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김경종
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<!-- source-page: 591 -->
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$$
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e (x) = e (h / 2) + \xi e ^ {\prime} (h / 2) + (\xi^ {2} / 2) e ^ {\prime \prime} (h / 2) + \dots \tag {d}
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$$
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and $e(h) = e'(h) = 0$ implies that $e''(h/2) = O(h^2)$ , from which the result follows.
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18.31 Assume that the loops in Fig. 18.6-1b are portions of parabolas. Hence, construct a geometric argument that shows that error $e_{max}$ is quartered if element length h is halved. Similarly, show that error $e'_{max}$ is halved if h is halved.
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# Section 18.7
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18.32 Imagine that stresses are to be calculated using the plane meshes shown in Fig. 18.7-2. Including meshes suggested by dashed lines, a total of four different meshes are indicated. For each of the four, what relative stress error is estimated by Eq. 18.7-3?
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18.33 Derive Eq. 18.7-4.
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18.34 Why would it be inadvisable to apply Eq. 18.7-4 to results obtained by use of the DKT plate element described in Section 11.4?
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18.35 For meshes $N = 1, 2$ , and 4, the respective computed values of displacement at a certain point in a plane mesh are 4.16, 4.64, and 4.76 units. What is the deflection predicted by extrapolation?
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18.36 Recompute the six “extrapolated results” in Table 18.7-1, this time using p = 1 in Eq. 18.7-4.
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18.37 What value of $\sigma_{max}$ is predicted by linear extrapolation in Fig. 18.7-6? Use all three possible combinations of two data points. What is the average of these three values?
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18.38 Verify that $\sigma_{max} = 93.3$ is the value predicted at $N = \infty$ in Fig. 18.7-6 by linear regression (least squares fit of a straight line).
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18.39 Use Eq. 18.7-4, with the appropriate value of p, to predict the converged value of stress at point A in Fig. 4.7-3a and in Fig. 4.7-3b.
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(a) Use the “standard” results in Table 4.7-1.
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(b) Use the iterated results in Table 4.7-1.
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18.40 Use Eq. 18.7-4, with a value of $p$ appropriate to the result to be extrapolated, to predict the converged values of $v_{C}, \sigma_{A}$ , and $\sigma_{B}$ in the following example problems.
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(a) “QM6” results in Fig. 8.3-3.
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(b) “Bilinear” results in Fig. 8.3-3.
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(c) “Equation 8.4-5” results in Fig. 8.6-2.
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(d) "Figure 8.6-1" results in Fig. 8.6-2.
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18.41 In Problem 13.17, axial vibrations of a bar were analyzed using three different mass matrices and meshes N = 1 and N = 2 for each mass matrix. The squared fundamental frequency is $\omega^{2} = cAE/mL$ , where $c = \pi^{2}/4$ for the exact result. For the respective mass matrices, the computations yield the following approximate values of c:
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Mesh $N = 1$ 3.000 2.000 2.400
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Mesh $N = 2$ 2.597 2.343 2.463
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<!-- source-page: 592 -->
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Use Eq. 18.7-4 to extrapolate to $N = \infty$ for each of the three sequences of two values.
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# Section 18.9
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18.42 Various situations that may promote numerical error are noted in the second paragraph of Section 18.9. For each of the four, cite an exception, for which the difficulty does not materialize.
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18.43 Imagine that you must model a very stiff beam on a very soft elastic foundation. What will you do to assure that deflections and bending moments of the beam are computed accurately?
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18.44 With mesh refinement, accuracy may improve, decline, or not change. For each of these three possibilities, give examples of problems (or situations, or meshes) that would behave in this way.
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<!-- source-page: 593 -->
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# MODELING, PROGRAMS, AND PROGRAMMING
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Guidelines are discussed for producing a finite element model of an actual structure. Advice is given regarding the writing and acquisition of computer programs.
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# 19.1 MODELING
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General. Modeling is an art based on the ability to visualize physical interactions. All basic and applied knowledge of physical problems, finite elements, and solution algorithms contributes to modeling expertise. Little is published regarding modeling. Practitioners tend to learn by doing and by talking with others. The documentation of a large general-purpose program usually contains some modeling advice in connection with specific example problems [e.g., 19.1].
