Bending and Buckling

Objective

In this laboratory, students will experimentally investigate the behavior of beams and columns subject to externally applied forces and displacements. First, students will explore the force-deflection and displacement-deflection behavior of a cantilever beam through surface strain measurements. Data from a rectangular rosette strain gauge will be used to determine the state of strain on the surface of a thin rectangular beam. Specifically, rosette strain analysis (strain transformation) theory can be used to determine the principal strain components of a beam under pure bending. Experiments will examine two types of boundary conditions: load-controlled, where prescribed loads are applied at the free end of the beam; and displacement-controlled, where known displacements (measured here with a precision micrometer) are applied to the free end. The measured beam behavior can then be compared to elementary Euler-Bernoulli beam theory predictions to determine Young’s modulus, Poisson’s ratio, and effective beam stiffness. Second, students will explore the buckling and post-buckling behavior of slender columns loaded axially along their length. This part of the experiment includes two tests: a long column and a short column buckling test. Each test will explore the change in critical buckling strength caused by changes in geometry and loading conditions. In addition, students will gain familiarity with buckling analysis techniques such as the Southwell plot.

Background

Simple Beam Theory

A beam is a structure whose length is much larger than its other two principal dimensions. In many cases, aircraft structures such as wings and fuselages can be treated as thin-walled beams. Simple beam theory is based on the Euler-Bernoulli assumptions (thus, simple beam theory is also known as Euler-Bernoulli beam theory). Specifically, those assumptions are: 1) the beam cross-section is rigid and does not (significantly) deform under the applied axial or transverse load; 2) the cross-section remains plane after deformation; and 3) the cross-section remains normal to the deformed axis of the beam. These assumptions have been extensively validated for long, thin beams formed from isotropic materials with solid cross-sections, as well as many other beams under “small” deflections

Applying these assumptions allows us to describe the behavior of the beam under load as a one-dimensional function, i.e., solely as a function of the distance along the beam length. For example, the deflection of a beam under an applied load is just a function of the load distribution, the Young’s modulus (E), and the geometry of the cross-section of the beam. The predictions of simple beam theory are available in numerous references, e.g. the text by Bauchau and Craig$^1$.

$^1$O. Bauchau and J. Craig, Structural Analysis: With Applications to Aerospace Structures, Springer, 2009.

Strain Transformation Theory

In this lab, we will study a cantilever beam that is subjected to applied loads and applied displacements. A rectangular rosette strain gauge will be used to measure the components of strain on the beam surface near the root in a reference frame that is not aligned with the beam axis. Therefore, it will be necessary to consider a transformation of the components of strain between coordinate frames. If we look at the stresses and strains in a beam subject to transverse load, as shown in Figure 4(a), a good approximation is that only the longitudinal (axial) components of stress exist along the length of the beam. If we use the coordinate frame defined in Figure 4(a), then the axial components of stress correspond to$\sigma_x$.

Figure 4. Cantilever beam schematic.

Using Hooke's law for an isotropic material, the non-zero observed components of strain, also with respect to the X-Y-Z coordinate frame defined in the figure, should be:

$$\large \epsilon_X=\frac{\sigma_x}{E} \\ \space \\ \space \\ \epsilon_Y=-\nu\frac{\sigma_x}{E} \\ \space \\ \space \\ \epsilon_Z=-\nu\frac{\sigma_x}{E}$$

(1)

where E is the elastic modulus and $v$ is the Poisson ratio. However, the rectangular rosette strain gauge that is used for these experiments will not be aligned with the X-Y-Z frame. Instead, it will be oriented at an angle. Looking at the beam, we will measure strains in the transformed coordinate system X*-Y*-Z*, shown in Figure 4(a), where we note that the Z axis is shared between the two coordinate frames, and the coordinate system is rotated about the Z axis by an unknown angle θ.

To relate the components of strain in these two coordinate frames, we will use strain transformations. The components of strain in the rotated reference frame X*-Y*-Z* are given as a function of the components of strain in the X-Y-Z frame through

$$\large \epsilon_X^*=\frac{\epsilon_X+\epsilon_Y}{2}+\frac{\epsilon_X-\epsilon_Y}{2}\cos(2\theta)+\frac{\gamma_{XY}}{2}\sin(2\theta) \\ \space \\ \space \\ \epsilon_Y^*=\frac{\epsilon_X+\epsilon_Y}{2}+\frac{\epsilon_X-\epsilon_Y}{2}\cos(2\theta)-\frac{\gamma_{XY}}{2}\sin(2\theta) \\ \space \\ \space \\ \frac{\gamma_{XY}^*}{2}=-\frac{\epsilon_X-\epsilon_Y}{2}\sin(2\theta)+\frac{\gamma_{XY}}{2}\cos(2\theta)$$

(2)

The strain components in the rotated frame X*-Y*-Z* can be determined with a rectangular strain rosette. For example, Figure 5. shows a strain gauge rosette mounted on the top surface of a beam, with the three gauges separated by 45° from each other.

