Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

6.6 Matrix Analysis of Space Frames 183

Finally,theforcesinthemembersarefoundfromEqs.(6.32),(vii),and(viii)

S 12 =

AE

L

[1 0]

{

u 2 −u 1

v 2 −v 1

}

=−W(compression)

S 13 =

AE

L

[0 1]

{

u 3 −u 1

v 3 −v 1

}

=0(asexpected)

S 23 =

AE

√

2 L

[

−

1

√

2

1

√

2

]{

u 3 −u 2

v 3 −v 2

}

=

√

2 W(tension)

6.5 ApplicationtoStaticallyIndeterminateFrameworks............................................

Thematrixmethodofsolutiondescribedintheprevioussectionsforspringandpin-jointedframework
assembliesiscompletelygeneralandisthereforeapplicabletoanystructuralproblem.Weobservethat
atnostageinExample6.1didthequestionofthedegreeofindeterminacyoftheframeworkarise.It
followsthatproblemsinvolvingstaticallyindeterminateframeworks(andotherstructures)aresolved
in an identical manner to that presented in Example 6.1, and the stiffness matrices for the redundant
membersbeingincludedinthecompletestiffnessmatrixasbefore.

6.6 MatrixAnalysisofSpaceFrames..................................................................

Theprocedureforthematrixanalysisofspaceframesissimilartothatforplanepin-jointedframeworks.
The main difference lies in the transformation of the member stiffness matrices from local to global
coordinates,since,asweseefromFig.6.5,axialnodalforcesFx,iandFx,jhaveeachnowthreeglobal

Fig.6.5

Local and global coordinate systems for a member in a pin-jointed space frame.