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BIOEN 4250: BIOMECHANICS I
Laboratory 4 – Principle Stress and Strain
November 13– 16, 2018
TAs: Allen Lin ([email protected]), Kelly Smith
([email protected])
Lab Quiz: A 10-point lab quiz, accounting for 10% of the lap
report grade, will be given at the beginning of
class. Be familiar with the entire protocol.
Objective: The objective of this experiment is to measure the
strains along three different axes surrounding
a point on a cantilever beam, calculate the principal strains and
stresses, and compare the result
with the stress calculated from the flexure formula for such a
beam.
Background: The ability to measure strain is critical to
materials testing as well as many other applications in
engineering. However, strain gages that adhere to a surface can
alter the local strain environment
if the material (or tissue) of interest is less stiff than the gage
itself. For this reason, contact strain
gages (or strain gages that attach directly to a surface) are not
typically used for the testing of soft
tissues such as ligament, arteries, or skin. However, when the
material is on the stiffer side, or
when the absolute value of the strain is less important than the
detection of the mere presence of
strain itself, contact strain gages are very useful. An example of
a stiffer biological material would
be bone. However, due to the porous nature of bone, one needs
to be extremely careful that the
strain gage is properly adhered to the material’s surface. Other
applications range from real world
stress analysis of a structure (e.g., a wing of an aircraft during
flight) to strain gages incorporated
into medical equipment to ensure proper function (e.g., gages
wrapped around the tubing in a
hospital infusion pump to detect blockages in the line – since
the tube swells more than it should
when the fluid path is occluded).
One common engineering loading case that involves a planar
stress field (i.e., the only non-zero
stresses are in the same plane), is that of beam bending. Beam
bending will be covered in greater
detail during lecture. However, in order to ensure you know the
basics of what is going on in this
lab, we will cover some fundamental topics. The simplest case
of beam loading is that of a
cantilever beam that is completely anchored at one end and
loaded at a point along its length
(Fig. 1). In Figure 1, � is the applied load, ℎ is the thickness of
the beam (with � as the half-
thickness), � is the distance from the fixed wall to the location
where we want to measure stress
and strain (point �), and � is the length of the beam. There are
a couple key points to know about
this loading scenario:
1. As the beam bends downward, the material above the midline
(the dashed line) is in
tension and the material below that line is in compression.
2. At the top and bottom free surfaces, there is only axial stress,
and zero shear stress.
3. At the midline (dashed line, also referred to as neutral axis)
there is zero axial stress and
it is the location of the maximum shear stress (as everything
above it is in tension and
everything below it is in
compression).
4. Theoretically, the only applied
non-zero stress on the upper
and lower free surfaces is the
longitudinal stress, which is the
stress component oriented
along the length of the beam.
However, in an experimental
measurement, due to various
factors (e.g., experimental or
Figure 1: Illustration of a cantilever beam fully supported at
one end.
BIOEN 4250: Laboratory 4 – Principal Stress and Strain
2
calculation error, noise in the signal) very small
transverse stresses in the plane of the free surface do
still typically appear. The equation for the longitudinal
stress (�); introduced as �++ in lecture) is known as the
beam flexure equation, and is as follows,
�) = −
��
�
where � is the bending moment, � is the distance from
neutral axis, and � is the second moment of area about
the neutral axis. To quantify to maximum/minimum value
for longitudinal stress, the equation can be evaluated at
the top and bottom of the beam (i.e., � = ±ℎ 23 ), as
follows,
�) = ±
6�(� − �)
�ℎ7
where � is the applied load, �, �, and ℎ are the beam
length, width, and thickness, respectively, and � is the
distance from the fixed wall to the location where we
want to measure stress.
In a more general sense, for a general biaxial stress or strain
field, three strains along different
axes at the same point must be measured to determine the
principal strains and stresses with
strain gages. While the stress field on the surface of a
symmetrically loaded cantilever beam is
uniaxial (except near the clamped end and loading point), the
stress at any point nevertheless
varies with angle about that point. The strain field (which, in
this case, is biaxial because of the
Poisson strain) varies similarly. Fig. 2 is a sketch showing a
polar plot of the normal stress and
strain at a point in a uniaxial stress field.
The three axes along which strains are to be measured can be
arbitrarily oriented about the point
of interest. For computational convenience, however, it is
preferable to space the measurement
axes apart by multiples of �, such as �/3 (60°) or �/4 (45°).
An integral array of strain gages
intended for simultaneous strain measurements about a point is
known as a “rosette”. Three-gage
strain rosettes are commercially available in two principal
forms corresponding to the above
angles. These are known as the “delta”, or equiangular rosette,
and the 45° rectangular rosette
(Fig. 3). The delta rosette is so-named because the strain-
sensitive elements are arranged in the
form of an equilateral triangle (i.e., two gages symmetrically
disposed 60° either side of a third
gage). Rectangular rosettes will be used in this experiment, and
the 3 gages are oriented 45° to
the adjacent gage (i.e., gage 1 is 45° from gage 2,
and gage 2 is 45° from gage 3).
Equipment: The following equipment will be required for each
group. Note that there are only three material
testing systems in the lab. Thus, three groups will
perform this experiment at one time.
• Micro-measurements pre-gaged cantilever
beam and associated mounting hardware Figure 3: Illustration of
2 typical strain gage
configurations.
Figure 2: Stress and strain
distributions about a given point in
loaded cantilever beam. Note how
values vary with the angle about
point.
BIOEN 4250: Laboratory 4 – Principal Stress and Strain
3
• digital calipers and ruler
• Model D4 data acquisition conditioner
• set of laboratory weights
Experimental
Procedure:
BE EXTREMELY CAREFUL WITH THE CANTELEVER
BEAM! IT CAN EASILY BE DAMAGED
OR DESTROYED VIA OVERLOADING OR APPLYING TOO
MUCH VIBRATION WHEN
PLACING THE WEIGHTS.
Sample Preparation and Mounting
1. The TAs will mount the cantilever beam to the frame on the
testing table breadboard and
connect the appropriate wires from the strain gage rosette to the
appropriate ports on the
Model D4 Data Acquisition Conditioner – jack #1 goes to gage
#1, so it gets connected to
port #1 on the D4 box, etc. Please review the setup before
proceeding.