Figure 5. Photograph of cantilever beam experimental configuration.

Labeling the gauges as A, B, and C as in Figure 4(b), the rosette can thus measure strain in these three directions. If we consider that the frame X*-Y*-Z* is aligned with the rosette such that X* is in the $\epsilon_A$ direction, then $\epsilon_X^*$ and $\epsilon_Y^*$ are simply given by $\epsilon_X^*=\epsilon_A$ and $\epsilon_Y^*=\epsilon_C$. To find $\gamma_{XY}^*$ we can use the strain gauge in the $\epsilon_B$ direction. We consider then a rotation of the X*-Y*-Z* frame to a frame that is aligned in the $\epsilon_B$ direction, and using equation (2) yields

$$\large \epsilon_B=\frac{\epsilon_X^*+\epsilon_Y^*}{2}+\frac{\epsilon_X^*-\epsilon_Y^*}{2}\cos(2\cdot45^{\circ})+\frac{\gamma_{XY}^*}{2}\sin(2\cdot45^{\circ}) \\ \space \\ \textrm{or} \\ \space \\ \epsilon_B=\frac{\epsilon_X^*+\epsilon_Y^*}{2}+\frac{\gamma_{XY}^*}{2}$$

Finally, solving for $\gamma_{XY}^*$ and using $\epsilon_X^*=\epsilon_A$ and $\epsilon_Y^*=\epsilon_C$, we can relate the strains measured from the gauges in the strain gauge rosette to the components of strain in the X*-Y*-Z* frame through

$$\large\epsilon_X^*=\epsilon_A \\ \space \\ \epsilon_Y^*=\epsilon_C \\ \space \\ \frac{\gamma_{XY}^*}{2}=\epsilon_B-\frac{\epsilon_A+\epsilon_C}{2}$$

(3)

With the strains known in the rotated frame X*-Y*-Z*, we can compute the strains in the reference frame aligned with the beam, that is the X-Y-Z frame. If the misalignment angle θ between the rosette and the beam was known, we could simply use the rotation equations (2). However, it is not necessary to know the misalignment angle since we can recognize that in the beam aligned X-Y-Z frame, all shear strain components are zero, and the strains are principal strains! That is $\epsilon_X$, $\epsilon_Y$, and $\epsilon_Z$ are principal strains under the Euler-Bernoulli beam bending assumption used here. For a two-dimensional state of strain, the principal strains can be computed from the strains in any arbitrary reference frame through

$$\large \epsilon_1=\frac{\epsilon_X^*+\epsilon_Y^*}{2}+\left[\left(\frac{\epsilon_X^*-\epsilon_Y^*}{2}\right)^2+\left(\frac{\gamma_{XY}^*}{2}\right)^2\right]^{1/2} \\ \space \\ \space \\ \epsilon_2=\frac{\epsilon_X^*+\epsilon_Y^*}{2}-\left[\left(\frac{\epsilon_X^*-\epsilon_Y^*}{2}\right)^2+\left(\frac{\gamma_{XY}^*}{2}\right)^2\right]^{1/2}$$

(4)

where $\epsilon_1$ and $\epsilon_2$ are the principal strains, and then angle from the given reference frame and the principal reference frame is given by

$$\large \tan(2\theta_p)=\frac{\gamma_{XY}^*}{\epsilon_X^*-\epsilon_Y^*}$$

(5)

We note again that the components $\epsilon_X^*$, $\epsilon_Y^*$, and $\gamma_{XY}^*$could be with respect to any Cartesian reference frame.

Figure 6. Mohr's circle for strain.

The strain transformation relations (2) can be visualized using Mohr’s circle as shown in Figure 6, which is a convenient graphical technique for transforming strains (and can also be applied to transforming stresses). All the components of strain in any rotated reference frame lie on Mohr's circle, where a rotation θ in physical space corresponds to a rotation of 2θ on Mohr's circle. In Mohr's circle, the shear strain axis is drawn in the reverse direction such that positive axis points downward. Similar to the equations in (2), Mohr's circle can be used to transform strain components $(\epsilon_X,\epsilon_Y,\gamma_{XY})$ in a X-Y coordinate frame to the corresponding components $(\epsilon_X^*, \epsilon_Y^*,\gamma_{XY}^*)$ in an X*-Y* coordinate frame, which is rotated about a common Z-axis by an angle To perform the transformation, first draw the points $(\epsilon_X,\gamma_{XY}/2)$ and $(\epsilon_Y,-\gamma_{XY}/2)$. Since all the strain states lie on the circle, straightforward geometry can be used to solve for the strain components in the rotated reference frame. Mohr’s circle also shows us graphically how one can rotate to a particular strain state where the shear strain is zero and there are two principal stresses $\epsilon_1$ and $\epsilon_2$.

