Clough_The FEM After Twenty-five Years - A Personal View

8/22/2019 Clough_The FEM After Twenty-five Years - A Personal View

1/10

Computers 1 Stnrctuws ol. 12. pp. 361470@ Pergamon Press Ltd., 1980. Printed in Great Britain

THE FINI TE ELEMENT METHOD AFTER TWENTY FIVE YEARS:A PERSONAL VIEW

RAY W. CLOUGHUniversity of California,Berkeley, CA 94720,U.S.A.(Received 20 September 1979; received for publ icati on 24 January 1980)

Abstract-The purpose of this paper is to examine the current state of development of the finite element methodwith regard to engineering applications. First is presented a personal view of the origins of the method, describingthe sequence of events at Berkeley. Next is a discussion of the state-of-the-art of structural dynamic analysis, withmention of important recent advances. Finally, two examples drawn from earthquake engineering experience arediscussed which demonstrate some limitations of present capabilities. Specific areas requiring new programdevelopment are mentioned; the need for a combined analytical-experimental approach i s emphasized.1. INTRODUCTION

It is now over twenty-five years since the finite elementmethod first was used in the solution of practical struc-tural engineering problems. Therefore, it may be ap-propriate at this time to examine the accqmplishments ofthis past quarter century of phenomenal development.This task has been undertaken with trepidation becausethe subject has grown in breadth and refinement to thepoint where it cannot be evaluated adequately by anyone observer. Nevertheless, it is hoped that this highlysubjective view may be of some value because it is aproduct of a continuing interest in and contact with thefinite element method during this entire period.Admittedly, taking a backward view like this suggeststhat one is no longer looking ahead, and this implicationis at least partially valid in the present case because thedirection of my research has changed. Almost a decadeage I became concerned that the advancement of struc-tural analysis capabilities was progressing much morerapidly than was knowledge of the basic material andstructural component behavior mechanisms, at least forthe nonlinear response range. This deficiency of experi-mental data was particularly evident in the field of ear-thquake resistant design, where the structural perfor-mance must be evaluated during large cyclic excursionsinto the nonlinear range. Therefore, during most of thepast decade I have followed this alternate path ofdynamic experimental research, and have been involvedonly peripherally with recent developments in the finiteelement field.At the outset of this review, it is important to expressmy concern over the tendency for users of the finiteelement method to become increasingly impressed by thesheer power of the computer procedure, and decreas-ingly concerned with relating the computer output to theexpected behavior of the real structure under in-vestigation. This concern is similar to that expressed lastAugust by Oden and Bathe in their Commentary on

Computational Mechanics[l], wherein they decried thecomplacency and overconfidence of number-crunchingexperts. They illustrated this attitude by the opinion ofone such expert that within the next decade the onlyuse aerodynamicists will have for wind tunnels is as aplace to store computer output. To the technologist whocommunicates only with a computer this may seem like areasonable assessment; but the opinion certainly is notshared, for example, by engineers designing for windloading on buildings who can estimate such loads only

from boundary layer wind tunnel experiments. In fact,experimental evidence on how structures actually behaveis usually the surest cure for overconfidence on the partof computer enthusiasts, and I am pleased to note thatthere is a growing trend toward experimental verificationof the results of extensive computer calculations.In order to provide a broad commentary on thedevelopment of the finite element method, this paper hasbeen divided into three parts. First is a look back to theearly days of the finite element development, when themethod was viewed in the context of an extension ofstandard methods of structural analysis. This emphasison application to the solution of real problems ofengineering practice is reflected in the remainder of thepaper as well. The second part is an assessment ofrecent advances in structural dynamic analysiscapabilities, where the point of reference is a pair ofstate-of-the-art papers written IO and 7 years ago.Emphasis is placed on dynamic analysis because thestatic response may be looked upon as a special case ofthe dynamic problem. In the third part of the paper,difficulties encountered in the analysis of two types ofstructures will be discussed. The purpose of this sectionis to demonstrate that major problems still remain tochallenge the ingenuity of the finite element researcher,and also that experimental research is essential to dis-covering, defining, and eventually solving many of theseproblems. In this latter opinion, I tend to differ from theviewpoint expressed in the paper by Oden and Bathe[l]which emphasized the necessity for fundamentalresearch in applied mechanics and mathematics as theprimary approach to a wide range of unsolved problemsin computational mechanics.

2. EARLYDEVELOPMENTSAs with any major development in engineering ormechanics, the early history of the finite element method(FEM) can be traced along several paths; certainly no

single view of the origins can cover all facets of thedevelopment. Moreover, as more individuals andorganizations began working with this engineering tool,the advances become increasingly diffuse-so detailingthe history for more than a few years quickly becomes atask for specialists in the history of engineering science.Accordingly, this account does not pretend to provide adefinitive history of the FEM; instead it merely givessome personal observations on the early days as seen byone of the participants.361


2/10

362 R. w. CLOUGHIt was the words Engineering Application in thename of this Conference that encouraged me to preparethis historic summary because the early development ofthe FEM with which I was involved was continuallydirected toward engineering applications. Initially, it wasa tool designed and developed to solve real engineeringproblems, even though that outlook is quite distinct from

