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ENGR20004 Engineering Mechanics
Assignment 2 – Stress and Deflection
Instructions for Reporting
Report submission Each group is to electronically submit the report (in PDF format, a single file,
no more than 10 pages, excluding appendix) to the LMS. Show all workings where applicable.
Additional note A short oral test (5 marks) will be conducted before your experiments. You are
expected to read the handouts and watch lab videos in advance. Demonstrators will ask you simple
questions related to experiments. If you fail to answer, you will be asked to return to your desk and
review the lab materials again.
The total marks for Assignment 2 is 25 marks ( Reports: 20 marks + oral test: 5 marks) excluding the
bonus 5 marks for Question 4 of the analysis of a simply supported beam (Question 4 is optional). If you
receive 30 marks acing the bonus question, your final mark will be adjusted to 25 marks.
1 Experimental part: Instron Compression Test (10 marks)
Essential personal protection equipment
1. Enclosed footwear (If not worn, you will not be admitted into the lab.)
2. Safety glasses (They will be provided and must be worn at all times.)
Recommended reading
1. “Static & Mechanics of Materials”, Ch. 9, R. C. Hibbeler, Prentice Hall
Samples
1. 10mm × 15mm aluminium alloy (6060 T5)
2. 20mm × 40mm PVC polymer
3. 11.5mm × 25mm (4mm inner diameters) alumina ceramic (Al2O3)
Overview
The experimental testing of materials is essential for Engineers for a wide variety of reasons, ranging
from validation of a component’s specifications, determination of a new material properties or assessing
if worn or corroded parts remain serviceable. This testing aims to determine the stress and strain rela-
tionship within a material under a known load. Ultimately conventional stress-strain diagrams are used
to determine such fundamental properties as
The Engineering Stress (σ)
σ =
P
A0
(1)
1
Assignment 2 – Stress and Deflection
where P is the applied load (N) and A0 is the original cross-sectional area of the test sample.
The Engineering Strain ()
=
δ
L0
=
Li ? L0
L0
(2)
where δ is the change in sample gauge length (also equal to the difference between the instantaneous
sample length, Li and the original sample length L0).
The Modulus of Elasticity or Young’s Modulus (E)
σ = E · (3)
Confirming a linear correlation between the stress and strain within a material when it remains within
the elastic range. N.B. if a plot of stress and strain were measured then the Modulus of Elasticity would
be represented by the gradient of the data.
Equation 3 is more commonly known as Hooke’s Law and relates to the deformation of material at
relatively low amounts of strain. Conceptually, the Modulus of Elasticity may be considered to be the
stiffness of a material, the higher value of which results in more resistance to elastic deformation. Im-
portantly, if an applied force (load) produces elastic deformation, once the force is removed then the
material will return to its original dimensions. If however, the force exceeds the Yield Strength (σyield) of
a material then plastic deformation occurs, resulting in permanent deformation of the test sample, that
is, the sample will not return to its original dimensions.
Overall, it is important to determine the yield strength of a material as most engineering materials
will only remain in service if their loading remains within the elastic range. If the yield strength is ex-
ceeded within a material and permanent, plastic deformation occurs then the component will most likely
become unserviceable due to changes in tolerances, volumetric dimensions and ultimately fail sooner and
unpredictably. Figure 2 illustrates a typical stress-strain diagram for a material in tension. The yield
strength has been estimated using the offset method by the determining the intersect of the 0.2% strain
line with the stress-strain curve, noting that the line has been drawn with the same gradient as the data
within the elastic range.
Generally, the properties of engineering materials differ if testing is conducted in tension (+ve load)
or compression (-ve load), while the inherent ductility or brittleness of a material defines which of these
methods would be the optimal test to perform.
This laboratory will aim to test three materials using 50kN Instron (5569a) testing machines: a polyner,
a metal and a ceramic.
Note carefully, as the compression platens are crushing the sample do not place your hands
within the machine while operating. Additionally, as the samples may fracture unpre-
dictably, ejecting material, wear safety glasses at all times during the laboratory.
Page 2 of 7
Assignment 2 – Stress and Deflection
Figure 1: Typical stress-strain curve, estimating the yield strength of the material by determining the
intersect of 0.2% strain line with the measured data.
During the lab session:
1. Record machine number.
2. Measure and record the initial inner diameter (of ceramic), outer diameters and length of the
samples using the digital calipers.
3. Make a prediction of what will happen to each of the samples.
4. Under the demonstrator’s supervision, perform the test in the following order: Aluminium alloy,
PVC polymer and alumina ceramic.
5. Observe the behavior of the material as it is compressed.
6. When choosing the filename for your data, please make it unique by including your name, machine
number, group number, workshop session and sample name.
After the lab session: You are expected to elaborate on any discussion points you find worthy and
interesting to point out beyond just the questions below. Summarise numerical results in a table on the
first page. Make sure you include axis labels and units on your plots. You should finish your report with
a conclusion on the key findings from the experiment.
Download the stress-strain data from LMS that corresponds to your Machine number and Day.
