김경종
318 lines
27 KiB
Markdown
318 lines
27 KiB
Markdown
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# 1.7 Computer Programs for the Finite Element Method
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There are two general computer methods of approach to the solution of problems by the finite element method. One is to use large commercial programs, many of which have been configured to run on personal computers (PCs); these general-purpose programs are designed to solve many types of problems. The other is to develop many small, special-purpose programs to solve specific problems. In this section, we will discuss the advantages and disadvantages of both methods. We will then list some of the available general-purpose programs and discuss some of their standard capabilities.
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Some advantages of general-purpose programs:
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1. The input is well organized and is developed with user ease in mind. Users do not need special knowledge of computer software or hardware. Preprocessors are readily available to help create the finite element model.
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2. The programs are large systems that often can solve many types of problems of large or small size with the same input format.
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3. Many of the programs can be expanded by adding new modules for new kinds of problems or new technology. Thus they may be kept current with a minimum of effort.
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4. With the increased storage capacity and computational efficiency of PCs, many general-purpose programs can now be run on PCs.
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5. Many of the commercially available programs have become very attractive in price and can solve a wide range of problems [45, 56].
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Some disadvantages of general-purpose programs:
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1. The initial cost of developing general-purpose programs is high.
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2. General-purpose programs are less efficient than special-purpose programs because the computer must make many checks for each problem, some of which would not be necessary if a special-purpose program were used.
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3. Many of the programs are proprietary. Hence the user has little access to the logic of the program. If a revision must be made, it often has to be done by the developers.
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Some advantages of special-purpose programs:
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1. The programs are usually relatively short, with low development costs.
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2. Small computers are able to run the programs.
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3. Additions can be made to the program quickly and at a low cost.
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4. The programs are efficient in solving the problems they were designed to solve.
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The major disadvantage of special-purpose programs is their inability to solve different classes of problems. Thus one must have as many programs as there are different classes of problems to be solved.
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There are numerous vendors supporting finite element programs, and the interested user should carefully consult the vendor before purchasing any software. However, to give you an idea about the various commercial personal computer programs now available for solving problems by the finite element method, we present a partial list of existing programs.
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1. Algor [46]
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2. Abaqus [47]
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3. ANSYS [48]
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4. COSMOS/M [49]
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5. GT-STRUDL [50]
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6. MARC [51]
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7. MSC/NASTRAN [52]
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8. NISA [53]
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9. Pro/MECHANICA [54]
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10. SAP2000 [55]
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11. STARDYNE [56]
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Standard capabilities of many of the listed programs are provided in the preceding references and in Reference [45]. These capabilities include information on
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1. Element types available, such as beam, plane stress, and threedimensional solid
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2. Type of analysis available, such as static and dynamic
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3. Material behavior, such as linear-elastic and nonlinear
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4. Load types, such as concentrated, distributed, thermal, and displacement (settlement)
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5. Data generation, such as automatic generation of nodes, elements, and restraints (most programs have preprocessors to generate the mesh for the model)
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6. Plotting, such as original and deformed geometry and stress and temperature contours (most programs have postprocessors to aid in interpreting results in graphical form)
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7. Displacement behavior, such as small and large displacement and buckling
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8. Selective output, such as at selected nodes, elements, and maximum or minimum values
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All programs include at least the bar, beam, plane stress, plate-bending, and threedimensional solid elements, and most now include heat-transfer analysis capabilities.
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Complete capabilities of the programs are best obtained through program reference manuals and websites, such as References [46–56].
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# References
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[1] Hrennikoff, A., ‘‘Solution of Problems in Elasticity by the Frame Work Method,’’ Journal of Applied Mechanics, Vol. 8, No. 4, pp. 169–175, Dec. 1941.
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[2] McHenry, D., ‘‘A Lattice Analogy for the Solution of Plane Stress Problems,’’ Journal of Institution of Civil Engineers, Vol. 21, pp. 59–82, Dec. 1943.
