A Historical Outline of Matrix Structural Analysis

8/17/2019 A Historical Outline of Matrix Structural Analysis

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A Historical Outline of Matrix Structural Analysis

Abstract

The evolution of Matrix Structural Analysis (MSA) from 1930 through 1970 is outlined

!ightlighted are ma"or contri#utions #y $ollar and %uncan& Argyris& and Turner& 'hichshaed this evolution To enliven the narrative the outline is configured as a threeact

lay Act * descri#es the re++** formative eriod Act ** sans a eriod of confusion

during 'hich matrix methods assumed #e'ildering comlexity in resonse to

conflicting demands and restrictions Act *** outlines the cleanu and consolidationdriven #y the aearance of the %irect Stiffness Method& through 'hich MSA

comleted morhing into the resent imlementation of the ,inite -lement Method

.ey'ords/ matrix structural analysis finite elements history dislacement method force

method direct stiffness method duality

1. INTRODUCTION

+ho first 'rote do'n a stiffness or flexi#ility matrixThe 2uestion 'as osed in a 199 aer 415 The educated guess 'as 6some#ody'oring in the aircraft industry of 8ritain or ermany& in the late 19:0s or early 1930s;

Since then the 'riter has examined reorts and u#lications of that time These trace the

origins of Matrix Structural Analysis to the aeroelasticity grou of the a#oratory () at Teddington& a to'n that has no' #ecome a su#ur# of greater >ondon

The resent aer is an exansion of the historical vignettes in Section ? of 415 *toutlines the ma"or stes in the evolution ofMSA#y highlighting the fundamental

contri#utions of four individuals/ $ollar& %uncan& Argyris and Turner Thesecontri#utions are lumed into three milestones/

Creation 8eginning in 1930 $ollar and %uncan formulated discrete aeroelasticity in

matrix form The first t'o "ournal aers on the toic aeared in 193?3 4:&35 and the

first #oo& couthored 'ith ,ra@er& in 193 4?5 The reresentation and terminology for discrete dynamical systems is essentially that used today

Unification *n a series of "ournal articles aearing in 19? and 19 45 Argyris resented a formal unification of ,orce and %islacement Methods using dual energy

theorems Although ractical alications of the duality roved ehemeral& this 'or

systemati@ed the concet of assem#ly of structural system e2uations from elemental

comonents FEMinization *n 199 Turner roosed 4B5 the %irect Stiffness Method (%SM) as an

efficient and general comuter imlementation of the then em#ryonic& and as yetunnamed& ,inite -lement Method


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This techni2ue& fully exlained in a follo'u article 475& naturally encomassed

structural and continuum models& as 'ell as nonlinear& sta#ility and dynamicsimulations 8y 1970 %SM had #rought a#out the demise of the $lassical ,orce Method($,M)& and #ecome the dominant imlementation in roductionlevel ,-M rograms

These milestones hel dividing MSA history into three eriods To enliven and focus theexosition these 'ill #e organi@ed as three acts of a lay& roerly sulemented 'ith a

6matrix overture; rologue& t'o interludes and a closing eilogue !ere is the rogram/

Prologue Victorian Artifacts/ 1C1930

Act I Gestation and Birth/ 1930C193Interlude I WWII Blackout / 193C19?7

Act II The Matrix Forest / 19?7C19B

Interlude II Questions/ 19BC199

Act III Ansers/ 199C1970Epilogue !e"isitin# the $ast / 1970date

Act *& as 'ell as most of the =rologue& taes lace in the D. The follo'ing eventsfeature a more international cast

2. BAC!ROUND AND TER"INO#O!$

8efore dearting for the theater& this Section offers some general #acground andexlains historical terminology Eeaders familiar 'ith the su#"ect should si to Section

3

The overall schematics of model#ased simulation (M8S) #y comuter is flo'charted in

