6.7Stiffness Matrix for a Uniform Beam 185
orinabbreviatedform
{F}=[T]{F}
Thederivationof[Kij]foramemberofaspaceframeproceedsonidenticallinestothatfortheplane
framemember.Thus,asbefore
[Kij]=[T]T[Kij][T]
Substitutingfor[T]and[Kij]fromEqs.(6.36)and(6.33)gives
[Kij]=
AE
L
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
λx ̄^2 λx ̄μ ̄x λx ̄νx ̄ −λx ̄^2 −λx ̄μ ̄x −λ ̄xν ̄x
λ ̄xμ ̄x μx^2 ̄ μx ̄νx ̄ −λ ̄xμ ̄x −μ^2 ̄x −μx ̄νx ̄
λ ̄xν ̄x μ ̄xνx ̄ νx^2 ̄ −λx ̄ν ̄x −μ ̄xν ̄x −νx^2 ̄
−λ^2 x ̄ −λ ̄xμ ̄x −λx ̄νx ̄ λx ̄^2 λx ̄μx ̄ λ ̄xν ̄x
−λ ̄xμ ̄x −μx ̄^2 −μx ̄ν ̄x λx ̄μx ̄ μx ̄^2 μ ̄xνx ̄
−λ ̄xνx ̄ −μx ̄νx ̄ −νx ̄^2 λx ̄ν ̄x μ ̄xν ̄x νx^2 ̄
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(6.37)
AllthesuffixesinEq.(6.37)are ̄xsothatwemayrewritetheequationinsimplerform,namely
[Kij]=
AE
L
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
λ^2
. SYM
λμ μ^2
λν μν ν^2
··········································
−λ^2 −λμ −λν
. λ^2
−λμ −μ^2 −μν
. λμ μ^2
−λν −μν −ν^2
. λν μν ν^2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(6.38)
whereλ,μ,andνarethedirectioncosinesbetweenthex,y,z,and ̄xaxes.
Thecompletestiffnessmatrixforaspaceframeisassembledfromthememberstiffnessmatricesin
asimilarmannertothatfortheplaneframeandthesolutioncompletedasbefore.
6.7 StiffnessMatrixforaUniformBeam..............................................................
Ourdiscussionsofarhasbeenrestrictedtostructurescomprisingmemberscapableofresistingaxial
loads only. Many structures, however, consist of beam assemblies in which the individual members
resistshearandbendingforces,inadditiontoaxialloads.Weshallnowderivethestiffnessmatrixfora
uniformbeamandconsiderthesolutionofrigid-jointedframeworksformedbyanassemblyofbeams
orbeamelementsastheyaresometimescalled.
Figure6.6showsauniformbeamijofflexuralrigidityEIandlengthLsubjectedtonodalforces
Fy,i,Fy,jandnodalmomentsMi,Mjinthexyplane.Thebeamsuffersnodaldisplacementsandrotations