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SPACE FRAMES --STATIC ANALYSIS
VIBRATION ANALYSIS AND DESIGN OPTINISATION
STUDIES OF SPACE FRAMES
I - STATIC ANALYSIS
BY
S. K. TIWARI
A Thesis
Submitted To The Faculty of Graduate Studies
in Partial Fulfilment of the Requirements
for the Degree ·
Master of Engineering ~
McMaster University
May, 1968
~~STER OF ENGINEERING (1968) (Mechanical Engineering)
McMASTER UNIVERSITY Hamilton~ Ontario.
TITLE: Vibration Analysis and Design Optimisation Studies
of Space .Frames - Static Analysis
AUTHOR: Sanat Kumar Tiwari, B.Sc. (Banaras Hindu University)
B.E. ·(Hons.) (Saugor University)
SUPERVISOR: Professor J. N. Siddall
NUf·1BER OF PAGES: vii 9 · 108
SCOPE AND CONTENT:
An oblique four bar structural model with fixed member ends,
being the most general building block for space frames, is analysed
for establishing its influence coefficients, using the Finite
Element i'~atrix ~1ethod.
Experimental techniques for measurement of the flexibility
influence c.oeffi ci ents of the mode 1 are described.
-Experimental results have been compared against analytical
ones ..
ii
ACKNOWLEDGEMENTS
It is a pleasure to the author to be able to record his
gratitude to Professor J. N. Siddall and Dr. M. A. Dokainish for
their interest, guidance and many valuable suggestions.
The author feels indebted to the Department of Mechanical
Engineering and the Canadian Armament Research & Development
Establishment for the opportunity to participate in the project.
Further~ the author wishes to express his appreciation to all
who directly or indirectly offered help at various stages.
Finally, the assistance of Mrs. Anne Woodrow in typing the
manuscript ts thankfully acknowledgedo
iii
TABLE OF CONTENTS
LIST OF ILLUSTRATIONS v
ABSTRACT vi
INTRODUCTION 1
THE PHYSICAL MODEL 12
THEORETICAL ANALYSIS 12
The Mathematic Model 14
Analysis 15
EXPERIMENTAL ANALYSIS 25
RESULTS AND CONCLUSIONS 35
FIGURES 43
APPENDICES
I. Computer Programmes 58
II., Analytical Flexibility Matrix 71
III .. Experimental Flexibility Matrix 85
IV. Percentage Error r~ati rx 99
iv
LIST OF ILLUSTRATIONS
FLO~~ CHART (Computer Programme)
HISTOGRAMS (Results)
FIGURES
21
37
43
1 Overall Picture of the Set Up 44
2 Details of Displacement Transducer 45
3 Details of Ball Joint for Loading Frame 45
4 Overall Picture of Angular Measurement Arrangement 46
· 5 Detail of Telescope 46
6 Installation of Devise to Eliminate Rotation 47 Error in Linear Displacement·
7 Devise to Eliminate Rotation Error in Linear 47 Displacement Measurements
8 Discretised Mathematical Model 48
8a Model Dimensions and Material Properties 49
9 Top Plate and the Composite Element 50
10 Error in Angular r~easurement 51
11 Sign Conventions 52
11 a Tenns in Matrix R 53
12 Resolution of Linear Displacement 54
v
ABSTRACT
This research p~ogramme has the general objective of
establishing analytical techniques for analysis of indeterminate
spatical frames and shells under dynamic loading, and the design
optimisation of these structures under the constraints of dynamic
loading. Although techniques developed should have wide
applicability, emp0asis will be placed, for experimental and .
illustrative purposes, on structural configurations common to
machine structures.
Recent success of the finite element matrix method and
progress in the field of nonlinear optimisation provide a rational
basis for the synthesis of space frames, with emphasis on configur
ations common to mechanical engineering structures.
For the initial stage of the project a highly redundant oblique
four bar space frame was selected to investi~ate into the nature of
·:.Problems involved in the optimisation of generalised space frames
subject to dynamic constraints.
The present work relates to the st~tic analysis of the frame
including a theoretical analysis based Qn the finite element approach
and experimental determination of the influence coefficients.
vi
The average percentage error ·;n actual measurement varies
approximately from four percent for larger flexibility influence
coefficients to about ten percent for smalller ones.
Related studies will examine the dynamic analysis of the
structure, and the optimisation prob1eme
vii
INTRODUCTION
This research programme has the general objective of establishing
analytical techniques for analysis of indeterminate spatial frames
and shells under dynamic loading, and the design optimisation of
these structures under the constraints of dynamic loading. Although
techniques developed should have wide applicability, emphasis will
be placed, for experimental and illustrative purposes, on structural
configurations common to machine structures.
This present work relates to the first stage of this programme
in which the problem. is explored by examining a simple discrete
element space frame with generalized characteristics. More
specifically, this thesis is concerned with the static analysis
of the structure, including a theoretical and experimental
determination of influence coefficients and stress-load transfer
matrices. Related studies· will examine the dynamic analysis of
the structure, and.the optimisation problem. The following
discussion reviews· the overall problem.
1
2
Design systhesis essentially is an evolutionary spiral process
involving a complex feed back interrelating the fields of creativity,
past experience and tools of analysis. The role of the designer
is to optimise the value of a synthesis on the basis of some
criteria through a balanced exploitation of the evergrowing
information from all the three fields. The basic techniques and
the criteria of evaluation themselves need refinement from time to
time in the light of achievements in the foregoing areas.
The process has been marked with a rather slow progress in the
field of mechanical engineering structures, mainly due to their
inherent nature. These have not received the intensive
investigation that civil and aerospace engineering configurations
have. Analysis of mechanical engineering structures has perhaps
1 agged behind because they are. much more di ffi cult to categorise
than in the· other fields where a few highly typical configurations
can be recognised~ modelled and studied .. in a concentrated way. In
addition, the analytical tools available until lately have had
their own limitations.
1 ,2 These methods can be broadly classified into two divisions,
(1) Methods based on exact solution of the differential equations
describing the structure.
Apart from the difficulties in se-tting up and solving the
equations subject often to awkward boundary conditions, in case of
3
. complex structures the basic assumptions proved too restrictive
for accurate solution.
(2) Approximate methods involving mathematical approximations
can be subclassified into -
(a) Those based on finite difference procedures. These
are unsatisfactory in their formulation of boundary
conditions and convergence characteristics, and
(b) Those which approximate the stress or displacement
distribution by a series of analytical expressions and
hence are unsuited for complex structures.
The classical ·analytical tools are thus incapable of providing
an integrated approach even for structures of moderate complexity.
Hence it i~ not surprising that the practical design of mechanical
engineering structures has relied more on past practical experience
supported by rough analytical checks wherever possible, rather than
on the analytical tools.
The need for a tool well suited to complex configurations was
most acute in the aircraft industry where the designer had to work
within extremely narrow margins of practical expediency3. The
extensive efforts over years by numerous and often isolated workers
culminated in the finite element approach which is a major break
through from the past.
4
Based on structural as against mathematical approximation, the
method essentially seeks to idealise the structure into an assembly
of a finite number of discrete elements connected at a finite
number of points, and then proceeds to solve for the system response
on an exact mathematical basis. It is the finite connectivity
which permits a complex continuous structure to be analysed by a
system of algebraic equations and forms the basis of the technique.
Although earlier work was restricted to the field of aeronautical
engineering, recently results of applications to nonaeronautical
problems 4' 5' 6 and extensions to three dimensional discrete elements7
have been reported.
It is realised that, although the finite element technique
is still developing, it provides a unified approach to the analysis
of any type of structural assembly, from any field and with any
combination of one, two or three dimensional elements of different
characteristics4• It thus provides a reliable analytical tool
which is a prerequisite for design systhesis.
A rather limited amount of work appears to have been done on
the general problem of elastic vibration of structures and the
problem of optimisation under vibrational constraints, although
techniques for calculating the natural modes and frequencies of
lumped mass spatial structures are fairly well established for
essentially beam like aircraft structures, and to a lesser extent
5
the rectangular frames of civil engineering. The significance
of rotary inertia in spatial frames does not appear to have
been studied. Archer8' 9 has provided two useful new papers
in this field and has related it to the finite element stiffness 10 matrix technique. Hurty has developed a method for analysing
complex structural systems that can be divided into interconnected
components.
The concept of optimum design has registered a drastic change
since the advent of high speed digital computers. Ear 1 i e r, the
magnitude of computation involved acted as a deterent and a feasible
solution w~s.accepted in lieu of the optimum. With computers
·-·to handle the arit~metic, systematic design synthesis has become
a reality.
Very many general techniques of optimisation appear in the
literature that might be applied to structural optimisation. Most
promising are the Direct Search Method first suggested by Hooke
and Jeeves and further deve 1 oped by Flood and Leon 11 , the r~ethod 12 of Successive Linear Approximation due to Griffith and Stewart ,
and the. Random Method of Di~kinson 13 •
Minimisation of weight, weight stiffness ratio, cost, volume
for a homogeneous structure, etc. have been suggested as criteria
for optimisation of structures, but mjnimisation of _weight appears
to have been accepted as the most satisfactory one even though the
G
minimum weight design is not always the minimum cost design.
The optimisation of a statically determinate truss subjected
to single loading is a problem in analysis rather than synthsise
For strength design, member cross sections are proportioned to
develop maximum allowable stress for the required failure mode.
For optimum stiffness design based on minimisation of weight
per unit stiffness, stiffness being defined as the reciprocal of
strain energy~ the members should carry stresses proportional
to the square root of the product of the modulus of elasticity and
specific 111eight. The constant of proportionality i's based on
stiffness requirements 14
For a. given determinate truss under multiple load condition
the problem essentially remains the same. All the member cross
sections carry the maximum allowable stress, based on strength
or stiffness design, at least under one load condition. The
optimum design has come to be recognised'as a fully stressed
design.
In the case of indeterminate trusses~ for a given configuration,
applied loading and allowable stress, the cross sectional area
of the members and hence the weight of the structure are functions
of forces in the redundant memberso Sved15 has shown analytically
that under single load conditions the minimum weight structure is
always determinate.
7
Using the Lagrange multiplier·technique~ L~ C. Schmidt16
has
shown that under alternative loads numerous fully stressed designs
of an indeterminate truss existo Due to the prohibitive nature
of computations involved in arriving at the minimum weight he has
suggested two complementary relaxation methods to arrive at a
fully stressed design.
The beginning of the present decade marked a radical departure
in the approach to structural optimisation. It came to be accepted
as a problem in mathematical programming with Schmit17 as the
pioneer. Utilising the joint force and displacement formulation
of structural analysis as first proposed by Klein 18 , he has optimised
a fixed configuration three bar truss subject to three alternate
1 oads. He treated it as a problem in nonlinear programming by
adopting a modified steepest descent method designated as the method
of alternate steps. On encountering an inequality constraint:. v1hich
must be convex, the ~earch moves along a constant weight plane in the
feasible region until the constraint is again contactedo It then
steps back halfway, and then continues.to move along the steepest
slope .. On the basis of numerical results he concludes that in
terms of design parameter space the minimum we.ight design need not
be a fully stressed design lying at the apex of constraint hyper
planes ..
8
19:.20 . Subsequently 1n collaberation with Mallett and Kicher
he extended the above to the problem of selecting a suitable
configuration and material for the three bar truss& Various
optimum designs were compiled by changing the material or config-
uration~ one at a time in discrete steps. The best of all these
design was chosen.
Dorn et a1 21 have proposed a linear programming method which
selects the optimum combination of configuration and member cross
section from a wide classes of admissible trusses defined by a given
number of admissible joints connected in all possible ways by linear
members. The optimisation is based on a modified simplex method
capable of handling large number of equationsQ The results provide
an interesting study in the behaviour of optima due to change in load
and the height-span ratio of the truss. The configuration remains
the same for the load for a certain change in height-span ratio~~
ahd then alters~ as c< continues to change. Thus a continuous
spectrum is provided from which the value of a< giving the absolute
minimum weight truss and the configuration itself could be selectedG
Best22 has optimised a cantilever box beam by the steepest
descent'method .. It has one unique featureQ The partical
derivatives of stress and deflections with respect to the design
parameters are calculated by the finite· difference approximation
using the stiffness 'matrix~ which must be inverted to obtain the
9
deflections. To avoid the time ·consuming process of inversion
at every step he adopts an interative scheme to obtain the deflections.
Only the incremental stiffness matrix for a given change in design
parameter is calculated which, in conjunction with the previously
inverted stiffness matrix, rapidly converges to the required dis-
placements on iterationso This feature is said to substantially
reduce the calculation time. Constraints on stresses and deflections
are handled by a version of tbe reduced gradient method. His
solution is a maximum stress solution~ and thus forced to be on a
boundary.
