Uniaxial Loading On A Bar With Circular Cross
03-08-2005, 11:46 PM
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City College of New York
CE 332
Mechanics of Materials
Professor Ghosn
Lab # 1
Bars under Uniaxial Loading
Jason Rodriguez
I.D 5111
Due: March 9th, 2005
TABLE OF CONTENTS
Abstract__________________________________________ _____________________3
Introduction______________________________________ ______________________4
Procedure_________________________________________ _____________________5
Result & Analysis__________________________________________ _____________7
Conclusion________________________________________ ____________________ 21
References________________________________________ _____________________22
Appendix A_________________________________________________ ___________22
Appendix B_________________________________________________ ___________22
LIST OF ILLUSTRATIONS
FIGURES
Figure 1: Normal stress due to centric load___________________________________ 4
Figure 2: Electro Mechanical Testing Machine________________________________ 5
Figure 3: Gage Length before Elongation____________________________________ 6
Figure 4: Steel: Stress versus Strain_________________________________________ 9
Figure 5: Steel: Loading and Unloading in Linear Elastic Range __________________ 9
Figure 6: Steel: Yielding__________________________________________ ________10
Figure 7: Steel: Strain-Hardening_________________________________________ __10
Figure 8: Steel: Necking___________________________________________ _______11
Figure 9: Steel: Yielding, Ultimate and Rupture Strength________________________11
Figure 10: Aluminum: Stress versus strain___________________________________ 15
Figure 11: Aluminum: Linear Elastic Range__________________________________15
Figure 12: Aluminum: Loading and Unloading________________________________16
Figure 13: Aluminum: Strain-Hardening ____________________________________ 16
Figure 14: Aluminum: Necking___________________________________________ _ 17
Figure 15: Aluminum: Yielding, Ultimate and Rupture Strength__________________ 17
Figure 16: Steel: Modulus of Elasticity from Slope of Stress versus Strain 1_________19
Figure 17: Steel: Modulus of Elasticity from Slope of Stress versus Strain 2_________19
Figure 18: Aluminum: Modulus of Elasticity from Slope of Stress versus Strain 1____ 20
Figure 19: Aluminum: Modulus of Elasticity from Slope of Stress versus Strain 2____ 20
TABLES
Table 1: Portion of Raw Data for Steel_______________________________________7
Table 2: Summary of Steel findings_________________________________________1 2
Table 3: Portion of Raw Data for Aluminum__________________________________13
Table 4: Summary of Aluminum findings ____________________________________18
ABSTRACT
INTRODUCTION
Material characteristics and properties of steel and aluminum are important when used in construction of buildings, bridges and various structures. Understanding how these materials react during applied loads is very important in construction and design. Characteristics that aid engineers in utilizing the right material for the specific tasks are yielding strengths, ultimate strength, and the rupture strength. These characteristics can be calculated from the data obtained from the engineering tension test.
The specimens used for the tensile test in this experiment are two bars each with a constant circular cross-sectional area, which is then put under an increasing axial load. The line of action of the load at both ends passes through the centroid of the bar. The load is then systematically applied at this centroid, where the loading is increased until the bar ruptures or fails. Since the loadings is at both ends of the bar and are directed away from the bar, this produces a tensile force acting longitudinally through the bar.
The centric loading produces normal stresses in the rod that acts perpendicular to its cross-section as seen in figure 1. From these normal stresses, and the strain in the bar one can accurately determine the yielding strength, ultimate strength, rupture strengths and the modulus of elasticity of the bar.
Figure 1: Normal stress due to centric load
PROCEDURE
An INSTRON 8500 PLUS Testing System was used for the tensile test. The bar is loaded into a machine where both ends of the specimen are held in position by powerful grips, see figure 2. These grips exert the simultaneous load at each end via an electronically driven gear (or hydraulics). The machine pulls at these grips in opposite direction, thus producing the tensile force. The load and strain are recorded for analysis.
Figure 2: Electro Mechanical Testing Machines
Two separate tests were performed in this experiment; the first test was steel, while the second test was aluminum. Each bar is comprised of the same cross-sectional area, and a given test length that is measured between two points, referred to as gage marks. The change in gage length is measured by mechanical extensometers, which are placed at the designated gage marks, see figure 3. As the load is applied at the ends of the bar, the bar begins to elongate. The elongation can be measured by the change between the gage marks compared to the original gage length. This strain caused by the elongation and the load at that specific time is recorded. The data obtained from the test is then saved onto a computer excel file, where the raw data may be used to calculate properties of the tested material.
Figure 3: Gage Length before Elongation
Using this information obtained from the INSTRON 8500, we can plot the relationship between stress and strain. Normal stress s is found by dividing the applied load P on the bar, by its cross-sectional area A0. (We can visually see that the area of the bar decreases as the load increases, we use the engineering stress, in which the original cross sectional area A0 is used). The elongation data provided by the extensometer one can calculate the strain e. The strain can be calculated by taking the change in elongation, L-Lo, divided by the original length Lo. Plotting a diagram of stress versus strain we can find the modulus of elasticity from the slope s versus e in its linear elastic stage (Linear portion of the graph) . Yield, ultimate and rupture strength is also obtained from the rest of the stress versus strain diagram.
RESULT & ANALYSIS
Using the raw data, a table was constructed; see Table 1, & Table 2, where the stress and strain were calculated from the data points obtained from the engineering tension test. Multiple diagrams were constructed to visually display the data and its defining characteristics.