Page 3 of 7
Assignment 2 – Stress and Deflection
For your data set:
1. Plot the stress-strain data for each sample.
2. Explain physically what causes the initial shallow gradient (if any) of the plots to justify why this
initial part of the graph (if any) can be ignored when computing Modulus of Elasticity and Yield
Strength.
3. Why do the samples bulge instead of uniformly expanding?
4. Measure the Modulus of Elasticity for each material from the plots.
5. Use the offset method to determine the yield strength of each material. Make sure you demonstrate
the method in the report.
6. Compare the Modulus of Elasticity of the 3 materials and comment.
7. Compare the Yield strengths of the 3 materials and comment.
8. Comment on the suitability of Yield strength as a useful means of predicting failure for each of the
3 materials.
9. Which material is brittle? and comment on it.
10. Calculate the Modulus of Resilience for the metal sample.
Page 4 of 7
Assignment 2 – Stress and Deflection
2 Experimental Part: Analysis of a Simply Supported Beam
(10 marks)
Description of the problem
A C
P1 P2
x
20mm
1mm
B ED
l l l l
Figure 2: Simply supported beam (Not drawn to scale).
For this experiment, you will be analysing the deflection of a simply supported beam AE illustrated
in Figure 2 after being subjected to two point loads P1 and P2. Similar to Assignment 1, the setup will
be modelled with the help of Pixie frames. Your task is to use a high-resolution camera to accurately
capture the deflection and compare your processed result to the analytical result of the loading scenario in
Figure 2. To compute the analytical result, you may assume that the length of the beam LAE = 292 mm,
l = 73 mm and the beam section is 20 mm×1 mm as shown in the figure. The beam material is aluminium
with E = 69 GPa.
The aim for this experiment is to
1. Measure and extract the deflection of the beam AE from the images taken.
2. Compute the analytical deflection of the beam AE using superposition.
3. Compare the measurement and analytical results, and discuss any differences.
Equipment
1. Pixie frames with a transparency grid of 5 mm × 5 mm.
2. Buckets with weights to apply the required point loads, P1 = 6.4 N and P2 = 2.2 N.
3. Digital weighing scales.
4. Measuring tape.
5. Camera (or a smartphone with a high-resolution in-built camera).
Experiment Procedures
These steps are to be completed on the day of your lab.
1. Starting with a beam with no loads (P1 = P2 = 0), take a photo of the beam from a distance
of 300 mm. Make sure the beam is horizontal to the eye as well as in the photo taken. Refer to
the section on Additional Notes for guides for taking good-quality photos. You may notice that
the beam may have a slight downward deflection due to its own weight, but we will attempt to
compensate for that during post-processing.
2. Repeat step 1 for
(a) when only P1 is applied (P2 = 0).
(b) when only P2 is applied (P1 = 0).
(c) when both P1 and P2 are applied simultaneously.
(N.B. For more accurate results, you should verify the weights of the loads using the digital weighing
scales provided.)
Page 5 of 7
Assignment 2 – Stress and Deflection
Question 1 (2 marks)
You should now have four images, one for the unloaded (original) beam and the remaining three for the
different loaded (deflected) cases. Using the function getDeflection.m, extract the original and deflected
plots of the beam by following the steps described in the function.
Now, you can calculate the deflection of the beam by subtracting the array of the original beam profile
from the array of the deflected beam profile. This roughly compensates for the slight downward deflection
of the beam due to its own weight.
(a) Calculate and plot the individual deflection curves ν versus horizontal distance x (both in mm) for
the cases when either P1 or P2 is applied.
(b) Repeat step (a) for the case when both P1 and P2 are applied. For this case, determine ν1 and ν2
from your data, where ν1 ≡ ν(x = l) and ν2 ≡ ν(x = 3l).
(c) Now, superpose your results in (a) and compare this deflection curve with your result in (b). Comment
on this result.
Hint: To subtract both profiles, you may need to use an interpolation function such as interp1 in
MATLAB to create equal-sized vector arrays.
Question 2 (7 marks)
Now, using superposition, determine the deflection curve of the setup in figure 2. To apply superposition,
you will need to derive 2 deflection curves. The first is when only P1 is applied and the second is when
only P2 is applied.
Plot the deflection ν versus horizontal distance x, both in mm. Determine ν1 and ν2 from this
analytical result.
Question 3 (1 mark)
Compare your analytical result in Question 2 with the experimental results from Question 1. Comment
on any differences and the accuracy of the approximation.
Question 4 (Bonus marks - 5 marks)
Using the superposition approach in Question 2, compute the horizontal location of the support C such
that the magnitude of the total deflection |(ν1 + ν2)| is minimum. You may use a computational method
(e.g. MATLAB) or an analytical approach.
Page 6 of 7
Assignment 2 – Stress and Deflection
Additional Notes
Taking the Photo
(a) Example of a good photo (b) Example of a bad photo
Figure 3
Similar to Assignment 1, when taking the photos, it is important to orient your camera parallel to
the beam. Make sure that the lines and the beam appear horizontal in your photo. You should only be
able to see the front edge of the beam as shown in Figure 3a and not the depth of the beam as shown in
Figure 3b.
Acknowledgements The Pixie frame setup is part of the TechnoLab – Basic Mechanics HomeLab
Series.
