates
out the values of e and t/for trimmed level flight (here taken as n = 1), cq and t/t, and the rates of variation
of ~ and t/with normal load factor, de/dn and &l/dn. Therefore we replace equations (28) and (29) above
by two sets of equations :

o~t Z~ + tl t Z~ = - Q (ZI + Ze) - Z c - Z w (31)

at M~ + ~h Mn = - ~ ( M I + M e ) - M c - M w (32)

and

do~Z~ + ~nnZn = --01 (z ' + Zv) - Zq

an (33)

-~nd°~M~ + ~nnMn = - Q (MI + M F)- Mq (34)

The above derivation of the conditions of overall equilibrium is strictly applicable to the flexible aircraft
with built-in compensatory warp. For the rigid aircraft the forces and moments become Z~ and M1,
for example, and in this case, and in the case of the flexible aircraft without compensatory warp, Zw
and Mw are omitted.

15
 3.4. The Resultant Load on the Aircraft.
When the equilibrium equations have been solved for e,, qt, da/dn and &l/dn the resultant load on the
aircraft may be found. The point loads Qi are given by

{Qi}I = n {Qi}~ +n {Qi}~+Q [{Q~}C+~t {Q~}~+t/t {Q~}~+ {Q,}W]

+Q(n-1) [dc~..
~n {Qi}l +~nn{Qi}~ +{Qi}~ ] •

The modifications necessary to make this equation applicable to either the rigid aircraft or the flexible
aircraft without compensatory warp are obvious.

3,5. Definition of the Design Point and other 'Standard' Conditions.

The design point was assumed for the purposes of this example to be a condition in approximately
mid-cruise : M - 2'2 at about 58 000 ft (17 700m). The weight assumed was consistent with data supplied
by B.A.C. The distribution of fuel was chosen to be in accordance with a feasible plan of fuel usage during
a complete flight and to be such that the rigid aircraft would be trimmed with virtually zero elevator
deflection (0.01%
In addition to the design point, two other 'standard' conditions were defined. These corresponded to
points at approximately the start and the end of cruise. The kinetic pressures and weights were again
in accordance with B.A.C.'s data while the fuel distribution was chosen to give small elevator angles
to trim and to bear a sensible relationship to the design-point distribution.
The detailed specifications of these conditions are

W lb Q lb/ft 2
(kg) (kN/m 2) kl k2 k3 k4 k5 k6 k7

Start of cruise 190 330 670 0 0.35 0 1.0 3.54 0"6 0'36
(86 330) (32.08)
Design point 170 840 55O 0 0.35 0 0.62 3"2 0.6 0"36
(77490) (26"33)
End of cruise 150 880 450 0.2 0'2 0 0.6 0'36
(68 440) (21-55)

3.6. Results.
3.6.1. General scope of calculations petformed and results presented. The calculations were per-
formed for a single Mach number, 2.2, corresponding to the cruise-climb phase of a typical flight plan.
This was in the interests of computational economy since the component aerodynamic loadings and
the aerodynamic influence matrix ~* (and its associated transforming matrices cg and o~) would have
had to have been recalculated for each Mach number considered. With economy again in mind, it was
decided to restrict the calculations of trim, manoeuvrability, resultant !~oaddistributions and deflections
to the three 'standard' weight distributions detailed in Section 3.5. In the practical application of the
method to aircraft design it would be necessary to consider a range of Mach numbers, with associated
altitudes and weight distributions, corresponding '.o various points of the flight envelope. Here we have
obtained some idea of the effects of operating away from the normal flight plan by making certain cal-
culations for kinetic pressures (and hence altitudes) other than the 'standard' (flight plan) value.
Accordingly, the aeroelastically modified component loading vectors {Q~}~, detailed in Section 3.2,

16
were calculated for kinetic pressures of 300, 450, 550, 670, 800 and 1000 lb/ft z (14.36, 21.55, 26.33, 32.08,
38.30 and 47.88 kN/m2), and used in balance calculations for each of the three weight distributions. The
altitude corresponding to a particular kinetic pressure may be found from Fig. 5. In all cases three com-
binations of the control gearings were used, viz.
91 = 02 = 93 = 1.0 : All surfaces moved together.
01 = 0, g2 = g3 = 1.0: The inboard surface undeflected, the outboard two moved together.
91 = 1.0, 92 = 03 = 0: Only the inboard surface deflected.
The incidences and elevator angles to trim and to sustain steady manoeuvres were determined for
each combination of the variations listed above (54 cases in all) and are plotted against kinetic pressure
Q in Figs. 6 to 17.
The resultant load distribution was calculated for the three 'standard' flight conditions and also for the
design-point fuel distribution at kinetic pressures of 450 and 670 lb/ft 2. In these cases the control gearings
were gl = ga = 93 = 1"0. F r o m these load distributions the longitudinal distributions of shear force and
bending moment were derived and are presented in Figs. 18 to 32.
The deflections due to elastic warp have been calculated for the design.-point flight condition at normal
accelerations of 1.0 and 2"59. For the former the control gearings were 91 = 92 = 93 = 1.0, while for
the latter the three combinations of gearings were considered. These deflections are presented in Figs. 33
to 36. Consideration of these figures (Section 3.6.6) provides some physical explanations of the results
of the trim and manoeuvrability calculations.
In the presentation of the trim data and the shear force and bending-moment distributions, three
curves are generally distinguishable in each case. These relate respectively to the rigid aircraft, the
flexible aircraft and the flexible aircraft with compensatory warp so that, in accordance with the definitions
of Section 2.3, the differences between the first and second represent the gross aeroelastic effects, while
the net aeroelastic effects are measured by the differences between the first and third curves. Incremental
incidences and elevator deflections for steady manoeuvres are independent of built-in warp so that
only two curves appear in the part of each figure presenting these data, and gross and net aeroelastic
effects are equal.

3.6.2. Incidences to trim and incremental incidences per g, ~ and ~nn(Figs. 6 to 8). Since for the rigid
aircraft the centres of pressure of the three control-surface loadings lie approximately on a spanwise line,
the incidences to trim and manoeuvre it are virtually independent (to within 0.01 °) of the control gearings
here considered. Consequently the 'rigid aircraft' curves in Figs. 6 to 8 can be taken as applying to any
of the three combinations of gearings. The effects of aeroelasticity on the required incidences are small.
The incidence to trim, ~t, is affected most in the 'start to cruise' fuel configuration with gx = 0, 92 = 93 =
1.0, while the greatest effects on d~/dn are in the same fuel configuration but with gl = 1.0, g2 = 93 = 0.
These effects are shown in Fig. 6. There are similar trends in the other two 'standard' configurations,
as shown in Figs. 7 and 8. (As already noted in Section 3.6.1 the incidence and elevator deflections to
manoeuvre the flexible aircraft are the same whether or not compensatory warp is included.)

3.6.3. Elevator deflections to trim, tlt.

(a) gl = g2 = 93 = 1.0 (Figs. 9 to 11).
With 'start of cruise' fuel, the gross effect of aeroelasticity is to change r/t by a positive increment, the
magnitude of which is reduced by compensatory warp for values of Q below about 800 lb/ft 2 but slightly
increased for greater values. With 'design point' fuel, the gross effect of aeroelasticity is again to cause
a positive increment in tit. However, this is small and is only 0.11 ° at the design point. Therefore the
inclusion of compensatory warp purely to avoid a trim drag penalty would not be worthwhile, although
other considerations could still lead a designer to incorporate it. In particular, he might wish to ensure
that there was no large increase in the drag of the wing itself. The net effect of aeroelasticity on tit with
this fuel distribution is generally smaller than the gross effect, the increment being negative for values

17
of Q less than the flight plan value (550) and positive for greater values. Aeroelasticity causes a negative
increment in ~/, with 'end of cruise' fuel, the net effect being greater than the gross except at values of
Q very much greater than the flight plan value (450).

