n5321 | 2025年6月14日 10:42
Since the introduction of the finite element method over thirty-five years ago, applications of this analytical procedure to problems of engineering practice have undergone phenomenal growth. Both the diversification in areas of application and the increasing complexity of the problems to which it has been applied could not have been imagined at the time when its basic concepts were first promulgated. One consequence of this rapid growth is that no single unique definition of the finite element method can be stated. In fact, it is evident that the name now means different things to different people, and it may be useful to mention two differing points of view.
One group of practitioners considers the finite element method to be merely an extension of standard methods of structural analysis [4] in which the structure is treated as an assemblage of discrete structural elements. From this point of view the finite element extension permits the use of two- and three-dimensional elements in addition to the structural beam and truss elements that had been used previously, as shown in Fig. 1 [4]. The alternative opinion, held by many other engineers and scientists, is that the finite element method is a procedure for a obtaining approximate solutions of problems in continuum mechanics based on applying assumed strain patterns independently in discrete regions of the system [16]. Figure 2 [3] shows a cross-section of a gravity dam, a classical problem of plane stress analysis that had been solved in the past by the finite difference method; this was one of the first practical problems of continuum mechanics that was solved using finite elements. As a matter of fact, the concept of regional discretization, which is a central feature of the finite element method, had been proposed many years earlier by Courant [8] and by Prager and Synge [14] for solving such plane stress problems. However, these early suggestions were not followed up, mainly because computers were not available to carry out the extensive numerical operations, and references to these ideas were only rediscovered long after the finite element method had been accepted as a practical engineering tool.
Both of these contrasting points of view are valid, and each presently has a group of staunch advocates. However, it is important to note in this description of the original formulation of the finite element method that the first of these concepts guided the development; that is, the new method was considered merely as an extension of standard analysis procedures that was introduced to solve a difficult problem. Admittedly this is only a subjective impression of the events that led to the development, and possibly others who were involved would remember it differently. However, this presentation has some validity because the author not only was a participant in developing the basic concept but also was responsible for coining the name of the method. For convenience, this description is presented in five parts: (a) prior state of the art of structural analysis; (b) the 1952 Boeing Summer Faculty Program; (c) the 1953 breakthrough; (d) matrix formulation of structural analysis theory; and (e) naming the finite element method.
In order to explain the development of the finite element method, it is useful to outline the state of the art of structural analysis that had been attained in the aircraft industry by the early 1950s. Generally speaking, at the end of World War II, the concepts of structural theory used in aircraft design were the same as those used by civil engineers in the analysis of buildings and bridges. In both fields, a structure was defined as an assemblage of discrete structural elements, and the analysis consisted of evaluating the joint displacements and element forces due to a specified applied load system. During the analysis the assemblage was required to satisfy three basic conditions: (a) equilibrium of the element forces with the external loads, (b) compatibility of the deformed elements so that continuity is maintained at the joints, and (c) force-deformation relationships in the elements that depend on the element properties.
It was recognized that the structural analysis might be done in either of two ways: the force method wherein the element forces were determined so that joint compatibility was achieved, or the displacement method in which joint displacements were calculated to satisfy equilibrium requirements. In the analysis of flexural frameworks, a special version of the displacement method, called moment distribution, usually was used in which the equilibrium conditions were satisfied by iterative adjustment. However, in the analysis of more general types of structures, the force method usually was preferred because it required the solution of a smaller set of simultaneous equations.
In the years immediately following World War II, the aircraft structural engineering profession began to move ahead of the civil engineers in the theory of structural analysis due to pressures resulting from the increasing complexity of airplane configurations and the compelling need to eliminate excess weight. An important advance initiated almost exclusively by aeronautical engineers was the introduction of matrix notation in formulating the analysis. A factor contributing to this step was that the major airplane design companies had access to the best computers available at that time, and the matrix formulation greatly facilitated the use of computers as well as the standardization and simplification of the calculations. One outcome of the matrix formulation of structural analysis was that it became apparent that either the force or the displacement method could be recognized as a coordinate transformation in which the element properties expressed in local coordinates were transformed to express the properties of the assembled structure in global coordinates. Among the first to recognize that structural analysis is essentially a coordinate transformation procedure were Falkenheimer [9] and Langefors [12]. In summary, by 1952 aircraft structural analysis had advanced to the point where a complex structure idealized as an assemblage of simple truss beam or shear panel elements could be analyzed by either the force or displacement method formulated as a series of matrix operations and using an automatic digital computer to carry out the calculations.