much of the research presently being done in the finiteelement field.It isnt possible to identify the exact starting point ofthe finite element method; and to spend time in con-jecture on this question is a meaningless exercise,because the method makes use of many theories andtechniques drawn from mathematics and continuummechanics. One aspect of the FEM, mathematical mode-ling of continua by discrete elements, can be related towork done independently in the 1930s by McHenry[2]and Hrennikoff [3]-formulating bar element assemblagesto simulate plane stress systems. Indeed, I spent thesummer of 1952 at the Boeing Airplane Company tryingto adapt this procedure to the analysis of a delta airplanewing, the problem which eventually led to the FEM; butbecause the technique could not effectively deal withplates of arbitrary configurations this effort was soonabandoned.A more significant preliminary to the development ofthe FEM was the matrix generalization of structuraltheory in which the analysis was formulated as a form ofcoordinate transformation. The earliest knownreferences to the assembly of structural elements by amatrix coordinate transformation were byFalkenheimer[4] and Langefors[5]. However, the classicwork which completely stated the matrix formulation ofstructural theory, and which clearly outlined the parallelprocedures of the force and displacement methods, wasthe series of articles first published in Aircraft Engineer-ing by Argyris et al.[6]. It was this work which demon-strated that the concepts of classical structural analysiscan be generalized for application to assemblages of anytype of structural elements, not only to the traditionalbeams, struts, etc.However, the true finite element concept is concernedprimarily with the discretization process, not with theprocedure used to analyze the system after the discreteelements have been identified and evaluated. Specifically,the FEM discretization involves the assumption of strainor stress fields defined on a regional basis, rather thanreplacement of the actual continuum by a set of sub-stitute elements. Of course, this general concept appliesto well known approximation methods of continuummechanics, such as the Rayleigh-Ritz method, and it istrue that the FEM may be looked upon as a special formof such methods. But the unique feature of the FEM wasthe idea of defining the strain field independently for thevarious regions or elements into which the continuumwas divided.Although this regional discretization concept had beenproposed earlier[7,8], it was only when it was used by anengineering organization as a means of avoiding thedifficulty of physical discretization by bar assemblagesthat the method really began to develop. In addition, theconcurrent availability of effective digital computers andof the matrix formulation of structural analysis wereessential factors in the early development. Also, tomaintain proper perspective on the early days of theFEM, it is important to realize that its utilization did nottake off explosively; during the first six or seven years

the application of the method spread very slowly indeed.The work which I associate with the beginning of thecomputerized FEM was done during summer 1953 whenI was again employed by Boeing Airplane Company ontheir summer faculty program. Again, I was assigned toMr. M. J. Turners Structural Dynamics Unit, to work onmethods of evaluating the stiffness of a delta airplanewing for use in flutter analysis. Because the bar assem-blage approach tried during the previous summer hadbeen unsatisfactory, Mr. Turner suggested that weshould merely try dividing the wing skin into appropriatetriangular segments. The stiffness properties of thesesegments were to be evaluated by assuming constantnormal and shear stress states within the triangles andapplying Castiglianos theorem; then the stiffness of thecomplete wing system (consisting of skin segments,spars, stringers, etc.) could by obtained by appropriateaddition of the component stiffnesses (the direct stiffnessmethod). Thus, at the beginning of the summer, 1953, Mr.Turner had completely outlined the FEM concept andthose of us working on the project merely had to carryout the details and test the results by numerical experi-ment.Our paper describing this initial effort was presentedat the New York meeting of the Institute of AeronauticalSciences in January 1954[9]. I have never known whythe decision was made not to submit the paper forpublication until 1955, so the publication date of Sep-tember 1956 was more than two years after the firstpresentation and over three years after the work wasdone. As was mentioned, this is graphic evidence that theFEM did not attain instant recognition. Undoubtedly, amajor factor which limited its acceptability was that theoriginal work was done in the Structural Dynamics Unit,where the objective was limited to stiffness anddeflection analysis; it was several years before theconcept was accepted and put to use by the stressanalysis groups at Boeing. Thus, it is possible that theorientation of this initial step toward a specific engineer-ing application tended to obscure the general ap-plicability of the FEM concept, even though the in-dividuals working with the development at Boeing werequite aware of its broader implications.Although I maintained close contact with several ofmy Boeing colleagues for many years after 1953, I didnot work there again and I had no opportunity for furtherstudy of the FEM until 195657, when I spent my firstsabbatical leave in Norway (with the Skipsteknisk For-sknings Institutt in Trondheim). This Norwegian con-nection also was a factor in my decision to prepare ahistorical summary for this Conference; it was thisperiod which made possible my continued contact withthe finite element concept. Lack of computer facilities inNorway limited the type of work I could do at this time,but I was intrigued by plane stress applications of themethod and I carried out some very simple analyses ofrectangular and triangular element assemblages using adesk calculator. Although this work was too trivial towarrant publication, it convinced me of the potential ofthe FEM for the solution of general continuum problems.About the time I returned to Berkeley from my sab-batical leave, the Engineering College acquired an IBM701 Computer (replacing the old Card Programmed Cal-culator) and we began to develop structural analysiscapabilities with this machine. For educational purposes,a Matrix Interpretive Program of the type pioneered inEngland[lO] offered the best means of making the com-


3/10

The finite element method after twenty-five years 363puter capabilities accessible to the students, and most ofmy early efforts went into developing such aprogram [111.Then it was possible to continue my workwith the FEM, which had been undergoing continuingdevelopment at Boeing but had attracted only littleattention elsewhere.Early FEM studies at Berkeley were greatly limited bythe two thousand word central processor capacity of theIBM 701, but by utilizing the seven 2000 word drumstorage units it was possible to carry out some creditableanalyses. Our first concentrated effort toward planestress analysis was in response to a challenge by one ofmy continuum mechanics colleagues who was scepticalof the validity of the procedure and wanted to see asolution of some classical problem. To me it seemedobvious that the method could solve any plane stressproblem to any desired accuracy-limited only by thetime and energy one wished to expend on the cal-culations. But in the hopes of attracting wider interesttoward the FEM concept, I allocated part of a small NSFresearch grant to the solution of a few sample planestress problems. The results were as good as I hadexpected, so a paper was prepared. The principal prob-lem that arose in writing the paper was choosing asuitable name for this analytical procedure and I decidedfinally on the Finite Element Method. This name firstappeared in that paper[l2], and I can only conclude fromsubsequent history that it was an apt choice.In retrospect, the next red letter event in my personalFEM history occurred in December 1960, when Profes-sor 0. C. Zienkiewicz invited me to Northwestern Uni-versity to give a seminar lecture on the new procedure.We were friends from previous meetings, and I knewthat he had been brought up in the Southwell finitedifference tradition, so it was apparent that his invitationwas prompted by scepticism and a desire to discuss therelative merits of finite elements vs finite differences.Certainly, we did have such discussions during my visit,but Professor Zienkiewicz obviously is a very intelligentperson and was quick to recognize the advantages of theFEM. During that short visit an illustrious convert waswon to the cause, and I think it is not coincidental thatrapid worldwide acceptance of the FEM started almostfrom that moment.

