MM/CV211 – Solid Mechanics 1
Lab 3 - Torsion Experiment
Objective
To observe the relationship between the torque and angular deflection of a specimen subjected to
torsional loading and to determine the Modulus of Rigidity of the shaft.
Introduction
The following assumptions are made when considering the shaft subject to pure torque as shown in the
figure below.
1. The twisting action is uniform along the whole length of the shaft.
2. Radii remain straight
3. Transverse planes remain parallel to each other.
.
Figure 1. Torsional load on a circular shaft.
These are reasonable assumptions if the deformation is elastic and the shaft is circular. Consider an
elemental tube of radius r and thickness δ in the shaft. The tangential displacement on the tube wall due to
angular displacement θ is 𝑟𝜃 (θ in radians) therefore
Shear strain around the wall of the tube =
𝑟𝜃
𝐿
(radians)
And the definition of the rigidity modulus 𝐺 =
Hence 𝜏 =
𝐺𝑟𝜃
𝐿
𝑁/𝑚2
𝑆𝑡𝑟𝑒𝑠𝑠
𝑆𝑡𝑟𝑎𝑖𝑛
=
𝜏
𝛿
Assuming this stress is uniform in the tube wall the torque producing the stress is given by
𝛿𝑇 = 𝑓𝑜𝑟𝑐𝑒 𝑖𝑛 𝑡𝑢𝑏𝑒 ∗ 𝑟𝑎𝑑𝑖𝑢𝑠
= (𝑆1 𝑠𝑡𝑟𝑒𝑠𝑠 ∗ 𝑎𝑟𝑒𝑎 𝑖𝑛 𝑠ℎ𝑒𝑎𝑟) ∗ 𝑟𝑎𝑑𝑖𝑢𝑠
= 𝜏 2𝜋𝑟𝛿𝑟 ∗ 𝑟
Substituting for τ
𝛿𝑇 = 2𝜋
𝐺𝜃 3
𝑟 𝛿𝑟
𝐿
Therefore, the torque in the whole shaft is given by
𝑇 = 2𝜋
𝐺𝜃 𝑅 3
∫ 𝑟 𝑑𝑟
𝐿 0
𝑅
𝐺𝜃 𝑟 4
= 2𝜋
[ ]
𝐿 4 0
=
𝐺𝜃 𝜋𝑅 4
𝐺𝜃 𝜋𝐷 4
[
]=
[
] (𝑁𝑚)
𝐿 2
𝐿 32
The terms in the brackets are the “Polar second moment of area” of the shaft denoted by J.
Part A
Methodology
1. Set up the specimen.
2. Position the deflection gauge at approximately three quarters of the length of the specimen,
measured from the fixed end.
3. Accurately measure the distance L before applying any load.
4. Apply a load of 0.2kg to the load hanger and record the deflection obtained.
5. Increase the load by increments of 0.2kg up to the maximum available each time recording the
corresponding total value of θ.
6. Tabulate the results in the table provided.
Results



Effective length of shaft _____________ m.
Diameter of shaft __________________m
Radius of torsion pulley _______________m.
Load Mass M (kg)
Torque T (Nm)
Angular Deflection θ (rad)
Results Analysis
Plot a graph of T vertically against angular deflection θ.
The graph should be a straight line of a slope
𝐺𝐽
.
𝐿
Measure the slope of the straight part of the graph ignoring any initial variations. Hence determine the
value of G.
Compare the value obtained for G with typical values for the material obtained from data books or
reference works.
Comment on any discrepancies, suggest reasons for its origin.
Comment also on the shape of the graph and the information it provides on torsion theory.
Part B
Methodology
1.
2.
3.
4.
Set up the specimen.
Keep the torque T constant with a value of 3kg.
Measure the angular deflection obtained over the same length L for a range of shaft diameters D.
Tabulate the results in the table below.
Results
Load mass M = ____________________kg
Radius of torsion pulley a = _______________m
Torque Mga = _________________Nm
Effective length L = ___________________m
Diameter D (m)
Angular Deflection θ
(rad)
Log θ
Log D
Analysis and Discussion
Plot the graph of log θ against log D.
1
𝐺𝜋 4
=
𝐷
𝜃 32𝐿𝑇
Taking logarithms on both sides we get
− log 𝜃 = 4 log 𝐷 + log
𝐺𝜋
32𝐿𝑇
log 𝜃 = −4 log 𝐷 − log
𝐺𝜋
32𝐿𝑇
Or
From the intercept the value of G can be determined.
Compare the value of G with the results obtained in Part A and other references.