,igure 1 ,or mechanical systems such as structures the ,inite -lement Method (,-M)is the most 'idely used discreti@ation and solution techni2ue !istorically the ancestor

of ,-M is MSA& as illustrated in ,igure : The morhing of the MSA from the recomuter eraFas descri#ed for examle in the first #oo 4?5 F into the first

rogramma#le comuters too lace& in 'o##ly gyrations& during the transition eriod

herein called Act ** ,ollo'ing a confusing interlude& the young ,-M #egin to settle&

during the early 19B0s& into the configuration sho'n on the right of ,igure : *ts #asiccomonents have not changed since 1970


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MSA and ,-M stand on three legs/ mathematical models& matrix formulation of the

discrete e2uations& and comuting tools to do the numerical 'or Gf the three legs thelatter is the one that has undergone the most dramatic changes The 6human comuters;

of the 1930s and 19?0s morhed #y stages into rogramma#le comuters of analog and

digital tye The matrix formulation moved lie a endulum *t #egan as a simle

dislacement method in Act *& reached #e'ildering comlexity in Act ** and 'ent #acto concetual simlicity in Act ***

Dnidimensional structural models have changed little/ a 1930 #eam is still the same #eam The most noticea#le advance is that re19 MSA& follo'ing classical

>agrangian mechanics& tended to use satially discrete energy forms from the start The

use of sacecontinuum forms as #asis for multidimensional element derivation 'as ioneered #y Argyris 45& successfully alied to triangular geometries #y Turner&$lough& Martin and To 45& and finali@ed #y Melosh 495 and *rons 410&115 'ith the

recise statement of comati#ility and comleteness re2uirements for ,-MMatrix formulations for MSA and ,-M have #een traditionally classified #y the choice

of %ri&ar' unknos These are those solved for #y the human or digital comuter todetermine the system state *n the %islacement Method (%M) these are hysical or

generali@ed dislacements *n the $lassical ,orce Method ($,M) these are amlitudes

of redundant force (or stress) atterns (The 2ualifier 6classical; is imortant #ecause

there are other versions of the ,orce Method& 'hich select for examle stress functionvalues or >agrange multiliers as unno'ns) There are additional methods that involve

com#inations of dislacements& forces andHor deformations as rimary unno'ns& #ut

these have no ractical imortance in the re1970 eriod covered hereAroriate mathematical names for the %M are ran#e(s%ace &ethod or %ri&al &ethod

This means that the rimary unno'ns are the same tye as the rimary varia#les of the

governing functional


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Aroriate names for the $,M are null(s%ace &ethod & ad)oint &ethod & or dual &ethod

This means that the rimary unno'ns are of the same tye of the ad"oint varia#les of the governing functional& 'hich in structural mechanics are forces These names are not

used in the historical outline& #ut are useful in lacing more recent develoments& as'ell as nonstructural ,-M alications& 'ithin a general frame'or

The terms Stiffness Method and ,lexi#ility Method are more diffuse names for the%islacement and ,orce Methods& resectively enerally seaing these aly 'hen

stiffness and flexi#ility matrices& resectively& are imortant art of the modeling and

solution rocess

%. PRO#O! & 'ICTORIAN ARTI(ACT)* 1+,+&1-%

Matrices F or 6determinants; as they 'ere initially called F 'ere invented in 1 #y$ayley at $am#ridge& although i##s (the coinventor& along 'ith !eaviside& of vector

calculus) claimed riority for the erman mathematician rassmann Matrix alge#ra

and matrix calculus 'ere develoed rimarily in the D. and ermany *ts original use

'as to rovide a comact language to suort investigations in mathematical toics

such as the theory of invariants and the solution of alge#raic and differential e2uations,or a history of these early develoments the monograh #y Muir 41:5 is unsurassedSeveral comrehensive treatises in matrix alge#ra aeared in the late 19:0s and early

1930s 413C15$omared to vector and tensor calculus& matrices had relatively fe' alications in

science and technology #efore 1930 !eisen#ergIs 19: matrix version of 2uantum

mechanics 'as a nota#le excetion& although technically it involved infinite matrices

