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					                              Young's Modulus and Stress Analysis

By the end of this practical you should have learned:
Understanding
 that the deformation (strain) of an elastic material is directly proportional to the amount of force
  applied to it (stress).
 that Young’s Modulus is a measure of how stiff a material is.
 that Poisson’s Ratio is a measure of the compressibility of a material in the direction perpendicular to
  an applied force or stress.
Practical skills
You will use strain gauges to determine the elastic properties of a material and use a spreadsheet to process
data and present your results graphically.


Overview
The objectives of this practical are
1. to demonstrate Hooke's law;
2. to determine the Young's modulus of the material, and hence identify it;
3. to determine Poisson’s ratio for the material;


In 1676, Robert Hooke stated, “The power (by which he meant applied force) of any springy body is in
the same proportion with the extension”.
This is often restated in a simple form as, “The extension of a springy body is proportional to the load
applied, unless the elastic limit is reached”. In engineering terms, this can be stated as a general law of
mechanics, “Stress is directly proportional to strain for elastic deformation”.
Elastic materials are ones which, if subjected to an applied force or stress, will return to their original
shape and size once the stress is removed, provided that the stress is below a characteristic threshold or
elastic limit. If this limit is exceeded, the material will undergo plastic deformation, in which a
rearrangement of the atomic or molecular structure occurs and the shape and size are changed
permanently.
Stiffer materials will deform less when subjected to an applied load and Young’s Modulus is a quantity
that allows a comparison of the stiffness of materials. A material with a high value of Young’s Modulus
will experience less deformation (strain) for a given force (stress) than one with a lower value of
Young’s Modulus.

When a sample of material is stretched in one direction, it tends to get thinner in the other two directions.
Poisson's ratio (ν) is a measure of this tendency. It is defined as the ratio of the strain in the direction of
the applied load to the strain normal to the load. For a perfectly incompressible material, the Poisson's
ratio would be exactly 0.5. Most practical engineering materials have ν between 0.0 and 0.5. Cork is close
to 0.0, most steels are around 0.3, and rubber is almost 0.5. Some materials, mostly polymer foams, have
a negative Poisson's ratio; if these materials are stretched in one direction, they become thicker in
perpendicular directions.




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Experimental details
The device used to determine Young’s Modulus and Poisson’s ratio is the electrical resistance strain
gauge. This is the most widely used device for measuring elastic strains. It is essentially a strip of metal
foil which is firmly glued to the surface where the strain is to be measured, so that when the material is
strained, the strain at the surface is fully transmitted to the metal foil. Elastic strain along the length of the
strip causes a small change in resistance of the gauge, largely because of the change in length and cross-
sectional area of the strip, although there is also a slight change in its resistivity. Small changes in
resistance are easy to measure accurately, and so the gauge gives an accurate reading of the small elastic
strain along the direction of the strip in the gauge. The change in resistance, and hence the voltage across
the strain gauge for a constant current, is proportional to the strain; the gauge manufacturer supplies the
value of the constant of proportionality.
For the strain gauges and current used here
                       ε = 3.803×10-7 V (a conversion factor provided by the manufacturer),
                       where ε = strain (No units – a dimensionless quantity)
and                    V = voltage across strain gauge (measured in microvolts).

The experimental work in this practical is very simple and proper working out of the results will take some
time. A results table is provided at the back of this booklet for you to record your results. To make
calculations easier, a spreadsheet is available to enter your data and process the results. Alternatively you
can use graph paper to plot graphs manually.
It will help to arrange members of the group to do specific tasks.
Background Information
This practical uses a simulation of the simple cantilever bending of the beam to which the strain gauges are
attached. The sidearm is not used in this experiment.

Identify the 3 strain gauges on the cantilever arm. For
this experiment you will use only two gauges; the one
that is lined up along the cantilever, the “x” direction and
the one lined up laterally across the cantilever, the “y”
direction. These are represented in the diagram, above,
by the strain gauges on the left and right respectively.

Measurements are taken using a single meter, which is
attached in turn to different strain gauges on the top and bottom of the beam. You will use four of these.
TOP – denotes the gauges measuring tension on the upper surface.
BOTTOM – denotes the gauges measuring compression on the lower surface.
ALONG – denotes the strain in the x direction, along the beam.
LATERAL – denotes the strain in the y direction, across the beam.

