Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

80 CHAPTER 3 Torsion of Solid Sections

Fig.3.9

Torsion of a narrow rectangular strip.

Althoughφdoesnotdisappearalongtheshortedgesofthestripandthereforedoesnotgiveanexact
solution, the actual volume of the membrane differs only slightly from the assumed volume so that
thecorrespondingtorqueandshearstressesarereasonablyaccurate.Also,themaximumshearstress
occursalongthelongsidesofthestripwherethecontoursarecloselyspaced,indicating,inanycase,
thatconditionsintheendregionofthestriparerelativelyunimportant.
ThestressdistributionisobtainedbysubstitutingEq.(3.26)inEqs.(3.2),andthen

τzy= 2 Gx

dθ

dz

τzx= 0 (3.27)

theshearstressvaryinglinearlyacrossthethicknessandattainingamaximum

τzy,max=±Gt

dθ

dz

(3.28)

attheoutsideofthelongedgesaspredicted.ThetorsionconstantJfollowsfromthesubstitutionof
Eq.(3.26)into(3.13),giving

J=

st^3

3

(3.29)

and

τzy,max=

3 T

st^3
These equations represent exact solutions when the assumed shape of the deflected membrane is
theactualshape.Thisconditionarisesonlywhentheratios/tapproachesinfinity;however,forratios
in excess of 10, the error is of the order of only 6 percent. Obviously, the approximate nature of the