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The documentation cannot be ignored: before all else fails, study the directions! Quite possibly, this study will not resolve all questions regarding terminology, symbols, assumptions, default conditions, and so on. Small test problems, sometimes involving a single element, can be run to clarify matters. Small test problems can help resolve questions about the sensitivity of an element type to aspect ratio and other shape distortions, whether beam and plate elements allow for transverse shear deformation, how distributed loads are treated, sensitivity to ill-conditioning, whether an eigensolver has trouble with zero or repeated eigenvalues (for the latter test one could use a three-dimensional beam of circular cross section [19.16]). These exercises will also increase confidence in use of the program and improve understanding of how various elements behave.
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In modeling, the principal difficulty faced by a typical user of a computer program is not understanding the physical action and boundary conditions of the actual structure, and the limitations of applicable theory, well enough to prepare a satisfactory model. Another difficulty is not understanding the behaviors of various elements, and the program's options and limitations, well enough to make an intelligent choice among them. The result may be a poor specification of the problem to be solved, a model that fails to reflect important features of the physical problem, fine detail irrelevant to the problem, a solution based on inappropriate loading or support conditions, and a surplus of computed results which are not properly examined and questioned $[19.2]$ . Automatic mesh generators make it easy to use too much fine detail. Powerful graphic postprocessors may smooth stress discontinuities that should warn of a need for local refinement or may hide questionable results by attractive display. It is possible that most finite element analyses are so flawed that they are worthless, and many experts feel that the situation is not improving. The reader is advised to review Section 1.8 of this book, “Warning: The Computed Answer May Be Wrong.”
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<!-- source-page: 594 -->
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Advice contained in the following discussion should not be regarded as a set of inflexible rules. An experienced and competent analyst may find exceptions that can sometimes be exploited to advantage.
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Cost and Dimensionality. Occasionally one may elect to analyze a one- or two-dimensional model rather than a two- or three-dimensional model, particularly when doing an initial simplified analysis. A rough guide to relative costs can be obtained by comparing the equation-solving costs of one-, two-, and three-dimensional meshes of comparable elements. For example, let us fill a square with $N^{2}$ elements and a cube with $N^{3}$ elements. Thus each mesh has N elements per side. If N is large, elements have corner nodes only, and each node has a single d.o.f., then the plane mesh produces rough $N^{2}$ equations having a semibandwidth of roughly N, for which the equation-solving expense is roughly proportional to $N^{2}(N)^{2} = N^{4}$ . The corresponding numbers for the solid mesh are $N^{3}$ , $N^{2}$ , and $N^{3}(N^{2})^{2} = N^{7}$ . Thus the rough measure of cost increases by the factor $N^{7}/N^{4} = N^{3}$ in going from two dimensions to three. Even for a mesh only ten elements on a side, this is a thousandfold increase.
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Which Element is Best? The appropriate answer is another question: Best for what? Element performance is problem-dependent. An element or mesh that works well in one situation may work badly in another. The analyst must understand how various elements behave in various situations, and must understand the physics of the problem well enough to make an intelligent choice of elements and mesh.
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A rough guideline, which falls well short of being a rule, is that elements of intermediate complexity work well for many problems. Thus, one would usually avoid using a great many of the simplest elements or a very few of the most complicated elements.
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Start Simply. A problem of moderate or large complexity should not be swallowed whole. One might begin with rough approximations from “back of the envelope” calculations. Numerically, a good beginning that often provides considerable insight is a “stick model,” which is a model built of a few bar and beam elements. A stick model is simple to prepare, cheap to run, and gives approximate results. If a more refined model gives greatly different results, the analyst should seek the reason for the discrepancy.
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A stick model, or a coarse-mesh model, can be used to guide subsequent refinement. If symmetry is to be exploited, a simple model serves to check anticipated symmetries and perhaps discover additional symmetries. A two-dimensional model may serve as an early analysis step in a three-dimensional problem, and may sometimes make a three-dimensional analysis unnecessary.
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If a dynamic or nonlinear analysis is contemplated, a linear static analysis of the proposed final mesh might be done beforehand as a relatively cheap test that may disclose flaws in the model. Loads applied in the static analysis should be contrived to produce strain distributions and gradients comparable to those anticipated in the subsequent dynamic or nonlinear analysis.
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Structure to Model. Modeling is more than just laying out a mesh. A focus on mesh layout or the minutiae of modeling may be at the expense of grasping important physical aspects that strongly affect the actual behavior.
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<!-- source-page: 595 -->
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In attempting to simulate reality with a mathematical model, the analyst must come to grips with the physics of the problem. What are the loads? What are the boundary conditions? Which actions are important and which are unimportant? Is the problem quasistatic or dynamic? If dynamic, is damping important? If so, how should it be represented? Is buckling a possibility? Is the material isotropic? Do properties depend on temperature or strain rate? Is plastic flow involved? If so, is it localized or widespread? Are there other nonlinearities that demand attention? These and other questions suggested by the problem at hand must be addressed before one can decide what element types and what arrangement of specific elements will produce a good model, and, if the problem is dynamic or nonlinear, what solution algorithm will produce reliable results at acceptable cost.