Table 1: Portion of Raw Data for Steel
Steel Bar Diameter = 0.335 in^2
Area = 0.0881
Gage Length = 2"
Load (Kip) Load (lbs) Stress (psi) Elongation (in) Strain(in/in)
1 0.1673 167.2948 1898.0292 0.0040 0.0020
2 0.1610 160.9634 1826.1975 0.0040 0.0020
3 0.1671 167.1139 1895.9769 0.0040 0.0020
4 0.1729 172.8623 1961.1955 0.0039 0.0019
5 0.4305 430.4976 4884.1757 0.0042 0.0021
6 0.5367 536.6833 6088.8960 0.0042 0.0021
7 0.7678 767.7671 8710.6392 0.0045 0.0022
8 1.1434 1143.3865 12972.1979 0.0046 0.0023
9 1.3101 1310.0649 14863.2340 0.0048 0.0024
10 1.4479 1447.9273 16427.3406 0.0049 0.0024
11 1.5932 1593.1662 18075.1366 0.0050 0.0025
12 1.7676 1767.5628 20053.7390 0.0051 0.0026
13 1.9758 1975.7601 22415.8244 0.0052 0.0026
14 2.1438 2143.8186 24322.5183 0.0054 0.0027
15 2.2863 2286.2972 25938.9976 0.0055 0.0028
16 2.3861 2386.0644 27070.8982 0.0055 0.0028
17 2.4811 2481.0546 28148.6017 0.0056 0.0028
18 2.5963 2596.3453 29456.6229 0.0058 0.0029
19 2.7342 2734.1608 31020.1974 0.0059 0.0029
20 2.9202 2920.1951 33130.8343 0.0059 0.0030
21 3.0671 3067.1491 34798.0889 0.0061 0.0030
22 3.0690 3068.9715 34818.7648 0.0061 0.0031
23 3.0683 3068.2814 34810.9353 0.0060 0.0030
24 3.0677 3067.7186 34804.5501 0.0060 0.0030
25 3.1732 3173.2410 36001.7458 0.0063 0.0031
26 3.1815 3181.4684 36095.0891 0.0061 0.0031
27 3.1860 3185.9773 36146.2445 0.0061 0.0030
28 3.0697 3069.6817 34826.8223 0.0060 0.0030
29 2.8576 2857.5985 32420.6497 0.0058 0.0029
30 2.6425 2642.4871 29980.1209 0.0058 0.0029
31 2.4531 2453.1163 27831.6300 0.0056 0.0028
32 2.3246 2324.6404 26374.0172 0.0055 0.0027
33 2.2557 2255.6991 25591.8494 0.0055 0.0027
34 2.1672 2167.1809 24587.5734 0.0054 0.0027
35 2.0902 2090.1865 23714.0397 0.0053 0.0027
36 2.0044 2004.3683 22740.3964 0.0053 0.0026
37 1.9176 1917.6256 21756.2642 0.0053 0.0026
38 1.7935 1793.5314 20348.3636 0.0052 0.0026
39 1.6810 1681.0010 19071.6591 0.0050 0.0025
40 1.5785 1578.5204 17908.9739 0.0050 0.0025
41 1.5109 1510.8789 17141.5528 0.0048 0.0024
42 1.4326 1432.5981 16253.4244 0.0048 0.0024
43 1.2448 1244.8487 14123.3290 0.0047 0.0024
44 1.1072 1107.2007 12561.6548 0.0046 0.0023
45 0.9914 991.4075 11247.9326 0.0045 0.0023
46 0.9433 943.2960 10702.0876 0.0045 0.0022
47 0.8523 852.3257 9669.9914 0.0044 0.0022
48 0.7257 725.6788 8233.1294 0.0042 0.0021
49 0.5898 589.8330 6691.9022 0.0042 0.0021
50 0.4814 481.4030 5461.7178 0.0042 0.0021
51 0.4485 448.4599 5087.9649 0.0042 0.0021
52 0.4384 438.4302 4974.1744 0.0040 0.0020
53 0.4306 430.6450 4885.8480 0.0041 0.0021
54 0.4322 432.1525 4902.9508 0.0041 0.0021
55 0.4318 431.7639 4898.5421 0.0041 0.0021
56 0.4133 413.2657 4688.6719 0.0041 0.0020
57 0.4139 413.8955 4695.8170 0.0041 0.0020
58 0.4255 425.5264 4827.7746 0.0040 0.0020
59 0.7692 769.2478 8727.4379 0.0043 0.0022
60 1.0934 1093.3655 12404.6887 0.0046 0.0023
61 1.2681 1268.1038 14387.1678 0.0047 0.0024
62 1.4474 1447.3578 16420.8794 0.0048 0.0024
63 1.6326 1632.5679 18522.1654 0.0049 0.0024
64 1.8071 1807.1186 20502.5160 0.0050 0.0025
65 2.0051 2005.0517 22748.1498 0.0053 0.0027
66 2.1742 2174.1555 24666.7031 0.0054 0.0027
67 2.3411 2341.0818 26560.5518 0.0055 0.0027
68 2.5291 2529.0991 28693.6867 0.0056 0.0028
69 2.6928 2692.7559 30550.4415 0.0056 0.0028
70 2.8557 2855.7494 32399.6709 0.0058 0.0029
71 3.0338 3033.7706 34419.3958 0.0060 0.0030
72 3.2178 3217.8417 36507.7594 0.0061 0.0031
73 3.3809 3380.8553 38357.2169 0.0062 0.0031
74 3.5594 3559.4058 40382.9469 0.0064 0.0032
75 3.7160 3715.9742 42159.2809 0.0065 0.0033
Figure 4: Steel: Stress versus Strain
Figure 5: Steel: Loading and Unloading in Linear Elastic Range
Figure 6: Steel: Yielding
Figure 7: Steel: Strain-Hardening
Figure 8: Steel: Necking
Figure 9: Steel: Yielding, Ultimate and Rupture Strength
Table 2: Summary of Steel findings
Steel Bar Diameter = 0.335 in^2
Area = 0.0881
Gage Length = 2"
Loading Unloading Load till Rupture
Load (Kip) Load (lbs) Stress (psi) Elongation (in) Strain(in/in)
1 0.1673 167.2948 1898.0292 0.0040 0.0020
27 3.1860 3185.9773 36146.2445 0.0061 0.0030
28 3.0697 3069.6817 34826.8223 0.0060 0.0030
58 0.4255 425.5264 4827.7746 0.0040 0.0020
59 0.7692 769.2478 8727.4379 0.0043 0.0022
87 5.5230 5522.9521 62660.2006 0.0078 0.0039
88 4.8327 4832.6752 54828.7206 0.0070 0.0035
89 4.8398 4839.8038 54909.5976 0.0071 0.0035
90 4.8546 4854.6038 55077.5097 0.0071 0.0035
111 4.9625 4962.4576 56301.1562 0.0071 0.0036
112 4.9154 4915.4449 55767.7775 0.0071 0.0035
182 0.3908 390.8011 4433.8023 0.0058 0.0029
183 0.6401 640.0751 7261.9192 0.0062 0.0031
221 4.4581 4458.0743 50578.7168 0.0090 0.0045
222 4.2869 4286.8668 48636.2961 0.0087 0.0044
271 0.4308 430.8125 4887.7483 0.0059 0.0029
272 2.5991 2599.1124 29488.0168 0.0075 0.0037
379 6.7051 6705.0837 76071.9779 0.3680 0.1840
380 6.6974 6697.3588 75984.3357 0.3746 0.1873
411 4.8176 4817.6341 54658.0730 0.5293 0.2646
1. Yielding Strength: 63,600 psi (Line item # 87)
2. Ultimate Strength: 76,072 psi (Line item # 379)
3. Rupture Strength: 54,658 psi (Line item # 411)
Table 3: Portion of Raw Data for Aluminum
Aluminum Bar Diameter = 0.335 in^2
Area = 0.0881
Gage Length = 2"
Load (kip) Load (lbs) Stress (psi) Strain (in) Strain (in/in)
1 0.0852 85.2286 966.9535 0.0097 0.0048
2 0.0824 82.3811 934.6483 0.0095 0.0048
3 0.0914 91.3991 1036.9609 0.0096 0.0048
4 0.0890 88.9872 1009.5965 0.0095 0.0047
5 0.1623 162.3101 1841.4761 0.0097 0.0049
6 0.3467 346.6560 3932.9568 0.0101 0.0051
7 0.5827 582.7044 6611.0250 0.0106 0.0053
8 0.8302 830.2230 9419.2266 0.0111 0.0056
9 1.0634 1063.3905 12064.6098 0.0116 0.0058
