Torsion Apparatus Experiment
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TORSION APPARATUS EXPERIMENT
(The Modulus of Rigidity)
Student's Guide
Laboratory Manual and Workbook
(2014)
Torsion Apparatus Experiment
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List of Equipment
1. Torsion Apparatus Machine
2. Metal Test Rods (Test Specimens with Different Diameters)
2.1. Steel (Diameter: 3.0mm, Length:100cm)
2.2. Steel (Diameter: 4.0mm, Length:100cm)
2.3. Brass (Diameter: 3.0mm, Length:100cm)
2.4. Brass (Diameter: 4.0mm, Length:100cm)
3. Wheel (Diameter:17cm)
4. Masses (8 Pieces)
4.1. Mass:4x400gr
4.2. Mass:2x200gr
4.3. Mass:1x100gr
4.4. Mass:1x50gr
5. String
Torsion Apparatus Experiment
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Purpose:
The objective of this experiment is to analyze and
determine the rigidity modulus of the various
metal rods.
Introduction:
The torsion apparatus is used to determine the
modulus of rigidity (or shear modulus) of the
materials. When under application of an external
force, the shape of a solid material changes
without change in its volume, and the solid
material is said to be shared. This deformation
happens when a tangential force )(F is applied
to one of the faces of the object.
In the case of the rigidity modulus, suppose a
rigid block whose lower face is fixed and a known
tangential force )(F is applied to its upper face
as seen in the Figure-(1). This external force
causes the parallel layers to the lower face to slip
through a distance of x from the fixed face of the
solid block such that the upper face shifts and the
solid material takes a new form while its volume
remaining unchanged. The strain produced in this
material by the stress is measured by the angle
)( called the shearing strain (or simply shear).
Due to this shearing of the solid material, a
tangential restoring force is developed in the solid
object which is equal and opposite to the external
applied force.
Since shear stress is a force per unit area, we
define the shear stress as the force )(F acting
tangent to the surface, divided by the area )(A on
which it acts:
A
FstressShear (1)
Here, A is the area of the upper face of the solid
block. The unit of stress is the Pascal (1 Pascal=1
Pa=1N/m2).
Figure-(1) shows that one side of the object under
shear stress is displaced by a distance x
relative to the opposite side. If the displacement
)(x is the deformation of the object, we can
define shear strain as the ratio of the
displacement x to the transverse dimension, h :
h
xstrainShear (2)
Shear strain is a dimensionless number with a
ratio of two lengths.
Figure-1: Deformation of a solid object under shear stress.
Torsion Apparatus Experiment
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The ratio of shear stress (tangential stress) to
shear strain is called rigidity modulus (or shear
modulus) :
hx
AF
/
/ (3)
Note that the concepts of the shear stress, shear
strain and shear modulus apply to solid
materials only. The reason is that the shear
force in the Figure-(1) is required to deform the
solid object, and the object tends to return to its
original shape if the shear force is removed. The
corresponding elastic modulus (ratio of shear
stress to shear strain) is given by the shear
modulus.
Similar to the solid block, when a torque )( is
applied to a rigid rod (shaft) with length )( and
radius )(r , the torque deforms the rod by twisting
it through a small angle )( . Torsion occurs
when a solid material (or shaft) is subjected to a
torque )( . Here, torsion refers to the twisting of a
shaft loaded by the applied torque.
Consider a horizontal line drawn along the length
of a given metal rod. The rod is fixed at one end
and twisted at the other end due to the action of
torque . This applied torque will cause the one
face of the circular cylinder (rigid rod) to be
twisted through an angle )( at the length .
When twisted, the horizontal line moves through
an angle and the line BC will be shifted to AC
through the angle )( . The length of the arc BA
produced is (Figure-2a).
In the geometry given by the Figure-(2), it can be
seen that one end of the rigid rod will rotate about
the longitudinal axis with respect to the other end
during twisting. The magnitude of this rotation is
measured in terms of the angle (in radians) by
which, one end rotates relative to the other end.
The angle is called the angle of twist.
Since the end of the metal rod is twisted, the
radial line on the end face of the rod rotates
through the angle . The length of the arc BA
produced at the outside edge of the rod is the
deformation distance (corresponding to x given
in the Figure-1) and will be also equal to r
(Figure-2b).
(a)
(b)
Figure-2: The shearing of a circular solid metal rod by the
applied torque.