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[3] Courant, R., ‘‘Variational Methods for the Solution of Problems of Equilibrium and Vibrations,’’ Bulletin of the American Mathematical Society, Vol. 49, pp. 1–23, 1943.
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[4] Levy, S., ‘‘Computation of Influence Coefficients for Aircraft Structures with Discontinuities and Sweepback,’’ Journal of Aeronautical Sciences, Vol. 14, No. 10, pp. 547–560, Oct. 1947.
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[5] Levy, S., ‘‘Structural Analysis and Influence Coefficients for Delta Wings,’’ Journal of Aeronautical Sciences, Vol. 20, No. 7, pp. 449–454, July 1953.
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[6] Argyris, J. H., ‘‘Energy Theorems and Structural Analysis,’’ Aircraft Engineering, Oct., Nov., Dec. 1954 and Feb., Mar., Apr., May 1955.
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[7] Argyris, J. H., and Kelsey, S., Energy Theorems and Structural Analysis, Butterworths, London, 1960 (collection of papers published in Aircraft Engineering in 1954 and 1955).
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[8] Turner, M. J., Clough, R. W., Martin, H. C., and Topp, L. J., ‘‘Stiffness and Deflection Analysis of Complex Structures,’’ Journal of Aeronautical Sciences, Vol. 23, No. 9, pp. 805–824, Sept. 1956.
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[9] Clough, R. W., ‘‘The Finite Element Method in Plane Stress Analysis,’’ Proceedings, American Society of Civil Engineers, 2nd Conference on Electronic Computation, Pittsburgh, PA, pp. 345–378, Sept. 1960.
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[10] Melosh, R. J., ‘‘A Stiffness Matrix for the Analysis of Thin Plates in Bending,’’ Journal of the Aerospace Sciences, Vol. 28, No. 1, pp. 34–42, Jan. 1961.
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[11] Grafton, P. E., and Strome, D. R., ‘‘Analysis of Axisymmetric Shells by the Direct Stiffness Method,’’ Journal of the American Institute of Aeronautics and Astronautics, Vol. 1, No. 10, pp. 2342–2347, 1963.
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[12] Martin, H. C., ‘‘Plane Elasticity Problems and the Direct Stiffness Method,’’ The Trend in Engineering, Vol. 13, pp. 5–19, Jan. 1961.
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[13] Gallagher, R. H., Padlog, J., and Bijlaard, P. P., ‘‘Stress Analysis of Heated Complex Shapes,’’ Journal of the American Rocket Society, Vol. 32, pp. 700–707, May 1962.
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[14] Melosh, R. J., ‘‘Structural Analysis of Solids,’’ Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, pp. 205–223, Aug. 1963.
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[15] Argyris, J. H., ‘‘Recent Advances in Matrix Methods of Structural Analysis,’’ Progress in Aeronautical Science, Vol. 4, Pergamon Press, New York, 1964.
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[16] Clough, R. W., and Rashid, Y., ‘‘Finite Element Analysis of Axisymmetric Solids,’’ Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, Vol. 91, pp. 71–85, Feb. 1965.
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[17] Wilson, E. L., ‘‘Structural Analysis of Axisymmetric Solids,’’ Journal of the American Institute of Aeronautics and Astronautics, Vol. 3, No. 12, pp. 2269–2274, Dec. 1965.
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[18] Turner, M. J., Dill, E. H., Martin, H. C., and Melosh, R. J., ‘‘Large Deflections of Structures Subjected to Heating and External Loads,’’ Journal of Aeronautical Sciences, Vol. 27, No. 2, pp. 97–107, Feb. 1960.
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[19] Gallagher, R. H., and Padlog, J., ‘‘Discrete Element Approach to Structural Stability Analysis,’’ Journal of the American Institute of Aeronautics and Astronautics, Vol. 1, No. 6, pp. 1437–1439, 1963.
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[20] Zienkiewicz, O. C., Watson, M., and King, I. P., ‘‘A Numerical Method of Visco-Elastic Stress Analysis,’’ International Journal of Mechanical Sciences, Vol. 10, pp. 807–827, 1968.