2. Using a pair of calipers, measure the height and width of the
beam. This should be
measured at a point that is NOT covered in the protective
polymer coating applied around
and on top of the rosette. Measure each of these dimensions (in
millimeters) three times
and report the average.
3. Using a ruler (because the calipers are not long enough)
measure the distance from the
line the gage is mounted to, and the dent at the end of the beam
where the weights will
be hung. Be EXTREMELY CAREFUL that you do not touch the
gage or the nearby
wires during this process. Measure each of these dimensions (in
millimeters) three
times and report the average.
Material Testing
1. Open the Micro-Measurements D4 Software. It should be
found on the desktop. Wait for
the program to recognize the strain gages that are connected and
open the program. If
they are not recognized, the program will not open. If this
happens, just close the error
box and try again.
2. In the Micro-Measurements D4 software, load the needed
configuration file for this rosette
gage. Click on “File” then select “Load Configuration”. Select
the file named
“Rosette_Beam_Config.md4”.
a. This will load the specific gage factors for each of the three
strain gages in the
rosette gage, as well as specify that they are each acting in a
quarter bridge
configuration (as in 1/4 of a typical Wheat-Stone bridge).
b. It will make it so that the Micro-Measurements D4 software
ignores channel 4 for
the onscreen display, since we only have three gages in the
rosette.
c. It will enable shunt calibration for the gages.
d. This will also set the units to be recorded from the strain
gages as microstrain (��)
which is essentially (∆�/� x 10-6).
3. Enable the real-time display of data. Click on “Hardware”
and ensure that there is a check
next to “Real Time Display”. You will know this was
successful when the values in the
display change, become non-zero, and are not grayed out.
4. Allow the beam to come to rest (don’t touch it until the
numbers stop changing). Then zero
out the strain gage values to account for non-zero strain due
to gravity acting on the
beam (yes, they are that sensitive, so please be careful not to
over strain the beams,
they are also quite pricey!).
a. To do this, click on “Channels” and select “Zero All”. This
should only be done at
the very beginning before you take data. If you take data and
then zero it again
later, you will need to start over.
5. Define the recording interval in the Micro-Measurements D4
Software to be 0.125 (this
means that there are 0.125 seconds between each data point, so
the actual data collection
BIOEN 4250: Laboratory 4 – Principal Stress and Strain
4
rate is 8 Hz). This is done by clicking on “File” and selecting
“Record Interval”, and select
“0.125 seconds”.
6. Define the file name for the data file you are going to take
shortly. To do this, click on
“Select File” and setting a file name unique to your group (this
will be a text file).
a. Be sure to use a unique file name in the Micro-Measurements
D4 Software
for each group so you do not overwrite someone else’s data
files!
7. READ THIS ENTIRE STEP (parts a-c) PRIOR TO
PROCEEDING: Now we are going to
take the data. To this there is a basic pattern you will want to
follow. The data collection in
the D4 Software can be paused while you are in the process of
taking the data (such as
when you want to change the weights).
a. In general, you want to follow this pattern and repeat:
i. Apply desired weight
ii. Give the beam a few seconds to equilibrate
iii. Start Recording, record for 10-15 data points, Pause
Recording
iv. Remove the weight(s)
b. The weight levels you will apply to the end of the beam
(which need to be applied
at the point where the dent is visible) are going to be:
i. 0 grams
ii. 50 grams
iii. 100 grams
iv. 200 grams
v. 300 grams
vi. 500 grams
c. When you are done with the entire series of weights, click on
“Close Capture” after
you click on “Pause” the last time. This will write your data file
to the specified
location and file name.
8. Save your data. (Make sure that all group members have
copies of the data prior to leaving
the lab.)
9. On a full-size piece of paper, write the following information
and give it to the TA. Do not
leave the lab until the TA has looked at it and has said you are
okay to leave.
a. Group identifier, include day, group number, and station
letter (e.g., Wed_G3_S1)
b. First and last names of everyone in the group
c. Average of three measurement for beam height, width, and
length
Data
Analysis:
The goal of the data analysis is to determine both the principle
strains and stresses in the
cantilever beam used in the lab. The follow parameters need to
be calculated.
1. values of the principle strains imposed on the beam
2. angle of rotation that would rotate the axis of gage #1 to be
parallel with the
direction of the largest principle strain
3. values (with units) of the principle in-plane stresses,
calculated from the principle
strains found above
4. longitudinal stress calculated from the beam flexure equation
5. the load cell calibration factor as if this cantilever beam was
a load cell (as was
done in Lab 1). Report this value in units of �/�
(Newtons/strain)
Based on these calculated values, please create the plots
indicated in the report
instructions. Those calculated values above that are not needed
for the plots (such as
calibration factor) should be included in the text of your results
section in the lab report.
All plots should have a title, labeled X and Y axes (with units),
and a figure legend, as
necessary.
BIOEN 4250: Laboratory 4 – Principal Stress and Strain
5
Tips for the Data Analysis
Notation convention to be used here:
- Consider the direction of the first principle strain to be the x-
axis. According to beam
theory, this should be along the long axis of the beam itself.
Thus, strain and stress
along this axis will be referred to as �AA and �AA,
respectively.
- Consider the y-axis to be perpendicular to the x-axis and
parallel to the top surface
of the beam (the other axis of the top plane). Thus, strain and
stress along this axis
will be referred to as �BB and �BB,respectively.
- Consider the z-axis to be perpendicular to the top surface of
the beam. Thus, strain
along this axis will be referred to as �++.
- Refer to the strains measured by the three strain gages in the
rosette as �C, �7, and
�D, associated with gages #1, #2, and #3, respectively.
- Shear strain in the xy-plane will be referred to as �AB
(remembering that the shear
strain component in the engineering strain tensor �AB =
C
7
�AB).
Values of the Principle Strains Imposed on the Beam
In the rectangular rosettes used in this lab, gage #2 (the one in
the middle) is needed in
order to calculate the shear strain and thus provide the needed
information to rotate the
strain field to be along the principle strain directions. If all we
had was gages #1 and #3,
we would not know if they were oriented along the principle
directions or not. The equation
needed to calculate the shear strain from these gage
measurements is as follows:
�AB = 2�7 − (�C + �D)
Using the same principles applied in previous homework to
diagonalize a tensor in order
to determine the principle stresses, we can rotate the 2�2 planer
strain tensor to find the
values of the principle strains in that plane. As before, this is
done with Eigen values. This
diagonalized planer strain tensor is defined from the measured
gage strains as,
� = H
�CC �C7
�7C �77
I = J
�C
1
2
�AB
1
2
�AB �D
L
Further, as a review, the basic method to determine the
principle strains will be
|� − ��| = 0
where this time, � indicates the principle strains rather than
principle stresses.