Torsion Apparatus Experiment
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As the external axial torque twists the rod, a
section perpendicular to the axis at a given
distance rotates through the angle . If we
assume that the two arcs are the same, it follows
that:
rBA (4)
r (5)
where is the shear strain (the angle of the
shear) on the outer surface of the solid rod.
The modulus of rigidity (or shear modulus) is
given by:
(6)
where,
: The applied shear stress,
: The resulting shear strain.
Substituting Equation-(5) into Equation-(6), we
get:
r
(7)
or,
r
(8)
The Equation-(8) uses the relationship in the
shear stress, shear modulus and angle of twist for
a given circular shaft.
In the case of the applied torque, the relationship
between the torque and the twist angle is given
by the torsion equation as:
J
(9)
where J is polar moment of inertia for a solid
cylindrical specimen. The polar moment of inertia
for a solid circular shaft with the diameter )(d
can be calculated as:
32
4dJ
(10)
By combining Equation-(8) and Equation-(9), we
get the general torsion equation as:
rJ
(11)
Using Equation-(11), we can rewrite the shear
modulus (or rigidity modulus) at the end of the
metal rod as follows:
J
(12)
where,
: Shear modulus (or rigidity modulus),
: Length of the metal test rod (shaft) over
which the angle of twist is measured,
J : Polar moment of inertia,
: Applied torque,
: Angle of twist in radians.
Torsion Apparatus Experiment
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Assume that the circular shaft has a uniform
cross-section along its length )( , it is straight
and the torque is constant along its length. Then
we can state that a cross-section normal to the
longitudinal axis is planar before the torque
application and it will remain planar after twisting
by the angle (Figure-3).
If the torque and the degree of rotation are
known in a given torsion experiment, the rigidity
modulus of the any test specimen can be
determined. In the torsion apparatus, the torque
)( is supplied by hanging masses )(M
attached to a string wrapped around a vertical
wheel with radius )(R . This radius is
perpendicular to the applied force )(F which is
the tension in the string wrapped around the
wheel. Therefore, the applied torque )( in the
torsion apparatus can be calculated as:
FR (13)
MgR (14)
where,
: Applied torque,
F : The constant applied force due to gravity acting on the hanging mass,
R : Radius of the wheel on the torsion apparatus,
M : Hanging mass.
If we use the applied torque and the polar
moment of inertia J for a given circular shaft in
the Equation-(12), we can also rewrite the rigidity
modulus )( as a function of the mass (load):
4
2
r
MgR (15)
where r is the radius of the circular shaft such
that rd 2 .
By plotting a graph of torque )( versus the
angle of twist )( , the rigidity modulus can be
determined from the slope of the graph. So,
Equation-(12) can be written as follows;
)(SlopeJ
(16)
Therefore, knowing )( and )(J of any metal
test rod for a particular experiment, and then
measuring the slope of corresponding
graph, Equation-(16) will give the rigidity modulus
)( of the metal rod experimentally.
Figure-3: The cross-section of a solid circular shaft.
Torsion Apparatus Experiment
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Description of the Torsion Apparatus
(a)
(b)
Figure-4: Schematic representation of the torsion apparatus (a) and the cylindrical shaft (metal rod) under applied
torsional loading (b).
Torsion Apparatus Experiment
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The torsion apparatus consists of a wheel, metal
test rods and masses with different weights. The
metal rods include brass with 3.0mm and 4.0mm
in diameter )(d ; the other test rods are steel with
the diameters of 3.0mm and 4.0mm.
The torsion test is performed by mounting the
metal rod specimen into a torsion testing machine
and then applying the twisting moment (or torque)
to the material. The metal rod is held horizontal
and rigidly mounted into the end sockets of
torsion apparatus (Figure-4a).
The large wheel contains a socket to hold one
end of the metal rod such that this side of the rod
is subjected to a torque produced by the vertical
wheel. A light string around the groove of the
wheel carries masses to produce the torque.
The Figure-(4b) shows a shaft fixed at one end
and twisted at the other end due to the action of a
torque . The masses )(M are chosen for each
rod so that the torsion angle (angle of twist)
never increases beyond 075 and the deformation
must remain linear elastic. Different cylindrical
rods are loaded with external torque and then the
torsion angle )( measured at the end )( of the
metal rod. Using the data set of the load (torque)
and the corresponding angular twist, the rigidity
modulus of the different cylindrical rods is
determined.
Experimental Procedure
Part-I
The torsion experiment looks at the shear
deformation of a cylindrical shaft (metal rod)
under applied torsional loading.