10 1.4155 1415.5336 16059.8205 0.0124 0.0062
11 1.5601 1560.1361 17700.3963 0.0126 0.0063
12 1.6886 1688.6254 19158.1612 0.0130 0.0065
13 1.8066 1806.6295 20496.9670 0.0132 0.0066
14 1.9352 1935.2261 21955.9492 0.0134 0.0067
15 2.0820 2082.0328 23621.5326 0.0137 0.0069
16 2.2032 2203.2461 24996.7481 0.0141 0.0071
17 2.3763 2376.3028 26960.1488 0.0144 0.0072
18 2.5707 2570.7453 29166.1802 0.0148 0.0074
19 2.7756 2775.5792 31490.1065 0.0153 0.0077
20 2.9708 2970.8324 33705.3356 0.0156 0.0078
21 2.9963 2996.3185 33994.4861 0.0158 0.0079
22 3.0625 3062.5464 34745.8693 0.0160 0.0080
23 3.0639 3063.8729 34760.9190 0.0159 0.0079
24 3.0696 3069.6214 34826.1381 0.0159 0.0079
25 3.0681 3068.0804 34808.6549 0.0159 0.0079
26 3.0668 3066.8008 34794.1373 0.0160 0.0080
27 3.1966 3196.5765 36266.4968 0.0163 0.0082
28 3.1974 3197.4274 36276.1506 0.0162 0.0081
29 3.2008 3200.8242 36314.6888 0.0162 0.0081
30 3.1190 3118.9656 35385.9687 0.0160 0.0080
31 2.9423 2942.3179 33381.8267 0.0157 0.0078
32 2.8209 2820.8902 32004.1787 0.0153 0.0077
33 2.5876 2587.5886 29357.2745 0.0148 0.0074
34 2.3675 2367.4523 26859.7361 0.0144 0.0072
35 2.1974 2197.4441 24930.9220 0.0141 0.0070
36 2.1073 2107.3381 23908.6318 0.0138 0.0069
37 2.0239 2023.9185 22962.2016 0.0136 0.0068
38 1.9680 1967.9548 22327.2700 0.0136 0.0068
39 1.9263 1926.2885 21854.5484 0.0133 0.0066
40 1.8779 1877.8956 21305.5107 0.0134 0.0067
41 1.8113 1811.2792 20549.7198 0.0131 0.0066
42 1.7294 1729.3738 19620.4688 0.0129 0.0064
43 1.6098 1609.7885 18263.7236 0.0127 0.0064
44 1.4754 1475.4301 16739.3713 0.0125 0.0062
45 1.3468 1346.8335 15280.3891 0.0122 0.0061
46 1.2189 1218.8601 13828.4774 0.0119 0.0059
47 1.1027 1102.7319 12510.9544 0.0117 0.0059
48 0.9988 998.8309 11332.1543 0.0115 0.0058
49 0.8985 898.5143 10194.0208 0.0113 0.0056
50 0.8122 812.1937 9214.6773 0.0110 0.0055
51 0.7331 733.1156 8317.5031 0.0110 0.0055
52 0.6456 645.5823 7324.4013 0.0107 0.0054
53 0.5552 555.2351 6299.3743 0.0106 0.0053
54 0.4564 456.3657 5177.6596 0.0103 0.0052
55 0.4373 437.2511 4960.7962 0.0101 0.0051
56 0.4371 437.1439 4959.5801 0.0102 0.0051
57 0.4307 430.7120 4886.6082 0.0102 0.0051
58 0.4336 433.5729 4919.0655 0.0102 0.0051
59 0.4341 434.1223 4925.2984 0.0103 0.0052
60 0.4242 424.2065 4812.8001 0.0101 0.0051
61 0.4182 418.2102 4744.7690 0.0102 0.0051
62 0.4213 421.2586 4779.3547 0.0103 0.0051
63 0.5644 564.4072 6403.4353 0.0104 0.0052
64 0.7071 707.1203 8022.5751 0.0108 0.0054
65 0.7982 798.2447 9056.4195 0.0110 0.0055
66 0.8795 879.5404 9978.7538 0.0112 0.0056
67 0.9596 959.5967 10887.0258 0.0113 0.0056
68 1.0393 1039.2510 11790.7371 0.0115 0.0057
69 1.1154 1115.4415 12655.1502 0.0117 0.0058
70 1.1982 1198.2246 13594.3590 0.0118 0.0059
71 1.2761 1276.1369 14478.3067 0.0121 0.0060
72 1.3547 1354.6723 15369.3236 0.0121 0.0061
73 1.4332 1433.1542 16259.7336 0.0123 0.0061
74 1.5104 1510.3898 17136.0038 0.0124 0.0062
75 1.5877 1587.6724 18012.8072 0.0127 0.0063
Figure 10: Aluminum: Stress versus strain
Figure 11: Aluminum: Linear Elastic Range
Figure 12: Aluminum: Loading and Unloading
Figure 13: Aluminum: Strain-Hardening
Figure 14: Aluminum: Necking
Figure 15: Aluminum: Yield, Ultimate and Rupture Strength
Table 4: Summary of Aluminum Findings
Aluminum Bar 0.335 in^2
Area = 0.0881
Gage Length = 2"
Loading Unloading Load till Rupture
Load (kip) Load (lbs) Stress (psi) Strain (in) Strain (in/in)
1 0.0852 85.2286 966.9535 0.0097 0.0048
29 3.2008 3200.8242 36314.6888 0.0162 0.0081
30 3.1190 3118.9656 35385.9687 0.0160 0.0080
57 0.4307 430.7120 4886.6082 0.0102 0.0051
58 0.4336 433.5729 4919.0655 0.0102 0.0051
157 4.4173 4417.2589 50115.6490 0.0312 0.0156
158 4.4041 4404.0803 49966.1323 0.0313 0.0156
208 0.3767 376.7248 4274.1004 0.0224 0.0112
209 0.4528 452.8081 5137.2970 0.0227 0.0114
266 4.6565 4656.5166 52830.1276 0.0704 0.0352
267 4.3913 4391.2770 49820.8735 0.0698 0.0349
310 0.4308 430.8259 4887.9003 0.0604 0.0302
311 1.3603 1360.2667 15432.7944 0.0625 0.0313
317 4.9908 4990.7778 56622.4607 0.0719 0.0360
*317.5 OFF SET METHOD 2% 56629.7201 0.0364
318 4.9921 4992.0575 56636.9794 0.0734 0.0367
409 5.8405 5840.5442 66263.4158 0.3603 0.1802
410 5.8321 5832.0890 66167.4880 0.3649 0.1825
422 5.0077 5007.6681 56814.0882 0.4512 0.2256
423 4.8861 4886.1399 55435.3000 0.4568 0.2284
1. Yielding Strength: 56,630 psi (Line item # 317.5)
2. Ultimate Strength: 66,623 psi (Line item # 409)
3. Rupture Strength: 55,435 psi (Line item # 423)
Figure 16: Steel: Modulus of Elasticity from Slope of Stress versus Strain 1
Figure 17: Steel: Modulus of Elasticity from Slope of Stress versus Strain 2
Figure 18: Aluminum: Modulus of Elasticity from Slope of Stress versus Strain 1
Figure 19: Aluminum: Modulus of Elasticity from Slope of Stress versus Strain 2
Using the data plotted for the stress strain diagrams for steel and aluminum, we can calculate the modulus of elasticity (E) for each of the tested materials. The modulus is the slope of the trend line for the material in the linear elastic range.
As you can see in figures 16 and 17, which were taken for two different loadings, both in the linear elastic range, and both yield the same modulus of elasticity. I choose to test the slope at two different loading points to double check that the modulus of elasticity is the same. The modulus of elasticity for the tested materials is listed below.
E (Steel) = 3*10^7 Pa or 30Mpa
E (Aluminum)= 1*10^7 or 10Mpa
CONCLUSION
Steel and aluminum are both classified as ductile materials due to the fact that the stress versus strain diagrams shares many similarities such as containing yielding point, stress hardening, necking and rupture. The biggest similarity between them is that ductile materials yield at normal temperature.