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[21] Archer, J. S., ‘‘Consistent Matrix Formulations for Structural Analysis Using Finite-Element Techniques,’’ Journal of the American Institute of Aeronautics and Astronautics, Vol. 3, No. 10, pp. 1910–1918, 1965.
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[22] Zienkiewicz, O. C., and Cheung, Y. K., ‘‘Finite Elements in the Solution of Field Problems,’’ The Engineer, pp. 507–510, Sept. 24, 1965.
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[23] Martin, H. C., ‘‘Finite Element Analysis of Fluid Flows,’’ Proceedings of the Second Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, pp. 517–535, Oct. 1968. (AFFDL-TR-68-150, Dec. 1969; AD-703-685, N.T.I.S.)
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[24] Wilson, E. L., and Nickel, R. E., ‘‘Application of the Finite Element Method to Heat Conduction Analysis,’’ Nuclear Engineering and Design, Vol. 4, pp. 276–286, 1966.
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[25] Szabo, B. A., and Lee, G. C., ‘‘Derivation of Stiffness Matrices for Problems in Plane Elasticity by Galerkin’s Method,’’ International Journal of Numerical Methods in Engineering, Vol. 1, pp. 301–310, 1969.
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[26] Zienkiewicz, O. C., and Parekh, C. J., ‘‘Transient Field Problems: Two-Dimensional and Three-Dimensional Analysis by Isoparametric Finite Elements,’’ International Journal of Numerical Methods in Engineering, Vol. 2, No. 1, pp. 61–71, 1970.
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[27] Lyness, J. F., Owen, D. R. J., and Zienkiewicz, O. C., ‘‘Three-Dimensional Magnetic Field Determination Using a Scalar Potential. A Finite Element Solution,’’ Transactions on Magnetics, Institute of Electrical and Electronics Engineers, pp. 1649–1656, 1977.
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[28] Belytschko, T., ‘‘A Survey of Numerical Methods and Computer Programs for Dynamic Structural Analysis,’’ Nuclear Engineering and Design, Vol. 37, No. 1, pp. 23–34, 1976.
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[29] Belytschko, T., ‘‘Efficient Large-Scale Nonlinear Transient Analysis by Finite Elements,’’ International Journal of Numerical Methods in Engineering, Vol. 10, No. 3, pp. 579–596, 1976.
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[30] Huiskies, R., and Chao, E. Y. S., ‘‘A Survey of Finite Element Analysis in Orthopedic Biomechanics: The First Decade,’’ Journal of Biomechanics, Vol. 16, No. 6, pp. 385–409, 1983.
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[31] Journal of Biomechanical Engineering, Transactions of the American Society of Mechanical Engineers, (published quarterly) (1st issue published 1977).
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[32] Kardestuncer, H., ed., Finite Element Handbook, McGraw-Hill, New York, 1987.
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[33] Clough, R. W., ‘‘The Finite Element Method After Twenty-Five Years: A Personal View,’’ Computers and Structures, Vol. 12, No. 4, pp. 361–370, 1980.
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[34] Kardestuncer, H., Elementary Matrix Analysis of Structures, McGraw-Hill, New York, 1974.
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[35] Oden, J. T., and Ripperger, E. A., Mechanics of Elastic Structures, 2nd ed., McGraw-Hill, New York, 1981.
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[36] Finlayson, B. A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972.
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[37] Zienkiewicz, O. C., The Finite Element Method, 3rd ed., McGraw-Hill, London, 1977.
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[38] Cook, R. D., Malkus, D. S., and Plesha, M. E., Concepts and Applications of Finite Element Analysis, 3rd ed., Wiley, New York, 1989.
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[39] Koswara, H., A Finite Element Analysis of Underground Shelter Subjected to Ground Shock Load, M.S. Thesis, Rose-Hulman Institute of Technology, 1983.