In the first part of the experiment, the deformation
test of the each metal rod is carried out to
determine if the deformation remains linear
elastic or not. In order to study the response of
materials under the torsional force, test
specimens are mounted between the two sockets
of the torsion testing machine and then twisted.
Each of the specimens should be elastically
deformed )750( 00 and the recovery of
the specimen to its original shape must be
possible if the specimen is unloaded )0( 0 .
The deformation is measured by the angle of
twist, at the length, of the metal rod.
To test the deformation of the specimen:
1. Measure the diameter of the first specimen
to be tested as 1dd .
2. Clamp the test specimen into the torsion
testing machine using the sockets and make
sure the specimen is firmly mounted and
twist angle, is zero )0( 0.
3. Load the masses one by one in order to
deform the test specimen elastically until
the rod is twisted about 075 of the
rotation. Note that the magnitude of torque
applied to the specimen must be such that
the resulting stress remains in the elastic
region.
Torsion Apparatus Experiment
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4. Remove the weights in the same order in
which they were added. Now, the reading of
twist angle should be again zero )0( 0. Note
that the twist angle must be zero each time
when the rod is relaxed (the torque
unloaded). This procedure is done both in
clockwise and anti-clockwise direction.
5. If the angle of twist is not zero after removing
the added weights, the rod is not deformed
elastically and the following conditions may
exist:
5.1. The fixed rod is not straight and
horizontal along the end sockets.
5.2. The rod is not firmly clamped into the
end sockets so it is slipping.
6. Before proceeding with the experiment,
repeat the deformation test for the each
specimen with different materials and
diameters.
Part-II
In this part of the experiment, different solid
cylindrical rods will be subjected to a torsional
load. Then, the modulus of rigidity will be
determined for the cylindrical materials.
1. For the first test piece, determine the
diameter )( 1d of the test rod. Also record the
length as m0.1 where the angle of twist
)( will be measured.
2. Measure the radius )(R of the loading
wheel in the torsion apparatus.
3. Calculate and record polar moment of inertia
by using the corresponding diameter )( 1d of
the test piece as 1JJ .
4. Before loading the masses for torque, it is
essential to make sure that the twist angle is
zero )0( 0. If not, set the angle to zero.
5. Now, load the rod with a torque by using
masses and measure the corresponding
torsion angle at the end )( of the rod.
6. Record the applied torque and the
corresponding angle of twist in the data
table. Note that for every load increment; you
should record the torque )( with the
corresponding angular displacement )( .
4.1. Repeat until enough measurements are
taken to draw a graph of applied torque
as a function of twist angle.
7. Plot a graph of the applied torque )(
against the twist angle
)( of rotation. In the
graph, angle of twist should be in radians.
8. Draw the best straight line through the points
in the graph and determine the slope.
6.1. By the slope of torque-twist angle
graph, find and record the ratio of the
applied torque to the angle of twist.
6.2. Using the slope, the rod diameter )( 1d
and rod length )( , determine the
rigidity modulus )( of the test
specimen from Equation-(16).
9. Compare the experimental value of the
rigidity modulus with the accepted value.
10. Carried out the experiment on the rods of the
steel and brass of the same diameter
)0.4( mm using the length, m5.0 until
the each rod is twisted about 045 .
11. Repeat the experiment to find the rigidity
modulus )( of the different metal rods
under applied torsional loading.
Torsion Apparatus Experiment
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LABORATORY REPORT
Material Type: . . . . .
Length of Metal Rod, )(m : . . . . .
Metal Rod Diameter, )(1 md : . . . . .
Polar Moment of Inertia, )( 4
1 mJ : . . . . .
)(m : Test length of the metal rod,
)(kgM : Hanging mass,
)(Deg : Twist angle in degrees measured
experimentally at the test length )( ,
)(Rad : Angle of twist in radians,
).( mN : Applied torque,
)/( 2mN : The modulus of rigidity.
Name
Department
Student No
Date
Table-1: Experimental data values for the rigidity modulus of the metal test rod in the diameter, 1d .
)(m )(kgM )(Deg )(Rad ).( mN Slope )/( 2mN
Measured Measured Measured Experimental Experimental Calculated Experimental
1.0
0.0 0.0 0.0 0.0
….. …..
…..
…..
…..
…..
…..
…..
…..
…..
…..
…..
Torsion Apparatus Experiment
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Graph-1: Applied torque as a function of twist angle for the metal rod with diameter
)( 1d at m0.1 .
