PEER-REVIEW ARTICLE PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8374 Improved Method to Determine Standard Values of Mechanical Properties of Original Bamboo Jianwei Li, Pengcheng Liu,* Haodi Chen, and Fangjie Cheng The standard values of mechanical properties are important performance indexes of original bamboo as a sustainable building material. Such values should be determined by combining the requirement of confidence level and the number of samples. In this paper, systematic tests of longitudinal compression, bending, longitudinal tensile, longitudinal shear, transverse compression, and transverse tensile of bamboo were performed. Based on parametric and non-parametric methods, the influencing factors of the standard values of mechanical properties of bamboo were analyzed. A calculation method and prediction formulas were proposed and the standard values of mechanical properties of bamboo were determined. The results show that the choice of parametric method to calculate the standard value of bamboo strength in the case of a small number of samples may lead to distortion of the results, and the use of non- parametric analysis can effectively reduce the error. DOI: 10.15376/biores.18.4.8374-8393 Keywords: Bamboo; Mechanical properties; Standard values; Calculation coefficient; Sampling sequence number Contact information: School of Civil Engineering, Central South University, Changsha 410075, China; *Corresponding author: pengchengliuwork@163.com INTRODUCTION Bamboo is a natural, green, renewable material that can be an ideal building component. Bamboo has a wide distribution range, many varieties, and is rich in resources. China is the most developed country in the bamboo industry and its bamboo planting area and bamboo product production rank first in the world. Bamboo has a close connection with Chinese culture and people’s life. Bamboo is found everywhere, in clothing, food, housing, transportation, history, and culture. With its excellent mechanical properties, bamboo is known as “plant reinforcement” and “ultimate green material” and is an ideal green building material (Sun et al. 2019; Tian et al. 2019a; Dauletbek et al. 2023; Zhou et al. 2023). At present, industrial bamboo building products have been developed, including cladding materials (Von Seidlein et al. 2017), bamboo plywood (Li et al. 2019), bamboo laminate (Guan et al. 2022b), bamboo fiberboard (Shi et al. 2023), bamboo scrap board (Guan et al. 2022a), bamboo decorative materials (Chen et al. 2019), etc. In 1984, a bamboo building (Fig. 1a) combining traditional Chinese construction techniques with modern science and technology was exhibited in Zurich, Switzerland. The building area was about 1200 m2, meeting the requirement load of 300 kN/m2 (Yang 2014). In 2008, the “Green School” (Fig. 1b) was built in Bali, Indonesia. The campus not only has the largest bamboo building complex in the world, but also adopts the green energy system for supporting facilities (Landwehr and Bambú 2016). PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8375 The German-Chinese Pavilion at the 2010 Shanghai World Expo (Fig. 1c) is a two- story building with original bamboo and bamboo laminated timber (Markus and Liu 2013), which has become a typical project symbolizing German-Chinese friendship. In 2011, the pure original bamboo bird wing structure (Fig. 1d) in Vinh Phuk Province, Vietnam was awarded the International Architecture Award by the Science Museum of Chicago as a typical example of eco-green architecture (Tian et al. 2019b). In 2019, the bamboo and rattan pavilion of Beijing International Horticultural Exposition (Fig. 1e) adopted the original bamboo arch architecture system, with the span of bamboo arch reaching 32 m, which is the largest non-fulcrum arch of original bamboo architecture in northern China (Su et al. 2020). In 2021, the original bamboo building (Fig. 1f) of Baizhi Mountain Tourist Reception Center in Zunyi, Guizhou, developed by China Metallurgical Construction Engineering Group, was completed. The scale of the project is 1000 m2, the span is 28 m, and the height is 15 m. Under the background of the strategic goal of “carbon peak and carbon neutrality” stated by the Chinese government, ten departments of China have presented opinions on accelerating the innovation and development of the bamboo industry. Under the condition of satisfying the quality and safety, bamboo structure building and bamboo building materials should