Structural Instability

Slender structures, like beams or columns, that are subject to compressive loads often exhibit elastic instabilities. One type of elastic stability is known as buckling. Aerospace structures are often slender and are subject to compressive loads. Therefore, understanding structural instability in general and buckling in particular is required for aerospace structural design.

Structural instability is a fundamentally nonlinear phenomena, therefore to predict buckling we must model the structural nonlinearity. The most common structural instabilities are due to geometric nonlinearity rather than nonlinear material behavior. Geometric nonlinearity accounts for the effect of deformation within an equilibrium analysis rather than assuming infinitesimal structural deflections.

Buckling of a Perfect Column (Euler Buckling)

Figure 1. Euler-Bernoulli beam subject to a compressive load, P.

Consider the buckling of a column loaded by opposing axial loads as shown in Figure 1. We can model this using an extension of Euler-Bernoulli beam theory. Using this theory, the transverse deformation, w(x), of a beam is governed by the equation (1)

$$\large EI\frac{d^4w}{dx^4}+P\frac{d^2w}{dx^2}=q(x)$$

(1)

where E and I are the elastic modulus and second moment of area, P is the axial compressive load and q(x) is the distributed load. At first glance, this equation may not appear to be nonlinear, however, the second term $P(d^2w/dx^2)$ is, in fact, a nonlinear term. If we consider a beam that is only subject to a compressive load, with q(x)=0, then the governing equation can be rewritten as

$$\large\frac{d^4w}{dx^4}+k^2\frac{d^2w}{dx^2}=0$$

(2)

where $k^2=P/EI$ .

Note that the governing equation (2) is now an eigenproblem. Using simply supported boundary conditions for the ends of the beam, we find the eigenvalues of (2) are given by equation (3).

$$\large k_nL=n\pi \space\space\space \textrm{for}\space\space\space n=1,2,3,...$$

(3)

Rearranging this expression gives the following equation for the loads corresponding to these eigenvalues

$$\large P_n=n^2\frac{\pi^2EI}{L^2}$$

(4)

We are most interested in the lowest load for which the beam becomes unstable, i.e., the critical load. This corresponds to the case with n = 1, i.e., .

$$\large P_{cr}=\frac{\pi^2EI}{L^2}$$

(5)

The critical load based on this theory is often referred to as the Euler buckling load. Using this load, we can compute the axial stress in the beam when it buckles

$$\large \sigma_{cr}=\frac{P_{cr}}{A}=\frac{\pi^2EI}{L^2A}$$

(6)

where A is the cross-sectional area of the beam. This critical stress is often written in terms of the radius of gyration ($R_g$) of the column about its weak axis, which is defined as $R_g\equiv\sqrt{I/A}$. With this definition, the critical buckling load stress and strain from this theory are given by

$$\large \sigma_{cr}=\left(\frac{\pi R_g}{L}\right)^2E \\ \space \\ \space \\ \varepsilon_{cr}=\left(\frac{\pi R_g}{L}\right)^2$$

(7)

Buckling of an Imperfect Column and the Southwell Plot

The model of Euler buckling is flawed. In particular, the beam is not perfectly straight and the axial compressive load is not applied through the center of the beam. As a result, the beam does not suddenly buckle at $P_{cr}$ but more gradually buckles in a direction determined by the imperfections in the beam. Such an imperfect beam is shown in Figure 2. Even though these imperfections can be quite small, they can have a large effect on the response of the structure. This is called imperfection sensitivity.

Figure 2. Imperfect Euler beam with initial curvature.

A more sophisticated analysis that takes into account an initial imperfection of the beam, where the imperfection causes an initial displacement with the form $a_1\sin(\pi x/L)$, results in the following expression for the transverse deflection of the beam at its mid-point ($x=L/2$) as a function of the Euler buckling load.

$$\large \delta=\frac{Pa_1}{P_{cr}-P}$$

(8)

This relationship forms the basis of the Southwell plot. Rearranging this relationship, one can create an expression for the deflection normalized by the load, i.e.,

$$\large \frac{\delta}{P}=\frac{\delta}{P_{cr}}+\frac{a_1}{P_{cr}}$$

(9)

A Southwell plot consists of a series of measurements plotted on a graph of δ/P versus δ. The slope of a linear fit to the data then provides $P_{cr}$ and the y-intercept provides a measure of the initial displacement magnitude (see Figure 3a).

Figure 3. Southwell plots for determining the critical load: (a) Southwell plot in original form, (b) Southwell plot for strain. The figures are not drawn to scale, and the slopes on the two graphs are the reciprocal of one another.