The 1960 paper was not a major work because itdescribed no new ideas; it was useful mainly in intro-ducing the FEM to the Civil Engineering profession. Myprincipal interest at that time was in developing themethod as a general purpose tool for the analysis ofarbitrary shell structures. One of my students alreadyhad worked on the FEM analysis of plate bending, butwhen we were about to prepare this paper for publicationI learned that my former associates at Boeing alreadyhad prepared an internal report on plate bending analysisusing finite elements. So this report was merely filed withmy sponsors at NSF[13] and my student continued withthe next step of combining plate bending and plane stressstiffness to obtain flat plate shell elements. Rectangularelements were derived for analysis of cylindrical shellsand triangular elements for shells of arbitrary geometry.Significant results were obtained[l4], but even in thisearly work it was apparent that an ad hoc approach toextending the FEM left much to be desired. Specificallythe triangular plate bending element was not performingsatisfactorily and it was necessary to go back for anotherlook at that problem. Results of a follow-on study[l5]were still less than satisfactory and it was not until 1964

that an adequate solution had been obtained for thisproblem.The other important FEM development at Berkeley inthe early 1960s was that we obtained a research contractfrom the U.S. Corps of Engineers for a practicalengineering investigation; specifically they were concer-ned with the state of stress in a concrete gravity dam thathad developed a major internal crack during constructiondue to temperature effects[l6]. This contract provided usfor the first time with enough money to support thedevelopment of a general purpose plane stress analysisprogram[l7]. It was this program that subsequentlyproved to many engineers the great power of the FEM;even more important, writing the program allowed mystudent at that time, E. L. Wilson, to develop anddemonstrate his great flair for finite element work. We atBerkeley are greatly indebted to the Corps of Engineersfor their contribution to this cause, as well as for supportof our finite element research through many years.One other aspect of the early development of the FEMshould be mentioned-the important role played by in-ternational conferences. To a large extent, the rapidworldwide expansion of interest and activities in themethod is attributable to these conferences, which al-lowed for formal presentations of ideas and for personalmeetings between the active researchers. To my know-ledge, the first truly international conference dealingextensively with computer analysis of structures washeld in Lisbon in September 1%2[18]. The subject of thisconference was Numerical Methods in Civil Engineering,and the FEM was the central theme of only one paper;but the conference provided a good forum for discussionof the relative merits of the finite difference and finiteelement procedures in structural analysis. By 1%5, whenthe first Conference on Matrix Methods in StructuralMechanics was held at the Wright Patterson Air ForceBase[l9], the expansion of interest in and activity withthe FEM was phenomenal. This milestone event broughttogether from all over the world nearly all researcherswho had done significant work with finite elements. Atthe conclusion of the Conference, it was evident that theFEM had come of age; its potential for solving practicalproblems had been demonstrated in many structural dis-ciplines, and powerful computer programs had beendescribed which could deal routinely with problems ofevery description. Of course tremendous advances inunderstanding and in computational capability have beenmade since 1%5, but this Conference, held only about adecade after the first preliminary applications, showedthat the FEM should be recognized as the major analy-tical tool in the field of structural mechanics.

3 RECENT ADVANCESAdvances in finite element methodology since 1%5have been so rapid and diverse that it is impossible tochronicle them here, even considering only a very limitedsegment of this rapidly broadening field of mechanics.However, it may be useful to continue this personal viewwith a very limited discussion of the directions in whichmore recent work has been headed. The point of depar-ture for this review is the 1%9-72 state-of-the-art ofstructural dynamic response analysis, as summarized intwo papers prepared for the U.S.-Japan Seminars onMatrix Methods of Structural Analysis[20,21]. As wasmentioned earlier, this is about the time that my interestturned toward experimental research: so it is evident thatmy comments on these recent developments present the


4/10

364 R. W. CLOUGHviews of an interested observer, but not an active parti-cipant in finite element research.The field of dynamic response analysis is appropriatefor the present discussion if the static case is consideredmerely as a special case of the dynamic problem, inwhich inertial and damping effects are not involved, andwhere equilibrium need be satisfied only at one timerather than at a succession of times during the response.In the 1%9 and 1972 papers, the response analysis pro-cedure was divided into two phases: (a) formulation ofthe equations of dynamic equilibrium, and (b) solution ofthese equations in response to the given condition ofdynamic loading. It will be convenient to discuss thesetwo phases separately here.3.1 Formulation of the equations of motionEstablishing the equations of equilibrium in a finiteelement analysis involves three essential steps: I) ideal-izing the actual structural system as an assemblage ofdiscrete elements, (2) evaluating the mechanical proper-ties of the elements, and (3) assembling the elementproperties to obtain the corresponding system properties.By 1%9, the concept of finite element discretization waswell developed, a wide range of element types wasavailable for providing reasonable idealizations of arbi-trary structures, and efficient coordinate systems andinterpolation functions had been established for evalu-ating the element mechanical properties.Moreover, procedures of assembling the element pro-perties to obtain the system properties were well under-stood in 1%9. To a great extent, these procedures aremore closely associated with computer program codingthan with finite element theory, and little need be saidabout them here. The only exception is the concept ofsubstructuring, or more specifically the recognition thatsubstructures can be looked upon as superelementsand thus may be incorporated directly into the finiteelement assembly sequence. The substructure idea waswell understood in 1%9, but great progress has beenmade since then in designing programs that use multi-level substructuring as a routine feature of the analysisprocedure [22].Thus the only phase of formulation of the equations ofmotion remaining to be mentioned is the evaluation ofthe element properties, which for a dynamic analysisinclude the stiffness, mass, damping and external load. Inthis discussion each of these properties will be treated inturn.3.1.1 Stifness. By far the most significant mechanicalproperty in most structural problems is the stiffness;accordingly the majority of finite element research hasbeen devoted to developing more efficient elementswhich provide a better approximation of the actualstructure stiffness at less computational cost. By 1%9great progress had been made in this direction, andalthough much research has gone into this area sincethen, I think it has been with considerably diminishingreturns. Although some significant improvements ofelement types have been made, these have had littleinfluence on most practical engineering applications. Arecent paper by Olsen(23] supports this opinion; henoted that the old triangular flat plate element stillremains competitive in analysis of general thin shells, inspite of extensive work done with refined shell elements.3.1.2 Mass. An effective description of the inertialresistance of a structure is fundamental to a dynamicresponse analysis, and the consistent mass concept[24]

