The finite element method is used to solve physical problems in engineering analysis and design. Figure 1. The physical problem typically involves an actual structure or structural component subjected to certain loads. The idealization of the physical problem to a mathematical model requires certain assump- tions that together lead to differential equations governing the mathematical model see Chapter 3.
The finite element analysis solves this mathematical model. Since the finite element solution technique is a numerical procedure, it is necessary to assess the solution accuracy. If the accuracy criteria are not met, the numerical i. It is clear that the finite element solution will solve only the selected mathematical model and that all assumptions in this model will be reflected in the predicted response.
We cannot expect any more information in the prediction of physical phenomena than the information contained in the mathematical model. Hence the choice of an appropriate mathematical model is crucial and completely determines the insight into the actual physical problem that we can obtain by the analysis. Let us emphasize that, by our analysis, we can of course only obtain insight into the physical problem considered: we cannot predict the response of the physical problem exactly because it is impossible to reproduce even in the most refined mathematical model all the information that is present in nature and therefore contained in the physical problem.
Furthermore, a change in the physical problem may be necessary, and this in turn will also lead to additional mathemat- ical models and finite element solutions see Fig.
The key step in engineering analysis is therefore choosing appropriate mathematical models. To define the reliability and effectiveness of a chosen model we think of a very- comprehensive mathematical model of the physical problem and measure the response of our chosen model against the response of the comprehensive model.
In general, the very- comprehensive mathematical model is a fully three-dimensional description that also in- cludes nonlinear effects. Effectiveness of a mathematical model The most effective mathematical model for the analysis is surely that one which yields the required response to a sufficient accuracy and at least cost.
Reliability of a mathematical model The chosen mathematical model is reliable if the required response is known to be predicted within a selected level of accuracy measured on the response of the very- comprehensive mathematical model. Hence to assess the results obtained by the solution of a chosen mathematical model, it may be necessary to also solve higher-order mathematical models, and we may well think of but of course not necessarily solve a sequence of mathematical models that include increasingly more complex effects.
For example, a beam structure using engineering termi- nology may first be analyzed using Bernoulli beam theory, then Timoshenko beam theory, then two-dimensional plane stress theory, and finally using a fully three-dimensional continuum model, and in each case nonlinear effects may be included. Such a sequence of models is referred to as a hierarchy of models seeK.
Bathe, N. Lee, and M. Bucalem [A]. Clearly, with these hierarchical models the analysis will include ever more complex response effects but will also lead to increasingly more costly solutions. As is well known, a fully three-dimensional analysis is about an order of magnitude more expensive in computer resources and engineering time used than a two-dimensional solution.
Let us consider a simple example to illustrate these ideas. For the analysis, we need to choose a mathematical model. This choice must clearly depend on what phenomena are to be predicted and on the geometry, material properties, loading, and support conditions of the bracket. We have indicated in Fig. The description "very thick" is of course relative to the thickness t and the height h of the bracket. We translate this statement into the assumption that the bracket is fastened to a practically rigid column.
Hence we can focus our attention on the bracket by applying a "rigid column boundary condition" to it. Of course, at a later time, an analysis of the column may be required, and then the loads carried by the two bolts, as a consequence of the load W, will need to be applied to the column. We also assume that the load W is applied very slowly.
The condition of time "very slowly" is relative to the largest natural period of the bracket; that is, the time span over which the load W is increased from zero to its full value is much longer than the fundamen- tal period of the bracket.
We translate this statement into requiring a static analysis and not a dynamic analysis. With these preliminary considerations we can now establish an appropriate mathe- matical model for the analysis of the bracket-depending on what phenomena are to be predicted. Let us assume, in the first instance, that only the total bending moment at section AA in the bracket and the deflection at the load application are sought.
To predict these quantities, we consider a beam mathematical model including shear deformations [see Fig. Pin w. Areas with imposed zero displacements u, v. Of course, the relations in 1. Let us now ask whether the mathematical model used in Fig. This model should include the two bolts fastening the bracket to the assumed rigid column as well as the pin through which the load W is applied.
The three-dimensional solution of this model using the appropriate geometry and material data would give the numbers against which we would compare the answers given in 1. Note that this three-dimensional mathematical model contains contact conditions the contact is between the bolts, the bracket, and the column, and between the pin carrying the load and the bracket and stress concentrations in the fillets and at the holes.
Also, if the stresses are high, nonlinear material conditions should be included in the model. Of course, an analytical solution of this mathematical model is not available, and all we can obtain is a numerical solution.
