김경종
525 lines
29 KiB
Markdown
525 lines
29 KiB
Markdown
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# INTRODUCTION
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A brief overview of the finite element method and its concepts is presented. Background information used for finite element applications in structural mechanics is discussed.
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# 1.1 THE FINITE ELEMENT METHOD
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The finite element method is a numerical procedure for analyzing structures and continua. Usually the problem addressed is too complicated to be solved satisfactorily by classical analytical methods. The problem may concern stress analysis, heat conduction, or any of several other areas. The finite element procedure produces many simultaneous algebraic equations, which are generated and solved on a digital computer. Finite element calculations are performed on personal computers, mainframes, and all sizes in between. Results are rarely exact. However, errors are decreased by processing more equations, and results accurate enough for engineering purposes are obtainable at reasonable cost.
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The finite element method originated as a method of stress analysis. Today finite elements are also used to analyze problems of heat transfer, fluid flow, lubrication, electric and magnetic fields, and many others. Problems that previously were utterly intractable are now solved routinely. Finite element procedures are used in the design of buildings, electric motors, heat engines, ships, airframes, and spacecraft. Manufacturing companies and large design offices typically have one or more large finite element programs in-house. Smaller companies usually have access to a large program through a commercial computing center or use a smaller program on a personal computer.
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Figure 1.1-1 shows a very simple problem that illustrates discretization, a basic finite element concept. Imagine that the displacement of the right end of the bar is required. The classical approach is to write the differential equation of the continuously tapered bar, solve this equation for axial displacement u as a function of x, and finally substitute $x = L_{T}$ to find the required end displacement. The finite element approach to this problem does not begin with a differential equation. Instead, the bar is discretized by modeling it as a series of finite elements, each uniform but of a different cross-sectional area A (Fig. 1.1-1b). In each element, u varies linearly with x; therefore, for $0 < x < L_{T}$ , u is a piecewise-smooth function of x. The elongation of each element can be determined from the elementary formula PL/AE. The end displacement, at $x = L_{T}$ , is the sum of the element elongations. Accuracy improves as more elements are used.
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In the foregoing example, and in general, the finite element method models a structure as an assemblage of small parts (elements). Each element is of simple geometry and therefore is much easier to analyze than the actual structure. In
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<details>
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<summary>text_image</summary>
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x,u
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L_T
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P
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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
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x
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P
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4@L = L_T
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</details>
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(b)
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Figure 1.1-1. (a) A tapered bar under end load P. (b) A model built of four uniform (nontapered) elements of equal length.
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essence, we approximate a complicated solution by a model that consists of piecewise-continuous simple solutions. Elements are called “finite” to distinguish them from differential elements used in calculus.
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In a heat transfer context, Fig. 1.1-1 might represent a bar with insulated sides, prescribed temperature at the left end, and prescribed heat flow at the right end. One might ask for the temperature in the bar as a function of x and time.
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Figure 1.1-2a shows a plane structure. Displacements and stresses caused by pressure p are required. The finite element model, Fig. 1.1-2b, consists of plane areas, some triangular and some quadrilateral (if done properly, there is no difficulty in combining the different element types). Black dots, called nodes or node points, indicate where elements are connected to one another. In this model each node has two degrees of freedom (d.o.f.): that is, each node can displace in both the x direction and the y direction. Thus, if there are n nodes in Fig. 1.1-2b, there are 2n d.o.f. in the model. (In the real structure there are infinitely many d.o.f. because the structure has infinitely many particles.) Algebraic equations that describe the finite element model are solved to determine the d.o.f. Use of only 2n d.o.f. in analysis is similar to use of the first 2n terms of a convergent infinite series. (In heat transfer, each node has only one d.o.f.—namely, the temperature of the node. Thus a finite element model of n nodes has n d.o.f.)
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We see that in going from Fig. 1.1-2a to 1.1-2b the distributed pressure p has been converted to concentrated forces at nodes. The analysis procedure gives a prescription for making conversion, as will be shown subsequently.
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From Fig. 1.1-2 it may appear that discretization is accomplished simply by
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<details>
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<summary>text_image</summary>
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y, v
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Pressure p
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Hole
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x, u
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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, v
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x, u
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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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y, v
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p4 → 4
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q4
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q3
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3
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p3
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p1 → 1
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1
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2
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p2
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q2
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x, u
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q1
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</details>
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{c}
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Figure 1.1-2. (a) A plane structure of arbitrary shape. (b) A possible finite element model of the structure. (c) A plane rectangular element showing nodal forces $p_{i}$ and $q_{i}$ . The dashed line shows the deformation mode associated with x-direction displacement of node 3.