The situation #egan to change 'ith the advent of electronic des calculators& #ecausematrix notation rovided a convenient 'ay to organi@e comlex calculation se2uences

Aeroelasticity 'as a natural alication #ecause the sta#ility analysis is naturally osed in terms of determinants of matrices that deend on a seed arameter

The nonmatrix formulation of %iscrete Structural Mechanics can #e traced #ac to the

1B0s 8y the early 1900s the essential develoments 'ere comlete A reada#le

historical account is given #y Timosheno 41B5 *nterestingly enough& the term 6matrix;never aears in this #oo

/. ACT I & !E)TATION AND BIRT0* 1-%&1-%+

*n the decade of +orld +ar * aircraft technology #egin moving to'ard monolanes8ilanes disaeared #y 1930 This evolution meant lo'er drag and faster seeds #ut

also increased disosition to flutter *n the 19:0s aeroelastic research #egan in an

international scale =ertinent develoments at the a#oratory ()

are 'ell chronicled in a 197 historical revie' article #y $ollar 4175& from 'hich thefollo'ing summary is extracted

/.1 Te )ource Papers

The aeroelastic 'or at the Aerodynamics %ivision of 'as initiated in 19: #y E

A ,ra@er !e 'as "oined in the follo'ing year #y + J %uncan T'o years later& in

August 19:& they u#lished a monograh on flutter 415& 'hich came to #e no'n as6The ,lutter 8i#le; #ecause of its comleteness *t laid out the rinciles on 'hich


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flutter investigations have #een #ased since *n January 1930 AE $ollar "oined ,ra@er

and %uncan to rovide more hel 'ith theoretical investigations Aeroelastice2uations 'ere tedious and error rone to 'or out in long hand !ere are $ollarIs o'n

'ords 417&age 175 on the motivation for introducing matrices/6,ra@er had studied matrices as a #ranch of alied mathematics under race at $am#ridge and

he recogni@ed that the statement of& for examle& a ternary flutter ro#lem in terms of matrices'as neat and comendious !e 'as& ho'ever& more concerned 'ith formal maniulation and

transformation to other coordinates than 'ith numerical results Gn the other hand& %uncan and *

'ere in search of numerical results for the vi#ration characteristics of airscre' #lades and 'e

recogni@ed that 'e could only advance #y #reaing the #lade into& say& 10 segments and treating

it as having 10 degrees of freedom This aroach also 'as more conveniently formulated in

matrix terms& and readily exressed numerically Then 'e found that if 'e ut an aroximate

mode into one side of the e2uation& 'e calculated a #etter aroximation on the other and the

matrix iteration rocedure 'as #orn +e u#lished our method in t'o aers in $hil* Ma#* 4:&35

the first& dealing 'ith conservative systems& in 193? and the second& treating damed systems& in

193 8y the time this had aeared& %uncan had gone to his $hair at !ull;

The aforementioned aers aear to #e the earliest )ournal %u+lications of MSA Theseare ama@ing documents/ clean and to the oint They do not feel outdated ,amiliar

names aear/ mass& flexi#ility& stiffness& and dynamical matrices The matrix sym#ols

used are 4&5& 4 f 5& 4c5 and 4 ,5 = 4c5−14&5 = 4 f 54&5& resectively& instead of the "& (& and D in common use today A general inertia matrix is called 4a5 As #efit the focus

on dynamics& the dislacement method is used =ointmass dislacement degrees of

freedom are collected in a vector { x} and corresonding forces in vector { $ } Theseare called 4-5 and 4Q5& resectively& 'hen translated to generali@ed coordinates

The notation 'as changed in the #oo 4?5 discussed #elo' *n articular matrices are

identified in 4?5 #y caital letters 'ithout surrounding #racets& in more agreement 'ith

the modern style for examle mass& daming and stiffness are usually denoted #y A& Band C & resectively