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Broad guidelines include the following. Include all real structure in the model; do not omit parts on the untested assumption that they carry little load or little stress. If a curved boundary is modeled as a polygon or a faceted surface, do so in a way that preserves the correct volume of the structure. In the analysis of thermally induced stresses, arrange element sizes and types so that the complexity of the temperature field can be approximately matched by the complexity of the strain field. If the temperature field is discontinuous across an interelement boundary, make sure the program does not “smear” the temperature change across elements by interpolation from nodal temperatures. Use consistent nodal loads rather than ad hoc lumping of loads. Use a relatively coarse mesh where gradients are known to be low and a relatively fine mesh where gradients are known to be high.
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In a coarse mesh, different element arrangements can produce a different model than intended. Consider, for example, a rectangular plate with clamped edges (Fig. 19.1-1). Unshaded elements are completely inactive because all their d.o.f. are set to zero. Thus the plate is modeled by only the shaded elements. The model differs from the structure in size and shape, and symmetries of behavior may be lost.
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Anticipate the Results and Know the Goal. If results were known in advance, it would be comparatively easy to prepare an adequate model. Similarly, if the probable results can be anticipated, a cheaper and better model will result. If locations of high stress are known in advance, it may be possible to model remote locations rather crudely. If the severity of stress gradients is anticipated, one can estimate the proper element size. If the goal of analysis is only to assess deflections, not to compute stresses, than a comparatively coarse mesh may suffice.
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Expect to Revise. Ideally, the original model is adequate and only a single analysis is performed. Far more often, the first analysis discloses inadequacies of the
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Figure 19.1-1. Four arrangements of eight triangular elements to model a rectangular plate with clamped edges.
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<details>
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<summary>text_image</summary>
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L_T
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P
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A
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B
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</details>
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{a}
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<details>
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<summary>text_image</summary>
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L_T
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P
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A
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B
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</details>
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(b)
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<details>
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<summary>text_image</summary>
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L_T
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P
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C
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A
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B
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</details>
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{c}
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Figure 19.1-2. Beam carrying load P, with elastic support at A and simple support at B. (a) Actual structure. (b) Simple model. (c) Model with elastic support at A.
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model, and one or more revisions are needed. Rather than regard the revisions as attempts to correct previous failures, it is better to regard them as expected steps in an investigation that proceeds from the overly simple to the adequate [19,14].
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For example, at the outset one may not know enough about stress gradients or the behavior of available elements to immediately generate an adequate model. Then, rather than attempting to overwhelm ignorance by a very refined initial model, it is usually easier and cheaper to start with a crude model and refine it in successive analyses until it is adequate. We may also consider gaining the necessary modeling insight by analysis of a different problem, related but simpler, and preferably one for which analytical or experimental results are known.
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Supports. Typically, in expositions of theory, structural supports are idealized as completely rigid or as ideally hinged (i.e., a simple support). Actual supports lie somewhere between completely fixed and ideally hinged. Consider, for example, Fig. 19.1-2. The left end of the beam is elastically supported. The simple model, Fig. 19.1-2b, might be analyzed twice, first with fixity at A and then with a hinge at A, in an attempt to bound the correct response. In Fig. 19.1-2c the actual support elasticity is modeled by elements. One might terminate the beam model at A and use constraint relations to couple the beam rotation $\theta_{A}$ to d.o.f. of the elastic support elements, as discussed in connection with Fig. 7.7-1a. Alternatively, the beam model might be extended into the support, as shown, with translational d.o.f. of the leftmost beam element coupled to translational d.o.f. of the elastic support elements. The associated incompatibilities of displacement along AC are ignored.
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Superficially innocuous changes in support conditions can substantially affect results. Consider Fig. 19.1-3. In Fig. 19.1-3a, strains $\epsilon_{y}$ associated with the Poisson effect are prohibited at the left end. In Fig. 19.1-3b, they are permitted, as is
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<details>
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<summary>text_image</summary>
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y
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P
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x
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</details>
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(a)
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<details>
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<summary>text_image</summary>
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y
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P
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x
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</details>
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(b)
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Figure 19.1-3. Two different models of a propped cantilever beam.
