Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

554 CHAPTER 19 Structural Idealization

FromEq.(15.18)andthethirdofEqs.(1.42),

σz,1=

(

My,1Ixx−Mx,1Ixy

IxxIyy−Ixy^2

)

x+

(

Mx,1Iyy−My,1Ixy

IxxIyy−Ixy^2

)

y

εz,0=

1

E

[(

My,0Ixx−Mx,0Ixy

IxxIyy−Ixy^2

)

x+

(

Mx,0Iyy−My,0Ixy

IxxIyy−Ixy^2

)

y

]

wherethesuffixes1and0refertotheunitandactualloadingsystems,andx,yarethecoordinatesof
anypointinthecrosssectionreferredtoacentroidalsystemofaxes.Substitutingforσz,1andεz,0in
Eq.(19.15)andrememberingthat

∫

Ax

(^2) dA=Iyy,∫
Ay
(^2) dA=Ixx,and∫
AxydA=Ixy,wehave
(^) M=

1

E(IxxIyy−Ixy^2 )^2

∫

L

{

(My,1Ixx−Mx,1Ixy)(My,0Ixx−Mx,0Ixy)Iyy

+(Mx,1Iyy−My,1Ixy)(Mx,0Iyy−My,0Ixy)Ixx

+[(My,1Ixx−Mx,1Ixy)(Mx,0Iyy−My,0Ixy)

+(Mx,1Iyy−My,1Ixy)(My,0Ixx−Mx,0Ixy)]Ixy

}

dz

(19.16)

Forasectionhavingeitherxoryaxisasanaxisofsymmetry,Ixy=0,andEq.(19.16)reducesto

(^) M=

1

E

∫

L

(

My,1My,0

Iyy

+

Mx,1Mx,0

Ixx

)

dz (19.17)

Thederivationofanexpressionforthesheardeflectionofthin-walledsectionsbytheunitloadmethod

isachievedinasimilarmanner.BycomparingEq.(19.15),wededucethatthedeflection (^) S,dueto
shearofathin-walledopenorclosedsectionbeamofthicknesst,isgivenby
(^) S=

∫

L

⎛

⎝

∫

sect

τ 1 γ 0 tds

⎞

⎠dz (19.18)

whereτ 1 istheshearstressatanarbitrarypointsaroundthesectionproducedbyaunitloadapplied

atthepointandinthedirection (^) S,andγ 0 istheshearstrainatthearbitrarypointcorrespondingto
theactualloadingsystem.Theintegralinparenthesesistakenoverallthewallsofthebeam.Infact,
boththeappliedandunitshearloadsmustactthroughtheshearcenterofthecrosssection;otherwise
additionaltorsionaldisplacementsoccur.Whereshearloadsactatotherpoints,thesemustbereplaced
byshearloadsattheshearcenterplusatorque.Thethicknesstistheactualskinthicknessandmay
vary around the cross section but is assumed to be constant along the length of the beam. Rewriting
Eq.(19.18)intermsofshearflowsq 1 andq 0 ,weobtain
(^) S=

∫

L

⎛

⎝

∫

sect

q 0 q 1

Gt

ds

⎞

⎠dz (19.19)