Belytschko T. - Introduction (779635), страница 100
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(ksi)10500.000.020.040.06TimeFigure 28. Load history0.080.10830.01Radial strain0.00-0.01elasticelastic-plastic-0.02-0.03-0.04104070100rFigure 29. Radial strain at t=0.09Table 13a. Normalized L2 norms of error in displacements for material 1 (elastic)θ0°45°QUAD4.014.022FB (0.1).014.022ASMD.014.019ASQBI.014.019ASOI.013.012ADS.014.021Table 13b. Normalized L2 norms of error in displacements for material 2 (elastic-plastic)θ0°45°QUAD4.0063.0069FB (0.1).0063.0069ASMD.0061.0086ASQBI.0061.0088ASOI.0061.0073ADS.0063.00881.6.5 Static Cantilever.
The solutions to the test problems of Sections 1.6.1 and 1.6.2were obtained using a local coordinate formulation of the stabilization matrix: Likewise,the solutions to the test problems of Sections 1.6.3 and 1.6.4 were obtained using acorotational coordinate formulation. The need for these local and corotational formulationsto obtain a frame invariant element is discussed in Section 1.4.6. The following solutionsto a static cantilever demonstrate this need.A cantilever with a shear load at its end was solved by two versions of the linearstatic finite element code using QBI stabilization.
One version had a local coordinateformulation, and the other did not. These are called the "local" and "global" formulationsrespectively. A total of seven solutions were obtained with three meshes as shown inFigs. (30a-c). Each was solved with the longitudinal axis of the undeformed beam alignedwith the global x axis, and also with the beam initially rotated before applying the load.84VV45oFigure 30a. 1x6 element meshVVV22.5o45oFigure 30b. 1x3 element meshVV45oFigure 30c.
4x12 element meshTable 14 lists the end displacement in the direction of the load for the seven solutionsnormalized by the solutions of the unrotated meshes. Therefore, these numbers do notdemonstrate absolute accuracy, but the variation in the element stiffness that occurs withrigid body rotation. The results show that the global formulation is sensitive to rigid bodyrotation when the elements are elongated and the mesh is coarse. When the aspect ratio 1,both formulations are frame invariant. Also, when the mesh is refined, the lack of frameinvariance is less noticeable. The local formulation is always frame invariant.85Table 14. End displacements in the direction of the applied load normalized by the 0°solutionMeshInitialGlobalLocalrotation(degrees)01.001.001x6451.001.0001.001.001x322.50.711.00450.491.0001.001.004x12450.941.001 .
7 Discussion and ConclusionsThe bilinear quadrilateral element is a good choice for solving two dimensionalcontinuum problems with explicit methods, because the mass matrix can be lumped withlittle loss of accuracy. There are two major benefits to 1-point integration with thequadrilateral. The first is the elimination of volumetric locking which plagues the fullyintegrated element. The second is a reduction in the computational effort for such elements.A drawback of 1-point integration is that spurious modes will occur if they are notstabilized.
We have examined some ways of stabilizing the spurious modes in thischapter.With all the methods considered, the stabilization forces are proportional to a g vectorwhich is orthogonal to the constant strain modes of deformation, so the stabilization forcesdo not contribute to the constant strain field. Therefore, all have a quadratic rate ofconvergence in the displacement error norm.
The major difference between the methods isin the way the evaluation of the magnitude of the stabilization forces.Flanagan and Belytschko (1981) were motivated by the desire to keep the stabilizationforces small so they would not interfere with the solution or cause locking. Thisstabilization has the drawback of requiring a user specified parameter. A bendingdominated solution can depend significantly on the value of the parameter which isundesirable.Using mixed methods, Belytschko and Bachrach (1986) chose strain and stress fieldsthat more closely resemble the strength of materials solution of elastic deformation.
Thus,they were able to use stabilization to improve to the accuracy of bending solutions. Theyobtain very accurate bending solutions with very few elements with elastic material. Mixedmethod stabilization is dependent only on material properties and element geometry; no userspecified parameter is needed.The Simo-Hughes form of the assumed strain method has also been used to developstabilization. The assumed strain fields are motivated in the same way as the mixed methodelements, and the resulting stabilization is nearly the same.
