526 CHAPTER 17 Torsion of Beams
P.17.11The thin-walled section shown in Fig. P.17.11 is symmetrical about thexaxis. The thicknesst 0 of the
centerweb34isconstant,whilethethicknessoftheotherwallsvarieslinearlyfromt 0 atpoints3and4tozeroat
theopenends1,6,7,and8.
Determine the St. Venant torsion constantJfor the section and also the maximum value of the shear stress
duetoatorqueT.IfthesectionisconstrainedtotwistaboutanaxisthroughtheoriginO,plottherelativewarping
displacementsofthesectionperunitrateoftwist.
Ans. J= 4 at^30 /3,τmax=± 3 T/ 4 at 02 ,w 1 =+a^2 ( 1 + 2√
2 ).
w 2 =+√
2 a^2 ,w 7 =−a^2 ,w 3 =0.Fig. P.17.11 Fig. P.17.12P.17.12Thethin-walledsectionshowninFig.P.17.12isconstrainedtotwistaboutanaxisthroughR,thecenter
of the semicircular wall 34. Calculate the maximum shear stress in the section per unit torque and the warping
distribution per unit rate of twist. Also compare the value of warping displacement at the point 1 with that
corresponding to the section being constrained to twist about an axis through the point O, and state what effect
thismovementhasonthemaximumshearstressandthetorsionalstiffnessofthesection.
Ans. Maximumshearstressis±0.42/rt^2 perunittorque.w 03 =+r^2 θ,w 32 =+
r
2(πr+ 2 s 1 ),w 21 =−
r
2( 2 s 2 −5.142r).WithcenteroftwistatO 1 w 1 =−0.43r^2 .Maximumshearstressisunchanged,torsionalstiffnessincreased,since
warpingreduced.