# Events / Finite Element Method

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**Introduction**

The analysis of structures, machines, fluids, solids, and natural and man-made products or systems can be modeled and analyzed because of advancing mathematics, equation solution techniques, finite elements, structure models, and sixty years of computer program development and rapid increase of computer capacity, power and speed. All these technical advances occurred in parallel from the 1940’s to today. In 1956, Ray Clough, a young University of California Berkeley Professor, and Jon Turner, the leader of Aeronautical Engineering at the Boeing Airplane Company in Seattle, developed a computer program and code to mathematically describe discrete elements that could be assembled to model a delta wing airplane and automate matrix mathematical solution procedures required to solve the structure behavior under constant forces or wing vibration and dynamic response subject to turbulent wind forces in flight. Calculating displacement and vibration behavior of multiple degree of freedom structure models boiled down to linear algebra matrix solution mathematics. Solution of multiple degree of freedom structure displacements at common element nodes then allowed the determination of associated material stresses and member forces. Ray Clough used a large number of discrete geometric elements, a connected series of two dimensional triangles or rectangles (plates), and/or beams, to describe and model the geometry and material of an airplane wing. This early work was done at Boeing Aircraft Company to understand airplane wing displacement and stresses during flight. The resulting mathematical model included steel and aluminum wing material strength and the stiffness. Different loads were applied, by the computer model, and the software Ray developed was able to generate the wing element displacements under different applied load conditions. Airplane wings are subject to varying air pressures due to the aerodynamic properties that make wings “fly” and air flow can be steady – state, uniform pressure, or turbulent, non—uniform and dynamic, rapidly changing over time. In 1953, Ray successfully completed the wing analysis and obtained displacements and vibration results from his new program.

From the 1930’s through 1950’s, manual calculations using moment distribution and slope-deflection methods were two classic manual structure mechanics analytical methods used to determine displacement, stresses and forces of structures. In the 1930’s, Charles Ellis, a Princeton University Professor and the “real” Structural Engineer of the Golden Gate Bridge, took months making manual calculations of a highly non-linear, cable supported structure, to determine the cables, deck truss and the bridge tower displacements and stresses, due to to high winds blowing on the proposed new 4,200 feet span suspension bridge. Starting in the 1950’s computers and automation changed all that.

**Who invented the Finite Element Method?**

Jon Turner, an engineer at the Boeing Aircraft Company from 1950 – 1962, decided to use the Direct Stiffness Method at Boeing for structure analysis, changing from the Force Method used for aircraft structures analysis at that time. At Boeing, Turner oversaw the development of the first continuum mechanics based, finite elements. Boeing owned and was using and operating the “best” main frame computers available. Turner was an international recognized aeroelasticity, dynamics and vibration expert in the aircraft industry.

Jon was the supervisor of the Structural Dynamics Unit at Boeing in Seattle in the 1950s, and brought the modern finite element method into everyday use. According to Carlos Felippa, who writes in the The Origins of the Finite Element Method, “Jon Turner forcefully got Boeing to commit resources to the Direct Stiffness Finite Element method while other aerospace companies were mired in the Force Method which eventually died out.” Boeing Aircraft Company was able to fund the expensive cost of a large 1950’s era mainframe digital computer and the airplane industry needed and supported bright young engineers who performed the research and developed the computer code for computational mechanics.

With the growing popularity of jet aircraft, and with demands for high performance military aircraft, delta wing structures presented new modeling and analysis problems. Existing unidirectional models, beam elements, did not provide sufficient accuracy. Instead, two-dimensional panel elements of arbitrary geometry were needed.

