Strain gauge rosette mohr's circle

Paraphrase

ENGR 2411 Mechanics of Materials Lab

Section No. 001

Lab No. 6

Lab Title: Principal Strains and Stresses

Submitted to:

Dr. Zahid Hossain

And

Mr. AM Feroze Rashid

College of Engineering

Submitted By:

Name: Kudakwashe Makuvire

Student ID: 50378716

04/18/2016

TABLE OF CONTENTS
Title

Pg. No.

List of Tables

3

List of Figures

4

Abstract

5

Introduction

5

Background and Methodology

5

Results and Discussions

8

Conclusions and Recommendations

9

Appendix A Raw Data

10

Appendix B Mohr’s circle

11

List of Tables
Table No. and Title

Pg. No

Table 1. Strain Raw data

8

Table 2. Principal Strains using Rosette Analysis

8

Table 3. Principal Stresses

8

Table 4. Results of the Flexure formula analysis

8

Table 5. Results from Mohr’s circle analysis

8

List of Figures
Figure No. and Title

Pg. No

Figure 1. Principle Stresses and Angle

5

Figure 2. Principle Strains and Angle

6

ABSTRACT

For this lab we compared a number of results for principal stresses, strains and angles, using different, methods. The methods used were the Rosette analysis, the Flexure formula and Mohr’s circle. For the Rosette analysis we found the principal stresses to be 14674.78 and -111.94 psi. For the Flexure formula the principal stresses were 1445 and 0 psi. For the Mohr’s circle the principal stresses were 14485.74 and 0 psi.

Keywords: Stress, Strain, Mohr’s circle, principal

INTRODUCTION

The experiment performed measured the strains along three different axes surrounding a point on a cantilever beam. The principle stresses, principle strains, and principle angles were then calculated given these strains, and the results were compared with the stress given by the flexure formula of the beam as well as a Mohr’s circle.

BACKGROUND AND METHODOLOGY

For comparison we used three different methods to obtain our principal stresses, strains and angles. The three methods used were the Rosette analysis, the flexure formula and Mohr’s circle. The Rosette analysis uses a triaxial rosette gauge described below. In general, three separate strains along different axes at the same point must be calculated by stain gages to determine the principle strains and stresses. Principal shear Stress is the maximum and minimum normal stress possible for a specific point on a structural element. Shear stress is 0 at the orientation where principal stresses occur. Principle shear strain is the maximum and minimum strain possible for a specific point. Principal Angle is the angle of orientation at which principal stresses occur for a specific point. The principle plane is the plane on which normal stress attains its maximum and minimum values. Figure 1 below shows a diagram of principle stresses along with the principle angle, and Figure 2 below shows a diagram of principle strains along with the principle angle.

http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/images/PrincipalStress.gif

Figure 1. Principle Stresses and Angle

To calculate these three strains, the gages were placed on the aluminum cantilever beam in the directions of 90 degrees, 60 degrees, and 45 degrees. These three measurements were taken around a point in the formation of a rectangular “rosette.” Longitudinal strain is the fractional deformation of a body when subjected to deforming forces that can elongate or compress the body lengthwise. Longitudinal stress is defined as the amount of force in a body that tends to cause lengthwise deformation, such as the bending of the aluminum beam in this given experiment. Lateral stress and strain are in definition the same, except perpendicular to the longitudinal stress and strain. The angle between Gage 1 and the principle axes sum to 90°, usually around 30° and 120° respectively.

Figure 2. Principle Strains and Angle

Equations

The following equations were used to calculate the principal stresses and strains using the Rosette analysis

· Principle Strain: ..................(1)

· Principle Strain: ..................(2)

· Principle Stress: ....................................................(3)

· Principle Stress: ....................................................(4)

· Principle Angle: ……………………………......(5)

·

Max Principle Longitudinal Stress: ……………………………(6)
For the Flexure formula analysis the following additional formulas were used

· ………………………………………………………………(7)

· …………………………………………………………..(8)

Where = 0.33, which we were given in the handout

The following steps were followed as per handout to obtain the raw data that was necessary for all the calculations.

1. We measured the distance from the center line of the rosette to the loading point on the free end of the beam (L).

2. Then we measured the width (b) and thickness (t) of the beam with a micrometer.

3. Using the cantilever beam flexure formulae, equation 6, we calculated the load P, to be applied at the free end of the beam; and we found a stress value of σ = 15000 psi.

4. The calibrated loading screw was backed and the beam was inserted into the Flexor with the gaged end in the clamp, and with the gage on the top surface.

5. Lead wires were connected from the rosette to the binding posts of the flexor as per the wiring diagram given in the handout.