Before a material reaches its yielding point, the relationship between it load and length increase linearly. Once the yielding point is reached, in all ductile materials, the specimen lengths changes at a larger rate for a small increase in the applied load. This is where we begin the signs of shear stress, which led to the large elongation, and will ultimately lead to rupture of the material after a maximum load of the material has been reached, we can visually see a large decrease in diameter, where the loads are needed to make the material fail. The main cause for rupture is due to shearing stresses in the bar. The shearing stresses are the greatest at an angle of 45 degrees, which can be visually seen in the specimen.
References
Beer, Johnston, DeWolf. Mechanics of Materials: Third Edition. McGraw-Hill 2002
Appendix A: Key Terms and Formulas Used In Calculations
Ultimate Strength: The stress corresponding to the maximum load reached in the tensile test.
Proportional Limit: The stress at which the strain ceases to be proportional to the stress (the limit of the straight part of the stress vs. strain diagram). This quantity indicates the range of stress in which the assumption of elastic action is valid.
Yield Strength: The stress determined by some arbitrary permanent strain. The most commonly used yield stress is the one determined by a line passing through a strain of 0.002, parallel with the elastic line. The yield stress represents a practical upper limit for the actual stress developed in a structure.
Ductility: The total normal strain that has occurred at failure (usually measured as the total permanent strain after failure). Elongation is generally specified in percent and is regarded as a measure of the ductility of a material.
Modulus of Elasticity: The ratio of stress to strain within the linear elastic range of a material. (The slope of the linear relationship between the stress vs. strain diagram)
Appendix B: Formulas used in calculations
Cross-sectional area of cylindrical rod = p (d2/4); where (d=diameter)
Stress = Force / Original Cross-sectional Area à s = P / A0
Strain = Deformation / Original Length à e = L-L0/ L0
Modulus of Elasticity, E = Stress/Strain à E = s/e
CE 332
Mechanics of Materials
Professor Ghosn
Lab # 1
Bars under Uniaxial Loading
Jason Rodriguez
I.D 5111
Due: March 9th, 2005
TABLE OF CONTENTS
Abstract__________________________________________ _____________________3
Introduction______________________________________ ______________________4
Procedure_________________________________________ _____________________5
Result & Analysis__________________________________________ _____________7
Conclusion________________________________________ ____________________ 21
References________________________________________ _____________________22
Appendix A_________________________________________________ ___________22
Appendix B_________________________________________________ ___________22
LIST OF ILLUSTRATIONS
FIGURES
Figure 1: Normal stress due to centric load___________________________________ 4
Figure 2: Electro Mechanical Testing Machine________________________________ 5
Figure 3: Gage Length before Elongation____________________________________ 6
Figure 4: Steel: Stress versus Strain_________________________________________ 9
Figure 5: Steel: Loading and Unloading in Linear Elastic Range __________________ 9
Figure 6: Steel: Yielding__________________________________________ ________10
Figure 7: Steel: Strain-Hardening_________________________________________ __10
Figure 8: Steel: Necking___________________________________________ _______11
Figure 9: Steel: Yielding, Ultimate and Rupture Strength________________________11
Figure 10: Aluminum: Stress versus strain___________________________________ 15
Figure 11: Aluminum: Linear Elastic Range__________________________________15
Figure 12: Aluminum: Loading and Unloading________________________________16
Figure 13: Aluminum: Strain-Hardening ____________________________________ 16
Figure 14: Aluminum: Necking___________________________________________ _ 17
Figure 15: Aluminum: Yielding, Ultimate and Rupture Strength__________________ 17
Figure 16: Steel: Modulus of Elasticity from Slope of Stress versus Strain 1_________19
Figure 17: Steel: Modulus of Elasticity from Slope of Stress versus Strain 2_________19
Figure 18: Aluminum: Modulus of Elasticity from Slope of Stress versus Strain 1____ 20
Figure 19: Aluminum: Modulus of Elasticity from Slope of Stress versus Strain 2____ 20
TABLES
Table 1: Portion of Raw Data for Steel_______________________________________7
Table 2: Summary of Steel findings_________________________________________1 2
Table 3: Portion of Raw Data for Aluminum__________________________________13
Table 4: Summary of Aluminum findings ____________________________________18
ABSTRACT
INTRODUCTION
Material characteristics and properties of steel and aluminum are important when used in construction of buildings, bridges and various structures. Understanding how these materials react during applied loads is very important in construction and design. Characteristics that aid engineers in utilizing the right material for the specific tasks are yielding strengths, ultimate strength, and the rupture strength. These characteristics can be calculated from the data obtained from the engineering tension test.