Turner’s application for the direct stiffness finite element method was the automation of vibration calculations to determine wing flutter and dynamic analysis. At this time, Boeing had a summer faculty program, whereby faculty members from universities were invited to work at Boeing over the summer. In the summers of 1952 and 1953 Jon Turner invited Ray Clough from U.C. Berkeley, and Harold Martin from the University of Washington to work for him on a method to calculate vibration properties for low-aspect ratio box beams, jet plane delta wings. This Boeing collaboration resulted in the seminal 1956 paper, *Stifffness and Deflection Analysis of Complex Structures, *by Turner, Clough, Martin and Topp which summarized an analysis procedure, called the Direct Stiffness Method (DSM) and described the derivation of a triangular Plate Element and a rectangular Plate Element used to model the wing. Ray Clough realized that this method of determining aircraft wing displacements and vibration could be extended to determine structure stresses associated with the displacements.

Ray Clough later wrote, “The job Jon Turner had for me was the analysis of the vibration properties of a fairly large model of a “delta” wing structure that had been fabricated at Boeing. This problem was quite different than the analysis of the typical airplane wing structure which could be done using standard beam theory. I spent the summer of 1952 trying to formulate a mathematical model of the delta wing represented by an assemblage of typical beam components. I was not successful. Jon suggested in come back in 1953 and continue my efforts to create a mathematical model and solve the question of the delta wing vibration behavior. In the 1953 effort, Jon suggested a model of 2-dimesional plate elements interconnected at their corners (nodes). With this suggestion, Jon has essentially defined the concept of the Finite Element Method. I began by developing in-plane stiffness matrices for 2D plates with corner connections. I derived these for both rectangular and triangular plates. The triangular plate had great advantage for modeling the delta wing. The derivation of the triangular plate element stiffness was much simpler than the rectangular plate element. I focused my emphasis to the study of assemblages of triangular plate elements. Using the triangular elements, my mathematical model, computer based vibration analysis, had rather good agreement with the measurement of experiential models and laboratory testing of physical models. I observed that the calculated results converged toward the physical model results as the mesh of the triangular elements in the wing mathematical model was refined. The DSM method included vibration calculations to facilitate flutter and dynamic analysis. Ray Clough realized that the DSM could be extended to stress analysis”. In 1960, Clough presented a research paper titled “Finite Elements for Plane Stress Analysis”. Which adopted the DSM method for stress analysis and he first used the term “Finite Element” in this paper to describe his method.

Besides the Boeing group in Seattle, many others have contributed to the development of modern Finite Element Analysis. Four academics are considered pioneers of the Finite Element Method. J.H. Argyris at London’s Imperial College, Ray W. Clough at U.C. Berkeley, H.C. Martin, and O.C. Zienkiewicz, Swansea College in Wales, are largely responsible for the research, development and transfer of FEM theory and knowledge to practical use in engineering and mechanics. Three centers of finite element research came into existence, University of Stuttgart, Germany under the leadership of Argyris; Swansea College, Wales, under Zienkiewicz; and U.C. Berkeley, where Ray Clough, attracted many bright students including Ed Wilson, and MIT’s, Jurgen Bathe. Argyris, Clough, and Martin learned and developed finite element solution methods while working at Boeing with Jon Turner. Argyris, a Force Method expert at Imperial College in England, was a consultant to Turner at Boeing. Argyris is credited with being the first in constructing a displacement-assumed continuum element. Ray Clough and H. C. Martin, a young University of Washington professor, spent 1952 and 1953 “faculty internships” in Jon Turner’s mechanics and analysis group at Boeing. Ray Cough was working in the dynamics analysis research group and modeled the Delta Wing with three-dimensional strut elements, (connected at two nodes), and triangular (connected to three nodes), or quadrilateral membrane elements (connected at four nodes). Ray’s research involved the development of elements that represent material properties and geometry behavior used in the mathematical models that calculate the structure displacements. Physical model testing and laboratory measurements was the aircraft industry standard engineering method from the 1930’s and continued until the 1970’s, when the reliability of mathematical models was widely accepted as equal to physical experimental model construction and laboratory tests.

In 1956, Ray spent his University sabbatical year in the Trondheim, Norway, Ship Research Institute developing methods to predict stress concentrations and resulting steel ship fatique failures. It was during this time in Norway, that Ray created the term, “Finite Element Method.” Clough continued to develop and advance the FEM into the 1960’s including forming a research group at U.C. Berkeley dedicated to Civil Engineering applications of the Finite Element Method. The 1956 paper by Turner, Clough, Martin and Topp, is recognized as “the paper” that describes the beginning of the Finite Element Method.