/.2 Te ")A )ource Boo

Several aers on matrices follo'ed& #ut aarently the traditional u#lication vehicles'ere not vie'ed as suita#le for descrition of the ne' methods At that stage $ollar

notes 417& age 15 that6South'ell 4Sir Eichard South'ell& the 6father; of relaxation methods5 suggested that the

authors of the various aers should #e ased to incororate them into a #oo& and this 'asagreed The result 'as the aearance in


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are o#solete That leaves the modeling and alication examles& 'hich are not

coherently inter'eaved


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The transition illustrated in ,igure : starts& driven #y t'o of the legs suorting MSA/

ne' comuting resources and ne' mathematical models The matrix formulationmerely reacts

5.1 Co7puters Beco7e "acines

The first electronic commercial comuter/ Dnivac *& manufactured #y a division of EemingtonEand& aeared during summer 191 The six initial machines 'ere

delivered to DS government agencies 4:15 *t 'as "oined in 19: #y the Dnivac 1103& a

scientificcomutation oriented machine #uilt #y -EA& a EE ac2uisition This 'as the

first comuter 'ith a drum memory T J +atson Sr& founder of *8M& had #een once2uoted as saying that six electronic comuters 'ould satisfy the needs of the lanet

Turning around from that rediction& *8M launched the cometing 701 model in 1938ig aircraft comanies #egan urchasing or leasing these exensive 'onders #y 19?

8ut this did not mean immediate access for every#ody The #ehemoths had to #e

rogrammed in machine or assem#ly code #y secialists& 'ho soon formed comuter

centers allocating and rioriti@ing cycles 8y 19B structural engineers 'ere still liely

to #e using their slides rules& Marchants and unched card e2uiment Gnly after the197 aearance of the first high level language (,ortran *& offered on the *8M 70?)'ere engineers and scientists a#le (and allo'ed) to 'rite their o'n rograms

5.2 Te "atri8 C(" Taes Center )tage

*n static analysis the nonmatrix version of the $lassical ,orce Method ($,M) had

en"oyed a distinguished reutation since the source contri#utions #y Max'ell& Mohr and

$astigliano The method rovides directly the internal forces& 'hich are of aramountinterest in stressdriven design *t offers considera#le scoe of ingenuity to exerienced

structural engineers through clever selection of redundant force systems *t 'asroutinely taught to Aerosace& $ivil and Mechanical -ngineering students

Success in handcomutation dynamics deends on 6a fe' good modes; >ie'ise& the

success of $,M deends crucially on the selection of good redundant force atterns

The structures of re190 aircraft 'ere a fairly regular lattice of ri#s& sars and anels&forming #eamlie configurations *f the anels are ignored& the selection of aroriate

redundants 'as 'ell understood =anels 'ere modeled conservatively as inlane shearforce carriers& circumventing the difficulties of t'odimensional elasticity +ith some

ad"ustments and exerimental validations& s'et#ac 'ings of high asect ratio'ere eventually fitted into these models

A matrix frame'or 'as found convenient to organi@e the calculations The first "ournal

article on the matrix $,M& 'hich focused on s'et#ac 'ing analysis& is #y >evy 4:05&

follo'ed #y u#lications of Eand 4::5& >angefors 4:35& +ehle and >ansing 4:?5 and%ene4:5 The develoment culminates in the article series of Argyris 45 discussed in

Section B

5.% Te Delta 3ing Callenge

The %islacement Method (%M) continued to #e used for vi#ration and aeroelastic

analysis& although as noted a#ove this 'as often done #y grous searated from stressand #ucling analysis A ne' modeling challenge entered in the early 190s/ delta 'ing

structures This reindled interest in stiffness methods


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The traditional aroach to o#tain flexi#ility and stiffness matrices of unidimensional

structural mem#ers such as #ars and shafts is illustrated in ,igure 3 The governingdifferential e2uations are integrated& analytically or numerically& from one end to the

other The end 2uantities& grouing forces and dislacements& are there#y connected #ya transition matrix Dsing simle alge#raic maniulations three more matrices sho'n in