The specimens used for the tensile test in this experiment are two bars each with a constant circular cross-sectional area, which is then put under an increasing axial load. The line of action of the load at both ends passes through the centroid of the bar. The load is then systematically applied at this centroid, where the loading is increased until the bar ruptures or fails. Since the loadings is at both ends of the bar and are directed away from the bar, this produces a tensile force acting longitudinally through the bar.
The centric loading produces normal stresses in the rod that acts perpendicular to its cross-section as seen in figure 1. From these normal stresses, and the strain in the bar one can accurately determine the yielding strength, ultimate strength, rupture strengths and the modulus of elasticity of the bar.
Figure 1: Normal stress due to centric load
PROCEDURE
An INSTRON 8500 PLUS Testing System was used for the tensile test. The bar is loaded into a machine where both ends of the specimen are held in position by powerful grips, see figure 2. These grips exert the simultaneous load at each end via an electronically driven gear (or hydraulics). The machine pulls at these grips in opposite direction, thus producing the tensile force. The load and strain are recorded for analysis.
Figure 2: Electro Mechanical Testing Machines
Two separate tests were performed in this experiment; the first test was steel, while the second test was aluminum. Each bar is comprised of the same cross-sectional area, and a given test length that is measured between two points, referred to as gage marks. The change in gage length is measured by mechanical extensometers, which are placed at the designated gage marks, see figure 3. As the load is applied at the ends of the bar, the bar begins to elongate. The elongation can be measured by the change between the gage marks compared to the original gage length. This strain caused by the elongation and the load at that specific time is recorded. The data obtained from the test is then saved onto a computer excel file, where the raw data may be used to calculate properties of the tested material.
Figure 3: Gage Length before Elongation
Using this information obtained from the INSTRON 8500, we can plot the relationship between stress and strain. Normal stress s is found by dividing the applied load P on the bar, by its cross-sectional area A0. (We can visually see that the area of the bar decreases as the load increases, we use the engineering stress, in which the original cross sectional area A0 is used). The elongation data provided by the extensometer one can calculate the strain e. The strain can be calculated by taking the change in elongation, L-Lo, divided by the original length Lo. Plotting a diagram of stress versus strain we can find the modulus of elasticity from the slope s versus e in its linear elastic stage (Linear portion of the graph) . Yield, ultimate and rupture strength is also obtained from the rest of the stress versus strain diagram.
RESULT & ANALYSIS
Using the raw data, a table was constructed; see Table 1, & Table 2, where the stress and strain were calculated from the data points obtained from the engineering tension test. Multiple diagrams were constructed to visually display the data and its defining characteristics.
Table 1: Portion of Raw Data for Steel
Steel Bar Diameter = 0.335 in^2
Area = 0.0881
Gage Length = 2"
Load (Kip) Load (lbs) Stress (psi) Elongation (in) Strain(in/in)
1 0.1673 167.2948 1898.0292 0.0040 0.0020
2 0.1610 160.9634 1826.1975 0.0040 0.0020
3 0.1671 167.1139 1895.9769 0.0040 0.0020
4 0.1729 172.8623 1961.1955 0.0039 0.0019
5 0.4305 430.4976 4884.1757 0.0042 0.0021
6 0.5367 536.6833 6088.8960 0.0042 0.0021
7 0.7678 767.7671 8710.6392 0.0045 0.0022
8 1.1434 1143.3865 12972.1979 0.0046 0.0023
9 1.3101 1310.0649 14863.2340 0.0048 0.0024
10 1.4479 1447.9273 16427.3406 0.0049 0.0024
11 1.5932 1593.1662 18075.1366 0.0050 0.0025
12 1.7676 1767.5628 20053.7390 0.0051 0.0026
13 1.9758 1975.7601 22415.8244 0.0052 0.0026
14 2.1438 2143.8186 24322.5183 0.0054 0.0027
15 2.2863 2286.2972 25938.9976 0.0055 0.0028
16 2.3861 2386.0644 27070.8982 0.0055 0.0028
17 2.4811 2481.0546 28148.6017 0.0056 0.0028
18 2.5963 2596.3453 29456.6229 0.0058 0.0029
19 2.7342 2734.1608 31020.1974 0.0059 0.0029
20 2.9202 2920.1951 33130.8343 0.0059 0.0030
21 3.0671 3067.1491 34798.0889 0.0061 0.0030
22 3.0690 3068.9715 34818.7648 0.0061 0.0031
23 3.0683 3068.2814 34810.9353 0.0060 0.0030
24 3.0677 3067.7186 34804.5501 0.0060 0.0030
25 3.1732 3173.2410 36001.7458 0.0063 0.0031
