Mechanical Engineering Principles

TWISTING OF SHAFTS 121

Rigid

2 R A

B

T

L

q

g

Figure 10.1

However, from equation (1.1, page 12),γ=

τ

G

Hence,

(τ

G

)

L=Rθ

or

τ

R

=

Gθ

L

( 10. 2 )

From equation (10.2), it can be seen that the shear
stressτ is dependent on the value ofRand it will
be a maximum on the outer surface of the shaft. On
the outer surface of the shaftτwill act as shown in
Figure 10.2.

AB

t

t

Figure 10.2

For any radiusr,

τ

r

=

Gθ

L

( 10. 3 )

The shaft in Figure 10.2 is said to be in a state of
pure shear on these planes, as these shear stresses
will not be accompanied by direct or normal stress.
Consider an annular element of the shaft, as shown
in Figure 10.3.
The torqueTcauses constant value shearing stresses
on the thin walled annular element shown in
Figure 10.4.

L

R

dr

r

Figure 10.3

t

t

t

t t

t

t

t

dr

r

Figure 10.4 Annular element

The elemental torque δT due to these shearing

stressesτat the radiusris given

by: δT=τ×( 2 πrdr)r

and the total torque T=

∑

δT

or T=

∫R

0

τ( 2 πr^2 )dr( 10. 4 )

However, from equation (10.3),

τ=

Gθ

L

r

Therefore, T=

∫R

0

Gθ

L

r( 2 πr^2 )dr

ButG,θ,Land 2πdo not vary withr,

hence, T=

Gθ

L

( 2 π)

∫R

0

r^3 dr

=

Gθ

L

( 2 π)

[

r^4

4

]R

0

=

Gθ

L

πR^4

2