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sawing the continuum into pieces and then pinning the pieces together again at node points. But such a model would not deform like the continuum. Under load, strain concentrations would appear at the nodes, and the elements would tend to overlap or separate along the saw cuts. Clearly, the actual structure does not behave in this way, so the elements must be restricted in their deformation patterns. For example, if elements are allowed to have only such deformation modes as will keep edges straight (Fig. 1.1-2c), then adjacent elements will neither overlap nor separate. In this way we satisfy the basic requirement that deformations of a continuous medium must be compatible.
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An important ingredient in a finite element analysis is the behavior of the individual elements. A few good elements may produce better results than many poorer elements. We can see that several element types are possible by considering Fig. 1.1-3. Function $\phi$ , which might represent any of several physical quantities, varies smoothly in the actual structure. A finite element model typically yields a piecewise-smooth representation of $\phi$ . Between elements there may be jumps in the $x$ and $y$ derivatives of $\phi$ . Within each element $\phi$ is a smooth function that is usually represented by a simple polynomial. What shall the polynomial be? For the triangular element, the linear polynomial
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$$
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\phi = a _ {1} + a _ {2} x + a _ {3} y \tag {1.1-1}
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$$
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is appropriate, where the $a_{i}$ are constants. These constants can be expressed in terms of $\phi_{1}$ , $\phi_{2}$ , and $\phi_{3}$ , which are the values of $\phi$ at the three nodes. Triangles model the actual $\phi$ by a surface of flat triangular facets. For the four-node quadrilateral, the “bilinear” function
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$$
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\phi = a _ {1} + a _ {2} x + a _ {3} y + a _ {4} x y \tag {1.1-2}
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$$
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is appropriate. The eight-node quadrilateral in Fig. 1.1-3 has eight $a_{i}$ in its polynomial expansion and can represent a parabolic surface.
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Equations 1.1-1 and 1.1-2 are interpolations of function $\phi$ in terms of the position $(x,y)$ within an element. That is, when the $a_{i}$ have been determined in terms of nodal values $\phi_{i}$ , Eqs. 1.1-1 and 1.1-2 define $\phi$ within an element in terms of the $\phi_{i}$ and the coordinates. Clearly, if the mesh of elements is not too coarse and if the $\phi_{i}$ happened to be exact, then $\phi$ away from nodes would be a good approx-
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<details>
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<summary>text_image</summary>
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Meaning
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Torsion: w
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Fluid flow
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Seepage f
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Magnetos
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Electric fi
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Heat cono
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</details>
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Meaning of $\phi$ in various problems:
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Torsion: warping function or stress function
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Fluid flow: stream function or velocity potential
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Seepage flow: hydraulic head
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Magnetostatic: magnetic potential
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Electric field: field potential (voltage)
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Heat conduction: temperature
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Figure 1.1-3. A function $\phi = \phi(x, y)$ that varies smoothly over a rectangular region in the xy plane, and typical elements that might be used to approximate it.
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imation. Nodal values $\phi_i$ are close to exact if the mesh is not too coarse and if element properties are properly formulated.
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How can the user decide which element to use? Unfortunately, the answer is not simple. An element that is good in one problem area (such as magnetic fields) may be poor in another (such as stress analysis). Even in a specific problem area an element may behave well or badly, depending on the particular geometry, loading, and boundary conditions. A competent user of finite elements must be familiar with how various elements behave under various conditions.
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We may now venture some definitions. The finite element method is a method of piecewise approximation in which the approximating function $\phi$ is formed by connecting simple functions, each defined over a small region (element). A finite element is a region in space in which a function $\phi$ is interpolated from nodal values of $\phi$ on the boundary of the region in such a way that interelement continuity of $\phi$ tends to be maintained in the assemblage.
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A finite element analysis typically involves the following steps. Again we will cite stress analysis and heat transfer as typical applications. Steps 1, 4, and 5 require decisions by the analyst and provide input data for the computer program. Steps 2, 3, 6, and 7 are carried out automatically by the computer program.