We describe in this book how such solutions can be calculated using finite element procedures, but we may note here already that the solution would be rela- tively expensive in terms of computer resources and engineering time used.
Since the three-dimensional comprehensive mathematical model is very likely too comprehensive a model for the analysis questions we have posed , we instead may consider a linear elastic two-dimensional plane stress model as shown in Fig. This mathemat- ical model represents the geometry of the bracket more accurately than the beam model and assumes a two-dimensional stress situation in the bracket see Section 4.
The bending moment at section AA and deflection under the load calculated with this model can be expected to be quite close to those calculated with the very-comprehensive three- dimensional model, and certainly this two-dimensional model represents a higher-order model against which we can measure the adequacy of the results given in 1. Of course, an analytical solution of the model is not available, and a numerical solution must be sought. Figures 1. Let us note the various assumptions of this math- ematical model when compared to the more comprehensive three-dimensional model dis- cussed earlier.
Also, the actual bolt fastening and contact conditions between the steel column and the bracket are not included. Deflections are drawn with a magnifi- cation factor of together with the original configura- tion.
Un- e Maximum principal stress near notch. The small Smoothed stress results. The averages of breaks in the bands indicate that a reasonably nodal point stresses are taken and interpo- accurate solution of the mathematical model lated over the elements.
However, since our objective is only to predict the bending moment at section AA and the deflection at point B, these assumptions are deemed reasonable and of relatively little influence. Let us assume that the results obtained in the finite element solution of the mathemat- ical model are sufficiently accurate that we can refer to the solution given in Fig.
Considering these results, we can say that the beam mathematical model in Fig. The beam model is of course also effective because the calculations are performed with very little effort. On the other hand, if we next ask for the maximum stress in the bracket, then the simple mathematical beam model in Fig.
Specifically, the beam model totally neglects the stress increase due to the fillets. The important points to note here are the following. The selection of the mathematical model must depend on the response to be predicted i. The most effective mathematical model is that one which delivers the answers to the questions in a reliable manner i.
A finite element solution can solve accurately only the chosen mathematical model e. The notion of reliability of the mathematical model hinges upon an accuracy assess- ment of the results obtained with the chosen mathematical model in response to the questions asked against the results obtained with the very-comprehensive mathemat- ical model.
However, in practice the very-comprehensive mathematical model is 1 The bending moment at section AA in the plane stress model is calculated here from the finite element. Finally, there is one further important general point. The chosen mathematical model may contain extremely high stresses because of sharp corners, concentrated loads, or other effects.
These high stresses may be due solely to the simplifications used in the model when compared with the very-comprehensive mathematical model or with nature. For example, the concentrated load in the plane stress model in Fig. This pressure would in nature be transmitted by the pin carrying the load into the bracket. The exact solution of the mathematical model in Fig.
Of course, this very large stress is an artifice of the chosen model, and the concentrated load should be replaced by a pressure load over a small area when a very fine discretization is used see further discussion. Furthermore, if the model then still predicts a very high stress, a nonlinear mathematical model would be appropriate. Note that the concentrated load in the beam model in Fig. Also, the right-angled sharp corners at the support of the beam model, of course, do not introduce any solution difficulties, whereas such corners in a plane stress model would introduce infinite stresses.
Hence, for the plane stress model, the corners have to be rounded to more accurately represent the geometry of the actual physical bracket. We thus realize that the solution of a mathematical model may result in artificial difficulties that are easily removed by an appropriate change in the mathematical model to more closely represent the actual physical situation.
Furthermore, the choice of a more encompassing mathematical model may result, in such cases, in a decrease in the required solution effort. While these observations are of a general nature, let us consider once again, specifically, the use of concentrated loads.
This idealization of load application is exten- sively used in engineering practice. We now realize that in many mathematical models and therefore also in the finite element solutions of these models , such loads create stresses of infinite value.
Hence, we may ask under what conditions in engineering practice solution difficulties may arise. We find that in practice solution difficulties usually arise only when the finite element discretization is very fine, and for this reason the matter of infinite stresses under concentrated loads is frequently ignored. As an example, Fig. The cantilever is subjected to a concentrated tip load.
In practice, the 6 X 1 mesh is usually considered sufficiently fine, and clearly, a much finer discretization would have to be used to accurately show the effects of the stress singularities at the point of load application and at the support. As already pointed out, if such a solution is pursued, it is necessary to change the mathematical model to more accurately represent the actual physical situation of the structure. This change in the mathematical model may be important in self-adaptive finite element analyses because in such analyses new meshes are generated automatically and artificial stress singularities cause-artificially-extremely fine discretizations.