In this lab, however, we will only be able to measure the bending strains using the strain gages mounted to the specimen. Therefore, we will modify the classical Southwell plot to use our strain measurements as follows. First, we note that the bending moment at the mid-point of the beam is $M=\delta P$. With the expression for the bending strain, $\varepsilon_{bend}=-yM/EI$, the deflection of the beam at the mid-span is

$$\large \delta=\frac{2EI}{Ph}\varepsilon_{bend}$$

(10)

where we have used $y= -h/2$ as the bending moment. Substitution of this value for δ into (9) yields

$$\large \varepsilon_{bend}=\frac{\varepsilon_{bend}}{P}P_{cr}-a_1\frac{Ph}{2EI}$$

(11)

This version of the Southwell plot for strain (see Figure 3b) is the one you will use for data reduction in this lab.

Experimental Setup: Cantilever Beam

The experiment involves a cantilever beam approximately 12 inches long, which is mounted within a support fixture, see Figure 5. Beam dimensions will be measured with a precision digital caliper. In general, calipers are devices used to measure thicknesses or distances between surfaces, usually having a screw-driven or sliding adjustable piece.

Figure 5. Photograph of cantilever beam experimental configuration.

The set-up allows either an applied load at the free end of the beam (by hanging dead weights) or a prescribed deflection of the beam using a precision micrometer (shown at right side of Figure 5). Precision micrometers typically use a finely threaded screw to accurately measure a linear distance. Turning the outer thimble moves a central rod, called the spindle. A given angular rotation corresponds to a linear distance traveled by the spindle based on the pitch of the screw. The distance the end of the spindle moves can be read from combined scales: a low-resolution length scale on the sleeve and a high-resolution rotating scale on the thimble (see Figure 7). On some micrometers, the readable resolution of the rotating scale is increased by the addition a fine vernier scale on the sleeve.

Figure 7. Micrometer scale.

A rectangular rosette strain gauge rosette is bonded on the top surface near the root of the beam (Figure 5). Each of the three individual grids in the rosette strain gauge will be connected to a dedicated 1/4 bridge circuit within the Vishay 7000 data acquisition system.

Experimental Setup: Column buckling

All the test subjects are composed of 2024-T3 aluminum. Each column will only be tested in the elastic regime to ensure accurate results with the strain gages and so as not to permanently deform the columns. There are two test specimens: a long and a short column; both have rectangular cross-sections (see Table 1 for dimensions).

	Long Column	Short Column
Length (in.)	29	24
Width (in.)	0.75	0.75
Thickness (in.)	0.50	0.50

Table 1. Dimension of aluminum columns.

When pure axial loads are applied to the nominally perfect columns, we have no advance knowledge of any preferential buckling direction; thus it is difficult to correctly place any lateral displacement gage. As a result, the columns are fitted with two strain gages at mid-span mounted on the top and bottom surfaces of the beam. Assuming that the strain varies linearly through the beam, the axial strain measured at the top ($\varepsilon_a$) and bottom ($\varepsilon_b$) surfaces of the beam would be:

$$\Large \varepsilon_a=\varepsilon_{axial}+\varepsilon_{bend} \\ \space \\ \varepsilon_b=\varepsilon_{axial}-\varepsilon_{bend}$$

(12)

where $\varepsilon_{axial}$ is the strain from the axial deformation and $\varepsilon_{bend}$ is the strain due to bending. To find these two strains, we could combine the measured strains to produce

$$\Large \varepsilon_{axial}=\frac{1}{2}(\varepsilon_a+\varepsilon_b) \\ \space \\ \varepsilon_{bend}=\frac{1}{2}(\varepsilon_a-\varepsilon_b)$$

(13)

Due to the fact that the direction of buckling is unknown, the strain reading may either be positive or negative.

Note that the bending strain is also given by

$$\large \varepsilon_{bend}=-\frac{h}{2}K=-\frac{hM}{2EI}$$

(14)

Procedure

Cantilever beam

Initial Measurements

1. Measure the beam dimensions

   1. Thickness (h) using the precise electronic caliper

   2. Width (b) using the precise electronic caliper

   3. The length $L_1$, from the application position of the micrometer to the clamped end of the beam (see Figure 8)

   4. The length $L_2$, from the point of application of the dead load to the clamped end of the beam (see Figure 8)

   5. Determine the distance $X_g$ between the end of the beam and the rosette center-line.

      1. $X_g$ = 0.8955 in

   6. Note that the accuracy of the distances $L_1-X_g$ and $L_2-X_g$ are critical to your data reduction

Figure 8. Cantilever beam dimensions.

Load Control Tests

In this experiment you will apply prescribed loads (P) by hanging dead weights from the point labeled “Applied load” in Figure 8, and record the strain readings indicated by each gauge within the rectangular rosette gauge. Follow the steps below:

1. Prepare the software:

   1. Power on the Vishay 8000 data acquisition unit if your TA has not already done so. There is a switch on the back right of the system.

   2. Open StrainSmart 8000 from the Start Menu

   3. Select New Project on the Top Left. When prompted to save changes to save the new project, Click NO. When prompted to use the New Project Wizard, Select YES.

