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CIVL222 STRENGTH OF MATERIALS
Chapter 6
Torsion
Definition
Torque is a moment that tends to
twist a member about its
longitudinal axis. Slender members
subjected to a twisting load are said
to be in torsion.
Example of Torsion
When opening the lid of a common plastic drinks bottle, a
torque T applied to the cap is gradually increased until the
plastic connectors between the cap and the bottle experience
shear failure.
Example of Torsion
Shafts are structural members with length significantly greater
than the largest cross-sectional dimension used in transmitting
torque from one plane to another.
Example of Torsion
Example of Torsion
For a non-circular section member or an open
section member subjected to torsion:
Plane cross sections of the member do not
remain plane and the cross sections distort in a
manner which is called warping. In other
words, the fibers in the longitudinal direction
deform unequally.
Example of Torsion
Example of Torsion
For a circular shaft or a closed circular section
member subjected to torsion:
Plane circular cross sections remain plane and
the cross sections at the ends of the member
remain flat.
The length and the radius of the member
remain unchanged.
Plane circular cross sections remain
perpendicular to the longitudinal axis.
Analogy Between Axial Deformation and Torsion
• Axial Force (P)
• Elongation (d)
• Normal Stress (s)
• Extensional Strain (e)
• Modulus of Elasticity (E)
• Torque (T)
• Twist Angle (f)
• Shear Stress (t)
• Shear Strain (g)
• Shear Modulus (G)
Torsion Theory for circular sections
Absence of Warping
Investigate Deformation
Investigate Deformation
State of Pure Shear
Shear Strain Relate to Angel of Twist
Linear Variation of Shear Stress
Linear Variation of Shear Stress
Shear Stress Surfaces
Shear Stress Surfaces
Moment “dM” developed on “dA”
Setup the Integration “dM” over the Area “A”
Relating Torque and Stress
Torsion Formula
“J” for Solid Circular Shape
“J” for Hollow Circular Shape
Angle of Twist Formula
Summery of Key Equations
Sign Conventions
Sign Conventions
Sign Conventions
Sign Conventions
Example #1
Example #2
Example #3
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 f (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 #4
A hollow steel shaft used in a construction
auger has outer diameter d2 =150 mm. and
inner diameter d1 = 115 mm. (see figure).
The steel has shear modulus of elasticity G
= 80 GPa
For an applied torque of 17 kN.m,
determine the following quantities:
(a) shear stress t2 at the outer surface of
the shaft,
(b) shear stress t1 at the inner surface, and
(c) rate of twist f (degrees per unit of
length).
Also, draw a diagram showing how the
shear stresses vary in magnitude along a
radial line in the cross section.
Example #5A hollow aluminum tube used in a roof structure has an outside diameter d2 =100
mm and an inside diameter d1 =80 mm (see figure). The tube is 2.5 m long, and
the aluminum has shear modulus G= 28 GPa.
(a) If the tube is twisted in pure torsion by torques acting at the ends, what is the
angle of twist f (in degrees) when the maximum shear stress is 50 MPa?
(b) What diameter d is required for a solid shaft (see figure) to resist the same
torque with the same maximum stress?
(c) What is the ratio of the weight of the hollow tube to the weight of the solid
shaft?
Example #6Four gears are attached to a circular shaft and transmit the torques shown in the
figure. The allowable shear stress in the shaft is 68 MPa.
(a) What is the required diameter d of the shaft if it has a solid cross section?
(a) What is the required outside diameter d if the shaft is hollow with an inside
diameter of 25 mm ?
Work Done and Power Transmitted
When a force moves in a straight line with constant
velocity the work done is given by the product the
magnitude of the force and the distance through
which it has moved.
Work done = force × distance
The power transmitted by this action is defined as
the rate at which this work is done, i.e. the work
done in unit time.
Power = work done
time
Work Done and Power TransmittedThe ‘distance’ travelled by a rotating body is
measured by the number of radians through which
it rotates. The work done by a torque acting on a
shaft is therefore given by the product of the
magnitude of the torque and the amount of rotation
in radians.
For one revolutions of the shaft:
The work done = T ×2p(since the shaft turns through 2p radians in one revolution)
If the shaft is rotating at N revolutions per minute,
then
work done = T ×2p N (units of work per minute)
Work Done and Power Transmitted
Usually the torque will measured in Newton meters
(N.m) and therefore the units of work will also be
N.m. However, it is more usual to give work in
joules (J) which are equal numerically to Newton
meters.
1 joule = 1 Newton meter
Power is measured in watts (W),
1 W = 1 J/s
= 1 N.m/s
=(1/60) N.m/min
Work Done and Power Transmitted
Hence, the power transmitted by a shaft rotating at
N revolutions per minute and subject to torque of T(N.m) will be given by:
WattsNT
Power60
2p
w: the shaft’s angular velocity (rad/s)
: the frequency of shaft’s rotation (Hz = 1 revolution/s)
rpm : revolutions per minute
hp: horsepower , 1 hp = 746 W
f
PPT
fTTP
pw
pw
2
2
fpw 2
Example #7A motor drives a shaft at 12 Hz and delivers 20 kW of power (see figure).
(a) If the shaft has a diameter of 30 mm, what is the maximum shear stress tmax
in the shaft?
(b) If the maximum allowable shear stress is 40 MPa, what is the minimum
permissible diameter dmin of the shaft?
Example #8The 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?
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