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<details>
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<summary>text_image</summary>
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Structure
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</details>
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<details>
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<summary>text_image</summary>
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Approximate
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model
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</details>
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(a)
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<details>
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<summary>text_image</summary>
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Structure
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</details>
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<details>
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<summary>text_image</summary>
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Approximate
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model
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</details>
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(b)
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Figure 19.1-4. Models that somewhat misrepresent the structure. (a) Welded connection of three angles whose axes are not concurrent. (b) Cylindrical vessel with step change in thickness and midsurfaces offset.
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presumed by elementary beam theory. This change in support conditions may produce large changes in $\sigma_{y}$ near the fixed end if Poisson's ratio is nonzero. In Fig. 19.1-3a the hinge support acts to resist rotation of the right end; in Fig. 19.1-3b it does not because it lies on the neutral surface of bending.
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Joints and Other Modifiers of Stiffness. Junctions between members may not have as much stiffness as a simple model attributes to them. For example, in Fig. 19.1-4, offsets produce significant local bending action in the actual structures, but this action is lacking in the approximate models that do not represent the offsets.
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The effect of joints is often underestimated but may have appreciable effect on global behavior. Stiffeners, swages, corrugations, and perforations also have an effect upon stiffness that cannot be ignored $[19.2]$ . However, detailed modeling—for example, of individual weld lines or of individual spot welds—is usually not appropriate unless the joint itself is the object of study.
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Some bodies have geometric irregularities that can be “smeared.” An example is a boiler tube sheet, which is a flat plate pierced by a regular pattern of identical holes. Effective elastic moduli and flexural rigidities for tube sheets have been established by analytical and experimental methods $[19.15]$ . Thus overall response can be analyzed as though the tube sheet were homogeneous. However, stress analysis must acknowledge the presence of individual holes.
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A related structure is a plate with a corrugated core, Fig. 19.1-5. If effective stiffnesses are known, the plate may be analyzed as if it were a homogeneous orthotropic plate. One way to obtain the effective stiffnesses is to model a small
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<details>
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<summary>text_image</summary>
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s
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</details>
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Figure 19.1-5. Cross section of a plate with thin facings and a corrugated core, viewed parallel to axes of the corrugations.
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<!-- source-page: 598 -->
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part of the structure in suitable fine detail (e.g., span s in Fig. 19.1-5), apply boundary displacements to this mesh consistent with a state of constant strain or constant curvature, compute the resulting boundary force or boundary moment, and finally obtain the required stiffness as the ratio of force to displacement (or of moment to rotation). If necessary, the same process can be applied to the aforementioned tube sheet to obtain its effective stiffness coefficients.
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Element Shapes, Connection, and Grading. An element performs best if its shape is compact and regular. An element tends to stiffen and lose accuracy as its aspect ratio increases, as its corner angles become markedly different from one another, as sides become curved, or as side nodes (if present) become nonuniformly spaced. Figure 19.1-6 shows element shapes that are usually undesirable.
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Different elements have different sensitivities to shape distortion. Accordingly, an all-purpose guideline must be vague: keep aspect ratios near unity, corner angles of quadrilaterals near $90^{\circ}$ , side nodes at midsides, and sides straight. Elements derived as planar may behave badly if warped to fit a curved surface [19.4,19.5].
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There are, of course, exceptions. Elements of large aspect ratio may be used in areas where the strain gradient is almost zero. Side nodes may be moved to quarter-points to produce crack-tip elements. Sides may be curved to fit a curved boundary (but sides of the element interior to the mesh should be straight).
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Poor elements (Fig. 19.1-6) and poor element connections (Fig. 19.1-7) may produce only locally poor results. Usually, if the surrounding mesh is satisfactory, spurious gradients caused by a local mesh error die out rather than propagate, in accord with Saint-Venant's principle. The same is usually true of errors caused by ad hoc (but statically equivalent) nodal lumping of distributed loads.
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If a mesh is graded rather than uniform, as is usually the case, grading should be done in a way that produces no great discrepancy in size between adjacent elements. Figure 19.1-8 shows three examples of mesh grading that use quadrilaterals. If triangles are also permitted, the range of possibilities increases. In general, adjacent elements should not differ greatly in stiffness. As a working rule, if $E$ and $V_{e}$ represent elastic modulus and element volume, the ratio $E / V_{e}$ should not change by more than a factor of roughly 3 in going from one element to the next.
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<details>
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||||
<summary>text_image</summary>
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a
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b
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a >> b
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</details>
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Large aspect ratio
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<details>
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<summary>text_image</summary>
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a >> b
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a
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b
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</details>
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Near-triangle
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Off-center node
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Highly skewed
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Triangular quadrilateral
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Curved side
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Figure 19.1-6. Elements having shape distortions that tend to promote poor results.
