Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

17.1 Torsion of Closed Section Beams 505

Fig.17.3

Sign convention for swept areas.

isvalidforthepuretorsioncase,whereqisconstant.Differentiatingthisexpressionwithrespecttoz,
wehave

∂q

∂z

=Gt

(

∂^2 w

∂z∂s

+

∂^2 vt

∂z^2

)

= 0

or

∂

∂s

(

∂w

∂z

)

+

∂^2 vt

∂z^2

= 0 (17.2)

Intheabsenceofdirectstresses,thelongitudinalstrain∂w/∂z(=εz)iszerosothat

∂^2 vt

∂z^2

= 0

Hence,fromEq.(16.7)

p

d^2 θ

dz^2

+

d^2 u

dz^2

cosψ+

d^2 v

dz^2

sinψ= 0 (17.3)

ForEq.(17.3)toholdforallpointsaroundthesectionwall,inotherwordsforallvaluesofψ

d^2 θ

dz^2

=0,

d^2 u

dz^2

=0,

d^2 v

dz^2

= 0