Hutton - Fundamentals of Finite Element Analysis, страница 6

Knowledge of the theory is necessary for bothproper modeling and evaluation of computational results.REFERENCES1.Bathe, K-J. Finite Element Procedures. Englewood Cliffs, NJ: Prentice-Hall,1996.2. Lord Rayleigh. “On the Theory of Resonance.” Transactions of the Royal Society(London) A161 (1870).3.

Ritz, W. “Uber eine neue Methode zur Losung gewissen Variations-Probleme dermathematischen Physik.” J. Reine Angew. Math. 135 (1909).4. Galerkin, B. G. “Series Solution of Some Problems of Elastic Equilibrium of Rodsand Plates” [in Russian]. Vestn. Inzh. Tekh. 19 (1915).5. Courant, R. “Variational Methods for the Solution of Problems of Equilibrium andVibrations.” Bulletin of the American Mathematical Society 49 (1943).6. Clough, R.

W. “The Finite Element Method in Plane Stress Analysis.”Proceedings, American Society of Civil Engineers, Second Conference onElectronic Computation, Pittsburgh, 1960.7. Melosh, R. J. “A Stiffness Method for the Analysis of Thin Plates in Bending.”Journal of Aerospace Sciences 28, no. 1 (1961).8. Grafton, P. E., and D. R. Strome. “Analysis of Axisymmetric Shells by the DirectStiffness Method.” Journal of the American Institute of Aeronautics andAstronautics 1, no. 10 (1963).9.

Gallagher, R. H. “Analysis of Plate and Shell Structures.” Proceedings,Symposium on the Application of Finite Element Methods in Civil Engineering,Vanderbilt University, Nashville, 1969.10. Wilson, E. L. “Structural Analysis of Axisymmetric Solids.” Journal of theAmerican Institute of Aeronautics and Astronautics 3, (1965).11. Melosh, R. J. “Structural Analysis of Solids.” Journal of the Structural Division,Proceedings of the American Society of Civil Engineers, August 1963.12. Martin, H. C. “Finite Element Analysis of Fluid Flows.” Proceedings of theSecond Conference on Matrix Methods in Structural Mechanics, Wright-PattersonAir Force Base, Kilborn, Ohio, October 1968.13. Wilson, E.

L., and R. E. Nickell. “Application of the Finite Element Method toHeat Conduction Analysis.” Nuclear Engineering Design 4 (1966).17Hutton: Fundamentals ofFinite Element Analysis181. Basic Concepts of theFinite Element MethodCHAPTER 1Text© The McGraw−HillCompanies, 2004Basic Concepts of the Finite Element Method14. Turner, M.

J., E. H. Dill, H. C. Martin, and R. J. Melosh. “Large Deflections ofStructures Subjected to Heating and External Loads.” Journal of AeronauticalSciences 27 (1960).15. Archer, J. S. “Consistent Mass Matrix Formulations for Structural Analysis UsingFinite Element Techniques.” Journal of the American Institute of Aeronautics andAstronautics 3, no. 10 (1965).16. Noor, A. K.

“Bibliography of Books and Monographs on Finite ElementTechnology.” Applied Mechanics Reviews 44, no. 6 (1991).17. MSC/NASTRAN. Lowell, MA: MacNeal-Schwindler Corp.18. ANSYS. Houston, PA: Swanson Analysis Systems Inc.19. ALGOR. Pittsburgh: Algor Interactive Systems.20. COSMOS/M. Los Angeles: Structural Research and Analysis Corp.21. Hutton, D. V. “Modal Analysis of a Deployable Truss Using the Finite ElementMethod.” Journal of Spacecraft and Rockets 21, no. 5 (1984).Hutton: Fundamentals ofFinite Element Analysis2. Stiffness Matrices,Spring and Bar ElementsText© The McGraw−HillCompanies, 2004C H A P T E R2Stiffness Matrices, Springand Bar Elements2.1 INTRODUCTIONThe primary characteristics of a finite element are embodied in the elementstiffness matrix. For a structural finite element, the stiffness matrix contains thegeometric and material behavior information that indicates the resistance ofthe element to deformation when subjected to loading.

