Hutton - Fundamentals of Finite Element Analysis (523155), страница 30
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Method of WeightedResidualsCHAPTER 5Text© The McGraw−HillCompanies, 2004Method of Weighted Residuals5.5(a)–(e) For each of the differential equations given in Problem 5.4, use themethod of Example 5.4 to determine the trial functions for a two-termapproximate solution using Galerkin’s method of weighted residuals.5.6 For the four-element solution of Example 5.5, verify the correctness of theassembled system equations. Apply the boundary conditions and solve thereduced system of equations. Compute the first derivatives at each node of eachelement. Are the derivatives continuous across the nodal connections betweenelements?5.7 Each of the following differential equations represents a physical problem, asindicated.
For each case given, formulate the finite element equations (that is,determine the stiffness matrix and load vectors) using Galerkin’s finite elementmethod for a two-node element of length L with the interpolation functionsN 1 (x ) = 1 −a.xLN 2 (x ) =xLOne-dimensional heat conduction with linearly varying internal heatgenerationkx Ad2 T+ Q 0 Ax = 0dx 2where kx, Q0, and A are constants.b. One-dimensional heat conduction with surface convectionkx Ac.d2 T− hPT = hPT adx 2where kx, Q0, A, h, P, and Ta are constants.Torsion of an elastic circular cylinderJGd2 =0dx 2where J and G are constants.A two-dimensional beam is subjected to a linearly varying distributed load givenby q (x ) = q0 x , 0 ≤ x ≤ L , where L is total beam length and q0 is a constant.For a finite element located between nodes at arbitrary positions xi and xj so thatxi < xj and Le = xj − xi is the element length, determine the components of theforce vector at the element nodes using Galerkin’s finite element method.
(Notethat this is simply the last term in Equation 5.47 adjusted appropriately forelement location.)5.9 Repeat Problem 5.8 for a quadratrically distributed load q (x ) = q0 x 2 .5.10 Considering the results of either Problem 5.8 or 5.9, are the distributed loadsallocated to element nodes on the basis of static equilibrium? If your answer isno, why not and how is the distribution made?5.11 A tapered cylinder that is perfectly insulated on its periphery is held at constanttemperature 212°F at x = 0 and at temperature 80°F at x = 4 in.
The cylinderdiameter varies from 2 in. at x = 0 to 1 in. at x = L = 4 in. per Figure P5.11.The conductance coefficient is kx = 64 Btu/hr – ft – °F. Formulate a four-elementfinite element model of this problem and solve for the nodal temperatures and5.8Hutton: Fundamentals ofFinite Element Analysis5. Method of WeightedResidualsText© The McGraw−HillCompanies, 2004Problems4 in.d 2 in.d 1 in.xT 80 FT 212 FInsulatedFigure P5.11the heat flux values at the element boundaries. Use Galerkin’s finite elementmethod.5.12 Consider a tapered uniaxial tension-compression member subjected toan axial load as shown in Figure P5.12. The cross-sectional area varies asA = A 0 (1 − x /2L ) , where L is the length of the member and A0 is the area atx = 0. Given the governing equationEd2 u=0dx 2as in Equation 5.31, obtain the Galerkin finite element equations perEquation 5.33.A A0(1 x)2LA0xLFigure P5.125.135.14Many finite element software systems have provision for a tapered beamelement.
Beginning with Equation 5.46, while noting that Iz is not constant,develop the finite element equations for a tapered beam element.Use the results of Problem 5.13 to determine the stiffness matrix for the taperedbeam element shown in Figure P5.14.y1 in.2 in.6 in.Uniform thickness t 0.75 in.E 30 106 lb/in.2Figure P5.14x161Hutton: Fundamentals ofFinite Element Analysis1625. Method of WeightedResidualsCHAPTER 55.15Text© The McGraw−HillCompanies, 2004Method of Weighted ResidualsConsider a two-dimensional problem governed by the differential equation∂ 2∂ 2+=02∂x∂ y2(this is Laplace’s equation) in a specified two-dimensional domain with specifiedboundary conditions. How would you apply the Galerkin finite element methodto this problem?5.16 Reconsider Equation 5.24.
