Mechanical Engineering Lab

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, 𝑃 is the applied load, ℎ is the thickness of the beam (with 𝑐 as the half- thickness), 𝑥 is the distance from the fixed wall to the location where we want to measure stress and strain (point 𝑎), and 𝐿 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.

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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 (𝜎); introduced as 𝑇++ in lecture) is known as the beam flexure equation, and is as follows,

𝜎) = − 𝑀𝑧 𝐼

where 𝑀 is the bending moment, 𝑧 is the distance from

neutral axis, and 𝐼 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., 𝑧 = ±ℎ 23 ), as follows,

𝜎) = ± 6𝑃(𝐿 − 𝑥)

𝑏ℎ7

where 𝑃 is the applied load, 𝐿, 𝑏, and ℎ are the beam length, width, and thickness, respectively, and 𝑥 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 𝜋, such as 𝜋/3 (60°) or 𝜋/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.

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• 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 (𝜇𝑒)

which is essentially (∆𝐿/𝐿 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

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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 𝑁/𝜀 (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.

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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 𝜀AA and 𝑇AA, 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 𝜀BB and 𝑇BB,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 𝜀++.

- Refer to the strains measured by the three strain gages in the rosette as 𝜀C, 𝜀7, and 𝜀D, associated with gages #1, #2, and #3, respectively.

- Shear strain in the xy-plane will be referred to as 𝛾AB (remembering that the shear strain component in the engineering strain tensor 𝜀AB =

C 7 𝛾AB).

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:

𝛾AB = 2𝜀7 − (𝜀C + 𝜀D) Using the same principles applied in previous homework to diagonalize a tensor in order to determine the principle stresses, we can rotate the 2𝑥2 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,

𝒆 = H 𝑒CC 𝑒C7 𝑒7C 𝑒77

I = J 𝜀C

1 2 𝛾AB

1 2 𝛾AB 𝜀D

L

Further, as a review, the basic method to determine the principle strains will be

|𝒆 − 𝜆𝑰| = 0 where this time, 𝜆 indicates the principle strains rather than principle stresses. Regarding the out of plane principle strain, 𝜀++, solve for 𝜀++ in terms of the principle strains (𝜀AA and 𝜀BB) ONLY, from the following equations. There should be no stress terms in the equation you use to find 𝜀++. These equations are from the generalized form of Hooke’s Law. As a hint, you can simplify the equations before solving for 𝜀++ by setting 𝑇++ equal to zero, as you know this to be the case from beam theory,

𝜀AA = 𝑇AA 𝐸 − 𝜈𝑇BB 𝐸

− 𝜈𝑇++ 𝐸

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𝜀BB = − 𝜈𝑇AA 𝐸

+ 𝑇BB 𝐸 − 𝜈𝑇++ 𝐸

𝜀++ = − 𝜈𝑇AA 𝐸

− 𝜈𝑇BB 𝐸

+ 𝑇++ 𝐸

where 𝜈 is the Poisson’s ratio and is defined as,

𝜈 = S 𝜀BB 𝜀AA

S

Angle of Rotation (𝜃U) 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 (𝜃U). 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.

𝑡𝑎𝑛X2𝜃UY = 𝛾AB

(𝜀C − 𝜀D)

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 𝐸 ≈71,700 MPa. Since there is no stress applied in the z-direction at the location of the strain gage rosette, we can assume 𝑇++ = 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:

𝑇AA = 𝐸

(1 − 𝜈7) (𝜀AA + 𝜈𝜀BB)

𝑇BB = 𝐸

(1 − 𝜈7) X𝜀BB + 𝜈𝜀AAY

As mentioned above our second stress here, 𝑇BB, 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 (𝑇BB) 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 𝑐 (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 (𝑇AA). The beam flexure equation is,

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𝜎) = − 𝑀𝑐 𝐼 = 6𝑃(𝐿 − 𝑥)

𝑏ℎ7

where,

𝑀 – bending moment at the rosette center line (𝑀 = 𝑃𝐿; N•m) 𝑐 – half of the height of the beam (𝑐 = [

7 = C

7 ∙ 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 ℎ𝑒𝑖𝑔ℎ𝑡; m)

𝐼 – 2nd moment of area for the beam cross section (m4)

𝐼 = 𝑏ℎD

12

𝑃 – applied load (N) 𝐿 – is the length of the beam (i.e. you average measured length; m) 𝑥 – is the distance between the location of beam fixing to the location where we want

to measure stress. Note that (𝐿 − 𝑥) 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)

𝑏 – 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 𝑁/𝜀 (Newtons/strain).