6. We then connected one of the common leads from the flexor (#1) to the S- binding post of the strain indicator.

7. The other common lead was connected from the flexor (#2) to the D-120 binding post of the strain indicator.

8. We connected the independent lead from Gage Element 1 (#3) to the P+ binding post of the strain indicator.

9. After balancing the strain indicator amplifier, we set the gage factor to the value given on the strain gauge.

10. With the beam unloaded, we set the instrument to RUN.

11. The balance control of the strain indicator was adjusted until the digital readout indicated precisely zero. The balance control was not adjusted again during the experiment.

12. The initial reading for the strain Gage Element 1 was recorded on the worksheet as 0με.

13. We turned the strain indicator off, and disconnected the independent Gage Element 1 (#3) lead from the P+ binding post; we left the common leads connected.

14. Next, we connected the cable lead from Gage Element 2 (#4) to the P+ binding post and then turned the instrument on.

15. Without adjusting the balance controls, we noted the reading on the indicator display.

16. This was the initial reading for Gage 2, and was recorded on the worksheet.

17. The strain indicator was turned off, and the independent Gage Element 2 (#4) lead was disconnected from the P+ binding post, we left the common leads connected.

18. We connected the cable lead from Gage Element 3 (#5) to the P+ binding post and turned the instrument on.

19. Without adjusting the balance controls, we noted the reading on the indicator display.

20. This was the initial reading for Gage 3, and we recorded it on the worksheet.

21. After recording the initial reading for Gage Element 3, we left the gage connected and applied the previously calculated load ‘P’; we hung the weights on the free end of the beam.

22. We recorded the exact weight ‘P’ on the work sheet and recorded the indicated strain for Gage Element 3 in the table.

23. With the load on the beam, we turned the strain indicator off and removed the Gage Element 3 (#5) and replaced it with the Gage Element 2 (#4) to the P+ binding post.

24. We turned the strain indicator on and recorded the indicated strain for Gage Element 2.

25. With the load on the beam, we turned off the strain indicator and removed the Gage Element 2 (#4) and replaced it with Gage Element 1 (#3) to the P+ binding post.

26. We turned the strain indicator on and recorded the indicated strain for Gage Element.

27. With the last gage still connected to the instrument, we removed the load from the beam. The strain indicator readout then indicated the same με as the initial reading for this gage.

28. Finally, we used a protractor to measure the counterclockwise angles between Gage 1 axis and lateral and longitudinal beam axis.

RESULTS AND DISCUSSIONS

Table one shows the strain results that were tabulated for group 2. We were able to find the three different strains, which we used to find the principal strains using equations 1 and 2.

We found the values of the principal strains as shown in table 2. For strain qr we applied a correction factor of 1.025. This was done to account for transverse sensitivity.

Table 1. Strain Raw data

Gage

Initial Reading (με)

Final Reading (με)

Strain (με)

1

0

944

944

2

-152

1212

1364

3

322

380

58

Table 2. Principal Strains using Rosette Analysis

Strain

εpr (με)

εqr (με)

1471.06

-480.7865

Next we calculated the principal angle θpr using equation 5. We found the value of the principal stress to be 31.41˚. Calculation of θqr was not necessary as we were instructed by the instructor that the value was to be zero.

The final step in the Rosette analysis was to calculate the principal stresses. We used equations 3 and 4 to get our values. Table 3 shows the Principal stresses σp and σq. For Poisson’s ratio υ, we used a value of .320, which we were given by our instructor.

Table 3. Principal Stresses

Stress

σ (psi)

σ (psi)

14674.78

-111.94

Table 4 shows the results of the principal stress and strains which were obtained using the Flexure formula. Paying close attention to the strains εpf and εqf, we can see that they are comparable to the principal strains in table 2, which were the strains obtained using the Rosette analysis.

Table 4. Results of the Flexure formula analysis

σpf (psi)

εpf (με)

εqf (με)

14445

1444.5

-476.69

For the Mohr’s circle analysis the construction of the circle was done by hand and is in appendix A of the lab report. Our angle for Mohr’s circle was compared to the angle for Rosette analysis, 31.41, and the difference is rather small.

Table 5. Results from Mohr’s circle analysis

ε (με)

ε (με)

σ (psi)

σ (psi)

θ˚

939.48

45.55

0

14485.74

31.41

CONCLUSIONS AND RECOMMENDATIONS

The overall experiment was a success, we had no difficulty in obtaining the required raw data with which to do and compare our different stresses, strains and angles. The stresses were generally within the same values, enough so to make a fairly confident assessment that the methods used could be considered to be accurate.

APPENDIX A RAW DATA and ORIGINAL HANDOUT

APPENDIX B Mohr’s circle

2

2

6

McPL

I