The presentation of the structural synthesis as an unconstrained
minimisation problem by Schmit and Fox23 is uniqueQ It is based on
the method of solving linear simultaneous equations by minimising
the sum of squares of the residuals to zero. This expression is
set up for·the equality co~straints defining the stresses~ To this
is added penalty terms for violated inequality constraintsg which are
.all simple upper and lower bounds. The actual quantity to be
optimised~ the weight~ is treated as an inequality constraint~
requiring that the weight be less than an arbitrarily defined draw
down \\Ieight$ The problem is now an unconstrained optimisation
problem solved by a gradient method. It is repeated using progress-
ively lower draw-down weights until the optimisation function cannot
be made zero. This indicates that the draw-down w~ight is lower
than the inherent minimum weight. 'rhe method thus actually requires
10
a series of optimisations. It does not seem too applicable
to complex probiems; as the constraints must be expressed explicitly
in order to set up the residualsQ
equality constraints are ruled outQ
The implicit matrix form of
Razani 24 has proposed an unconventional approach using an iterative
technique ·in which areas are changed by successive increments from
an initial feasible solution so that each member is fully stressed
in at least one of the several possible load conditions. This
gives a feasible solution forced to be on a boundarya The true
minimum. may not be on a boundary if the stress is indeterminate.
The gradient projection technique has been successfully adopted 25
by Brown and Alfredo to optimise a portal frame and a two storey
single bay frame. The search begins at a feasible starting.point
until constraints are encountered& At this point the constraint
hypersurfaces are approximated by hyperplanes and gradient of the
objective function is projected on the line of intersection of these
planes. After a move along the indicated direction a correction
is indicated due to the nonlinearity of the constraint hypersurfaceso
The authors have proposed the use of only one design parameter for a
member as variable vJhi le the rest of the parameters for the same member
are expressed as functions of the selected one. As moment of
inertia of the members has a predominant effect on the behaviour of
the structure~ other. parameters are expressed as functions of moment
of inertia. Inspite of this simplification the procedure seems
11
too invloved for complex structures.
Young and Christiansen 14 have provided the first known optimal
structural design technique using vibrational constraints using
an iterative technique. Adjustment of the member area to achieve
a fully stressed design simultaneously with the required resonant
frequency characteristic is the main feature. An application to
pin jointed space .truss is includedo
12
THE PHYSICAL MODEL
For the first stage of the project it was decided to examine
a simple but highly redundant space frame with generalised
characteristics. An oblique four bar frame without symmetry and
~vith fixed member ends illustrated in Figure 1 ~"as selected. The
obliquity of the bars ensured assymmetry of static and dynamic
response. The four bars are welded at the base to a half inch
thick aluminum alloy plate at a~24 inch square spacing. The top
ends of the bars are brought close together and welded to another
half an inch thick aluminum plate at a square spacing of 2.5 inches.
It is extremely important for the accuracy of the results that
deflection of the bars at the ends fixed to the base plate are small
as compared to the relative displacement between points on the
structure due to applied load. To ensure this the base plate is
bolted firmly around each leg of the structure between 1.25 inches
thick steel plates. The point at which the external loads are
applied is located at the centre of the top plate.
THEORETICAL ANALYSIS
The decision was made to first examine the problem using
the finite element - matrix approach.
by this method is well established.
Static analysis of structures
During the earlier stages of
development the literature v1as in the form of a number of short
individual publications marked by variety of notations and seemingly
13
different approaches to the same basic technique, often applied
to specific situationsQ The most oft quoted work as the first
comprehensive presentation of the subject is by Argyris and Kelsey26
.
Results of a more recent survey of the finite element method with a
vie\-J to unifying the techniques and classifying the approaches to
derivation of element properties has been presented by Gallagher
2 7 b k b z . k . . 30 . t . 1 1 et al . The very recent oo y 1en 1ew1cz 1s par 1cu ar y
comprehensive and useful.
The finite element analysis of a general structure consists of
three distinct phases4 -
(1) Structural idealisation wherein the original structure
is represented as an assembly of discrete elements,
(2) Evaluation of element properties,
(3) Analysis of the discretised structure.
Phase 1 introduces into the anlaysis the first of the approx-
imations. Judgement is required to provide a discretisation capable
of reasonably accurate results but in general is not a difficult
prob 1 em.
The success of the method and its extension to new areas depend
almost entirely on the second phase. It is therefore not surprising
that the derivation of element properties of various shapes has
received such wide attention. TvJo .levels of approximations are
involved in the development of these properties - assumptions about
14
the essential element behaviour and secondly representation of
distributed stress and displacements in terms of nodal forces and
displacements 27.
The third phase like any other method of structural analysis
seeks to satisfy the conditions of equilibrium, compatibility and
force displacement relationship~ simultaneously. Depending on
\'lhether compatibility or equilibrium equations are utilised first,
the methods can be classified into the equilibrium or displacement
method and the compatibility or force method. The procedural
details of these methods are readily available in a number of excellent
1. . 28~29,30.
recent pub 1cat1ons •
The Mathematical Model:
The analytical procedure uses two mathematical models - static ..
and dynamic. The later is usually an extension of the static
model. The accuracy of analytical dynamic response depends on the
number of nodes selected for formulation of the mass matrix. The
amount of experimental work involved in determination of the influence
coefficients, however, increases in proportion to the number of nodes.
As a compromise each member was discretised into three equal lenghts.
For the static model the top plate was treated as rigid due to
its relatively small size and comparatively l~rge thickness. It \-Jas
ide ali sed into four rigid bars each jo"i n·i ng the centre of the top
plate to the point of intersection of a member axis with the plate.
15
The combination of the rigid element and the corresponding flexible
member element was treated as an integral finite element, and the
element property was formulated as such.
The static mathematical model thus consisted of t~tJelve discrete
elements. The arrangement of the elements in the physical
model is shown in Figure 8 •
Analysis:
The analysis was subject to the usual limitations of small
displacements and linearity of stress-strain and force-displacement
relationshipso The weight of the structural elements was neglected
as being small compared to their load capacity. In evaluating the
element characteristie:.matrix the deflection due to shear was neglected
The dfsplacement method was adopted as it is simple and
strai ghtfonvard to programme.
The basic assumptions used in this method are -
(1) Boundary displacements of adjacent elements are mutually
compatiblell
(2) Stresses in the elements due to the boundary displacements are
equilibrated by a set of forces at the element boundary in
the direction of the displacements, and
(3) Element forces are related tb the corresponding element
displacements by an element stiffness matrix expressed by
16
the matrix equatione
{ pi} :. [kil{Si} ( 1 )
where { Pl·} is the element boundary forces vector,
[ ki] is the element stiffness matrix,
{ b£} is the element boundary displacement vector~ ..
refers to the element number l
For all the elements treated as unconnected Equation (1) can be
vJri tten as { P} = [k] \S"} (2)
\·Jhere t P) r P£_}
= t ~.J PVII
and n is the total number of
elements in the structure
Also
[ ~] =
and { s}
S·t: 1 d-j .
b,J When elements are connected together to form a structure
Equations (l) can combined into a singl~ relationship. .. _-.:..
{ F} (3)
\vhere {F} is the structDral joint forces vector
tj~] is the structural assembly stiffness matrix
17
and is the structural joint displacement vector.
Element boundaries have the same displacements as the structural
joints to which they are connected if both are referred to a common
coordinate system. This can be expressed by
(4)
where [ ~ ] is the displacement transformation matrix. It consists
of zero and one as elements. If the displacements are referred to
different coordinate systems the relation (4) still holds but
e 1 ements of rna tri x [f.'] contain e 1 ements of the coordinate trans-
formation matrix. The work done in loading the structure by the
applied loads{F} must be equal to the internal energy. Therefore
from Equation (2) and (3)
But
Therefore
or
2 {SiH P}
2 { oT}[ ~~J{S}
~ { bT\ l ~] [~] {u} 2
{u.T}[~T]
~ { ll}[~T]f ~] [£3]{ u}
[ ~-r][ V<.] [fl { u.1
(5)
(6)
(7)
18
Comparing Equations (3) and (7) we see that
[ K] (8)
[ ~] is a sparsely populated matri xQ The matrix product
indicated by Equation (8) 11 therefore,·can be a very time consuming
step particularly for large structures. However, the structural
stiffness matrix [~] can be generated· directly by assembling the
element stiffness matrices already transformed into the structural
coordinate system. This method has come to be known as the 3:~33
Direct Stiffness Method
The assumption of rigidity of the plate and the selection of the
displacement method precluded the choice of more than one node on
the top plate. If more than one node is selected the flexibility
matrix will hav~ dependent displacement rows. Hence its inverse, 28
the stiffness matrix, does not exist Imposition of a one node
condition dictated the use of integral elements made up of flexible
and rigid components. The combination shown in Figure (8) was
achieved by the transformation of the stiffness matrix of the flexible
component BC by the equilibrium matrix of the rigid component CD
according to the expressions given beleN 3J~ 32 ...
[~ul - [K,,Jec - BD
' -\ T
[l~t2.] BD - [K12) .. [ Hco] (9) ec.
[ l/\22) BD - [1-\:o] [,, K '-2 j sc [ i1 ~~ f
19
where
BD is the composite element,
[ H c 0 ] i s the eq u i 1 i b ri urn rna t ri x of CD
·K K etc. are 6 x 6 submatrices of the element stiffnesses ~~.? B a.
in system coordinates.
The following computer programmes were used in the analysis.
Subroutine TRANF:
With position coordinates of member ends and components of a
unit vector along the member Y axis as the input, the subroutine
calculates the coordinate transformation matrix and checks it for
orthogonality. If this condition is not satisfied execution is
terminated after printing out an error message.
Subrountine STIFCO:
Area of cross section and moment of inertia are read in along
with the elastic modulii~ Depending upon the value of an index,
the output is a stiffness matrix in membe~·.coordinates or in system
coordinates, or is the stiffness matrix of a composite element.
Subroutine TRANQL:
This subroutine transforms the stiffness matrix by the equilibrium
matrix to provide the stiffness matrix of the composite element. In
20
case of shortage of storage space it can be used to provide the
12 x 12 stiffness matrix of a free element from the 6 x 6 stiffness
matrix of the same element when fixed at one end.
Subroutine ASSEMGL:
Once the stiffness matrix of all the elements has been worked
out the subroutine can assemble the structural stiffness matrix.
Main Programme:
The main programme utilises the subroutines to obtain the
structural stiffness matrix, inverts it to supply the flexibility
matrix, and checks the validity of inversion
The subroutine approach provides the more flexible and more
general progra~mes.
The scheme of computations is indicated as follows on the next
page.
b. I
ASEMBL . '
CHECK
I C{H~t~CS~TE ~-=:.;:.. - ~,----·
'E.LE.ME.NT
~ TR~~NQLI
'-~------~---------~b---------~1
1. S_.T.n.~ - O!JT!='UT ~ : - , uc~
22
For optimisation analysis a relationship between external loads
and resultant forces on member ends was required. The external
loads were to act at the apex of the structure. Hence only one
node at the top plate was required. The structure was therefore
idealised into four flexible members integral with a short rigid
element contributed by the top plate. Further the lower ends of
all the four members are fixed to the foundation, hence the structural
assembly matrix was a superimposition of submatrices for each
composite element. The structural stiffness matrix can be
expressed as
~~ -n . • T [ K] - L . [ HiJ [Ti}[ ~~2·1 L Ti}[ Hi']
i =a where [ ~22) is the element stiffness submatrix in member
6 X 6,
( 10)
coordinates
[ T] is the transformation matrix to system coordinate, 6 x 6,
[_ H J i s the eq u i 1 i b ri u m rna t r i x , 6 x 6 . 2 is the member number
If the displacements at the node are given by { u..} load by { f}
then -~
[~~]tF} 6.·~6 6)'\
The member end disp 1 a cements_ .. {d.} ~re given by
{d\}
and· exte rna 1
( ll )
( 12)
where {~]is a transformation matrix defined by expression (4)
23
For the present structure it is a column vector of four 6 x 6 indentity
matrices. Substituting (ll}in(l2) gives
[ (3] [ ~~-~] { F} ( 13) 2.4~6 G.r.G 6)'.~
Due to the special nature of [ (3 J~ for the present case we can
\vri te
{ d~i \ : . [ K .. e] { F} ( 14)
6)(.~ 6)(.G 6,..J
\vhere fdl}is the end displacement of the :/A member in system coordinates.
Transforming these displac~ments to the ends of the flexible component
( 15)
vJhere t d. if\ represents the di sp 1 a cement at the f1 exi b 1 e end of the
it~-J member, connected to the rigid part. Transforming { di-5} to
member coordinates to give { dif~"ryl. we have
t d. if }--m = [ T i 1 { Ji~} (16)
Substituting (14) and (15) into (16) w~ get
( 17)
Forces { Pa:} m at the ·free fl exi b 1 e end of each e 1 ement are given by
{ Pi}m -
Substituting (9) in (10)
..
[ ~~u] { o\if} m ( 18)
24
= 'J. T -a -~
[ k 2 &.] [ T i] ( Hi] ( K] { F} ( 19)
6')(..~ ~~~ 6~6 i&i'b '":;..t
The stresses in the member can be obtained by transforming
{ P2• }.~''~\. by a genera 1 transformation matrix [ R ] ~ Let i Si} be a
column vector of axial stress S~i and transverse shear stresses
S>'-yz and s.~zi ~ v-1here xvz. refer to member coordinates.
