184 CHAPTER 6 Matrix Methods
componentsFx,i,Fy,i,Fz,iandFx,j,Fy,j,Fz,j,respectively.Thememberstiffnessmatrixreferredtoglobal
coordinatesisthereforeoftheorder6×6sothat[Kij]ofEq.(6.22)mustbeexpandedtothesameorder
toallowforthis.Hence,
[Kij]=AE
L
⎡u ̄i v ̄i w ̄i u ̄j ̄vjw ̄j
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣100 − 100
000 000
000 000
−100 100
000 000
000 000
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(6.33)
InFig.6.5,thememberijisoflengthL,cross-sectionalareaA,andmodulusofelasticityE.Global
andlocalcoordinatesystemsaredesignatedasforthetwo-dimensionalcase.Further,wesupposethat
θxx ̄=anglebetweenxandx ̄
θxy ̄=anglebetweenxandy ̄
..
.
θzy ̄=anglebetweenzand ̄y
..
.Therefore,nodalforcesreferredtothetwosystemsofaxesarerelatedasfollows:
Fx=Fxcosθxx ̄+Fycosθx ̄y+Fzcosθx ̄z
Fy=Fxcosθy ̄x+Fycosθyy ̄+Fzcosθyz ̄
Fz=Fxcosθzx ̄+Fycosθzy ̄+Fzcosθz ̄z⎫
⎪⎬
⎪⎭
(6.34)
Writing
λx ̄=cosθx ̄x, λ ̄y=cosθxy ̄, λz ̄=cosθxz ̄
μx ̄=cosθy ̄x, μ ̄y=cosθyy ̄, μz ̄=cosθyz ̄
ν ̄x=cosθz ̄x, νy ̄=cosθzy ̄, ν ̄z=cosθzz ̄⎫
⎪⎬
⎪⎭
(6.35)
wemayexpressEq.(6.34)fornodesiandjinmatrixformas
⎧
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩
Fx,i
Fy,i
Fz,i
Fx,j
Fy,j
Fz,j⎫
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎭
=
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
λ ̄x μ ̄x νx ̄ 000
λ ̄y μ ̄y νy ̄ 000
λ ̄z μ ̄z νz ̄ 000
000 λx ̄ μx ̄ ν ̄x
000 λ ̄y μy ̄ ν ̄y
000 λ ̄z μz ̄ ν ̄z⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎧
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩
Fx,i
Fy,i
Fz,i
Fx,j
Fy,j
Fz,j