Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

6.2 Stiffness Matrix for an Elastic Spring 171

where[K]isasymmetricmatrixoftheform

[K]=

⎡

⎢

⎢

⎣

k 11 k 12 ··· k 1 n

k 21 k 22 ··· k 2 n

··· ··· ··· ···

kn 1 kn 2 ··· knn

⎤

⎥

⎥

⎦ (6.2)

Theelementkij(i.e.,theelementlocatedinrowiandincolumnj)isknownasthestiffnessinfluence
coefficient(notekij=kji). Once the stiffness matrix [K] has been formed, the complete solution to a
problem follows from routine numerical calculations that are carried out, in most practical cases, by
computer.

6.2 StiffnessMatrixforanElasticSpring.............................................................

Theformationofthestiffnessmatrix[K]isthemostcrucialstepinthematrixsolutionofanystructural
problem.Weshallshowinthesubsequentworkhowthestiffnessmatrixforacompletestructuremay
be built up from a consideration of the stiffness of its individual elements. First, however, we shall
investigatetheformationof[K]forasimplespringelementwhichexhibitsmanyofthecharacteristics
ofanactualstructuralmember.
ThespringofstiffnesskshowninFig.6.1isalignedwiththexaxisandsupportsforcesFx,1and
Fx,2atitsnodes1and2wherethedisplacementsareu 1 andu 2 .Webuildupthestiffnessmatrixfor
thissimplecasebyexaminingdifferentstatesofnodaldisplacement.First,weassumethatnode2is
preventedfrommovingsuchthatu 1 =u 1 andu 2 =0.Hence,

Fx,1=ku 1

andfromequilibrium,weseethat

Fx,2=−Fx,1=−ku 1 (6.3)

whichindicatesthatFx,2hasbecomeareactiveforceintheoppositedirectiontoFx,1.Second,wetake
thereversecasewhereu 1 =0andu 2 =u 2 andobtain

Fx,2=ku 2 =−Fx,1 (6.4)

Fig.6.1

Determination of stiffness matrix for a single spring.