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)
WuhanUniversityof Technology14.2 Kinematic constraintsAdditionalkinematicconstraintssometimeshavebeenassumedinboththestaticandthedynamicanalysisofbuildingframes,suchasthatthecolumnsareinextensibleand/orthatthefloorslabsarerigidoutofplaneaswellasin-plane.However,theseassumptionsseldomarejustifiedbytheactualstiffnessproperties of the components of which the building is assembled and theyshouldnotbeemployedexceptinspecialcircumstances.·It is important to recognize that all members are free to distort in flexureand thatall columns haveaxial flexibility inthetypeof modeldescribedabove.14-6
14-6 Wuhan University of Technology 14.2 Kinematic constraints Additional kinematic constraints sometimes have been assumed in both the static and the dynamic analysis of building frames, such as that the columns are inextensible and/or that the floor slabs are rigidoutofplane as well as inplane. However, these assumptions seldom are justified by the actual stiffness properties of the components of which the building is assembled and they should not be employed except in special circumstances. It is important to recognize that all members are free to distort in flexure and that all columns have axial flexibility in the type of model described above
Wuhan University of Technology14.3 StaticcondensationIncontrasttothekinematicconstraintideadescribedabove,theconceptofstaticcondensationisbasedonstaticequilibriumconstraints.hencethenameoftheprocedure.To apply this principle, the degrees of freedom of the structural system aredivided into two categories:those in which no mass participates so that inertialforcesarenotdeveloped andthosehavingmassthat induces inertialforces.AstheprocedurewasdescribedinSection106,thedegreesoffreedomwereclassified aseitherrotational ortranslational because itwas assumedthatthemasswasconcentratedinpointlumpswhichhadno inertial resistancetorotation.However,thefundamentalconceptinvolvesmerelytherecognitionofthosedegreesoffreedomthatcandevelopinertialforcesasdistinguishedfromthosethat cannot.14-7
14-7 Wuhan University of Technology 14.3 Static condensation In contrast to the kinematic constraint idea described above, the concept of static condensation is based on static equilibrium constraints . hence the name of the procedure. To apply this principle, the degrees of freedom of the structural system are divided into two categories: those in which no mass participates so that inertial forces are not developed and those having mass that induces inertial forces. As the procedure was described in Section 106, the degrees of freedom were classified as either rotational or translational because it was assumed that the mass was concentrated in point lumps which had no inertial resistance to rotation. However, the fundamental concept involves merely the recognition of those degrees of freedom that can develop inertial forces as distinguished from those that cannot
Wuhan University of Technology14.3 Static condensationk=w?mvko0kotvo00kooVo+kotVt=0Vo = -koo kot Vtkt Vt = wmtVtkt = ktt - kto koo kot14-8
14-8 Wuhan University of Technology 14.3 Static condensation
Wuhan University of Technology14.3 Static condensationThis static condensation procedure can be used to effect a very considerablereductioninthenumberofdegreesoffreedomtobeused inadynamicanalysis,suchasthereductionfrom1500toonly60inthebuildingframeexamplediscussedabove;However,thereductioninactualcomputationaleffortmaybemuchlesssignificant than these data suggest.This isbecause the narrow banding of thestiffnessmatrixkinEq.(141)makespossibleaveryefficient solutionprocedure when the analysisis performed in the original coordinates,whereastheanalysis using Eq.(144a)ismuchmore expensiveper degreeoffreedombecause the reduced stiffnessk, becomes fully populated as a result of thecondensationprocedure.Forthisreason,theadvisabilityofusingstaticcondensationshouldbeevaluated carefully on a casebycase basis.14-9
14-9 Wuhan University of Technology 14.3 Static condensation This static condensation procedure can be used to effect a very considerable reduction in the number of degrees of freedom to be used in a dynamic analysis, such as the reduction from 1500 to only 60 in the building frame example discussed above; However, the reduction in actual computational effort may be much less significant than these data suggest. This is because the narrow banding of the stiffness matrix k in Eq. (141) makes possible a very efficient solution procedure when the analysis is performed in the original coordinates, whereas the analysis using Eq. (144a) is much more expensive per degree of freedom because the reduced stiffness kt becomes fully populated as a result of the condensation procedure. For this reason, the advisability of using static condensation should be evaluated carefully on a casebycase basis
WuhanUniversityof Technology14.4 Rayleigh method in discrete coordinatesInmatrixnotation,theassumedfreevibrationdisplacementsmaybeexpressed[comparewith Eq. (825)]v(t)=bZ(t)=bZo sinwtv(t)=bwZocoswt1ITTmaxVmax m Vmax121TVmaxkVmaxmax121-Zg w267myTmax-21-2ZUTVmaxkb14-10
14-10 Wuhan University of Technology 14.4 Rayleigh method in discrete coordinates In matrix notation, the assumed freevibration displacements may be expressed [compare with Eq. (825)]