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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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