INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2004; 60:283–287 (DOI: 10.1002/nme.962)

SPEECH BY PROFESSOR R. W. CLOUGH

Early history of the finite element method from theview point of a pioneer

I am happy to take part in WCCM V because this meeting is making it possible for me to meetagain with a number of my European structural engineering friends with whom I have beenassociated for more than 40 years. During my teaching career at Berkeley, which extended from1949 to 1987, I observed the development and growth of the field of computational mechanicsfrom the earliest days when the name ‘Computational Mechanics’ did not yet exist, to thepresent when the scope of the field is almost unlimited.

The subject of my talk, as you know, is the Early History of the Finite Element Method(FEM). In any comprehensive discussion of this subject, four names should be mentioned, asfollows:

John H. Argyris (John) [1]Ray W. Clough (Ray) [2]M. J. Turner (Jon) [3]O. C. Zienkeiwicz (Olek) [4]They are shown here in alphabetical order to avoid establishing any sense of priority among

them. With each name, I also show a number from the reference list at the end of this paper.For present purposes, that reference best characterizes the contribution each man has made toFEM history, in my opinion.

At the start of this talk, I must point out that I have presented several papers on this subjectbefore, as is evident from the reference list. However, before I get into the details of theFEM history, I think it will be useful for me to say a few words about my personal historyduring that period, because it was that history which got me pointed in the direction of thisnew approach to structural analysis. When I was in my final year of Civil Engineering at theUniversity of Washington, my structural engineering professor, Prof. C. C. More, suggestedthat I should go on to Graduate School after I finished my B.S. C.E. degree. This certainlywas a new idea to me because in those days very few students went on to do graduate studies.But Professor More made a strong case for my going to M.I.T., and he helped me prepare myapplication for the M.I.T. Graduate School. Professor More had graduated from Cornell, butfor some reason he was pushing M.I.T. strongly at that time.

Within a couple of months after I had submitted my application, I was surprised and pleasedto receive notice that I had been awarded a tuition fellowship from M.I.T. for the term startingin September 1942. To have such a fellowship was very important because the tuition at M.I.T.

Correspondence to: R. W. Clough, Earthquake Engineering Research Center, University of California, 1306 South46th street, Richmond, CA 94804, U.S.A.

Received 10 May 2003Copyright � 2004 John Wiley & Sons, Ltd. Accepted 25 July 2003

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was many times greater than the resident’s fee I had been paying at Washington. However, themost important aspect of the situation at that time was that the U.S. was at war, so I had togo to my Draft Board to request a deferment that would permit me to continue my studiesat M.I.T. I was shocked and very disappointed when the Draft Board told me they could notdefer my military service any longer just so I could go to Graduate School. In fact they saidI was lucky that already I had been deferred long enough to complete my Bachelor’s Degree.

After the disappointment of not being able to go to Graduate School, I took a job in theStress analysis unit at the Boeing Airplane Company—working on the design of the B-47airplane. This was a position that I was sure would keep me out of military service for theduration of the war, and it had the additional advantage that I could continue living in myparents’ home in Seattle. However, in spite of this very favourable situation, I soon began toexplore the possibilities that might be open to me in military service because I was boredwith the stress analysis job at Boeing. My first military job choice at that time was to jointhe Navy Civil Engineer Corps as a ‘Sea Bee’ Officer, but I was rejected from that possibilitybecause I could not pass the Navy eye exam. They required 20/20 vision in those days, andI had been wearing glasses for several years. So I had to consider some other possibility, andwhen I learned that the Air Force was accepting candidates for their Aviation Cadet programin Meteorology, which did not require 20/20 vision, I decided to apply for that program. Theidea was that in about 9 months I would receive a commission as a Weather Officer, and thenwould be sent to an Air Force base to predict weather for the pilots.

I was accepted as a Weather Aviation Cadet in December 1942 and was assigned to theWeather School at the California Institute of Technology. Then in September 1943 I receivedmy Commission as a Weather Officer, as well as a Master’s Degree in Meteorology from CalTech. Rather than sending me into the field to practice the art of weather forecasting, the AirForce decided that the best plan for me was to stay at Cal Tech to be an instructor for the nextclass of weather cadets. That class was graduated in September 1944, but by that time (with fivedifferent weather schools producing weather officers) the Air Force realized that they had farmore weather officers than they could use. So at that point they went through their personnelrecords and decided that any weather officer who already had an engineering degree couldapply for a position in the Air Force that would take advantage of that previous education. Ihad never liked being a Weather Officer because I liked to do work in which I could expectto get the right answer somewhat more than half of the time. Parenthetically, I may remarkthat while I was serving as a Weather Officer, I was disappointed to note that the best forecastI could make usually was a ‘persistence forecast’—that is to predict that tomorrow’s weatherwill be just like today’s. Of course, that was decades before weather satellites were givingbeautiful images of cloud cover over most of the surface of the earth, and I might not havemade the same decision if the tools of today were available then. In any case, I seized on theopportunity to apply for transfer to the Air Force Aviation Engineers; and shortly thereafter Iwas assigned to the 1870th Engineer Aviation Battalion and was sent to MacDill Field, Floridafor unit training.