(b) 91 = 0, 02 = 9a = 1.0 (Figs. 12 to 14).

With this combination of gearings (corresponding to use of the two outboard pairs of controls only)
the elevator angles, negative at low Q and positive at high Q, required to trim the rigid aircraft are larger
than in case (a). Whereas in that case the gross effect of aeroelasticity was to produce an increment in
rh which was approximately constant throughout the Q range, in the present case the increment increases
algebraically with increasing Q. Thus, for example, we see from Fig. 12 that with 'start of cruise' fue!
the increment is small and positive for Q = 300 but increases such that at Q = 1000 the value of~h for
the flexible aircraft is about i~ times that for the rigid aircraft. A similar trend in the increment in ~/t
is found for the other two fuel distributions. Differences between gross and net effects of aeroelasticity
on qt are generally small but can be of either sign depending on the particular combination of fuel distribu-
tion and kinetic pressure.

(C) 91 = i"0,92 = 93 -= O(Figs. 15to 17).

With these gearings (corresponding to use of inboard controls only) the effects of aeroelasticity on
qt show opposite trends when compared with those in (b). Thus the increment in q, becomes increasingly
negative at high Q whereas in case (b) it became increasingly positive.

d~/
3.6.4. Incremental elevator deflections per g'd-n " On the 'classical' rigid tailed aircraft, d~I/dn is
proportional to the manoeuvre margin and inversely proportional to the lift curve slope with respect to
elevator angle of the tailplane plus elevator. The effects ofaeroelasticity are (largely) a change in manoeuvre
margin and a change in the power of the elevator to produce lift on the tail. Now, as Taylor has pointed
out 1, when calculations which introduce arbitrary constraints are performed for slender flexible aircraft,
variations in quantities which correspond to these 'classical' concepts of manoeuvre margin and elevator
power do not necessarily have any physical significance. Rather, the behaviour of the unconstrained
flexible aircraft has to be considered as a whole. Nevertheless one may still talk in a somewhat loose
way about certain effects of aeroelasticity as stemming from a change in one or other of these parameters,
by analogy with the 'classical' aircraft, depending on the way in which the aircraft behaves.
In this connection it is worth remarking that in Part 2 of Ref. 1 (Section 4(a)) intuitive reasoning based
on these 'classical' concepts was used to predict the variations in elevator effectiveness with elevator
location which are discussed in (a) to (c) below.
(a) 91 = 92 = 93 = 1'0 (Figs. 9 to 11).
The values of d~l/dn for the flexible aircraft are lower in absolute value than those for the rigid aircraft
for 'start of cruise' fuel but higher for 'design point' and 'end of cruise' fuel. Hence the decrease in
manoeuvre margin is dominant for the 'start of cruise' configuration but dominated by the decrease
in elevator power in the other two cases. The effects remain largely 'in step' in that the differences between
the rigid and flexible aircraft values of dq/dn do not vary markedly as Q varies.
--- 93 = l"O(Figs 12 to 14).
( b ) 91 = 0 , 9 2
Here, with only the controls on the more flexible outer portion of *.he wing being deflected, the loss
in elevator power is more marked but the difference between the flexible and rigid aircraft values of
dq/dn still shows no very rapid increase with increasing Q.
(C) g l = 1"0, 92 = 93 ----- 0 (Figs. 15 to 17).
Since aeroelasticity causes little loss of power for the inboard control the decrease in manoeuvre
margin is dominant and the absolute value of d11/dn is lower for the flexible aircraft: the difference from
the rigid aircraft value decreases somewhat as Q is increased.

18
 3.6.5. Longitudinal distributions of shecu'force and bending moment (Figs. 18 to 32). It was decided
to restrict calculations of load distributions to the cases with control gearings gl = g 2 = g 3 = 1"0. This
decision rested on three points:
(i) It was recognised that a reliable reconstruction of the continuous loading was not practicable
over the rearmost parts of the aircraft.
(ii) It was assumed that the 'carry-forward' aeroelastic effects on the control-surface loadings would
be fairly loca!ised and not greatly affect the loading over most of the aircraft.
(iii) The use of differing control gearings has, as was seen in Section 3.6.2, little effect on the incidences
required, even for the flexible aircraft.
From the point loads Qi it has been possible to derive (approximately) the continuous longitudinal
distributions (over the complete aircraft) of shear force S(x) and bending moment M(x). The approxima-
tions made in this reconstruction were consistent with the methods used to derive the individual loading
vectors. Further, it was assumed that the nose undercarriage could be represented by a point mass of
1150 Ib, 56 ft from the nose, and that each main undercarriage could be represented by a point mass of
3750 lb, 109 ft from the nose. S(x) and M(x) are presented, for n = 1"0, 2.5 and -0.5, in Figs. 18 to 32.
The distributions could not be derived for the foremost parts of the fuselage. Likewise, a region rearward
of the last row of grid points in the case of M(x), or a station somewhat ahead of this in the case of S(x),
is not covered.
The greatest positive value Of M(x) is increased by aeroe!asticity, but in the overriding 2.5g case this
increase never exceeds 5 per cent. In the -0.5g cases, the considerable negative bending moment which
occurs about 130 ft from the nose is little affected. Generally, then, the effects of aeroelasticity on S(x)
and M(x) are small. However, one should not assume that more localised loading actions quantities
would be as little affected.

3.6.6. Elastic warp (Figs 33 to 36). The elastic warp acquired by the flexible aircraft (without
compensatory warp) is shown for a representative set of cases in Figs. 33 to 36. The fight and fuel con-
ditions are those of the design point : the elastic warp is shown for trimmed lg flight, with gl = g2 = ga =
1.0, in Fig. 33 and for 2.5g flight, with the three combinations of control gearings here considered, in
Figs. 34 to 36. The projection of the drawings is isometric, with the deflections being increased in the ratio
of 10:1 over distances in the reference plane. The undeflected shape of the aircraft in the reference plane
is indicated by dashed lines. In the present case, the deflections of the rearmost parts of the aircraft could
not be obtained and the elastic warp can therefore be presented only for the parts forward of the rearmost
row of grid points. The chordwise and spanwise variations of the deflections can be judged from the method
of presentation adopted for the starboard half of the aircraft, while a more immediate overall impression
of the elastic warp can perhaps be obtained from the port half.
In Fig. 33 it is seen that the deflections in lg flight, with gl = g2 = g3 = 1-0, are such that the wing
undergoes mainly spanwise bending. There is a small amount of wash-out, which leads one to surmise
that the elastic warp of the wing produces a positive increment in the value of rh. This effect is, however,
opposed by the effect resulting from the downward deflection of the nose. The forward parts of the
aircraft carry a negative increment of lift, which demands a compensating negative increment in ~h.
In this case the combined effect of these opposing trends is to produce a small positive increment in ~h
(0.11°).
In Fig. 34 it is seen that when all three controls are used the deflections for a steady 2.5g manoeuvre
result in mainly spanwise bending, though there is a certain amount of wash-out, especially near the tips.
Because of Zhe opposing influence of the negative incremental lift on the nose, however, the elevator
angle required on the flexible aircraft ( - 7.47 °) is close to that on the rigid aircraft (-7.30°).
When only the outer controls are used there is virtually no wash-in or wash-out (Fig. 35). In the absence
of incremental lift on the nose, one would expect almost equal elevator angles on the rigid and the flexible
aircraft. Therefore the fact that a more negative angle is required on the flexible aircraft, -14.41 ° as
against -11'88 °, can probably be attributed largely to the negative incremental lift on the nose.
The use of only the inboard controls results in a mode of deflection consisting of spanwise bending
together with considerable wash-out (Fig. 36). In this case the influence of the downward deflected nose

19
is overcome by the influence of the wash-out on the wing and the elevator angle for the flexible aircraft
is - 15.56°, compared with the rigid aircraft value of - 18-93°.