,igure 3 can #e o#tained/ deformational flexi#ility& deformational stiffness and freefreestiffness This 'ell no'n techni2ue has the virtue of reducing the num#er of unno'ns

since the integration rocess can a#sor# structural details that are handled in the resent

,-M 'ith multile elements

evy 4:5 this 'as only artly successful #ut 'as a#le to

illuminate the advantages of the stiffness aroach

The article series #y Argyris 45 contains the derivation of the × freefree stiffnessof a flat rectangular anel using #ilinear dislacement interolation in $artesian

coordinates 8ut that geometry 'as o#viously inade2uate to model delta 'ings The


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landmar contri#ution of Turner& $lough& Martin and To 45 finally succeeded in

directly deriving the stiffness of a triangular anel $lough 4:95 o#serves that this aer reresents the delayed u#lication of 19:3 'or at 8oeing *t is recogni@ed as one of

the t'o sources of resent ,-M imlementations& the second #eing the %SM discussedlater 8ecause of the larger num#er of unno'ns comared to $,M& cometitive use of

the %M in stress analysis had necessarily to 'ait until comuters #ecome sufficiently o'erful to handle hundreds of simultaneous e2uations

5./ Reduction (osters Co7ple8it9

,or efficient digital comutation on resent comuters& data organi@ation (in terms of fast access as 'ell as exloitation of sarseness& vectori@ation and arallelism) is of

rimary concern 'hereas ra' ro#lem si@e& u to certain comuterdeendent #ounds&is secondary 8ut for hand calculations minimal ro#lem si@e is a ey asect Most

humans cannot comforta#ly solve #y hand linear systems of more than or B unno'ns

#y direct elimination methods& and C10 times that through ro#lemoriented

6relaxation; methods The firstgeneration digital comuters imroved seed and

relia#ility& #ut 'ere memory straed ,or examle the Dnivac * had 1000 ?#it 'ordsand the *8M 701& :0? 3B#it 'ords $learly solving a full system of 100 e2uations 'asstill a ma"or challenge

*t should come as no surrise that ro#lem reduction techni2ues 'ere aramount

throughout this eriod& and exerted noticea#le influence until the early 1970s *n staticanalysis reduction 'as achieved #y ela#orated functional #rou%in#s of static and

inematic varia#les Most schemes of the time can #e understood in terms of the


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follo'ing classification/

!ere a%%lied forces are those acting 'ith non@ero values& that is& the ones visi#ly dra'n

as arro's #y an engineer or instructor *n reductionoriented thining @ero forces onunloaded degrees of freedom are classified as condensa+le #ecause they can #e removed

through static condensation techni2ues Similarly& non@ero a%%lied dis%lace&ents 'ere

clearly differentiated from @erodislacements arising from suort conditions #ecausethe latter can #e thro'n out 'hile the former must #e retained Eedundantdislacements& 'hich are the counterart of redundant forces& have #een givenmany

names& among them 6inematically indeterminate dislacements; and 6inematic

deficiencies;

Matrix formulation evolved so that the unno'ns 'ere the force redundants 9 in the$,M and the dislacement redundants : in the %M =artitioning matrices in accordance

to (1) fostered exu#erant gro'th culminating in the &atrix forest that characteri@es'ors of this eriod

To a resent day ,-M rogrammer familiar 'ith the %SM& the comlexity of the matrix

forest 'ould strie as madness The %SM master e2uations can #e assem#led 'ithout

functional la#els 8oundary conditions are alied on the fly #y the solver 8ut the

comuting limitations of the time must #e et in mind to see the method in themadness

5., T;o Pats Troug te (orest

A series of articles u#lished #y J ! Argyris in four issues of Aircraft En#r#* during