was introduced early in the history of finite elements toprovide a rational basis for forming mass matrices.However, a strong case was made in the I%9 paper forlumped mass representation rather than consistent mass,on the basis that inertial effects required a less refineddiscretization than did the elastic resistance. Subsequentresearch has supported this opinion, and some recentresearch has been concerned with developing proceduresfor effectively lumping the mass of arbitraryelements [25].3.1.3 Damping. In contrast to the other mechanicalproperties, definition of damping or energy loss charac-teristics continues to be an elusive problem. Because solittle is known about the actual damping processes, thedamping property generally is defined at the assemblagelevel rather than for individual elements. In most cases aviscous damping mechanism seems to give adequatecorrelation between observed structural behavior andanalytical predictions; experimental data usually is toolimited to warrant development of more refined mathe-matical formulations.Effective procedures were described in the I%9 paperand subsequently[26] for explicit definition of a propor-tional viscous damping matrix. Considerable interest hasarisen since then in non-proportional damping, due inpart to concern over radiation damping effects asso-ciated with soil-structure interaction. The classical tech-niques of treating non-proportional damping (based onmode-superposition using the complex mode shapes)have been employed in some finite element analyses [27];but a preferable approach in most earthquake engineer-ing problems is to transform to a reduced set of un-damped modal coordinates, and then to solve the cou-pled modal equations[28]. The cost of solving the com-plex eigenproblem associated with non-proportionaldamping, and the subsequent cost of expressing theresponse in complex modal coordinates far exceeds thecost of calculating the response with a coupled set ofcoordinates.Another major recent development in treating dam-ping, which will be mentioned later, is through the use ofa frequency domain solution. But the major fact whichcontinues to limit the treatment of damping in dynamicsystems is the paucity of experimental data concerningthe actual energy loss mechanisms in real structuralsystems.3.1.4 Load. Definition of the load terms in the equa-tions of motion also generally is done at the level of theassembled structure rather than with respect to the ele-ments, and the principal problems with this factor usu-ally are due to lack of knowledge of the true loadingmechanisms rather than with the discretization process.Therefore, progress in this area depends more on obtain-ing experimental data than on finite element research,and little change can be reported since 1969. Significantadvances have been made recently in related areas in-volving fluid-structure interaction and foundation-struc-ture interaction, especially in response to earthquakeexcitation; these topics will be mentioned briefly later,but generally they are outside the scope of the presentdiscussion.3.2 Solution of the equilibrium equationsIn discussing the second phase of the finite elementanalysis, the solution of the equations of equilibrium, it isuseful to divide the subject into several types of cate-gories. First, static and dynamic problems will be


5/10

The finiteelement method after twenty-five years 365separated because of the radically different analyticalprocedures that may be employed in the solution of thedynamic problem. Then the dynamic analyses may beseparated according to the type of coordinates used inthe solution: modal or discrete system coordinates. Ano-ther classification concerns the domain of the analysisprocedure: time domain or frequency domain.

An alternative approach to categorization would be toseparate linear problems from nonlinear. In discussingthe nonlinear analyses, a generally similar approachcould be adopted for both static and dynamic cases, thelatter merely requiring the inclusion of extra terms torepresent the effects of inertia and damping. However,the first approach is more satisfactory for the purpose ofthis paper.3.2.1 Static analysis. Because the development ofefficient algorithms for the solution of large equation setshas been a primary objective for finite element resear-chers since the method first was put to practical use, thecapability in this area already was quite advanced by1%9. Powerful equation solvers were available then, andextensive research since 1%9 has further advanced thestate-of-the-art. Undoubtedly many of the mostsignificant recent advances have pertained to analysis onnonlinear structures because the nonlinear analysis costfor large practical systems has been almost prohibitive.The continuing active interest in this subject is evidencedby the many major international conferences recentlydevoted to it[29,30]; also it is significant that two ses-sions of this present conference deal with nonlinearanalysis.3.2.2 Dynamic coordinates. The first major decision tobe made in planning a dynamic response analysis is thetype of coordinates to be used. One option is merely tosolve directly the equations of motion expressed in theoriginal system coordinates, but this has the disad-vantage that the original equation set may involve hun-dreds or thousands of degrees of freedom. An alternativeis to transform to the natural or modal coordinates of thestructure which describe the dynamic response moreefficiently, and therefore, may be reduced in number.Each of these approaches will be discussed briefly.

(a) Modal coordinatesIf the structure is linearly elastic, it is usually desirableto transform the equations of motion to modal coor-dinates and then to evaluate the response in terms of atruncated modal set. Solution of the structure eigen-problem to obtain the modal coordinates is a majorcomputational task, and very effective eigenproblemsolvers were developed quite early, as is described onthe I%9 and 1972 papers. Research in this area hascontinued since then, and further refinements of tech-nique are being made, but it is not likely that a majorbreakthrough will result.The principal decision facing the analyst in a modalcoordinate solution is the number of modes to be in-

cluded. Clearly this depends on both the spatial dis-tribution and the frequency content of the applied load-ing. If the load pattern is distributed widely over thestructure, it tends to excite mainly the lower modes ofvibration which may have distribution patterns similar tothe loading. However, if only a small portion of thestructure is attacked by the external load, many modeswill be excited.In the past it generally has been assumed that allmodes which are excited significantly must be included