Regarding the out of plane principle strain, �++, solve for �++
in terms of the principle strains
(�AA and �BB) ONLY, from the following equations. There
should be no stress terms in the
equation you use to find �++. These equations are from the
generalized form of Hooke’s
Law. As a hint, you can simplify the equations before solving
for �++ by setting �++ equal
to zero, as you know this to be the case from beam theory,
�AA =
�AA
�
−
��BB
�
−
��++
�
BIOEN 4250: Laboratory 4 – Principal Stress and Strain
6
�BB = −
��AA
�
+
�BB
�
−
��++
�
�++ = −
��AA
�
−
��BB
�
+
�++
�
where � is the Poisson’s ratio and is defined as,
� = S
�BB
�AA
S
Angle of Rotation (�U) to Rotate Axis of Gage #1 to be Parallel
with Direction of Largest
Principle Strain
This angle is the angle theta contained in the following equation
(�U). If you were to rotate
the rosette by this angle, the axis of gage #1 would be along the
direction of the largest
principle strain, gage #3 would be along the axis of the smaller
planer principle strain, and
gage #2 would be theoretically zero as there are no shear strain
on the top and bottom
free surfaces of a cantilever beam.
���X2�UY =
�AB
(�C − �D)
Values of Principle In-plane Stresses
The principle stresses will be (by definition) along the same
directions as the principle
strains. These are found using the generalized form of Hooke’s
Law, as shown in the set
of 3 equations above. Additionally, the Young’s Modulus for
the aluminum in these beams
is � ≈71,700 MPa. Since there is no stress applied in the z-
direction at the location of the
strain gage rosette, we can assume �++ = 0. Thus, the set of
equations simplify
significantly and can be re-written in the following form in
order to solve for the stresses of
interest:
�AA =
�
(1 − �7)
(�AA + ��BB)
�BB =
�
(1 − �7)
X�BB + ��AAY
As mentioned above our second stress here, �BB, will be in the
upper plane of the beam,
and transverse to the long axis of the beam. In a cantilever beam
such as the one we are
using today, this stress (�BB) is zero, and the calculated value
you get for it here should
be close to that value. Further, the stresses on the lower surface
of the beam would be
exactly opposite those on the top surface. This is why the beam
flexure equation in the
next section uses � (the half height of the beam) when finding
the longitudinal stress.
When a load is placed on the top of the beam, the result is that
the top surface is in tension
and the bottom surface is in compression in relation to the long
axis of the beam.
Longitudinal Stress Calculated from the Beam Flexure Equation
The longitudinal stress is the theoretical stress along the axis of
the beam, and should
correspond (in theory) with larger of the two principle stresses
calculated above (�AA). The
beam flexure equation is,
BIOEN 4250: Laboratory 4 – Principal Stress and Strain
7
�) = −
��
�
=
6�(� − �)
�ℎ7
where,
� – bending moment at the rosette center line (� = ��; N•m)
� – half of the height of the beam (� = [
7
= C
7
∙ �������� ������� ℎ���ℎ�; m)
� – 2nd moment of area for the beam cross section (m4)
� =
�ℎD
12
� – applied load (N)
� – is the length of the beam (i.e. you average measured length;
m)
� – is the distance between the location of beam fixing to the
location where we want
to measure stress. Note that (� − �) is your average measured
distance from the
line the gauge is mounted to and the dent at the end of the beam
where the weights
will be hung (in units m)
� – width of the beam (i.e., your average measured width; m)
ℎ – thickness of the beam (i.e., your average measured height of
beam; m)
Calculate Load Cell Calibration Factor
Base this calculation on the largest principle strain. See Lab 1
protocol if you still are
unsure how to do this procedure. It is just the same, but using
strain rather than voltage.
Please calculate the calibration factor and report it in units of
�/� (Newtons/strain).
BIOEN 4250: BIOMECHANICS I
Laboratory 4 – Principal Stress and Strain
Report Guide
Due Date: Tuesday, December 4, 2018 by 10:45am (submit via
Canvas)
Instructor: Lucas Timmins ([email protected])
TAs: Allen Lin ([email protected]), Kelly Smith
([email protected])
Each student must turn in a separate laboratory report
representing his or her own work. The report should be
prepared using MS Word or an equivalent word processor.
Grammar and style of the written will be evaluated
and included in the grading, so please proof your report, rewrite
the initial draft as necessary, and check for
spelling and other grammatical errors before submission. The
report should contain the following sections:
Title/Name:
Your report must include the following information (in the
following format) in the upper-left corner of the first
page:
BIOEN 4250 – Laboratory 4, Fall 2018: Principal Stress and
Strain
<YOUR NAME HERE>
<YOUR GROUP ID>
<DATE HERE>
Objective (1 paragraph):
State the purpose of the lab measurements and analysis.
Motivate the need for the measurements. State your
perception of the intended educational goals of the laboratory in
terms of learning new measurement and
analysis techniques. The objective section should be one
paragraph.
Methods: (no longer than 2 pages)
a) Describe the methods and step-by step procedure to perform
the measurements. For example,
• Describe the strain gage rosette configuration and orientation
of how it was mounted to the beam
• Describe how the beam was mounted
• State the material the beam was constructed from and provide
the elastic modulus
• Provide the beam dimensions (e.g., average beam width and
thickness, length between rosette
centerline and application of point load
• Describe the procedure for application of loads to the beam
and capturing strains
b) Describe the methods employed to analyze the data. For
example,
• Provide an overview of the analysis (i.e., step-by-step
description), including explicitly stating all
equations
• Provide an overview of the code/processes used to analyze the
data (Matlab is encouraged)
• State how the plots were created (e.g., Matlab, Sigmaplot)
Note: You do not need to include your analysis code in your lab
report. However, you do need to
provide a clear and concise explanation of what you did. Do not
give a line by line description of your
code.