Then { s;} - [ R 1 { P·} l m (20).
~ )f..l 3xG 6 )(.a
Substituting ( 19) into (20) gives
= [ R] [ ~ ~ ._] [ "T i 1 [ H J r' [ K-1 ] { "F \ ~i'(G 6'li'tG ~-::tG, 6x~ 6~6 ~-.r.\
(21)
The exact nature of e 1 ements, in [ R] wi 11 depend on the member
cross-section. For a hollow circular cross-section it is given
by the following matrix.
' . l·R~CosS l • R~ S LV\ 9· ~Sine l
A I:z. Iy 0 R9
cos 0 -l-y l;z.
0 -· MA:t Sa-n 6 ~l~.y si'Y\ e ..,. R.@ St't,~ C) 0 (22) I-z · t Iy·t r~
MAz·Cose MAy (o~e RoCose 0 0
J:z• t Iy ·t. IIJ'o.
where the significance and sign convention for the terms in matrix
R are shown in Figures 11 and lla.
25
EXPERIMENTAL ANALYSIS
Both static and dynamic analysis may require the stiffness
or the flexibility matrix of the entire structure~ depending upon
the approach selected for subsequent analysis~ Experimental
compilation of the stiffness matrix would require measurement of
forces or moments at all the coordinates~ for a given displacement
at one of them~ while the displacements at all the rest are of zero
magnitude" It is obvious that direct measurement of the stiffness
coefficients would be extremely inconveniento Hence the only
a1ternative is the determination of the flexibility coefficients.
The experimental procedure generally would be to apply a known
force or a moment at a given coordinate i and measure the corresponding
linear or angular displacements at coordinate j Q The magnitude
of the displacement at j per unit force or moment at z gives the
coefficient zj of the flexibility matrix .. Inversion of the
flexibility matrix thus. obtained cans if desired~ provide the stiffness
matrix. Instrumentation was, therefore~ designed to measure flex
ibilities.
The loading technique was to transmit sufficient force at a node
to cause displacements of convenient magnitude without appreciably
stiffening the structure. The requirement was fulfilled by two
· parts of a split collar~ held together by countersunk screws, clamped
on to the member over a 3/4 inch length, at the point of load application.
26
A small gap between the split parts· ensured a friction grip between
the member and the collar which had an outer spherical diameter
of 2 inches. A split spherical shell having 2 inches inner and
3 inches outer spherical diameters~ was locked on to the collar so
that its axis was in any position within 30° of the member axis.
It carried two pins screwed at diametrically opposite locations on
the outer surfacee Load was applied to the pins. by the ends of a
· fine steel wire which supports a pulley in a vertical plane. A load
supported by the pulley gives rise to identical tensions equal to
half the applied load~ in each section of the steel wire leaving the
pulley vertically. These tensions were transmitted through an
intermediate system of pulleyss supported by an independent tubular
framework, to the pins in the same or opposite sense according as a
force or a moment was being applied. Theoretically, the arrange-
ment ensured that the two equal tensions· would give rise, at the node,
to a resultant force or a pure moment passing through the inaccessible
member axis. The cross hairs of a levelling instrument were used
as a reference to align the sections of the steel wire in the required
horizontal or vertical directions.
The load application to the pulley was achieved by means of a
pneumatic cylinder working off 100 psi air, through a sprang balance
calibrated at Oo2 lbs per division up to 100 lbs. Alternatively the
load could be applied directly by steel weights supported on a pan
held by the pulley.
27
The objective of the measurment of displacement was to
isolate the required component~ angular or linear, for measurement
while suppressing all the other componentso The points at which
displacements were desired were inaccessible~ The situation is
represented in Figure 12o The fixed point 0 is lying on a member
axis when the structure is unloadede An orthogonal right handed
coordinate system ijk parallel to the structural coordinate system
is eatablished with origin at 0 G is a point on the member axis
coinciding with 0 before displacement. Linear and angular displace
ments of G are defined respectively by vectors I and A with subscripts
indicating their components along corresponding coordinate direction.
The components are positive along positive coordinates. As point G I
is inaccessible, measurements are to be made at P fixed with respect
to ijk GH is a rigid linke
A plane ABCD always passing through H can be oriented perpendicular
to the desired coordinate axise H coincides with P before displace-
ment. The displacement of H from P to Q is given by~ where
-+ A~ R (23)
where R ( R.i , R~ $ Rk ) is the position vector of
PandA is small.
In terms of scaler components
28
o\i -:::. L . + A··R - At<: Rj ~ .J k
dj = L" J ·~ Ak Ri-At•Rk (24)
dk -:.:. Lk + A··R· 2 J - Aj · Ri
Suppose Li is to· be measured~ The surface ABCD is oriented
perpendicularly to the i axis at P and locked to H. ABCD has a
rigid body motion with GH but it can be ·considered first to move
parallel to itself from P to Q with displacement d and then at Q
take the required orientation A. The displacement transducer axis
is fixed and passes through P along the i axise The moving element
of the transducer derives its displacement from each phase of move-
ment of the surface ABCD, referred to above. The recorded
displacement d1 i can therefore be expressed as the sum
L i, -t A.j R k - A k RJ
+ Ak ( l. j i- A k · R i - A i · R k)
+ A. j ( s.. k +- A4.. ~j - Aj. ~i.)
(25)
It is noted that displacements due to phase II are of higher order
and can be neglected, generally
= (26)
29
It is obvious from Equation (26)that if the transducer axis coincides
with the i axis R ..0 and Rf~ wi 11 be zero and thus the effects of a 11
the other components are suppressed.
This may not always be possible. Care must be taken to
maintain the instrument either in the zi plane or the ilt plane so
that only one error term is effectiveo Further~ if surface ABCD
is not axactly perpendicular to the i axis~ phase I will contribute
two more error terms -tA~0 ~jand -Aj 0 ·dk ~where Aj"and Ak.o are the
components in initial angular displacement. of the perpendicular to
ABCD from i.
By cyclic permutation of the indices~ the expressions for the
other two coordinate directions can be obtained. The problem
of resolution of angular displacements will be discussed along with
the method.of angular measuremente
To achieve the requirements for linear measurements as discussed
above two clamps were designed as shown in Figure 6. The inner
clamp grips the member along a 1/2 inch length while providing an
outer spherical surface having a 2 inch diameter over which the
outer clamp can be fixed in any position~ The outer clamp carries
a ball-and-socket joint which in turn carries an optically flat first
surface mirror~off which the linear displacements were measured. The
mirror thus corresponds to plane ABCD~ The ball-and-socket joint
permitted the morror surface to be aligned perpendicular to the
30
required coordinate~ while the outer clamp allowed the transducer
axis to coincide with the coordinate axis along which measurements
were desirede The mirror was aligned perpendicular to the
coordinate axis by ensuring simultaneously that
(1) Reference lines on the base ot the structure and parallel
to the coordinate axis were in line with their image in
the mirror~ and
(2) A vertical plumb line was parallel to its own image in the
mirror~
The mirror provided an optically flat surf~ce and was used to align
the transducer axis with its image~ thus putting th~ transducer
perpendicular to the mirror and parallel to the desired coordinate
axis.
A capacitance type proximity transducer coupled through an
oscillator and reactance convertor to a cathode ray oscilloscope
was adopted as the linear measuring device. The transducer consists
of a fixed electrode. Any flat conducting surface parallel to
the fixed electrode can act as the moving electrode. Normally,
the moving electrode is fastened to the component whose displacement
is to be measured. In the present application the structure
had both linear as well as angular displacements whereas the proximity
transducer is designed to work when electrodes remain parallel while
31
moving towards or away from each other; When angular displace-
ments are present too high results are registered~
· To eliminate this defect the moving electrode was mounted on
·the transducer itselfo The spindle of the moving electrode was
supported jointly by two 2-1/2 inch x 1/2 inch x 5/1000 inch thick
stainless steel strips, parallel to each other~ which forced the
electrode surfaces to remain parallel during relative motion. This is
illustrated in Figures 6 and 7 o The spindle on the moving
electrode was kept in positive contact with the mirror during dis
placement by the steel strips~ which exerted a force on the structure
of the order of Oo02 lbs or less as against 0.2 to 0.4 lbs exerted
by an average dial indicator with a least count of 080001 inches.
The errors introduced due to friction were thus minimised.
The tr.ansducer system is based on frequency modulation of a
carrier wave. The capacitance of the electrodes is in parallel
with another fixed capacitance. The combination forms a series
resonant circuit with an inductanceQ The change in distance
between the electrodes~ due to loading of the structure caused a
c~ange in reactance in the resonant circuit which is used to change
the frequency of the signal delivered by the oscillator·. The
signal is amplified and detected to provide a proportional· D.C.
voltage which was metered on the oscilloscope. Unloading the
structure restores the initial. gap between electrodes. The transducer
32
can then be calibrated by the integral micrometer producing a
deflection on the oscilloscope of the same order as that obtained
due to the load~
to be eva1uatedc
The calibration enabled the displacement
The least count of the micrometer was 0~01
wm which could be further subdivided by the oscilloscope.
of 0.5 to 1.5 mm between the electrodes was usede
A initial gap
The resolution of angular displacement into its components was
achieved by a simple optical deviceo As the magnitude of rotations
involved was small~ vectorial resolution was possible. The optically
flat first surface mirror used for linear measurements was also
· used to view the image of an illuminated transparent grid of l/40
inch squares through a levelling telescope equipped·with optical
axis and base spirit levels together with vertical and horizontal
cross hairs. The grid and the telescope were so placed that the
line of sig~t was within 5° of the perpendicular to the mirror at
the point of incidence.
The cross hairs were made to coincide with a pair of lines at
right angles to each other on the grid. · Rotation of the mirror
due to loading of the structure had three components - one about an
axis perpendicular to the mirror and two about axes in its plane. The
first had no effect on the grid image as viewed through the telescope.
The other two caused a displacement of the grid image with respect
to the cross hairs. The vertical and horizontal displacement of
33
the grid was directly proportional to twice the rotation of the
mirror about the horizontal and vertical axes respectively.
Thus viewing the mirror along the i axis provides rotations
along j and k axes and that along j axis those about i and k axes.
Readings with these two settings provided all the three rotations
and values about z provided a checko
An additional optically flat first surface mirror, mounted
close to the objective of the telescope, could be used to view the
grid reflected by both the mirrorsg thereby increasing the optical
arm. The arrangement is shown in Figures 4 & 5 The
telescope "'as further equipped with a lateral displacement type optical
vernier , with a least count of 0.001 inches which could be used to
further subdivide the 1/40 inch grid. The set up could accurately
· 1 o 1 1 o-4 · prov1de measurements with a east count of • x rad1ans up to a
distance of 200 inches between the grid and the telescope along the
line of sight.
The linear displacement of the mirror which shifted the point
of incidence along the line of sight affected the readings as
represented in Figure 10 !l which is se 1 f exp 1 ana tory. The error
~ in the reading recorded for rotation of the mirror about an
axis in its own plane but perpendicular to the plane containing the
line of sight is given by
(27)
34
ignoring the change in the included angle ~ due to mirror rotation.
t·Je can determine d i from the equation (26). The correct reading
without the linear displacement of the mirror is
= 2. .. lL · A"k (28)
Therefore the percentage error
where LL is the distance bet\'/een the mirror and the grid
In the ltJorst case oti and LL'"Ak are of the same order. Hence the
percentage error is determined by 100 sin ct appr~ximately.
It is thus obvious that by increasing the distance between
the mirror and grid and decreasing the angle between the line of
sight and the perpendicular to the mirror, the percentage error can be
decreased. In· the event that errors are still large, the reading can
be repeated after shifting the mirror~ mounted on the telescope,
through 90°.
35
RESULTS AND CONCLUSIOr!S
The analytical influence coefficients, their experimental
values and the percentage error are listed in the Apnendices II-IV
in the form of a 1m,;er triangular matrix~ No attempts have been
made to measure the coefficients if the theoretical influence
coefficients were below the absolute value of 0.25 x 10-5 for
angular ones and OQ5 X 10-5 for the linear oneso
The coefficients are classified into linear and angular
measurements. Each of these are further subdivided into three
groups each according to the order of absolute magnitude of the
analytical coefficientse The results have been presented in
the form of his tog rams for each of the above groups.
Although some of the results are higher, in general the
experimental values are lower than the theoretical ones. The
largest single reason is apparently the error in load application.
For larger values the scatter in the readings is small. The
scatter in error for smaller values of influence coefficients
would indicate that limits of instrumentation have been reached.
The resultant error is dependent upon a number of other
independent factors like error in location of the nodes, deviation
of the instrument axis from the coordinate direction, errors in
taking observations, other personal errors - to name .a few.
Effect of each of these factors vary"about some average value in a
36
random fashion during experimentation~ Hence it is natural to
expect a more or less nol"mal distribution for the resultant error
\vhich is a function of all the independent factorsQ The tendency
is brought out clearly by the histograms.
