508 CHAPTER 17 Torsion of Beams
Fig.17.4
Torsion of a rectangular section beam for Example 17.2.
where
δ=∮
ds
tand δOs=∫s0ds
tInEq.(i),
w 0 =0, δ= 2(
b
tb+
a
ta)
and A=abFrom0to1,0≤s 1 ≤b/2andδOs=∫s^10ds 1
tb=
s 1
tbAOs=as 1
4(ii)NotethatδOsandAOsarebothpositive.
SubstitutionforδOsandAOsfromEq.(ii)in(i)showsthatthewarpingdistributioninthewall01,
w 01 ,islinear.Also,
w 1 =T
2 abG2
(
b
tb+
a
ta)[
b/ 2 tb
2 (b/tb+a/ta)−
ab/ 8
ab]
whichgives
w 1 =T
8 abG(
b
tb−
a
ta)
(iii)Theremainderofthewarpingdistributionmaybededucedfromsymmetryandthefactthatthewarping
mustbezeroatpointswheretheaxesofsymmetryandthewallsofthecrosssectionintersect.Itfollows
that
w 2 =−w 1 =−w 3 =w 4