510 CHAPTER 17 Torsion of Beams
SubstitutinginEq.(i)andsettingw 0 = 0w′ 12 =Tδ
2 abG(
s 1
δta−
s 1
4 a)
(iv)sothatw′ 12 varieslinearlyfromzeroat1to
w′ 2 =T
2 abG2
(
b
tb+
a
ta)[
a
2 (b/tb+a/ta)ta−
1
4
]
at2.Thus,
w′ 2 =T
4 abG(
a
ta−
b
tb)
or
w′ 2 =−T
4 abG(
b
tb−
a
ta)
(v)Similarly,
w′ 23 =Tδ
2 abG[
1
δ(
a
ta+
s 2
tb)
−
1
4 b(b+s 2 )]
(vi)Thewarpingdistributionthereforevarieslinearlyfromavalue−T(b/tb−a/ta)/ 4 abGat2tozero
at3.Theremainingdistributionfollowsfromsymmetrysothatthecompletedistributiontakestheform
showninFig.17.7.
Fig.17.7
Warping distribution produced by selecting an arbitrary origin fors.