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[40] Greer, R. D., ‘‘The Analysis of a Film Tower Die Utilizing the ANSYS Finite Element Package,’’ M.S. Thesis, Rose-Hulman Institute of Technology, Terre Haute, Indiana, May 1989.
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[41] Koeneman, J. B., Hansen, T. M., and Beres, K., ‘‘The Effect of Hip Stem Elastic Modulus and Cement/Stem Bond on Cement Stresses,’’ 36th Annual Meeting, Orthopaedic Research Society, Feb. 5–8, 1990, New Orleans, Louisiana.
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[42] Girijavallabham, C. V., and Reese, L. C., ‘‘Finite-Element Method for Problems in Soil Mechanics,’’ Journal of the Structural Division, American Society of Civil Engineers, No. Sm2, pp. 473–497, Mar. 1968.
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[43] Young, C., and Crocker, M., ‘‘Transmission Loss by Finite-Element Method,’’ Journal of the Acoustical Society of America, Vol. 57, No. 1, pp. 144–148, Jan. 1975.
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[44] Silvester, P. P., and Ferrari, R. L., Finite Elements for Electrical Engineers, Cambridge University Press, Cambridge, England, 1983.
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[45] Falk, H., and Beardsley, C. W., ‘‘Finite Element Analysis Packages for Personal Computers,’’ Mechanical Engineering, pp. 54–71, Jan. 1985.
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[46] Algor Interactive Systems, 150 Beta Drive, Pittsburgh, PA 15238.
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[47] Web site http://www.abaqus.com.
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[48] Swanson, J. A., ANSYS-Engineering Analysis Systems User’s Manual, Swanson Analysis Systems, Inc., Johnson Rd., P.O. Box 65, Houston, PA 15342.
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[49] COSMOS/M, Structural Research & Analysis Corp., 12121 Wilshire Blvd., Los Angeles, CA 90025.
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[50] web site http://ce6000.cegatech.edu.
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[51] web site http://www.mscsoftware.com.
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[52] MSC/NASTRAN, MacNeal-Schwendler Corp., 600 Suffolk St., Lowell, MA, 01854.
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[53] web site http://emrc.com.
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[54] Toogood, Roger, Pro/MECHANICA Structure Tutorial, SDC Publications, 2001.
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[55] Computers & Structures, Inc., 1995 University Ave., Berkeley, CA 94704.
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[56] STARDYNE, Research Engineers, Inc., 22700 Savi Ranch Pkwy, Yorba Linda, CA 92687.
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[57] Noor, A. K., ‘‘Bibliography of Books and Monographs on Finite Element Technology,’’ Applied Mechanics Reviews, Vol. 44, No. 6, pp. 307–317, June 1991.
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[57] Belytschko, T., Liu W. K., and Moran, B., Nonlinear Finite Elements For Continua and Structures, John Wiley, 1996.
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# Problems
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1.1 Define the term finite element.
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1.2 What does discretization mean in the finite element method?
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1.3 In what year did the modern development of the finite element method begin?
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1.4 In what year was the direct stiffness method introduced?
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1.5 Define the term matrix.
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1.6 What role did the computer play in the use of the finite element method?
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1.7 List and briefly describe the general steps of the finite element method.
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1.8 What is the displacement method?
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1.9 List four common types of finite elements.
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1.10 Name three commonly used methods for deriving the element stiffness matrix and element equations. Briefly describe each method.
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1.11 To what does the term degrees of freedom refer?
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1.12 List five typical areas of engineering where the finite element method is applied.
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1.13 List five advantages of the finite element method.
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# Introduction to the Stiffness (Displacement) Method
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# Introduction
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This chapter introduces some of the basic concepts on which the direct stiffness method is founded. The linear spring is introduced first because it provides a simple yet generally instructive tool to illustrate the basic concepts. We begin with a general definition of the stiffness matrix and then consider the derivation of the stiffness matrix for a linear-elastic spring element. We next illustrate how to assemble the total stiffness matrix for a structure comprising an assemblage of spring elements by using elementary concepts of equilibrium and compatibility. We then show how the total stiffness matrix for an assemblage can be obtained by superimposing the stiffness matrices of the individual elements in a direct manner. The term direct stiffness method evolved in reference to this technique.