Torsion Apparatus Experiment
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Material Type: . . . . .
Length of Metal Rod, )(m : . . . . .
Metal Rod Diameter, )(2 md : . . . . .
Polar Moment of Inertia, )( 4
2 mJ : . . . . .
Table-2: The rigidity modulus of the metal test rod with the diameter, 2d .
)(m )(kgM )(Deg )(Rad ).( mN Slope )/( 2mN
Measured Measured Measured Experimental Experimental Calculated Experimental
1.0
0.0 0.0 0.0 0.0
….. …..
…..
…..
…..
…..
…..
…..
…..
…..
…..
Graph-2: Applied torque as a function of twist angle for the
metal rod diameter )( 2d at m0.1 .
Torsion Apparatus Experiment
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Material Type: . . . . .
Length of Metal Rod, )(m : . . . . .
Metal Rod Diameter, )(3 md : . . . . .
Polar Moment of Inertia, )( 4
3 mJ : . . . . .
Table-3: The rigidity modulus of the metal test rod with the diameter, 3d .
)(m )(kgM )(Deg )(Rad ).( mN Slope )/( 2mN
Measured Measured Measured Experimental Experimental Calculated Experimental
1.0
0.0 0.0 0.0 0.0
….. …..
…..
…..
…..
…..
…..
…..
…..
…..
…..
…..
Graph-3: Applied torque as a function of twist angle for the
metal rod diameter )( 3d at .0.1 m
Torsion Apparatus Experiment
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Material Type: . . . . .
Length of Metal Rod, )(m : . . . . .
Metal Rod Diameter, )(4 md : . . . . .
Polar Moment of Inertia, )( 4
4 mJ : . . . . .
Table-4: The rigidity modulus of the metal test rod with the diameter, 4d .
)(m )(kgM )(Deg )(Rad ).( mN Slope )/( 2mN
Measured Measured Measured Experimental Experimental Calculated Experimental
1.0
0.0 0.0 0.0 0.0
….. …..
…..
…..
…..
…..
…..
…..
…..
…..
…..
Graph-4: Applied torque as a function of twist angle for the
metal rod diameter )( 4d at m0.1 .
Torsion Apparatus Experiment
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Material Type: . . . . .
Length of Metal Rod, )(m : . . . . .
Metal Rod Diameter, )(md : . . . . .
Polar Moment of Inertia, )( 4mJ : . . . . .
Table-5: The rigidity modulus of the steel with the diameter, mmd 0.4 at m5.0 .
)(m )(kgM )(Deg )(Rad ).( mN Slope )/( 2mN
Measured Measured Measured Experimental Experimental Calculated Experimental
0.5
0.0 0.0 0.0 0.0
….. …..
…..
…..
…..
…..
…..
…..
…..
…..
…..
Graph-5: Applied torque as a function of twist angle for the
steel rod diameter )0.4( mmd at m5.0 .
Torsion Apparatus Experiment
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Material Type: . . . . .
Length of Metal Rod, )(m : . . . . .
Metal Rod Diameter, )(md : . . . . .
Polar Moment of Inertia, )( 4mJ : . . . . .
Table-6: The rigidity modulus of the brass with the diameter, mmd 0.4 at m5.0 .
)(m )(kgM )(Deg )(Rad ).( mN Slope )/( 2mN
Measured Measured Measured Experimental Experimental Calculated Experimental
0.5
0.0 0.0 0.0 0.0
….. …..
…..
…..
…..
…..
…..
…..
…..
…..
…..
Graph-6: Applied torque as a function of twist angle for the
brass rod diameter )0.4( mmd at m5.0 .
Torsion Apparatus Experiment
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Table-7: Comparison of the experimental and expected values of the rigidity modulus corresponding
to material type.
Material Diameter
Rigidity Modulus Rigidity Modulus
)(% Error )/( 2mN )/( 2mN
Experimental Expected
Steel …..
Steel …..
Brass …..
Brass …..
The test length )( of the each metal rod is set to 1.0m ± 0.001m in the torsion apparatus.
Table-8: Comparison of the experimental rigidity modulus with the expected value of the metal rods
(steel and brass).
Material Diameter
Rigidity Modulus Rigidity Modulus
)(% Error )/( 2mN )/( 2mN
Experimental Expected
Steel …..
Brass …..
The test length )( of the each metal rod is set to 0.5m ± 0.001m in the torsion apparatus.