Thoughts about the origin of the finite element method

R.W Clough, P.O. Box 4625, Bend, OR 97707, USA (Computers & Structures, 2001, 79(22):2029-2030.)

I am very pleased to take part in ECCM'99. As I told Dr. Wunderlich when he asked me to participate, I certainly am not in a position to make any technical contribution to the field of Computational Mechanics. However, I have worked actively with the field that is called the finite element method (FEM) for over 45 years, both in graduate level teaching and in my consulting practice during the 12 years since my retirement. The emphasis in most of this consulting has been on the earthquake performance of concrete dams—especially concrete arch dams.

My involvement with the FEM began when I was employed by the Boeing Airplane Company in Seattle during summer 1952 as a member of their summer faculty program. When I had joined the civil engineering faculty at Berkeley in 1949, I decided to take advantage of my MIT structural dynamics background by taking up the field of Earthquake Engineering. So because the Boeing summer faculty program offered positions with their structural dynamics unit, I seized on that as the best means of advancing my preparation for the earthquake engineering field. I was particularly fortunate in this choice of summer work at Boeing because the head of their structural dynamics unit was Mr. M.J. Turner—a very capable man in dealing with problems of structural vibrations and flutter.

When I arrived for the summer of 1952, Jon Turner asked me to work on the vibration analysis of a delta wing structure. Because of its triangular plan form, this problem could not be solved by procedures based on standard beam theory; so I spent the summer of 1952 trying to formulate a delta wing model built up as an assemblage of one-dimensional beams and struts. However, the results of deflection analyses based on this type of mathematical model were in very poor agreement with data obtained from laboratory tests of a scale model of a delta wing. My final conclusion was that my summer's work was a total failure—however, at least I learned what did not work.

Spurred by this disappointment, I decided to return to Boeing for the summer faculty program in 1953. During the winter, I stayed in touch with Jon Turner so I was able to rejoin the structural dynamics unit in June. The most important development during the winter was that Jon suggested we try to formulate the stiffness property of the wing by assembling plane stress plates of either triangular or rectangular shapes. So I developed stiffness matrices for plates of both shapes, but I decided the triangular form was much more useful because such plates could be assembled to approximate structures of any configuration. Moreover, the stiffness properties of the individual triangular plates could be calculated easily based on assumptions of uniform states of normal stress in the X and the Y directions combined with an uniform state of shear stress. Then the stiffness of the complete structure was obtained by appropriate addition of the contributions from the individual pieces. The Boeing group called this procedure the direct stiffness method.

The remainder of the summer of 1953 was spent in demonstrating that deflections calculated for structures formed as assemblages of triangular elements agreed well with laboratory measurements on the actual physical models. Also, it became apparent that the precision of the calculated results could be improved asymptotically by continued refinement of the finite element mesh. The conclusions drawn from that summer's work were presented in a paper given by Jon Turner at the annual meeting of the Institute of Aeronautical Sciences in January 1954. However, for reasons I never understood Jon did not submit the paper for publication until many months later. So this paper, which often is considered to be the first published description of the FEM, was not published until September 1956 [1]—more than two years after the verbal presentation.

It is important to note that the basic purpose of the work done by Jon Turner's structural dynamics unit was vibration and flutter analysis. They were not concerned with stress analysis because that was the responsibility of the stress analysis unit. However, it was apparent that the model formed by the direct stiffness method could be used for stress analysis as well as for vibration analysis, and I made plans to investigate this stress analysis application as soon as possible. However, because of my other research responsibilities, I was not able to spend any significant time on the stress analysis question until I went on my sabbatical leave to Trondheim, Norway in September 1956. Then, when I arrived in Norway all I could do was to outline the procedures for carrying out the analysis, and to do calculations for very small systems using a desk calculator because the Norwegian Institute of Technology did not yet have an automatic digital computer.

The presentation of the paper to the Institute of Aeronautical Sciences [1] was the first introduction of the principles of the FEM to a technical audience; although some of the basic concepts of the method were stated a short time later in a series of articles published in Aircraft Engineering by Dr. John H. Argyris during October 1954 to May 1955 [2]. However, the rectangular element presented in those articles is only a minor part of that contribution. The Argyris work came to my attention during my sabbatical leave in Norway, and I considered it then (as I still do now) to be the most important series of papers ever published in the field of Structural Mechanics. I credit that work for extending the scope of my understanding of structural theory to the level it eventually attained.

From my personal point of view, the next important event in finite element history was the coining of the name FEM. My purpose in choosing that name was to distinguish clearly the relatively large size pieces of the structure that make up a finite element assemblage as contrasted with the infinitesimal contributions that go into evaluation of the displacements of a structure in a typical virtual work analysis. The name first appeared in a publication [3] that was written to demonstrate the finite element procedure for the civil engineering profession. A much more significant application of the method was presented at the Symposium on the use of Computers in Civil Engineering, held in Lisbon, Portugal in 1962 [4] where it was used to evaluate the stress concentrations developed in a gravity dam that had cracked at its mid-section.

