UFDC Home   Help 
Material Information
Subjects
Notes
Record Information

Material Information
Subjects
Notes
Record Information

This item is only available as the following downloads:  
Full Text  
PAGE 1 A MEMOIR ON ANALYSIS OF STRESS/STRAIN RELATIONS ON WOODEN DOWEL RODS Allen R. Overman Agricultural and Biological Engineering University of Florida Copyright 2013 Allen R. Overman PAGE 2 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods i ABSTRACT This memoir is focused o n stress/strain relationships for wooden dowel rods (round, inch diameter). Data are adapted from an honors thesis by Mary Grace Danao (2000) conducted under my supervision. To qualify for this designation in the University of Florida College of Enginee ring requires: (1) a supervisory committee, (2) a research project, (3) a written report, and (4) completion of an examination by the committee. Measurements were carried out on an Instron (Model 5566, Serial No. 1663). Output was in analog format to a s trip chart recorder. The initial linear portion of the recording was used to calculate Youngs modulus ( E ). The process was then carried out until the rod broke under load and the peak breaking force ( F ) and peak extension ( x ) were recorded. The ultimate g oal of the project was to examine frequency distributions to test whether these results followed Gaussian distributions. A total of 165 samples were examined in the study. It shown that measurements of frequency f vs. E were described very accurately by a Gaussian distribution with a very high correlation coefficient ( r = 0.99472). Measurements of f vs. F and f vs. x exhibited much lower correlation coefficients ( r = 0.9665 and r = 0.9502, respectively, for peak force and peak extension). Finally, analysi s was performed on correlation between x and F which led to a very low correlation coefficient ( r = 0.6234). Apparently, measurements of F does not provide a very reliable estimator for x For this and other reasons engineers incorporate safety factors in their design work. Keywords : Stress/Strain relations, Gaussian distributions, statistical analysis. Acknowledgement: The author expresses appreciation to Amy G. Buhler, Associate University Librarian, University of Florida, for assistance with preparati on of this memoir as part of the UF digital library. PAGE 3 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 1 Analysis of Stress/Strain Relationships in Wooden Dowel Rods Introduction This analysis is based on an Engineering Honors Project conducted in 2000 (Mary Grace Danao, Variability in Peak Force, Peak Extension, and Youngs Modulus of Wooden Dowel Rods:An Engineering Honors Project, 28 p., 14 figures, 1 table, 8 references). The samples consisted of 165 round 1/4 inch rods selected from a local hardware store. These were cut into pieces 8 inches long. Stress/Strain measurements were conducted on an Instron (Model 5566, Serial No. 1663). Samples were mounted in the support shown in Figure 1. To qualify for the honors designation in the College of Engineering the student must have a supervisory committee, conduct a research project, write a project report, and pass an exam by the committee. In this case the committee consisted of Allen