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√1. Divide the structure or continuum into finite elements. Mesh generation programs, called preprocessors, help the user in doing this work.
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√2. Formulate the properties of each element. In stress analysis, this means determining nodal loads associated with all element deformation states that are allowed. In heat transfer, it means determining nodal heat fluxes associated with all element temperature fields that are allowed.
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3. Assemble elements to obtain the finite element model of the structure.
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√4. Apply the known loads: nodal forces and/or moments in stress analysis, nodal heat fluxes in heat transfer.
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5. In stress analysis, specify how the structure is supported. This step involves setting several nodal displacements to known values (which often are zero). In heat transfer, where typically certain temperatures are known, impose all known values of nodal temperature.
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6. Solve simultaneous linear-algebraic equations to determine nodal d.o.f. (nodal displacements in stress analysis, nodal temperatures in heat transfer).
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7. In stress analysis, calculate element strains from the nodal d.o.f. and the element displacement field interpolation, and finally calculate stresses from strains. In heat transfer, calculate element heat fluxes from the nodal temperatures and the element temperature field interpolation. Output interpretation programs, called postprocessors, help the user sort the output and display it in graphical form.
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The power of the finite element method resides principally in its versatility. The method can be applied to various physical problems. The body analyzed can have arbitrary shape, loads, and support conditions. The mesh can mix elements of different types, shapes, and physical properties. This great versatility is contained within a single computer program. User-prepared input data controls the selection of problem type, geometry, boundary conditions, element selection, and so on.
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Another attractive feature of finite elements is the close physical resemblance
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between the actual structure and its finite element model. The model is not simply an abstraction. This seems especially true in structural mechanics, and may account for the finite element method having its origins there.
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The finite element method also has disadvantages. A specific numerical result is found for a specific problem: a finite element analysis provides no closed-form solution that permits analytical study of the effects of changing various parameters. A computer, a reliable program, and intelligent use are essential. A general-purpose program has extensive documentation, which cannot be ignored. Experience and good engineering judgment are needed in order to define a good model. Many input data are required and voluminous output must be sorted and understood.
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Example Applications. Figure 1.1-4 shows a finite element model of an axially symmetric rocket nozzle [10.1].¹ The axis, not shown, is horizontal and lies above the cross section in Fig. 1.1-4. Each element is a toroidal ring of triangular cross section. Each element has a node (actually a nodal circle) at each vertex. Each nodal circle has axial and radial displacements as d.o.f. Stresses caused by temperature gradient and internal pressure are desired.
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Figure 1.1-5 shows three ways of modeling an arch dam using “isoparametric” solid elements (discussed in Chapter 6). One might ask for the stresses produced by hydrostatic and gravity loads. Or, the response to earthquake motion might be required, in which fluid-structure interaction is taken into account.
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Figures 1.1-6 and 1.1-7 show typical problems in structural mechanics. The structure in Fig. 1.1-6 consists primarily of plate-bending elements. The structure in Fig. 1.1-7 consists of three-dimensional solid elements. The postprocessor has removed hidden lines. The deformation and stress plots display the results of analysis.
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Figure 1.1-8 shows a nonstructural problem. Lines of magnetic flux are crowded
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<details>
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<summary>text_image</summary>
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21.37 Diameter
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Graphite
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Insulator
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Glass filament
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Glass fabric
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Steel shell
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Asbestos
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</details>
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Figure 1.1-4. Cross section of a multimaterial rocket nozzle, showing construction (left portion) and possible finite element mesh (right portion). This problem was solved in the early days of finite element technology [10.1].
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$^{1}$ Numbers within brackets indicate references listed at the back of the book.
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<details>
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<summary>natural_image</summary>
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Pure geometric line drawing of a 3D curved structure with internal grid lines (no text or symbols)
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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 line drawing of a 3D geometric shape with grid lines, no text or symbols present
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</details>
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(b)
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<details>
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<summary>natural_image</summary>
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Simple line drawing of a curved, segmented shape with internal lines and dots, no text or symbols present.
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</details>
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{c}
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Figure 1.1-5. Half of an arch dam, modeled by $(a,b)$ quadrilateral and triangular “quadratic” elements, and $(c)$ a single “cubic” element [1.1]. Nodes of a typical element are shown by dots.
