Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

196 CHAPTER 6 Matrix Methods

whichwewriteas

{ε}=[C]{α} (6.65)

Substitutingfor{α}inEq.(6.65)fromEq.(6.60),wehave

{ε}=[C][A−^1 ]{δe} (6.66)

Stepfiverelatestheinternalstressesintheelementtothestrain{ε}andhence,usingEq.(6.66),to
thenodaldisplacements{δe}.Inourbeamelement,thestressdistributionatanysectiondependsentirely
onthevalueofthebendingmomentMatthatsection.Thus,wemayrepresenta“stateofstress”{σ}
atanysectionbythebendingmomentM,which,fromsimplebeamtheory,isgivenby

M=EI

∂^2 v

∂x^2

or

{σ}=[EI]{ε} (6.67)

whichwewriteas

{σ}=[D]{ε} (6.68)

Thematrix[D]inEq.(6.68)isthe“elasticity”matrixrelating“stress”and“strain.”Inthiscase,[D]
consists of a single term, the flexural rigidityEIof the beam. Generally, however, [D] is of a higher
order.Ifwenowsubstitutefor{ε}inEq.(6.68)fromEq.(6.66),weobtainthe“stress”intermsofthe
nodaldisplacements,thatis,

{σ}=[D][C][A−^1 ]{δe} (6.69)

Theelementstiffnessmatrixisfinallyobtainedinstepsixinwhichwereplacetheinternal“stresses”
{σ}byastaticallyequivalentnodalloadsystem{Fe},therebyrelatingnodalloadstonodaldisplacements
(fromEq.(6.69))anddefiningtheelementstiffnessmatrix[Ke].Thisisachievedbyusingtheprinciple
ofthestationaryvalueofthetotalpotentialenergyofthebeam(seeSection5.8)whichcomprisesthe
internalstrainenergyUandthepotentialenergyVofthenodalloads.Thus,

U+V=

1

2

∫

vol

{ε}T{σ}d(vol)−{δe}T{Fe} (6.70)

SubstitutinginEq.(6.70)for{ε}fromEq.(6.66)and{σ}fromEq.(6.69),wehave

U+V=

1

2

∫

vol

{δe}T[A−^1 ]T[C]T[D][C][A−^1 ]{δe}d(vol)−{δe}T{Fe} (6.71)