We refer to these considerations in Section 4. In summary, we should keep firmly in mind that the crucial step in any finite element analysis is always choosing an appropriate mathematical model since a finite element solution solves only this model.
Furthermore, the mathematical model must depend on the analysis questions asked and should be reliable and effective as defined earlier. In the process of analysis, the engineer has to judge whether the chosen mathematical model has been solved to a sufficient accuracy and whether the chosen mathematical model was appropriate i. Choosing the mathematical model, solving the model by appropriate finite element procedures, and judging the results are the fundamental ingredients of an engineering analysis using finite element methods.
Although a most exciting field of activity, engineering analysis is clearly only a support activity in the larger field of engineering design. The analysis process helps to identify good new designs and can be used to improve a design with respect to performance and cost. In the early use of finite element methods, only specific structures were analyzed, mainly in the aerospace and civil engineering industries.
However, once the full potential of finite element methods was realized and the use of computers increased in engineering design environments, emphasis in research and development was placed upon making the use of finite element methods an integral part of the design process in mechanical, civil, and aeronautical engineering.
Bathe [C, D, H]. Finite element analysis is only a part of the complete process, but it is an important part. We note that the first step in Figure 1. Many different computer programs can be employed e. In this step, the material properties, the applied loading and boundary conditions on the geometry also need to be defined.
Given this information, a finite element analysis may proceed. Since the geometry and other data of the actual physical part may be quite complex, it is usually necessary to simplify the geometry and loading in order to reach a tractable mathematical model. Of course, the mathematical model should be reliable and effective for the analysis questions posed, as discussed in the preceding section.
The finite element analysis solves the chosen mathematical model, which may be changed and evolve depending on the purpose of the analysis see Fig. Considering this process-which generally is and should be performed by engineer- ing designers and not only specialists in analysis-we recognize that the finite element methods must be very reliable and robust.
By reliability of the finite element methods we now 3 mean that in the solution of a well-posed mathematical model, the finite element procedures should always for a reasonable finite element mesh give a reasonable solution, 3 Note that this meaning of "reliability of finite element methods" is different from that of "reliability of a.
Open navigation menu. Close suggestions Search Search. User Settings. Skip carousel. Carousel Previous. Carousel Next. What is Scribd? Explore Ebooks. Bestsellers Editors' Picks All Ebooks. Explore Audiobooks. Bestsellers Editors' Picks All audiobooks. Explore Magazines.
Editors' Picks All magazines. It has been a great privilege to be their teacher and work with them. I would like to thank all my students, colleagues and friends who contributed to my knowledge and understanding of finite element methods. My interaction with them, given in papers referred to in the book, has resulted in achievements that I am very proud of, see also my book "To Enrich Life".
I humbly hope that this book will be of value to many students and professionals to increase their understanding of finite element procedures. In that sense, I would like to close this Preface by quoting Shakespeare:.
Klaus-Jiirgen Bathe MI. Finite element procedures are at present very widely used in engineering analysis, and we can expect this use to increase significantly in the years to come. The procedures are employed extensively in the analysis of solids and structures and of heat transfer and fluids, and indeed, finite element methods are useful in virtually every field of engineering analysis. The development of finite element methods for the solution of practical engineering problems began with the advent of the digital computer.
That is, the essence of a finite element solution of an engineering problem is that a set of governing algebraic equations is established and solved, and it was only through the use of the digital computer that this process could be rendered effective and given general applicability. These two properties- effectiveness and general applicability in engineering analysis-are inherent in the theory used and have been developed to a high degree for practical computations, so that finite element methods have found wide appeal in engineering practice.
As is often the case with original developments, it is rather difficult to quote an exact "date of invention," but the roots of the finite element method can be traced back to three separate research groups: applied mathematicians-seeR. Courant [A]; physicists-see J. Synge [A]; and engineers-see J. Argyris and S.
Kelsey [A]. Although in principle published already, the finite element method obtained its real impetus from the develop- ments of engineers. The original contributions appeared in the papers by J. Kelsey [A]; M. Turner, R. Clough, H. Martin, and L. Topp [A]; and R. Clough [A]. The name "finite element" was coined in the paper by R. Important early contributions were those of J.
Argyris [A] and 0. Zienkiewicz and Y. Cheung [A]. Since the early s, a large amount of research has been devoted to the technique, and a very large number of publications on the finite element method is.
Kardestuncer and D. Norrie [A]. The finite element method in engineering was initially developed on a physical basis for the analysis of problems in structural mechanics. However, it was soon recognized that the technique could be applied equally well to the solution of many other classes of problems. The objective of this book is to present finite element procedures comprehen- sively and in a broad context for solids and structures, field problems specifically heat transfer , and fluid flows.
