paraphrase

Spring 2016

ENGR 2411 Mechanics of Materials Lab

Section No. 001

Lab No. 7

Lab Title: Stress Concentrations

Submitted to:

Dr. Zahid Hossain

And

Mr. AM Feroze Rashid

College of Engineering

Submitted By:

Name: Kudakwashe Makuvire

Student ID: 50378716

04/25/2016

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	7
Conclusions and Recommendations	8
Appendix A Raw Data	10
Appendix B sample calculations	11

List of Tables

Table No. and Title	Pg. No
Table 1. Data table	7
Table 2. Corrected strain for gage 4	7
Table 3. Extrapolation Coefficients	7
Table 4. ε0 and Kt	8
Table 5. Strain and X/R	8

List of Figures

Figure No. and Title	Pg. No
Figure 1. Stress Concentration in a Cantilever Beam	5
Figure 2. Strain Distribution Cantilever Beam with Hole	8

ABSTRACT

For this lab we determined the stress concentration value for a beam with a hole along its centerline and found the value to 1.14. We also found the peak strain, ε0, to be 2375 με.

Keywords: Stress, Strain, Kt, ε0

INTRODUCTION

This experiment was performed to show that stress and strain concentrations exist around discontinuities in a cantilever beam. The results recorded were used to approximate the stress concentration factor, Kt. A circular hole along the centerline was the discontinuity for the given cantilever beam. The maximum stress at the discontinuity, or stress-raiser, was to be determined from the experiment data as well as a graph of the stress concentration factor. Lastly, the theoretical value of stress concentration factor for a 0.25 inch hole in a beam with 1 inch width was to be determined.

BACKGROUND AND METHODOLOGY

Any irregularity in a loaded member effects the way stress will flow, causing a ‘concentration’ of the stress at certain points around the irregularity. This strain distribution can be approximated by the expression in equation 1, where R is the radius of the hole, X is the distance from the center of the hole to any point on the transverse centerline, and A, B, and C are coefficients to be determined. The coefficients A, B, and C were found using three strain gages and equations 5, 6 and 7 respectively. Figure 1 below shows the stress concentration in a cantilever beam. The maximum stress is uniform across the width of the beam and can be calculated using equation 12. The nominal stress which is based upon the net area of the section can be calculated using equation 11.

Figure 1. Stress Concentration in a Cantilever Beam

To fully understand the experiment and its purpose we had to define the following terms.

· Stress Concentration[1] is localized stress that is considerably higher than average due to abrupt changes in geometry or localized loading. In figure one you would expect to see a stress concentration around the hole that is length l from the load P.

· Stress Concentration Factor is the ratio of the maximum stress at the discontinuity to the nominal stress at the same point, Kt= and when further simplified equation 10.

Equations

The following equations were used throughout the course of this experiment with the end goal being to find the stress concentration factor with which we could compare with a published value.

……………………………………(1)

…………………………………(2)

…………………………………(3)

…………………………………(4)

C = 5.86*(ϵ1 - ϵ2) – 5.44*( ϵ2 - ϵ3)………………………………(5)

B = 3.49*( ϵ1 - ϵ2) – 1.20*C …………………………………(6)

A = ϵ1 – 0.743*B – 0.522*C …………………………………(7)

Corrected Strain for Gage 4 = = 2000*( ….………(8)

…………………………………………..(9)

Stress Concentration Factor: Kt = ………...……………(10)

………………………………………………(11)

………………………………………..……..(12)

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

1. We back calibrated the screw of the beam clamp and carefully placed the beam as a cantilever.

2. We connected the common leads 1 and 2 with S- and d120. For gage 1, we connected leads 3 to P+.

3. The P-3500 strain indicator was turned and the device was set for initial reading for gage 1 as 0 micro-strains.

4. We turned off the device and disconnected lead 3 and connect 4.

5. We turned it back on and got the strain for gage 2.

6. We turned off the device and disconnect lead 4 and connect 5.

7. We turned it on again and got strain for gage 3.

8. We turned off the device and disconnected lead 5 and connected 6.

9. The device was turned on once more and we got strain for gage 4.

10. Next, we rotated the screw in clockwise direction until we got 2000 plus strain reading for gage 4 at initial stage.

11. Once the required strain was achieved, we turned off the device and went backward from step 9 to step 3.

12. We turned off the device and disconnected all the cables and took it back to the cabinet.

RESULTS AND DISCUSSIONS

Following the procedure, we were able to retrieve values of strain for gages 1 through 4. These strain values were tabulated and are shown in table one. The units of strain are micro strain as these were the units the machine was using.

Table 1. Data table

Gage	Initial Reading (με)	Final Reading (με)	Strain (με)
1	0	2070	2070
2	-167	1726	1893
3	-168	1668	1836
4	253	2253	2000

Once we had collected the strain data, we had to apply a correction factor for strain 8 and we used equation 8. Table 2 shows the corrected value of ε’4.

Table 2. Corrected strain for gage 4

ε’4 (με)

2085

Next we used equations 5-7 to compute our extrapolation coefficients A, B and C.

Table 3. Extrapolation Coefficients

Extrapolation Coefficients
A	1968.809
B	-646.518
C	1053.54

Using the extrapolation coefficients we calculated the maximum strain at the hole, ε0 which was measured in micro strain. We also calculated the stress concentration factor Kt using equation 10. Both these values were calculated and tabulated in table 4.

Table 4. ε0 and Kt

ε0 and Kt
ε0	2375.813
Kt	1.14

We looked up the theoretical concentration stress factor. The theoretical value was 2.4, which just about double our concentration stress factor value.

We plotted the strains ε0, ε1, ε2, and ε3 versus the corresponding dimensionless distance X/R as shown in table 5 and figure 2. The smooth curve was of the second polynomial. The figure suggests that at a certain relationship of X and R stress is at a minimum and a maximum at the extreme ends.

Table 5. Strain and X/R

	strain	X/R
ε0	2375	1
ε1	2070	1.16
ε2	1893	1.275862069
ε3	1836	2.6

Figure 2. Strain Distribution Cantilever Beam with Hole

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 the stress concentration factor and maximum strain. Our Kt however, was only half the theoretical value and this posed a concern. This suggests that there might have been human error in performing the experiment as the calculations were done repeatedly to ensure accuracy. Despite this fact the overall principle of the lab was fully understood.

APPENDIX A RAW DATA and ORIGINAL HANDOUT

APPENDIX B Sample Calculations

Strain Distribution Cantilever Beam with Hole

1 1.1599999999999999 1.2758620689655173 2.6 2375 2070 1893 1836

X/R

Strain (με)

2