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After establishing the total structure stiffness matrix, we illustrate how to impose boundary conditions—both homogeneous and nonhomogeneous. A complete solution including the nodal displacements and reactions is thus obtained. (The determination of internal forces is discussed in Chapter 3 in connection with the bar element.)
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We then introduce the principle of minimum potential energy, apply it to derive the spring element equations, and use it to solve a spring assemblage problem. We will illustrate this principle for the simplest of elements (those with small numbers of degrees of freedom) so that it will be a more readily understood concept when applied, of necessity, to elements with large numbers of degrees of freedom in subsequent chapters.
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# 2.1 Definition of the Stiffness Matrix
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Familiarity with the stiffness matrix is essential to understanding the stiffness method. We define the stiffness matrix as follows: For an element, a stiffness matrix $\underline { { \hat { k } } }$ is a matrix such that $\underline { { \hat { f } } } = \underline { { \hat { k } } } \underline { { \hat { d } } } .$ , where $\underline { { \hat { k } } }$ relates local-coordinate $( \hat { x } , \hat { y } , \hat { z } )$ nodal displacements $\hat { \underline { d } }$ t o local forces ^f of a single element. (Throughout this text, the underline notation denotes
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<details>
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<summary>text_image</summary>
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y
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ŷ
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1
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2
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x
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x
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z
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ẑ
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</details>
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Figure 2–1 Local $( \hat { x } , \hat { y } , \hat { z } )$ and global $( x , y , z )$ coordinate systems
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a matrix, and the ^ symbol denotes quantities referred to a local-coordinate system set up to be convenient for the element as shown in Figure 2–1.)
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For a continuous medium or structure comprising a series of elements, a stiffness matrix K relates global-coordinate $( x , y , z )$ nodal displacements d to global forces $\underline { { F } }$ of the whole medium or structure. (Lowercase letters such as $x , y ,$ , and z without the ^ symbol denote global-coordinate variables.)
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# 2.2 Derivation of the Stiffness Matrix for a Spring Element
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Using the direct equilibrium approach, we will now derive the stiffness matrix for a one-dimensional linear spring—that is, a spring that obeys Hooke’s law and resists forces only in the direction of the spring. Consider the linear spring element shown in Figure 2–2. Reference points 1 and 2 are located at the ends of the element. These reference points are called the nodes of the spring element. The local nodal forces are $\hat { f } _ { 1 x }$ and $\hat { f } _ { 2 x } ^ { \mathrm { ~ ~ } }$ for the spring element associated with the local axis x^. The local axis acts in the direction of the spring so that we can directly measure displacements and forces along the spring. The local nodal displacements are $\hat { d } _ { 1 x }$ and $\hat { d } _ { 2 x }$ for the spring element. These nodal displacements are called the degrees of freedom at each node. Positive directions for the forces and displacements at each node are taken in the positive x^ direction as shown from node 1 to node 2 in the figure. The symbol k is called the spring constant or stiffness of the spring.
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Analogies to actual spring constants arise in numerous engineering problems. In Chapter 3, we see that a prismatic uniaxial bar has a spring constant $k = A E / L$ , where A represents the cross-sectional area of the bar, E is the modulus of elasticity, and L is the bar length. Similarly, in Chapter 5, we show that a prismatic circularcross-section bar in torsion has a spring constant $k = J G / L$ , where J is the polar moment of inertia and G is the shear modulus of the material. For one-dimensional heat conduction (Chapter 13), $k = A K _ { x x } / L$ , where $K _ { x x }$ is the thermal conductivity of
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<details>
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<summary>text_image</summary>
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1
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k
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2
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f̂₁ₓ, d̂₁ₓ
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f̂₂ₓ, d̂₂ₓ
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x̂
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L
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</details>
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Figure 2–2 Linear spring element with positive nodal displacement and force conventions
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the material, and for one-dimensional fluid flow through a porous medium (Chapter 14), $k = A K _ { x x } / L ,$ where $K _ { x x }$ is the permeability coefficient of the material.