To introduce the topics of this book we consider three important items in the following sections of this chapter. First, we discuss the important point that in any analysis we always select a mathematical model of a physical problem, and then we solve that model. The finite element method is employed to solve very complex mathematical models, but it is important to realize that the finite element solution can never give more information than that contained in the mathematical model.
Then we discuss the importance of finite element analysis in the complete process of computer-aided design CAD. This is where finite element analysis procedures have their greatest utility and where an engineer is most likely to encounter the use of finite element methods.
In the last section of this chapter we mention some recent important research accom- plishments that have been reached since the first publication of this book in These achievements have been published in numerous papers, of which we can only mention some, but it is important to note that these research efforts build, to a large degree, upon the fundamental finite element procedures focused upon in this book.
The finite element method is used to solve physical problems in engineering analysis and design. Figure 1. The physical problem typically involves an actual structure or structural component subjected to certain loads. The idealization of the physical problem to a mathematical model requires certain assump- tions that together lead to differential equations governing the mathematical model see Chapter 3. The finite element analysis solves this mathematical model.
Since the finite element solution technique is a numerical procedure, it is necessary to assess the solution accuracy. If the accuracy criteria are not met, the numerical i. It is clear that the finite element solution will solve only the selected mathematical model and that all assumptions in this model will be reflected in the predicted response.
We cannot expect any more information in the prediction of physical phenomena than the information contained in the mathematical model. Hence the choice of an appropriate mathematical model is crucial and completely determines the insight into the actual physical problem that we can obtain by the analysis. Let us emphasize that, by our analysis, we can of course only obtain insight into the physical problem considered: we cannot predict the response of the physical problem exactly because it is impossible to reproduce even in the most refined mathematical model all the information that is present in nature and therefore contained in the physical problem.
Furthermore, a change in the physical problem may be necessary, and this in turn will also lead to additional mathemat- ical models and finite element solutions see Fig.
The key step in engineering analysis is therefore choosing appropriate mathematical models. To define the reliability and effectiveness of a chosen model we think of a very- comprehensive mathematical model of the physical problem and measure the response of our chosen model against the response of the comprehensive model.
In general, the very- comprehensive mathematical model is a fully three-dimensional description that also in- cludes nonlinear effects. Effectiveness of a mathematical model The most effective mathematical model for the analysis is surely that one which yields the required response to a sufficient accuracy and at least cost.
Reliability of a mathematical model The chosen mathematical model is reliable if the required response is known to be predicted within a selected level of accuracy measured on the response of the very- comprehensive mathematical model. Hence to assess the results obtained by the solution of a chosen mathematical model, it may be necessary to also solve higher-order mathematical models, and we may well think of but of course not necessarily solve a sequence of mathematical models that include increasingly more complex effects.
For example, a beam structure using engineering termi- nology may first be analyzed using Bernoulli beam theory, then Timoshenko beam theory, then two-dimensional plane stress theory, and finally using a fully three-dimensional continuum model, and in each case nonlinear effects may be included. Such a sequence of models is referred to as a hierarchy of models seeK. Bathe, N. Lee, and M. Bucalem [A]. Clearly, with these hierarchical models the analysis will include ever more complex response effects but will also lead to increasingly more costly solutions.
As is well known, a fully three-dimensional analysis is about an order of magnitude more expensive in computer resources and engineering time used than a two-dimensional solution. Let us consider a simple example to illustrate these ideas. For the analysis, we need to choose a mathematical model. This choice must clearly depend on what phenomena are to be predicted and on the geometry, material properties, loading, and support conditions of the bracket. We have indicated in Fig. The description "very thick" is of course relative to the thickness t and the height h of the bracket.
We translate this statement into the assumption that the bracket is fastened to a practically rigid column. Hence we can focus our attention on the bracket by applying a "rigid column boundary condition" to it. Of course, at a later time, an analysis of the column may be required, and then the loads carried by the two bolts, as a consequence of the load W, will need to be applied to the column.
We also assume that the load W is applied very slowly. The condition of time "very slowly" is relative to the largest natural period of the bracket; that is, the time span over which the load W is increased from zero to its full value is much longer than the fundamen- tal period of the bracket.
We translate this statement into requiring a static analysis and not a dynamic analysis. With these preliminary considerations we can now establish an appropriate mathe- matical model for the analysis of the bracket-depending on what phenomena are to be predicted. Let us assume, in the first instance, that only the total bending moment at section AA in the bracket and the deflection at the load application are sought.