Results/Discussion (2.5 pages, including a 1 page limit for
plots):
Discuss the following points, regarding the distribution of stress
in the cantilever beam, in connection to the plots
requested below. The description of your results and
interpretation (excluding plots) should be a maximum of
1.5 pages.
BIOEN 4250: Laboratory 4 – Report Guide
2
Basic Discussion Points
1) Where (through the thickness of the beam) is the maximum
axial (longitudinal) stress (longitudinal)?
Where is the minimum value? Provide an explanation why.
2) Where (through the thickness of the beam) is the maximum
shear stress? Where is it the minimum?
Provide an explanation why.
3) Describe why points 1 and 2 are what they are, and how they
are related.
4) If you did not know the elastic modulus of the beam, which
was given to you in this case, how would you
go about calculate it from the data you collected in this lab?
5) In the text of the results section, provide the resulting
“calibration constant” that you determined in units
of Newtons/strain, as well as your calculated angle of rotation
that would bring the axis of gage #1 in line
with the axis of the largest principal stress.
6) Address the discussion points specific to each plot as listed
below.
7) Interpret your results. Are they what you expected? Why or
why not?
Plot 1: Principal Strains vs. Applies Load. Determine all three
principal strains in the location of the strain gage
rosette and then plot these strains (in micro-strain) as a function
of applied load (in Newtons). You should have
a single value for each measurement (as you have 6 different
loading levels, you should have 6 data points per
line in your plot). See the tips below for how to do this in a
manner that will be consistent with the answer key.
Discuss: How do the strains differ from each other? Are the
values and magnitudes what you expected to see?
Plot 2: Principal Stress vs. Applied Load. Calculate the
principal stresses (there are only 2 that are non-zero)
based on the measured principal strains. Plot these principal
stresses (in MPa) as a function of the applied load
(in Newtons). You should have a single value for each
measurement (as you have 6 different loading levels, you
should have 6 data points per line in your plot). See the tips
below for how to do this in a manner that will be
consistent with the answer key.
Discuss: How do the stresses differ from each other? Are the
values and magnitudes what you expected to see,
and why? Explain why there is not three principal stresses on
this plot.
Plot 3: Comparison of Measured and Theoretical Longitudinal
Stress. On the same set of axes, plot both the
largest principal stress (measured longitudinal stress) and the
theoretical longitudinal stress as calculated from
the beam flexure equation (both in MPa) as functions of applied
load. You should have a single value for each
measurement (as you have 6 different loading levels, you should
have 6 data points per line in your plot). See
the tips below for how to do this in a manner that will be
consistent with the answer key.
Discuss: How do these two measures of the same quantity
compare? Explain any differences that you see, and
what could have caused them. Is this what you expected to see?
Notes and Tips:
• Regarding the calculation of strains, you will likely have some
noise in the strain gage measurements.
Also, as you will be manually starting and pausing the data
recording, each group will likely have a
different number of data points at each loading case. For
example, one group may have 15 data points
for the 100 g loading for gage #1, while another has 17 or 20
data points for that same measurement. To
ensure that you have a single data point for each loading case
for the plots above, take the average of a
given measurement rather than dealing with every data point.
With 6 loading levels (0 grams through 500
grams) you will have 6 averaged measured values for each of
the three strain gages in the strain gage
rosette. Please contact your TA if this is not clear.
• Regarding the calculation of Poisson’s ratio, calculate it
separately for each of the 5 non-zero loading
levels and use the overall average Poisson’s ratio for your
calculations when it is needed.
• When item 1 is plotted as a function of item 2, item 2 is on the
X-axis and item 1 is on the Y-axis (i.e.,
� = �(�) denotes � is a function of �, and � would be on the
Y-axis).
Date Time200245-Ch 1 ue200245-Ch 2 ue200245-Ch 3
ue11/15/18 02:17:13:855000011/15/18
02:17:13:995400011/15/18 02:17:14:126000011/15/18
02:17:14:256000011/15/18 02:17:14:375900011/15/18
02:17:14:506000011/15/18 02:17:14:636700011/15/18
02:17:14:766500011/15/18 02:17:14:897500011/15/18
02:17:15:017500011/15/18 02:17:15:137500011/15/18
02:17:15:267500011/15/18 02:17:15:387500011/15/18
02:17:15:527500011/15/18 02:17:15:667800011/15/18
02:17:15:797600011/15/18 02:17:15:938100011/15/18
02:17:16:068700011/15/18 02:17:40:9941-6232611/15/18
02:17:41:1241-6232611/15/18 02:17:41:2541-6232711/15/18
02:17:41:3844-6232711/15/18 02:17:41:5046-6232711/15/18
02:17:41:6250-6232711/15/18 02:17:41:7549-6232711/15/18
02:17:41:8849-6232711/15/18 02:17:42:0255-6232711/15/18
02:17:42:1555-6232711/15/18 02:17:42:2858-6232711/15/18
02:17:42:4165-6232711/15/18 02:17:42:5466-6232711/15/18
02:17:42:6866-6232711/15/18 02:17:42:8165-6232711/15/18
02:17:42:9466-6232711/15/18 02:17:43:0666-6232711/15/18
02:17:43:1869-6242711/15/18 02:17:43:3171-6242711/15/18
02:18:17:8356-12485311/15/18 02:18:17:9556-11485511/15/18
02:18:18:0757-11485511/15/18 02:18:18:2057-11455411/15/18
02:18:18:3356-11455411/15/18 02:18:18:4658-12465211/15/18
02:18:18:6065-12465211/15/18 02:18:18:7468-12485311/15/18
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BIOEN 4250 BIOMECHANICS I Laboratory 4 – Principle Stres.docx

  • 1. BIOEN 4250: BIOMECHANICS I Laboratory 4 – Principle Stress and Strain November 13– 16, 2018 TAs: Allen Lin ([email protected]), Kelly Smith ([email protected]) Lab Quiz: A 10-point lab quiz, accounting for 10% of the lap report grade, will be given at the beginning of class. Be familiar with the entire protocol. Objective: The objective of this experiment is to measure the strains along three different axes surrounding a point on a cantilever beam, calculate the principal strains and stresses, and compare the result with the stress calculated from the flexure formula for such a beam. Background: The ability to measure strain is critical to materials testing as well as many other applications in engineering. However, strain gages that adhere to a surface can alter the local strain environment if the material (or tissue) of interest is less stiff than the gage itself. For this reason, contact strain gages (or strain gages that attach directly to a surface) are not typically used for the testing of soft