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near the gap between rotor and stator, which means that gradients are large in this region. Areas of large gradient are areas of particular interest. The analyst places more elements there in order to calculate the magnetic field in greater detail. The mesh shown is adequate for the analysis of magnetic flux, but is probably too crude to be used for stress analysis.
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Clearly some problems use a great many d.o.f. How many d.o.f. must a problem have to be considered “large”? In 1960, perhaps 1000; in 1980, over 10,000. Improvements in hardware and software have made this increase possible.
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Why Study the Theory of Finite Elements? Many satisfactory elements have already been formulated and reside in popular computer programs. The practitioner desires to understand how various elements behave. Clearly, engineers who understand analysis tools will be able to use them to better advantage and will be less likely to misuse them. Such an understanding cannot be achieved if theory is ignored. In this book we intend to aid the eventual practitioner and to present theory to an adequate but not excessive degree. We recognize that for engineers the study of finite elements is more than a theoretical study of mathematical foundations and formulation procedures for various types of finite elements.
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Complete computer codes need not be studied in detail, but concepts and assumptions' behind the coding should be mastered. Otherwise, the treatment of loads and boundary conditions may be confusing, the variety of program options and element types may be baffling, and error messages may provide no clue as to the source of difficulty or how to correct it.
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<details>
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<summary>natural_image</summary>
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3D wireframe diagram of a twisted rope or cable structure (no text or symbols)
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</details>
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Figure 1.1-6. A detailed model of half of an automobile frame, used to find deformations, stresses, natural frequencies, and mode shapes. (Courtesy of A. O. Smith Corp., Data Systems Division, Milwaukee, Wisconsin.)
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<details>
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<summary>natural_image</summary>
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3D wireframe model of a curved surface with grid lines, no text or symbols present
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</details>
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BEARING HOUSING MODEL
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<details>
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<summary>natural_image</summary>
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3D wireframe model of a curved, folded structure (no text or symbols)
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</details>
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DEFORMED MODEL
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<details>
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<summary>text_image</summary>
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100
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300
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100
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300
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500
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700
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900
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</details>
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STRESS CONTOUR
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Figure 1.1-7. Finite element mesh and computed deformations and stresses in a portion of a bearing housing. (Courtesy of Algor Interactive Systems Inc., Pittsburgh, Pennsylvania.)
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# 1.2 THE ELEMENT CHARACTERISTIC MATRIX
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The element characteristic matrix has different names in different problem areas. In structural mechanics it is called a stiffness matrix: it relates nodal displacements to nodal forces. In heat conduction it is called a conductivity matrix: it relates nodal temperatures to nodal fluxes. There are three important ways to derive an element characteristic matrix.
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1. The direct method is based on physical reasoning. It is limited to very simple elements, but is worth studying because it enhances our physical understanding of the finite element method.
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<details>
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<summary>text_image</summary>
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stator
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rotor
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</details>
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<details>
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<summary>natural_image</summary>
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Topographic contour lines diagram of a building structure (no text or labels)
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</details>
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Figure 1.1-8. Part of an induction motor. Elements model the solid parts as well as the spaces between them. Symmetry is exploited by modeling only a half-pole. The computed magnetic flux contours for zero rotor speed are shown by the right-hand figure. (Courtesy of A. O. Smith Corp., Data Systems Division, Milwaukee, Wisconsin.)
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2. The variational method is applicable to problems that can be stated by certain integral expressions such as the expression for potential energy. This method is discussed in Chapters 3 and 4.
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③. Weighted residual methods are particularly suited to problems for which differential equations are known but no variational statement is available. For stress analysis and some other problem areas, the variational method and the most popular weighted residual method (the Galerkin method) yield identical finite element formulations. Weighted residual methods are discussed in Chapter 15.
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In the present section we consider applications of the direct method.