Method – Measurements
  1. Select the strain gauge labeled TOP, ALONG – x direction in tension
  2. Take a reading with no load. There may be some drift in this value. Wait until a steady reading is
      obtained, or estimate the average value.
  3. Suspend weights from point “A”, and apply successively larger loads to the end of the beam.
      Record the strain gauge outputs (in μV) produced in each case. It may be useful to remove some
      weights from time to time to check that you get the same (or almost the same) readings.
  4. Repeat the method selecting the following strain gauges:
TOP, LATERAL – y direction in tension.
BOTTOM, ALONG – x direction in compression.
BOTTOM, LATERAL – y direction in compression.

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Treatment of Results
To verify Hooke’s Law for this material

Plot suitable graphs of the gauge readings as a function of the applied. The linearity of the graphs will
demonstrate the validity of Hooke's law.

To determine Young’s Modulus for this material

Convert the strain gauge readings to strains εx and εy, (Use the conversion factor provided by the
manufacturer) and the loads to Newtons (1 kg = 9.81N). From the εx and εy readings (no units), the loads
applied (in Newtons) and the dimensions of the beam (measure these yourself – each apparatus is
different), calculate the Young's modulus E and Poisson's ratio v of the beam. Use the gradients of the
stress/strain lines, rather than individual readings.

NOTE: The LINEST function in Excel is useful here; or you could plot the graphs on graph paper.
Make an estimate of the accuracy of your values. The theory for this part of the practical is given in
appendix 1.

Formulae to use:
                                                 For Young' s Modulus
                                                      6 Fl
                                           E 
                                                     x wh 2
                                                         6l
                                                
                                                    gradient.wh 2

                                                 For Poisson' s Ratio...
                                                    y
                                          
                                                    x

Where          E       =       Young’s Modulus
               F       =       Force (N)
               l       =       Distance between the strain gauge and the eye holding the weights (m)
        εx     =       Strain in the x direction (no units)
               εy      =       Strain in the y direction (no units)
               w       =       Width of the beam (m)
               h       =       Thickness of the beam (m)
               ν       =       Poisson’s ratio

                       Material                Young’s Modulus (Pa)            Poisson’s Ratio
                      Magnesium                      45 x 109                       0.29
                      Aluminium                      69 x 109                       0.35
                        Brass                       103 x 109                       0.34
                       Titanium                     105 x 109                       0.32
                         Steel                      200 x 109                       0.29
                       Tungsten                     400 x 109                       0.28

What material do you think the beam is made of?



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Appendix: Elasticity theory
Cantilever beam theory




Due to the applied force F, a couple of moment F l (the bending moment) acts on the beam cross section at the gauge position.

The bending stress σx varies linearly from a maximum σmax at the top surface to a minimum - σmax
(compressive) at the bottom surface. Hence at a height z from the centre line of the beam, the stress σ
is:



This stress acts on an area = w.dz at a distance from the centre line (moment arm) = z. The total
moments from all of these stresses internal to the beam must balance the moment F l applied
externally.




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Results Tables – Use the Interactive Young’s Modulus simulation to collect data. Enter your readings in
these tables, then transfer them to the Excel Spreadsheet

           Dimensions of material tested                        1    (cm)       2   (cm)      3   (cm)

           Distance of weights to strain gauge     =        l

           Width of the beam                       =    w

           Thickness of the beam                   =    h



Direct Readings from Strain Gauges (microvolts)
                                     TOP                                        BOTTOM
  Load / g           ALONG (x)              LATERAL (y)             ALONG (x)              LATERAL (y)
      0
     100
     150
     200
     250
     300
     350
     400
     450
     500

To convert μV to measurements of elastic strain (ε), these readings are multiplied by 3.803 x 10-7

NOTE BENE
Enter your data in the corresponding blue columns in the Excel spreadsheet. As time is very short, much
of the routine calculation is done for you in the spreadsheet – this includes the plotting of the graphs. You
will, however, need to print your graphs.
Explain from the shape of the graphs whether your material demonstrates the validity of Hooke’s Law.
You then need to calculate Young’s Modulus and Poisson’s ratio for your material, using the formulae
shown on page 3, and by comparing your value to those in the table on page 3 state which material you
have been using.
A website you may find useful :- http://schools.matter.org.uk/




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