Ed Wilson said, “in 1956, Ray, Shirley, and three small children spent a year in Norway at the Ship Research Institute in Trondheim. The engineers at the institute were calculating stresses due to ship vibrations in order to predict fatigue failures at the stress concentrations. This is when Ray realized his element research should be called the Finite Element Method which could solve many different types of problems in continuum mechanics. Ray realized the FEM was a direct competitor to the Finite Difference Method. At that time FDM was being used to solve many problems in continuum mechanics. His previous work at Boeing, the Direct Stiffness Method, was only used to calculate displacements not stresses.

In the fall semester of 1957, Ray returned from his sabbatical leave in Norway and immediately posted a page on the student bulletin board asking students to contact him if they were interested in conducting finite element research for the analysis of membrane, plate, shell, and solid structures. Although Ray did not have funding for finite element research, a few graduate students who had other sources of funds responded. At that time, the only digital computer in the College of Engineering was an IBM 701 that was produced in 1951 and was based on vacuum tube technology. The maximum number of linear equations that it could solve was 40. Consequently, when Ray presented his first FEM paper in September 1960, “The Finite Element Method in Plane Stress Analysis,” at the ASCE 2^{nd} Conference on Electronic Computation in Pittsburgh, Pennsylvania, the course-mesh stress-distribution obtained was not very accurate. Therefore, most of the attendees at the conference were not impressed.

After the improvement of the speed and capacity of the computers on the Berkeley campus, Professor Clough’s paper was a very fine mesh analysis of an existing concrete dam. The paper was first presented in September 1962 at a NATO conference in Lisbon, Portugal. Within a few months, the paper was republished in an international Bulletin, which had a very large circulation, as “Stress Analysis of a Gravity Dam by the Finite Element Method”, (with E. Wilson), International Bulletin RILEM, No. 10, June 1963.

The Lisbon paper reported on the finite element analysis of the 250-foot-high Norfork Dam in Arkansas, which had developed a vertical crack during construction in 1942. The FEM analysis correctly predicted the location and size of the crack due to the temperature changes and produced realistic displacements and stresses within the dam and foundation for both gravity and several hydrostatic load conditions.

Due to this publication, many international students and visiting scholars came to Berkeley to work with Professor Clough. Also, he freely gave the FORTRAN listing of their finite element analysis computer program to be used to evaluated displacement and stresses in other two-dimensional plane structures with different geometry, materials and loading.

Therefore, professional engineers could immediately use this powerful new FEM to solve for the stress distributions in their structural engineering problems in continuum mechanics. However, Ray Clough did not capitalize on his success in the development of the FEM. Ray returned to the task of building the Earthquake Engineering program at Berkeley – the task he was given when he was hired in 1949.

**Stiffness and Deflection Analysis **

In the 1950’s, the method of calculating stiffness coefficients of structures was automated. At that time, the objective was to develop a method that will yield accurate structure displacements, and associated strains and forces, for linear elastic material displacements and vibration and dynamic behavior.

Stiffness of a complete structure is obtained by summing the stiffness of individual elements. Basic conditions of continuity and equilibrium are established at the nodes of the elements of the structure model. Increasing the number of nodes, a finer assemblage of elements to describe the global structure, increases the accuracy of the analysis results. Any physically possible supports or boundary condition of the structure can be included in the model. Calculations were carried out on digital main frame computers

The stiffness matrix of the entire structure is computed by simple summation of the stiffness stiffness matrices of the elements of the structure. The matrix of deflection influence coefficients is obtained by inversion of the stiffness matrix. The class of structures considered includes idealized structures using combined individual elements to analyze the “wing” that has behavior between thin stiffened shells and solid plates. These are hollow structures having a large share of the bending material located in the skin, which is relatively thick, but still thin enough so plate bending stiffness can be “ignored” in the model elements. The method includes shear lag, torsion-bending, and Poisson’s ration effects to sufficient approximation for reliable prediction of vibration modes and natural frequencies. What is required, is an approximate numerical method of analysis which avoids modification of the structure geometry or artificial constraints of the elastic elements.

In the displacement method, joint displacements and rotations are the unknowns, and solution generates a system of joint equilibrium equations. The global stiffness matrix is the sum of element stiffness matrices. Supports are required for a stable system that prevents rigid body translation or rotation. A large number of significant digits in the computation are required for accuracy. Both bending and axial deformation are included in the analysis of frame structures. Connected parts or elements of a continuous structure transfer forces, and the nodes or joints at the boundary of the connected elements, must have compatible translations and rotations. The connected element is a “force – transducer” or transmitter. This thinking supported the theory of flux assumptions that led to development of element(s) stiffness, stiffness equations, and matrix mathematics solutions methods that led more efficient stiffness equation solvers.

Professor Olek Zienkiewicz is a FEM expert. Zienkiewicz’s graduate studies and early academic focus was research and development of the Finite Difference Method (FDM). In 1964, Ray Clough convinced Zienkiewicz, to try his Finite Element Method. Zienkiewicz became convinced of the elegance and potential of this method. Zienkiewicz, is the first academic to author and publish a now classic book on FEM. In 1966, he completed writing the first landmark book, The Finite Element Method, now in the Sixth Edition. Zienkiewicz graduate students, at the University of Wales, at Swansea, continue the development of the FEM.

Jon Turner is recognized as a scientist, an engineer, a mathematician, and an innovator. He was a visionary as exemplified by his continued leadership in advancing Boeing’s development of the integrated multi-disiplinary structure design and analysis system called ATLAS. The ATLAS System was a large – scale Finite Element based computing system for linear and nonlinear modeling and analysis of metal and composite materials and structure optimization. The roots of ATLAS and NASTRAN date back to the 1960’S when Boeing was a FEM leader. NASTRAN, the next generation FEM program developed for the NASA space program, is an acronym for __N__ASA __STR__uctural __AN__alyisis.

ATLAS included several “state-of-the-art” sub-structuring techniques including Component Mode Reductions. This method reduces a finite element model down to a set of boundary matrices that represent the dynamic characteristics of the structure. One of the methods, “Craig-Bampton” component mode reduction is incorporated in MSC Nastran, NX Nastran, ABAQUS and ANSYS.

Natural frequencies are one of the fundamental dynamic characteristic of a structure. Engineers use the natural frequencies and associated mode shapes to understand dynamic response. From a mathematical perspective, the calculation of natural frequencies and mode shapes is an eigenvalue extraction problem in which roots (eigenvalues) and associated mode shapes (eigenvectors) are computed from the dynamic equation of motion, with the assumption of harmonic motion, while neglecting damping and applying no loading. The Lanczos eigenvalue extraction method is the most prevalent extraction method used in FEM programs today. Prior to the commercial availability of the Lanczos method in the mid 1980’s, engineers spent much time in determining ways to reduce the model size and still obtain accurate and reliable results. Cornelius Lanczos was a colleague of Albert Einstein.

From 1962 – 1972, FEM research and development was accelerated by the work of Melosh who showed that conforming displacement models are a form of the Rayleigh-Ritz minimum potential energy principle. Melosh integrated three areas of research, Argyris’s dual formulation of energy methods, Turners direct stiffness method, and the ideas of interelement compatibility as the basis of error bounding and convergence. The pioneers thought of Finite Elements as idealizations of structural components. By the early 1960’s, FEM began to expand into Civil Engineering through Clough’s Boeing – Berkeley connection. From 1962 onwards, the displacement formulation of structure analysis dominates research and development and professional engineers use for structure analysis. The displacement formulation was supported and improved because of the invention of the isoparametric formulation and related tools including; numerical integration, fitted natural coordinates, shape functions, and patch test by Irons and his research assistants.