3.6.7. Comparison with results of other investigations. The present investigation is by no means
the first to seek to determine the effects of aeroelasticity on slender wing aircraft. Of those that have
been conducted in the past, that reported in R. & M. 34261 has been referred to several times and it has
been intimated that calculations have been made by B.A.C. and Sud Aviation. We now discuss how the
results of this Report relate to these previous investigations.
In the two-dimensional method of R. & M. 3426, spanwise deformability was ignored. It would be
hoped, then, that consideration of the present results would indicate She extent to which spanwise de-
formability does in fact influence the behaviour ofa 'Concorde'-like slender aircraft and also the variations
in the effects of aeroelasticity with variations in the spanwise locations of the control surfaces. However,
it must be remembered that the two sets of results relate not only to somewhat different aerodynamic
configurations but also to two fundamentally dissimilar aircraft as regards structural stiffness character-
istics. A quantitatively meaningful comparison is therefore not possible and we must be content with a
discussion in qualitative terms.
It has already been seen that the effects of aeroelasticity vary markedly with the spanwise positions
of the control surfaces and in Section 3.6.6 some physical explanation of this variation has been furnished
by consideration of the elastic warp produced. This warp consists of spanwise bending together with
an amount of wash-in or wash-out which differs with differing control gearings. There is only a small
amount of curvature of chordwise sections. By contrast, the fact that in the two-dimensional approach
there is no distortion of spanwise sections implies that the mode of deflection consists of curvature of
chordwise sections together with wash-in or wash-out, of which the spanwise variation is, for practical
layouts, quite gradual. There is, then, little similarity between the deflections calculated here and those
presented in R. & M. 3426 and it seems that if an attempt were made to apply the two-dimensional
approach to a 'Concorde'-like design there could be little confidence in the results obtained.
The two-dimensional approach predicted an increase due to aeroelasticity in the magnitude of the
elevator angle per 9 ; the magnitude of this increase became larger at higher kinetic pressures. The present
results exhibit similar trends when only outboard controls are used. When all three controls are used
the increment can be of either sign, depending on the mass distribution, and its magnitude can increase
or decrease with increasing Q. The use of only inboard controls produces an increment such that the
magnitude of the elevator angle per g is decreased by aeroelasticity: the increment itself decreases in
magnitude with increasing Q. This progressive divergence from the trends exhibited by the two-dimen-
sional results, as the control gearings are varied to decrease the contribution of the outboard controls,
is consistent with the progressive development of elastic warp of a character very different from that
of the two-dimensional mode of deflection.
A sounder basis for a direct comparison of the present results with earlier work is provided by an
unpublished report of the British Aircraft Corporation which presents results of aeroelastic calculations
for a configuration substantially the same as that considered here. In the B.A.C. investigation an 'assumed
mode' approach was adopted, with aerodynamic loads due to distortion specified by influence coefficients
derived by the method of Pines, Dugundji and Neuringer. The rectangular grid employed divided the
planform into about 100 areas but did not extend over the whole of the forward fuselage. Structural
characteristics were defined by a matrix of structural-stiffness influence coefficients for 121 points on the
half-aircraft. Both the aerodynamic loads due to distortion and the structural characteristics were
therefore defined in rather greater detail in the B.A.C. calculations than in those of this Report. In problems
as complex as these, where accuracies are inherently difficult to determine, it is not very profitable to
attempt to explain every apparent discrepancy between results achieved by different people using different
methods. One can hope, however, for the absence of any gross discrepancies which would suggest a
fundamental error in one or other of the methods, or some incidental errors of computation. Bearing
such considerations in mind we may compare some of the results for the control angles to trim and to
manoeuvre, as determined here and as determined by B.A.C. for the same rigid aircraft cg margins and
(approximately) the same weights. (The fuel distributions assumed by B.A.C. are not known in detail.)

20
The results to be compared are presented in Tables 1 and 2.
The results for control angles per 9 (Table 2) are considered to indicate a good measure of agreement
between the two investigations. In general, the B.A.C. results show a somewhat larger variation of the
aeroelastic effects with control location.
The results for control angles to trim (Table 1) do not compare so satisfactorily, although it is to be
noted that all the angles involved are quite small. The differences in the values of 11t for the rigid aircraft
are largely attributable to a difference in the pitching moment at zero lift. The R.A.E. results consistently
indicate smaller increases due to aeroelasticity in rh. A possible explanation is that the Lees method,
applied to the complete planform, overestimates the down load due to deflection of the nose and thus also
overestimates the extent to which the destabilising moment due to elastic warp of the wing is counteracted
by incremental down load on the nose. Such considerations underline the need for experimental sub-
stantiation of methods of determining the loading on slender configurations due to arbitrary distributions
of elastic warp.

4. The State of Development of the Method.

The method presented here was first conceived when it became apparent that the neglect of spanwise
deformability, implicit in the method of R. & M. 3426, was invalid for a 'Concorde'-like slender layout
incorporating a thin wing in conjunction with a discrete fuselage. To provide a direct follow-up to the
results of that report, the present method has been used to determine static aeroelastic effects on an early
projected version of the 'Concorde' at its cruising Mach number of 2-2.
It has seemed worthwhile to develop the method in a general form and, with the help of the experience
gained in an initial application, to consider its potentialities and the difficulties which must be overcome
before it can become a practical design technique.
The principle of superposition has been employed so that linearity of the aircraft's structural and
aerodynamic properties has been tacitly assumed. A fully three-dimensional description of these pro-
perties can be achieved by the matrix approach and so, in principle, the method is applicable to any
aircraft, the structure of which is analysable in such a way that a flexibility influence matrix can be found,
and for' which there exist theories to provide the aerodynamic influence matrices. The theory has so
far been developed in detail only for symmetric flight conditions; while there are apparently no difficulties
of principle which would prevent its extension to more general conditions it is felt that the practical
problems in its application would be quite formidable. In the subsequent discussion of the developments
which are necessary we will therefore assume that we are seeking to establish the method as a design
technique for symmetric flight conditions only.
In any method which employs discrete-element approximations for continuously varying quantities
it is desirable to know the number and distribution of elements necessary to obtain an adequate approxi-
mation. Thus, in the present method, one wishes to be able to make appropriate choices of the sets of
points to be used in the calculation of the structural and aerodynamic influence coefficients and the
component loading systems. Some basic considerations are discussed in Appendix A, where it is suggested
that extensive numerical investigations would be necessary to determine the optimum relationship
between the sets and that, further, the most profitable choices for any particular problem might have
to be found by numerical experiment. In the single application of the method conducted so far the matrix
of aerodynamic coefficients was derived by 'hand' computation using the method of Lees 2. Even for
the modestly sized matrix used this was a tedious process so that although Lees's method appears potenti-
ally well suited to the task one would wish to programme it for automatic computation before attempting
to use it systematically in the present method. A similar proviso could be attached to the incorporation
of most other aerodynamic theories. Some of the other computations, involved in the derivation of the
transformation matrices cg and ~ and in the conversion of distributed loadings to poinMoad vectors,
also proved rather laborious when performed by hand.
In view of the above, it appears that although the numerical work described in this Report has
established the general practicability of the method it has not been extensive enough to develop it to the
stage where it could be offered as a routine design procedure of immediate and universal applicability.

