Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.4Application to the Solution of Statically Indeterminate Systems 135

Assumingthatthefuselageframeislinearlyelastic,wehave,fromEqs.(ii)and(iii),

2

∫π

0

M

EI

∂M

∂MA

Rdθ= 2

∫π

0

M

EI

∂M

∂NA

Rdθ= 0 (vi)

SubstitutingfromEqs.(iv)and(v)intoEq.(vi)givestwosimultaneousequations

−

PR

2 π

=MA+NAR (vii)

−

7 PR

8 π

=MA+

3

2

NAR (viii)

Theseequationsmaybewritteninmatrixformasfollows:

PR

π

{

− 1 / 2

− 7 / 8

}

=

[

1 R

13 R/ 2

]{

MA

NA

}

(ix)

sothat
{
MA
NA

}

=

PR

π

[

1 R

13 R/ 2

]− 1 {

− 1 / 2

− 7 / 8

}

or
{
MA
NA

}

=

PR

π

[

3 − 2

− 2 /R 2 /R

]{

− 1 / 2

− 7 / 8

}

whichgives

MA=

PR

4 π

NA=

− 3 P

4 π
ThebendingmomentdistributionfollowsfromEq.(iv)andis

M=

PR

2 π

(

1 −

1

2

cosθ−θsinθ

)

(x)

ThesolutionofEq.(ix)involvestheinversionofthematrix
[
1 R
13 R/ 2

]

which may be carried out using any of the standard methods detailed in texts on matrix analysis. In
thisexample,Eqs.(vii)and(viii)areclearlymosteasilysolveddirectly;however,thematrixapproach
illustratesthetechniqueandservesasausefulintroductiontothemoredetaileddiscussioninChapter6.

Example 5.6
Atwo-cellfuselagehascircularframeswitharigidlyattachedstraightmemberacrossthemiddle.The
bendingstiffnessofthelowerhalfoftheframeis2EI,whilethatoftheupperhalfandalsothestraight
memberisEI.
Calculatethedistributionofthebendingmomentineachpartoftheframefortheloadingsystem
showninFig.5.17(a).Illustrateyouranswerbymeansofasketchandshowclearlythebendingmoment