   4. In the Project Settings, you may title the project with your section number (e.g. A12-X). Click NEXT.

   5. Duplicate/extend the desktop to the large portable monitor

   6. Arrange all relevant windows so everything can be seen optimally (e.g. side-by-side)

2. Configure the project and scan session:

   1. You should now be on the "Add Sensors" page. From the list on the left, select Rectangular Rosette. This is the type of strain gauge we will be using for the Cantilever Beam Experiment.

   2. Enter the Gage Factor, Kt value (Transverse Sensitivity) found on the sticker on the beam.

   3. Select a resistance of 120 V from the drop down menu (this is the base resistance of our strain gauge under no excitation). Leave the Default Excitation at 2V.

   4. Click ADD and when the window closes, click NEXT.

   5. On the Material Selection page, again click NEXT.

   6. On the Controller Hardware page, select Connect & Detect Devices from the list on the left. Click NEXT.

   7. On the Assignments page, move the cursor over the image of the hardware. You should see the individual ports light up in yellow as the cursor hovers over them. Right click on port 1 and click Add Rectangular Rosette Assignment.

   8. Verify that Grid 1 is on S1-Ch1, Grid 2 is on S1-Ch2, and Grid 3 is on S1-Ch3. If so, click SAVE. The first three ports should now have a red box around them.

   9. Click FINISH.

   10. When prompted to create a scan session, select YES.

   11. Under session assignments, verify the Rectangular Rosette box is checked and then click NEXT.

   12. Under scan rate, select 10 and ensure that Time Based Recording is Enabled under Continuous Mode. Click NEXT.

   13. Click NEXT. Click NEXT again.

   14. Under relay output, click NEXT.

   15. The scan session creation is now complete. Click FINISH.

   16. Click the red DISCONNECTED button on the bottom left corner of the screen. The red square should turn green and it should now read CONNECTED. This means the connection between the StrainSmart software and the data acquisition hardware is online and active.

       1. If the system does not connect, ask your TA for help.

3. Zero the strain gauges:

   1. Ensure the Micrometer is not touching the beam surface

   2. Ensure the hook is on the "Applied Load" part of the beam (the hook is placed in the dimple on the beam)

      1. SKIP THIS STEP FOR THE DISPLACEMENT CONTROLLED EXPERIMENT

   3. Click on Zero Balance from the choices in the toolbar neat the top of the screen. Click Set Zero on the bottom of the screen. Ensure there are no errors.

   4. Click on the Shunt Calibration button near the top of the window and then click Perform Shunt Calibrate on the bottom of the screen. Ensure there are no errors.

   5. If there was no error message, calibration was successful and click Close. Note that the ethernet connectors in the Vishay box can be a little finicky... if you have trouble with errors being thrown when you zero, or with strain readings drifting in later steps, it is more than likely these connections. To fix this, manually adjust the connectors until the problem stops then take care to not move the connectors afterwards. You may also try different ethernet ports but you will need to change the channel assignment accordingly.

      1. You may have to start a new project if the problem persists. Sometimes errors can be thrown if the settings and numbers you innput do not make sense. Ensure all steps are followed properly.

      2. If problems persist, call the Head TA or Lab Manager for help.

4. Configure the live display:

   1. Click on Arm and Scan. From the choices on the left of the window, select Validate Project.

   2. Click Arm. This gets the software and hardware ready to acquire data.

   3. On the top toolbar, select Open Online Viewer. Select "Grid 1 Microstrain", "Grid 2 Microstrain", and "Grid 3 Microstrain" from the checkboxes. Ensure everything else is unchecked.

   4. Click Apply Selected Assignments. Click View Strip Chart. On the bottom of the screen, ensure the "Auto Y Fit" checkbox is selected. This shows you a graph of Strain vs. Time for each of the 3 strain channels.

   5. Repeat steps 4.3 and 4.4 above but this time, select View Table Data instead of the Strip Chart. This will open another instance so you can see the actual strain values in real time.

   6. Position both Displays such that you can see them clearly on the TV Monitor

5. Perform your experiment:

   1. As a group, decide on 5 masses between 100 and 1000 grams, where 1000g must be one of the five.

      1. Don't arbitrarily pick these! Try to discuss with your groupmates and have a reason for selecting the loads you select.

   2. READ STEPS 3-6 BELOW COMPLETELY BEFORE PROCEEDING

   3. When you have your values and justification, click Scan. You should see your strip chart and table data viewer populated with values.

      1. Ensure that there is no drift in the microstrain values.

         1. If there is, stop the scan, and ensure the strain gauges are properly connected to the Vishay. You will then have to zero the balance again, shunt calibrate, and restart the scan.

   4. For each mass, apply the load at the point labeled “Applied load” in Figure 8 above (there is a dimple on the beam where you apply load), and record the three strains manually.