R. Overman, Chair (Prof. of Agricultural Engineering), Ray Bucklin (Prof. of Agricultural Enginee ring), and Frank G. Martin (Prof. of Statistics) at the University of Florida. Ms Danao excelled in all aspects of her project. She received a National Science Foundation PhD fellowship (in national competition), which was later completed at another university. Measurements of stress and strain were recorded in analog form on a strip chart recorder. The initial linear portion of the output was used to estimate Youngs modulus. Ranking of the 165 values is given in Table 1. The frequency distribution is t hen given in Table 2. The test for a Gaussian distribution must follow 2 erf 1 2 1 E f (1) in which f is the frequency, E is Youngs modulus, is the mean of the distribution, and 2 is the spread i n the distribution around the mean. Equation (1) can be easily rearranged to the linear form Efz21212erf1 (2) If f follows a Gaussian distribution then Eq. (2) must follow a straight line. This test is carried out in Table 2, which leads to EEfz744.1484.321212erf1 r = 0.99472 (3) 998 1 5734 0 2 Correlation of z with E is shown in Figure 2, where the line is drawn from Eq. (3) PAGE 4 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 2 5734 0 00 2 erf 1 2 1 2 erf 1 2 1 E E f (4) Note the high correlation coefficient ( r = 0.99472), which confirms the Gaussian distribution for Youngs modulus. The Gaussian curve for dependence of number frequency ( n ) vs. Youngs modulus ( E ) is calculated from 2225734.000.2exp27.165734.000.2exp)5734.0()1.0)(160(2exp2))((EEEENn (5) Results are shown in Figure 3, where the curve is drawn from Eq. (5). Uncertainty in the z distribution is then given by 2 / 1 2 2 / 1 2 2 / 1 2 2 80 2 ) 00 2 ( 056 1 176 0 80 2 ) 00 2 ( 056 1 ) 0830 0 )( 12 2 ( ) ( ) ( 1 1 E E E E E E n s t zx y (6) where n is the number of observations (18) and the Student t statistic, the 95% level of confidence is given by t (16,95%) = 2.12, and the variance of the estimate is estimated from 0830 0 00689 0 16 11023 0 2 ) ( 2 2 x y x ys n z z s (7) The stress test was then carried out until the dowel rod broke under load. The point of breakage was recorded as peak force and peak extension. Results for peak force are given in Table 3, with FFfz174.077.3212)12(erf1 r = 0.9665 (8) 75.52,7.21 Results are shown in Figure 4, where the line is drawn from Eq. (8). This result leads immediately to PAGE 5 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 3 75 5 7 21 erf 1 2 1 F f (9) The correlation coefficient ( r = 0.9665) is much lower than for Youngs modulus. Results for peak extension are given in Table 4, with x x f z 00 12 93 3 2 1 2 ) 1 2 ( erf 1 r = 0.9502 (10) 0833 0 2 327 0 Results are shown in Figure 5, where the line is drawn from Eq. (10). This result leads immediately to 0833.0327.0erf121xf (11) Note the low correlation coefficient ( r = 0.9502). This analysis leads naturally to the question of any correlation between peak extension ( x ) and peak breaking force ( F ). This point is examined in Table 5. Linear r egression of x vs. F leads to FbFax00768.0169.0 r = 0.6234 (12) with a very low correlation coefficient ( r = 0.6234). The significance of the correlation coefficient is illustrated by