19? and 19

collectively reresents the second ma"or milestone in MSA *n 19B0 the articles 'ere

collected in a #oo& entitled 6-nergy Theorems and Structural Analysis; 45 =art *& su#entitled

eneral Theory&rerints the four articles& 'hereas =art **& 'hich covers additional material on thermal

analysis andtorsion& is coauthored #y Argyris and .elsey 8oth authors are listed as affiliated 'ith

the Aerosace%eartment of the *merial $ollege at >ondon

The dual o#"ectives of the 'or& stated in the =reface& are 6to generali@e& extend andunify the fundamental

energy rinciles of elastic structures; and 6to descri#e in detail ractical methods ofanalysis


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of comlex structuresFin articular for aeronautical alications; The first o#"ective

succeeds 'ell&and reresents a ey contri#ution to'ard the develoment of continuum#ased models

=art * carefullymerges classical contri#utions in energy and 'or methods 'ith matrix methods of

discrete structuralsystems The coverage is methodical& 'ith numerous illustrative examles The

exosition of the

,orce Method for 'ing structures reaches a level of detail une2ualed for the time

The %islacement Method is then introduced #y dualityFcalled 6analogy; in this 'or/6The analogy #et'een the develoments for the flexi#ilities and stiffnesses sho's clearly that

arallel to the analysis of structures 'ith forces as unno'ns there must #e a corresonding

theory

'ith deformations as unno'ns;

This section credits Gstenfeld 4305 'ith #eing the first to dra' attention to the arallel

develoment

The duality is exhi#ited in a striing ,orm in Ta#le **& in 'hich #oth methods are resented side #y side'ith simly an exchange of sym#ols and aroriate re'ording The stes are #ased on

the follo'ing

decomosition of internal deformation states g and force atterns p/

p = B0


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of continuum#ased finite elements through stiffness methods These are naturally

derived directlyfrom the total otential energy rincile via shae functions& a techni2ue not fully

develoed until themid 19B0s

The side #y side resentation of Ta#le ** of 45 tried to sho' that $,M and %M 'eregoing through

exactly the same se2uence of stes Some engineers& eventually a#le to 'rite ,ortran

rograms&

concluded that the methods had similar caa#ilities and selecting one or the other 'as amatter of

taste (Most structures grous& uholding tradition& oted for the $,M) 8ut the fe'engineers 'ho

tried imlementing #oth noticed a #ig difference And that 'as #efore the %SM& 'hich

has no dual

counterart under the decomosition (:)& aeared

The aradox is exlained in Section ? of 415 *t is also noted there that (:) is not a articularly usefulstate decomosition A #etter choice is studied in Section : of that aer this one

ermits all no'nmethods of $lassical MSA& including the %SM& to #e derived for seletal structures as

'ell as for a

su#set of continuum models

4 INTER#UDE II & =UE)TION)* 1-,5&1-,-

*nterlude * 'as a silent eriod dominated #y the 'ar #lacout *nterlude ** is more

vocal/ a time of 2uestions An array of methods& models& tools and alications is no' on the ta#le& and

gro'ing

Solidstate comuters& ,ortran& *$8Ms& the first satellites So many otions Stiffness or

flexi#ility,orces or dislacements %o transition matrix methods have a future *s the $,M%M

duality a recursor to generalurose rograms that 'ill simulate everything +ill engineers #e

allo'ed to'rite those rograms

As convenient milestone this outline taes 199& the year of the first %SM aer& as the

#eginning of

Act *** Arguments and counterarguments raised #y the foregoing 2uestions 'ill linger&ho'ever& for

t'o more decades into diminishing circles of the aerosace community+ ACT III & AN)3ER)* 1-,-&1-4

The curtain of Act *** lifts in Aachen& ermany Gn B


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%irect Stiffness Method to anAAE%Structures and Materials =anel meeting 4B5

(AAE%is


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that is comutationally unnecessary A more suggestive notation used in resent %SM

exositions is / = _ ( 0e )T / e 0e& in 'hich 0e are 8oolean locali@ation matrices