      1. You will need to hang weights using the hook provided. The hook will rest on the dimple in the beam.

      2. Before recording the strain values, ensure there are no oscillations in the beam.

   5. Discuss all the data you see with your groupmates and TA. Is there anything interesting? Is it what you would expect? Why are you seeing what you are seeing?

   6. Once you have your data, click Stop.

   7. Remove all masses and the mass application hook and return them to the holder.

   8. When asked if you would like to duplicate the scan session, select YES. This saves us from redoing all the settings for the following experiment.

Displacement Control Tests

In this experiment you will apply prescribed displacements using the precision micrometer used to measure the beam deflection $\delta$ and record the subsequent strain readings indicated by each gauge within the rectangular rosette gauge. Follow these steps:

1. Because we duplicated the session from above, we will not need to re-enter all the system and strain gauge settings!

2. Zero the balance and shunt calibrate according to the steps in Step 3 from the Load controlled experiment above.

3. Perform your experiment:

   1. Familiarize yourself with the micrometer markings and how they relate to displacement

   2. Click Arm and Scan, and then click Validate Project.

   3. On your two Online Viewer windows (which should still be open), set one of them to View Table Data and the other to View Strip Chart same as you did for the Load Controlled experiment.

   4. Click Scan in the main window. You should see your strip chart and table data viewer populated with values.

      1. Ensure that there is no drift in the microstrain values.

         1. If there is, stop the scan, and ensure the strain gauges are properly connected to the Vishay. You will then have to zero the balance again, shunt calibrate, and restart the scan.

   5. Zero the micrometer by screwing it down until it just starts to cause a strain, then back off just a touch until it releases the strain

   6. Note down the displacement value on the micrometer. This will be your "zero displacement" value.

   7. Displace the beam to the following positions, measuring and manually recording the subsequent three strains at each position: 0.1", 0.2", 0.3", 0,4", 0.5", 0.6"

   8. Discuss all the data you see with your groupmates and TA. Is there anything interesting? Is it what you would expect? Why are you seeing what you are seeing?

   9. Once you have your data, click Stop.

   10. When asked if you would like to duplicate the scan session, select NO. This is your last experiment with this software!

   11. Unload the micrometer so that it is no longer touching the beam

4. Close down the experiment:

   1. Close any Online Viewer displays that are open.

   2. On the top of the screen, go to the Completed Sessions tab.

   3. For each of the two sessions, click View Data. Select Grid 1, Grid 2, and Grid 3 and ensure all the other boxes are unchecked. Click Select.

      1. Scroll through the Raw Data and look at the Strip Chart. Does this data make sense? Are you satisfied with using this data for your lab report? Verify with your TA if you would like.

   4. If you are satisfied with your data, click the Export Data button on the top of the main window

      1. Select both sessions from the window to ensure you export both sets of data.

      2. Select "microstrain (Grid 1)", "microstrain (Grid 2)", and "microstrain (Grid 3)" and ensure all the other variables are unchecked.

      3. Export it as an Excel file and have your TA save it to a destination of their choosing.

      4. Click Export.

      5. Open the Excel file and verify all data was exported properly.

   5. Go to the Configuration tab near the top of the main window. Click EXIT. When prompted to archive the project, select NO.

   6. Power down the Vishay data acquisition unit by flipping the switch on the back of the box.

   7. Have your TA upload your data to Canvas.

Column buckling

Prepare the Instron Load Frame and Bluehill Universal software

1. Turn on the load frame using the power button on the right hand side of the base

2. Open Bluehill Universal and Click "Test"

3. Under "New Sample" select "Browse Methods"

4. Navigate to C:\Users\Public\Documents\Instron\Bluehill Universal\Templates\AE_Labs\AE3610\Bending Buckling

5. Open the file titled AE_3610Buckling.im_tens and click OK

6. Familiarize yourself with the software, shown below. This experiment will be performed manually using the jog function and the fine position dial on the Instron control panel. Look for the following:

   • Displacement (as measured by the load frame's motor controller)

   • Force (as measured by the load cell on the crosshead)

   • Strain Gauge Adapter Module A and B (strain measured by the strain gauges wired into strain gauge adapters A and B)

   The Zero Displacement and Balance All buttons enable you to balance (aka zero) the sensors, which will not be balanced when you first open Bluehill Universal

Bluehill Universal Software

Long Column Tests

1. Locate and install the long column:

   1. With nothing installed in the load frame, hit Balance All in Bluehill to zero the load cell.

   2. Jog the load frame manually, using the jog buttons and fine adjustment wheel on the control panel, until the gap between the v-grooves is just larger than the long column.

      1. Sometimes the Jog Down button doesn't work; if this happens, have a TA jog the crosshead down using a quick compression method. Do not do this yourselves! the TAs are trained to handle this scenario!

