CHAPTER 5
TORSION
CONTENTS
3.1 Introduction
3.2 Deformations in a Circular Shaft
3.3 Stresses in the Elastic Range
3.4 Angle of Twist in the Elastic Range
3.5 Power Transmission
3.1 Introduction
Torsion
o It refers to the twisting of a straight bar when it is loaded by moments (or torques) that tend to produce
rotation about the longitudinal axis of the bar.
o Cylindrical members that are subjected to torques and transmit power through rotation are called shafts;
for instance, the drive shaft of an automobile or the propeller shaft of a ship.
o The most common application is provided by transmission shafts in automobiles, axles, propeller shafts,
steering rods, and drill bits.
o The moment of a couple vector may be represented either by double-headed couple vectors as in
Fig.(a), or in the form of curved arrows as in Fig. (b).
• The SI unit for torsion is newton meter (N.m).
3.2 Deformations in a Circular Shaft
o Consider a circular shaft of radius r and length L that is attached to a fixed support at the left end.
o When a torque T is applied at the right end, the shaft will twist, with its free end rotating through an
angle ϕ called the angle of twist. The left fixed end will react equal and opposite torque T.
o When a torque T is applied at the right end, the straight line pq, before applying the torque, will also
change to pq’, after the torque is applied, forming an angle
γ ,called shearing strain or angular
distortion, with the horizontal line pq.
o Within a certain range of values of T, the angle of twist ϕ is proportional to the applied torque T and the
length L of the shaft.
o To derive a simple theory which describe the behavior of shafts subjected to torque, it is necessary to
make the following assumptions
Assumptions
1.
The material is homogeneous, i.e. uniform elastic properties throughout
2.
The material is elastic following Hook’s law with shear stress proportional to shear strain
3.
Circular sections remain circular
4.
Cross sections rotate as if rigid, i.e. every diameter rotates through the same angle
Shear Strain within the bar
➢ Referring the above figure, for small values of γ we can express the arc length qq’ as
qq' = Lγ ………. (i)
• But, on the other hand, we have
qq' = ϕr ……… (ii)
• It follows that
Lγ = rϕ ……….. (iii)
• or shear strain with in the bar is
Where: γ and ϕ are both expressed in radians
and r is interior radius of the shaft
 =
r
…………. (3.1)
L
Shear Strain at the outer surface
➢ As Eqn. (3.1) signifies, the shearing strain in a circular shaft varies linearly with r, the distance from the
axis of the shaft, and it becomes maximum on the surface of the shaft, where r = R.
 max =
R
L ……………. (3.2)
➢ Eliminating ϕ from Eqs. (3.1) and (3.2), we can express the shearing strain γ at a distance r from the axis
of the shaft as
r
 =  max ……………….. (iv)
R
Shear Strain for hollow shaft
➢ The shear strain equations for hollow circular shafts are as follows
r2
 max =
L
 min
…………. (v)
r1 r1
=
=  max
L r2
…………. (vi)
Example 3.1
1. A plastic bar of diameter d = 50 mm is to be twisted by torques T (see figure) until the angle of
rotation between the ends of the bar is 5.0°. If the allowable shear strain in the plastic is 0.012 rad, what
is the minimum permissible length of the bar?
Example 3.2
2. A circular steel tube of length L = 0.90 m is loaded in torsion by torques T (see figure).
(a) If the inner radius of the tube is r1 = 40 mm and the measured angle of twist between the ends is 0.5°, what is
the shear strain γ1 (in radians) at the inner surface?
(b) If the maximum allowable shear strain is 0.0005 rad and the angle of twist is to be kept at 0.5° by adjusting the
torque T, what is the maximum permissible outer radius (r2)max?
3.3 Stresses in the Elastic Range
o For a linear elastic shaft the relation between shear stress and shear strain can be expressed using
Hooke's law as
 = G
…………. (3.3)
o Multiplying both sides of Eq. (iv) by G, we write and equating with eqn. (3.3)
=
r
 max
R
…………. (3.4)
❖ This eqn. shows the distribution of a shear stress on a solid circular shaft of radius r
varying from r = 0, at the center of the shaft, to r = R, at the outer surface of the shaft as
shown in the figure on the left
Torsion Formula
➢ Consider an element of area dA located at radial distance r from the axis of the bar (see fig.).
➢ The shear force acting on this element is equal to τdA, where τ is the shear stress at radius r.
➢ The moment of this force about the axis of the bar is equal to the force times its distance from
the centre, or (τrdA).