My participation in the work that led to the development of the finite element method began in 1952 when I joined the Boeing Summer Faculty Program; this was a program in which young engineering professors were hired from all over the country to work on various special research projects. I was attracted to the program because it offered an opportunity to work in the field of structural dynamics, and I was particularly fortunate to be assigned to work directly under the head of the Structural Dynamics Unit, Mr. M.J. Turner.
The specific problem that Jon Turner asked me to work on in 1952 was the calculation of flexibility influence coefficients for low aspect ratio wing structures having either straight or swept-back configurations. Such influence coefficients were needed to predict flutter and other aeroelastic effects that might influence flight stability and control, and it was evident that ordinary beam theory was not suitable for such calculations on low aspect ratio wings, even if torsion bending and shear lag effects were considered. Accordingly I was first asked to review the recent literature on the subject, and it was during this time that I became familiar with the work of Falkenheimer and Langefors on matrix transformation concepts applied to structural analysis [9,12].
Making use of such matrix procedures, I attempted to calculate the flexibility influence coefficients for a 45° swept-back box beam that had been tested in the laboratory so that experimental values of the coefficients were available. Figure 3 is a sketch of the test structure, which was modeled as an assemblage of rectangular and triangular skin shear elements together with axial force elements (spar caps and stringers). In addition, a portion of the wing skin was combined with the spar caps to account for its contribution to the flexural rigidity.
Results of the analysis for the simple bending load condition (Case 1) showed that the calculated deflections exceeded the measured values by 13-65%. The results for the twist loading (Case 2) were somewhat better, especially for the structure with the refined model of the root region (NACA-2), but still were not of acceptable quality. These generally poor results were not surprising, but they clearly demonstrated that improvements were needed in the modeling concepts; it was believed that the stiffness contributions due to the wing skin were the probable cause of most of the discrepancy and an effort was made to formulate an improved model of the wing skin using the "lattice analogy" as developed by Hrennikoff [10] and McHenry [13]. For simple rectangular panels, it is easy to define a lattice arrangement of truss bars that will represent exactly the deformations of the panel induced by simple patterns of applied normal and shear stresses. However, no lattice analogy procedure could be developed to model the stiffness of panels of arbitrary quadrilateral or triangular shape. Hence, at the end of the 1952 Summer Faculty Program, it was concluded that the essential features of the flexibility influence coefficient problem had been identified, but that little progress had been made toward the proper representation of the wing skin in the mathematical modeling of low aspect ratio wing structures.
Because of the unfinished state of my work at the termination of the 1952 Summer Faculty Program, I was pleased to be employed again with M.J. Turner's Structural Dynamics Unit in 1953. During the 1952-53 winter, Jon Turner had conceived a better way to model the skin panels of a low aspect ratio wing. Rather than representing the skin as shear panels which also made axial stiffness contributions to the spar and rib caps, he proposed that the skin should contribute its full normal and shear stress resistance in response to any applied loads.
The essential idea in the proposed Turner procedure was that the deformations of any plane stress element be approximated by assuming a combination of simple strain fields acting within the element. The idea is applicable to both rectangular and triangular elements, but the use of triangular elements was given greater emphasis because an assemblage of triangular elements could serve to approximate plates of any shape. In modeling a triangular plate, the deformations were approximated by three constant strain fields: uniform normal strains in the x and the y directions combined with a uniform x-y shear strain. Based on these strain patterns, the force-displacement relationships for the corner nodal points could be calculated using Castigliano's theorem, or the equivalent principle of virtual displacements. An important feature of the assumed constant strain condition within the triangle was that each side of the element would remain straight during deformation; thus full continuity between elements of an assemblage was assured if continuity were maintained at the corner nodes.