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The Elastic Bar: Direct Method. Consider a weightless straight bar of length L, elastic modulus E, and cross-sectional area A. We regard the bar as a finite element and place a node at each end. If only axial loads and axial displacements are allowed, nodal d.o.f. are displacements $u_{i}$ and $u_{j}$ (Fig. 1.2-1). The element stiffness matrix is formulated by determining the nodal forces that must be applied in order to produce nodal displacements $u_{i}$ and $u_{j}$ . Our sign convention is that both force and displacement are positive when directed toward the right. Accordingly, when $u_{i} > 0$ but $u_{j} = 0$ (Fig. 1.2-1a), nodal forces consistent with static equilibrium and a linearly elastic material are
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$$
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F _ {i} = \frac {A E}{L} u _ {i} \quad \text { and } \quad F _ {j} = - \frac {A E}{L} u _ {i} \tag {1.2-1}
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$$
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Similarly, when $u_{i} = 0$ but $u_{j} > 0$ (Fig. 1.2-1b),
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$$
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F _ {i} = - \frac {A E}{L} u _ {j} \quad \text { and } \quad F _ {j} = \frac {A E}{L} u _ {j} \tag {1.2-2}
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$$
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If both $u_{i}$ and $u_{j}$ can be simultaneously nonzero, then nodal forces are $F_{i} = (AE/L)(u_{i} - u_{j})$ and $F_{j} = (AE/L)(-u_{i} + u_{j})$ . In matrix format these two equations are
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$$
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\left[ \begin{array}{c c} A E / L & - A E / L \\ - A E / L & A E / L \end{array} \right] \left\{ \begin{array}{l} u _ {i} \\ u _ {j} \end{array} \right\} = \left\{ \begin{array}{l} F _ {i} \\ F _ {j} \end{array} \right\} \tag {1.2-3}
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$$
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or
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$$
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[ \mathbf {k} ] \{\mathbf {d} \} = \{\overline {{\mathbf {r}}} \} \tag {1.2-4}
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$$
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where [k] is the element stiffness matrix. By considering first $u_{i}=1$ and $u_{j}=0$ , then $u_{i}=0$ and $u_{j}=1$ , each time computing nodal forces $F_{i}$ and $F_{j}$ by the matrix-
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<details>
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<summary>text_image</summary>
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F_i
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u_i
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F_j (= -F_i)
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i
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L
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j
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(a)
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</details>
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<details>
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<summary>text_image</summary>
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F_i (= -F_j)
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i
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L
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u_j
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j
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F_j
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(b)
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</details>
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Figure 1.2-1. A uniform bar, showing nodal forces associated with nodal displacements $u_{i}$ and $u_{j}$ . Displacements $u_{i}$ and $u_{j}$ are greatly exaggerated in the drawing.
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times-vector multiplication indicated in Eq. 1.2-3, we reach the following conclusion: a column of [k] lists nodal loads that must be applied to nodal d.o.f. in order to create the deformation state associated with unit value of the corresponding element d.o.f. while all other element d.o.f. are zero. Later in this section we will use this procedure in calculating [k] for a beam element, Fig. 1.2-2.
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Remarks. No approximation was used in deriving the foregoing stiffness matrix. It is exact. Of course, if used to construct a stepwise model of a continuously tapered bar, as in Fig. 1.1-1, the stepwise model is inexact and computed displacements will differ from those of the actual structure.
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The stiffness matrix is symmetric. This is always true when displacements are directly proportional to applied loads.
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By trying to apply the direct method of element derivation to Fig. 1.1-2c, we can see that the direct method is limited to simple elements: physical reasoning, based on elementary mechanics of materials, does not quantify the eight nodal forces associated with a nodal displacement such as $u_{3}$ .
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Heat or Current Conduction. A uniform bar element for heat conduction analysis has an element characteristic matrix that resembles stiffness matrix [k] of the elastic bar. With s a coordinate along the bar, the Fourier heat conduction equation becomes
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$$
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q = - k \frac {d T}{d s} = - k \frac {T _ {j} - T _ {i}}{L} \tag {1.2-5}
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$$
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where $q =$ heat flux per unit area, $k_{\perp} =$ thermal conductivity, and $T =$ temperature. If $A$ is the cross-sectional area and nodal heat flux is considered to be positive when directed into the bar at either end, then the equation analogous to Eq. 1.2-3 is
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$$
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\frac {A k}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] \left\{ \begin{array}{l} T _ {i} \\ T _ {j} \end{array} \right\} = \left\{ \begin{array}{l} A q _ {i} \\ A q _ {j} \end{array} \right\} \tag {1.2-6}
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$$
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Similarly, if the bar is regarded as an electrical resistor, Ohm's law becomes $I = (V_{i} - V_{j}) / r$ , where $r$ is the resistance of the bar. In matrix form,
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$$
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\left[ \begin{array}{c c} 1 / r & - 1 / r \\ - 1 / r & 1 / r \end{array} \right] \left\{ \begin{array}{l} V _ {i} \\ V _ {j} \end{array} \right\} = \left\{ \begin{array}{l} I _ {i} \\ I _ {j} \end{array} \right\} \tag {1.2-7}
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$$
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where $V_{i}$ and $V_{j}$ are nodal voltages. Nodal currents $I_{i}$ and $I_{j}$ are considered positive when flowing into the element.