Considering the order of maqniturles of the displacements
invloved~ it can be concluded that a reasonably accurate technique
fot"' measurement of the ·f"iexibility influence coefficients, both
linear and angular, for a generalised space frame has been established.
Proximity transducer has been successfully adopted for the measurement
of linear displacements in a component having linear displacement
coupled with rotationo Large frictional and application error
inherrent in the use of a dial indicator for measurement of linear
displacements have been eliminated.
---~.~-~~·-·~-·
-
l ! I I I i I I l l c
.. _ ~~--··-··~-------
------~·--~ ·-----
L_~l .~
~
37
' ! , .
..--;···~' IY
' 1,
~---~j r:t.
l 0
1----i
r;! !
: ;_n t(
I +
UJ ! i I
,__
__
__
_•i I
UJ ~
i :-0
:!. i
t-!
z !
UJ
j_ 0
~~
I I
!J tt 'UJ 0.
,..,. \.J
u z Ul :J _j lJ.. z a: 4 w
z
j\i') I Ct
0
L--t--0 ~-
·-1·-·----~---r
0 ~
i'.fJ
38
Li ;
()
tn
39
~----------------------------------------------~~
tU v
1(/l
z l-
)Jj 2
:> w
_
!
lL u
2
--
u.. rJ..
tL <
( U
i U1
0 2
u \
:J j I
~
'q
~
"!' 0 e r-t 0 -~
~
.. 0 lU ._, 2 ( rt
I 0 ! i_ 'J
lt\J
! I I I
l~ ,54~~
i ri
! w
I I
YJ ·~
~
f (
l \-
.j "Z
i w
l
v t~
<! l'
w
l 0-
1 1 l I~ ... ~
I I
I l l I ·~
.{-t-.') i
I I ! I~
L------------------------~~-------------~------~~------------~~------~~~ 'Z
" ~
('-\ -
I.·~
L..;, ~ --, __ I
->< .... ··0
\ C
'· U
) .
[-~
0 -~·-~..,
--/
"·' [.._
_
~
~~('J
(__) 1 ~-::~
L!_ ><
LL .
UJ 0
0 l;
l '-'-' <!J
u z
7 -4. ~
_j
::::J
\ 40
I I I i I r ! -1
_
j ~ ? i
I I I
a "\["
0 Nl
0 C\!
0 -0 0 -I
0 ('\j I 0 !'-f)
0 "\[
I
[(
c [1 .'\1 LL
w
w
0 ~ ~
z w
u (I(
w
Q...
! I I ! I iS' ~ C
• -
! X:
I UJ
-~
I u
(]c
z •.l)
0
! LLi
! I 0
; :)
L
t-_
j 1.1
!..'1 LL
u -
I
z l)
0 -.... :'* ,. ....
-u_ "!"
-
rr 0
<
li. c
l w
-.I
0 LJ
:._] 0
0)
v z
z <t
4 f{
I I ! I I I
0 --2-
··.,._j
0 M
0 ru 0 0
C)
I
0 (\]
I
{'_~tv~
-0 • ...... i
u: r{ j• I jJ
L.J 1"'1 ·\...;
/
-<', r ~-
L w
u C
( w
Q_
FIGURES
43
z
48
tOLA.N ( NOT TO SCALE )
X
FIGURE 8 DISCRETISED MATHEMATICAL MODEL
I ETC (j) ETC
TP
BP "AY z
0 1/il ..
" .... " G •
" ...
NODE POH~TS
D~SCRETE ELEMENTS
TOP PL.A..TE
BOTTOM PLATt:..
SYSTEM COORD~N/~TE.
A){E:S
49
,, ~---24 1 ~----31·35.. I PLAN
3=5"'6'-" ----~ l ( NOT TO SCALE )
-------:ss·es'-' --"~i
FIGURE 8a MODEL DIMENSIONS AND MATERIAL PROPERTIES
AREA OF CROSS SEC.TQON OfF
'ELEMENTS
MOMENT OF DNEFCnA or.:
'El'E M ENT cR6 SS .'.;\H2C.T~ O'-''
.....
.. .. ..
.. .,o
go" ..
1•0S11
II
O•S24
0 • ~2~ SQ.\N.
41 O· 0'3~~15 ~I"\~ •
4 o·o7&.73. iN .
iD ·;;oo ooo , , i..SS/IN:--
3,950,000
l6S/ H~~ I I
I t
50
""i:""'j'..,-,--::"'
)dq_ J ~\
\.
\ '\@
\
\
PART i~ L ;.0~t~
( NOT TO SCAi-E)
FIGURE 9 · TOP PLATE AND THE C0!'-1POSITE ELEt,1ENT
.,... _ _.___._.
---VT ETC
(2) ETC
" Q •
COMDOStTE ELEMENTS
TOP PLJ~TE BOtH-..JDARY
R~G~D cLE~~ENT
FLE r-~ BlE ELE~\ENT
NODE POiNTS
D~SCRETE ELEMENTS
I l
·I 1 I
-~----}) ~ --- 0
--- - -- 2 --Dz=-
-·--
FIGURE lO ERROR IN ANGULAR MEASUREMENT
....,-a
~4
·trvu a
M2
M3
C~ ~ETC
Ca C 2
c 2 c3
0) <) "
Q '"' Q
" .. " " .. 0
~VHRF::QR
~N~ f~AL ~o S\TlON
D~SPLACED PcS~TiON
P\S\Pl-ACE'D ANG\ULJ\t< PDSiT~ON tr~rH·luuT LUNEAR
D fi ~ PLAC.E" MEN-~-
Ch~OSS HA~R ~ NTt=RSECT!ON
.0,'~ Gf,n:> CCR~;;.(~5PO N D~NG
TO M 1 ETc
RE.(.ORCED l~~EAR. GR~O
D ~ 'SPLA.C.c M ~NT
, E RR.OR DUE TiJ "t'H E: LH,HEAR
D ~:SPLACt=M Ei'\ T OF T~t=:
\V\1 ~ R. RO R
LH~E"tk.R M ~~ReF~ D~SPLA<-EMEI'\T
ALONG £ A)'\5 II EGUAL.~ Li
51
I !
!
I ~ ~
I I
I I I I
I \
I I I I
I .I I
I •
e . P--~ ~z k .d-.
tl
~~~~,~~z !
FIGURE It SIGN CONVENTIONS
2
p)C. ETC. FORCES. AT MEM8Ei-<. END 2
!N X ETC DH<.E<.TQCN
M~t ETc ......
.... 1 2.
CORRESPONDiNG '"'40ME~T~
A-r €\"\ E~H3~~ END~
D q ~;~ E C -r 5 T R E .5 S E 'S A Lu ~~ G
c Cor~ PI N' AT E A "'E 5
T RAN S V E RS E: S T ~ E. S S E s
52
t
e
MAy
53
FIGURE 11 a TERHS IN f"1ATRIX R
.. Q"'
......
.......
Tt~& c l<(.N E' SS OF THE
t~ o E.. L 0\.~ C V l. a N DR U CAL
ElJE["-1 E~~T
.~ ~~ G L E B 'E ~r ~\ E EN TH E
POS~TqON V E~TOR OF
AN\f PO~NT ~~NO "V A')(tS
~~C MEN T OF AR'k?A GET ~4J 'E EN
THE ~AQ~~S VE"CTOR AND Al'\5
\{ A ~OUT T~~ E Z Ajt.U S
\
54
FIGURE 12 RESOLUTION OF LINEAR DISPLACEMENT
• • '1 0 ?..J!~
GH ......
ABCDH
p " ....
..... Q
R~GHT HANDED COORDaNATE:
S't STEM~ F~~ED BN SPACE
R~GiO LiNK Fi~ED 1"0 THE -OOt: PO~NT &: EQUAL TO R
FLAT SURFACE:, CAN BE lOCKED TO GH
COMPONENTS OF POSIT(ON
V Ec T<)R R OF H \M·R~T" G
A PO~NT F!/<.ED H~ SPACE,
CO~NCU)E NT ~~~~TH H BEFoRE
~ r's D « S PL.. A c E M E. NT
A7·qS PARALLeL To 2 A~\S.
~l
I
I
I J
REFERENCES
1. Przemieniecki ~ J. S.,: ur1atr·ix Structural ~.nalysis of Substtuctures~~~ AIAA JourG~ Vol$ 1:. No. l, Jan .. 1963, pp., 138-1470
2. Argyris~ J. H. 3 S. Kelsey: Modern Fuselage Analysis and Elastic Aircraft, ButtervJorths, 1963.
3. Turner~ t· .. :. J., R~ vJ. Clough, H. C. i·1artin~ Lo ~L Topp~ 11 Stiff'ness and Deflection Analysis of Complex StructureS 11
3 Jour~ of Aerono Sci.~ Vol. 23, No.9, Sept. 1956~ pp. BOb-824.
4. Clough!/ R. 1~ .. : 11 Finite Element :·1ethod in Structural t .. iechanics", Stress Ana lys ·is 9 ~·Ji 1 ey, 1966, pp. 85-119 ..
5. Zienkie~.:vicz~ O.C.: "Finite Element Procedures in the Solution of Plate ano Shell Problems 11
, Stress f\na·lysis 51 ~~i-ley:~ 1966, pp. 120-1440
6. Per·cy. J. H., T.H.HQ Pian 9 S. Klein~ D., R. Navratna: 11 Application of Matrix Disp~acement Method to Linear Elastic Analysis of She 11 s of Rev o 1 uti on 11 ~ A II\A 1J our I) Vo 1 . 3 , No . ll , Nov~ 1965~ pp. 2138-21450
7. Argyris~ J. H.: 11 f-latrix Analysis of Three Dimensional Elastic Media 11 ~ AI/\A Jour. Vol. 3 51 No. 1 ~ Jan. 1965$ pp 45-51 .
. 8. Archer, J.S.: 11 Consistent r'latr·ix Formulation for Structural Analysis Using Finite Element Techniques''~ AIAA Jour. Vol. 3~ No. 10, Oct. 1965, pp. 1~10-1918.
9. Archer:~ J. S.,: "Consistent f'iass r•1atrix for Distributed r,1ass Systems .. :. Jour .. Str" Div. 3 Proc. ,1\SCE, Vol. 89, St. 4, Pt. I~ August l963~ pp. 161-178.
l 0. Hurty, t~ ~ C.: " Dynamic Analysis of Structura 1 Sys terns Using Component Modes~·~ AIAA Jour., Vol. 3, No. 4, April 1965, pp. 678.
ll. Flood, i·1. t~. and A. Leon: "A Generalised Direct Search Code for Optimisation'1
, Mental Health Research Institute, University of f~1ichigan, Preprints 109, and 129~ f'·1ay 1963 and June 1964.
55
12.
56
Griffith, R. E. and R. A. Stewart,: 11 A Nonlinear Programmin~ Technique for the Optimisation of Continuous Process1ng Sys terns 11
, r·1anagement Science, Vo 1. 7, 1961 , pp. 379.
13. Gallagher, P .. J.: 11 t·1op-l, An Optimising Routine for the I.B.~1. 650", Can. G.E. Civilian Atomic Power Dept. Report No. R60CAP35, 1960.
14. Young, Jo W. and H. N. Christiansen: "Synthesis of Space Truss Based on Dynamic Criteria 11
, Jour. Str. Di v. , Proc. ASCE, ST 6, Dec. 1966, pp. 425-441.
15. Sved, G.: 11 The t,1inimum Height of Certain Redundant Structures .. , Austr. Jour. of App. Sci., Vol. 5, 1954, pp. 1-9.
16. Schmidt, L. C.: 11 Fully Stressed Design of Elastic Redundant Trusses Under Alternate Load Systems .. , Austr. Jour. of App. Sci., Vol. 9, 1958, pp. 337-348.
17. Schmit, L.A. Jr.: 11 Structural Design By Systematic Synthesis .. , Proc. ASCE, 2nd Conference on Electr. Comput., Pitts, Pa. , Sept. 8-9, 1960, pp. 105-132.
18. Klein, B.: 11 A Simple Method of ~1atrix Structural Analysis 11,
Jour. Aeron. Sci., Jan. 1957, pp. 39-46.
19. Schmit, L.A. Jr., and R. H. Nallett: "Design Synthesis in Multidimensional Space with Automated Material Selection••, Research Report, Engg. Design Centre, Case Inst. of Tech. Clev., Ohio, Aug. 1962.
20. Schmit, L.A. and T. P. Kicher: "Synthesis. of Haterial and Configuration Selection", Jour. Str. Div., Proc.· ASCE, ST 3, June 1963, pp. 79-10 .
21. Dorn, vJ. S., R. E. Gomory, H. J. Greenberg: "Automatic Design of Optimal Structures'', Jour. de Mecanique, Vol. 3, No. 1, March 1964, pp. 25-52.
22._ Best, G. C.: "Completely Automated Height Minimisation ~1ethod for High Speed Digital Computers••, Jour. Aircraft, Vol. 9, No. 3, May-June 1964, pp. 129-133.
