Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

176 CHAPTER 6 Matrix Methods

6.4 MatrixAnalysisofPin-jointedFrameworks......................................................

The formation of stiffness matrices for pin-jointed frameworks and the subsequent determination of
nodaldisplacementsfollowasimilarpatterntothatdescribedforaspringassembly.Amemberinsuch
aframeworkisassumedtobecapableofcarryingaxialforcesonlyandobeysauniqueforce–deformation
relationshipgivenby

F=

AE

L

δ

whereFistheforceinthemember,δitschangeinlength,Aitscross-sectionalarea,Litsunstrained
length,andEitsmodulusofelasticity.Thisexpressionisseentobeequivalenttothespring–displacement
relationshipsofEqs.(6.3)and(6.4)sothatwemayimmediatelywritedownthestiffnessmatrixfora
memberbyreplacingkbyAE/LinEq.(6.7).Thus,

[K]=

[

AE/L −AE/L

−AE/LAE/L

]

or

[K]=

AE

L

[

1 − 1

− 11

]

(6.20)

sothatforamemberalignedwiththexaxis,joiningnodesiandjsubjectedtonodalforcesFx,iand
Fx,j,wehave
{
Fx,i
Fx,j

}

=

AE

L

[

1 − 1

− 11

]{

ui

uj

}

(6.21)

Thesolutionproceedsinasimilarmannertothatgivenintheprevioussectionforaspringorspring
assembly.However,somemodificationisnecessary,sinceframeworksconsistofmemberssetatvarious
angles to one another. Figure 6.3 shows a member of a framework inclined at an angleθto a set of

Fig.6.3

Local and global coordinate systems for a member of a plane pin-jointed framework.