If we do not integrate by parts and simply substitutethe discretized solution form, what is the result? Explain.5.17 Given the differential equationd2 y+ 4y = xdx 2Assume the solution as a power seriesy(x ) =nai x i = a0 + a1 x + a2 x 2 + · · ·i =05.18and obtain the relations governing the coefficients of the power series solution.How does this procedure compare to the Galerkin method?The differential equationdy+y=3dx0≤x ≤1has the exact solutiony(x ) = 3 + C e −xwhere C = constant. Assume that the domain is 0 ≤ x ≤ 1 and the specifiedboundary condition is y(0) = 0. Show that, if the procedure of Example 5.4 isfollowed, the exact solution is obtained.Hutton: Fundamentals ofFinite Element Analysis6.
Interpolation Functionsfor General ElementFormulation© The McGraw−HillCompanies, 2004TextC H A P T E R6Interpolation Functionsfor General ElementFormulation6.1 INTRODUCTIONThe structural elements introduced in the previous chapters were formulated onthe basis of known principles from elementary strength of materials theory. Wehave also shown, by example, how Galerkin’s method can be applied to a heatconduction problem. This chapter examines the requirements for interpolationfunctions in terms of solution accuracy and convergence of a finite elementanalysis to the exact solution of a general field problem.
Interpolation functionsfor various common element shapes in one, two, and three dimensions are developed, and these functions are used to formulate finite element equations forvarious types of physical problems in the remainder of the text.With the exception of the beam element, all the interpolation functions discussed in this chapter are applicable to finite elements used to obtain solutionsto problems that are said to be C 0 -continuous. This terminology means that,across element boundaries, only the zeroth-order derivatives of the fieldvariable (i.e., the field variable itself) are continuous. On the other hand, thebeam element formulation is such that the element exhibits C1-continuity, sincethe first derivative of the transverse displacement (i.e., slope) is continuousacross element boundaries, as discussed previously and repeated later for emphasis.
In general, in a problem having C n-continuity, derivatives of the fieldvariable up to and including nth-order derivatives are continuous across element boundaries.163Hutton: Fundamentals ofFinite Element Analysis1646. Interpolation Functionsfor General ElementFormulationCHAPTER 6© The McGraw−HillCompanies, 2004TextInterpolation Functions for General Element Formulation6.2 COMPATIBILITY AND COMPLETENESSREQUIREMENTSThe line elements (spring, truss, beam) illustrate the general procedures used toformulate and solve a finite element problem and are quite useful in analyzingtruss and frame structures. Such structures, however, tend to be well defined interms of the number and type of elements used. In most engineering problems, thedomain of interest is a continuous solid body, often of irregular shape, in whichthe behavior of one or more field variables is governed by one or more partial differential equations.
The objective of the finite element method is to discretize thedomain into a number of finite elements for which the governing equations arealgebraic equations. Solution of the resulting system of algebraic equations thengives an approximate solution to the problem. As with any approximate technique, the question, How accurate is the solution? must be addressed.In finite element analysis, solution accuracy is judged in terms of convergence as the element “mesh” is refined. There are two major methods of meshrefinement. In the first, known as h-refinement, mesh refinement refers to theprocess of increasing the number of elements used to model a given domain, consequently, reducing individual element size. In the second method, p-refinement,element size is unchanged but the order of the polynomials used as interpolationfunctions is increased.
The objective of mesh refinement in either method is toobtain sequential solutions that exhibit asymptotic convergence to values representing the exact solution. While the theory is beyond the scope of this book,mathematical proofs of convergence of finite element solutions to correct solutions are based on a specific, regular mesh refinement procedure defined in [1].Although the proofs are based on regular meshes of elements, irregular or unstructured meshes (such as in Figure 1.7) can give very good results.
In fact, useof unstructured meshes is more often the case, since (1) the geometries beingmodeled are most often irregular and (2) the automeshing features of most finiteelement software packages produce irregular meshes.An example illustrating regular h-refinement as well as solution convergenceis shown in Figure 6.1a, which depicts a rectangular elastic plate of uniformthickness fixed on one edge and subjected to a concentrated load on one corner.maxExactxxFx41664Number of elements(a)(b)(c)(d)Figure 6.1 Example showing convergence as element mesh is refined.(e)Hutton: Fundamentals ofFinite Element Analysis6. Interpolation Functionsfor General ElementFormulation© The McGraw−HillCompanies, 2004Text6.2Compatibility and Completeness RequirementsThis problem is modeled using rectangular plane stress elements (Chapter 9) andthree meshes used in sequence, as shown (Figure 6.1b–6.1d).
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