   3. With your hands in a safe position so they can't get pinched, use the fine adjustment jog wheel to jog the top frame down until the column is held in place. Find the exact correct position by observing the Force reading in Bluehill - it will start to shoot up once the upper v-groove makes contact. At this point, back off slowly until the column is just unloaded. Ensure the column doesnt slip out of the grooves.

2. Install and calibrate the strain gauges:

   1. Connect the left-hand strain gauge to the left-hand strain gauge adapter module (120 Ohm A) and the right-hand strain gauge to the right-hand strain gauge adapter module (120 Ohm B). In each case, connect Port 1 of the adapter module to one side of the strain gauge, Port 2 to the other side of the strain gauge, and install a jumper wire between Ports 2 and 3. See the image below for a reference. Gently tug on each wire to ensure a good secure connection is made.
   2. Click on the System Details tab in the top right corner of the screen
   3. Open the Strain Gauge Transducer A settings by clicking the button under System Settings. Ensure the following settings:

      1. Full scale: 5%

      2. Gauge Length: 1 mm

      3. Calibration Type: Automatic

      4. Calibration point--Values below as of 10/17/2023 (Calculated by 10/Gauge Factor)

         1. Long Column Calibration Point = 4.93%

         2. Short Column Calibration Point = Written on the Column itself

            1. The two gauges have different gauge factors for each gauge since the strain gauges are from two different batches

            2. Since the gauge factors are different, the two gauges have different calibration point values. These calibration point values are written on the column itself for each side

            3. Make sure you know which gauge is plugged into Transducer A/1 and Transducer B/2 so you can enter the appropriate value for each transducer

      5. Offset: 0%

      6. Click Calibrate and then Click OK

         1. If the calibration fails, check the wiring at the strain gauge completion box - usually it fails when there is an open circuit which can come from a faulty connection or damaged strain gauge

         2. Note: If the strain gauge transducer icon turns gray, it means that the transducer did not calibrate properly

      7. Once calibrated, click Balance and then click Close

   4. Repeat the above step 3 for Strain Gauge Transducer B
   5. Close the system details window

   6. Click Balance All in the home screen

   7. Observe the strain gauge readings in Bluehill - if they are erratic or drifting, you possibly have a bad electrical connection. In this case, adjust the wires and re-balance as necessary until you have the required behavior.

   8. Without touching the strain gauges directly, perform a sanity check on the setup by forcing the column to the left and right with your hands. Verify what you expect to happen against what you observe on the live Bluehill strain readings.

3. LOAD STAGNATION--Read this step carefully before continuing!

   1. In the steps below, you will load and unload the column to 9 separate values.

   2. Sometimes, the load stagnates despite increasing crosshead displacement--this means that even though you increase displacement using the fine adjustment wheel and the column continues to buckle, the load reading stays constant.

   3. To prevent damage to the column and strain gauges, please following these steps:

      1. Continuously ensure that the applied load is increasing as the fine adjustment wheel increases crosshead displacement

      2. Keep an eye on the columns to make sure it is not visibly buckling too much. These columns can and do yield if too much load is applied and the columns then become useless! It can also cause the strain gauges to fail or pop off!

   4. If the load does stagnate during your experiment, follow these steps:

      1. Unload the column completely

      2. Visually inspect the column to make sure there is no plastic deformation

      3. Hit Balance All to zero everything

      4. Re-insert the column into the grooves

      5. Hit Balance All again

      6. Continue the experiment from where you left off

   5. A TA will help with this process.

4. Commence the loading experiment:

   1. You will load the column to five values of load between 100lbf and 900lbf (inclusive)

      1. 100 lbf

      2. 300 lbf

      3. 500 lbf

      4. 700 lbf

      5. 900 lbf

   2. Have one student open a blank spreadsheet and be ready to record the data

   3. Using the fine adjustment jog wheel, load the column to each of the five load values. At each point:

      1. Visually observe the column for buckling

      2. Log the exact load and both strain gauge readings into your spreadsheet

5. Commence the unloading experiment:

   1. You will unload the column to four different values of load and one same value of load between 100lbf and 900lbf (inclusive)

      1. 800 lbf

      2. 600 lbf

      3. 500 lbf

      4. 400 lbf

      5. 200 lbf

   2. Using the fine adjustment jog wheel, unload the column to each of the five load values. At each point, log the load and both strain gauge readings into your spreadsheet

6. Verify your data:

   1. In your spreadsheet, add columns for, and calculate, axial and bending strain from your recorded strain gauge data

   2. Make a quick scatter plot of bending strain to sanity check the data with the help of the TAs. If anything looks off, now is the time to repeat the experiment.

Short Column Tests

1. Unhook the strain gauges from the strain gauge adapter module. Make sure to save the jumper wires!

2. Remove the long column by manually adjusting the fine adjustment wheel until it can be freed.

3. REMEMBER TO KEEP AN EYE OUT FOR LOAD STAGNATION

4. Repeat all steps from the long column test, but this time use loading/unloading values in the range of 500lbf to 1300lbf (inclusive).