26 3.1815 3181.4684 36095.0891 0.0061 0.0031
27 3.1860 3185.9773 36146.2445 0.0061 0.0030
28 3.0697 3069.6817 34826.8223 0.0060 0.0030
29 2.8576 2857.5985 32420.6497 0.0058 0.0029
30 2.6425 2642.4871 29980.1209 0.0058 0.0029
31 2.4531 2453.1163 27831.6300 0.0056 0.0028
32 2.3246 2324.6404 26374.0172 0.0055 0.0027
33 2.2557 2255.6991 25591.8494 0.0055 0.0027
34 2.1672 2167.1809 24587.5734 0.0054 0.0027
35 2.0902 2090.1865 23714.0397 0.0053 0.0027
36 2.0044 2004.3683 22740.3964 0.0053 0.0026
37 1.9176 1917.6256 21756.2642 0.0053 0.0026
38 1.7935 1793.5314 20348.3636 0.0052 0.0026
39 1.6810 1681.0010 19071.6591 0.0050 0.0025
40 1.5785 1578.5204 17908.9739 0.0050 0.0025
41 1.5109 1510.8789 17141.5528 0.0048 0.0024
42 1.4326 1432.5981 16253.4244 0.0048 0.0024
43 1.2448 1244.8487 14123.3290 0.0047 0.0024
44 1.1072 1107.2007 12561.6548 0.0046 0.0023
45 0.9914 991.4075 11247.9326 0.0045 0.0023
46 0.9433 943.2960 10702.0876 0.0045 0.0022
47 0.8523 852.3257 9669.9914 0.0044 0.0022
48 0.7257 725.6788 8233.1294 0.0042 0.0021
49 0.5898 589.8330 6691.9022 0.0042 0.0021
50 0.4814 481.4030 5461.7178 0.0042 0.0021
51 0.4485 448.4599 5087.9649 0.0042 0.0021
52 0.4384 438.4302 4974.1744 0.0040 0.0020
53 0.4306 430.6450 4885.8480 0.0041 0.0021
54 0.4322 432.1525 4902.9508 0.0041 0.0021
55 0.4318 431.7639 4898.5421 0.0041 0.0021
56 0.4133 413.2657 4688.6719 0.0041 0.0020
57 0.4139 413.8955 4695.8170 0.0041 0.0020
58 0.4255 425.5264 4827.7746 0.0040 0.0020
59 0.7692 769.2478 8727.4379 0.0043 0.0022
60 1.0934 1093.3655 12404.6887 0.0046 0.0023
61 1.2681 1268.1038 14387.1678 0.0047 0.0024
62 1.4474 1447.3578 16420.8794 0.0048 0.0024
63 1.6326 1632.5679 18522.1654 0.0049 0.0024
64 1.8071 1807.1186 20502.5160 0.0050 0.0025
65 2.0051 2005.0517 22748.1498 0.0053 0.0027
66 2.1742 2174.1555 24666.7031 0.0054 0.0027
67 2.3411 2341.0818 26560.5518 0.0055 0.0027
68 2.5291 2529.0991 28693.6867 0.0056 0.0028
69 2.6928 2692.7559 30550.4415 0.0056 0.0028
70 2.8557 2855.7494 32399.6709 0.0058 0.0029
71 3.0338 3033.7706 34419.3958 0.0060 0.0030
72 3.2178 3217.8417 36507.7594 0.0061 0.0031
73 3.3809 3380.8553 38357.2169 0.0062 0.0031
74 3.5594 3559.4058 40382.9469 0.0064 0.0032
75 3.7160 3715.9742 42159.2809 0.0065 0.0033
Figure 4: Steel: Stress versus Strain
Figure 5: Steel: Loading and Unloading in Linear Elastic Range
Figure 6: Steel: Yielding
Figure 7: Steel: Strain-Hardening
Figure 8: Steel: Necking
Figure 9: Steel: Yielding, Ultimate and Rupture Strength
Table 2: Summary of Steel findings
Steel Bar Diameter = 0.335 in^2
Area = 0.0881
Gage Length = 2"
Loading Unloading Load till Rupture
Load (Kip) Load (lbs) Stress (psi) Elongation (in) Strain(in/in)
1 0.1673 167.2948 1898.0292 0.0040 0.0020
27 3.1860 3185.9773 36146.2445 0.0061 0.0030
28 3.0697 3069.6817 34826.8223 0.0060 0.0030
58 0.4255 425.5264 4827.7746 0.0040 0.0020
59 0.7692 769.2478 8727.4379 0.0043 0.0022
87 5.5230 5522.9521 62660.2006 0.0078 0.0039
88 4.8327 4832.6752 54828.7206 0.0070 0.0035
89 4.8398 4839.8038 54909.5976 0.0071 0.0035
90 4.8546 4854.6038 55077.5097 0.0071 0.0035
111 4.9625 4962.4576 56301.1562 0.0071 0.0036
112 4.9154 4915.4449 55767.7775 0.0071 0.0035
182 0.3908 390.8011 4433.8023 0.0058 0.0029
183 0.6401 640.0751 7261.9192 0.0062 0.0031
221 4.4581 4458.0743 50578.7168 0.0090 0.0045
222 4.2869 4286.8668 48636.2961 0.0087 0.0044
271 0.4308 430.8125 4887.7483 0.0059 0.0029
272 2.5991 2599.1124 29488.0168 0.0075 0.0037
379 6.7051 6705.0837 76071.9779 0.3680 0.1840
380 6.6974 6697.3588 75984.3357 0.3746 0.1873
411 4.8176 4817.6341 54658.0730 0.5293 0.2646
1. Yielding Strength: 63,600 psi (Line item # 87)
2. Ultimate Strength: 76,072 psi (Line item # 379)
3. Rupture Strength: 54,658 psi (Line item # 411)
Table 3: Portion of Raw Data for Aluminum
Aluminum Bar Diameter = 0.335 in^2
Area = 0.0881
Gage Length = 2"
Load (kip) Load (lbs) Stress (psi) Strain (in) Strain (in/in)
1 0.0852 85.2286 966.9535 0.0097 0.0048
2 0.0824 82.3811 934.6483 0.0095 0.0048
3 0.0914 91.3991 1036.9609 0.0096 0.0048
4 0.0890 88.9872 1009.5965 0.0095 0.0047
5 0.1623 162.3101 1841.4761 0.0097 0.0049
6 0.3467 346.6560 3932.9568 0.0101 0.0051
7 0.5827 582.7044 6611.0250 0.0106 0.0053
8 0.8302 830.2230 9419.2266 0.0111 0.0056
9 1.0634 1063.3905 12064.6098 0.0116 0.0058
10 1.4155 1415.5336 16059.8205 0.0124 0.0062
11 1.5601 1560.1361 17700.3963 0.0126 0.0063
12 1.6886 1688.6254 19158.1612 0.0130 0.0065
13 1.8066 1806.6295 20496.9670 0.0132 0.0066
14 1.9352 1935.2261 21955.9492 0.0134 0.0067
15 2.0820 2082.0328 23621.5326 0.0137 0.0069
16 2.2032 2203.2461 24996.7481 0.0141 0.0071
17 2.3763 2376.3028 26960.1488 0.0144 0.0072