In about 4 months our unit was put on a troop ship and sent to Okinawa. Fortunately, wehad the good luck to be landing on Okinawa about the time that the Japanese surrendered.Because of this I have always been an enthusiastic supporter of President Truman’s decision touse the atomic bombs at Hiroshima and Nagasaki. My return from Okinawa was delayed byseveral months because I had not yet accumulated enough ‘points’ to be eligible for dischargefrom the Air Force, but finally I was shipped back to the States. Then I was able to make use

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of my tuition fellowship at M.I.T., starting in September 1946 and completing my doctorate inJune 1949. At that time I accepted a position as Assistant Professor with the Civil EngineeringDepartment at U.C. Berkeley.

All of this is a preamble to the subject of my talk this evening on the ‘Early History of theFinite Element Method’. My studies at M.I.T. had included a course in Dynamics of AircraftStructures, and this fitted well with the interest of the Berkeley Civil Engineering Departmentthat my teaching should emphasize the dynamic response of structures to earthquake excitation.So Earthquake Engineering became my chosen field at Berkeley. As a matter of fact, therewere no programs on that subject at any school in the United States at that time, so I hadto develop my lectures from the ground up. Also, the National Science Foundation had noexperience with providing research support in this field. In these circumstances, the closestI could get to academic support for my work was the Boeing Summer Faculty Program, inwhich Boeing would provide the equivalent of an academic salary for work that was of interestto Boeing.

When I applied for the Boeing Summer Faculty job in June 1952, I was assigned to theStructural Dynamics Unit under the supervision of Mr M. J. Turner. He was a very competentengineer with a background in applied mathematics, and several years of experience withBoeing. The job that Jon Turner had for me was the analysis of the vibration properties of afairly large model of a ‘delta’ wing structure that had been fabricated in the Boeing shop. Thisproblem was quite different from the analysis of a typical wing structure which could be doneusing standard beam theory, and I spent the summer of 1942 trying to formulate a mathematicalmodel of the delta wing representing it as an assemblage of typical 1D beam components. Theresults I was able to obtain by the end of the summer were very disappointing, and I wasquite discouraged when I went to say goodbye to my boss, Jon Turner. But he suggested thatI come back in Summer 1953. In this new effort to evaluate the vibration properties of a deltawing model, he suggested I should formulate the mathematical model as an assemblage of 2Dplate elements interconnected at their corners. With this suggestion, Jon had essentially definedthe concept of the finite element method.

So I began my work in Summer 1953 developing in-plane stiffness matrices for 2D plateswith corner connections. I derived these both for rectangular and for triangular plates, butthe assembly of triangular plates had great advantages in modeling a delta wing. Moreover,the derivation of the in-plane stiffness of a triangular plate was far simpler than that for arectangular plate, so very soon I shifted the emphasis of my work to the study of assemblagesof triangular plate ‘elements’, as I called them. With an assemblage of such triangular elements,I was able to get rather good agreement between the results of a mathematical model vibrationanalysis and those measured with the physical model in the laboratory. Of special interest wasthe fact that the calculated results converged toward those of the physical model as the meshof the triangular elements in the mathematical model was refined.

It should be emphasized now that the work I was doing for Jon Turner had as its objectivethe analysis of vibrations, and that objective is reflected in the title of Reference [3], whichpaper often is taken as the first paper in the history of FEM. It is of interest to note that JonTurner presented that paper at the annual meeting of the Institute of Aeronautical Sciences inJanuary 1954 but, for reasons I have never understood, it was not immediately submitted forpublication. So the 1956 publication date of the paper is nearly 3 years after the work wasdone at Boeing. That work was described in the report I submitted to Jon Turner at the endof my 1953 Summer Faculty Employment.

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Although my Boeing summer work was never directed toward the analysis of stresses, itwas apparent that the Boeing Direct Stiffness Method could be used for stress analysis as wellas for calculating vibrations, and I decided I would investigate the stress analysis applicationas soon as possible. However, because of my other research interests at Berkeley, I did notbegin to look into the stress analysis question until I went on my first sabbatical leave. Thiswas when I went to Norges Tekniske Hogskole in Trondheim, Norway in September 1956.