  • 2. tissues such as ligament, arteries, or skin. However, when the material is on the stiffer side, or when the absolute value of the strain is less important than the detection of the mere presence of strain itself, contact strain gages are very useful. An example of a stiffer biological material would be bone. However, due to the porous nature of bone, one needs to be extremely careful that the strain gage is properly adhered to the material’s surface. Other applications range from real world stress analysis of a structure (e.g., a wing of an aircraft during flight) to strain gages incorporated into medical equipment to ensure proper function (e.g., gages wrapped around the tubing in a hospital infusion pump to detect blockages in the line – since the tube swells more than it should when the fluid path is occluded). One common engineering loading case that involves a planar stress field (i.e., the only non-zero stresses are in the same plane), is that of beam bending. Beam bending will be covered in greater detail during lecture. However, in order to ensure you know the basics of what is going on in this lab, we will cover some fundamental topics. The simplest case of beam loading is that of a cantilever beam that is completely anchored at one end and loaded at a point along its length (Fig. 1). In Figure 1, � is the applied load, ℎ is the thickness of the beam (with � as the half- thickness), � is the distance from the fixed wall to the location where we want to measure stress and strain (point �), and � is the length of the beam. There are a couple key points to know about
  • 3. this loading scenario: 1. As the beam bends downward, the material above the midline (the dashed line) is in tension and the material below that line is in compression. 2. At the top and bottom free surfaces, there is only axial stress, and zero shear stress. 3. At the midline (dashed line, also referred to as neutral axis) there is zero axial stress and it is the location of the maximum shear stress (as everything above it is in tension and everything below it is in compression). 4. Theoretically, the only applied non-zero stress on the upper and lower free surfaces is the longitudinal stress, which is the stress component oriented along the length of the beam. However, in an experimental measurement, due to various factors (e.g., experimental or Figure 1: Illustration of a cantilever beam fully supported at one end. BIOEN 4250: Laboratory 4 – Principal Stress and Strain 2
  • 4. calculation error, noise in the signal) very small transverse stresses in the plane of the free surface do still typically appear. The equation for the longitudinal stress (�); introduced as �++ in lecture) is known as the beam flexure equation, and is as follows, �) = − �� � where � is the bending moment, � is the distance from neutral axis, and � is the second moment of area about the neutral axis. To quantify to maximum/minimum value for longitudinal stress, the equation can be evaluated at the top and bottom of the beam (i.e., � = ±ℎ 23 ), as follows, �) = ± 6�(� − �) �ℎ7 where � is the applied load, �, �, and ℎ are the beam length, width, and thickness, respectively, and � is the distance from the fixed wall to the location where we want to measure stress. In a more general sense, for a general biaxial stress or strain
  • 5. field, three strains along different axes at the same point must be measured to determine the principal strains and stresses with strain gages. While the stress field on the surface of a symmetrically loaded cantilever beam is uniaxial (except near the clamped end and loading point), the stress at any point nevertheless varies with angle about that point. The strain field (which, in this case, is biaxial because of the Poisson strain) varies similarly. Fig. 2 is a sketch showing a polar plot of the normal stress and strain at a point in a uniaxial stress field. The three axes along which strains are to be measured can be arbitrarily oriented about the point of interest. For computational convenience, however, it is preferable to space the measurement axes apart by multiples of �, such as �/3 (60°) or �/4 (45°). An integral array of strain gages intended for simultaneous strain measurements about a point is known as a “rosette”. Three-gage strain rosettes are commercially available in two principal forms corresponding to the above angles. These are known as the “delta”, or equiangular rosette, and the 45° rectangular rosette (Fig. 3). The delta rosette is so-named because the strain- sensitive elements are arranged in the form of an equilateral triangle (i.e., two gages symmetrically disposed 60° either side of a third gage). Rectangular rosettes will be used in this experiment, and the 3 gages are oriented 45° to the adjacent gage (i.e., gage 1 is 45° from gage 2, and gage 2 is 45° from gage 3).
  • 6. Equipment: The following equipment will be required for each group. Note that there are only three material testing systems in the lab. Thus, three groups will perform this experiment at one time. • Micro-measurements pre-gaged cantilever beam and associated mounting hardware Figure 3: Illustration of 2 typical strain gage configurations. Figure 2: Stress and strain distributions about a given point in loaded cantilever beam. Note how values vary with the angle about point. BIOEN 4250: Laboratory 4 – Principal Stress and Strain 3 • digital calipers and ruler • Model D4 data acquisition conditioner • set of laboratory weights Experimental Procedure:
  • 7. BE EXTREMELY CAREFUL WITH THE CANTELEVER BEAM! IT CAN EASILY BE DAMAGED OR DESTROYED VIA OVERLOADING OR APPLYING TOO MUCH VIBRATION WHEN PLACING THE WEIGHTS. Sample Preparation and Mounting 1. The TAs will mount the cantilever beam to the frame on the testing table breadboard and connect the appropriate wires from the strain gage rosette to the appropriate ports on the Model D4 Data Acquisition Conditioner – jack #1 goes to gage #1, so it gets connected to port #1 on the D4 box, etc. Please review the setup before proceeding. 2. Using a pair of calipers, measure the height and width of the beam. This should be measured at a point that is NOT covered in the protective polymer coating applied around and on top of the rosette. Measure each of these dimensions (in millimeters) three times and report the average. 3. Using a ruler (because the calipers are not long enough) measure the distance from the line the gage is mounted to, and the dent at the end of the beam where the weights will be hung. Be EXTREMELY CAREFUL that you do not touch the gage or the nearby wires during this process. Measure each of these dimensions (in millimeters) three times and report the average.