To predict these quantities, we consider a beam mathematical model including shear deformations [see Fig. Pin w. Areas with imposed zero displacements u, v. Of course, the relations in 1. Let us now ask whether the mathematical model used in Fig. This model should include the two bolts fastening the bracket to the assumed rigid column as well as the pin through which the load W is applied. The three-dimensional solution of this model using the appropriate geometry and material data would give the numbers against which we would compare the answers given in 1.
Note that this three-dimensional mathematical model contains contact conditions the contact is between the bolts, the bracket, and the column, and between the pin carrying the load and the bracket and stress concentrations in the fillets and at the holes.
Also, if the stresses are high, nonlinear material conditions should be included in the model. Of course, an analytical solution of this mathematical model is not available, and all we can obtain is a numerical solution. We describe in this book how such solutions can be calculated using finite element procedures, but we may note here already that the solution would be rela- tively expensive in terms of computer resources and engineering time used.
Since the three-dimensional comprehensive mathematical model is very likely too comprehensive a model for the analysis questions we have posed , we instead may consider a linear elastic two-dimensional plane stress model as shown in Fig. This mathemat- ical model represents the geometry of the bracket more accurately than the beam model and assumes a two-dimensional stress situation in the bracket see Section 4.
The bending moment at section AA and deflection under the load calculated with this model can be expected to be quite close to those calculated with the very-comprehensive three- dimensional model, and certainly this two-dimensional model represents a higher-order model against which we can measure the adequacy of the results given in 1. Of course, an analytical solution of the model is not available, and a numerical solution must be sought. Figures 1. Let us note the various assumptions of this math- ematical model when compared to the more comprehensive three-dimensional model dis- cussed earlier.
Also, the actual bolt fastening and contact conditions between the steel column and the bracket are not included. Deflections are drawn with a magnifi- cation factor of together with the original configura- tion. Un- e Maximum principal stress near notch. The small Smoothed stress results. The averages of breaks in the bands indicate that a reasonably nodal point stresses are taken and interpo- accurate solution of the mathematical model lated over the elements.
However, since our objective is only to predict the bending moment at section AA and the deflection at point B, these assumptions are deemed reasonable and of relatively little influence. Let us assume that the results obtained in the finite element solution of the mathemat- ical model are sufficiently accurate that we can refer to the solution given in Fig. Considering these results, we can say that the beam mathematical model in Fig.
The beam model is of course also effective because the calculations are performed with very little effort. On the other hand, if we next ask for the maximum stress in the bracket, then the simple mathematical beam model in Fig.
Specifically, the beam model totally neglects the stress increase due to the fillets. The important points to note here are the following. The selection of the mathematical model must depend on the response to be predicted i. The most effective mathematical model is that one which delivers the answers to the questions in a reliable manner i. A finite element solution can solve accurately only the chosen mathematical model e.
The notion of reliability of the mathematical model hinges upon an accuracy assess- ment of the results obtained with the chosen mathematical model in response to the questions asked against the results obtained with the very-comprehensive mathemat- ical model.
However, in practice the very-comprehensive mathematical model is 1 The bending moment at section AA in the plane stress model is calculated here from the finite element. Finally, there is one further important general point. The chosen mathematical model may contain extremely high stresses because of sharp corners, concentrated loads, or other effects. These high stresses may be due solely to the simplifications used in the model when compared with the very-comprehensive mathematical model or with nature.
For example, the concentrated load in the plane stress model in Fig. This pressure would in nature be transmitted by the pin carrying the load into the bracket.
The exact solution of the mathematical model in Fig. Of course, this very large stress is an artifice of the chosen model, and the concentrated load should be replaced by a pressure load over a small area when a very fine discretization is used see further discussion.
Furthermore, if the model then still predicts a very high stress, a nonlinear mathematical model would be appropriate. Note that the concentrated load in the beam model in Fig. Also, the right-angled sharp corners at the support of the beam model, of course, do not introduce any solution difficulties, whereas such corners in a plane stress model would introduce infinite stresses.
Hence, for the plane stress model, the corners have to be rounded to more accurately represent the geometry of the actual physical bracket.
We thus realize that the solution of a mathematical model may result in artificial difficulties that are easily removed by an appropriate change in the mathematical model to more closely represent the actual physical situation. Furthermore, the choice of a more encompassing mathematical model may result, in such cases, in a decrease in the required solution effort. While these observations are of a general nature, let us consider once again, specifically, the use of concentrated loads.
0コメント