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We will then observe that the stiffness method can be applied to nonstructural problems, such as heat transfer, fluid flow, and electrical networks, as well as structural problems by simply applying the proper constitutive law (such as Hooke’s law for structural problems, Fourier’s law for heat transfer, Darcy’s law for fluid flow and Ohm’s law for electrical networks) and a conservation principle such as nodal equilibrium or conservation of energy.
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We now want to develop a relationship between nodal forces and nodal displacements for a spring element. This relationship will be the stiffness matrix. Therefore, we want to relate the nodal force matrix to the nodal displacement matrix as follows:
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$$
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\left\{ \begin{array}{l} \hat {f} _ {1 x} \\ \hat {f} _ {2 x} \end{array} \right\} = \left[ \begin{array}{l l} k _ {1 1} & k _ {1 2} \\ k _ {2 1} & k _ {2 2} \end{array} \right] \left\{ \begin{array}{l} \hat {d} _ {1 x} \\ \hat {d} _ {2 x} \end{array} \right\} \tag {2.2.1}
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$$
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where the element stiffness coefficients $k _ { i j }$ of the $\underline { { \hat { k } } }$ matrix in Eq. (2.2.1) are to be determined. Recall from Eqs. (1.2.5) and (1.2.6) that $k _ { i j }$ represent the force $F _ { i }$ in the ith degree of freedom due to a unit displacement $d _ { j }$ in the jth degree of freedom while all other displacements are zero. That is, when we let $d _ { j } = 1$ and $d _ { k } = 0$ for $k \neq j ,$ , force $F _ { i } = k _ { i j }$ .
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We now use the general steps outlined in Section 1.4 to derive the stiffness matrix for the spring element in this section (while keeping in mind that these same steps will be applicable later in the derivation of stiffness matrices of more general elements) and then to illustrate a complete solution of a spring assemblage in Section 2.3. Because our approach throughout this text is to derive various element stiffness matrices and then to illustrate how to solve engineering problems with the elements, step 1 now involves only selecting the element type.
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# Step 1 Select the Element Type
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Consider the linear spring element (which can be an element in a system of springs) subjected to resulting nodal tensile forces T (which may result from the action of adjacent springs) directed along the spring axial direction x^ as shown in Figure 2–3, so as to be in equilibrium. The local x^ axis is directed from node 1 to node 2. We represent the spring by labeling nodes at each end and by labeling the element number. The original distance between nodes before deformation is denoted by L. The material property (spring constant) of the element is k.
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<details>
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<summary>text_image</summary>
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1
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k
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2
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x̂
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L
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T
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1
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k
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2
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T
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x̂
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d̂₁ₓ
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d̂₂ₓ
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</details>
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Figure 2–3 Linear spring subjected to tensile forces
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# Step 2 Select a Displacement Function
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We must choose in advance the mathematical function to represent the deformed shape of the spring element under loading. Because it is difficult, if not impossible at times, to obtain a closed form or exact solution, we assume a solution shape or distribution of displacement within the element by using an appropriate mathematical function. The most common functions used are polynomials.
