Wuhan Universityof Technology110.1 Elastic propertiesIn principle, theflexibility or stiffness coefficients associated with anyprescribed set of nodal displacementscanbeobtainedbydirectapplicationoftheir definitions.Inpractice,however,thefiniteelementconcept,describedinChapter1,freguentlyprovidesthemostconvenientmeansforevaluatingtheelasticproperties.Bythisapproachthestructureisassumedtobedividedintoasystemofdiscreteelementswhichareinterconnectedonlyatafinitenumberof nodalpoints.Thepropertiesofthecompletestructurearethenfoundbyevaluatingtheproperties of the individual finite elements and superposing them appropriately.Theproblemofdefiningthestiffnesspropertiesofanystructureisthusreducedbasicallytotheevaluationofthestiffnessof atypical element.10-11
10-11 Wuhan University of Technology 10.1 Elastic properties In principle, the flexibility or stiffness coefficients associated with any prescribed set of nodal displacements can be obtained by direct application of their definitions. In practice, however, the finiteelement concept, described in Chapter 1, frequently provides the most convenient means for evaluating the elastic properties. By this approach the structure is assumed to be divided into a system of discrete elements which are interconnected only at a finite number of nodal points. The properties of the complete structure are then found by evaluating the properties of the individual finite elements and superposing them appropriately. The problem of defining the stiffness properties of any structure is thus reduced basically to the evaluation of the stiffness of a typical element
Wuhan UniversityofTechnology10.1ElasticpropertiesU(x)EI(x)0.=0,=1Y, (x)W3(x)FIGURE10-4Beamdeflectionsduetounitnodaldisplacementsatleftend.10-12
10-12 Wuhan University of Technology 10.1 Elastic properties FIGURE 10-4 Beam deflections due to unit nodal displacements at left end
Wuhan Universityof Technology10.1 Elastic propertiesThesedisplacementfunctionscouldbetakenasanyarbitraryshapeswhichsatisfynodalandinternal continuityrequirements,buttheygenerallyareassumedtobetheshapesdevelopedinauniformbeamsubjectedtothesenodaldisplacements.Thesearecubichermitianpolynomialswhichmaybeexpressedasb1() =1-3Tb3(r) =a(三) -2 ()2(c) = 3(a)= (-1)10-13
10-13 Wuhan University of Technology 10.1 Elastic properties These displacement functions could be taken as any arbitrary shapes which satisfy nodal and internal continuity requirements, but they generally are assumed to be the shapes developed in a uniform beam subjected to these nodal displacements. These are cubic hermitian polynomials which may be expressed as
Wuhan Universityof Technology10.1 Elastic propertiesWiththesefourinterpolationfunctions,thedeflectedshapeoftheelementcannowbeexpressedintermsofitsnodaldisplacements:v() = 1()Vi+2(α)V2+ 3()V3+4()Vwherethenumbereddegreesoffreedomareasfollows:01Ua02Ub00V310-14
10-14 Wuhan University of Technology 10.1 Elastic properties With these four interpolation functions, the deflected shape of the element can now be expressed in terms of its nodal displacements: where the numbered degrees of freedom are as follows: