Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

17.1 Torsion of Closed Section Beams 513

TheshearstrainsareobtainedfromEq.(17.1)andare

γa=

T

2 abGta

, γb=

T

2 abGtb

fromwhich

θ=

TL

4 a^2 b^2 G

(

a

ta

+

b

tb

)

Thetotalangleoftwistfromendtoendofthebeamis2θ,therefore,

2 θ

L

=

TL

4 a^2 b^2 G

(

2 a

ta

+

2 b

tb

)

or

dθ

dz

=

T

4 A^2 G

∮

ds

t

asinEq.(17.4).
Substitutingforθineitheroftheexpressionsfortheaxialdisplacementofthecorner1givesthe
warpingw 1 at1.Thus,

w 1 =

a

2

b

L

TL

4 a^2 b^2 G

(

a

ta

+

b

tb

)

−

T

2 abGta

a

2

thatis,

w 1 =

T

8 abG

(

b

tb

−

a

ta

)

as before. It can be seen that the warping of the cross section is produced by a combination of the
displacements caused by twisting and the displacements due to the shear strains; these shear strains
correspondtotheshearstresseswhosevaluesarefixedbystatics.Theangleoftwistmustthereforebe
suchastoensurecompatibilityofdisplacementbetweenthewebsandcovers.

17.1.2 Condition for Zero Warping at a Section

Thegeometryofthecrosssectionofaclosedsectionbeamsubjectedtotorsionmaybesuchthatno
warpingofthecrosssectionoccurs.FromEq.(17.5),weseethatthisconditionariseswhen

δOs

δ

=

AOs

A

or

1

δ

∫s

0

ds

Gt

=

1

2 A

∫s

0

pRds (17.6)