6.3Stiffness Matrix for Two Elastic Springs in Line 173Finally,wesetu 1 =u 2 =0,u 3 =u 3 andobtainFx,3=kbu 3 =−Fx,2
Fx,1= 0}
(6.10)
Superimposingthesethreedisplacementstates,wehave,fortheconditionu 1 =u 1 ,u 2 =u 2 ,u 3 =u 3 ,Fx,1=kau 1 −kau 2
Fx,2=−kau 1 +(ka+kb)u 2 −kbu 3
Fx,3=−kbu 2 +kbu 3⎫
⎬
⎭
(6.11)
WritingEqs.(6.11)inmatrixformgives
⎧
⎨
⎩
Fx,1
Fx,2
Fx,3⎫
⎬
⎭
=
⎡
⎣
ka −ka 0
−ka ka+kb −kb
0 −kb kb⎤
⎦
⎧
⎨
⎩
u 1
u 2
u 3⎫
⎬
⎭
(6.12)
ComparingEq.(6.12)withEq.(6.1)showsthatthestiffnessmatrix[K]ofthistwo-springsystemis
[K]=
⎡
⎣
ka −ka 0
−ka ka+kb −kb
0 −kb kb⎤
⎦ (6.13)
Equation(6.13)isasymmetricmatrixoforder3×3.
It is important to note that the order of a stiffness matrix may be predicted from a knowledge of
thenumberofnodalforcesanddisplacements.Forexample,Eq.(6.7)isa2×2matrixconnectingtwo
nodalforceswithtwonodaldisplacements;Eq.(6.13)isa3×3matrixrelatingthreenodalforcesto
threenodaldisplacements.Wededucethatastiffnessmatrixforastructureinwhichnnodalforcesrelate
tonnodaldisplacementswillbeofordern×n.Theorderofthestiffnessmatrixdoesnot,however,
bearadirectrelationtothenumberofnodesinastructure,sinceitispossibleformorethanoneforce
tobeactingatanyonenode.
Sofarwehavebuiltupthestiffnessmatricesforthesingle-andtwo-springassembliesbyconsidering
various states of displacement in each case. Such a process would clearly become tedious for more
complexassembliesinvolvingalargenumberofsprings,soashorter,alternativeprocedureisdesirable.
FromourremarksintheprecedingparagraphandbyreferencetoEq.(6.2),wecouldhavededucedat
theoutsetoftheanalysisthatthestiffnessmatrixforthetwo-springassemblywouldbeoftheform
[K]=
⎡
⎣
k 11 k 12 k 13
k 21 k 22 k 23
k 31 k 32 k 33⎤
⎦ (6.14)
Theelementk 11 ofthismatrixrelatestheforceatnode1tothedisplacementatnode1andsoon.Hence,
rememberingthestiffnessmatrixforthesinglespring(Eq.(6.7)),wemaywritedownthestiffnessmatrix
foranelasticelementconnectingnodes1and2inastructureas
[K 12 ]=
[
k 11 k 12
k 21 k 22