21
However, as far as the authors are aware, this stage has not been reached by any method currently
available (in the U.K. at least): the tendency has been for designers to develop ad hoc procedures appro-
priate to their own particular problems. Thus, in an assessment of relative merits the present method
would not suffer, on the score of incomplete development, in comparison with those used elsewhere.
Indeed, it should have some advantage in that the general framework here described affords considerable
scope for developing a fully automated and widely applicable method of assessing quasi-steady aeroelastic
effects.

5. Concluding Discussion.
Early concepts of a slender near-delta supersonic transport aircraft involved designs which were to
a large extent 'integrated', with the wing providing considerable volume and the fuselage being regarded
as a lifting component. For this type of design the stiffness of spanwise sections would be large compared
with that of lengthwise sections ; therefore it could be considered plausible to ignore spanwise deformations
and to make aeroelastic calculations on the assumption that the aircraft behaved as a beam, subject
only to longitudinal bending. A method based upon this assumption was described in R. & M. 34261.
However, the layout selected for the Anglo-French 'Concorde' project features a thin wing in conjunction
with a discrete fuselage and it was clear that for such an aircraft the neglect of spanwise deformations
was not justified. Accordingly, it was necessary to formulate a 'three-dimensional' method of determining
static aeroelastic effects. A matrix formulation of the problem was selected and the method was first
developed in a general form and then somewhat specialised foi"its application to a 'Concorde'-like slender
aircraft in symmetric flight at M = 2.2.
The method is intended primarily for application at a fairly advanced stage of design. I't is assumed that
the stiffness characteristics of the structure and the mass distribution will then be specified in considerable
detail, and that correspondingly detailed basic aerodynamic load distribution data will be available from
experiment or theory. It is further assumed that the incremental aerodynamic loading due to deformation
will be calculable, in the form of a matrix of influence coefficients, from a linearised theory appropriate
to the configuration and flow regime under consideration. One may then make suitable choices of the
various sets of points to be employed and derive the rigid aircraft load vectors, the structural flexibility
and aerodynamic influence matrices, and their associated transformation matrices. Starting from this
representation of the problem, the method is able to provide direct solutions which are 'exact' within
the limits of accuracy of the computing processes employed. In this respect it is superior to alternative
methods which usually involve iteration or the adoption of a limited number of assumed modes of
deformation. As regards computational economy the present method, in a fully automated form, should
compare favourably with its rivals.
Results of the numerical application to a slender supersonic transport aircraft have been presented
and discussed. They include the calculated incidences and control angles to trim and to sustain quasi-
steady manoeuvres, the longitudinal distributions of shear force and bending moment, and the deforma-
tion (elastic warp) of the aircraft.
As calculated by the present method, all of the effects in question are quite unspectacular at flight-plan
values of the kinetic pressure. The trim and manoeuvrability results reveal no tendency towards control
reversal up to kinetic pressures well in excess of these values. The spanwise position of the control surfaces
has a marked effect; in the case of outboard controls the elevator angle per g is increased by aeroelasticity
whereas in the case of inboard controls it is reduced. These results are ~onsistent with the elastic warp
produced. When only outboard controls are used there is little wash-in or wash-out, the mode of deflection
consisting mainly of spanwise bending together with a downward deflection of the nose, whilst the use
of only inboard controls produces a pronounced curling-up at the wing tips, i.e. the spanwise bending
is combined with considerable wash-out of the outer parts of the wing. The longitudinal distributions
of shear force and bending moment are little affected by aeroelasticity: the maximum value of the latter
has been increased in the 2.50 case by up to about 5 per cent.
Comparison with the results of R. & M. 3426 shows that the 'two-dimensional' method presented
there is not applicable to a 'Concorde'-like design and may give completely misleading results, especially

22
when inboard control surfaces are used. A comparison with the results of an investigation by B.A.C.
for a very similar design, but using a different technique, shows that the two sets of results agree well
as regards the elevator angles per g but exhibit differences in the elevator angles to trim.
This Report, wherein the principles behind the method are expounded and a preliminary application
is described, marks the completion of the first phase of the method's development. From the remarks
made earlier in this discussion and in Section 4 it is concluded that the potentialities of the method are
sufficient to justify the expenditure of the further effort which would be necessary to develop it into a
routine design tool with wide applicability. Automation of many of the processes involved would be the
next obvious step, since this would make possible an extensive application of the method to a variety of
designs and flight conditions. Only in this way could the potentialities of the method be assessed.

23
 LIST OF SYMBOLS
Note:
(i) Loads, incidences etc. are defined at discrete points. Suppose that all sets of such points are numbered
as they are introduced ; then we denote the rth set by Y~ and the number of points in Zr by hr. A typical
member of any set of quantities is indicated by the suffix i. Hence from the symbol A~, say, we cannot
deduce which set of points contains the point where A~is defined, nor does the suffix i indicate any particu-
lar point in a set. Thus the various suffices i in any sentence or matrix equation are unrelated.
(ii) A column vector of quantities defined at the points of the set I~, (or 'in Er') is denoted by {Ai}~,
for example.
(iii) Matrices, with the exception of column vectors, are denoted by script characters, e.g. 6e.
(iv) In the following list all symbols refer to quantities appropriate to the flexible aircraft: the corre-
sponding rigid aircraft quantities are denoted by a superscribed bar, e.g. ~9~ from Q~.

CL Lift coefficient
Li Load at a point of E 2 : lb
M Mach number
M(x) Longitudinal distribution of bending moment : lb ft
Mo MF etc. Moments about moment reference station corresponding to component loadings
{Qi}c, (Qi}f, etc.
Pi Incremental load due to elastic warp at a point of E 3 : lb
9_ ½p Vz Kinetic pressure: lb/ft 2
Oi Load at a point of E 1 : lb
Q? Value of Qi at the design point
Ag-~ Q i - ~?i Increment in load due to aeroelasticity : lb
Vector of aerodynamic loads for CL = 0.1: Ib/Q
{Q,}f Vector of inertial loads due to fuel (per 9): lb
Vector of inertial loads for the aircraft with zero fuel (per 9): lb
Vector of aerodynamic loads due to compensatory warp: lb/Q
Vector of aerodynamic loads due to rate of pitch (per incremental 9) : lb/Q
Vector of aerodynamic loads due to incidence (per radian): lb/Q
Vector of aerodynamic loads due to elevator deflection (per radian): lb/Q
S(x) Longitudinal distribution of shear force : lb
V Speed of flight : ft/sec
W Aircraft mass: lb
Z o Zv, etc. Total loads due to loading systems {Qi}c, {Qi}~ etc.
ar Coefficient of rth component loading (equation (13))
co Reference chord : ft
9, Gearing of rth control
k, Ratio of amount of fuel in rth tank to maximum capacity of tank

24
 LIST OF SYMBOLS--continued

n Total normal-acceleration coefficient

/'/r Number of points in 12r

q Rate of pitch: rad/sec
x Distance aft of nose of aircraft: ft
xi Distance of a general point aft of nose of aircraft : ft
x', Distance of a general point ahead of moment reference station : ft
Distance of centre of gravity ahead of moment reference station: ft
wi Deflection at a point of Z2 : ft
w? Value of w~ for design-point loading
Transformatlon matrix between ei and w~ (equation (6))
g Transformation matrix between L~ and Qi (equation (3))
Trxnsformation matrix between AQ~ and P~ (equation (7))
J Unit matrix
Unit column vector in I24
Matrix product ~NcgSeg
Aerodynamic influence matrix

Structural flexibility influence matrix

Zr rth set of points
Incidence: rad
Elastic incidence at a point of 124
Value of c~for n = 1.0
q Elevator deflection : rad
th Value of t/for n = 1-0

25
 REFERENCES

No. Author(s) Title, etc.

1 A.S. Taylor and •. Effects of longitudinal elastic camber on slender aircraft in steady
W. F. W. Urich symmetrical flight: Parts I and II.
A.R.C.R. & M. 3426 (1964).
2 D.E. Lees .. .. Aerodynamic influence coefficients for delta wings in steady
supersonic flow.
Aircraft Engineering, 31, No. 359, 9-12 (1959).
3 D. Williams .... •. Solution ofaeroelastic problems by means of influence coefficients.
R.A.E. Report Structures 169 (1954). A.R.C. 17439.
4 E . G . Broadbent .. •. The rolling power of an elastic swept wing.
A.R.C.R. & M. 2857 (1954).
5 J.J. Foody and L. Reid •. The solution of aeroelastic problems on wings of arbitrary plan-
form by matrix methods.
J. R. aero. Soc., 59, No. 540, 843-846 (1955).
6 W.A. Tucker and •. Characteristics of thin triangular wings with constant-chord
R. L. Nelson partial-span control surfaces at supersonic speeds•
NACA T N 1660 (1948).
7 G . T . S . Done . . . . .. Interpolation of mode shapes: a matrix scheme using two-way
spline curves.
Aeronautical Quarterly, X V l , Part 4, 333-348 (1965).

26
 APPENDIX A

On Choosin 9 the Sets of Points lEr.

Fundamentally, one desires to choose the sets of points lEt so that a certain accuracy may be achieved
with the greatest overall computational economy. In practice, the pursuance of this goal in any given
case is made difficult by the existence of features peculiar to that case, including the aircraft's configuration
and the methods selected for performing aerodynamic and structural analyses, as well as by the freedom
of choice which one has at several points in the procedure. Detailed discussion of such topics is beyond
the scope of this Report and we content ourselves with discussing in this Appendix some of the factors
which determine the accuracy of the present method and deriving some basic principles which guide the
choices of the various sets of points employed.
Since the net loading on a free-flying aircraft is a combination of various locally additive or subtractive
load systems, it is virtually impossible to assess the general standard of accuracy attainable in its estima-
tion. Clearly, however, the accuracy of this net loading will improve as the accuracies of its components
improve and therefore it is both convenient and instructive to consider the accuracy of an arbitrary
individual loading system. In all that follows, three assumptions are made:
(i) The load system on the hypothetical rigid aircraft is known precisely.
(ii) By taking a sufficiently large aerodynamic influence matrix, the load due to elastic warp may be
derived as accurately as we please.
(iii) An analysis of the aircraft structure, idealised in some way, has provided a matrix 6a* of flexibility
influence coefficients for a set of points Z2*. The matrix 6e will be formed by selecting a sub-matrix of
6a* comprising the rows and columns corresponding to a subset, Z2, of E2*.
We recall that an individual loading system for the flexible aircraft, {Q~}], is derived from the corre-
sponding system for the rigid aircraft, {~9~}], by the addition of a system of aerodynamic loads due to
elastic warp, {AQ~)] (equation (1)). The accuracy in (Qi}] is most easily discussed by retracing the steps
through which {AQ~}] is determined. That is, we recapitulate the steps of the method, as given in Section
2, but in 'reverse' order. The steps are
(a) The derivation of loads in ZI from loads in Za, using matrix ~ .
(b) The derivation of loads in Za from incidences in E4, using matrix ~.
(c) The derivation of incidences in E4 from deflections in Z2, using matrix ft.
(d) The derivation of deflections in E2 from loads in Z2, using matrix 6a.
(e) The derivation of loads in Z2 from loads in El, using matrix 8.
We now consider each of these in turn.
(a) A fundamental point is that Z1 must cover the whole of the region of the aircraft which is covered
by Za. In the example presented in this Report the various sets of points are common to all the calculations
of the aeroelastic modifications to individual loading systems, and each set covers the whole aircraft.
However, in cases where an aircraft could be considered as consisting of discrete components, such as
the 'classical' fairly high aspect ratio, tailed aircraft, certain load systems would act over only a part of
the aircraft and cause significant load due to elastic warp over only a part. It might then be profitable to
use different sets of points for different loading systems. For any system, Za must cover the part of the
aircraft where deflections cause significant loads and so E1 must also cover this area. Thus, for instance,
for the aircraft considered here, the rigid aircraft load due to control deflection acts only on the part
rearward of the hinge line (at supersonic speeds) but a set Z, which covered only this area would be
inadequate.
Also, it can be seen that the density of points in Za should be at least as high as in Z1 since otherwise
there is a degree of indeterminateness in transforming the loads in lE3 to loads in El.
(b) The set IEa which is necessary to determine the load due to elastic warp with the required accuracy
and detail is dependent on the aerodynamic theory employed, the aircraft configuration and, possibly,
on the mode of deflection. The choice of E3 is the central step which determines the accuracy which will
be obtained, and which guides the choices of lE1 and lE2.

27
 (c) The choice of a •3, associated with a particular aerodynamic theory, fixes a corresponding Y'4.
Then the choice of Y2 depends on the requirement that one can determine the incidences in 524 sufficiently
accurately from the deflections in 522. This may be influenced by the technique employed to derive the
incidences. In the method used here they are derived from the deflections at neighbouring points and
there is no attempt to deduce the overall deflected shape of the aircraft during this process. An alternative
technique would be to find this overall shape by curve or surface fitting, by a method such as that of Done 7,
and from it deduce the local incidences.
(d) Consider two influence matrices 5e and .9°' corresponding to sets of points E 2 and E~, such that
Y'~ contains 522 as a subset. Then if the loading were in fact a set of point loads in Y~2 the deflections at
the point~ of Ig2 would, by assumption (iii) above, be the same whether they were calculated by using
5¢ or ,5"'. We may, then, say that this step alone does not introduce any errors in addition to those inherent
in the structural analysis for the aircraft and those produced by the approximation to the distributed
loading by a set of point loads in a particular Z2.
(e) To avoid introducing indeterminateness in the transformation of loads in Z 1 to loads in £a, the
density of points in £~ should be at least as high as in 212.
From the foregoing discussion we see that, from considerations of the transformation of loadings,
Ig3 should be at least as dense as Z1 which should itself be at least as dense as Z 2. However, the density
of 22 has a lower limit dictated by step (c).
It is clear that, even for a given configuration, in a fairly narrow band of flight conditions, much work
is needed before one can determine the most appropriate choices of the various sets of points. As regards
the choice of 2;3, a large body of calculations performed for various 23's (within a single aerodynamic
theory) and for various configurations could indicate the accuracy in step (b) alone. Although in principle
this step determines the accuracy of the whole process, one still has a good deal of freedom in choosing
Ex and Ez. It is possible that an extensive investigation on fairly general lines could provide a clearer
indication of the optimum relationships between the sets of points. However, it is suspected that one
would still be only part way to determining the most suitable sets for any particular problem. The feasibi-
lity of the routine use of the method may well depend on the success of attempts to provide a highly
automated computational process which would allow one to investigate, by simple numerical experiment,
the most profitable choices.