in the dynamic response analysis. However, it is evidentthat the amplitude of the dynamic response in any modedepends on the frequency content of the loading. Thehigher mode response to a low frequency excitation may beessentially static in character, and recently [3 ] it has beendemonstrated that these modal contributions can be ac-counted for adequately by a static correction term. Thus astandard mode superposition response is performed foronly those modes which are subject to dynamic am-plification: the response to the load associated with thehigher modes is evaluated by a static analysis, and thisstatic correction is added to the dynamic response.In general, the modal coordinate approach is notrecommended for nonlinear analysis; however, somestudies have shown that efficient nonlinear solutions canbe obtained with modal coordinates [32,33]. Clearly thisapproach can be effective only if the dynamic responsecan be expressed conveniently by superposition of themode shapes, and therefore it is applicable to systems inwhich the nonlinearity does not drastically modify thevibration shapes. In principle, one would expect goodperformance where the nonlinearity is widely distributedover the structure, but not where local concentrationssuch as plastic hinges are involved.

(b) System coordinatesIntegration of the equations of motion expressed insystem coordinates is the standard approach to nonlinearanalysis, but significant savings may be effected if thestructure has only localized nonlinearity[34]. In this casea substructure analysis procedure may be adoptedwherein all nonlinear parts of the structure are includedin a single substructure. The elastic portions of thesystem are included in other substructures, and thedegrees of freedom not required for interconnection tothe nonlinear component are removed by condensation.Thus the number of degrees of freedom required forconsideration in the nonlinear analysis may be greatlyreduced from the original coordinate set.3.2.3 Dynamic domain. The other basic option opento the dynamic analyst is the domain in which theanalysis will be performed. In 1%9-72 this questionseldom arose-step-by-step integration of the equationsof motion was the only technique capable of dealing withlarge scale dynamic applications of the FEM. During thepast decade, however, the extremely efficient FastFourier Transform (FFT) programs for numerical evalu-ation of Fourier transforms have been discovered bystructural engineers, and have proven to be advan-tageous in many practical cases. Comments on bothfrequency domain and time domain analyses follow.

(a) Frequency domainFrequency domain analyses involve the superpositionof the response of a structure to various harmonic exci-tations; therefore, they are only applicable to linearsystems. Moreover, because the analysis must be per-formed for many hundreds of frequencies it will beextremely expensive when dealing with systems havingmany coupled degrees of freedom. Therefore, frequencydomain calculations generally are performed upon un-coupled modal coordinate sets.One of the principal areas of application of thefrequency domain approach to dynamic structuralanalysis is in situations involving liquid-structure or soil-structure interaction, because the continuous mediumcan be treated without discretization and coupled


6/10

366 R W CLOUCH

directly to the finite element model of the structure.Many important applications of this type of analysishave been reported in recent years[35,36], and this isone of the significant areas of advances since l%9.Another practical application of frequency domainanalysis is deconvolution, i.e. evaluating the input load-ing history from measured response data[37]. Suchanalyses can be performed easily in the frequencydomain but are very difficult to do satisfactorily in thetime domain. Except for special types of problems suchas those mentioned above, however, frequency domainanalyses seldom are superior to direct time domain in-tegration of the equations of motion.

Improvement in ~nderstandi~ of the process of directintegration of the equations of motion is one of the mostsignificant achievements of structural dynamics researchduring the past decade. The step-by-step analysis pro-cedures described in the 1%9 and 1972 papers are stillvalid and widely used, but research donesubsequently[3&40] shows that these methods aremerely members of much broader families of analysistechniques. These more recent studies (and thereferences cited above are merely arbitrary examples ofnumerous papers in this field) show that it is possible todesign a step-by-step procedure to have desired ap-proximation characteristics with respect to such factorsas period elongation and artificial damping. However, theolder methods, even going back 20 years to the pioneer-ingwork by Newmark[41] still serve well in many cur-rent applications.

4 EXAMPLES OF CURRENT PROBLEMSAlthough remarkable progress has been made with theFEM during its first quarter century, and the basic pro-cedures of structural dynamic analysis have advanced tothe point where solution of arbitrary dynamic problemsappears to be routine, it is obvious to those working withgeneral engineering applications of the method that newproblems and difficulties are encountered regularly. Inthis section of the paper two examples of problems thathave confronted the author during the past year aredescribed. These problems are viewed from the stand-

point of an FEM user rather than a program developer;they illustrate limitations of analysis capabilitiespresently available to the user, and therefore, are offeredas challenges to those involved in program development.The two examples are drawn from earthquakeengineering and concern the seismic response of (a) thinshell metal cylindrical liquid storage tanks, and (b) thinshell concrete arch dams. Although both involve liquid-structure interaction, the problems identified in the twocases are fundamentally different. However, bothdemonstrate that num~r-cruncher complacency ispremature at this stage of development of the FEM, andthat ex~rimental research is essential to further pro-gress-both in understanding of mechanical processesand in verification of analysis procedures.4.1 Response o liquid storage tanksExamples of damage to ground supported liquidstorage tanks are noted after nearly every major ear-thquake, and designer efforts to improve their ear-thquake resistance have been under way for over twentyyears. The present design procedure is based on anapproximate evaluation of the hydrodynamic pressures