be gradually promoted. For wood used in construction, the strength standard value is usually selected to determine its strength grade. Therefore, the reasonable determination of the standard value of structural wood strength is one of the keys to its reliable application in building structures. The Chinese standard “Specification for Wood Structure Design” (GB 50005 2017) does not mention any method of obtaining values and formulas used to predict standard values of bamboo mechanical properties. It is assumed that the index values of mechanical properties obey a normal distribution, and that the calculation coefficient of standard values is uniformly stipulated as 1.645. This method has major disadvantages. Not only does it not consider the confidence level, but also ignores the influence of the number of samples on the results, which may lead to a large error in the results. At present, other national specifications (ASTM D2915-17; EN 1058-2009) all take 5% quantile value under 75% confidence level as their standard strength value, and the methods to determine it include parametric and non-parametric methods. The key of parameter method analysis is to determine the calculation coefficient of standard value according to the number of samples. The key to non-parametric analysis is to determine the sampling sequence number of the standard value according to the number of samples. Based on three probability distribution models, this study derived the calculation formula of the standard value of bamboo, respectively, based on the parametric method and non- parametric method. Additionally, it provided the value requirements of the value coefficient of the standard value (parametric method) and the sampling sequence number of the standard value (non-parametric method) under different strength variation coefficient, sample number, and confidence level. The calculation method of strength standard value proposed in this paper can provide reference for material analysis and structural design. PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8376 (a) (b) (c) (d) (e) (f) Fig. 1. Application of original bamboo structures: (a) The Zulli Bamboo Tower in Switzerland; (b) Green School, Bali, Indonesia; (c) German-Chinese pavilions at the Shanghai World Expo; (d) Vietnamese bird-winged bamboo structure; (e) Bamboo and rattan Pavilion of Beijing World Horticultural Exhibition; (f) Baizhi Mountain Tourist Reception Center EXPERIMENTAL Materials and Test Methods The bamboo species considered in this paper is P. edulis bamboo, which was collected from Hunan Province, China. The bamboo was 3 to 4 years old, and the average moisture content was approximately 11%. Multiple mechanical properties tests were performed on the bamboo with reference to JG/T199-2007 (2007) and ISO 22157-1-2019 (2019), and the following mechanical properties were obtained: longitudinal compressive strength (UCS), longitudinal compressive elastic modulus (UCE), bending strength (MOR), bending elastic modulus (MOE), longitudinal tensile strength (UTS), longitudinal PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8377 tensile elastic modulus (UTE), longitudinal shear strength (USS), transverse compressive strength (CCS), and transverse tensile strength (CTS). The design of the specimens is shown in Table 1 and Fig. 2. The size ratio H/D (height/diameter) of the longitudinal compression and longitudinal shear specimens was 1, and the loading rate was 0.01 mm/s. The bending specimen size was 220 mm × 15 mm × t mm (t refers to the wall thickness), and the loading rate was 150 N/mm2 per minute. The size of the longitudinal tensile specimen was 330 mm × 15 mm × t mm, and the tensile rate was 0.01 mm/s. The size of the transverse compressive specimen was 15 mm × 15 mm × t mm, and the compressive rate was 20 N/mm2/min. The length of the transverse tensile specimen was 100 mm, and the tensile rate was 0.005 mm/s. And the mechanical properties were calculated using Eqs. 1 through 4, max W P f A = (1) W 20 P E A l  =  (2) max W 2 150P MOR tb = (3) W 3 m 1920000 P MOE 8 tb  = (4) where fW is the strength of the specimen under the moisture content W with UCS, USS, UTS, CCS, and CTS (MPa). EW is the elastic modulus along the grain direction under the moisture content W (MPa); MORW is the flexural strength under the moisture content W (MPa); MOEW is the flexural elastic modulus under the moisture content W (MPa); Pmax is failure load (N); A is the force area (mm2); t is