28
 TABLE 1

Comparison of R.A.E. and B.A.C. Results for Elevator Angle to Trim at M = 2.2.

F l i g h t c o n d i t i o n s : s t a r t of c r u i s e ; W g 190 000 lb, Q ~ 670 lb/ft 2, rigid a i r c r a f t cg m a r g i n 0-049c0.

Control gearings B.A.C. R.A.E.

g l = g2 = g3 = 1"0 (q~) rigid - 0-75 ° 0.80 °

(All c o n t r o l s ) (q,) flexible +0"75 ° 1"36 °

ANt d u e to flexibility 1.50 ° 0-56 °

91 = 0, g2 = g3 = 1.0 (Nt) rigid -1.15 ° 1"30 °

( O u t b o a r d c o n t r o l s only) (N,) flexible + 1"43 ° 2.75 °

ANt d u e to flexibility 2-58 ° 1"45 °

91 = 1'0, 92 = 93 = 0 (~h) rigid - 2.30 ° 2.10 °

( I n b o a r d c o n t r o l s only) (Nt) flexible + 1'26 ° 2.75 °

Aqt d u e to flexibility 3.56 ° 0.65 °

29
 TABLE 2

Comparison of R.A.E. and B.A.C. Results for Elevator Angle per g at M = 2"2.

Flight Start of cruise: W ~ 190000 Ib, End of cruise: W ,,~ 152 000 lb,
conditions: Q ,~ 670 lb/ft 2 Q ~ 455 lb/ft 2
Rigid aircraft cg margin 0.049c o Rigid aircraft cg margin 0.052c o
Control gearings
B.A.C. R.A.E. B.A.C. R.A.E.

91 = 02 = 93 = 1"0 (d~l°/dn) rigid -4.47 - 4.07 -5"7 - 5.08

(All controls) (d~l°/dn) flexible -4-30 -3.83 -6"18 - 5.52
t.,o Ratio (flexible/rigid) 0-962 0.94 1"084 1.087

0~ = 0, 02 = ga = 1"0 (drl°/dn) rigid - 6-88 -6-6 - 8.77 -8.25

(Outboard controls only) (dq°/dn) flexible -9.17 -7.7 - 11-58 - 10.3
Ratio (flexible/rigid) 1.33 1-165 1"32 1.249

91 = 1"0, 02 ----ga = 0 (d~l°/dn) rigid - 12.2 - 10.6 - 15"37 - 13.2

I n b o a r d controls only) (drl°/dn) flexible -8.13 -7-6 -13"1 - 11.9

Ratio (flexible/rigid) 0-667 0.717 0"852 0.901
 i loft
i,,, Sm

<..._

~ ~
FIG. 1. General arrangement of the aircraft.
i I¸
t IOCti
Srn

+ 5trucJcttr~l grid points +/~* ++~ ~--~

@Suppor~po,nt, /¢'~++ i 1 1 ~I

I /
CO

I/
'FIG. 3. Arrangement and numbering of the fuel tanks and elevators.
31
 5 /\
/
2

/ Altitude
~{ x 103
70
km \
\
-Zo
6S
30 40

0 4i 60 :x: f.~ $0 iO0 IZO 14(

60 %
.IB
-I
-,.,,
55
t~ 150 -16

/ 50

/
t~L(x) .14
4s
I00 I
300 400 ~ 500 GO0 700 800
I ~OO IOOO
Q Ib/f'L z

/
75

50

?5

~ ~
/ 2p ~ 30 40
FIG. 5. Altitude versus kinetic pressure, M = 2.2.

ZO 4.0 60 SO lO0 120 14.0

el{.

FIG. 4. Lengthwise distribution of aerodynamic

loading components.
 7
I I I t

]~
.....
.....
Rigid czircrefL
Fltxible alrcr<tfl.
Flexible aircraft with compe.sa+.ory wcrp

5 I I I
4.

FlighL
I
plan

ta.J
"~'x I 30 4-0
L"-, I i
30o 400 500"~...600 700 SO0 900 IOI
Ib/ff. z '':''" ...

-I -!

-z

FIG. 6. 2, a n d dct/dn. ' S t a r t of c r u i s e ' fuel, 91 = 0, FIG. 7. st a n d d~/dn. ' D e s i g n p o i n t ' fuel, gz = 0,
92 = Y,~ = 1.0 for 2," gl = 1-0, g2 = 93 = 0 for g2 = 93 = 1"0, f o r s t ; 91 = 1"0, 92 = g3 = 0 f o r
dct/dn. do~/dn.
 Rigid aircraft J
..... Flcxibl~ ~ircr~ft
I Flexible aircref~ wif.h compensatorywQrp I ° f ° 1
f
fJ J
I I

Riqid aircrofL
I I I J
5 I

.... Flexible aircraCt

Flexible aircraft with compense~oryw~rp ~0 4O
! I

I I S00 400 /., g00 900 I000

d~c°/dn
Flight
II I
plan
votive t I

I
~~'x Flight
plan
VoIU.~

0 _A .I
300 400 ~O0 700 800 SOO 10)0
Q
500
-I

FIG. 8. ~t a n d d~/dn. ' E n d o f c r u i s e ' fuel, 9z = Oo

92 = ga = 1"0, f o r ~t; 91 = 1.0, 92 = 93 = 0 f o r
do~/dn.

FIG. 9. ~h a n d
I
d~l/dn, g l = g2 = ga = 1.0, ' s t a r t o f
c r u i s e ' fuel.
t
iI
 I I I I

Rigid aircr~f~
" Flexible aircraft I I I i I

..... Flexible aircraft, with ¢ompensa(or¥ warp I Rigid aircro~t ~

Flexible aircrQf~.
Flexible aircraft wit.h compensa~.or~jwa~p ~.
°a I

01--~
-I I I I~,° I z.'/~l 3? KH~mz 4.p
300 400 so~" 600 700 800 soo I000
zoo 400//,:,'5001 600 700 800 900 iooo
-I
-I
-2
-Z
/// ¢,gh~: I
//; plan I i /
-3
S -3
,,,"
,
wlue I
I t
!
-
/ ~ -
I--
-4
-4.

-5
-5

-6 /./

-6 I /~'/

/
-7
//
-7
'// I I
¢s
-8 i''
¢
-8
-9

FIG. 10. ~/t a n d d~l/dn. 01 = g2 = g3 = 1-0, ' d e s i g n FIG. 11. qt a n d dq/dn. 01 = g2 = g3 = 1.0, ' e n d o f
p o i n t ' fuel. c r u i s e ' fuel.
 "2

-2
-4

-4

L~ -6
-8

-8
-tO

-IO

-IZ

-14

-i4
-16

-16

FIG. 12. r/t a n d d~l/dn. 91 = 0, 92 = 93 = 1"0, ' s t a r t FIG. 13. r/t a n d d~l/dn, g l = 0, g2 = g3 = 1"0,
o f c r u i s e ' fuel. ' d e s i g n p o i n t ' fuel.
 0
18

-?_

-d-

-6

-8

-LO

-I~-

-14

-16

FIG. 14. r/, a n d drl/dn, g l = 0, g2 = g3 = 1.0, ' e n d of c r u i s e ' fuel.