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one adds a geometric stiffness and solves the sta#ility eigenro#lem& a techni2ue first

exlained in4335 To do nonlinear analysis one modifies the stiffness in each incremental ste To

aly multiointconstraints the aer 475 advocates a masterslave reduction method

Some comutational asects are missing from this aer& nota#ly the treatment ofsimle dislacement

#oundary conditions& and the use of sarse matrix assem#ly and solution techni2ues

The latter 'ere

first addressed in +ilsonIs thesis 'or 43?&35+.2 Te (ire )preads

%SM is a aragon of elegance and simlicity The 'riter is a#le to teach the essentialsof the method

in three lectures to graduate and undergraduate students alie Through this ath the old

MSA and the

young,-Machieved smooth confluence The matrix formulation returned to the

crisness of the source aers 4:&35 A 'idely referenced MSA correlation study #y allagher 43B5 heleddissemination

$omuters of the early 19B0s 'ere finally a#le to solve hundreds of e2uations *n anideal 'orld&

structural engineers should have 2uicly ra@ed the forest and em#raced %SM

*t did not haen that 'ay The 'orld of aerosace structures slit %SM advanced first

#y 'ord of mouth Among the aerosace comanies& only 8oeing and 8ell (influenced #y Turner

and allagher&resectively) had made ma"or investments in %SM #y 19B Among academia the $ivil

-ngineering

%eartment at 8ereley #ecome a%SMevangelist through $lough& 'homade his

studentsFincludingthe 'riterFuse%SMin their thesis'or These codes 'ere freely disseminated into the

nonaerosace'orld since 19B3 Martin esta#lished similar traditions at +ashington Dniversity& and

ienie'ic@&influenced #y $lough& at S'ansea The first text#oo on ,-M 4375& 'hich aeared in

19B7& maes

no mention of force methods 8y then the alication to nonstructural field ro#lems

(thermal& fluids&electromagnetics& ) had #egun& and again the %SM scaled 'ell into the #rave ne'

'orld+.% Te (inal Test

>egacy $,M codes continued& ho'ever& to #e used at many aerosace comanies The

slit reminds

one of -insteinIs ans'er 'hen he 'as ased a#out the reaction of the oldguard schoolto the ne'


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hysics/ 6'e did not convince them 'e outlived them; Structural engineers hired in

the 19?0s and190s 'ere often in managerial ositions in the 19B0s They 'ere set in their 'ays

!o' can dualityfail All that is needed are algorithms for having the comuter select good redundants

automaticallySu#stantial effort 'as sent in those 6structural cutters; during the 19B0s 43:&35

That tenacity 'as eventually ut to a severe test The 19B


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a hy#rid variational formulation for this com#ined aroach

The true duality for structural mechanics is no' no'n to involve dislacements andstress functions&

rather than dislacements and forces This 'as discovered #y ,raei"s de Neu#ee in the1970s 4?5

Although extendi#le #eyond structures& the otential of this idea remains largelyunexlored

1 CONC#UDIN! RE"AR)

The atient reader 'ho has reached this final section may have noticed that this is a

critical overvie'of MSA history& rather than a recital of events *t reflects ersonal interretations and

oinions Thereis no attemt at comleteness Gnly 'hat are regarded as ma"or milestones are covered

in some

detail ,urthermore there is only sotty coverage of the history of ,-M itself as 'ell as

its comuter

imlementation this is the toic of an article under rearation for Alied MechanicsEevie'sThis outline can #e hoefully instructive in t'o resects ,irst& matrix methods no' in

disfavor maycome #ac in resonse to ne' circumstances An examle is the resurgence of

flexi#ility methods in

massively arallel rocessing A general a'areness of the older literature hels Second&

the s'eeingvictory of %SM over the #efuddling comlexity of the 6matrix forest; eriod illustrates

the virtue of GccamIs roscrition against multilying entities/ 'hen in dou#t chose simlicity This

dictum is

relevant to the resent confused state of comutational mechanicsAcno;ledge7ents

The resent 'or has #een suorted #y the

Documents

A Historical Outline of Matrix Structural Analysis