the dimensionless plot 86 3 60 21 6234 0 0475 0 335 0 F x s F F r s x xf x (13) in which r represents the slope of the plot. In Eq. (13) xsxand represent mean and standard deviation of the distribution, respectively, for peak extension; and fs F and represent mean and standard deviation of the distribut ion for, respectively, for peak force. Results are shown in Figure 6, which confirms the poor correlation between x and F Reference 1. Danao Mary Grace (2000) Variability in Peak Force, Peak Extension, and Youngs Modulus of Wooden Dowel Rods: An Engi neering Honors Project. 28 p. PAGE 6 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 4 Table 1. Ranked data for Youngs modulus for wooden dowel rods. E Rank E Rank E Rank E Rank 106 psi 106 psi 106 psi 106 psi 0.95 1 1.76 43 1.97 86 2.33 129 0.97 2 1.76 44 1.98 87 2.33 130 1.06 3 1.77 45 1.99 88 2.33 131 1.28 4 1.77 46 1.99 89 2.33 132 1.29 5 1.77 47 1.99 90 2.34 133 1.35 6 1.79 48 1.99 91 2.34 134 1.35 7 1.79 49 1.99 92 2.35 135 1.36 8 1.79 50 2.00 93 2.38 136 1.41 9 1.79 51 2.01 94 2.38 137 1.42 10 1.79 52 2.03 95 2.45 138 1.43 11 1.80 53 2.04 96 2.46 139 1.43 12 1.80 54 2.04 97 2.48 140 1.44 13 1.81 55 2.05 98 2.49 141 1.44 14 1.82 56 2.05 99 2.50 142 1.45 15 1.83 57 2.06 100 2.51 143 1.45 16 1.83 58 2.06 101 2.52 144 1.45 17 1.84 59 2.08 102 2.53 145 1.48 18 1.84 60 2.08 103 2.55 146 1.49 19 1.84 61 2.09 104 2.57 147 1.50 20 1.85 62 2.09 105 2.58 148 1.54 21 1.85 63 2.10 106 2.59 149 1.55 22 1.85 64 2.10 107 2.59 150 1.56 23 1.86 65 2.12 108 2.61 151 1.56 24 1.86 66 2.12 109 2.64 152 1.58 25 1.87 67 2.12 110 2.65 153 1.59 26 1.87 68 2.13 111 2.65 154 1.60 27 1.88 69 2.14 112 2.66 155 1.63 28 1.88 70 2.16 113 2.68 156 1.69 29 1.88 71 2.16 114 2.68 157 1.69 30 1.89 72 2.17 115 2.69 158 1.70 31 1.89 73 2.17 116 2.71 159 1.71 32 1.89 74 2.19 117 2.78 160 1.71 33 1.89 75 2.20 118 2.78 161 1.71 34 1.90 76 2.20 119 2.81 162 1.72 35 1.91 77 2.24 120 2.81 163 1.72 36 1.92 78 2.24 121 2.81 164 1.74 37 1.93 79 2.25 122 2.91 165 1.75 38 1.94 80 2.26 123 1.75 39 1.95 81 2.26 124 1.75 40 1.95 82 2.27 125 1.75 41 1.95 83 2.30 126 1.75 42 1.95 84 2.31 127 1.95 85 2.31 128 PAGE 7 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 5 Table 2. Frequency distribution for Youngs modulus ( E ) for wooden dowel rods. E n f z z 2 z z z f 106 psi 1.3 5 0.0312 1.318 1.217 0.01020 0.195 0.042 1.4 8 0.0500 1.163 1.042 0.01464 0.192 0.070 1.5 20 0.125 0.814 0.868 0.00292 0.188 0.109 1.6 27 0.1669 0.678 0.694 0.00036 0.186 0.161 1.7 31 0.194 0.610 0.519 0.00828 0.184 0.230 1.8 54 0.338 0.295 0.345 0.00250 0.182 0.311 1.9 76 0.475 0.045 0.170 0.01563 0.181 0.403 2.0 92 0.575 0.135 0.004 0.01716 0.181 0.500 2.1 107 0.669 0.283 0.178 0.01103 0.181 0.597 2.2 117 0.731 0.435 0.353 0.00672 0.182 0.689 2.3 123 0.769 0.508 0.527 0.00005 0.184 0.770 2.4 133 0.831 0.678 0.702 0.00058 0.186 0.839 2.5 138 0.862 0.770 0.876 0.01124 0.188 0.891 2.6 146 0.912 0.958 1.050 0.00846 0.192 0.930 2.7 153 0.9562 1.208 1.225 0.00029 0.195 0.958 2.8 156 0.9750 1.386 1.399 0.00017 0.199 0.9757 2.9 156 0.9938 3.0 160 1.0000 E E f z 744 1 484 3 2 1 2 1 2 erf 1 r = 0.99472 998.1,5734.02 5734 0 00 2 erf 1 2 1 2 erf 1 2 1 E E f 2225734.000.2exp27.165734.000.2exp)5734.0()1.0)(160(2exp2))((EEEENn 