37
 I I I I T
Rigid aircraft
.... Flexible aircraft
Flexible aircraft with cotnpcnsatorv wcrp
4

0 A
2O
I
~f~/3,0 k,/." 4O
I
300 400 •/~ 600 700 8OO 900 I000
Q Ib/fL z

-2 I I

7
Fright
/'/~ plan
V~lUe
,/
-4
//
ii I f°.

-6 /// /
°J
.f
I

f
/
/

'i/
-8 d~7O/dn /' /"
/
/
-I0
/ J
/
/
/
/
-12

i,I'//
/
-14-
/
/
-IG /
/
/
-is1
;/ /
-ZO

-22 /
-?-4

FIG. 15. q~ a n d d~/dn, gz = 1.0, g2 ~--- 93 ---- 0, ' s t a r t of cruise' fuel.

38
 6 I I I I I
R~gid aircraf~
Flexib e aircraf~ /
..... Flexible ~ircraft with caxpensa~ory warp , /
4

0 ~ - '
~,o //~,o kN,/,,,~ 4o
300 400 so{ 600 700 . aOO soo I000
Q Ib/~c~,z
-2

{ I
Fiigh~
ph~n
vaLu~
-6
I
f
i,*

,.'1' dq°/dn .,"1"" /

y
-8 d

-lO
// /
-IZ

-14 /
/
/ "
/
/
-18 ./

-20

-22

-24.

FIG. 16. qt and dq/dn, at = 1.0, 92 = 9a = 0, 'design point' fuel.

39
 ca

-4

-6

-8

-a0

-[2

-14

-18

-20

FtG. 17. qt a n d drl/dn. Ox = 1"0, g2 = g3 = 0, 'end of cruise' fuel.

40
 4'0-
10. - 160
s®
s~ Ib ,xlO z
lbxl0 ~ ZO-
lo z,o ~ ~ 3o 40 -80
0

' , , ' , \ ', I~L, ' I

-10, ~o 60 _.so ,\ ,oo I '~.-_'~ 140

--80
• - 80 -P.O
-Z0

Rigid ¢ircr~ft ] I~ Ri9id aircraft [

.... Flexible aircraft
..... Flexible aircraft Flexible¢ircraft with compen~¢{orywarp
Flexible aircrGft with compensatorywarp

800 12 1600
M(:¢) ~.,, ~C~:)
600 ~ ",~. 1200 ~,
4-
Ib ~ctxIOs KNm / "~, Ib f~x,O: IcN "at
• 800

Z-
:00/ -400
/
lO 20 ~ 30 40 zo "., 30 40
Ii i l -.,%~
~'o ' 4~ ~ ' 8~ ,oo ,ao ~,~o o ZO 4.0' 6; 8'0 100 IZ0
~ '
140
~FE ~c 4:£

FIG. 18. L o n g i t u d i n a l d i s t r i b u t i o n s of shear force FIG. 19. L o n g i t u d i n a l d i s t r i b u t i o n s of shear force

a n d b e n d i n g m o m e n t . ' S t a r t of cruise' fuel, gl = a n d b e n d i n g m o m e n t . ' S t a r t of cruise' fuel,
g2 = g3 = 1.0, Q = 670 lb/ft 2, n = 1'0. gl = gz = g3 = 1.0, Q = 670 lb/ft 2, n = 2.5.
 s~
ZO-

Ib x I0 ~'
8o / I0.
• 40
s~
I0-
- 4.0 Ib xl03 kN
"~N.,

0
"'" "-""=" EO "m 30 I160 -v' ~'o :o
" I I I II I ?r/I I i

'
zo 40 " ~ , "~.'-.
6o 8o x ~', Io0 I~o/,/'
,~/' 140
~.~- -
- -40
-10
~::-~ .. .... /7
-I0

I Rigid aircraft
--$0
Flexible aircraft
- ~ - - ~ Flexible aircrcft with compensatoryWarp
-ZO

I Rigid aircr~f£
..... Flexible ¢ircrafL
Flexible aircraft with cornptns~tor~
I,o 6" ,800
4
M~
m~
-400 ----
I[', ftxlO s
kNm . ~ . 4
Z N"X'- \ ibf~xlOS

,o
T .
20I ~n, "-'~,~,0
"~ "1 i
~0 I
a-
~N¢ ~0
, 140 00
40' 60' BO' ~ \ I?-0

-?.
--200 ,\ /
~ '\ o ~"
I
20
" I0
I I
40
I
60
zO
I
80
"m
I
30
II
I00
I
120
40
"~
140

- - 400 ~ '

-4 FIG. 21. Longitudinal distributions of shear force

FIG. 20. Longitudinal distributions of shear force
and bending moment. 'Design point' fuel,
and bending moment. 'Start of cruise' fuel,
gl = 92 = 93 = 1.0, Q = 4501b/ft z, n = 1.0.
9,_ = 92 = 93 = 1.0, Q = 670 lb/ft 2, n = - 0 . 5 .
 40-
-160 ;'0
s~ s~ - 80

!bxlO ~ kN
Ib x 103
20- -80
10
-40

..,p I I I ~ ~ 0 .~ ZO ~n 30 /40
2O 140 i , ~u.~. , ! , 11 i ~' i ,
- " zb 40 "~. 60 80 ,oo ,~/ ,,o
--80
-Z0-
-I0-

--160
-40 -SO
"ZO -
[ Z Rigid °'ircrafL ]
Flexible aircraft
Flexible aircraTL with compen~i.oryw~rp ~ Rigid etircrcLft I
t.~

I~ - -1600
i
.... Flexible eircrctf+.
Flexible.aircr~f(, wi~ compcns¢~ryw~p

8-
Ib FLxI0~
/ x, M
Ib ~¢£xl0s kN mc,,f "---,~. ,.

0 - % . i . I ~I ~
-900 I I I I I I
~'0 40 60 80 "¢~.~ I00 120 140
4.. "~'~.
- 400 -2oo "-~..~.
-2 ".,~ ,
zO ~ 3.0 "9
0 ..~ I t I I I l i I I I
ZO 40 60 80 [00 120 140
.j

FIG. 22. Longitudinal distributions of shear force FIG. 23. Longitudinal distributions of shear force
and bending moment• 'Design point' fuel, and bending m o m e n t . 'Design point' fuel,
gl = g2 = g3 = 1.0, Q = 4501b/ft 2, n = 2.5. g l = g2 = g3 = 1.0, Q = 450 lb/ft 2, n = - 0 . 5 .
 40-
-160
s~
Ib x I0z ~H

20-
IO- 80
- 40

lb x 103 KH J 20 " ~
I0 30 J~, 40
~ . , 30 40 "/' zb ' , ' -; \ ', I ~k ,
, I I I II I I 60 8o Uoo I ~ ,zO i~.o
~o ~o ~o ~0 '% ,~o ,~o id0

-I0
--40

l--Rigid (zircrcz~t
. . . . . Flexible czircrcf~. I
-&0
-80

-160
-\
-40
-800 /.~--~..
M®
J. . . . . . Rigid aircraFE
Flcxibia aircraf{
r
4.-
+o / Flexible aircraf~ wiLh compons~t.oryw~rp

