Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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