Hutton - Fundamentals of Finite Element Analysis (523155), страница 68
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Withthe exception of strain energy, the stress data are available on either a nodal orelement basis. The distinction is significant, and the analyst must be acutelyaware of the distinction. Since strain components (therefore, stress components)are not in general continuous across element boundaries, nodal stresses are computed as average values based on all elements connected to a specific node. Onthe other hand, element stresses represent values computed at the element centroid.
Hence, element stress data are more accurate and should be used in making engineering judgments. To illustrate, we present some of the stress dataobtained in the solution of Example 9.4 based on two-dimensional, four-nodequadrilateral elements. In the model, node 107 (selected randomly) is common tofour elements.
Table 9.1 lists the stresses computed at this node in terms of thefour connected elements. The values are obtained by computing the nodalstresses for each of the four elements independently, then extracting the valuesTable 9.1 Stress Values (psi) Computed at Node 107 of Example 9.4Element 1Element 2Element 12Element 99Averagexyxy2049.32149.41987.31853.82009.8187.36315.59322.72186.88253.14118.491.89204.13378.36198.19371Hutton: Fundamentals ofFinite Element Analysis3729.
Applications in SolidMechanicsCHAPTER 9Text© The McGraw−HillCompanies, 2004Applications in Solid MechanicsTable 9.2 Element Stress Components (psi) for Four Elements Sharing a Common Nodein Example 9.4Element 1Element 2Element 12Element 99xyxye2553.51922.71827.52189.0209.71351.69264.42249.14179.8743.55154.44480.572475.81774.81731.52236.4for the common node. The last row of the table lists the average values of thethree stress components at the common node. Clearly, the nodal stresses are notcontinuous from element to element at the common node.
As previously discussed, the magnitudes of the discontinuities should decrease as the elementmesh is refined.In contrast, the element stress components for the same four elements areshown in Table 9.2. The values listed in the table are computed at the elementcentroid and include the equivalent (Von Mises) stress as defined previously.While not included in the table, the principal stress components are also available from the solution. In general, the element stresses should be used in resultsevaluation, especially in terms of application of failure theories.9.8 PRACTICAL CONSIDERATIONSzp(x,y)xFigure 9.12Example of a thinplate subjected tobending.yProbably the most critical step in application of the finite element method is thechoice of element type for a given problem.
The solid elements discussed in thischapter are among the simplest elements available for use in stress analysis.Many more element types are available to the finite element analyst. (One commercial software system has no fewer than 141 element types.) The differencesin elements for stress analysis fall into three categories: (1) number of nodes,hence, polynomial order of interpolation functions; (2) type of material behavior(elastic, plastic, thermal stress, for example); and (3) loading and geometry of thestructure to be modeled (plane stress, plane strain, axisymmetric, general threedimensional, bending, torsion).As an example, consider Figure 9.12, which shows a flat plate supported atthe corners and loaded by a pressure distribution p(x, y) acting in the negative zdirection. The primary mode of deformation of the plate is bending in the z direction.
To adequately describe the behavior, a finite element used to model the platemust be such that continuity of slope in both xz and yz planes is ensured. Therefore, a three-dimensional solid element as described in Section 9.8 would not beappropriate as only the displacement components are included as nodal variables.Instead, an element that includes partial derivatives representing the slopes mustbe included as nodal variables. Plate elements have been developed on the basisof the theory of thin plates (usually only covered in graduate programs) in whichHutton: Fundamentals ofFinite Element Analysis9. Applications in SolidMechanicsText© The McGraw−HillCompanies, 20049.8 Practical ConsiderationsyzydSP⬘TPzx(a)(b)xyxy␣xzdS(c)(x, y)xzyOz(d)Figure 9.13(a) A general, noncircular section in torsion.
(b) Motion of a point from P to P as aresult of cross-section rotation. (c) A differential element at the surface of a torsionmember. (d) A differential element showing the contribution of shear stress to torque.the bending deflection is governed by a fourth-order partial differential equation.The simplest such element is a four-node element using cubic interpolation functions and having 4 degrees of freedom (displacement, two slopes, and a mixedsecond derivative) at each node [4].