the thickness of the specimen (mm); b is the height of the specimen (mm); ∆P is the difference between upper and lower loads (N); ∆l is the difference between the deformation value of the specimen under the upper and lower loads (mm); and δm is the pure deflection of the specimen under the action of ∆P (mm). Table 1. Test Specimen Design Number of specimens Specimen size Rate of loading UCS 321 height/diameter=1 0.01 mm/s UCE 314 height/diameter=1 0.01 mm/s MOR 171 220 mm × 15 mm × t mm 150 N/mm2/min MOE 167 220 mm × 15 mm × t mm 150 N/mm2/min UTS 307 330 mm × 15 mm × t mm 0.01 mm/s UTE 325 330 mm × 15 mm × t mm 0.01 mm/s USS 216 height/diameter=1 0.01 mm/s CCS 292 15 mm × 15 mm × t mm 20 N/mm2/min CTS 120 length is 100 mm 0.005 mm/s Note: In the table, t refers to the wall thickness. PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8378 Fig. 2. Specimen preparation After the failure of the specimen, the small specimen was immediately cut off at the failure site for the determination of moisture content. Equation 5 is the calculation method of moisture content. The mechanical properties of bamboo were adjusted to the values under the standard moisture content (12%), and the calculation formula is Eq. 6, 1 0 0 m m W 100 m − =  (5) W W12M K M= (6) where W is the air-dried moisture content (%); m1 and m0 are the mass of air dry and full dry, respectively (g); M12 is the strength or elastic modulus of the specimen under the standard moisture content (12%); MW is the strength or elastic modulus of the specimen when the moisture content is W; and KW is the moisture content correction factor. The Calculation of Confidence of Standard Value under Parameter Method Under the parameter method, the calculation coefficient K of the standard value of the mechanical properties of bamboo has nothing to do with the true mean value of the material (Zhong et al. 2018). Using MATLAB (MathWorks, R2022a, U.S.) programming, suppose the mean value of mechanical properties μ = 200 MPa, the range of coefficient of variation Cv is [0.05, 0.5], and the range of standard deviation δ is [10 MPa, 100 MPa]. The 5% quantile value P of the original bamboo strength sample data can be calculated according to the mean μ, standard deviation δ, and probability distribution model. First, it is assumed that the mechanical properties of bamboo obey the Normal distribution, Lognormal distribution, or 2-P-Weibull distribution, and a random sequence of columns I and rows J is generated according to the assumed μ and δ, where I = 10000. Assume that the mean value of each column of the resulting matrix is μi and the standard deviation is δi. If the calculation coefficient of the standard value is K, the standard value Xi of the mechanical properties of original bamboo can be calculated according to the following Eq. 7, i i i-X K = (7) Compare the Xi of each column with the 5% percentile value P, and if the value of column Z is not greater than P, the confidence level r for the calculation coefficient K is calculated as follows: /r Z I= (8) The calculation flow chart of the confidence level r of the standard value of mechanical properties of the original bamboo based on the parameter method is shown in Fig. 3a. PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8379 Calculation of Confidence of Standard Value under Non-parameter Method Under the non-parametric method, the calculation coefficient K of the standard value of the mechanical properties of bamboo has nothing to do with the true mean value of the material (Zhong et al. 2018). When analyzing the standard value of bamboo strength through the nonparametric method, using MATLAB (MathWorks, R2022a, U.S.) programming, each sample is sorted according to the value size from small to large, and then the data in the row of each sample is compared with the 5% quantile value P. If the value of W samples is less than or equal to P, the corresponding confidence level r can be calculated as follows for S = i: /r W I= (9) The calculation flow chart of the confidence level r of the standard value of the mechanical properties of the original bamboo based on the non-parametric method is shown in Fig. 3b. (a) (b) Fig. 3. Calculation flow chart: (a) Parameter method; (b) Non-parametric method RESULTS AND DISCUSSION Analysis of Test Results The failure modes of mechanical properties of bamboo can be divided into ductile failure and brittle failure. The longitudinal compressive specimens have a long yield platform in