  • 8. Material Testing 1. Open the Micro-Measurements D4 Software. It should be found on the desktop. Wait for the program to recognize the strain gages that are connected and open the program. If they are not recognized, the program will not open. If this happens, just close the error box and try again. 2. In the Micro-Measurements D4 software, load the needed configuration file for this rosette gage. Click on “File” then select “Load Configuration”. Select the file named “Rosette_Beam_Config.md4”. a. This will load the specific gage factors for each of the three strain gages in the rosette gage, as well as specify that they are each acting in a quarter bridge configuration (as in 1/4 of a typical Wheat-Stone bridge). b. It will make it so that the Micro-Measurements D4 software ignores channel 4 for the onscreen display, since we only have three gages in the rosette. c. It will enable shunt calibration for the gages. d. This will also set the units to be recorded from the strain gages as microstrain (��) which is essentially (∆�/� x 10-6). 3. Enable the real-time display of data. Click on “Hardware” and ensure that there is a check
  • 9. next to “Real Time Display”. You will know this was successful when the values in the display change, become non-zero, and are not grayed out. 4. Allow the beam to come to rest (don’t touch it until the numbers stop changing). Then zero out the strain gage values to account for non-zero strain due to gravity acting on the beam (yes, they are that sensitive, so please be careful not to over strain the beams, they are also quite pricey!). a. To do this, click on “Channels” and select “Zero All”. This should only be done at the very beginning before you take data. If you take data and then zero it again later, you will need to start over. 5. Define the recording interval in the Micro-Measurements D4 Software to be 0.125 (this means that there are 0.125 seconds between each data point, so the actual data collection BIOEN 4250: Laboratory 4 – Principal Stress and Strain 4 rate is 8 Hz). This is done by clicking on “File” and selecting “Record Interval”, and select “0.125 seconds”. 6. Define the file name for the data file you are going to take
  • 10. shortly. To do this, click on “Select File” and setting a file name unique to your group (this will be a text file). a. Be sure to use a unique file name in the Micro-Measurements D4 Software for each group so you do not overwrite someone else’s data files! 7. READ THIS ENTIRE STEP (parts a-c) PRIOR TO PROCEEDING: Now we are going to take the data. To this there is a basic pattern you will want to follow. The data collection in the D4 Software can be paused while you are in the process of taking the data (such as when you want to change the weights). a. In general, you want to follow this pattern and repeat: i. Apply desired weight ii. Give the beam a few seconds to equilibrate iii. Start Recording, record for 10-15 data points, Pause Recording iv. Remove the weight(s) b. The weight levels you will apply to the end of the beam (which need to be applied at the point where the dent is visible) are going to be: i. 0 grams ii. 50 grams iii. 100 grams iv. 200 grams v. 300 grams vi. 500 grams c. When you are done with the entire series of weights, click on
  • 11. “Close Capture” after you click on “Pause” the last time. This will write your data file to the specified location and file name. 8. Save your data. (Make sure that all group members have copies of the data prior to leaving the lab.) 9. On a full-size piece of paper, write the following information and give it to the TA. Do not leave the lab until the TA has looked at it and has said you are okay to leave. a. Group identifier, include day, group number, and station letter (e.g., Wed_G3_S1) b. First and last names of everyone in the group c. Average of three measurement for beam height, width, and length Data Analysis: The goal of the data analysis is to determine both the principle strains and stresses in the cantilever beam used in the lab. The follow parameters need to be calculated. 1. values of the principle strains imposed on the beam 2. angle of rotation that would rotate the axis of gage #1 to be parallel with the direction of the largest principle strain 3. values (with units) of the principle in-plane stresses,
  • 12. calculated from the principle strains found above 4. longitudinal stress calculated from the beam flexure equation 5. the load cell calibration factor as if this cantilever beam was a load cell (as was done in Lab 1). Report this value in units of �/� (Newtons/strain) Based on these calculated values, please create the plots indicated in the report instructions. Those calculated values above that are not needed for the plots (such as calibration factor) should be included in the text of your results section in the lab report. All plots should have a title, labeled X and Y axes (with units), and a figure legend, as necessary. BIOEN 4250: Laboratory 4 – Principal Stress and Strain 5 Tips for the Data Analysis Notation convention to be used here: - Consider the direction of the first principle strain to be the x- axis. According to beam theory, this should be along the long axis of the beam itself. Thus, strain and stress
  • 13. along this axis will be referred to as �AA and �AA, respectively. - Consider the y-axis to be perpendicular to the x-axis and parallel to the top surface of the beam (the other axis of the top plane). Thus, strain and stress along this axis will be referred to as �BB and �BB,respectively. - Consider the z-axis to be perpendicular to the top surface of the beam. Thus, strain along this axis will be referred to as �++. - Refer to the strains measured by the three strain gages in the rosette as �C, �7, and �D, associated with gages #1, #2, and #3, respectively. - Shear strain in the xy-plane will be referred to as �AB (remembering that the shear strain component in the engineering strain tensor �AB = C 7 �AB). Values of the Principle Strains Imposed on the Beam In the rectangular rosettes used in this lab, gage #2 (the one in the middle) is needed in order to calculate the shear strain and thus provide the needed information to rotate the strain field to be along the principle strain directions. If all we had was gages #1 and #3, we would not know if they were oriented along the principle directions or not. The equation needed to calculate the shear strain from these gage
  • 14. measurements is as follows: �AB = 2�7 − (�C + �D) Using the same principles applied in previous homework to diagonalize a tensor in order to determine the principle stresses, we can rotate the 2�2 planer strain tensor to find the values of the principle strains in that plane. As before, this is done with Eigen values. This diagonalized planer strain tensor is defined from the measured gage strains as, � = H �CC �C7 �7C �77 I = J �C 1 2 �AB 1 2 �AB �D L Further, as a review, the basic method to determine the principle strains will be