23. Schmit, L.A., R. L. Fox: "An Integrated Approach to Structural Synthesis and Analysis 11
, AIAA Jour., Vol. 3, No. 6, June 1964, pp. 1104-1112.
57
. 24. Razani , R.: 11 Behaviour of Fully Stressed Design of Structures and It's Relationship to i"iinimum ~~eight Design", AIAA Jour., Vol. 3, No. 12, Dec. 1965, pp. 2262-2268.
25. Brown, D. M., H. S. Alfredo,: 11 Structural Optimisation by Non 1 i near Programming 11
, Jour. Str. Di v. Proc. ASCE, ST6, Dec. 1966, pp. 319-340 •.
26. Argyris, J. H., S. Kelsey,: Energy Theorems and Structural Analysis, Butterworths, 1960.
27. Gallagher, R. H., I. Rattinger, J. S. Archer: A Correlation Study of Nethods of Structural Analysis, AGARDograph 69, Pergamon, 1964.
28. Rubinstein, M. F.: Matrix Computer Analysis of Structures, Prentice & Hall, 1966.
29. Robinson, J.: Structural Matrix Analysis for the Engineer, Hi 1 ey, 1966.
30. Zienkiewicz, O.C., Y. K. Cheung: The Finite Element Nethod in Structural and Continuum Mechanics, McGraw-Hill, 1967.
31. Lives1ey, R. K.: ~1atrix t~ethods of Structural Analysis, Pergamon, 1964.
32. Tezcan,, S. S.: 11 Computer Analysis of Plane and Space Structures 11,
Jour. Str. Div. Proc. ASCE, ST2, April 1966, pp.l43-173.
'33. Turner, M.J., H. C. Martin, R. C.· Weikel, "Further Development and Applications of the Stiffness Method, AGARD Meeting, Paris, France, July 1962.
APPENDIX I
COMPUTER PROGRAt1t·1ES
58
NELEM
NJOIN
MI
MJ
SAM
SAMT
STIFF
TRANS
XP
XQ
YM
XL
ALONG
AREA
PINERT
YINERT
ZINERT
EMOD
GMOD
NO
IND
INDEX
NI
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
e••
•••
•••
FORTRAN PARAMETERS
NO. OF ELEMENTS
NO. OF JOINTS
JOINT NO. TO WHICH THE FIRST END OF THE ELEMENT IS CONNECTED
JOINT NO. TO WHICH SECOND END OF THE ELEMENT IS CONNECTED
STRUCTURAL ASSEMBLY MATRIX
TEMPORARY STORAGE FOR SAM FOR INVERSION
ELEMENT STIFFNESS MATRIX
ELEMENT COORDINATE TRAVSFORMATION MATRIX
COORDINATES OF THE FIRST END OF THE ELEMENT
COORDINATES OF THE SECOND END OF THE ELEMENT
COORDINATES OF THE ELEMENT Y AXIS
RELATIVE COORDINATES OF END TWO WITH RESPECT TO END ONE OF AN ELEMENT
ELEMENT LENGTH
AREA OF CROSSSECTION OF A MEMBER
MOMENT OF INRTIA ABOUT THE ELEMENT X AXIS
MOMENT OF INRTIA ABOUT THE ELEMENT Y AXIS
MOMENT OF lNRTIA ABOUT THE ELEMENT l AXIS
MODULUS OF ELASTICITY
MODULUS OF RIGIDITY
TRANSFER OF CONTROL PARAMETER
ADDITIONAL TRANSFER OF CONTROL PARAMETER
CONTROL PARAMETER FOR COORDINATE TEANSFORMATION OF THE ELEMENT STIFFNESS MATRIX
TEMPORARY STORAGE REQUIRED BY THE INVERSION SUBROUTINE
59
$JOB $IBJOB $IBFTC MOD c
· 003102S.K.TIWARI DECK
COMMON STIFF<20tl2tl2)t TRANS<50t3t3lt XP(3), XQ(3)t
60
1 YM(3lt ALONGC5Q), EMODt GMODt NOt INDt SAM( 54t54 ), 2 SAMT( 54t54 ), Mt( 20 ), MJ( 20 ), Nl( 54), MEM
c DIMENSION AA( 54 )
c C READ THE NO. OF ELEMENTS AND NO. OF JOINTS.
c c c
READ (5tl00) NELEMt NJOIN
READ THE NO. OF JOINT TO WHICH FIRST IS CONNECTED READ (5tl00) ( MI(I), I = lf> NELEM )
END OF ELEMENT
c c c
READ THE NO. OF JOINT TO WHICH SECOND END OF ELEMENT IS CONNECTED READ (5tl00) ( MJ ( I ) , I = lt NELEM )
c C READ THE ELASTIC CONSTANTS.
READ (5tl01) EMODt GMOD c C CALCULATE THE COORDINATE TRANSFORMATION AND THE C STIFFNESS MATRIX FOR ALL THE ELEMENTS c
DO 10 MEM = lt NELEM C CALCULATE THE COORDINATE TRANSFORMATION MATRIXo
CALL TRANF<MEM> c C CALCULATE THE ELEMENT STIFFNESS MATRIX IN C SYSTEM COORDINATES
c
CALL STIFCO<MEMt INDEX ) 10 CONTINUE
C START ASSEMBLY. CALL ASEMBL
c C INVERT THE ASSEMBLED STIFFNESS MATRIX FOR C THE STRUCTURE c
CALL INVMAT ( SAMTt NMt NMt l.OE-08t IERRt Nl )
c C CHECK THE ACCURACY OF THE INVERSION c
DO 13 I = lt NM DO 11 J = lt NM
11 AA ( J ) = SAMT< J' I )
DO 13 J = lt> NM X = 0.0 D 0 12 K = 1 ' N~~
12 X = X + SAM< JtK ) * AA( K )
SAMT( J t I ) = X 13 CONTINUE
DO 15 I = lt NM X = 0.0 DO 14 J = lt NM
14 X = X + SAMT( I 'J > IF ( ASS( SAMT< I t I l-1.0 } .LT. Oe1E=03 .AND.
1 ASS( X) eLTf/1 OelE-03 ) GO TO 12 WRITE ( 6' 110 ) It SAMT( I t I ) ' X
15 CONTINUE c C OUTPUT
WRITE ( 6t 115 ) DO 13 I = lt NM WRITE ( 6t 120 ) { SAMT< ItJ ), J = 1t I ) WRITE ( 6t 200 )
13 CONTINUE STOP
100 FORMAT ( 1615 ) 101 FORMAT ( 4F20.8 >
110 FORMAT ( lXt 22HINVERSION UNSUCCESSFUL t//t 1 1Xt 15HROW NUMBER = tl3t /It 9HDIAGONALt 2 12HELEMENT = tEl5e8t /It lXt lOHSUM OF NON , 3 19HDIAGONAL TERMS = , El5e8t // )
115 FORMAT ( lXt 22HTHE FLEXIBILITY MATRIX • II 120 FORMAT ( lXt 6E20.5 > 200 FORMAT ( // )
END
61
$lBFTC TRAN c
SUBROUTINE TRANFCil c
COMMON STIFF(20tl2tl2l' TRANS<50t3t3lt XP(3), XQC3lt
62
1 YMC3)t ALONGC5Q), EMOD' GMODt NOt INDt SAM( 54t54 lt 2 SAMT( 54t54 lt MIC 20 ), MJ< 20 ), Nl( 54), MEM
c READ (5t100) (XP<IL>~ IL=lt3)t CXQ(J), J=lt3),
1 ( YM ( K ) ' K= 1 t 3 ) ' NO 9 I ND IF ( NO .Ea. 7 ) GO TO 22
c C CALCULATION OF THE FIRST ROW OF THE MATRIX.
c
T = 0.0 DO 10 J = 1t3 XQ(J) = XQ(J) - XP(J) TL(J) = TL(J) + XQ(J) IF ( I ND • EQ • 1 ) GO TO 10 T = T + XQ(J)**2
10 CONTINUE IF ( IND .Ea. 1 ) RETURN ALONGCll = SQRTCT) DO 11 J = lt3
11 TRANSCltltJ) = XQ(J) I ALONG(!) IF ( XQ(3).EO.O.O ) GO TO 20
C CALCULATION OF THE SECOND ROW OF THE MATRIX.
c
T = 0.0 DO 12 J = lt3
12 T = T + YMCJ)**2 T = SQRT<T) DO 13 J = 1t3
13 TRANSCit2tJ) = YM(J) IT
C CALCULATION OF THE THIRD ROW OF THE MATRIX.
c
XP(ll = - XQ(3) * YMC2) + XQ(2) * YMC3) XP(2) = - XQ(l) * YMC3) + XQC3) * YM(l) XP(3} = - XQ(2) * YM(l) + XQ(l) * YM(2) T = 0.0 DO 14 J = lt3
14 T = T + XP<Jl**2 T = SQRT(T) DO 15 J = lt3
15 TRANSCit3tJl = XP(J) IT
C CHECK IF THE SUM OF SQUARES OF ELEMENTS IN THE SAME C ROW OR COLUMN IS EQUAL TO UNITY
DO 19 L = lt2
c
DO 19 K = lt3 T = 0.0 DO 18 J = lt3 GO TO ( 16tl7 lt L
16 T = T + TRANS(ItKtJ>**2 GO TO 18
17 T = T + TRANS<ItJtK)**2 18 CONTINUE
IF ( ABS( T-1.0 >.GT.l.OE-03 ) GO TO 21 19 CONTINUE
RETURN
C SECOND AND THIRD ROW OF THE MEMBER IN THE XY PLANE•
c
20 TRANS<It2tl) =- TRANS<Itlt2) TRANS(It2t2} = TRANSCitltl) TRANS<It2t3) = 0.0 TRANS(lt3tl) = 0.0 TRANS(It3t2) = OeO TRANS(lt3t3) = 1.0 RETURN
C THE CHECK HAS FAILEDt STOP.
c
21 WRITE<6t200) I WRITE ( 6t300) ((TRANS( I~ IL~ IM )t IM=lt3 ),
1 IL=lt3 ) STOP
22 DO 23 II = lt 3 DO 23 JJ = 1• 3
23 TRANS ( It Ilt JJ ) =TRANS ( I-1• !It JJ 1 RETURN
100 FORMAT { 8FlO.O/ FlO.Q, 2110 ) 200 FORMAT ( /It lXt 31HCHECK THE COORDINATES OF MEMBERt
1 7H NUMBERt 13 // 300 FORMAT ( 3F20e5 )
END
63
64
$IBFTC STIFF SUBROUTINE STIFCO (MEM >
c COMMON STIFF<20tl2•12)t TRAN5(50t3t3)t XP(3)t XQ(3),
1 YM(3)t ALONG(50)t EMODt GMODt NOt INDt SAM( 54t54 ), 2 SAMT« 54t54 )t Ml( 20 lt MJ( 20 ), Nl( 54 ), MEM
c IF ( NO .Ea. 7 ) GO TO 22
C READ IN THE MEMBER INCIDENCES. READ (5tll AREAt PINERT, YINERTt ZINERTt INDEX
C LET ALL THE ELEMENTS BELOW THE MAIN DIAGONAL BE ZERO.
c
DO 50 I = 2t12 IM = I - 1 DO 50 J = ltiM
50 STIFF<MEMtltJ) = 0.0
C CALCULATE THE VALUES OF THE ELEMENTS.
c
TEMP = EMOD I ALONG<MEM1 STIFFCMEMtltll = AREA* TEMP STIFF<MEMtllt5l = 2. * TEMP * YINERT STIFF<MEMtl2t6) = 2. * TEMP * ZINERT STIFFCMEMt5t5) = 2. * STIFFCMEMtllt5l STIFFCMEMt6t6) = 2. * STIFFCMEMtl2t6) STIFF<MEMt5t3) = - 3~ * STIFF(MEMtllt5) I ALONG(MEMl STIFF<MEMt6t2l = 3. * STIFFCMEMtl2t6) I ALONGCMEMl STIFF<MEMt8t6l = - STIFFCMEMt6t2) STIFF<MEMtl2t8) = STIFFCMEMt8t6) STIFF<MEMt9t5l = - ST!FF<MEMt5t3l STIFFCMEMtllt9) = STIFF(MEMt9t5) STIFF<MEMtllt3l = STIFFCMEMt5t3) STIFFCMEMtl2t2l = STIFF<MEMt6t2) STIFF<MEMt2t2l = 2. * STIFF<MEMt6t2l I ALONG(MEM) STIFF<MEMt3t3l = 2s * STIFFCMEMt9t5) I ALONG<MEMl STIFF<MEMt4t4) = GMOD * PINERT I ALONG(MEM) STIFF(MEMtl0t4) = - STIFF<MEMt4t4l
C FILL IN THE VALUES OF ELEMENTS THAT REPEAT. DO 151 I = lt6
151 STIFF<MEMti+6tl+6) = STIFF<MEMtiti> DO 52 I = lt3
52 STIFF<MEMtl+6ti) =- STIFF<MEM,Itll c C IF K-BAR MATRIX IS NOT REQUIRED' RETURN C TO MAIN PROGRAMME
IF ( INDEX.EQ.O ) GO TO 70 c C TRANSFORM THE THREE BY THREE SUBMATRICES C WITH DIAGONAL ELEMENTS
M = 0 N = 0 DO 59 L = lt3
c C STORE THE DIAGONAL ELEMENTS TEMPORARILY.
