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MEE 322 STRUCTURAL MECHANICS
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Laboratory Experiment #2: Bending and Torsion of Cantilever Structures.
1. OBJECTIVES
 Carry out simultaneous bending and torsion tests on a 2024T3 Aluminum structure in cantilever configuration that has been equipped with strain gauges at a particular location along its length, where the crosssection is tubular.
 Obtain loadstrain data at one of the locations within the crosssection with maximum normal stress due to bending, where one strain gage is located, for increasing values of the applied load and use these data to calculate Young’s Modulus (E) for the material of the structure.
3. Obtain loadstrain data at one of the locations within the crosssection with maximum shear stress due to torsion, where one strain gage is located, for increasing values of the applied load and use these data to calculate Shear Modulus (G) for the material of the structure.
 Using the calculated values of E and G, draw the graph for the variation of the normal stresses due to bending as well as the shear stresses due to torsion over the crosssection where the measurements are made. Also compare the values of E and G obtained from the experiment with the literature values.
 Prepare a technical report where the experiment and the findings are described, comparisons between the experimental findings and theoretical results are made, discussion explaining the reasons for the different behaviors is offered and conclusions regarding these behaviors are reached. The technical report must be properly formatted according to the guidelines provided.
IMPORTANT NOTES:
Make sure you sign the attendance sheet!
One of the mobile demonstration carts available in the Mechanical Testing Lab will be used to carry out the experiments on a cantilever structure designed in such a way that a single load can produce both bending and torsion simultaneously on a chosen location. The overall geometry of this structure is shown in Fig. 1.
Vertical loads of increasing amplitude will be applied at a chosen location along the torsion arm, which is marked with a ruler that indicates the length of the arm for the applied load that produces a torsion moment in the section of the structure with a tubular crosssection. Note that the structure has a section with reduced outside diameter. An analysis of internal reactions would show that a vertical load applied anywhere along the torsion arm will also lead to a bending moment in the section with reduced outside diameter.

Fig. 1. Exploded view of cantilever structure.
Therefore, the section with reduced diameter will undergo bending and torsion simultaneously, but we will not address their combined effects, and they will be treated individually. In particular, strain gauges will be placed at chosen locations at the middle of the section with reduced diameter, as shown in Fig. 2.


Fig. 2. Strain gauge locations at the section of reduced outside diameter.
The strain gauges labeled “2” and “3’ will be used in this lab. Please read sections 8.3 to 8.6.2 in the textbook, which describe how strain gauges work and how to use them, as that material will be extremely helpful to understand the measurements and to write the lab report. The strains measured by the strain gauges will be obtained for several values of the applied load (P) on the torsion arm (see Fig. 1), making sure that the material stays within the elastic regime. The lab supervisor will announce the value of the maximum load during the lab session as well as the value of the torsion arm length to be used[1]. Make sure you take notes carefully of the length of the torsion arm. The bending arm length, LBGage in Fig. 2, is fixed and equal to 0.1125 meters, see Fig. 3 for additional dimensions.
Fig. 3. Dimensions of the cantilever structure, in mm.
Strain gauge 3 will be used to record the maximum normal strain due to bending (why?) at the chosen crosssection, whereas strain gauge 2 will be used to record the maximum shear strain due to torsion (why?). This setup will make it possible to obtain both the Young’s Modulus E and Shear Modulus G of the material using the formulae given in the next section. The readings from the strain gauges will need to be recorded for at least 5 values of P.
Theory
The main goal here is to calculate E and G. The axial stress of the bottom fiber of the crosssection can be related to the strain reading from strain gauge 3 and to the value of the moment at that location using the following formulas:
(1)
(2)
Where the inertia I and y_{max} can be calculated with the geometry shown in Fig. 3, whereas M needs to be calculated as a function of the applied load and the bending arm length. Make sure you do not get confused with sign conventions[2]. Hence the combination of (1) and (2) leads to the relation:
(3)
A stressstrain (σ_{xx}e_{xx}) diagram can be obtained from the experiments, i.e., stress calculated from equation (2), where all terms are known, versus the reading of the strain gauge 3. The plot should be linear, according to equation (1). Hence, E can be estimated through a linear regression of the collected stressstrain data, where E will be given by the slope of the line that best fits the stressstrain data.