  • 15. |� − ��| = 0 where this time, � indicates the principle strains rather than principle stresses. Regarding the out of plane principle strain, �++, solve for �++ in terms of the principle strains (�AA and �BB) ONLY, from the following equations. There should be no stress terms in the equation you use to find �++. These equations are from the generalized form of Hooke’s Law. As a hint, you can simplify the equations before solving for �++ by setting �++ equal to zero, as you know this to be the case from beam theory, �AA = �AA � − ��BB � − ��++ � BIOEN 4250: Laboratory 4 – Principal Stress and Strain 6
  • 16. �BB = − ��AA � + �BB � − ��++ � �++ = − ��AA � − ��BB � + �++ � where � is the Poisson’s ratio and is defined as, � = S �BB �AA S
  • 17. Angle of Rotation (�U) to Rotate Axis of Gage #1 to be Parallel with Direction of Largest Principle Strain This angle is the angle theta contained in the following equation (�U). If you were to rotate the rosette by this angle, the axis of gage #1 would be along the direction of the largest principle strain, gage #3 would be along the axis of the smaller planer principle strain, and gage #2 would be theoretically zero as there are no shear strain on the top and bottom free surfaces of a cantilever beam. ���X2�UY = �AB (�C − �D) Values of Principle In-plane Stresses The principle stresses will be (by definition) along the same directions as the principle strains. These are found using the generalized form of Hooke’s Law, as shown in the set of 3 equations above. Additionally, the Young’s Modulus for the aluminum in these beams is � ≈71,700 MPa. Since there is no stress applied in the z- direction at the location of the strain gage rosette, we can assume �++ = 0. Thus, the set of equations simplify significantly and can be re-written in the following form in order to solve for the stresses of interest:
  • 18. �AA = � (1 − �7) (�AA + ��BB) �BB = � (1 − �7) X�BB + ��AAY As mentioned above our second stress here, �BB, will be in the upper plane of the beam, and transverse to the long axis of the beam. In a cantilever beam such as the one we are using today, this stress (�BB) is zero, and the calculated value you get for it here should be close to that value. Further, the stresses on the lower surface of the beam would be exactly opposite those on the top surface. This is why the beam flexure equation in the next section uses � (the half height of the beam) when finding the longitudinal stress. When a load is placed on the top of the beam, the result is that the top surface is in tension and the bottom surface is in compression in relation to the long axis of the beam. Longitudinal Stress Calculated from the Beam Flexure Equation The longitudinal stress is the theoretical stress along the axis of the beam, and should
  • 19. correspond (in theory) with larger of the two principle stresses calculated above (�AA). The beam flexure equation is, BIOEN 4250: Laboratory 4 – Principal Stress and Strain 7 �) = − �� � = 6�(� − �) �ℎ7 where, � – bending moment at the rosette center line (� = ��; N•m) � – half of the height of the beam (� = [ 7 = C 7 ∙ �������� ������� ℎ���ℎ�; m) � – 2nd moment of area for the beam cross section (m4)
  • 20. � = �ℎD 12 � – applied load (N) � – is the length of the beam (i.e. you average measured length; m) � – is the distance between the location of beam fixing to the location where we want to measure stress. Note that (� − �) is your average measured distance from the line the gauge is mounted to and the dent at the end of the beam where the weights will be hung (in units m) � – width of the beam (i.e., your average measured width; m) ℎ – thickness of the beam (i.e., your average measured height of beam; m) Calculate Load Cell Calibration Factor Base this calculation on the largest principle strain. See Lab 1 protocol if you still are unsure how to do this procedure. It is just the same, but using strain rather than voltage. Please calculate the calibration factor and report it in units of �/� (Newtons/strain).
  • 21. BIOEN 4250: BIOMECHANICS I Laboratory 4 – Principal Stress and Strain Report Guide Due Date: Tuesday, December 4, 2018 by 10:45am (submit via Canvas) Instructor: Lucas Timmins ([email protected]) TAs: Allen Lin ([email protected]), Kelly Smith ([email protected]) Each student must turn in a separate laboratory report representing his or her own work. The report should be prepared using MS Word or an equivalent word processor. Grammar and style of the written will be evaluated and included in the grading, so please proof your report, rewrite the initial draft as necessary, and check for spelling and other grammatical errors before submission. The report should contain the following sections: Title/Name: Your report must include the following information (in the following format) in the upper-left corner of the first page: BIOEN 4250 – Laboratory 4, Fall 2018: Principal Stress and Strain <YOUR NAME HERE> <YOUR GROUP ID> <DATE HERE> Objective (1 paragraph):
  • 22. State the purpose of the lab measurements and analysis. Motivate the need for the measurements. State your perception of the intended educational goals of the laboratory in terms of learning new measurement and analysis techniques. The objective section should be one paragraph. Methods: (no longer than 2 pages) a) Describe the methods and step-by step procedure to perform the measurements. For example, • Describe the strain gage rosette configuration and orientation of how it was mounted to the beam • Describe how the beam was mounted • State the material the beam was constructed from and provide the elastic modulus • Provide the beam dimensions (e.g., average beam width and thickness, length between rosette centerline and application of point load • Describe the procedure for application of loads to the beam and capturing strains b) Describe the methods employed to analyze the data. For example, • Provide an overview of the analysis (i.e., step-by-step description), including explicitly stating all equations • Provide an overview of the code/processes used to analyze the data (Matlab is encouraged) • State how the plots were created (e.g., Matlab, Sigmaplot)
  • 23. Note: You do not need to include your analysis code in your lab report. However, you do need to provide a clear and concise explanation of what you did. Do not give a line by line description of your code. Results/Discussion (2.5 pages, including a 1 page limit for plots): Discuss the following points, regarding the distribution of stress in the cantilever beam, in connection to the plots requested below. The description of your results and interpretation (excluding plots) should be a maximum of 1.5 pages. BIOEN 4250: Laboratory 4 – Report Guide 2 Basic Discussion Points 1) Where (through the thickness of the beam) is the maximum axial (longitudinal) stress (longitudinal)? Where is the minimum value? Provide an explanation why. 2) Where (through the thickness of the beam) is the maximum shear stress? Where is it the minimum? Provide an explanation why. 3) Describe why points 1 and 2 are what they are, and how they are related. 4) If you did not know the elastic modulus of the beam, which
