Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

6.7Stiffness Matrix for a Uniform Beam 187

whichisoftheform

{F}=[Kij]{δ}

where[Kij]isthestiffnessmatrixforthebeam.
It is possible to write Eq. (6.44) in an alternative form such that the elements of [Kij] are pure
numbers.Thus,
⎧
⎪⎪
⎨
⎪⎪
⎩

Fy,i

Mi/L

Fy,j

Mj/L

⎫

⎪⎪

⎬

⎪⎪

⎭

=

EI

L^3

⎡

⎢

⎢

⎣

12 − 6 − 12 − 6

−64 62

−12 6 12 6

−62 64

⎤

⎥

⎥

⎦

⎧

⎪⎪

⎨

⎪⎪

⎩

vi

θiL

vj

θjL

⎫

⎪⎪

⎬

⎪⎪

⎭

ThisformofEq.(6.44)isparticularlyusefulinnumericalcalculationsforanassemblageofbeamsin
whichEI/L^3 isconstant.
Equation(6.44)isderivedforabeamwhoseaxisisalignedwiththexaxissothatthestiffnessmatrix
definedbyEq.(6.44)isactually[Kij]thestiffnessmatrixreferredtoalocalcoordinatesystem.Ifthe
beamispositionedinthexyplanewithitsaxisarbitrarilyinclinedtothexaxis,thenthexandyaxes
formaglobalcoordinatesystemanditbecomesnecessarytotransformEq.(6.44)toallowforthis.The
procedureissimilartothatforthepin-jointedframeworkmemberofSection6.4inthat[Kij]mustbe
expandedtoallowforthefactthatnodaldisplacements ̄uiandu ̄j,whichareirrelevantforthebeamin
localcoordinates,havecomponentsui,vianduj,vjinglobalcoordinates.Thus,

[Kij]=EI

ui vi θi uj vj θj

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

00 000 0

012 /L^3 − 6 /L^20 − 12 /L^3 − 6 /L^2

0 − 6 /L^24 /L 06 /L^22 /L

00 000 0

0 − 12 /L^36 /L^2012 /L^36 /L^2

0 − 6 /L^22 /L 06 /L^24 /L

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(6.45)

Wemaydeducethetransformationmatrix[T]fromEq.(6.24)ifwerememberthatalthoughuandv
transforminexactlythesamewayasinthecaseofapin-jointedmember,therotationsθremainthe
sameineitherlocalorglobalcoordinates.
Hence,

[T]=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

λμ 0000

−μλ 0000

001000

000 λμ 0

000 −μλ 0

000001

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(6.46)

whereλandμhavepreviouslybeendefined.Thus,

[Kij]=[T]T[Kij][T] (seeSection6.4)