Strain gauge 2 provides a reading that is numerically equal to the shear strain at the point where the strain gauge is installed, since it is installed at 45° from the axis of the structure. The reasons for that are described in the textbook and will also be covered in class later on; however, for now it is sufficient to know that the output of the strain gauge is actually the shear strain g/2 (note that g is called the engineering shear strain). Recall that:
(4)
The shear stress due to torsion at the location of strain gauge 2 is given by
(5)
Where the polar moment of inertia I_{p} (often also called J) and r_{max} can be calculated with the geometry shown in Fig. 3, whereas T needs to be calculated as a function of the applied load and the torsion arm length. Make sure you do not get confused with sign conventions[3]. Hence the combination of (4) and (5) leads to the relation:
(6)
A stressstrain (s_{xy}e_{xy}) diagram can be obtained from the experiments, i.e., stress calculated from equation (5), where all terms are known, versus the reading of the strain gauge 2. The plot should be linear, according to equation (4). Hence, G can be estimated through a linear regression of the collected stressstrain data, where 2G will be given by the slope of the line that best fits the stressstrain data.
Lab 2 report guidelines
The report should be no longer than 10 pages, from the title to the references.
Title page: [2 points]
Must contain lab number and title; full name; date of the experiment; and due date
Objectives: [2 points]
Abstract [8 points]
No more than 250 words. It must contain a brief description of what was done, the main results and the conclusions, in this case the nature of the relationship among load, position, moment of inertia, stress and strain tested experimentally for both bending and torsion and how they compare to those predicted by beam and torsion theory and the values of elastic properties and how they compare to literature results. No raw data should be presented here
Data analysis and discussions [80 points]
 Record the values of P and readings from the strain gauges, and present them in a table (Table 1). [5 pts]
 Calculate the bending and torsion moments as internal reactions as algebraic functions of P at the location where the strain gauges are located using the method of sections. Show a carefully drafted diagram, as close to scale as possible, with your Free Body Diagram (FBD) and also show your statics calculations. [5 pts]
 Plot axial strain e_{xx} (e_{gauge3}) vs. P in Fig. 1 and shear strain e_{xy} (e_{gauge2}) vs. P in Fig. 2. [10 pts]
 Determine the bending stress σ_{xx} from P using bending theories, and plot σ_{xx} vs. P (Fig. 3). [5 pts]
 Determine the shear stress σ_{xy} from P using torsion theories, and plot σ_{xy} vs. P (Fig. 4). [5 pts]
 Plot a “σ_{xx}e_{xx}” diagram (Fig. 5) and perform linear regression over the data points to estimate E. The figure should include the original experimental data and the line fitted through linear regression for comparison. Include the value of R^{2} in your plot as well as the equation of the fit. [15 pts]
 Plot a “σ_{xy}e_{xy}” diagram (Fig. 6) and perform linear regression over the data points to estimate G. The figure should include the original experimental data and the line fitted through linear regression for comparison. Include the value of R^{2} in your plot as well as the equation of the fit. [15 pts]
 Compare the “experimental” values of E and G and the ones in the “literature” (internet, book), and so on [5 pts]
 Sketch the variation of shear stress due to torsion along the radius and the variation of normal stress due to bending along y for the cross section where the measurements were made (use arrows to show the direction of the stresses). Use symbolic math software to plot these stresses as 3D surfaces (Fig. 7 and Fig. 8) over the domain defined by the crosssection of the structure where the measurements were made [15 pts].
Conclusions [6 points]
References [2 points]
[1] Nonobservance of these limits may result in irreversible damage of the lab equipment.
[2] The expressions given in equations 1 and 2 show the absolute values of the maximum normal stress and strain at the location of the strain gauge.
[3] The expressions given in equations 4 and 5 show the absolute values of the maximum shear stress and strain at the location of the strain gauge.