DO 54 I = lt3 MI = M + I NI = N + I
54 ST(I) = STIFF(MEMtMitNil c C TRANSFORMATION PROPER.
c
DO 56 I = lt3 MI = M + I DO 56 J = ltl TEMP = o.o DO 55 K = lt 3
55 TEMP= TEMP+ STCK) * TRANSCMEMtK,Jl * TRANSCMEMtKtil NJ = N + J STIFF(MEMtMitNJ) =TEMP
56 CONTINUE GO TO ( 57t58t59 )t L
57 M ::&: 3 N = 3 GO TO 59
58 M = 9 59 CONTINUE
DO 61 L = lt2 IL = L * l LL = IL + 2 DO 61 I = ILtLL DO 61 J = I L' I STIFFCMEMti+6,J+6) - STIFFCMEMtitJl GO TO C 60t61 lt L
60 STIFF(MEMt!+6tJl = - STIFF<MEMt!tJl 61 CONTINUE
M = 6 N = 0 DO 64 L = lt2 DO 62 I = lt2 MI = M + I NI = N + I II = I + 1 DO 62 J = IIt3 MJ = M + J NJ = N + J
62 STIFF<MEMtMitNJl = STIFF(MEMtMJtNI) 'GO TO ( 63t64 ), L
63 M = M + 3 64 N = N + 3
C TRANSFORMATION OF THE SUBMATRICES WITH C TWO NONDIAGONAL ELEMENTS
M = 3 N = 0 DO 68 L = lt2
65
c C STORE THE TWO ELEMENTS TEMPORARILY.
c
DO 65 I = 2t3 MI = M + I NN = N + 5 - I
65 ST(I) = STIFF<MEMtMitNN)
C TRANSFORMATION PROPER. DO 67 I = lt3 NI = N + I MI = M + I DO 67 J = lti TEMP = 0.0 TEM = 0.0 DO 66 K = 2t 3 KK = 5 - K TEMP= TEMP+ ST(K) * TRANS(MEMtKKtJl * TRANS(MEMtKti) IF « I.EQ.J ) GO TO 66 TEM = TEM + ST(KK) * TRANS<MEMtKK,J) * TRANSCMEMtKtl)
66 CONTINUE MJ = M + J NJ = N + J STIFF<MEMtMitNJ> =TEMP I F ( I • EQ • J ) GO T 0 6 7 STIFF<MEMtMJtNI> = TEM
67 CONTINUE M = M + 3 N = N + 3
68 CONTINUE DO 69 I = 4t6 DO 69 J = lt3 STIFF(MEMti+6tJ) = STIFF(MEMtitJ) STIFF<MEMti+6t J+6~ = = STIFFCMEMtitJ)
69 CONTINUE 70 DO 111 JL = ltll
JLP = JL + 1 DO 111 KL = JLPt12
111 STIFF<MEMtJLtKL} = STIFFCMEMtKLtJL) IF ( INDEX .EOe 2 ) CALL TRANQL RETURN
22 DO 23 II = lt 12 DO 23 JJ = 1t 12
23 STIFF ( MEMt lit JJ ) =STIFF ( MEM-lt lit JJ ) RETURN
1 FORMAT 4FlO.Ot 4110 100 FORMAT 8F10.8 ) 601 FORMAT 6E20.5 ) 602 FORMAT I )
END
66
67
$IBFTC TRANQ
c
c
c
SUBROUTINE TRAMQL
COMMON STIFF{20tl2tl2Jt TRANS<50t3t3Jt XPC3lt XQ(3Jt 1 YM{3), ALONG(50)t EMODt GMODt NOt INDt SAM( 54t54 j, 2 SAMT( 54t54 )t MIC 20 ), MJ{ 20 lt Nl< 54), MEM
112 READ C 5tl00 > XLCilt I = lt3 DO 20 L = lt2 pp = -1.0 IP = C L-1 ) * 6 + 1 IQ = L * 6 DO 2 0 I = I P' I Q
F = pp DO 20 J :::: lt3 F = - F JJ = 6 - J SUM = 0.0 DO 19 K =·lt3 IF ( K .EQ. J ) GO TO 19 KJ = JJ - K SUM= SUM+ STIFF ( MEMt It K+6 l *XL( KJ } * F F = - F
19 CONTINUE STIFF C MEMt It J+9 ) = STIFF ( MEMt It J+9 ) + SUM
20 CONTINUE pp = - 1.0 DO 10 I = 1t 6 F = pp DO 10 J = lt 3 F = - F JJ ::: 6 - J SUM = .0.0 DO 9 K = lt> 3 IF ( K .Ea. J > GO TO 9 KJ = JJ - K SUM = SUM + STIFF ( MEMt K+6t !+6 ) * XL ( KJ ) * F F = - F
9 CONTINUE STIFF ( MEM' J+9t I+6 ) = STIFF { MEMt J+9t !+6 ) + SUM
10 CONTINUE DO 21 I = lt 6 DO 21 J = 1t 6
21 STIFF ( MEMt J+6t I } =STIFF ( MEMt It J+6 > RETURN
END
68
$!8FTC ASMBL SUBROUTINE ASEMBL
c COMMON STIFFC20t12tl2)t TRANSC5Q,3,3}t XP<3lt XQ(3)t
1 YMC3), ALONGC5Q), EMODt GMODt NOt INDt SAM( 54t54 )t 2 SAMTC 54t54 )t MIC 20 ), MJC 20 ), NlC 54), ME~1
c C INITIALISE THE STRUCTURAL ASSEMBLY MATRIX.
NM = 6 * NJOIN DO 549 I = lt NM DO 549 J = 1 t I
549 SAMCitJ) = 0.0 c C START ASSEMBLY. c
DO 552 MEM = 1t NELEM c C CALCULATE LOCATION OF ELEMENT STIFFNESS C SUBMATRICES IN SAM
c
53 M = C MICMEM> - 1 ) * 6 N = ( MJCMEM) - 1 ) * 6
C TRANSFER OF ELEMENT STIFFNESS SUBMATRICES TO SAMe c
c
DO 552 K = lt 6 MK = M + K NK = N + K
C TRANSFER SUBMATRIX K22.
c
DO 550 L = lt K NL = N + L
550 SAM( NKtNL ) = SAM( NKtNL + STIFF( MEMtK+6tL+6
C IF FIRST END OF THE ELEMENT IS FIXEDt C TAKE UP THE NEXT ELEMENT
IF C MICMEM> .EQ. 0 l GO TO 552 c C TRANSFER OF SUBMATRIX K11•
DO 551 Ll = 1t K ML = M + LI
551 SAM( MKtML ) = SAM< MKtML ) + STIFF( MEMtKtLI ) c C TRANSFER OF SUBMATRIX Kl2•
DO 570 LJ = lt 6 ML = M + LJ
570 SAM< NKtML ) = SAMC NKtML l + STIFF( MEMtK+6tLJ ) 552 CONTINUE
RETURN END
$ENTRY c c NELEM AND NJOIN c
12 9 c M ~ I c
1 c M J c
1 5 c EMOD AND GMOD c
5
9
2
2 6
10300000.0 3850000.0 c
6 3 7
9 3 7 9 4
THE DATA INCLUDED BETWEEN STARRED LINES IS REPEATED IN THE DO LOOP NUMBER TEN OF THE MAIN PROGRAMME. FOR INDEX EQUAL TO TWO SUBROUTINE TRANQL IS CALLED. IN TRANQL RELATIVE COORDINATES OF END TWO OF THE RIGID ELEMENT WITH RESPECT TO END ONE ARE READ AS XL.
69
4 8
8 9
WHEN NO IS EQUAL TO SEVEN TRANS AND STIFF FOR THE ELEMENT UNDER CONSIDERATION IS EQUAL TO THAT OF THE PREVIOUS ELEMENT$ A LINE WITHOUT A C IN THE FIRST GOLUMN IS A BLANK DATA CARD ************************************************************ XP { IL )t XQ ( J ), YM ( K )~NO AND IND c o.o o.o o.o 5.5 10.7833 15.92 10.7833 -5.5 o.o c AREAt PINERT, YINERT, ZINERT AND INDEX c 0.326 0.07272 0.03635 0.03635 1 ************************************************************ c NO IS SEVEN c
7 *************************************************************
70
o.o o.o o.o 5e5 10.7833 51.92 10.7833 -5.5 o.o 0.326 0.07272 0.03635 Oe03635 2 c XL « I IS READ c 1.25 1.25 ***********************************************************•** 24.0 o.o o.o 22.6666 10.7833 15.92 10.7833 1.3333 o.o Oe326 Oe07272 0.03635 Oe03635 1 *************************************************************
7 ************************************************************* 24.0 o.o o.o 22.6666 10.7833 15.92 10.7833 1.3333 o.o Oe326 0.07272 Oe03635 Oe03635 2 -1.25 1e25 ************************************************************* 24.0 24.0 o.o 22.6666 27.6166 15.92 3.6166 1.3333 o.o 0.326 0.07272 0.03635 0.03635 1 *************************************************************
7 ************************************************************* 24.0 24.0 o.o 22.6666 27.6166 15.92 3.6166 1.3333 o.o Oe326 0.07272 Oe03635 Oe03635 2 -1.25 -1.25 ************************************************************* o.o 3.6166 0.326
24.0 -5.5 0.07272
o.o o.o 0.03635
5.5 27.6166 15.92
Oe03635 1 *************************************************************
7 ************************************************************* o.o 24.0 o.o 5.5 27.6166 15.92 3.6166 -5.5 o.o 0.326 0.07272 0.03635 0.03635 2 1.25 -1.25 ************************************************************* $IBSYS
APPENDIX II
ANALYTICAL FLEXIBILITY MATRIX
71
f~OD E I
0.223~3[-02
-0.5G9:32E-03
().524')7[-05 0el6867[-0L+
C.87321E-C4 0.6246SE-05
-O.Zt8015E-0Lt C.84578E-C5
NODE II
0.28443[-03 0.52565E-C;5 0 • 2 1 s Lt 3 E- C1 2
0.40040E-C4 :...0.96963[-05
O.ll292E-03
-0.38088[-0S 0.6938SE-OC::, 0.lC389C.-C3
-0.3571!+[-Cl(:: 0.82843[-06
-u.35655E-Js 0.13549[-04
0 • 2 t1- f:, 0 6 E - 0 ~~ o.l49:!3E-os o.s5512E-J4
-0.46632[-05 uel1-t52')[-U5
-0.457SlE-C4 -l:.2S3C-3E-C~s
t1•J71 5~E-n?
- C • P. 4 1 5 5 E - C' l;
~~ • 7 ':=:) 2 0 6 E- C .'t-
n.?227GE-()4 UelC15')f-U4
c • 1 J 7 2 6 [- (' 3 .0 • 1 1 2 lt 5 E- 0 4 0 • 1 4 9.3 3 E- 0 2
-11. 07864F-0t.;. -0.5Al 0 0[-0r; -0.99766[-03
-0.1 ?756E-0t+ -C.ll74r:;E-:-05 - 0 • s J 0 t; 3 [- ~ 4
- ( 1 • 1 -:?. 6 r-. t: F- r 1·,
n.2S457E-05 0.65685[-('>5 0 • 2 5 3 6. 7 [- (il+
-o. ~ 1'?1 Je-er;. J.31835F-~J6
-J.ll?::C'[-:i:::
72
0 • 8 7 /; C:: ;:) [ -- •J J
~~.sr::-c::lE-04
- () • 2 5 r; 7 5 E- () It-
.n .14:.:-·~+ 7F-rr::) 0 • ? 6 ') (ll+ [ - () !t
-0 .105')'][-8 ~; 0 • l 0 ~J 4 ':.; E- c; C...
0e677L;(~F-rl+
-().77?::'0[-()C:: 0.6,'3'):)/j-[-~3
Ce07P07[-Qr:. -0 .l228•j[-05 C.53~?7E-Clt
n.lr,--:l_:J<J[-rl7
0.::-- 07/J-[-0 6
C. 7C?n .~E-O::,
f~ODE I I I
C.l68::·~·E-Dl
-0.66001E-Ur;C.l6350E-03 0.32525[-(!5 O.l3514E-02
O.t+0041E-0l+ -0.96866E-n::,
0.39968E-04 -O.l9338E-04
J.l8260E·-·05
\..! • .39869E-:15 0. 1 7('64[-0 5 0.40205·E-05 G.47705E-05 () • 1 1 1 :- 8 E - 0 ~
-U.4B676E-G5 0.776~;GC:-G6
-C.48587C:-J:) o.l976SF-05 u.Z7956r::-os 0.11935[-06.
