186 CHAPTER 6 Matrix Methods
Fig.6.6
Forces and moments on a beam element.
vi,vj,andθi,θj.Wedonotincludeaxialforceshere,sincetheireffectshavealreadybeendetermined
inourinvestigationofpin-jointedframeworks.
The stiffness matrix [Kij] may be built up by considering various deflected states for the
beamandsuperimposingtheresults,aswedidinitiallyforthespringassembliesshowninFigs.6.1and
6.2,oritmaybewrittendowndirectlyfromthewell-knownbeamslope–deflectionequations[Ref.3].
Weshalladoptthelatterprocedure.Fromslope–deflectiontheory,wehave
Mi=−6 EI
L^2
vi+4 EI
L
θi+6 EI
L^2
vj+2 EI
L
θj (6.39)and
Mj=−6 EI
L^2
vi+2 EI
L
θi+6 EI
L^2
vj+4 EI
L
θj (6.40)Also,consideringverticalequilibrium,weobtain
Fy,i+Fy,j= 0 (6.41)andfrommomentequilibriumaboutnodej,wehave
Fy,iL+Mi+Mj= 0 (6.42)Hence,thesolutionofEqs.(6.39)through(6.42)gives
−Fy,i=Fy,j=−12 EI
L^3
vi+6 EI
L^2
θi+12 EI
L^3
vj+6 EI
L^2
θj (6.43)ExpressingEqs.(6.39),(6.40),and(6.43)inmatrixformyields
⎧
⎪⎪
⎪⎨⎪⎪
⎪⎩Fy,i
Mi
Fy,j
Mj⎫
⎪⎪
⎪⎬
⎪⎪
⎪⎭
=EI
⎡
⎢
⎢
⎢
⎣
12 /L^3 − 6 /L^2 − 12 /L^3 − 6 /L^2
− 6 /L^24 /L 6 /L^22 /L
− 12 /L^36 /L^212 /L^36 /L^2
− 6 /L^22 /L 6 /L^24 /L
⎤
⎥
⎥
⎥
⎦
⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩
vi
θi
vj
θj