2/122/122/12280.2)00.2(056.1176.080.2)00.2(056.1)0830.0)(12.2()()(11EEEEEEnstzxy PAGE 8 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 6 Table 3 Frequency distribution of peak force for 165 wooden dowel rods F n f erf 1 (2 f 1) lb 15 0 16 1 0.059 1.105 17 1 0.059 1.105 18 2 0.118 0.839 19 6 0.353 0.265 20 7 0.412 0.158 21 10 0.588 +0.158 22 10 0.588 +0.158 23 10 0.588 +0.158 24 12 0.706 +0.383 25 13 0.765 +0.511 26 16 0.941 +1.105 27 16 0.941 +1.105 28 16 0.941 +1.105 29 16 0.941 +1.105 30 16 0.941 +1.105 31 17 FFfz174.077.3212)12(erf1 r = 0.9665 75.52,7.21 75.57.21erf121Ff PAGE 9 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 7 Table 4 Fr equency distribution of peak extension for 165 wooden dowel rods. x n f erf 1 (2 f 1) in. 0.27 0 0 0.28 2 0.118 0.839 0.29 2 0.118 0.839 0.30 4 0.235 0.510 0.31 7 0.412 0.158 0.32 10 0.588 +0.158 0.33 11 0.647 +0.265 0.34 12 0.706 +0.383 0.35 13 0.765 +0.511 0.36 13 0.765 +0.511 0.37 14 0.824 +0.658 0.38 14 0.824 +0.658 0.39 15 0.882 +0.839 0.40 15 0.882 +0.839 0.41 15 0.882 +0.839 0.42 16 0.941 +1.105 0.43 16 0.941 +1.105 0.44 16 0.941 +1.105 0.45 17 1 xxfz00.1293.3212)12(erf1 r = 0.9502 0833.02,327.0 0833.0327.0erf121xf PAGE 10 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 8 Table 5. Mechanical properties of wooden dowel rods.1 Sample F x E 86 3 60 21 F 0475.0335.0x lb in. 106 psi 1010 19.95 0.32 1.95 0.427 0.316 1020 25.83 0.32 2.81 +1.096 0.316 1030 18.82 0.35 1.87 0.720 +0.316 1040 25.24 0.45 1.74 +0.943 +2.421 1050 20.13 0.28 1.93 0.381 1.158 1060 17.75 0.31 1.44 0.997 0.526 1070 23.12 0.39 2.12 +0.394 +1.158 1080 15.43 0.31 1.28 1.598 0.526 1090 24.83 0.37 2.38 +0.837 +0.737 2000 23.46 0.30 2.52 +0.482 0.737 2010 25.91 0.33 2.59 +1.117 0.105 2020 18.33 0.30 1.79 0.847 0.737 2030 20.15 0.28 2.33 0.376 1.158 2040 18.94 0.31 1.90 0.689 0.526 2050 18.92 0.32 1.77 0.694 0.316 2060 30.39 0.42 2.27 +2.277 +1.789 2070 20.44 0.34 2.12 0.300 +0.105 avg 21.60 0.335 2.05 std dev 3.86 0.0475 0.405 rel error 0.179 0.142 0.197 1F = peak force, x = peak extension, E = Young modulus FbFax00768.0169.0 r = 0.6234 86.360.216234.00475.0335.0FxsFFrsxxfx PAGE 11 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 9 Figure 1. Support device for stress/strain measurements of wooden dowel rods. All dimensions are in inches. PAGE 12 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 10 Figure 2. Frequency distribution ( z vs. E ) for Youngs modulus for wooden dowel rods. The line is drawn from Eq. (3). PAGE 13 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 11 Figure 3. The Gaussian curve for dependence of number frequency ( n ) vs. Youngs modulus ( E ) The curve is calculated from Eq. (5). PAGE 14 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 12 F igure 4 Frequency distribution ( z vs. F ) for peak breaking force for wooden dowel rods. The line is drawn from Eq. (8 ). PAGE 15 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 13 Figure 5. Frequency distribution ( z vs. x ) for peak extension for wooden dowel rods. The line is drawn from Eq. (10). PAGE 16 A.R. Overman Stress/Strain Relations on Wooden Dowel Rods 14 Figure 6. Correlation of dimensionless extension vs. dimensionless force. The line is drawn from Eq. (13). 