   1. Use these loads for loading:

      1. 500 lbf

      2. 700 lbf

      3. 900 lbf

      4. 1100 lbf

      5. 1300 lbf

   2. Use these loads for unloading:

      1. 1200 lbf

      2. 1000 lbf

      3. 900 lbf

      4. 800 lbf

      5. 600 lbf

5. Verify your data! Redo the experiment if anything looks off.

6. As before, unhook the strain gauge from the strain gauge adapter module and remove the short column from the load frame by manually adjusting the fine adjustment wheel until it can be freed.

7. Return the jumper wires to the green/black toolbox on the table!

8. Close Bluehill. If promped to save the sample, select "No". Turn off the Computer.

9. If you are the 8:25am section or the last section of the day, power down the Instron. Otherwise, keep it on.

*These instructions are for the linear strain gauges currently installed on the columns, which are 2-wire Omega KFH-3-120-C1-11L1M2R strain gauges (as of February 2022). Consult the strain gauge adapter module manual for new values/wiring instructions if these ever change.

Data to be Taken

1. Cantilever beam dimensions and locations of rosette, micrometer and load along beam.

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2. Strain readings for known loads from the cantilever experiment.

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3. Strain readings for known deflections from the cantilever experiment.

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4. Axial loads and bending strains for loading and unloading runs for long column buckling.

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5. Axial loads and bending strains for loading and unloading runs for short column buckling.

Data Reduction

For the load-controlled experiments:

1. Using the rectangular rosette strain gauge data, determine the strains $\epsilon_X^*$, $\epsilon_Y^*$, and $\gamma_{XY}^*$ and that resulted from each of the applied loads.

2. Based on $\epsilon_X^*$, $\epsilon_Y^*$, and $\gamma_{XY}^*$ estimate for each of the applied loads: the maximum and minimum principal strains ($\epsilon_1$ and $\epsilon_2$).

3. Determine the slope of the best fit linear relationship between $\epsilon_1$ and P and use it, along with cantilever beam theory for the relationship between the axial strain $\epsilon_X$ and P at the free end of the beam, to estimate the elastic modulus E in MPa of the cantilever beam material. This relationship will also depend on the geometric quantities $(L_2-X_g),b$ and $h$

For the displacement-controlled experiments:

1. Using the rectangular rosette strain gauge data, determine the strains $\epsilon_X^*$, $\epsilon_Y^*$, and $\gamma_{XY}^*$ that resulted from each of the applied deflections.

2. Based on $\epsilon_X^*$, $\epsilon_Y^*$, and $\gamma_{XY}^*$, estimate for each applied deflection: the maximum and minimum principal strains ($\epsilon_1$ and $\epsilon_2$).

3. Determine the slope of the best fit linear relationship between $\epsilon_1$ and $\delta$ , and use it, along with the relationship between$P$and the beam deflection $\delta$ from cantilever beam theory to estimate the “effective bending spring stiffness” (in N/m) of the cantilever beam. To accomplish this, you will also need to use your estimate of $E$ from the load-controlled data

For the column buckling experiments:

1. Using a Southwell plot for each of the columns, determine the column’s buckling load and buckling (axial) stress.

2. Calculate the theoretical buckling load and stress for each column based on column theory.

Results Needed for Data Report

1. Table of measured rosette strains $(\epsilon_A,\epsilon_B,\epsilon_C)$ and inferred local strains $(\epsilon_X^*,\epsilon_Y^*,\gamma_{XY}^*)$ for each of the applied loads.

2. Table of measured rosette strains $(\epsilon_A,\epsilon_B,\epsilon_C)$ and inferred local strains $(\epsilon_X^*,\epsilon_Y^*,\gamma_{XY}^*)$ for each of the applied deflections.

3. A combined plot of $\epsilon_X^*, \space \epsilon_Y^*$ and $\gamma_{XY}^*$ as a function of the applied load (in Newtons).

4. A combined plot of $\epsilon_X^*, \space \epsilon_Y^*$ and $\gamma_{XY}^*$ as a function of the applied deflection $\delta$ (in meters).

5. Plot of $\epsilon_1$ as a function of $P$ (in Newtons), including the best-fit line.

6. Plot of $\epsilon_1$ as function of $\delta$ (in meters), including the best-fit line.

7. Table of linear proportionality constants for $\epsilon_1$ vs. P and $\epsilon_1$ vs. $\delta$, and inferred beam elastic modulus and effective bending stiffness.

8. Table of measured strains for each of the applied (measured) axial loads for long column.

9. Table of measured strains for each of the applied (measured) axial loads for short column.

10. Individual Southwell plots for each of the columns.

11. Table of experimentally determined and predicted buckling loads for each column.