Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

6.4Matrix Analysis of Pin-jointed Frameworks 177

arbitraryreferenceaxesx,y.Weshallrefereverymemberoftheframeworktothisglobalcoordinate
system,asitisknown,whenweareconsideringthecompletestructure,butweshalluseamemberor
localcoordinate system ̄x,y ̄when considering individual members. Nodal forces and displacements
referredtolocalcoordinatesarewrittenasF ̄, ̄u,andsoonsothatEq.(6.21)becomes,intermsoflocal
coordinates,
{
Fx,i
Fx,j

}

=

AE

L

[

1 − 1

− 11

]{

ui

uj

}

(6.22)

wheretheelementstiffnessmatrixiswritten[Kij].
In Fig. 6.3, externalforcesFx,iandFx,jare applied to nodesiandj. It should be notedthatFy,i
andFy,jdo not exist, since the member can only support axial forces. However,Fx,iandFx,jhave
componentsFx,i,Fy,iandFx,j,Fy,j, respectively, so that only two force components appear for the
memberintermsoflocalcoordinates,whereasfourcomponentsarepresentwhenglobalcoordinates
areused.Therefore,ifwearetotransferfromlocaltoglobalcoordinates,Eq.(6.22)mustbeexpanded
toanorderconsistentwiththeuseofglobalcoordinates:

⎧

⎪⎪

⎨

⎪⎪

⎩

Fx,i

Fy,i

Fx,j

Fy,j

⎫

⎪⎪

⎬

⎪⎪

⎭

=

AE

L

⎡

⎢

⎢

⎣

10 − 10

00 00

−10 10

00 00

⎤

⎥

⎥

⎦

⎧

⎪⎪

⎨

⎪⎪

⎩

ui

vi

uj

vj

⎫

⎪⎪

⎬

⎪⎪

⎭

(6.23)

Equation (6.23) does not change the basic relationship betweenFx,i,Fx,jandui,ujas defined in
Eq.(6.22).
FromFig.6.3,weseethat

Fx,i=Fx,icosθ+Fy,isinθ

Fy,i=−Fx,isinθ+Fy,icosθ

and

Fx,j=Fx,jcosθ+Fy,jsinθ

Fy,j=−Fx,jsinθ+Fy,jcosθ

Writingλforcosθandμforsinθ,weexpresstheprecedingequationsinmatrixformas
⎧
⎪⎪
⎨
⎪⎪
⎩

Fx,i

Fy,i

Fx,j

Fy,j

⎫

⎪⎪

⎬

⎪⎪

⎭

=

⎡

⎢

⎢

⎣

λμ 00

−μλ 00

00 λμ

00 −μλ

⎤

⎥

⎥

⎦

⎧

⎪⎪

⎨

⎪⎪

⎩

Fx,i

Fy,i

Fx,j

Fy,j

⎫

⎪⎪

⎬

⎪⎪

⎭

(6.24)

or,inabbreviatedform,

{F}=[T]{F} (6.25)