Ib F~.xlOs - 1600
4.00 " "
m~
Z- + \ 8
-1200

IbCq:xl0: kN "m
I0 ZO m 30 40 -800
I I [ I II I I I
zo 4'o d so too ~zo t4o 4-
-400

e0 m 30 40
1 t 1 i
i i i
2O
•,r .FE

FIG. 24. Longitudinal distributions of shear force FIG. 25. Longitudinal distributions of shear force
and bending moment. 'Design point' fuel, and bending moment. 'Design point' fuel,
gz = g2 = g3 = 1.0, Q = 5501b/ft z, n = 1.0. gl = g2 = g3 = 1.0, Q = 550 lb/ft 2, n = 2.5.
 ;'O-
lO- -40
s~
S(X)
Ib x IO] kN Ib x 103 KN
I0 I0 20 m ~ 30~ 40
, I I I "~ I I I I

-40, ~10 ~,~%~." ~0

20 rn :30
|
--4.0
/,/
20 40 60 80 IO0 12 140 -I0-

-I0' -SO
- ?.0

--80 I Rigid aircraft I

-?.0 . . . . . Flexible aircrM~.
Flexibleaircraft with compens¢torywarp
--Ri9id ~ircraft I
. . . . . Flexible oircraf£
. . . . Flexible~rcrMt with compensatoryWarp

4.
6 SO0 . / ~ " - .
M® -
- 400 M~
Ibf~xlO-~ KNrn
it-
4
Ib f~.xlO' kN,. / '~.
1 0 2 0 \~\ 30 4-0
20 40 60 80 ~ I00 I,P--O 14.0 2-
-ZOO
--200 ~
"2-
"\. / I0
I I I
zO
I
"m 30
II I
4-0
I

--4.00 \ ~ 40 ,o 8; ,oo ,to

-Z

FIG. 26. Longitudinal distributions of shear force FIG. 27. Longitudinal distributions o f shear force
and bending m o m e n t . 'Design p o i n t ' fuel, and bending m o m e n t . 'Design p o i n t ' fuel,
91 = 92 = g3 = 1.0, Q = 5501b/ft 2, n = - 0 . 5 . 9z = 92 = gs = 1.0, Q = 670 lb/ft 2, n = 1.0.
 40-

~0-
.160

kN
s~
?O-

IbxlO3 KN
I0.
/
-80 -40

20 -m 30
0 % ,
2O i~o 20 +o ~.~ 60 so too IZOl/ i~,o
~. x~t //

-'80 --+0 -~ ~.. .7

-~0- -lO-
+N.

.'160 --80
-+0 -ZO
I Rigid aircraft l
Flexible aircraft
- - Rigid ~ircraft warp ..... Fle~ibl=oircraft with compensatorywarp
..... Flexible aircraft
I ..... Fle~ibe aircraft with compens¢tory 4-
c~

M ( . ~ -400
. > s-~ __.
- 1600 Ibft,ld KN'm f ~ ~ . . ,,

- I~'00 d
8-
Ibf£xlOs KN "m
- 800 ~0 40 60 80 ~'\I00 IPO 14-0

--:~00 "~.~
-+00

~o ~ ao +o - -400 "~,,. /
I --tl I
2O 20 +; 8; ,oo llo ,40 -t

FIG. 29. Longitudinal distributions of shear force

FIG. 28. Longitudinal distributions of shear force
and bending moment. 'Design point' fuel,
and bending moment. 'Design point' fuel,
gz = 92 = #3 = 1.0, Q = 6701b/ft 2, n = - 0 . 5 .
91 = 92 = 93 = 1.0, Q = 6701b/ft 2, n = 2.5.
 •" ' ~" 40 -
i0-
_,0 - 160
s~ s(,~)
Ib x 10 3 kH ~ "~'~, Ib x i 0 3 kN
• ~p zo ~, "~, 3o 4o 2o- .J "%.
0 iI~, I I .I I_ I I ~\ I I I
ZO 40 60 80 \\ I00 leo 14.0

-10-
,0 \,k 0 ' , ~? 7
i 410
,
-%a; 4~ 60 so ~ a to0 t40
--80 x .F.E I 0 0 ~ ~

--80 -P.O -
-~0
, %
I Rigid aircraft I
. . . . . FlexibleaircraT~ - -160 " ~
Flexible~ircr~f~with compens~£orywarp -4o \
~igid ~ir,ro~t
i - L - ---Flexible ~ircr~f£
Flexible aircrafLwiLh compensatory warp
• /"L--- "-
--..I 800 " ~ ~" -2000 . ~. , = ~ .

M~
-1600 ~f '~,
4~
IbfLxl0" kN m Ibf'LxIOs
- 400
/7 \,
Z-
-P.O0 - 800 /
/ \
ZO rn 30 4-0 4
I 4-; 6; t 8; I, , I , -400 /
I00 IZO 140
J,'CTO 20 m 30 qO
0
20 40 60 80 I00 IZO 140

FIG. 30. L o n g i t u d i n a l d i s t r i b u t i o n s o f s h e a r force FIG. 31. L o n g i t u d i n a l d i s t r i b u t i o n s o f s h e a r force

a n d b e n d i n g m o m e n t . ' E n d of cruise' fuel, a n d b e n d i n g m o m e n t . ' E n d of c r u i s e ' fuel,
01 = 92 = 93 = 1.0, Q = 4501b/ft z, n = 1.0. 91 = 92 = g3 = 1.0, Q = 4501b/ft 2, n = 2-5.
 ?...0-
-8O

I0_
kN

-40
/
ZO m 30

-I0

--~0
-Z(

I Rigid aircraft
Flexible aircraft. I
~-'--'-- Flexibleaircvaf+,wl;~hcompens*to,y warp

-z00 ~ ¢ - ~ - ~ . ~ . . . ~
M~
IbftxlQs k Nrn " ZO
" I0 ZO~%~" 'm 30 40
0
ZO 4.0 60 8~'~. i00 tZ0 t40

--200 x ~ /
"Z-

--4°° j_/
-4

FIG. 32. Longitudinal distributions of shear force and bending moment, 'end of cruise' fuel,
91 = 92 = 93 = 1.0, Q = 4501b/ft2, n = -0"5.

48
 \
\
\

I ~ i / // 1

J
/

\
\
\
\

FIG. 33. Elastic warp, 'design point' fuel, gl = g2 = 03 ~--- 1.0, Q = 550 lb/ft 2, n = 1"0.

.~ I ~ j r /

- \

FI~. 34. Elastic warp, 'design point' fuel, 91 = 92 = 93 = 1.0, Q = 550 lb/ft 2, n = 2.5.

49
 ~'N x

\N

FIG. 35. Elastic warp, 'design point' fuel, 9a = 0, 92 = g3 = 1.0, Q = 550 lb/ft 2, n = 2.5.

\
\
\
b. \
~ ,..i- j"
ft

\
\

•,,. \
\ N
\ \
...... ~3

FIG. 36. Elastic warp, 'design point' fuel, g, = 1.0, g2 = 93 = 0, Q = 550 lb/ft 2, n = 2'5.

50
P r i n t e d in W a l e s f o r H e r M a j e s t y ' s S t a t i o n e r y Office by Aliens P r i n t e r s (Wales) L i m i t e d

D d . 135646 K.5.
 R. & M, No° 35,

(C) Crown copyright 1969

Published by
HER MAJESTY'S STATIONERY OFFICE

To be purchased from
49 High Holborn, London w.c.l
13A Castle Street, Edinburgh EH2 3AJe
109 St. Mary Street, Cardiff CFI 1JW
Brazennose Street, Manchester M60 8As
50 Fairfax Street, Bristol nsl 3DE
258 Broad Street, Birmingham 1
7 Linenhall Street, Belfast BT2 8hV
or through any bookseller

R. & 1VL No. 3573

SBN 11 470232 2