the loading process, which provides obvious warning before failure. The bending deformation of the flexural specimen is visible to the naked eye in the loading process, and finally the specimen suddenly fails. The transverse tensile, transverse shear, and transverse tensile specimens all fail without warning during the test. The transverse compressive specimens are similar to the transverse compressive specimens in that the PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8380 deformation varies greatly with loading. To summarize, the tests for ductile failure include longitudinal compression and transverse compression, while the tests for brittle failure include flexural, longitudinal tensile, longitudinal shear, and transverse tensile. Figure 4 shows the photos of the specimens when they are damaged. The mechanical properties of bamboo were statistically analyzed, and the results as shown in Table 2 and Fig. 5 were obtained. The average UCS, UCE, MOR, MOE, UTS, UTE, USS, CCS, and CTS of bamboo were 57.48 MPa, 13.92 GPa, 132.36 MPa, 17.84 GPa, 145.08 MPa, 16.44 GPa, 15.92 MPa, 27.29 MPa, and 4.70 MPa, respectively. Bamboo has good longitudinal tensile, compressive, and flexural properties, but relatively poor longitudinal shear and transverse mechanical properties. Fig. 4. Damage forms of bamboo Table 2. Mechanical Properties Statistics μ δ Cv UCS 57.48 MPa 7.30 MPa 0.127 UCE 13.92 GPa 1.41 GPa 0.101 MOR 132.36 MPa 7.02 MPa 0.053 MOE 17.84 GPa 1.29 GPa 0.074 UTS 145.08 MPa 35.54 MPa 0.245 UTE 16.44 GPa 1.17 GPa 0.071 USS 15.92 MPa 1.26 MPa 0.079 CCS 27.29 MPa 5.35 MPa 0.196 CTS 4.70 MPa 1.80 MPa 0.383 Note: μ is the average value, δ is the standard deviation, and Cv is the coefficient of variation. (a) (b) Fig. 5. Statistics of mechanical properties of bamboo: (a) Strength; (b) Elastic modulus PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8381 Analysis of Influencing Factors of Standard Values under Parametric Method The influencing factors of confidence The calculation coefficient K of the characteristic values of the mechanical properties has nothing to do with the material mechanics performance of the real average. Therefore, the assumption is the average size of various mechanical properties of original bamboo (UCS, UCE, UTS, UTE, MOR, MOE, USS, CCS, and CTS) is 200 MPa. The range of variation coefficient Cv of various mechanical properties of bamboo is [0.05, 0.50]; that is, the range of standard deviation δ is [10 MPa, 100 MPa]. Normal model, Lognormal model, and 2-P-Weibull model are generally used as the probability distribution models of original bamboo. Different probability distribution models have certain influence on the value of mechanical properties characteristic values of original bamboo. The results under Normal model, Lognormal model, and 2-P-Weibull model are obtained through MATLAB programming and calculation. Figures 6 through 8 show the results of calculation and analysis under Normal model, Lognormal model, and 2-P-Weibull model, respectively, in which the coefficient of variation is 0.2 and 0.4, the number of samples n is 50, and the calculation coefficient K of parameter method is 1.5, 2.0, and 2.5, respectively. It can be seen from Fig. 6 that under Normal probability distribution model, when the coefficient of variation is constant, confidence r increases with the increase of calculation coefficient K. When Cv = 0.2, the confidence r corresponding to K 1.5, 2.0, and 2.5 are 0.2331, 0.9232, and 0.9985, respectively, that is, 2331, 9232, and 9985 mechanical properties characteristic values of original bamboo in 10,000 samples are less than 5% of their true values P. When Cv = 0.4, there are 2323, 9231, and 9989 mechanical properties of bamboo with K are 1.5, 2.0, and 2.5, respectively, which are less than 5% of their true values P. Through comparing the results of different coefficient of variation, it can be seen that Cv increases from 0.2 to 0.4, and the change of confidence r is extremely small and negligible. Fig. 6. Influence of K and Cv on r under Normal distribution model PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8382 According to Fig. 7, under the Lognormal probability distribution model, confidence r gradually increases with the increase of K under the same Cv. For the same K, the confidence r increases with the increase of Cv. Through comparing Figs. 6 and 7, it can be seen that under the same conditions, the confidence r of Lognormal probability distribution