Professor Ed Wilson is credited with huge contributions in developing and broadening the scope of Finite Elements beyond aerospace applications. Much of the 1960’s and 1970’s research and development, including the work at U.C. Berkeley by Ed Wilson, involved optimizing equation solvers, and the development of new and improved 2 dimensional elements and 3 dimensional solid elements that capture linear and non-linear material behavior. Further FEM advances by Clough and Wilson at U.C. Berkeley included dynamic analysis and the ability to model and analyze structures for linear and recently non linear, inelastic behavior, subjected to earthquake generated motions.

The U.C. Berkeley Structural Engineering and Structural Mechanics Department (SESM), was the “center of the universe” for FEM Dynamic Analysis, Structure Dynamics, and Earthquake Engineering in the 50’s, 60’s, and 70’s. Ray Clough shifted his engineering and academic focus from the FEM to structure dynamics and earthquake engineering. In 1969 he published his landmark book, Dynamics of Structures and he is universally recognized and his reputation is for his work in Structure Dynamics. His early work and influence in creating the modern methods of the Finite Element Method was in the shadow of his stature in structure dynamics and linear and non-linear structure response to earthquake ground motions.

A quote from Clough’s 1956 reference paper, *Stiffness and Deflection Analysis of Complex Structures,* “It is to be expected that modern developments in high-speed digital computing machines will make possible a more fundamental approach to the problems of Structural Analysis.” The paper presents a method of theoretical analysis and outlines the development of a method that is well adapted to the use of high-speed digital computing machinery.

- A complex structure must first be described by an equivalent idealized “model” consisting of structure parts [Elements] geometry and material properties, that are connected to each other and at “Supports” at selected nodes.
- The stiffness matrices for each Element must be determined.
- While all other nodes are fixed. A given node is displaced in one selected coordinate direction. The forces required to do this, and the reactions at neighboring nodes are then known for the various individual member stiffness matrices. When all components of the displacement at all the nodes are considered in this manner, the complete stiffness matrix can be developed.
- Desired support conditions are imposed by striking out columns and rows, in the stiffness Matrix, for which zero displacements have been specified by the engineer.
- For any set of external forces at the nodes, matrix calculations applied to the stiffness matrix yields all components of node displacement, plus the external reactions.
- Forces in internal members are calculated by applying the appropriate force-deflection relationship.

Professor Ray Clough invented and developed the application of the Finite Element Method to determine structure displacement, calculate stresses, vibration, and dynamic response from external loads. The FEM method evolution continues to be supported by many technical developments, from Ray Clough colleagues at Boeing Aircraft Company and at Universities in Europe, Asia and the Americas.

**Finite Element Method Computer Programs **** **

1967 – STRUDL __STRU__ctural Design __L__anguage – a subsystem of the Integrated Civil Engineering System [ICES] Developed by M.I.T Civil Engineering System Laboratory for analysis ad design in structural engineering. Lincoln Laboratory Version of STRUDL – Department of the Air Force Contract F19668-76-C-0002

1967 – NASTRAN __NA__sa __STR__uctural ANalysis a nonlinear structure analysis program. R H MacNeal, C W McCormick A new general purpose structural analysis program, solve static and dynamic structural problems. Developed by Computer Science Corporation [CSC] , the Martin Corporation and MacNeal-Schwendler Corporation

1973 – SAP A __S__tructural __A__nalysis Program for linear systems, linear static and dynamic capability, Solid SAP only linear analysis, EASE -1972 dynamic analysis and non-linear analysis , SAP IV, & NONSAP nonlinear dynamic analysis , E L Wilson, K J Bathe, F E, Peterson, H H Dovey

1974 – NONSAP released – NONlinear Structural Analysis Program developed nonlinear analysis program for the Bureau of Mines – incremental solution was needed with iteration in each load or time step

1974 – ANSR D P Mondkar, G H Powell General purpose computer program for static and dynamic analysis of non-linear structures ANSR is a general purpose computer program for static and dynamic analysis of nonlinear structures. This report documents the features and organization of the current version of the program. The theoretical formulations and solution schemes used in the program are described, and details are given about the structure and organization of the auxiliary program for adding new finite elements to the program. Several examples are presented to illustrate the scope of ANSR. The user’s manual for the program is described.