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Because the spring element resists axial loading only with the local degrees of freedom for the element being displacements $\hat { d } _ { 1 x }$ and $\hat { d } _ { 2 x }$ along the x^ direction, we choose a displacement function u^ to represent the axial displacement throughout the element. Here a linear displacement variation along the x^ axis of the spring is assumed [Figure 2–4(b)], because a linear function with specified endpoints has a unique path. Therefore,
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$$
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\hat {u} = a _ {1} + a _ {2} \hat {x} \tag {2.2.2}
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$$
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In general, the total number of coefficients a is equal to the total number of degrees of freedom associated with the element. Here the total number of degrees of freedom is
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Figure 2–4 (a) Spring element showing plots of (b) displacement function u^ and shape functions (c) $N _ { 1 }$ and (d) $N _ { 2 }$ over domain of element
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two—an axial displacement at each of the two nodes of the element (we present further discussion regarding the choice of displacement functions in Section 3.2). In matrix form, Eq. (2.2.2) becomes
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$$
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\hat {u} = \left[ \begin{array}{l l} 1 & \hat {x} \end{array} \right] \left\{ \begin{array}{l} a _ {1} \\ a _ {2} \end{array} \right\} \tag {2.2.3}
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$$
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We now want to express u^ as a function of the nodal displacements $\hat { d } _ { 1 x }$ and $\hat { d } _ { 2 x }$ . as this will allow us to apply the physical boundary conditions on nodal displacements directly as indicated in Step 3 and to then relate the nodal displacements to the nodal forces in Step 4. We achieve this by evaluating u^ at each node and solving for $a _ { 1 }$ and $a _ { 2 }$ from Eq. (2.2.2) as follows:
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$$
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\hat {u} (0) = \hat {d} _ {1 x} = a _ {1} \tag {2.2.4}
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$$
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$$
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\hat {u} (L) = \hat {d} _ {2 x} = a _ {2} L + \hat {d} _ {1 x} \tag {2.2.5}
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$$
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$_ { \mathrm { o r , } }$ solving Eq. (2.2.5) for $a _ { 2 } .$ ,
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$$
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a _ {2} = \frac {\hat {d} _ {2 x} - \hat {d} _ {1 x}}{L} \tag {2.2.6}
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$$
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Upon substituting Eqs. (2.2.4) and (2.2.6) into Eq. (2.2.2), we have
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$$
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\hat {u} = \left(\frac {\hat {d} _ {2 x} - \hat {d} _ {1 x}}{L}\right) \hat {x} + \hat {d} _ {1 x} \tag {2.2.7}
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$$
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In matrix form, we express Eq. (2.2.7) as
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$$
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\hat {u} = \left[ \begin{array}{l l} 1 - \frac {\hat {x}}{L} & \frac {\hat {x}}{L} \end{array} \right] \left\{ \begin{array}{l} \hat {d} _ {1 x} \\ \hat {d} _ {2 x} \end{array} \right\} \tag {2.2.8}
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$$
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or
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$$
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\hat {u} = \left[ \begin{array}{l l} N _ {1} & N _ {2} \end{array} \right] \left\{ \begin{array}{l} \hat {d} _ {1 x} \\ \hat {d} _ {2 x} \end{array} \right\} \tag {2.2.9}
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$$
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Here
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$$
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N _ {1} = 1 - \frac {\hat {x}}{L} \quad \text { and } \quad N _ {2} = \frac {\hat {x}}{L} \tag {2.2.10}
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$$
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are called the shape functions because the $N _ { i } { ' } \mathrm { s }$ express the shape of the assumed displacement function over the domain (x^ coordinate) of the element when the ith element degree of freedom has unit value and all other degrees of freedom are zero. In this case, $N _ { 1 }$ and $N _ { 2 }$ are linear functions that have the properties that $N _ { 1 } = 1$ a t node 1 and $N _ { 1 } = 0$ at node 2, whereas $N _ { 2 } = 1$ at node 2 and $N _ { 2 } = 0$ at node 1. See Figure $2 { \ - } 4 ( \mathrm { c } )$ and (d) for plots of these shape functions over the domain of the spring element. Also, $N _ { 1 } + N _ { 2 } = 1$ for any axial coordinate along the bar. (Section 3.2 further explores this important relationship.) In addition, the $\boldsymbol { N _ { i } } ^ { \prime } \boldsymbol { \mathrm { s } }$ are often called interpolation functions because we are interpolating to find the value of a function between given nodal values. The interpolation function may be different from the actual function except at the endpoints or nodes, where the interpolation function and actual function must be equal to specified nodal values.
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