Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

17.1 Torsion of Closed Section Beams 511

ComparingFigs.17.5and17.7,itcanbeseenthattheformofthewarpingdistributionisthesame
butthatinthelattercasethecompletedistributionhasbeendisplacedaxially.Theactualvalueofthe
warpingattheoriginforsisfoundusingEq.(16.26).
Thus,

w 0 =

2

2 (ata+btb)

⎛

⎝

∫a

0

w′ 12 tads 1 +

∫b

0

w′ 23 tbds 2

⎞

⎠ (vii)

SubstitutinginEq.(vii)forw′ 12 andw′ 23 fromEqs.(iv)and(vi),respectively,andevaluatinggive

w 0 =−

T

8 abG

(

b

tb

−

a

ta

)

(viii)

Subtractingthisvaluefromthevaluesofw′ 1 (= 0 )andw′ 2 (=−T(b/tb−a/ta)/ 4 abG),wehave

w 1 =

T

8 abG

(

b

tb

−

a

ta

)

, w 2 =−

T

8 abG

(

b

tb

−

a

ta

)

asbefore.Notethatsettingw 0 =0inEq.(i)impliesthatw 0 ,theactualvalueofwarpingattheorigin
fors,hasbeenaddedtoallwarpingdisplacements.Thisvaluemustthereforebesubtractedfromthe
calculated warping displacements (i.e., those based on an arbitrary choice of origin) to obtain true
values.
Itisinstructiveatthisstagetoexaminethemechanicsofwarpingtoseehowitarises.Supposethat
eachendoftherectangularsectionbeamofExample17.2rotatesthroughoppositeanglesθ,givinga
totalangleoftwist2θalongitslengthL.Thecorner1atoneendofthebeamisdisplacedbyamounts
aθ/2 verticallyandbθ/2horizontally,asshowninFig.17.8.Considernowthedisplacementsofthe
webandcoverofthebeamduetorotation.FromFigs.17.8and17.9(a)and(b),itcanbeseenthatthe
anglesofrotationofthewebandthecoverare,respectively,

φb=(aθ/ 2 )/(L/ 2 )=aθ/L

and

φa=(bθ/ 2 )/(L/ 2 )=bθ/L

Theaxialdisplacementsofthecorner1inthewebandcoverarethen

b

2

aθ

L

,

a

2

bθ

L

respectively,asshowninFig.17.9(a)and(b).Inadditiontodisplacementsproducedbytwisting,the
websandcoversaresubjectedtoshearstrainsγbandγacorrespondingtotheshearstresssystemgiven
byEq.(17.1).Duetoγb,theaxialdisplacementofcorner1inthewebisγbb/2inthepositivezdirection,
whileinthecoverthedisplacementisγaa/2inthenegativezdirection.Notethattheshearstrainsγb