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<!-- source-page: 599 -->
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<details>
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<summary>natural_image</summary>
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Pure geometric diagram of a square with internal lines and dots, no text or symbols present
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</details>
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{a}
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<details>
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<summary>natural_image</summary>
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Pure geometric diagram of two connected squares with dots at vertices (no text or symbols)
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</details>
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{b}
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|
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<details>
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||||
<summary>text_image</summary>
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||||
|
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A B C
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</details>
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{c}
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Figure 19.1-7. Poor element connections. (a) Two bilinear elements and one quadratic element. (b) Two quadratic elements. (c) Two quadratic elements, connected at A and B but not at C (as if to model a crack from B to C).
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Checking the Model. A model should be checked before results are computed; afterward, there will be even greater reluctance to do the job. Ideally, a model is checked by an analyst who was not directly involved in its preparation and is therefore more likely to be objective.
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Graphical display makes it comparatively easy to detect gross errors, such as a misplaced node or a missing element. Preprocessors offer color, shrink plots, rotation, sectioning, exploded views, and removal of hidden lines as aids in the checking process (Figs. 19.1-9 and 19.1-10).
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Tests and warnings should already be coded into the program and should be exercised by a “check run” that precedes actual solution. Such tests may include examination for overly distorted element shapes, checking that adequate supports are provided, and searching for poor element connections like those in Fig. 19.1-7. The check run may also estimate the time required for the actual solution. All error messages and warnings produced by the program should be investigated, whether they appear during the check run or later.
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Built-in error tests cannot be given all responsibility for the success of an analysis. The program cannot know whether the element type is appropriate, if supports are properly located, whether data have been supplied using consistent units, and so on. Responsibility resides with the user.
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|
||||
Ill-Conditioning, Locking, and Instability. Great stiffness discrepancies between elements, poor choice of quadrature rule, and a Poisson ratio near 0.5 in plane strain and solid problems may provoke ill-conditioning, locking, or instability. These are dangerous difficulties because they can seriously degrade results rather than making results so peculiar that it becomes obvious that something is wrong.
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<details>
|
||||
<summary>natural_image</summary>
|
||||
|
||||
Geometric diagram of a square divided into smaller triangles and smaller squares (no text or symbols)
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</details>
|
||||
|
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(a)
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|
||||
|
||||
|
||||
<details>
|
||||
<summary>natural_image</summary>
|
||||
|
||||
Geometric pattern of interconnected squares and triangles (no text or symbols)
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||||
</details>
|
||||
|
||||
(b)
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||||
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(c)
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Figure 19.1-8. (a, b) Transitions from coarse to finer mesh that avoid abrupt size changes. (c) Possible mesh of quadrilaterals on one quadrant of a circular plate.
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|
||||
<!-- source-page: 600 -->
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>natural_image</summary>
|
||||
|
||||
Technical line drawing of a mechanical component with a central protrusion (no text or symbols)
|
||||
</details>
|
||||
|
||||
Figure 19.1-9. The intersection of two cylindrical shells. The “shrink plot” shows elements at about 75% of their actual sizes.
|
||||
|
||||
Some important “don’ts” are as follows. Do not support a stiff element by flexible elements; instead, impose rigid-body constraints on the stiff element. Do not “fake” a skew support (Fig. 18.2-3). Do not let Poisson’s ratio approach 0.5 in plane strain and solid problems unless a special formulation is used. Do not let three-dimensional elements or Mindlin plate and shell elements become extremely thin. Do not use a minimal integration rule without being aware of possible mechanisms.
|
||||
|
||||
Some of these difficulties can be detected by error tests in the coding, such as a test for the condition number of the structure stiffness matrix or a test for diagonal decay during equation solving. Such tests are usually a posteriori rather than a priori and may be optimistic or pessimistic.
|
||||
|
||||
Stresses. At optimal stress points, computed stresses may be as accurate as displacements. Usually, however, stresses are less accurate than displacements. Accordingly, a finer mesh is needed for stress analysis than for displacement
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>natural_image</summary>
|
||||
|
||||
3D wireframe model of mechanical components (no text or symbols)
|
||||
</details>
|
||||
|
||||
Figure 19.1-10. An exploded view of a machine part. (Courtesy of Algor Interactive Systems Inc., Pittsburgh, Pennsylvania.)
|
||||
Reference in New Issue
Block a user