Such deformation mayinclude axial, bending, shear, and torsional effects. For finite elements used innonstructural analyses, such as fluid flow and heat transfer, the term stiffnessmatrix is also used, since the matrix represents the resistance of the element tochange when subjected to external influences.This chapter develops the finite element characteristics of two relativelysimple, one-dimensional structural elements, a linearly elastic spring and an elastic tension-compression member. These are selected as introductory elements because the behavior of each is relatively well-known from the commonly studiedengineering subjects of statics and strength of materials.

Thus, the “bridge” to thefinite element method is not obscured by theories new to the engineering student.Rather, we build on known engineering principles to introduce finite elementconcepts. The linear spring and the tension-compression member (hereafter referred to as a bar element and also known in the finite element literature as a spar,link, or truss element) are also used to introduce the concept of interpolationfunctions. As mentioned briefly in Chapter 1, the basic premise of the finite element method is to describe the continuous variation of the field variable (in thischapter, physical displacement) in terms of discrete values at the finite elementnodes. In the interior of a finite element, as well as along the boundaries (applicable to two- and three-dimensional problems), the field variable is described viainterpolation functions (Chapter 6) that must satisfy prescribed conditions.Finite element analysis is based, dependent on the type of problem, on several mathematic/physical principles.

In the present introduction to the method,19202. Stiffness Matrices,Spring and Bar ElementsCHAPTER 2Text© The McGraw−HillCompanies, 2004Stiffness Matrices, Spring and Bar Elementswe present several such principles applicable to finite element analysis. First, andforemost, for spring and bar systems, we utilize the principle of static equilibrium but—and this is essential—we include deformation in the development;that is, we are not dealing with rigid body mechanics.

For extension of the finiteelement method to more complicated elastic structural systems, we also state andapply the first theorem of Castigliano [1] and the more widely used principle ofminimum potential energy [2]. Castigliano’s first theorem, in the form presented,may be new to the reader. The first theorem is the counterpart of Castigliano’ssecond theorem, which is more often encountered in the study of elementarystrength of materials [3].

Both theorems relate displacements and applied forcesto the equilibrium conditions of a mechanical system in terms of mechanicalenergy. The use here of Castigliano’s first theorem is for the distinct purpose ofintroducing the concept of minimum potential energy without resort to the highermathematic principles of the calculus of variations, which is beyond the mathematical level intended for this text.2.2 LINEAR SPRING AS A FINITE ELEMENTA linear elastic spring is a mechanical device capable of supporting axial loadingonly and constructed such that, over a reasonable operating range (meaning extension or compression beyond undeformed length), the elongation or contraction of the spring is directly proportional to the applied axial load.

The constantof proportionality between deformation and load is referred to as the spring constant, spring rate, or spring stiffness [4], generally denoted as k, and has unitsof force per unit length. Formulation of the linear spring as a finite element isaccomplished with reference to Figure 2.1a. As an elastic spring supports axialloading only, we select an element coordinate system (also known as a local coordinate system) as an x axis oriented along the length of the spring, as shown.The element coordinate system is embedded in the element and chosen, by geometric convenience, for simplicity in describing element behavior. The elementu2u1f11Force, fHutton: Fundamentals ofFinite Element Analysis2k(a)f2k1xDeflection, ␦ u2 u1(b)Figure 2.1(a) Linear spring element with nodes, nodal displacements, and nodal forces.(b) Load-deflection curve.Hutton: Fundamentals ofFinite Element Analysis2.

Stiffness Matrices,Spring and Bar ElementsText© The McGraw−HillCompanies, 20042.2 Linear Spring as a Finite Elementor local coordinate system is contrasted with the global coordinate system. Theglobal coordinate system is that system in which the behavior of a completestructure is to be described. By complete structure is meant the assembly ofmany finite elements (at this point, several springs) for which we desire to compute response to loading conditions.

In this chapter, we deal with cases in whichthe local and global coordinate systems are essentially the same except for translation of origin. In two- and three-dimensional cases, however, the distinctionsare quite different and require mathematical rectification of element coordinatesystems to a common basis. The common basis is the global coordinate system.Returning attention to Figure 2.1a, the ends of the spring are the nodes andthe nodal displacements are denoted by u1 and u2 and are shown in the positivesense.