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We specialize in performing industry leading tensile tests for a wide range of metallic and polymer materials. Our tensile test expertise allows us to assimilate critical information relating to the strength, stiffness and ductility of materials in line with your production requirements and business needs.
From these measurements, the following properties can also be determined: Young's modulus, Poisson's ratio, yield strength, and strain-hardening characteristics. The yield strength is the stress at which a prescribed amount of deformation commonly 0. The resulting elongation of materials is measured by an extensometer or strain gauge, where the stress obtained at the highest level of applied force determines the tensile strength of the specimen in question.
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In addition, the maximum COV discussed by the authors is as high as 0. In order to facilitate further discussion, it is necessary to first present an introduction to the Weibull distribution. The probability density function PDF of a two-parameter Weibull distribution is:.
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The corresponding cumulative distribution function CDF is:. The expression for x p is:. Bain proposed a method through which the sample size depends only on the p-percentile, the confidence level, and the desired confidence interval [ 31 ], as mentioned in the Introduction. Since we focus on the mean value, it is necessary to ascertain the percentile of the mean value. This is illustrated in Figure 1. Bain constructs a pivotal quantity to solve such a problem [ 31 ]. The pivotal quantity U R is defined as follows:. The results of two-side confidence limits are shown Table 2.
U R can also be expressed as a form related to x p :.
For two-side confidence limit estimation, the lower and upper confidence limit are denoted as x p, L and x p, U. By transforming Equation 10 , the following formula can be derived:. Table 5 presents the sample size corresponding to varied COVs, with a confidence level of 0. It is noted that all the values in Table 5 are higher than five, which is the value used in current guidelines. This result validates the risk in using only five coupons to derive the properties of FRP in tensile coupon test.
It is especially interesting to note that even for a COV value of 0. Theoretically, the sample size based on a COV of 0. A step-by-step explanation will be presented in Appendix A. To reveal the prediction error if five coupons are used, Equation 13 is rearranged as the following:. Based on Equation 14 , the relative error limit of the derived mean value with various sample COVs can be illustrated see Table 6 :. By comparing Table 4 with Table 5 , it is found that the sample sizes based on the Weibull distribution and the normal distribution are almost the same, with the values based on the normal distribution being slightly larger.
The similarity in the sample sizes based on the Weibull distribution and the normal distribution shows that the Weibull distribution does not lead to a more conservative estimation on the sample size, and that sample size based on a normal distribution is applicable. For FRP coupon test, it is recommended that the sample sizes listed in Table 5 be used, i. It is worth mentioning that if the coupons are with large COVs, the researchers should carefully check the fabrication and test procedure, rather than simply increasing the sample size to derive a more accurate property value.
The large COVs can indicate problems with respect to the quality of fibers or resins, the impregnation procedure, the preparation and curation of specimens, the test setup, etc. Certain measures must be taken to correct the errors.
Sample Sizes Based on Weibull Distribution and Normal Distribution for FRP Tensile Coupon Test
The fabrication and test of samples should follow the procedure recommended by current guidelines [ 20 , 21 , 22 , 23 ]. This paper presents an analysis on the sample size for FRP coupon test. Both Weibull distribution and normal distribution were discussed with respect to the sample sizes corresponding to varied COVs. It was found that the sample size based on a Weibull distribution is almost the same as that based on a normal distribution see Table 4 and Table 5. In other words, the Weibull distribution does not lead to a more conservative result with respect to the sample size for derived property with required accuracy and confidence level.
If only five specimens are used for tensile coupon test of FRP composites, the possible prediction error ranges from 6. Both the current guidelines and the authors assume a confidence level of 0. The derivation by guidelines is based on a formula similar to but different from Equation 13 , which is as follows:. T , which is 1. By using a COV value of 0. Therefore, the guidelines believe it is reasonable to use a sample size of 5 for tensile coupon test. In comparison, the derivation by the authors strictly follows Equation The sample size n is determined through trial and error method.
To ensure the precision and reliability of the mechanical property based on sample size n , the integer n should be higher than or equal to the value calculated in the right-hand side of Equation Conceptualization, W. The authors would like to appreciate the financial support by the National Natural Science Foundation of China and the National industrial building diagnosis and Reconstruction Engineering Technology Research Center open fund. National Center for Biotechnology Information , U.
Journal List Materials Basel v. Materials Basel. Published online Jan 2. Find articles by Yongxin Yang. Find articles by Weijie Li.
Find articles by Wenshui Tang. Find articles by Biao Li. Find articles by Dengfeng Zhang. Author information Article notes Copyright and License information Disclaimer. Received Dec 5; Accepted Dec Abstract Current guidelines stipulate a sample size of five for a tensile coupon test of fiber reinforced polymer FRP composites based on the assumption of a normal distribution and a sample coefficient of variation COV of 0. Keywords: sample size, Weibull distribution, normal distribution, fiber reinforced polymer FRP , tensile coupon test. Introduction Fiber reinforced polymer FRP composites have been increasingly used in the strengthening of engineering structures due to their advantages of high tensile strength, excellent corrosion resistance, light weight, and flexibility in shape [ 1 , 2 , 3 , 4 , 5 , 6 , 7 ].
Sample Size Based on Weibull Distribution 2. Introduction to Weibull Distribution In order to facilitate further discussion, it is necessary to first present an introduction to the Weibull distribution.
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Percentile of the Mean Value Bain proposed a method through which the sample size depends only on the p-percentile, the confidence level, and the desired confidence interval [ 31 ], as mentioned in the Introduction. Open in a separate window. Figure 1. Illustration of the population distribution and the sampling distribution. Table 4 Sample size from the Weibull distribution.
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Table 5 Sample size from the normal distribution. Sample COV 0. Table 6 Relative error limit in using five coupons for tensile test. Comparison and Recommendation By comparing Table 4 with Table 5 , it is found that the sample sizes based on the Weibull distribution and the normal distribution are almost the same, with the values based on the normal distribution being slightly larger.
Conclusions This paper presents an analysis on the sample size for FRP coupon test. Appendix A.
Explanation for the Determination of Sample Size 5 and 7 Both the current guidelines and the authors assume a confidence level of 0. Author Contributions Conceptualization, W. Funding The authors would like to appreciate the financial support by the National Natural Science Foundation of China and the National industrial building diagnosis and Reconstruction Engineering Technology Research Center open fund. Conflicts of Interest The authors declare no conflict of interest.
References 1. Zhao X.