O.l9049E-04 -O.l6497E-06
O.l9045E-04 O.ll995E-06 0.68654[-04
-O.t+7451E-06
O.l0829E-G4 0.21227[-0:· O.liJ832E-C14
-0.87937[-06 -0.23116E-0Lt -0.330C3E-DS
O.l6861E-03 -0.65923E-05
0.16856[-03 o.32586E-o:. 0.19979[-03 0.28036E-Cr:) 0 .14114[-;'J;~
8 • l 1 !+ (\ ? F - (\ L~ 1) • ~ 7 1 9 ~) E - C s
-0 .l ')019[-0!+ CJ.49706E-C15
0.13711[-(~'?,
0.1124!1-E-04 O.l9417E-03 0.65657E-O~
O.l2969E-02
-0.21782[-rJ!; - 0 • !.+ 6 :3 2 3 E - n 4 --O.l131Sr-os -0. 2G4t\UF-U3
- 0 • :;_ 2 2 7 7 [ - 0 ~t -0.13097[-()':· - c; • 1 9 6 r 7 F. - n !; -fl.87371E-06 - C • 7 1 2 5 :: E - 8 ~~
-J.21980E-Ot; 0.11156[-(15
-0.30128E-06 OellC36E-G5
-0.29556E-06 0 • 1 ?. 7 f. 1 E - 0 L.j.
- 0 • 1 !1. 5 8 l [ - (I Lt-0 • ? ~~ 2 50 E- 0 5 C.69302E-05 C.2695~~·E-t•5
0. 2600/+E-05 0.5986?-E-n:;
0 • 1 l 3 4 6 E - C !4
0.'::.72SOE-05 -J .1 sos::·E-C4
n.t+()765F-C'5 -0.1'/i'!lOF-04
0 • l 7 0 6 ° F- n 1t
J ·73 " ·. '
_n. 64 7LL(-:.f::,-n;!
-() el ( .. (j?lJ[-QLl
n.zc:,cJl~IE-04
-0 ·1 7151E-:)h
-O.JGr:"-lE-0? 0 el\J.({4?[-()L~
-0.12A60E-03 0 • ? 8 ;::; It 1 E- 0 5
0.1 ??'"· lE-0{+ -0.37Q00F-0t;
~"• 3l80f-,[-()L+
-Oe2?()8r:;E-Q5
Ce98C05E-:J5 -').~2??1~-!)h
n • 1 2 7 h L, =:-o 1..:
- n • 5 e. ,, c; 7 E- 0 A
-0.!t-977PE-n:; -n.11176"C-r;:. Oelt<~77[-05
-0. 11 3 7 7 E.-n 5 · 0.57101.[-05
0 • 6 () If 6 7 [- () t:;
O.?U107[-U5 -0.370'3.?.E-OS n.241JJ~E-n?
-0.25'?~CE-05 0 • 41 (I 0 1 [ -(YI+
- D • f- ~~ h 0 6 E - Ci 4
1 - 0 • 1 6 n 2 1 E-n li ('.25°?7[·-0!+
-n.17JC:.2[-n4 0.2ClJ 77E~-04
- 0 • l ]_ :? 0 S [- 0 L;
-0.71917E-04 - 0 • 2 1 14 o E- C1 4 -0.72009E-C4 -O.l2744E-04
0.11337E-04 -O.l2260E-C)4 -0.7S884E-04
- 0 • 4 C 7 0 1 E - 0 t+ Oe71851E-05
-0.4C68SE-04 Oel8329E-05
-0.70483E-04 O.l9072E-05
-0.46728E-03
U.99959E-~5
c.24092E-n~
C .1 QC106E-·::'4 Oe65469E-06
-0.46920E-05 0.57987E-06
-0.46834E-05 U .15l.783E-04
O.l8479E-04 -0.43410E-06
O.l8474E-04 -0.76806E-07
O.l7261E-n4 -U.l4110E-U6 0.7152~[-04
Cel8885E-05
C.B4192E-C!5 0.95095E-06 0.84204E-05
-0.17355E-05 -O.l0423t-04 -O.l8195E-05 -0.22751E-04
O.lr'351E-04
:·:.21772E-t:?, -0 • 6 7L+09E-05
O.l3711E-03 -0.12603E-04
C.13698E-03 -· 0 • 2 2 C 3 1 E- o 5
O.l5069E-02
-CJ.54075E-04 -0.34013E-06 -0.25315E-04
O.l2634E-05 -0.25232E-Cl4 -0.52632E-05 -0.31499[-03
- 0 • 2 2 1 8 8 E - C1 4 O.l4471E-O:J
-0.79714E-C5 0 e199L~9E-05
-0. 79568E-C;5 0.973SISE-U7
-C) • 76426E-04
0.62911E-06 0.10771F-05 O.l7836E-05 Oe98537E-06 O.l7868E-U5 C.l5970E-05 0.62228E-06 O.l3203E-04
-0.22318E-t)5 0.22642E-C5 O.l60Cl5E-G4 0.21347E-05 o.l5995E-04
-0.35110E-06 O.l6518E-04 0.5638BF-05
74
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o.z3:·21E=:J::. U.26316E=04 0.635CL+E~D6
=0.299BOE=06 0. 77 324E-O r;:,
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0.25659E=04 CJ. 2 71Lt,~, E=C 5 0.25663[=04
=0.27~'02E-C•5
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o.33795E=06 O.ll011E=04 0.~7598E-OLJ-
0.38422E=C5 C.57595E-C4-
=0.l6225E-GC:, =0.33977[-04 =O.lS?llE=:Js =0.21650E=04
u .13665E-GL~
:J@ 1 '~1 6 J. ?·E=C:5
c;., 211:-:.1 E=•l ~-:.
0 = l:. 0 It .2 2 E = (} r.:,
0.5L~·,..)92E-1~7
rJ4!1 C)'::":0,P3F~r:t.:-
0 "' l r, L~ 3 1 E - 0 ~::,
- 0 8 2 7 7 :: .{.: c:- = 01':. 0.2B977E=05 o.875B8E-05 0.311L:-5E-C5 n 8 .? 7'? 3 P-E=r '"l G.S00L,.6E=n6 Cle26L;.72[·=04
-Oo29L:22E=05 U.13::~34E=CC::)
O. 5C)251E~O~ 0. 12· ':J~6E=('5
c. 5 (151 7t:=06 =0• ~o5n~r::-~nr:, =0.3/l-?6?!:-C.'r-:, ~-0.66677E~C~·
()o l4~·99E-O:.
0111 114f12E=r'L: •).lt+611E~,c::.
0 "' 1 1 3 q 3 E ~ C· ~Oe 6716LJ-E=f'i(:.
- 0 • 6 f, 5 9 l1F = (! 5 r).J.2236E~Oli-
=0 ellj-63L:-E=GL:. o.50325E=05 C.2t., .. 724E=::-~1 Lr.
o.sl?·53E=05 0 e~ ? 6.- 7 6 !1. E = C, tf
~o. 56026E~~07
~o. 3805:::,E=OL:C eLH=)l5BE=n5
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==C: 6) ;) l7:3~3F.=CJ4
Cl.319L.,-lE-C'J~,
81
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-Cio1SI~26C:-t!5
~~.l:•')C~3E~o:·
=OmJ?t~71[-(iCi
nIB 2 f-, '), 1 1 E -r) ::1 -010)1 CJr;t;r.;[-nL·-
0 <ill c;r-,L:.R 7E-0 ~· ()Gn~f..rl4F-115
n.,37r::·76[-0!:) -(). 41 P. ~- c::. E -0 ~
n e~ L·. o q r: 3 E-o r;
-n.?7~:::7nE-05
f"\@II:::J:-'1 !~[~0C:,
=Oel ':lr.:,? 0 E=nr.·
-r)® 18322[-0~
0.32320[-!)6 -o,.?r:663L:-us
0 ~»? ') 1 t:() E-n r-.
'J@ 10C:77E-C15 r.e?'-J.l~7F:::-0h
0 ,, r:;, (") ~· () 1.: [- C1 t:;
~ 0 f,} L;. = r: t:: :~ E- 0 ~~ 0 • 1 5 r:. ?· ?· E-U r:.
-n~~5775~':1c-n:::,
0el:~r::16E-r;r:,
0.i:?o.72[-0~.
n.J7Cl1 ,q[-nt.·
0 ~ r.12 7 7 ~' E -1'16 o@\~:711JE-n~,
-0 tt 141., 6S [-\Jl;
n ~ :?o.:?.S50[-0 5 =0. (:,01 r, 1[-0 c:;
0. t.2f<"7E-0 ~, Oe;=\524/C:-C':' 0 • 226C~ lE-OL~ o.22Dl2E-o5 0. r;'77J nE-n 5
=0.32671E-0Lj. n.CJ92~ .. '3E-05
~o. 15r.7,9nE-rJL:-0 e 1 2 ? 2 '} E ·- C' l;.
0 e 19226E-8~-
0<ll~3L.:.l6[-0Lc-
f·~CD E I X
0.27011E-03 -0.78179[~06
0.26985E-03 =0.25379E-06
0.22302E-03 -0.11516[-05
0.22339E-03 Oe29472E=O:) 0.62278E-03
-0.61983[-06 0.62304E-03
-0.91769E-07 v.51768E-03
-0.98965E-06 0.51730[-:Jl o.3l091E-05 0.35COCE-03
=O.l7444E-04 -0.18032[-04 -0.17963[-04 -CJ.l8769E-04 -0.43969E-Or; -0.19032E=04 - 0 • 4 8 ~ 0 5 t: - i~l '3 -O.l7942E-04 -U.40762C.--(J4 =0.64174[~-0t;
-().40245[-04 =0.71541E-05 ~O.l0390E-04
-(). 74174[~~\..)~) -Q.!)'J::J75E~C5
·-0.6~)272[-05
~ 0. 1 3 Lt 3 !+ E- 0 4
~o. 3 3 4 J 1 E- u Lt
o.lsG04E-n4 0 • 1 0 5 7 1 E = 0 t+ C' • 1 7 9 n 4 E - (1 1}
O.l0911E-(')4 Ce21925E-OL~
-0.33784[-01+ Ue2l607E-04
-0. 77664E-01+ 0.64962E-U5 0.25.:383[-04 o. 6 3 9 6L+ E -c :, 0.25541E-C4 O.l0417E-C•Ll-
-0.77~3l1F-0L}
0.1 009 1JE-Cl+
~. 2 0 p, 13 E -r'i 3 0.2Ll·679E~05
,J.l9523F-O:?, -0.3J3:,4F-0S
O.l()416E-03 - 0 • 1 l-t /1 6 5 [- 0 ~I
C.20710E-03 ~") • 1 1+ 6 9 0 E - C :_, 0.47965E-0J 0. 2 ~~~2PE.-05 'J • 4 5 0 I} 2 E - ~ 3
-0. 3:; lCSF-C::, o.45153E-o::::
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82
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-O.l26')3,C:-OLr C.G2J.6':::·E-05
- n • 1 2 9 8 (: E- 0 t+ 1).99?·?<':l[-05
- 0 • 7 3 !..f 2 7E-0 C ~Q.()2159E-O~+
-0 el ':.}4-L,L[-[1 5 -O.lL+R~·lE-n3
-o.:l7223E-os o.22203E-04·
-c.60J C:·6E-o:. Oe25Cl~J[-04
c.6236~E-o:)
~ 0 • 1 L1 (, 4 7 [- 0 3 o.!s4.:::·63E~os
-o.J 2r:.~~1 E-n? o.646Jlt.:-os
-0.12:;2()[--03
= 0 • 4- t'j -::, ::, ~~ [ -· 0 t+
- 0 • 1+ 2 (, 6 7 E - 0 5 - 0 • 5 0 6 2 l E- 0 t:
o. 7DL,.f:,7[-Q5 -0.29:">i3E-o:. ('1.1~474Lf[-(15
-0.:29147E-S3 -0.43Bl6E-05 -8.11 ~:,6CE-O :;
=0.27C82E=0L:. 0.82385t.=J5
~C.2G954E=0Lj.
0.8/.1997[=(1 5· ~0.269SJE~04
Cf)867721::-C~.