➢ Substituting for the shear stress τ from Eq. (3.4), we can express this elemental moment as
dM = rdA =
Where: r is interior radius of the shaft
 max
R
r 2 dA
…………. (vii)
➢ The resultant moment (equal to the torque T ) is the summation over the entire cross-sectional area of all
such elemental moments:
T =  dM =
A
 max
R
2
r
 dA =
 max
A
R
J ……………….. (viii)
2
• In which J =  r dA is the polar moment of inertia of the circular cross section
A
• For a circle of radius r and diameter d, the polar moment of inertia is
J=
r 4
2
=
d 4
……….. (ix)
32
• An expression for the maximum shear stress can be obtained by rearranging Eq. (viii), as follows:
 max =
TR
………… (3.5)
J
➢ Substituting R = d/2 and J = πd4/32 into the torsion formula, we get the following equation
for the maximum stress:
 max
16T
=
3
d
………… (3.6)
Example 3.3
3. When drilling a hole in a table leg, a furniture maker uses a hand-operated drill (see
figure) with a bit of diameter d = 4.0 mm.
(a) If the resisting torque supplied by the table leg is equal to 0.3 Nm, what is the maximum
shear stress in the drill bit?
(b) If the shear modulus of elasticity of the steel is G = 75 GPa, what is the rate of twist of
the drill bit (degrees per meter)?
Example 3.4
4. The steel shaft of a socket wrench has a diameter of 8.0 mm. and a length of 200 mm (see figure).
If the allowable stress in shear is 60 MPa, what is the maximum permissible torque Tmax that may be
exerted with the wrench? Through what angle ϕ (in degrees) will the shaft twist under the action of
the maximum torque? (Assume G = 78 GPa and disregard any bending of the shaft.)
Example 3.5
5. Under normal operating conditions, the electric motor exerts a torque of 2.8 kN • m on shaft AB. Knowing
that each shaft is solid, determine the maximum shearing stress in
(a) shaft AB, (b) shaft BC, (c) shaft CD
Example 3.6
6. Shaft BC is hollow with inner and outer diameters of 90 mm and 120 mm. respectively. Shafts AB and CD are
solid and of diameter d. For the loading shown, determine
(a) The maximum and minimum shearing stress in shaft BC.
(b) The required diameter d of shafts AB and CD if the allowable shearing stress
in these shafts is 65 MPa.
3.4 Angle of Twist in the Elastic Range
➢ For a shaft of length L and of uniform cross section of radius R subjected to a torque
T at its free end, we recall from Sec. 3.2 that the angle of twist ϕ and the maximum
shearing strain γmax are related as follows
 max =
R
L
……………… (3.2) (repeated)
• But, in the elastic range, the yield stress is not exceeded anywhere in the shaft.
Hooke's law applies, and we have
 max =
 max
G
…………………..(x)
And using  max = TR from eqn. (2.5) and substituting it in eqn.(x),we get
J
 max =
TR
GJ
………..…………..………….. (xi)
➢ Equating eqn. (3.2) and eqn. (xi) and solving for ϕ, we write
=
TL
GJ
………….. (2.7)
Example 3.7
7. The horizontal shaft AD is attached to a fixed base at D and is subjected to
the torques shown. A 44-mm-diameter hole has been drilled into portion CD
of the shaft. Knowing that the entire shaft is made of steel for which G = 77
GPa. determine the angle of twist at end A.
Example 3.8
8. For the aluminum pipe shown (G = 27 GPa).
(a) determine the torque To causing an angle of twist of 2°.
(b) Determine the angle of twist if the same torque To is applied to a solid cylindrical shaft of the same length
and cross-sectional area.
3.5 Power Transmission
➢ The most important use of circular shafts is to transmit mechanical
power from one device or machine to another, as in the drive shaft of an
automobile, the propeller shaft of a ship, or the axle of a bicycle.
➢ The power is transmitted through the rotary motion of the shaft, and the amount of power
transmitted depends upon the magnitude of the torque and the speed of rotation.
➢ The torque T exerted on the shaft and the power P associated with the rotation of a rigid body is
related as
2nT
P = T =
60
………… (3.9)
Where Ω = Angular Speed (Rad/S)
= 2πf
= 2πn/60
F = Frequency (Cycle/S Or Hz)
N (Rpm)
Example 3.9
9. A solid steel shaft ABC of 50 mm diameter is driven at A by a motor that transmits 50 kW to the shaft at 10
Hz. The gears at B and C drive machinery requiring power equal to 35 kW and 15 kW, respectively. Compute
the maximum shear stress τmax in the shaft and the angle of twist ϕAC between the motor at A and the gear at C.
(Use G = 80 GPa.)
Example 3.10
10. The drive shaft for a truck (outer diameter 60 mm and inner diameter 40 mm) is running at 2500 rpm (see
figure).
(a) If the shaft transmits 150 kW, what is the maximum shear stress in the shaft?
(b) If the allowable shear stress is 30 MPa, what is the maximum power that can be transmitted?