induced by the earthquake, assuming that the tank isrigid. From these pressures the base shear and overturn-ing moment are determined, and then the resulting shellstresses are calculated by elementary beam theory,treating the tank as a vertical cantilever.An obvious limitation of this approach is the rigid tankassumption; a thin metal tank can be expected to deformsufficiently during the seismic response to permitsignificant dynamic amplification of the liquid pressures,Accordingly, several investigators have developed finiteelement programs[42,43] which are intended to accountfor this seismic fluid-structure interaction. The basicassumption of these programs. drawn from hydro-dynamic theory, is that a horizontal base motion alongone axis will produce only a first Fourier harmonic(cos 8) dis~ibution of pressure and deformation. Thus asingle term axisymmetri~ shell analysis procedure hasbeen adopted.In order to verify results of such analyses (which areapplicable only to fixed base tanks) and also to providequantative information on the response of unanchoredtanks, a group of tank manufacturers and designerssponsored an experimental study at the University ofCalifornia using the 20 x 20ft shaking table facility[44].Figure 1 shows a tall tank on the shaking table, sur-rounded by (but not in contact with) its reference frame.It may be noted here that unanchored tanks are usedfrequently because they do not require the expensivefoundations needed for anchorage, but even moderateearthquake motions will induce rocking uplift of the tankwalls from the foundation and only very crude estimatesof this dynamic response mechanism are presentlyavailable to designers.Probably the most significant and surprising result ofthis test program was the observation that fixed basetanks did not respond only in the first Fourier harmonic;important cos20 and cos38 components of pressure,displacement and stress also were observed in tests ofboth short [44] and tall[45] tanks. Some of the tall tankresults are included here to illustrate this unexpectedresponse behavior. Figure 2 shows deflected shapes ofthe tank cross section at several times during a typicaltest, and Fig. 3 depicts the relative maximum amplitudesof the Fourier components contained in various responsequantities, These figures clearly show that the standardfinite element analysis, which recognizes only the cos 0component, cannot provide much understanding of theactual fixed-base tank response.In the case of the free-base tank, out-of-round res-ponse was expected because the axial symmetry of thebase support condition is destroyed by uplift. However,comparison of the axial stress distribution about the baseof the unanchored tank with an estimate based on acurrent design procedure, shown in Fig. 3, demonstratesthe need for complete revision of the present designconcept. Clearly the rocking mechanism concent~tes thebase section axial stress in a much narrower arc than hadbeen expected, with the result that the measured stressexceeds the design estimate by a factor of about 4. Thebehavior in this case is highly nonlinear with regard tothe support forces about the base, but no effort yet hasbeen directed toward developing a nonlinear analysisprocedure to deal with this important problem.In the context of this paper, the first conclusion drawnfrom this example is that experimental verification isessential for ensuring reliability of finite element pro-grams. The test of the anchored tanks had been looked


7/10

The finit e element method after twenty-five years 361

Fig. I. 7 -3/4x I5 ft tank with reference frame on shaking table.

I 89 5 et 1.9673 et

2.0075 Set 2.0441 Set 2.0825 SetScale H 1.97E-01 In)

Fig. 2. Deflected shape of tank top during shaking table test.


8/10

368 R. W. CLOUGH

^ +-+ -za 05

0 nnr-lrlrlnHYDRO-DYNAMIC PRESSURE AT 9HEIGHT0 15 1 - I

: 01052 005.;0 -

: SHELL DEFORMATION AT ISHEIGHTIf 4000 ri: 3000aY 2 2000aaw-a 1000

0 r l nt7llllAXIAL MEMBRANE STRESS AT 5A BOVE BASE2000 -=,Jl IcaO~ nP

nil--n-HOOP MEMBRANE STRESS AT 5A BOVE BASE

SHEAR MEMBRANE STRESS AT 5 ABOVE BASE

Fig. 3. Peak amplitu de of Fourier componentsshaking able testof tank.

upon only as a routine check of the finite element ap-proximation; the observed out-of-round response wastotally unexpected. Theoretical mechanics has not yetprovided a basis for predicting these response com-ponents, and of course, the finite element analysis cannotgo beyond its theoretical origins no matter how refinedthe mesh or capable the computer. A second closelyrelated conclusion is that experimental data is essentialfor developing adequate numerical approximations whentheory is lacking. Certainly, future improvements inanalysis techniques for either anchored or unanchoredtanks will depend heavily on experimental results such asthose mentioned here, and development of a finite ele-

ment program which can deal effectively with either ofthese design problems is a major challenge to the pro-fession of computational mechanics.4.2 Concrete arch damsThe seismic performance of concrete arch dams is aproblem of even more critical concern to engineers andthe general public because a large reservoir could con-stitute a major hazard to the people living below it.Although the past earthquake performance of arch damshas been excellent, non-earthquake failures such as thoseat Malpasset, France and Vaiont, Italy demonstrate thatdams embody a significant potential for disaster. There-fore, great emphasis presently is being placed on seismicsafety evaluations of major dams in many regions havingactive seismicity.Superficially the analysis of earthquake stresses in aconcrete arch dam would appear to be well within thepresent state-of-the-art of dynamic finite element analy-sis. Indeed, computer programs have been written forthat specific purpose [46], and general purpose programssuch as SAPIV also have been used extensively in suchstudies. However, when these analyses are examined indetail their serious limitations become apparent. Forexample, reservoir interaction is treated only ap-proximately at best-using added masses derived bymeans of incompressible liquid elements. However, two-dimensional gravity dam analyses have demonstrated theimportance of liquid compressibility in the earthquakeanalysis of such structures[44], and it should be equallyimportant in a three-dimensional arch dam analysis; yetno program accounting for arch dam-compressible liquidinteraction is presently available.Possibly even more critical in this arch dam-reservoirinteraction behavior is the fact that added mass analysesindicate negative net fluid pressures in response to asevere earthquake. In other words, the peak negativedynamic pressures momentarily exceed the hydrostaticpressures, and therefore cavitation should result.However, no analysis procedure is available to deal withthis nonlinear interaction mechanism, either for com-pressible or incompressible liquids.The other major limitation of present seismic responsestudies for concrete arch dams is that they all treat thestructure as a linearly elastic system. For the staticdesign conditions this assumption is reasonable, but theresponse to a major earthquake is expected to includetwo significant forms of structural nonlinearity. The firstresults from the fact that typical arch dams are con-

SCALE W 7.09lN SCALE c--i 2279PSl0) UPLIFT OF BOTTOM PLATE b ) AXIAL STRESS DISTRIBUTIONFig. 4. Comparison of observed and predicted performance during test of unanchored tank


