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Torsion
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Circular Shaft
For the shaft shown below ,Due to the puretorque about its polar axis through point C,
In the transverse direction:
Take an element at distance r from the centerC there will be shear stresses perpendicular to
the radius r, so point A will move to point Bthrough an angle ,we assume that points lingon particular radius will remain in the sameradius after twisting which represents the
angle of twist , so!""#$%
&here
r
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In the longitudinal direction :
Due to the complementar shear stress A (ineoriginall )A will twist to )B through an angle whichrepresents the shear strain so ,
!!"#*%
&here
From #$% +#*%
But
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The torque is equal to the sum of the
moments of the tangential stress onthe element
T
&here - polar moment of inertia
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Solid Shaft
.olar moment of inertia ,
/aximum shear stress ,
at r-
Hollow shaft.olar moment of inertia ,
/aximum shear stress ,
at r-
Torsional stiness ,k
0t1s the torque per unit radian twist-
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Strain energy in torsion
Total strain energ of a shaft of alength ( under the action of a torque
Tis the work done in twisting
Solid shaft
0n terms of maximum shear stress
2 volume
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• Hollow shaft•
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Torsion of Thin TubularSections
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Bredt-Batho theory
From the equilibrium of the complementar shear stress on
.s and 34
0f dis an element around the circumference
T
k
&here
h- is the perpendicular distance from o to
T-*kA
&hereA- area enclosed b the mean circumference
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The strain energ of a length ( of the tube is
k-T5*A
0f t is constant
&here 6-mean circumference
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7xample8
9how that for uniform hollowtube of outside and inside Dand d ,Bredt: Batho theor
under estimate the maximumshear stress due to a giventorque b ;"
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?ormal theor for hollow shaft
i"e that is < = than
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@sing Bredt : Batho
#
!!""#i%
@sing the normal method
!!!"#ii%
The ratio between #i%+#ii% is ;"< i"ethe dierence of about ;"
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Torsion of Thin alled Cellular
Section
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Consider the twin celled shown below ,
the mean area of the two cells A$+A*"ifthe length ABC is of uniform thickness t$and stress , CDA of thickness t* and stress
From the equilibrium of complementarshear stress
!!"#$%
Total torque adding for two cells
!!#*%
For each cell to nd
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