While I was in Norway on this leave, I became aware of the very important work thathad been done by Dr John Argyris in the field of airplane structural analysis. This work waspresented in a series of articles published in ‘Aircraft Engineering’ between October 1954and May 1955. It was published later by Butterworths of London as a single volume entitled‘Energy Theorems and Structural Analysis’. In my opinion, this monograph (listed here asReference [1]) certainly is the most important work ever written on the theory of structuralanalysis, and when I read those articles during my sabbatical leave I immediately concludedthat there was no need for me to deal with the subject of Structural Analysis Theory duringmy stay in Trondheim.

From my point of view, the next important event in the finite element history was mycoining the name ‘Finite Element Method’. I never thought that the name used by Boeing fortheir procedure, the ‘Direct Stiffness Method’, was at all descriptive of the concept involvedin the method. So when I later wrote the stress analysis paper that is listed as Reference [2],I had to choose a new name for the procedure. On the basis that a deflection analysis donewith these new ‘pieces’ (or elements) of the structure is equivalent to the formal integrationprocedure of integral calculus, I decided to call the procedure the FEM because it deals withfinite components rather than differential slices.

A ‘red letter’ event that occurred during this very early history of FEM was my visitingNorthwestern University to give a seminar lecture on finite elements. When I received thisinvitation from Olek Zienkiewicz, who was teaching at Northwestern at that time, I expected wewould have some arguments about the relative merits of finite elements versus finite differencesbecause Olek had been brought up in the tradition of Professor Southwell. It is true that wedid have some such discussions, but Olek recognized very quickly the advantages of the finiteelement approach. In fact I would say that my visit to Northwestern yielded a tremendousdividend in the conversion of Olek from finite differences to finite elements.

It should be noted that the finite element name had been established by the paper I gave atthe ASCE Conference on Electronic Computation in Pittsburgh in 1960. On the other hand, itwill be observed that the name ‘Computational Mechanics’ had not yet been adopted at thattime. That first FEM paper (Reference [2]) attracted very little attention, but Reference [5],(to which my colleague of many years at Berkeley, Professor E. L. Wilson, made a significantcontribution) did attract some attention. We presented that paper at a Symposium in Lisbon,Portugal, and in a relatively short time the name FEM came to be in common usage. I mustemphasize here the pivotal role that Ed Wilson played in the development of FEM. When hewas working as my doctoral student in 1962, I realized that I no longer had to worry aboutthe details of computer program development—Ed was so much more effective in that partof the work than I ever could be. So from that time on, I used the method continually togain understanding of the behaviour to be expected in a given structural system, but there wasno need for me to try to improve on the program development work that was being doneby Ed Wilson. His recollections of that early work are well described in his paper cited hereas Reference [6].

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EARLY HISTORY OF THE FINITE ELEMENT METHOD 287

REFERENCES

1. Argyris JH. Energy Theorems and Structural Analysis. Butterworths: Washington, DC.2. Clough RW. The finite element method in plane stress analysis. Proceedings of the Second ASCE Conference

on Electronic Computation, Pittsburgh, PA, 1960.3. Turner MJ, Clough RW, Martin HC, Topp L. Stiffness and deflection analysis of complex structures. Journal

of Aeronautical Sciences 1956; 23.4. Zienkiewicz OC. The Finite Element Method (3rd edn). McGraw-Hill: New York, 1977.5. Clough RW. Stress analysis of a gravity dam by the finite element method. Proceedings of the Symposium

on the Use of Computers in Civil Engineering, Laboratorio Nacional de Engenharia Civil, Lisbon, Portugal,1962 (see also RILEM Bull. No. 19; June 1963).

6. Wilson EL. Automation of the finite element method, a personal historical view. Finite Elements in Analysisand Design, vol. 13. Elsevier: Amsterdam, 1993; 91–104.

7. Clough RW. The finite element method in structural mechanics. Stress Analysis, Chapter 7. Wiley:New York, 1965.

8. Clough RW. The finite element method after 25 years. In Engineering Applications of the Finite ElementMethod, A. S. Computas, Det Norske Veritas, Hovik, Norway, 1979.

9. Clough RW. Original formulation of the finite element method. ASCE Structure Congress, San Francisco,CA, May 1989 (also published in Finite Elements in Analysis and Design, vol. 10, 1990).

10. Clough RW. FEM—a personal view of its original formulation (in the special volume published to celebratethe 70th Birthday of Ivar Holand, see also Proceedings of the U.S. Conference on Computational Mechanics,Boulder, CO, 1993).

11. Clough RW. Thoughts about the original formulation of the FEM—a personal view. Proceedings of theEuropean Conference on Computational Mechanics, Munich, 1999.

Copyright � 2004 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 60:283–287