For many years my contact with FEM research was maintained mainly through the efforts of my doctoral students carrying out their dissertation research—Ref. [4] is an example of such work. More recently, however, I have contributed to the finite element field only through retrospective papers such as Refs. [5] ;  [6] and this present paper.



[1]Turner MJ, Clough RW, Martin HC, Topp LJ. Stiffness and deflection analysis of complex structures. J Aero Sci 1956;23:805-23.

[2]Argysis J. Energy theorems and structural analysis. London: Butterworth; 1954.

[3]Clough RW. The finite element method in plane stress analysis. Proc ASCE Conf Eletron Computat, Pittsburg, PA, 1960.

[4]Clough RW, Wilson EL. Stress analysis of a gravity dam by the finite element method. Proc Symposium on the Use of Computers in Civil Engineering, Lisbon , Portugal, 1962.

[5]Clough RW, The finite element after 25 years – a personal view. Int Conf on Applications of the Finite Element Method. Veritas Center, Hovik, Norway : Computas; 1979.

[6]Clough RW, Original formulation of the finite element method. Finite Elem Anal Desig 1990;7:89-101.




我与FEM结缘是在1952年夏天,我被西雅图波音飞机公司聘请为他们summer faculty项目的成员。那时我已经在1949年加入Berkeley土木系。由于波音summer faculty项目提供了结构动力学小组的位置,我抓住了这个最好机会,做为我在地震工程领域发展的准备。我很幸运地选择了波音公司的那次夏季工作机会,因为他们结构动力学小组的领导人是M.J. Turner先生–一个在处理结构振动与颤振问题非常有能力的人。

当我在1952年夏天参加的时候,Jon Turner让我从事一种delta翼结构的振动分析工作。由于它是三角平面形状,这个问题不能用基于标准梁理论的方法解决;于是我花了1952年一个夏天时间来建立由一维梁与桁架组拼成的一个delta翼模型。然而基于这种类型的数学模型得到的变形分析结果与delta翼比例模型试验数据吻合很差。我最后的结论是我一个夏天的工作是一个彻底的失败–然而,至少我知道了什么是不可行的。受这次失败挫折的刺激,我决定重新回到波音参加1953年的夏季faculty项目。在冬季期间,我和Jon Tuner一直保持联系,这样我能够重新参加六月的结构动力学小组。冬季期间最重要的进展是,Jon建议我尝试通过组装三角或矩形形状的平面应力小块来建立机翼的刚度特性列式,但我确定三角形状会更有用,因为这样一种小块可以组拼近似任意结构形状。况且,单个三角块的刚度特性可以在假定X与Y方向主应力的均匀分布结合剪应力均匀分布状态的基础上很容易的计算。于是整个结构的刚度由单个子块的贡献相应叠加得到。波音小组称这种方法为直接刚度法。1953年夏天的剩余时间花在演示用三角单元组装成结构的模型的变形计算结果和实际结构的试验室测量结果非常吻合。同样,也很显然的发现计算结果的精度可以通过连续细分有限元网格渐进提高。那个夏天工作得到的结论由Jon Turner发表在1954年一月的Institue of Aeronautical Sciences年度会议上。然而至今我仍不明白为什么Jon直到许多月以后才把这篇论文拿去发表。因此,这篇被认为是有限元第一篇论文的文章直到1956年9月才发表[1]–在它被口头提出两年多后。

值得强调的是Jon Turner的结构动力学小组做这个工作的基本目的是为了振动和颤振分析。他们并不关心应力分析,因为那是应力分析小组的任务。然而,很显然的是通过直接刚度法建立的模型除了用于振动分析,同样可以用于应力分析。因此我计划一旦可能,马上调查它在应力分析中的应用。然而当时我有其它研究任务,我无法抽出有用的时间在应力分析问题上,直到1956年9月我申请到挪威Trondheim进行休假。于是,当我到达挪威时,我所能做的只是拟定了一个研究方案,并且用一台台式计算器计算一些很小的系统,因为挪威理工学院那时还没有一台自动电子计算机。

在Institute of Aeronautical Sciences上发表的那篇文章[1]第一次向技术人员引入了有限元的原理;尽管很短时间后这种方法的一些基本概念在1954年八月到1955年五月期间被John H. Argyris博士[2]在Aircraft Engineering发表的一系列论文提出。然而,在这些论文中出现的矩形单元仅仅是所做贡献的一小部分。在我在挪威的休假期间 ,Argyris的工作吸引了我的注意,那时我把它认为是(现在我仍这么认为)曾经在结构力 学领域发表的最重要的一系列文章。我认为那些工作把我对结构理论的理解扩展到它最终达到的层次。

根据我个人的观点,有限元历史上另一个重要事件是创造了有限元(FEM)这个名字。 我选择这个名字的目的是为了将组成有限元整体的尺寸相对较大的结构小块,与在结构位移计算典型虚功分析中的无穷小量明确区分。这个名字最早出现在一篇向土木工程界演示有限元方法的文章中。这个方法的一个更重要的应用发表在1962年在波兰Lisbon的一个有关计算机在土木工程中的应用的会议上[3],它用有限元进行一个已经在中截面开裂的重力大坝集中应力分析。