torsion of circular shaft

8/13/2019 Torsion of Circular Shaft

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NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY

BHOPAL (M.P)

COMPILED BY AMIT SINGH 9827740442 ([email protected])

TORSION OF CIRCULAR SHAFT

When a circular shaft is subjected to torsion, shear stresses are set up in the

material of the shaft. To determine the magnitude of shear stress at any point onthe shaft, consider a shaft fixed at one end AA and free at the end BB as shown.

Let CD is any line on the outer surface of the shaft. Now let the shaft is subjected

to a torque T at the end BB as shown. As a result of this torque T, the shaft at the

end BB will rotate clockwise and every cross-section of the shaft will be subjected

to shear stresses. The point D will shift to D’ and hence line CD will be deflected to

CD’ and the line OD’ will be shifted to OD’.

R = Radius of the shaft.

L = Length of the shaft.

T = torque applied at the end BB.

τ = Shear stress induced at the surface of the

shaft due to torque T.

C = Modulus of rigidity of the material of the

shaft.

Also equal to shear strain.

And is also called angle of twist.

If is very small than




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So

Now in the next figure we find the arc

So now the

Now the modulus of rigidity C of the material of the shaft is given as

So we will get

Now for a given shaft subjected to a given torque T, the value of C, ϴ and L are

constant. Hence shear stress produced is proportional to the radius R.

If q is the shear stress induced at a radius r from the center of the shaft then

The above equation is called torsion equation.




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From the above equation it is clear that shear stress at a point in the shaft is

proportional to the distance of the point from the axis of the shaft. Hence the

shear stress is Maximum at the outer surface and shear stress is zero at the axis of

the shaft.

MAXIMUM TORQUE TRANSMITTED BY A CIRCULAR SOLID SHAFT

The maximum torque transmitted by a circular solid shaft, is obtained from the

maximum shear stress induced at the outer surface of the solid shaft. Consider a

shaft subjected to a torque T as shown

τ=Maximum shear stress induced at the outer surface

R=Radius of the shaft

q=Shear stress at a radius r from the center

Consider an elementary circular ring of thickness ‘dr’ at a distance r from the

center as shown

Then the area of the ring




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From the equation

Shear stress at the radius r

Turning force on the elementary circular ring=

= Shear stress acting on the ring Area of the ring

Now turning moment due to turning force on the elementary ring

The total turning moment (or total torque) is obtained by integrating the above

equation between 0 to R

∫

∫




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Torque transmitted by a hollow circular shaft

POWER TRANSMITTED BY THE SHAFTS

Once the expression for torque T for a solid or a hollow shaft is obtained, power

transmitted by the shafts can be determined.

Let

EXPRESSION FOR TORQUE IN TERM OF POLAR MOMENT OF INERTIA




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POINT TO REMEMBER

(i) When two shafts are connected in series each shaft transmits the same

torque; the angle of twist is the sum of the angles of twist of the two

shafts.

(ii) The shafts are said to be in parallel when the driving torque is applied at

the junction of the shafts and resisting torque is at the other ends of the

shafts. Here the angle of twist is same for each shaft, but the applied

torque is divided between the two.




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STRAIN ENERGY STORED IN A BODY DUE TO TORSION

Consider a solid shaft which is in torsion. Take an elementary ring of width ‘dr’ at

the radius r as shown.

Then shear stress due to torsion at a radius r from the center is given by




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Total strain energy stored in the shaft is obtained by integrating the above

equation between 0 to R.

∫

∫

∫

∫




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Total strain energy in the hollow shaft due to torsion

COMBINED BENDING & TORSION

When a shaft is transmitting torque or power, it is subjected to shear stresses. At

the same time the shaft is also subjected to bending moment due to gravity or

inertia loads. Due to bending moment, bending stresses are also set up in the

shaft. Hence each particle in a shaft is subjected to shear stress and bending

stress. For design purpose it is necessary to find the principal stresses, maximumshear stresses and strain energy. The principal stresses and maximum shear stress

when a shaft is subjected to bending and torsion, are obtained as

Consider any point on the cross-section of a shaft

Let

The torque T will produce shear stress at the point whereas the B.M. will produce

bending stress.

Let q = Shear stress at the point produced by the torque T and




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Maximum shear stress i.e. on the surface of the shaft

()

√

√




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√

For a hollow shaft

[ ] √

[

] √

[ ] √

torsion of circular shaft

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