9/10

The finite element methodafter twenty-fiveyears 369strutted as a series of independent monoliths separatedby joints extending from foundation to crest and fromupstream to downstream face. These joints are intendedto minimize cracking due to shrinkage and temperaturechange during construction; after construction they areclosed by grouting and/or the reservoir pressure on theupstream face, and no movement of the joints occursduring normal static load conditions. But during a severeearthquake, tension and bending effects are induced inthe arch rings which can overcome the static com-pression and cause joint opening. Consideration of thisjoint behavior is essential to understanding of theseismic response; the tensile arch stresses indicated bylinear dynamic analyses obviously are inconsistent withthe actual jointed construction of the dam. inclusion ofthis type of nonlinearity is well within present finiteelement analysis capabilities, but it has not yet beenapplied in any significant practical investigation.The second type of structural nonlinearity expectedduring a major earthquake involves cracking of the con-crete. Because the tensile strength of concrete is onlyabout IO percent of its compressive strength, stressesindicated by linear response analyses for a major ear-thquake often exceed the cracking limit. As was men-tioned above, the vertical construction joints wouldprevent development of arch tensile stresses, but crack-ing would be expected on horizontal planes due to can-tilever tensile stresses in the monoliths. Thus, during asevere earthquake an extensive network of horizontalcracks might develop, which could combine with thevertical joints to divide the dam into a system of separateblocks. Because the cracking tendencies are associatedwith high frequency vibrations, little relative displace-ment would be expected between the blocks; thus onewould intuitively expect that the arch mechanism of thedam would not be adversely affected by such damageand that it would continue to support the gravity andwater loads after the seismic input ended. However, suchintuitive estimates seldom are convincing to Boards res-ponsible for public safety; analytical procedures whichcan follow the history of cracking, joint opening, andblock displacement, and can then evaluate the stability ofthe system, are urgently needed.In concluding this discussion, it may be noted that thethree types of nonlinearity identified in the seismic res-ponse of an arch dam-cavitation, joint opening, andcantilever cracking-each present formidable obstaclesto a finite element analysis, and experimental data on theactual behavior would be required before any such pro-gram could be accepted for routine use. On the otherhand, the difficulties of conducting a valid experimentalstudy of these mechanisms also are great, becausedynamic similitude requirements cannot easily besatisfied with the very small scale models that could beemployed. Consequently, a combined analytical-ex~ri-mental approach probably offers the best prospects fordealing with this important engineering problem.

5. CONCLUDING REMARKSThe purpose of this paper has been to provide someperspective on the background of the FEM and on itscurrent capabilities in application to engineering prob-lems. It is well to recall that the emphasis during earlydevelopment of the method was oriented totally towardpractical application. At present it probably is fair tosay that the state-of-the-art has advanced to the pointwhere solution of any structural engineering problem can

be contemplated, but there may be a wide variation inthe quality of the result obtained. Depending on thevalidity of the assumptions made in reducing the physicalproblem to a numerical ~go~thrn~ the computer outputmay provide a detailed picture of the true physicalbehavior or it may not even remotely resemble it. Acontrolling influence on where the final result lies alongthis scale is the skill of the engineer who prepares themathematical idealization; when dealing with complexand unusual structures, this phase of the analysis is anart and the program cannot be treated merely as a blackbox.Because of the significant possibility that the analysismay have totally overlooked or misjudged some im-portant aspects of the mechanical behaviour, experi-mental veri~cation should be inco~orated into theanalytjcal process whenever it steps beyond the bordersof experience and established practice. In addition,experimental studies often will be required when theanalysis is breaking new ground in order to define andquantify the parameters which characterize the system.For these reasons, investigations of new structural sys-tems should be planned by means of a combined analy-tical-experimental approach.

REFERENCESI. J. T. Oden and K. J. Bathe, A commentary on computationalmechanics. App~.Meek. Reu.31(8), 1053-1058 1978).2. D. McHenry, A lattice analogy for the solution of planestress problems. J. Ins&. Citiif Engrs.Z(2), (1943-44).3. A. Hrennikoff, Solution of problems in elasticity by the

framework method. I. Appl. Meek. 8, Al@-175 (1941).4. H. Falkenheimer, Systematic calculation of the elastic charac-teristics of hyperstatic systems. La Reckeroke AerorrautiqueNo. 17 (Sept.-Oct. 1950 ,No. 23 (Sept.-Oct. 1951).S. B. Langefors, Analysis of elastic structures by matrix trans-formation, with special regard to semi-monocoque structures.I; Aeron. Sci. 19(7), 1952).6. J. H. Argyris, Energy theorems and structural analysis. Air-craft Engineetitrg 1954-55, eprinted by Butterworths, Lon-don 1960).7. R. Courant, Variational methods for the solution of problemsof equilibrium and vib~~ion. quiz. Am. Math. Sot. 49, I-43(19431.8. W. Prager and J. L. Synge, Approximation in elasticity basedon the concept of function space. Q. J. Appl. Math. 5,241-269 1947).9. M. J. Turner, R. W. Ciough, H. C. Martin and J. I,. Topp,Stiffness and deflection analysis of complex structures. .I.Aeron. Sci. 23(9), 805-823 1956).IO. C. P. Hunt, The electronic digital computer in aircraft struc-tural analysis. Aircraft Engineering,Vol. 28, March (p. 40)Aoril lo. Ill Mav CD 155).Butterworths. London (1956).I I. R. W.Clo&h, t&ural analysis by means of a matrixalgebra program. Proc. ASCE Conf. Hectronic Compufarion,pp. 109-132,Kansas City (Nov. 1958).12. R. W. Clough, The finite element method in plane stressanalysis. Proc. 2nd ASCE Conf. E~ect~nic Co~putution,Pit&burg,Pennsylvania(Sept. 1960).

13. A Adini and R. W. Clough.Analysis of Dbte bendinnby thefinite element method. R&&t submittedto National-ScienceFoundation, Washington, D.C. (Grant G7337) 1960).14. A. Adini, Analysis of shell structures by the finite elementmethod. Ph.D. Dissertation, University of California, Ber-keley (1961).15. J. L. Tocher, Analysis of plate bending using triangularelements. Ph.D. Dissertation, University of California, Ber-keley (1962).16. R. W. Ciough and E. L. Wilson, Stress analysis of a gravitydam by the finite element method. Prt~c.Symp. Use Cornput.


10/10

370 R. W. CLOUGHCioilEngng, Laboratorio National de Engenharia Civil, Lis-bon, Portugal (1%2).17. E. L. Wilson, Finite element analysis of two-dimensionalstructures. Structural Engineering Lab. Rep. No. 63-2, Uni-versity of California, Berkeley (June 1%3).18. Symp. on the Use of Computers in CivilEngineering Labora-torio National de Engenharia Civil, Lisbon, Portugal(I-5 Oct. 1%2).

19. Conf. Matrix Methods Structural Mech. Air Force Instituteof Technology and Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio (Oct. 1%5).20. R. W. Clough, Analysis of structural vibrations and dynamic

21.

22.

23.24.25.

26.27.

28.

29.30.31.

response. Proc. U.S.-Japan Seminar on Recent Advances inMatrix Methods of Structural Analysis and Design, Tokyo,Japan, August 1979. pp. 441-486. University of AlabamaPress, Alabama (1971).R. W. Clough and K. J. Bathe, Finite element analysis ofdynamic response. Proc. 2nd U.S.-Japan Seminar on MatrixMethods of Structural Analysis, Berkeley, California, July1972.University of Alabama Press, Alabama (1972).K. Bell et al. NORSAM-A Proarammina Svstem for theFinite Element Method. University of Tyondheim-NTH,SINTEF (1973).M. D. Olson and T. W. Bearden, A simple flat triangular plateelement revisited. Int. J. Num. Meth. Emma 14G). 51-68.J. S. Archer, Consistent mass matrix for-distributed masssystems. Proc. ASCE. J. Struct. Div. 89, 161-178 (1 3).D. Shantaram, D. R. J. Owen and 0. C. Zienkiewicz,Dynamic transient behavior of two- and three-dimensionalstructures, including plasticity, large deformation effects, andfluid interaction. Eq. Engng Structural Dyn. 4(6), 561-578(1976).E. L. Wilson and J. Penzien, Evaluation of orthogonal dam-nine matrices. Int. L Num. Meth. Emma 4(l). 5-10 (1972).

32. R. E. Nickell, Nonlinear dynamics by mode superposition.Camp. Meth. Appt. Mech. Engng 7, (1976).

33. S. N. Remseth, Nonlinear static and dynamic analysis ofspace structures. Rep. No. 78-2, Institute for Statikk, Uni-versity of Trondheim-NTH (Apr. 1978).34. R. W. Clough and E. L. Wilson, Dynamic analysis of largestructural systems with local nonlinearities. Proc.FENOMECH-78, Stuttgart (Sept. 1978).

35. P. Chakrabarti and A. K. Chopra, Earthquake analysis ofgravity dams including hydrodynamic interaction. Int. I.Earthq. Engng Struct. Dyn. 2, 143-H (1973).

36. A. K. Chopra and J. B. Gutierrez, Earthquake responseanalysis of multiatory buildings including foundation inter-action. Jut. L Earthq. Engng Struct. Dyn. 3,65-78 (1974).37. R. Reimer Deconvolution of seismic response for linearsystems, Uttiuersity of California, Earthquake Engineering

Research Center, Rep. No. EERC73-IO (1973).38. G. L. Goudreau and R. L. Taylor, Evaluation of numericalintegration methods in elastodynamics. Camp. Meth. Appl.Mech. Engng 2.69-97 (1972).39. H. M. Hilber and T. J. R. Hughes, Collocation, dissipation,and overshoot for time integration schemes in structuraldynamics. Inf. J. Earthq. Engng Struct. Dyn. 6(l), 99-118(1978).

40. T. Belytschko et al., Efficient large scale nonlinear transientanalysis by finite elements. Ittt. .I. Num. Meth. Engng 10,5795% (1976).41. N. M. Newmark, A method of computation for structuraldynamics. J. Engng Mech. Div. ASCE 85(EM3) (July 1959).42. N. W. Edwards, A procedure for dynamic analysis of thin-walled cylindrical liquid storage tanks subjected to lateralground motions. Ph.D. Dissertation, University of Michigan,Ann Arbor. Michigan (1%9).43. S. H. Shaaban and W. A. Nash. Finite element analvsis of aYoihikazu Yamada, Dynamic analy&-of structures. Proc. seismically excited cylindrical storage tank. University of2nd U.S.-Japan Seminar on Matrix Methods of Structural Massachusetts Rep. (Aug. 1975).Analysis. University of Alabama Press, Alabama (1972). 44. D. P. Clough, Experimental evaluation of seismic designR. W. Clough and S. Mojtahedi, Earthquake response analy- methods for broad cylindrical tanks. Universityof California,sis considering non-proportional damping. ht. J. Earthq. Earthquake Engineering Research Center, Rep. No.Engng Struct . Dyn. 4, 489-4 (1976). UCBIEERC-77/10 May 1977).Int. Conf. Finite Elements in Nonlinear Solid and Structural 45. A. Niwa, Seismic behavior of tall liquid storage tanks. Uni-Mechanics, Geilo, Norway (Aug. 1977). versity of California. Earthquake Engineeri ng ResearchTICOM-2, 2nd Int. Conf. Computational Methods in Non- Center. Reo. No. UCBIEERC-78/&i Oct. 1978).linear Mechanics. University of Texas, Austin (Mar. 177). 46. R. W.Clough, J. M. Raphael and S. Mojtahedi, ADAP: aK. Bell, DYNOGS-a computer program for dynamic computer program for static and cynamic analysis of archanalysis of offshore gravity platforms. SZNTEF. Norwegian dams. Unioersity of California, Earthquake EngineeringInstitute of Technology, Trondheim, Norway (1978). Research Center, Rep. No. EERC 73-14(June 1973).

Documents

Clough_The FEM After Twenty-five Years - A Personal View