Wuhan University of Technology10.1 Elastic propertiesMatriceswhich satisfythis condition,where v or pis any arbitrary nonzero vectoraresaidtobepositivedefinite;positivedefinitematrices(andconsequentlytheflexibilityandstiffnessmatricesofa stable structure)arenonsingularand canbeinverted.InvertingthestiffnessmatrixandpremultiplyingbothsidesofEq.(96)bytheinverseleadstoKwhichuponcomparisonwithEg.(105)demonstratesthattheflexibilitymatrixistheinverseofthestiffnessmatrixk-1 =f10-6
10-6 Wuhan University of Technology 10.1 Elastic properties Matrices which satisfy this condition, where v or p is any arbitrary nonzero vector, are said to be positive definite; positive definite matrices (and consequently the flexibility and stiffness matrices of a stable structure) are nonsingular and can be inverted. Inverting the stiffness matrix and premultiplying both sides of Eq. (96) by the inverse leads to which upon comparison with Eq. (105) demonstrates that the flexibility matrix is the inverse of the stiffness matrix:
WuhanUniversityof Technology1o.1 ElasticpropertiesBetti'slaw.Apropertywhichisveryimportant instructuraldynamicsanalysiscanbederivedbyapplyingtwosetsof loadstoastructureinreversesequenceandcomparingexpressionsfortheworkdoneinthetwocases.Consider,forexample,thetwodifferentloadsystemsandtheirresultingdisplacementsshownin Fig. 103. If the loads a are applied first followed by loads b, the work done willbeasfollows:Load system a:Load system b:PlaP2aP3aPib-P2b-P3bDeflections b:Deflections a:U3bUlaU2aU3aUih-U2b10-7
10-7 Wuhan University of Technology 10.1 Elastic properties Betti's law. A property which is very important in structuraldynamics analysis can be derived by applying two sets of loads to a structure in reverse sequence and comparing expressions for the work done in the two cases. Consider, for example, the two different load systems and their resulting displacements shown in Fig. 103. If the loads a are applied first followed by loads b, the work done will be as follows:
Wuhan University of TechnologyC10.1 Elastic propertiesCase 1:Waa=EpiaUia=IpTvaWhb + Wab = p,Tv+p.TvbWi = Waa +Wbb+Wab=p.Tva+p,Tv+ pTvCase 2:Wab=p,TvWaa+Wba=p.Tva+pTvaW2=Whb+ Waa+Wba=pTv +pTva+p,Tv10-8
10-8 Wuhan University of Technology 10.1 Elastic properties Case 1: Case 2:
WuhanUniversityof Technology10.1 ElasticpropertiesThedeformationofthe structureisindependentof theloadingsequence,however;thereforethestrainenergyandhencealsotheworkdonebytheloadsis the same in both these cases; that is, Wi = W2. From a comparison of Eqs.(1011)and (1012) it maybe concluded that Wab= Wbai thusTVp=PbEquation(1013)isanexpressionofBetti'slaw;itstatesthattheworkdonebyonesetof loadsonthedeflectionsduetoasecondsetofloadsisequaltotheworkofthesecondsetofloadsactingonthedeflectionsduetothefirst.=pTfpp.Tfp.10-9
10-9 Wuhan University of Technology 10.1 Elastic properties The deformation of the structure is independent of the loading sequence, however; therefore the strain energy and hence also the work done by the loads is the same in both these cases; that is, W1 = W 2. From a comparison of Eqs. (1011) and (1012) it may be concluded that Wab = Wba; thus Equation (1013) is an expression of Betti's law; it states that the work done by one set of loads on the deflections due to a second set of loads is equal to the work of the second set of loads acting on the deflections due to the first
Wuhan University of Technology10.1 Elastic propertiesItis evidentthatf=fThus the flexibility matrix must be symmetric; that is, f, = fi,.. This is anexpressionofMaxwell'slawofreciprocaldeflections.SubstitutingsimilarlywithEq.(96)(andnotingthatp=f.)leadstok = kT10-10
10-10 Wuhan University of Technology 10.1 Elastic properties It is evident that Thus the flexibility matrix must be symmetric; that is, . This is an expression of Maxwell's law of reciprocal deflections. Substituting similarly with Eq. (96) (and noting that p = fS) leads to