172 CHAPTER 6 Matrix Methods
Bysuperpositionofthesetwoconditions,weobtainrelationshipsbetweentheappliedforcesandthe
nodaldisplacementsforthestatewhenu 1 =u 1 andu 2 =u2.Thus,
Fx,1=ku 1 −ku 2
Fx,2=−ku 1 +ku 2}
(6.5)
WritingEq.(6.5)inmatrixform,wehave
{
Fx,1
Fx,2}
=
[
k −k
−kk]{
u 1
u 2}
(6.6)
andbycomparingwithEq.(6.1),weseethatthestiffnessmatrixforthisspringelementis
[K]=
[
k −k
−kk]
(6.7)
whichisasymmetricmatrixoforder2×2.
6.3 StiffnessMatrixforTwoElasticSpringsinLine.................................................
Bearinginmindtheresultsoftheprevioussection,weshallnowproceed,initiallybyasimilarprocess,
toobtainthestiffnessmatrixofthecompositetwo-springsystemshowninFig.6.2.Thenotationand
signconventionfortheforcesandnodaldisplacementsareidenticaltothosespecifiedinSection6.1.
First,letussupposethatu 1 =u 1 andu 2 =u 3 =0.Bycomparingthesingle-springcase,wehave
Fx,1=kau 1 =−Fx,2 (6.8)but,inaddition,Fx,3=0,sinceu 2 =u 3 =0.
Second,weputu 1 =u 3 =0andu 2 =u 2 .Clearly,inthiscase,themovementofnode2takesplace
againstthecombinedspringstiffnesseskaandkb.Hence,
Fx,2=(ka+kb)u 2
Fx,1=−kau 2 , Fx,3=−kbu 2}
(6.9)
Hence,thereactiveforceFx,1(=−kau 2 )isnotdirectlyaffectedbythefactthatnode2isconnectedto
node3,butitisdeterminedsolelybythedisplacementofnode2.Similarconclusionsaredrawnforthe
reactiveforceFx,3.
Fig.6.2
Stiffness matrix for a two-spring system.