During the summer of 1953, procedures for calculating the plane stress stiffness of triangular (as well as rectangular) plates using the Turner concept were formulated, and the effectiveness of the concept was evaluated by analysis of the deflections of various plate systems subjected to load. In these analyses, the direct stiffness method was used to obtain the stiffness of the assemblage rather than the matrix transformation approach; the direct stiffness procedure was more efficient because it involved only the appropriate summation of the individual plate stiffnesses. One study case was a rectangular plate clamped at one edge and subjected to uniform tensile stress applied at the opposite edge, as shown in Fig. 4. This structure was first modeled as a single quadrilateral plate element and then as assemblages of various shapes and numbers of quadrilateral elements, as shown in Fig. 5. The stiffness of each quadrilateral element was obtained by assembling four triangular elements, as shown in Fig. 6(c), and then eliminating the central node by static condensation. The alternative methods of assembling triangles to form a quadrilateral, shown in Figs. 6(a, b), also were studied, but they led to results that were less accurate and consistent. The number of triangular elements used in modeling the various cases (a) through (d) of Fig. 5 ranged from 4 to 64. The results of the analyses showed that the deflections calculated at the corners of the plate converged toward those obtained from a refined finite difference analysis, and that the values obtained using the finest mesh of quadrilateral plates gave excellent agreement with the finite difference solution.
When the feasibility of evaluating the behavior of plane stress plates had been demonstrated in this way, it became possible to model a complete wing structure using assemblages of two-dimensional elements to represent the wing skin and the spar and rib webs, combined with one-dimensional spar and rib flange elements. This type of modelling of a low aspect ratio wing is illustrated in Fig. 7. The modeling concept was applied to the simple box cantilever shown in Fig. 8 to provide a numerical demonstration of the procedure. The convergence with increasing mesh refinement was studied by varying the number of rectangular plates used to model the top wing skin; between 1 and 18 elements were used in the wing skin (Fig. 9) together with a corresponding mesh of plate and flange elements in the other surfaces of the box beam. The deflections produced by various loads applied at the tip of the box were observed to converge consistently as the number of elements increased, thus demonstrating that the finer element meshes provided better approximations of shear lag and other secondary distortion affects.
To describe the excellent performance achieved by the Turner plane stress triangles in these and other analytical studies done during the 1953 Summer Faculty Program, a paper was prepared that was presented by Jon Turner at the January 1954 meeting of the Institute of Aeronautical Sciences in New York. For reasons I never understood, the paper was not submitted for publication until June 1955, and due to normal publication delays it did not appear in print until September 1956 [15]. However, it is important to note that this 1956 paper, which generally is recognized to have introduced the finite element method as a tool for structural analysis, describes the work done during the 1953 Boeing Summer Faculty Program. Also, it should be recognized that the principal credit for conceiving the procedure should go to M.J. Turner, who not only led the developmental effort for the two critical years of 1952-53, but who also provided the inspiration to use assumed strain patterns in defining the stiffness of triangular plane stress elements.
For various reasons, I was not able to schedule another summer work period at Boeing after 1953; however, I kept in touch with Jon Turner and others involved in the project while the paper [15] was being prepared and for the next several years. Also, I retained a strong interest in matrix methods of structural analysis, and my sabbatical leave in Trondheim, Norway, during 1956-57 gave me time to think further about the analytical work we had done at Boeing in 1953. During the stay in Norway, my attention at first was directed toward the matrix analysis of structures, because great advances had been made in that subject since 1953. By far the most significant contribution was the classic work by Dr. J.H. Argyris and his coworkers first published as a series of articles in Aircraft Engineering [1], which completely stated the matrix formulation of structural theory and clearly outlined the parallel transformation procedures involved in the force and the displacement methods. It was this work that demonstrated that the concepts of classical structural analysis can be generalized for application to assemblages of any types of structural elements, not only to the traditional beams, struts, etc.; however, the Argyris presentation did not discuss the finite element concept for plane stress analysis. A closely related development that also originated in England was a computer program specifically intended to deal with the sequences of matrix operations that performed the structural analysis expressed in matrix form. This pioneering matrix interpretive code by Hunt [11] was of great interest to me, but unfortunately no computer facilities were available in Trondheim in 1956-57 so I was not able to pursue such research at that time.
Instead I turned my attention back to the finite element method, and in particular to the stress analysis of plane stress systems using discrete triangular plate elements because the work at Boeing had been concerned only with deflections and displacement influence coefficients. It was evident that the strain patterns used in defining the in-plane plate stiffness also could be related to the state of stress developed in the plate, and I studied this stress analysis problem in principle while I was at Trondheim. However, the lack of automatic computer facilities made it impossible for me to do any significant analyses of this type while I was there.