1986 – ADINA __A__utomatic __D__ynamic __I__ncremental __N__onlinear __A__nalysis Klaus-Jurgen Bathe, ADINA Structures for linear and nonlinear analysis of solids and structures, for thermal analysis of heat transfer in solids and field problems, for analysis of compressible and incompressible flows, including heat transfer, and analysis of electromagnetic phenomena

1983 – ANSYS J A Swanson, G R. Cameron, J C Haberland, Dynamic analysis, soil – structure interaction

References

- J.H. Argyris, Energy Theorems and Structural Analysis, Aircraft Engineering, 1956
- M.J. Turner, R.W. Clough, H.C. Martin, and L.J. Topp,
*“Stifffness and Deflection Analysis of Complex Structures,*” Journal of the Aeronautical Sciences, 1956 - R.W. Clough presentation,
*“Early History of the Finite Element Method from the view point of a pioneer”*, International Journal for Numerical Methods in Engineering, 2004 - R.W. Clough, “
*The Finite Element Method in Plane Stress Analysis*,” Proceedings of the Second ASCE Conference on Electronic Computation, Pittsburgh, PA 1960 - R.W. Clough,
*“The Finite Element Method is Structural Mechanics”*, Stress Analysis, Chapter 7, Wiley: New York, 1965 - R.W. Clough,
*“The Finite Element Method after 25 years”*, Engineering Applications of the Finite Element Method, A. S. Computas, Dt Norske Veritas, Hovik, Norway, 1979 - R.W. Clough,
*“Original Formulation of the Finite Element Method”*, ASCE Structures Congress, San Francisco, CA, May 1989. - R.W. Clough,
*“Thoughts about the Original Formulation of the FEM – a Personal View”*, Proceedings of the European Conference on Computational Mechanics, Munich 1999 - R.W. Clough, E.L. Wilson,
*“Stress Analysis of a Gravity Dam by the Finite Element Method”*, BULLETIN RILEM No. 19, report by University of California, Institute of Engineering Research Contract for the Little Rock District, U.S. Army Corps of Engineers, June 1963 - E.L. Wilson,
*“Structural Analysis of Axisymmetric Solids”,*AIAA Journal 3:12, 2269-2274, 1965 - E.L. Wilson,
*“Automation of the Finite Element Method, a Personal Historical View”*, Finite Elements in Analysis and Design, volume 13, 91-104 Elevier: Amsterdam, 1993 - Carlos A. Felippa,
*“The Origins of the Finite Element Method”*, Appendix O - Carlos A. Felippa,
*“A Historical Outline of Matrix Structural Analysis: a Play in Three Acts”*, Department of Aerospace Engineering Sciences and Center for Aerospace Structures, University of Colorado, Boulder, CO 2001

Textbooks

- O.C. Zienkiewicz, Y. K. Cheung,
*The Finite Element Method in Structural and Continuum Mechanics, McGraw Hill, 1967*and further editions. - Klaus J. Bathe,
*Finite Element Procedures in Engineering Analysis,*1982 - Klaus J. Bathe,
*To Enrich Life*

Contributors to article: Reinhard Ludke, Ed Wilson

**Impact to Structural Engineering History in Northern California**

The Finite Element Method, created by Ray Clough in the 1950’s is used by engineers, scientists, and many professionals and in many disciplines to model and analyse static and dynamic behavior of structures and soils and water. Computers and this tool fundamentally changed the practice of structural engineering. The evolution of FEM to non-linear dynamic analysis of soil – structure interaction to earthquake ground motions would not be possible without Ray Clough’s vision for FEM.

**Related Organization(s)**

- University of California, Berkeley
- University of Washington

**Related Engineer(s)**

**Related Structure(s)**