  • 24. was given to you in this case, how would you go about calculate it from the data you collected in this lab? 5) In the text of the results section, provide the resulting “calibration constant” that you determined in units of Newtons/strain, as well as your calculated angle of rotation that would bring the axis of gage #1 in line with the axis of the largest principal stress. 6) Address the discussion points specific to each plot as listed below. 7) Interpret your results. Are they what you expected? Why or why not? Plot 1: Principal Strains vs. Applies Load. Determine all three principal strains in the location of the strain gage rosette and then plot these strains (in micro-strain) as a function of applied load (in Newtons). You should have a single value for each measurement (as you have 6 different loading levels, you should have 6 data points per line in your plot). See the tips below for how to do this in a manner that will be consistent with the answer key. Discuss: How do the strains differ from each other? Are the values and magnitudes what you expected to see? Plot 2: Principal Stress vs. Applied Load. Calculate the principal stresses (there are only 2 that are non-zero) based on the measured principal strains. Plot these principal stresses (in MPa) as a function of the applied load (in Newtons). You should have a single value for each measurement (as you have 6 different loading levels, you should have 6 data points per line in your plot). See the tips
  • 25. below for how to do this in a manner that will be consistent with the answer key. Discuss: How do the stresses differ from each other? Are the values and magnitudes what you expected to see, and why? Explain why there is not three principal stresses on this plot. Plot 3: Comparison of Measured and Theoretical Longitudinal Stress. On the same set of axes, plot both the largest principal stress (measured longitudinal stress) and the theoretical longitudinal stress as calculated from the beam flexure equation (both in MPa) as functions of applied load. You should have a single value for each measurement (as you have 6 different loading levels, you should have 6 data points per line in your plot). See the tips below for how to do this in a manner that will be consistent with the answer key. Discuss: How do these two measures of the same quantity compare? Explain any differences that you see, and what could have caused them. Is this what you expected to see? Notes and Tips: • Regarding the calculation of strains, you will likely have some noise in the strain gage measurements. Also, as you will be manually starting and pausing the data recording, each group will likely have a different number of data points at each loading case. For example, one group may have 15 data points for the 100 g loading for gage #1, while another has 17 or 20 data points for that same measurement. To ensure that you have a single data point for each loading case
  • 26. for the plots above, take the average of a given measurement rather than dealing with every data point. With 6 loading levels (0 grams through 500 grams) you will have 6 averaged measured values for each of the three strain gages in the strain gage rosette. Please contact your TA if this is not clear. • Regarding the calculation of Poisson’s ratio, calculate it separately for each of the 5 non-zero loading levels and use the overall average Poisson’s ratio for your calculations when it is needed. • When item 1 is plotted as a function of item 2, item 2 is on the X-axis and item 1 is on the Y-axis (i.e., � = �(�) denotes � is a function of �, and � would be on the Y-axis). Date Time200245-Ch 1 ue200245-Ch 2 ue200245-Ch 3 ue11/15/18 02:17:13:855000011/15/18 02:17:13:995400011/15/18 02:17:14:126000011/15/18 02:17:14:256000011/15/18 02:17:14:375900011/15/18 02:17:14:506000011/15/18 02:17:14:636700011/15/18 02:17:14:766500011/15/18 02:17:14:897500011/15/18 02:17:15:017500011/15/18 02:17:15:137500011/15/18 02:17:15:267500011/15/18 02:17:15:387500011/15/18 02:17:15:527500011/15/18 02:17:15:667800011/15/18 02:17:15:797600011/15/18 02:17:15:938100011/15/18 02:17:16:068700011/15/18 02:17:40:9941-6232611/15/18 02:17:41:1241-6232611/15/18 02:17:41:2541-6232711/15/18 02:17:41:3844-6232711/15/18 02:17:41:5046-6232711/15/18 02:17:41:6250-6232711/15/18 02:17:41:7549-6232711/15/18 02:17:41:8849-6232711/15/18 02:17:42:0255-6232711/15/18 02:17:42:1555-6232711/15/18 02:17:42:2858-6232711/15/18 02:17:42:4165-6232711/15/18 02:17:42:5466-6232711/15/18 02:17:42:6866-6232711/15/18 02:17:42:8165-6232711/15/18 02:17:42:9466-6232711/15/18 02:17:43:0666-6232711/15/18
  • 27. 02:17:43:1869-6242711/15/18 02:17:43:3171-6242711/15/18 02:18:17:8356-12485311/15/18 02:18:17:9556-11485511/15/18 02:18:18:0757-11485511/15/18 02:18:18:2057-11455411/15/18 02:18:18:3356-11455411/15/18 02:18:18:4658-12465211/15/18 02:18:18:6065-12465211/15/18 02:18:18:7468-12485311/15/18 02:18:18:8871-12485311/15/18 02:18:19:0174-11475411/15/18 02:18:19:1477-11475411/15/18 02:18:19:2782-11465311/15/18 02:18:19:4088-11465311/15/18 02:18:19:5387-11475311/15/18 02:18:35:0362-219410811/15/18 02:18:35:1669- 219410811/15/18 02:18:35:2976-219410811/15/18 02:18:35:4276-219410811/15/18 02:18:35:5581- 219410711/15/18 02:18:35:6787-219410711/15/18 02:18:35:7989-219410711/15/18 02:18:35:9290- 219410811/15/18 02:18:36:0595-219410811/15/18 02:18:36:1900-219310711/15/18 02:18:36:3203- 219310711/15/18 02:18:36:4507-219410711/15/18 02:18:36:5808-219410711/15/18 02:18:36:7114- 219410711/15/18 02:18:36:8318-219410711/15/18 02:18:56:4700-3114016211/15/18 02:18:56:6000- 3114016111/15/18 02:18:56:7303-3114016111/15/18 02:18:56:8603-3114016111/15/18 02:18:56:9903- 3114016111/15/18 02:18:57:1203-3114116111/15/18 02:18:57:2403-3114116111/15/18 02:18:57:3702- 3114116011/15/18 02:18:57:4906-3114116011/15/18 02:18:57:6205-3114116011/15/18 02:18:57:7506- 3114116011/15/18 02:18:57:8906-3114116011/15/18 02:18:58:0206-3114216011/15/18 02:18:58:1508- 3114216011/15/18 02:18:58:2810-3114216011/15/18 02:18:58:4113-3114216011/15/18 02:18:58:5415- 3114116011/15/18 02:18:58:6720-3114116011/15/18 02:18:58:8020-3114116111/15/18 02:18:58:9321- 3114116111/15/18 02:19:23:0232-5523427211/15/18 02:19:23:1437-5423626911/15/18 02:19:23:2737- 5423626911/15/18 02:19:23:3940-5423427211/15/18 02:19:23:5242-5423427211/15/18 02:19:23:6546- 5523626911/15/18 02:19:23:7946-5523626911/15/18
  • 28. 02:19:23:9246-5423427211/15/18 02:19:24:0546- 5423427211/15/18 02:19:24:1850-5523527011/15/18 02:19:24:3152-5523527011/15/18 02:19:24:4452- 5523527211/15/18 02:19:24:5757-5523527211/15/18 02:19:24:7156-5523427011/15/18 02:19:24:8357- 5523427011/15/18 02:19:24:9560-5523427011/15/18 02:19:25:0860-5423627011/15/18 02:19:25:2166- 5423627011/15/18 02:19:25:3472-55233271