170 CHAPTER 6 Matrix Methods
Amoredifficulttypeofstructuretoidealizeisthecontinuumstructure;inthiscategoryaredams,
plates,shells,and,obviously,aircraftfuselageandwingskins.Amethod,extendingthematrixtechnique
forskeletalstructures,ofrepresentingcontinuabyanydesirednumberofelementsconnectedattheir
nodeswasdevelopedbyCloughetal.[Ref.2]attheBoeingAircraftCompanyandtheUniversityof
BerkeleyinCalifornia.Theelementsmaybeofanydesiredshape,butthesimplest,usedinplanestress
problems,arethetriangularandquadrilateralelements.Weshalldiscussthefiniteelementmethod,as
itisknown,ingreaterdetaillater.
Initially, we shall develop the matrix stiffness method of solution for simple skeletal and beam
structures.Thefundamentalsofmatrixalgebraareassumed.
6.1 Notation...............................................................................................
Generally, we shall consider structures subjected to forces, Fx,1,Fy,1,Fz,1,Fx,2,Fy,2,Fz,2,...,
Fx,n,Fy,n,Fz,n,atnodes1,2,...,natwhichthedisplacementsareu 1 ,v 1 ,w 1 ,u 2 ,v 2 ,w 2 ,...,un,vn,wn.
The numerical suffixes specify nodes, while the algebraic suffixes relate the direction of the forces
toanarbitrarysetofaxes,x,y,z.Nodaldisplacementsu,v,wrepresentdisplacementsinthepositive
directionsofthex,y,andzaxes,respectively.Theforcesandnodaldisplacementsarewrittenascolumn
matrices(alternativelyknownascolumnvectors)
⎧
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩Fx,1
Fy,1
Fz,1
Fx,2
Fy,2
Fz,2
..
.
Fx,n
Fy,n
Fz,n⎫
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎭
⎧
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
u 1
v 1
w 1
u 2
v 2
w 2
..
.
un
vn
wn⎫
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎭
which,whenonceestablishedforaparticularproblem,maybeabbreviatedto
{F}{δ}Thegeneralizedforcesystem{F}cancontainmomentsMandtorquesTinadditiontodirectforces,
in which case{δ}includes rotationsθ. Therefore, in referring simply to a nodal force system, we
imply the possible presence of direct forces, moments, and torques, while the corresponding nodal
displacementscanbetranslationsandrotations.Foracompletestructure,thenodalforcesandnodal
displacementsarerelatedthroughastiffnessmatrix[K].Weshallseethat,ingeneral,
{F}=[K]{δ} (6.1)