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0.2S311E=·05 ,~G • 6246'JE=Our
0.31921E=r)5 ~0.6212.0[=0~-
0.33693[=05 ~u. 62 4~-3 E-04
0.23463[=05 =0.38324E=04
C.3:JL:-llE=05 - 0 • 2 3 4 5 1 E ~· 0 5
0.35567E-C5 =0.35582[=0:) =0.25787E=·-J::; =U.35657E-O~)
~G.Z56~·8E=C:l5
=0.208~14[=05
O. 79<'~.0.6E=C5 0.347L~0E=CJ7
c. 79727E=C:, =G.ll782L-U5 ~ 0 • 5 8 C1 0 1 [ = 0 : =0.11858[-CS =0.58152[=05
0.29448[=06 C.L~S593E-06
0.713S:6[=05
== ~) & 3 [', 5 r;, ') E ~ C.· 1+
--0iii34810E=C~·
-n.e.e55·.'JC=OL:~u • f.()l:-/Lf[-C6 ~0.332!.:--7E~0Lr
-().2l7t~~o[~Cl::-)
~Ci • (?, 8 ~ /} ~j E ~ C! l+
~o. 3l-: s s :. c = c: ~~
~0.2U4J2E=C3
- n • 1 8 8 6 6 E-n 5·
n.7aon6E-06 -0. ?0L~F.·5E-0?. -o. 534:)6E-06 -0 • 2CL;.3 7E=03 -O.l9013E-~0~5
- 0 • 1 2 3 8 ~j E ~ ~) 3
!J.?D0?7E-04 'J.15L:-25E=05 C • 3 ~-·S66E=·:~1 1J,
O. 5L:-95:3E-06 U.J5997E~U4
-0.1 ~; 145E~06 O.JOC70E=C1.tiO • 3 OL,LOOE=06 n.67132E=C't.: Dsl~206t~C5
n • c; o 5 n L:- E- (It:
J.'?277UE=U6 0 • s n t: 7 2 c = \1 ~~
J. 2L;-672E==C6 0.67C99E"=04
83
0 e 6 t, ; 7 2 E -- 0 !.: ~ 0 • 1 2 ~· 5 ~;- E - ~~ S·
(\ • ::· (-, p (l 9 [- 0 !.: C:1 il27A7SC-n:·
n ') n 1 r., ) c- - n [, ._, 6 ,_ '·-' .L -· ..... c. _, :.J
Oil3.2l.'35[~00
- n • 1 2 t<: 7 E-o :, 0.1313:·E--03 0 • 2 7 ~· L: ~1 E - n ~) O.t;(l(':POE-(1,(~
0.20Cl9E-05 o. 738o;-2C:-Ol:
~ 0 e 3 ='./+ ~~:, 1 [ - G 5 0.6t+C;::::,E-OL:-
- () e 2 1 q ? ~ [- 0 1:.
o.l7~-~~6E-~=·s
= 0 ~~ 2 .t+ :? .r: '7 E - rJ !.~
0 e 2 9 ?. t.1 l E- 0 6 ~O.C.219SE-U::,
0.2·4713[-05 = ·'! • t: o 7 t., ~ E - n L:.
o.2fJ,7t.;E-cr::, ~n.,5L<::·l lE-OL:.
= n • 1 e ~~ 1 1-=; E - 0 ~~~
G.l-~- 14t::-C'::
-U.20?,JlE-C:4 0.25253t.::·=U5
-G • 2033 7E~UL~ -O.l3S·l2E-U5 ·= U • 4 1 9 ·J 9 E - 0 t1
=O.l3385E-Cr:; ~0.41988[-04
0.24310[-05 -0.46732E~D4 o.29034E-o::~
-0.46726E-04 -0.97300[-06 -0 • 94306[-QLi-~~(J • 96C'36E-Cl6 ~O • 9431 7E-Ctl-
0.28091E-05 - 0 • 3 4 5 2 lt 1.:- G 4 o.llS45E-05
Oe5172r}[-Cll+ 0.52905E-C5 o.sl72oE-04
=0. 3 04,+ 1 E--n r:, -O.l3937E·=()4 -0. 34178E~O~J ~U.l3937E~C4
O.G:;5a4E-05 0 .ll626E-CJ3 Oe62023E-C:;. o.ll625E-::3
-U • 2132 3 E -0:-. ~D. 31 ;,71 E-'}~I=0.25C:60E-05 ~0.3l372E-04
0.74702E-U5 0.2C912E~04
0.27::?-SSE~C':.
(\ • 1 1 r'J 7 7 [ = (', 5 tJ.•J7270E-05
~n.'>~9:·6F.-')~:
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().1+3}01E-O[: n. 2 J 9?4[-CLJ-0 • 4 6 z~ 8 8 E- 0 5 o • 2 1 9 9 1 E ~ n 1t
0.44CJ5)E-06 ~ 0 • 2 4 2 0 6 E - 0 l-1
0.319~-4E=C:6
-0 • 5 zr:-, 2 7E~OC> O.l0623F-C4
-0. 372G6E-0Lr 0 • P ? Ci 5 1 E - C1 c; C. 26:Y?.:iE~01l n.C!p;lqE_nr::; o.;;:(,30'1E-C~t+
0 • 8 8 ') 2 1 [- CJ 6 - ,J • 3 7 1 8 ;. E- C' 4
0.54lnOE-06 - 0 • 8 3 8 1 9 E - C L;-
0 • 1 l L;- ':':· ~~- E - 0 /\ n • r:. G t._ 1 5 E - n LJ_
o. 5Q4Lj0F._-nL~ :.:./~0337F-I!C·
~ o • c~ :::; e 4 c F ~ c Lr
J.36';0JE-C~
-0. 59663E-~OC· C'.nt~~~ 5'::[-n"'
84
n • l .~ (, 0 1 E - r-: ''
(1 • ~ 2 tt ::;, 7 E ~ 0 :5
n.4S~lnc-0:.
-u. ~·,·!':'(juc:-u :;, 11.4 ~ 1 2!.~E~(' ~; C:.l716':,E-04 o.:::~17 ~CE-o:
o. 7?· 0 2?[-!ir', - o • 1 '? 1.) :=. :? E - n Lt
0 • 7 C.: n C' 1+ C: - n 5 ~O.J3J12E-O"-l
() • 7 6 (. 0 '0 E- 0 :. o.?.2r::,_...;;E-0!+ 8e7l:?4l[-QS
0.71244E-O:• 0.L~7~~':\E-~~
- 0 • l ~~ ;:~ 7 7 E - C' r: n • c; r; 7 u_ t ... E - 0 r;
·~ • r;c/20,4E-0 ::;, C'.l~~~2~f:E-C)t+
0 • (; I+ l 0 3 [- 0 5 0.16102[-U 0. 1/.J,_ c::.? 2 [ -0 l:
- 0 • ~ :J n r:::, n E-n 11
-n.1F-..??7E-0L, c • J. '? .s c. ~-, [- '>''
0. l ;~. J. 7 ') [ -() ~ -=()e Jr;.~2?()[--(}7
APPENDIX III
EXPERIMENTAL FLEXIBILITY ~~TRIX
85
.. ,
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C. 3962nE=nt:1-=C.2CllCE-0~
=0 •
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=' ....... ·.j e
C' ~a 4CC 1 OE='~'5 n.lOllC'F-113
=:J. 42 39"'C:-!!r.;, =0. =C.4981°E=·:ls
0 •. 2221nE·="S OolC4'~r.E=•':)L~
iJ ol7840E=r_:LI
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=0. O.lC35 1JE=04
~(J.
= () • 2 2 n 4 ,...... [ ~ r. tf=0. 3C01 ~·F=C';;
;-....:co E i v
O.l504';E=r'3 =D.55690E-nf:1
:')el6l?'"'E-,...,--:J, ~c: •
~::.lp,c:., r:t-_-0'?,
~o.
=· ., • 1 7 ,; ?· 11 F. = ~~~
ri@j4JJ3nF=~"''-;
n !!I 1 ,~ (') r;:, n F- = n ·::, n • 1 nL~ S nE~~f:L~n.l ?snn~=:-=CI3 n.686J.0[=ti5 n.1 C·30E= 1i2
~n.2P.J !JF=r·,~
~n(!)
= n • 4 1 s ,q n r- = n L': =r) 0
~~n.1 J ~:=.nr=rH~ =n. "=n. 1 n2.9C:F=n~f =n.
_t .....
='l.
0 ol ;"2900c_.=()l\
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~n.
=n. ,,..., e :? ~) 1 !.:. r. F~- ('· r.,
0 .CJ7Sr1r'IF=rl~,
n 8 t: •) (; 1 n E = n c;
= i'"'l o } ·=-· (-;. '.· (1 F =· rl ~-
= n • 1 7 <'+ l o r=_ ~- n /,, n.J t;61!iF--,1,4
87
= r'l • 1 :.. ' :·. n F -··· r: n,.::::.2~ '":'IJ[~n
=n., lnr·,~:r. 1~-.-,n:
n.l:-1;''-:Jrq=·=n~.
= n ~ 1 1 ~~ r·) r:- - o ·7, ne 37'""'7nE---n:=-
n • 1 7 n .) n F ~ n .::~ - n e ~· 3 1 :-- n ;::: ~ n r-.
rJi\!~ .. 1 1UC"~ur:.
~·n el
n ~ 1. ; 1 n [ ··- r: ": ~- (') ~
n ., r:-, ·' f!!l J. '
~n.L~~-.~:.nE~n
=0. =!lo
=0\l)
n 0 L; . .... : 1 l il E ~. (1 r
=n., = r: • .=; '~·' 6 ::::. n [ - 0 ::::
=~ ·": !)
=n(! nit rn] r]r:r-n
= n !) 5 ·-~ r, n F- n ~ =f"'~ e Jr .. "]4!l[·~f1
I") • ? .:.:. 1 ...... n r -- n ',c
""'-' n @ l. :: ;' l ~·~ :=- ~ ~~~ '-:~
O('J:':':-'.~'"'·1 UF·~O
= n e 1 l · ·~· ..... n F -- n ·1i
- ~~ • b :~ ';1 l n t. - (' 4 ~ c • 1 ;~ r:\ 1 ("\ r = n 4.
Cl • 1 1 7 ~) n E ~ (', i+
~-C~.l227'.~E--t'lL~
-c. 12 rs1 ,-, E -04
-C.4192 11 E-t14 c.s7~~3C'E-05
- C) • 3 7 7 2 11 E- 'J 4 -c. - C' • 7 r 1 1 n t - :~ <'1
-o. ~'1.4373r'f-r'l3
C:. 91.2 8 n E -r-· S -o. -0.43l4,E-nc:. ..... ().
= 0 • 4 C 5 c1 0 E - n ~ Oel5C;zoE-G4
O.l68~0f:.-04
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0.71680E-t15 -G.
0.769]0[-115 -C) • - C • 1 C C: 9 n E ~ 1~1 !+
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0.9186nE-r··5
n • 2 r) l r::, n E - i""' ::;
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· :-·, • J .2 ~. n r F "~ r'- !;.
~~: • j_ 2 ;7' ('o () r:- - (\ ~·
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~ :~ • t+ -r;; 1 r: ::--- ·= ~--.~ 4 -n. -n.2412nr--n4 -n. "- n • ? r., n J n F - n L:.
-'l.439r-nF-n) -n.2912C'tF-n3
- n • ~? n ~1 ~:, n F- r: l+ r, .. ~ : .
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88
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L=->0 ,'; • ·~ ~. 1 ? !,..I l - r) ~
~ (' • '-~ "'· '"', J (I :? - 11 '.:,
- n • l -~ " 1 0 F ~ 11 c.:c.
c. 4'''= 1 (if_-04 n.~.-9.?2nc-nr::.
n.lJP.l r1~--11l, 0.60J3nF-n'~
-0. n • 6 .~ r /+ ,-: F _ n r·,
n.2227C'[-n;.
n • 1 o -:-; ? 1 :=.: ~ n ,., -0.,
u.r)4tJ'lriE=ur..,
-n. - r~ • -n.
n • 1 7 7 n n ~- r) 1.~
~·o (J.
-n. -n. ~(I.
-!J.
- n • c; 1 2 r-, () F - r l :,
~n.
=n.4n<+nn~~-r~,
0 • 26SFJIE-n S c l • '] 6 3 '~ u E ~~ (J S () •--:: 7,".LruF~U!+
f\lODF V
:-, • 1 9 3 n r· E - n .? 0 • 1 \~~ 7 t1- CJ E - D ~C.61110E-rl:~
~o. 6393C·E-() r::,
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Q. 3613ilE.-n·~.
0 • 2115·'lE-"4 0.33~,9nE-r12
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0. 2 2 54 11 E- 1'~LJ
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- 0 • 4 7 9 0 ~; E - tJ ~~ -o. 6320'"'1 E-n~
- 0 • 3 9 2 1 rl E-.') 3 0.7678CE-C'4
-0.10~~20[-'-,3
C.2072CE-C'4 -C.l361nE-0~
0 • 2 0 1 o :' E - :~: t+ ~-C.l4L4-2flE-f1 ~
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0.549lr"E-C5 ~" 0. o.7n6lr1 E-ns " ----}.
--0.6742nE-ns -o. -:J. 66 7 5 n E-n 5 =-0. 0el351'~'[-fiL~
0.14~·1 OE-r14
n.l2? 1 llF~0':', () o r:.: 9 2 II() F - n L+
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~n.l4E~.znF-n3 n • 4 3 8 3 n c ·= 0 t 1_
0 • 1 :5 6 :=, () F ~ 0 ? - 0 • 1 3 L: 6 0 F - (")/+
n. 34-410E-n:::: -n • .2 t:; 3 J !!F··-r1 4
n.3192nr::-::3 -n. n. l1-- 3 CJ? n F- (', ·?
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n.6229CE-r,4 n. 539nnr=~n:,
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n.34.?1C:E-"'4
89
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n e 1 7 1 ,J. n r- n ,cf
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= n • 6 7 ·':~ ":: n E-n ~
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PERCENTAGE ERROR MATRIX
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