174 CHAPTER 6 Matrix Methods
andfortheelementconnectingnodes2and3as
[K 23 ]=
[
k 22 k 23
k 32 k 33]
(6.16)
Inourtwo-springsystem,thestiffnessofthespringjoiningnodes1and2iskaandthatofthespring
joining nodes 2 and 3 iskb. Therefore, by comparing with Eq. (6.7), we may rewrite Eqs. (6.15)
and(6.16)as
[K 12 ]=
[
ka −ka
−ka ka]
[K 23 ]=
[
kb −kb
−kb kb]
(6.17)
SubstitutinginEq.(6.14)gives
[K]=
⎡
⎢
⎣
ka −ka 0
−ka ka+kb −kb
0 −kb kb⎤
⎥
⎦
which is identical to Eq. (6.13). We see that only thek 22 term (linking the force at node 2 to the
displacementatnode2)receivescontributionsfrombothsprings.Thisresultsfromthefactthatnode2
isdirectlyconnectedtobothnodes1and3,whilenodes1and3areeachjoineddirectlyonlytonode2.
Also,theelementsk 13 andk 31 of[K]arezero,sincenodes1and3arenotdirectlyconnectedandare
thereforenotaffectedbyeachother’sdisplacement.
Theformationofastiffnessmatrixforacompletestructurethusbecomesarelativelysimplematter
of the superposition of individual or element stiffness matrices. The procedure may be summarized
asfollows:termsoftheformkiionthemaindiagonalconsistofthesumofthestiffnessesofallthe
structuralelementsmeetingatnodei,whileoff-diagonaltermsoftheformkijconsistofthesumofthe
stiffnessesofalltheelementsconnectingnodeitonodej.
Anexaminationofthestiffnessmatrixrevealsthatitpossessescertainproperties.Forexample,the
sumoftheelementsinanycolumniszero,indicatingthattheconditionsofequilibriumaresatisfied.
Also,thenonzerotermsareconcentratedneartheleadingdiagonal,whileallthetermsintheleading
diagonalarepositive;thelatterpropertyderivesfromthephysicalbehaviorofanyactualstructurein
whichpositivenodalforcesproducepositivenodaldisplacements.
Further inspection of Eq. (6.13) shows that its determinant vanishes. As a result the stiffness
matrix [K] is singular and its inverse does not exist. We shall see that this means that the associ-
ated set of simultaneous equations for the unknown nodal displacements cannot be solved for the
simple reason that we have placed no limitation on any of the displacementsu 1 ,u 2 ,oru 3. Thus,
the application of external loads results in the system moving as a rigid body. Sufficient bound-
ary conditions must therefore be specified to enable the system to remain stable under load. In this
particular problem, we shall demonstrate the solution procedure by assuming that node 1 is fixed—
thatis,u 1 =0.