510 CHAPTER 17 Torsion of Beams
SubstitutinginEq.(i)andsettingw 0 = 0
w′ 12 =
Tδ
2 abG
(
s 1
δta
−
s 1
4 a
)
(iv)
sothatw′ 12 varieslinearlyfromzeroat1to
w′ 2 =
T
2 abG
2
(
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.