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Torsion Apparatus Experiment

1

TORSION APPARATUS EXPERIMENT

(The Modulus of Rigidity)

Student's Guide

Laboratory Manual and Workbook

(2014)


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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


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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.


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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.


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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.


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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.


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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).


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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.


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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.


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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

….. …..

…..

…..

…..

…..

…..

…..

…..

…..

…..

…..


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Graph-1: Applied torque as a function of twist angle for the metal rod with diameter

)( 1d at m0.1 .


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2 mJ : . . . . .

Table-2: The rigidity modulus of the metal test rod with the diameter, 2d .



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 .


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3 mJ : . . . . .




1.0

0.0 0.0 0.0 0.0

….. …..

…..

…..

…..

…..

…..

…..

…..

…..

…..

…..


metal rod diameter )( 3d at .0.1 m


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4 mJ : . . . . .




1.0

0.0 0.0 0.0 0.0

….. …..

…..

…..

…..

…..

…..

…..

…..

…..

…..


metal rod diameter )( 4d at m0.1 .


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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 .



0.5

0.0 0.0 0.0 0.0

….. …..

…..

…..

…..

…..

…..

…..

…..

…..

…..


steel rod diameter )0.4( mmd at m5.0 .


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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 .



0.5

0.0 0.0 0.0 0.0

….. …..

…..

…..

…..

…..

…..

…..

…..

…..

…..


brass rod diameter )0.4( mmd at m5.0 .


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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.

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