194 CHAPTER 6 Matrix Methods
Thesolutionprocedureisidenticalinoutlinetothatdescribedintheprevioussectionsforskeletal
structures;thedifferenceslieintheidealizationofthestructureintofiniteelementsandthecalculation
ofthestiffnessmatrixforeachelement.Thelatterprocedure,whichingeneraltermsisapplicableto
all finite elements, may be specified in a number of distinct steps. We shall illustrate the method by
establishingthestiffnessmatrixforthesimpleone-dimensionalbeamelementofFig.6.6forwhichwe
havealreadyderivedthestiffnessmatrixusingslope–deflection.
6.8.1 Stiffness Matrix for a Beam Element
Thefirststepistochooseasuitablecoordinateandnodenumberingsystemfortheelementanddefine
itsnodaldisplacementvector{δe}andnodalloadvector{Fe}.Useismadehereofthesuperscripteto
denoteelementvectors,since,ingeneral,afiniteelementpossessesmorethantwonodes.Again,we
arenotconcernedwithaxialorsheardisplacementssothatforthebeamelementofFig.6.6,wehave
{δe}=⎧
⎪⎪
⎨
⎪⎪
⎩
vi
θi
vj
θj⎫
⎪⎪
⎬
⎪⎪
⎭
{Fe}=⎧
⎪⎪
⎨
⎪⎪
⎩
Fy,i
Mi
Fy,j
Mj⎫
⎪⎪
⎬
⎪⎪
⎭
Sinceeachofthesevectorscontainsfourterms,theelementstiffnessmatrix[Ke]willbeoforder4×4.
Inthesecondstep,weselectadisplacementfunctionwhichuniquelydefinesthedisplacementof
all points in the beam element in terms of the nodal displacements. This displacement function may
betakenasapolynomialwhichmustincludefourarbitraryconstantscorrespondingtothefournodal
degreesoffreedomoftheelement.Thus,
v(x)=α 1 +α 2 x+α 3 x^2 +α 4 x^3 (6.54)Equation(6.54)isofthesameformasthatderivedfromelementarybendingtheoryforabeamsubjected
toconcentratedloadsandmomentsandmaybewritteninmatrixformas
{v(x)}=[
1 xx^2 x^3]
⎧
⎪⎪
⎨
⎪⎪
⎩
α 1
α 2
α 3
α 4⎫
⎪⎪
⎬
⎪⎪
⎭
orinabbreviatedformas
{v(x)}=[f(x)]{α} (6.55)Therotationθatanysectionofthebeamelementisgivenby∂v/∂x;therefore,
θ=α 2 + 2 α 3 x+ 3 α 4 x^2 (6.56)