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One can envision networks of the foregoing heat conduction or current flow elements, in which a single node may be shared by several elements. If no external source supplies heat or current to such a node, the net flow into the node is zero.
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The Elastic Beam: Direct Method. Consider a uniform beam that deforms in the plane of the paper and undergoes no axial deformation. This element has four d.o.f.: a lateral displacement w and a rotation $\theta$ at each end (Fig. 1.2-2). Nodal forces $F_{i}$ and $F_{j}$ correspond to nodal displacements $w_{i}$ and $w_{j}$ . Nodal moments $M_{i}$
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$$
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\vert C C (W) [ H ] / U _ {P}
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INTRODUCTION
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<details>
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<summary>text_image</summary>
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θi
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θj
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wi
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L
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wj
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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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M_i
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M_j
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F_j
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F_i
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L
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(b)
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<summary>text_image</summary>
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w_i = 1
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k_{11}
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k_{21}
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k_{31}
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k_{41}
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</details>
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(r)
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<summary>text_image</summary>
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k₂₂
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θᵢ=1
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k₁₂
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k₃₂
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k₄₂
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</details>
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(d)
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<summary>text_image</summary>
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k₂₃
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k₁₃
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k₃₃
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k₄₃
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wⱼ=1
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</details>
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(e)
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k24
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k14
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θj=1
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k34
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k44
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</details>
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(1)
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Figure 1.2-2. (a) A uniform beam element and its four nodal d.o.f. (b) Associated nodal forces and moments. (c-e) Deformation states associated with activation of each d.o.f. in turn, showing the required nodal forces and moments labeled according to their position in [k].
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and $M_{j}$ correspond to nodal rotations $\theta_{i}$ and $\theta_{j}$ . Positive directions of these quantities are shown in Figs. 1.2-2a and 1.2-2b.
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The element equation is $[\mathbf{k}]\{\mathbf{d}\} = \{\overline{\mathbf{r}}\}$ , where $[\mathbf{k}]$ is a 4 by 4 stiffness matrix for the present beam element. The element nodal displacement vector is
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$$
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\{\mathbf {d} \} = \left[ \begin{array}{l l l l} w _ {i} & \theta_ {i} & w _ {j} & \theta_ {j} \end{array} \right] ^ {T} \tag {1.2-8}
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$$
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(A different ordering of d.o.f. in $\{d\}$ would change the ordering of coefficients in $[k]$ but would not change their numerical values.) Vector $\{\bar{r}\}$ contains nodal forces and moments applied to the element to maintain the deformation state $\{d\}$ . Written out, the element stiffness equation is
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$$
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\left. \begin{array}{l} \text {Apo} \left[ \begin{array}{l l l l} k _ {1 1} & k _ {1 2} & k _ {1 3} & k _ {1 4} \\ k _ {2 1} & k _ {2 2} & k _ {2 3} & k _ {2 4} \\ k _ {3 1} & k _ {3 2} & k _ {3 3} & k _ {3 4} \\ k _ {4 1} & k _ {4 2} & k _ {4 3} & k _ {4 4} \end{array} \right] \left\{ \begin{array}{l} w _ {i} \\ \theta_ {i} \\ w _ {j} \\ \theta_ {j} \end{array} \right\} = \left\{ \begin{array}{l} F _ {i} \\ M _ {i} \\ F _ {j} \\ M _ {j} \end{array} \right\} \end{array} \right\} \tag {1.2-9}
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$$
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We must express each stiffness coefficient in [k] in terms of element geometry and elastic modulus. To thus determine the stiffness coefficients in a single column of [k], we set the corresponding d.o.f. to unity while keeping all other d.o.f. zero, and calculate the values of $F_{i}$ , $M_{i}$ , $F_{j}$ , and $M_{j}$ needed in order to produce this deformation state. Thus, the first column of [k] is determined by activating only the first d.o.f. Specifically, for the case $w_{i} = 1$ and $\theta_{i} = w_{j} = \theta_{j} = 0$ , we see
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