As with mixed methodstabilization, no user specified parameter is needed. The most noticeable differencebetween assumed strain and mixed-method stabilization is in the derivation. Assumedstrain stabilization is much simpler. As we will see in Chapter 2, a major benefit of thissimplification is the ability to derive stabilization for the three dimensional 8 nodehexahedral element.The relative performance of these elements is problem dependent; thus QBI andASQBI are very accurate for elastic bending, but they do not perform as well for elastic-86plastic problems.
Although it is not so accurate for elastic bending, ADS may be a goodchoice since it is very simple to implement and does not require knowledge of the material'sPoisson's ratio. It's performance should exceed that of the other 1-point elements forelastic-plastic solutions.
If the Poisson's ratio of the material is known, the ASQBI strainfield with 2-point integration will provide both accurate elastic bending and reasonableelastic-plastic performance at a slightly higher cost.T. Belytschko, Chapter 9, Shells and Structures, December 16, 1998CHAPTER 9SHELLS AND STRUCTURESDRAFTby Ted BelytschkoNorthwestern UniversityCopyright 19979.1 INTRODUCTIONShell elements and other structural elements are invaluable in the modeling ofmany engineered components and natural structures. Thin shells appear in manyproducts, such as the sheet metal in an automobile, the fuselage, wings and rudder of anairplane, the housings of products such as cell phones, washing machines, computers.Modeling these items with continuum elements would require a huge number of elementsand lead to extremely expensive computations. As we have seen in Chapter 8, modeling abeam with hexahedral continuum elements requires a minimum of about 5 elementsthrough the thickness.
Thus even a low order shell element can replace 5 or morecontinuum elements, which improves computational efficiency immensely. Furthermore,modeling thin structures with continuum elements often leads to elements with highaspect ratios, which degrades the conditioning of the equations and the accuracy of thesolution.
In explicit methods, continuum element models of shells are restricited to verysmall stable time steps. Thus it can be seen that structural elements are very useful inengineering analysis.Structural elements are classified as:1. beams, in which the motion is described as the function of a single independentvariable;2. shells, where the motion is described as a function of two independentvariables;3. plates, which are flat shells.Plates are usually modeled by shell elements in computer software. Since they are justflat shells, we will not consider plate elements separately. Beams on the other hand,require some additional theoretical considerations and provide simple models for learningthe fundamentals of structural elements, so we will devote a substantial part of thischapter to beams.There are two approaches to developing shell finite elements:1.
develop the formulation for shell elements by using classical straindisplacement and momentum (or equilibrium) equations for shells to developa weak form of the momentum (or equilibrium) equations;2. develop the element directly from a continuum element by imposing thestructural assumptions on the weak form or on the discrete equation; this iscalled the continuum based (CB) approach.The first approach is difficult, particularly for nonlinear shells, since the governingequations for nonlinear shells are very complex and awkward to deal with; they areusually formulated in terms of curvilinear components of tensors, and features such as9-1T. Belytschko, Chapter 9, Shells and Structures, December 16, 1998variations in thickness, junctions and stiffeners are generally difficult to incorporate.There is still disagreement as to what are the best nonlinear classical shell equations.
TheCB (continuum-based) approach, on the other hand, is straightforward, yields excellentresults, is applicable to arbitrarily llarge deformations and is widely used in commercialsoftware and research. Therefore we will concentrate on the CB methodology. It is alsocalled the degenerated continuum approach; we prefer the appellation continuum based,coined by Stanley(1985), since there is nothing degenerate about it.The CB methodology is not only simpler, but intellectually a more appealingappraoch than classical shell theories for developing shell elements. In most plate andshell theories, the equilibrium or momentum equations are developed by imposing thestructural assumptions on the motion and then using the principle of virtual work todevelop the partial differential equations for momentum balance or equilibrium. Thedevelopment of a weak form for these shell momentum equations than entails going backto the principle of virtual work.
In the CB approach, the kinematic assumptions are either1. imposed on the motion in the weak form of the momentum equations forcontinua or2. imposed directly on the discrete equations for continua.Thus the CB shell formulation is a more straightforward way of obtaining the discreteequations for shells and structures.We will begin with a description of beams in two dimensions. This will provide asetting for clearly and easily describing the assumptions of various structural theories andcomparing them with CB beam elements. In contrast to the schema in previous Chapters,we will begin with the implementation, for in the implementation the simplicity and keyfeatures of the CB approach are most transparent.
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