We were discussing the concept of Power transmitted by a circular solid shaft, Expression
for bending stress and derivation of flexural formula or bending
equation in our previous posts.

Now we are
going further to start a new topic i.e. Power transmitted by a circular hollow
shaft with the help of this post. We recommend to first review the recent post
“Torsion or twisting moment” for having basic information about
the concept of torsion or twisting moment.

###
**So,
what is torsion or twisting moment?**

A shaft will said to be in torsion, if it will be
subjected with two equal and opposite torques applied at its two ends.

When a shaft will be subjected to torsion or
twisting moment, there will be developed shear stress and shear strain in the
shaft material.

We will discuss here one case of a hollow circular
shaft which will be subjected to torsion and we will secure here the expression
for maximum torque transmitted by a hollow circular shaft.

*We have following information from above figure*

Ro = Outer radius of the hollow circular shaft

Ri = Inner radius of the hollow circular shaft

Do = Outer diameter of the hollow circular shaft

Di = Inner diameter of the hollow circular shaft

dr = Thickness of small elementary circular ring

r = Radius of the small elementary of circular ring

q = Shear stress at a radius r from the centre of
the hollow circular shaft

τ = Maximum shear stress at outer surface of shaft

dA = Area of the small elementary of circular ring

dA = 2П x r x dr

Shear stress, at a radius r from the centre, could
be determined as mentioned here

q/r = τ / Ro

q = τ x r/ Ro

Turning force due to shear stress at a radius r from
the centre could be determined as mentioned here

dF = q x dA

dF = τ x r/ Ro x 2П x r x dr

dF = τ/ Ro x 2П r

^{2}dr
Twisting moment at the circular elementary ring
could be determined as mentioned here

dT = Turning force x r

dT = τ/ Ro x 2П r

^{3}dr
Total torque could be easily determined by
integrating the above equation between limits Ri and Ro.

Therefore total torque transmitted by a hollow
circular shaft will be given by following formula

Let us
consider that N is the R.P.M of the shaft and ω is the angular velocity of the
shaft. We have already secured here the torque transmitted by circular solid
shaft and it is mentioned above.
Now we will find here the expression for power
transmitted by solid circular shaft as mentioned here.

Do you have suggestions? Please write in comment box.

We will now discuss the Torque in terms of polar moment of inertia, in the category of strength of material, in our next post.

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

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