When I returned to Berkeley in September 1957, I found that a new IBM 701 computer had been installed in the College of Engineering, and I immediately began to develop a Matrix Algebra Program [2] for that facility similar to the one described by Hunt. This program would carry out any specified sequence of matrix operations, thus it made it possible for me to study the use of the Turner triangular plate elements in solving practical plane stress problems. After a few trial analyses I became convinced that this was a tool capable of solving any plane stress problem to any desired degree of accuracy. However, when I commented to this effect to my Berkeley colleague (now Dean) K.S. Pister, he was very skeptical and challenged me to solve some of the classical problems of plane stress analysis by this method. Much to his surprise and to my satisfaction, excellent results were obtained, as shown by Fig. 10, which depicts one of the earliest test cases. Of course the stress accuracy obtained in this case and in the dam section analysis of Fig. 2 (which was done at the same time) was limited by the large size of the elements used. Particularly in the Fig. 2 analysis of an assemblage of triangular elements, the assumed constant state of stress in each element leads to significant stress discontinuities, as shown in the figure. However, by using small elements in regions of steep stress gradients, and by drawing smooth curves to depict the stress distribution resulting from the analysis, it was possible to get excellent agreement with theory. More refined analyses that were done a short time later further demonstrated this fact, as shown in Figs. 11 and 12 [4].
Because the results of these studies were so encouraging, I thought it was important to prepare a paper for the structural engineering profession on the use of Turner triangular elements in stress analysis. Probably the potential for such analyses was recognized by Jon Turner and his group, but their mission was concerned with stiffness and deflection as required in structural dynamics, rather than with stress analysis. The forthcoming 2nd ASCE Conference on Electronic Computation, to be held in Pittsburgh in September 1960, offered the ideal occasion to present the paper because the triangular element concept had been discussed only in the aeronautical industry up to that time. In writing the paper, the principal problem turned out to be the selection of a suitable name for the method. At Boeing, the term "direct stiffness method" tended to be associated with the solution of structural problems using assemblages of plane stress elements. However, in my opinion this name merely described the element assembly procedure, and did not relate to the essential problem of evaluating the stiffness of the plane stress elements. Noting the analogy to the finite difference procedure for solving plane stress problems, but recognizing that in the new method the equations of elasticity were applied independently in discrete regions, it seemed to me that those building blocks of the structural model should be called finite elements. Consequently, the 1960 paper was called "The Finite Element Method in Plane Stress Analysis" [3].
When the paper was presented, it had essentially no impact on the civil engineering profession, mainly because the method could be applied effectively only by means of an automatic digital computer, and these were not readily available to typical structural engineers. However, we had a suitable computer at Berkeley, and we were fortunate to obtain a research contract from the U.S. Corps of Engineers at this time which enabled us to make a significant advance in the application of the finite element method to practical civil engineering problems. The objective of this research was to evaluate the safety of a concrete gravity dam that had developed a major interior crack as a result of temperature changes during construction, as depicted by Fig. 13(a). This plane stress problem was an ideal example for study by the finite element method, and provided an opportunity for my doctoral student (now colleague) E.L. Wilson to write the first computer program for finite element plane stress analysis. The program calculated the stiffness properties for any specified mesh of triangular elements, and then used Gauss-Seidel iteration to solve the equilibrium equations so that very large equation sets could be considered. The finite element meshes used in this study are shown in Fig. 13(b) and some of the stress results are shown in Fig. 14. The results of the research effort were very satisfactory to the sponsors and were presented at the Symposium on the Use of Computers in Civil Engineering that was held in Lisbon, Portugal, in 1962; this was the second time that the finite element name appeared in the title of a paper [6].
In my opinion, that paper and the computer program used in its calculations marked the end of the "Original Formulation" period in the history of the finite element method. The rapid world-wide acceptance of the method was very evident at the 1965 Conference on Matrix Methods in Structural Mechanics held at Wright-Patterson Air Force Base [7], at which many papers were presented involving a wide range of applications of the finite element method.
In closing this paper, I will reiterate some concerns that I expressed in a paper written over ten years ago [5], in which I deplored the excessive confidence that some engineers had in the results of computer analyses of complex structures. The major point of those comments was that the results of a finite element analysis cannot be better than the data and the judgment used in formulating the mathematical model, regardless of the refinement of the computer program that performs the analysis. The main purpose of that word of caution was to emphasize the continuing need for experimental observations of structural behavior; it is only with such experimental evidence that computer analysis procedures can be validated, and there is no question that the need for such validation is as great now as it was ten years ago.
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