18 2.5707 2570.7453 29166.1802 0.0148 0.0074
19 2.7756 2775.5792 31490.1065 0.0153 0.0077
20 2.9708 2970.8324 33705.3356 0.0156 0.0078
21 2.9963 2996.3185 33994.4861 0.0158 0.0079
22 3.0625 3062.5464 34745.8693 0.0160 0.0080
23 3.0639 3063.8729 34760.9190 0.0159 0.0079
24 3.0696 3069.6214 34826.1381 0.0159 0.0079
25 3.0681 3068.0804 34808.6549 0.0159 0.0079
26 3.0668 3066.8008 34794.1373 0.0160 0.0080
27 3.1966 3196.5765 36266.4968 0.0163 0.0082
28 3.1974 3197.4274 36276.1506 0.0162 0.0081
29 3.2008 3200.8242 36314.6888 0.0162 0.0081
30 3.1190 3118.9656 35385.9687 0.0160 0.0080
31 2.9423 2942.3179 33381.8267 0.0157 0.0078
32 2.8209 2820.8902 32004.1787 0.0153 0.0077
33 2.5876 2587.5886 29357.2745 0.0148 0.0074
34 2.3675 2367.4523 26859.7361 0.0144 0.0072
35 2.1974 2197.4441 24930.9220 0.0141 0.0070
36 2.1073 2107.3381 23908.6318 0.0138 0.0069
37 2.0239 2023.9185 22962.2016 0.0136 0.0068
38 1.9680 1967.9548 22327.2700 0.0136 0.0068
39 1.9263 1926.2885 21854.5484 0.0133 0.0066
40 1.8779 1877.8956 21305.5107 0.0134 0.0067
41 1.8113 1811.2792 20549.7198 0.0131 0.0066
42 1.7294 1729.3738 19620.4688 0.0129 0.0064
43 1.6098 1609.7885 18263.7236 0.0127 0.0064
44 1.4754 1475.4301 16739.3713 0.0125 0.0062
45 1.3468 1346.8335 15280.3891 0.0122 0.0061
46 1.2189 1218.8601 13828.4774 0.0119 0.0059
47 1.1027 1102.7319 12510.9544 0.0117 0.0059
48 0.9988 998.8309 11332.1543 0.0115 0.0058
49 0.8985 898.5143 10194.0208 0.0113 0.0056
50 0.8122 812.1937 9214.6773 0.0110 0.0055
51 0.7331 733.1156 8317.5031 0.0110 0.0055
52 0.6456 645.5823 7324.4013 0.0107 0.0054
53 0.5552 555.2351 6299.3743 0.0106 0.0053
54 0.4564 456.3657 5177.6596 0.0103 0.0052
55 0.4373 437.2511 4960.7962 0.0101 0.0051
56 0.4371 437.1439 4959.5801 0.0102 0.0051
57 0.4307 430.7120 4886.6082 0.0102 0.0051
58 0.4336 433.5729 4919.0655 0.0102 0.0051
59 0.4341 434.1223 4925.2984 0.0103 0.0052
60 0.4242 424.2065 4812.8001 0.0101 0.0051
61 0.4182 418.2102 4744.7690 0.0102 0.0051
62 0.4213 421.2586 4779.3547 0.0103 0.0051
63 0.5644 564.4072 6403.4353 0.0104 0.0052
64 0.7071 707.1203 8022.5751 0.0108 0.0054
65 0.7982 798.2447 9056.4195 0.0110 0.0055
66 0.8795 879.5404 9978.7538 0.0112 0.0056
67 0.9596 959.5967 10887.0258 0.0113 0.0056
68 1.0393 1039.2510 11790.7371 0.0115 0.0057
69 1.1154 1115.4415 12655.1502 0.0117 0.0058
70 1.1982 1198.2246 13594.3590 0.0118 0.0059
71 1.2761 1276.1369 14478.3067 0.0121 0.0060
72 1.3547 1354.6723 15369.3236 0.0121 0.0061
73 1.4332 1433.1542 16259.7336 0.0123 0.0061
74 1.5104 1510.3898 17136.0038 0.0124 0.0062
75 1.5877 1587.6724 18012.8072 0.0127 0.0063
Figure 10: Aluminum: Stress versus strain
Figure 11: Aluminum: Linear Elastic Range
Figure 12: Aluminum: Loading and Unloading
Figure 13: Aluminum: Strain-Hardening
Figure 14: Aluminum: Necking
Figure 15: Aluminum: Yield, Ultimate and Rupture Strength
Table 4: Summary of Aluminum Findings
Aluminum Bar 0.335 in^2
Area = 0.0881
Gage Length = 2"
Loading Unloading Load till Rupture
Load (kip) Load (lbs) Stress (psi) Strain (in) Strain (in/in)
1 0.0852 85.2286 966.9535 0.0097 0.0048
29 3.2008 3200.8242 36314.6888 0.0162 0.0081
30 3.1190 3118.9656 35385.9687 0.0160 0.0080
57 0.4307 430.7120 4886.6082 0.0102 0.0051
58 0.4336 433.5729 4919.0655 0.0102 0.0051
157 4.4173 4417.2589 50115.6490 0.0312 0.0156
158 4.4041 4404.0803 49966.1323 0.0313 0.0156
208 0.3767 376.7248 4274.1004 0.0224 0.0112
209 0.4528 452.8081 5137.2970 0.0227 0.0114
266 4.6565 4656.5166 52830.1276 0.0704 0.0352
267 4.3913 4391.2770 49820.8735 0.0698 0.0349
310 0.4308 430.8259 4887.9003 0.0604 0.0302
311 1.3603 1360.2667 15432.7944 0.0625 0.0313
317 4.9908 4990.7778 56622.4607 0.0719 0.0360
*317.5 OFF SET METHOD 2% 56629.7201 0.0364
318 4.9921 4992.0575 56636.9794 0.0734 0.0367
409 5.8405 5840.5442 66263.4158 0.3603 0.1802
410 5.8321 5832.0890 66167.4880 0.3649 0.1825
422 5.0077 5007.6681 56814.0882 0.4512 0.2256
423 4.8861 4886.1399 55435.3000 0.4568 0.2284
1. Yielding Strength: 56,630 psi (Line item # 317.5)
2. Ultimate Strength: 66,623 psi (Line item # 409)
3. Rupture Strength: 55,435 psi (Line item # 423)
Figure 16: Steel: Modulus of Elasticity from Slope of Stress versus Strain 1
Figure 17: Steel: Modulus of Elasticity from Slope of Stress versus Strain 2
Figure 18: Aluminum: Modulus of Elasticity from Slope of Stress versus Strain 1
Figure 19: Aluminum: Modulus of Elasticity from Slope of Stress versus Strain 2
Using the data plotted for the stress strain diagrams for steel and aluminum, we can calculate the modulus of elasticity (E) for each of the tested materials. The modulus is the slope of the trend line for the material in the linear elastic range.