A similar situation exists with shell (thincurved plate) structures. Specialized elements are required (and available) forstructural analysis of shell structures. The major point here is that a breadth ofknowledge and experience is required for a finite element analyst to become trulyproficient at selecting the correct element type(s) for a finite element model and,subsequently interpreting the results of the analysis.Once the element type has been selected, the task becomes that of definingthe model geometry as a mesh of finite elements.
In its most rudimentary form,this task involves defining the coordinate location of every node in the model373Hutton: Fundamentals ofFinite Element Analysis3749. Applications in SolidMechanicsCHAPTER 9Text© The McGraw−HillCompanies, 2004Applications in Solid Mechanics(note that, by default, the nodes define the geometry) followed by definition of allelements in terms of nodes.
Many years ago, in the early development of thefinite element method, the tasks of node and element definition were laborintensive, as the definitions required use of the specific language statements of aparticular finite element software system. The tasks were laborious, to say theleast, and prone to error. With currently technology, especially graphical userinterfaces and portability of computer-aided design (CAD) databases, thesetasks have been greatly simplified. It is now possible, with many FEA programs,to “import” the geometry of a component, structure, or assembly directly from aCAD system, so that geometry does not need to be defined. The finite elementsoftware can then automatically create a mesh (automeshing) of finite elementsto represent the geometry.
The advantages of this capability include (1) the finiteelement analyst need not redefine the geometry; consequently, (2) the designer’sintent is not changed inadvertently; and (3) the finite element analyst is relievedof the burden of specifying the details of the node and element definitions. Themajor disadvantage is that the analyst is not in direct control of the meshing operation.The word direct is emphasized. In automeshing, the software user has somecontrol over the meshing process. There are two general types of automeshingsoftware, generally referred to as free meshing and mapped meshing. In freemeshing, the user specifies a general, qualitative mesh description, ranging fromcoarse to fine, with 10 or more gradations between the extremes.
The softwarethen generates the mesh accordingly. In mapped meshing, the user specifies quantitative information regarding node spacing, hence, element size, and the software uses the prescribed information to generate nodes and elements. In eithermethod, the software user has some degree of control over the element mesh.A very important aspect of meshing a model with elements is to ensure that,in regions of geometric discontinuity, a finer mesh (smaller elements) is definedin the region.
This is true in all finite element analyses (structural, thermal, andfluid), because it is known that gradients are higher in such areas and finermeshes are required to adequately describe the physical behavior. In mappedmeshing, this is defined by the software user. Fortunately, in free meshing, thisaspect is accounted for in the software. As an example, refer back to Example 9.4, in which we examined the stress concentration factor for a hole in a thinplate subjected to tension.
The solution was modeled using the free mesh featureof a finite element software system. Figure 9.8b is a coarse mesh as generated bythe software. Geometry is defined by four lines and a quarter-circular arc; these,in turn, define a single area of interest. Having specified the element type (in thiscase a plane stress, elastic, quadrilateral), the area meshing feature is used to generate the elements as shown. It is important to note the relatively fine mesh in thevicinity of the arc representing the hole.
This is generated by the software automatically in recognition of the geometry. The mesh-refined models of Figure 9.8c and 9.8d are also generated by the free meshing routine. From each ofthese cases, we see that, not only does the number of elements increase, but theHutton: Fundamentals ofFinite Element Analysis9. Applications in SolidMechanicsText© The McGraw−HillCompanies, 20049.9 Torsionrelative size of the elements in the vicinity of the hole is maintained relative toelements far removed from the discontinuity.The automeshing capabilities of finite software as briefly described here areextremely important in reducing the burden of defining a finite element model ofany geometric situation and should be used to the maximum extent. However,recall that the results of a finite element analysis must be judged by humanknowledge of engineering principles.
Automated model definition is a nicety ofmodern finite element software; automated analysis of results is not.Analysis of results is the postprocessing phase of finite element analysis.Practical models contain hundreds, if not thousands, of elements, and the computed displacements, strains, stresses, and so forth are available for every element. Poring through the data can be a seemingly endless task. Fortunately, finiteelement software has, as part of the postprocessing phase, routines for sortingthe results data in many ways. Of particular importance in stress analysis, thedata can be sorted in ascending or descending order of essentially any stresscomponent chosen by the user. Hence, one can readily determine the maximumequivalent stress, for example, and determine the location of that stress by theassociated element location.
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