Wuhan University of TechnologyChapter14Selection of dynamicdegrees offreedom14-1
14-1 Wuhan University of Technology Chapter 14 Selection of dynamic degrees of freedom
Wuhan University of Technology1Contents14.1 Finite-element degrees of freedom14.2 Kinematic constraints14.3 Static condensation14.4 Rayleigh method in discrete coordinates14.5 Rayleigh-ritz method14.6Subspace iteration14-2
14-2 Wuhan University of Technology 14.1 Finite-element degrees of freedom 14.2 Kinematic constraints 14.3 Static condensation 14.4 Rayleigh method in discrete coordinates 14.5 Rayleigh-ritz method 14.6 Subspace iteration Contents
Wuhan Universityof Technology14.1 Finite-element degrees of freedomOneDimensionalElementsAfiniteelementmodelofaframed structuretypicallyisformedbyassemblinga set of onedimensional elements which are in onetoone correspondence withthe beams, struts, girders, etc., that make up the actual structure.Thenumberofdegreesoffreedominthemodel,therefore,isfixedbythephysical arrangementofthe structure,andingeneral all of thedegreesoffreedomwouldbeinvolvedintheanalysisofstressesanddisplacementsresultingfromapplicationofageneralstaticloaddistribution.Ontheotherhand,notallofthedegreesoffreedomneedbeconsideredasindependent variables inanalysisof theresponsetoanarbitrarydynamicloadingDependingonboththetimevariationaswell asthespatialdistributionoftheload, the dynamic analysis often may be performed effectively with a muchsmallernumberofindependentdegreesoffreedomusingprocedurestobeexplainedlaterinthischapter.14-3
14-3 Wuhan University of Technology 14.1 Finite-element degrees of freedom OneDimensional Elements A finiteelement model of a framed structure typically is formed by assembling a set of onedimensional elements which are in onetoone correspondence with the beams, struts, girders, etc., that make up the actual structure. The number of degrees of freedom in the model, therefore, is fixed by the physical arrangement of the structure, and in general all of the degrees of freedom would be involved in the analysis of stresses and displacements resulting from application of a general static load distribution. On the other hand, not all of the degrees of freedom need be considered as independent variables in analysis of the response to an arbitrary dynamic loading. Depending on both the time variation as well as the spatial distribution of the load, the dynamic analysis often may be performed effectively with a much smaller number of independent degrees of freedom using procedures to be explained later in this chapter
Wuhan University of Technology14.1 Finite-element degrees of freedomTwoandThreeDimensionalElementsManystructurescanbetreated astwoorthreedimensional continuaorascombinationsofsuchcontinuumcomponents,andappropriatetwoorthree-dimensionalelementsaremosteffectiveinmodelingsuchstructures.In formulating models of this type, the number of degrees of freedom to beusedisnotdictated justbytheconfigurationofthestructure;inadditionthedegreeofmeshrefinementthatisrequiredtoobtainareasonableapproximationoftheactualstraindistributionisanimportantconsideration.Thebasicfactorthatcontrolsthestiffnesspropertiesoftheindividualfiniteelements isthevariationofdisplacementswithintheelementsasexpressedbytheassumeddisplacementinterpolationfunctions.14-4
14-4 Wuhan University of Technology Two and ThreeDimensional Elements Many structures can be treated as two or threedimensional continua or as combinations of such continuum components, and appropriate two or threedimensional elements are most effective in modeling such structures. In formulating models of this type, the number of degrees of freedom to be used is not dictated just by the configuration of the structure; in addition the degree of mesh refinement that is required to obtain a reasonable approximation of the actual strain distribution is an important consideration. The basic factor that controls the stiffness properties of the individual finite elements is the variation of displacements within the elements as expressed by the assumed displacement interpolation functions. 14.1 Finite-element degrees of freedom
Wuhan University of Technology14.2 Kinematic constraints1FIGURE14-1Twenty-storybuildingframe(2880degreesoffreedom)14-5
14-5 Wuhan University of Technology 14.2 Kinematic constraints FIGURE 14-1 Twenty-story building frame (2880 degrees of freedom)