As you can see in figures 16 and 17, which were taken for two different loadings, both in the linear elastic range, and both yield the same modulus of elasticity. I choose to test the slope at two different loading points to double check that the modulus of elasticity is the same. The modulus of elasticity for the tested materials is listed below.
E (Steel) = 3*10^7 Pa or 30Mpa
E (Aluminum)= 1*10^7 or 10Mpa
CONCLUSION
Steel and aluminum are both classified as ductile materials due to the fact that the stress versus strain diagrams shares many similarities such as containing yielding point, stress hardening, necking and rupture. The biggest similarity between them is that ductile materials yield at normal temperature.
Before a material reaches its yielding point, the relationship between it load and length increase linearly. Once the yielding point is reached, in all ductile materials, the specimen lengths changes at a larger rate for a small increase in the applied load. This is where we begin the signs of shear stress, which led to the large elongation, and will ultimately lead to rupture of the material after a maximum load of the material has been reached, we can visually see a large decrease in diameter, where the loads are needed to make the material fail. The main cause for rupture is due to shearing stresses in the bar. The shearing stresses are the greatest at an angle of 45 degrees, which can be visually seen in the specimen.
References
Beer, Johnston, DeWolf. Mechanics of Materials: Third Edition. McGraw-Hill 2002
Appendix A: Key Terms and Formulas Used In Calculations
Ultimate Strength: The stress corresponding to the maximum load reached in the tensile test.
Proportional Limit: The stress at which the strain ceases to be proportional to the stress (the limit of the straight part of the stress vs. strain diagram). This quantity indicates the range of stress in which the assumption of elastic action is valid.
Yield Strength: The stress determined by some arbitrary permanent strain. The most commonly used yield stress is the one determined by a line passing through a strain of 0.002, parallel with the elastic line. The yield stress represents a practical upper limit for the actual stress developed in a structure.
Ductility: The total normal strain that has occurred at failure (usually measured as the total permanent strain after failure). Elongation is generally specified in percent and is regarded as a measure of the ductility of a material.
Modulus of Elasticity: The ratio of stress to strain within the linear elastic range of a material. (The slope of the linear relationship between the stress vs. strain diagram)
Appendix B: Formulas used in calculations
Cross-sectional area of cylindrical rod = p (d2/4); where (d=diameter)
Stress = Force / Original Cross-sectional Area à s = P / A0
Strain = Deformation / Original Length à e = L-L0/ L0
Modulus of Elasticity, E = Stress/Strain à E = s/e
03-09-2005, 12:11 AM
#4
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Join Date: Dec 2002
Location: QUEENS NYC
Posts: 1,687
Originally Posted by Srce' date='Mar 8 2005, 09:57 PM
yes that is mines but um.. dunno how to upload an excel file so your missing all the kewl graphs i made. Just cuase im ghetto doesnt mean im dumb lol
Or does it YO
03-09-2005, 12:34 AM
#6
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Thread Starter
Join Date: Dec 2002
Location: QUEENS NYC
Posts: 1,687
Originally Posted by Dramon_Killer' date='Mar 8 2005, 10:31 PM
Thats nto **** homie, one time I tagged that **** on a fuckinb building, took 20 cans of ****** paint *****...
Westsiiiide!!
Westsiiiide!!
hmm jew so funny dude.
Thats a report for my Mechanics of Materials classs. This weeks lab is Torsion and Pure Bending moment
03-09-2005, 06:59 AM
#7
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Join Date: Nov 2002
Posts: 3,108
Here's a tip. I don'tk now how your school works, but at my school and at my work... all the raw data is shoved into the appendix. Only graphs and final calculated results (with a very very small amount of raw data is presented). You are better off defining the equations and labeling them equation 1.0, 1.2, 2.1 etc and then sticking the graph after it and referencing the raw data in teh appendix.
03-09-2005, 08:31 AM
#8
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Join Date: Dec 2004
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Posts: 177
Wow Cheers, you're almost...ALLLLLmost U Waterloo material
You are right though. Stick raw data in the appendix.
Also, I found that it was helpful to leave printing and binding until 1 hour before the report was due. And try to be hung over (or better yet, still drunk) while you are trying to print, assemble different files, and do the self-binding ('cause it's cheaper). The dash to the dropbox is good exercise, as is knocking over the TA as he tries to open the box and collect the reports.
You are right though. Stick raw data in the appendix.
Also, I found that it was helpful to leave printing and binding until 1 hour before the report was due. And try to be hung over (or better yet, still drunk) while you are trying to print, assemble different files, and do the self-binding ('cause it's cheaper). The dash to the dropbox is good exercise, as is knocking over the TA as he tries to open the box and collect the reports.
03-09-2005, 06:33 PM
#9
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Join Date: Dec 2002
Location: QUEENS NYC
Posts: 1,687
Originally Posted by Feds' date='Mar 9 2005, 06:31 AM
Wow Cheers, you're almost...ALLLLLmost U Waterloo material
You are right though. Stick raw data in the appendix.
Also, I found that it was helpful to leave printing and binding until 1 hour before the report was due. And try to be hung over (or better yet, still drunk) while you are trying to print, assemble different files, and do the self-binding ('cause it's cheaper). The dash to the dropbox is good exercise, as is knocking over the TA as he tries to open the box and collect the reports.
You are right though. Stick raw data in the appendix.
Also, I found that it was helpful to leave printing and binding until 1 hour before the report was due. And try to be hung over (or better yet, still drunk) while you are trying to print, assemble different files, and do the self-binding ('cause it's cheaper). The dash to the dropbox is good exercise, as is knocking over the TA as he tries to open the box and collect the reports.
yeah i was thinking bout putting in the back but i decided to leave it there =) ,l I know how to write them just dont agree with all of them and plus is didn have massive equations they were simple enuff and used in the text lloll
anyone here good with fluid mechanics..
Please explain surface tension to me, i thought i had it down pat but i guess i was wrong. i remeber someone saying they were good in fluids dunno who it is
Jason NYC