model is higher than that of Normal probability distribution model. Fig. 7. Influence of K and Cv on r under Lognormal distribution model Fig. 8. Influence of K and Cv on r under 2-P-Weibull distribution model PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8383 As shown in Fig. 8, the confidence r of 2-P-Weibull model is smaller than that of Normal model and Lognormal model. When the calculation coefficient K and coefficient of variation Cv are constant, the confidence r of the three probability distribution models is sorted as Lognormal model > Normal Model > 2-P-Weibull model. Analysis of K values The relationship between r, K, and n was obtained through MATLAB (MathWorks, R2022a, Natick, MA, USA) programming. The results are shown in Figs. 9 through 11 and Tables 3 through 5. As shown in Figs. 9 through 11 and Tables 3 through 5, the calculation coefficient K presents a nonlinear increase trend with the increase of confidence r, and the slope of the curve gradually increases. The calculation coefficient K decreases with the increase of the number of samples n, and the slope of the curve gradually decreases. When the number of samples exceeds a certain value, K tends to coincide. The rules under Normal model, Lognormal model, and 2-P-Weibull model are consistent. (a) (b) Fig. 9. Relationship between K, r, and n under Normal distribution model: (a) relationship between K and r; and (b) relationship between K and n (a) (b) Fig. 10. Relationship between K, r, and n under Lognormal distribution model: (a) relationship between K and r; and (b) relationship between K and n PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8384 · Fig. 11. Relationship between K, r, and n under 2-P-Weibull distribution model: (a) relationship between K and r and (b) relationship between K and n Table 3. Calculation Coefficient K (Normal Model, Cv = 0.2) r n 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.99 5 1.761 1.870 1.994 2.128 2.284 2.455 2.667 2.965 3.420 4.244 6.922 10 1.688 1.759 1.832 1.913 2.006 2.107 2.217 2.368 2.568 2.904 3.690 20 1.678 1.722 1.770 1.817 1.873 1.936 2.007 2.092 2.215 2.409 2.825 40 1.658 1.687 1.720 1.757 1.795 1.837 1.882 1.938 2.015 2.128 2.374 60 1.656 1.680 1.708 1.737 1.766 1.796 1.831 1.878 1.934 2.025 2.202 80 1.655 1.677 1.699 1.724 1.748 1.777 1.807 1.842 1.887 1.960 2.100 100 1.650 1.667 1.687 1.708 1.731 1.755 1.783 1.818 1.862 1.929 2.054 120 1.649 1.667 1.686 1.707 1.726 1.749 1.776 1.805 1.844 1.903 2.027 140 1.648 1.664 1.681 1.699 1.717 1.738 1.760 1.787 1.820 1.875 1.977 160 1.647 1.662 1.678 1.696 1.715 1.734 1.754 1.780 1.812 1.862 1.969 180 1.650 1.664 1.679 1.694 1.709 1.727 1.748 1.773 1.804 1.849 1.937 200 1.646 1.660 1.674 1.688 1.704 1.721 1.739 1.763 1.794 1.840 1.925 300 1.645 1.657 1.668 1.681 1.694 1.709 1.724 1.743 1.765 1.801 1.870 400 1.645 1.655 1.666 1.676 1.688 1.700 1.715 1.730 1.750 1.778 1.834 500 1.646 1.655 1.664 1.674 1.684 1.696 1.707 1.722 1.739 1.766 1.816 Table 4. Calculation Coefficient K (Lognormal Model, Cv = 0.2) r n 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.99 5 1.575 1.658 1.751 1.861 1.980 2.138 2.326 2.553 2.889 3.478 5.517 10 1.521 1.575 1.632 1.692 1.766 1.840 1.925 2.032 2.209 2.486 3.123 20 1.489 1.526 1.562 1.602 1.647 1.702 1.757 1.826 1.915 2.074 2.428 40 1.472 1.498 1.523 1.550 1.579 1.611 1.647 1.689 1.746 1.835 2.019 60 1.471 1.491 1.512 1.535 1.558 1.583 1.613 1.649 1.691 1.769 1.900 80 1.467 1.484 1.501 1.522 1.544 1.565 1.590 1.623 1.659 1.719 1.843 100 1.466 1.482 1.497 1.514 1.532 1.552 1.572 1.597 1.630 1.684 1.782 120 1.464 1.478 1.493 1.509 1.526 1.545 1.565 1.589 1.621 1.667 1.755 140 1.465 1.479 1.492 1.507 1.523 1.539 1.557 1.578 1.603 1.648 1.735 160 1.463 1.475 1.488 1.502 1.517 1.532 1.550 1.570 1.596 1.637 1.714 180 1.462 1.475 1.486 1.499 1.512 1.527 1.543 1.563 1.589 1.628 1.700 200 1.463 1.474 1.485 1.498 1.510 1.525 1.540 1.556 1.580 1.619 1.677 300 1.462 1.471 1.480 1.490 1.499 1.510 1.523 1.538 1.558 1.586 1.637 400 1.464 1.471 1.479 1.487 1.496 1.506 1.516 1.530 1.546 1.569 1.619 500 1.462 1.469 1.476 1.483 1.491 1.499 1.509 1.520 1.534 1.555 1.593 PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8385 Table 5. Calculation Coefficient K (2-P-Weibull Model, Cv = 0.2) r n 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.99 5 2.167 2.325 2.502 2.700 2.915 3.195 3.530 3.972 4.585 5.799 9.334 10 1.992 2.105 2.213 2.343 2.477 2.629 2.813 3.045 3.338 3.882 5.008 20 1.939 2.010 2.081 2.159 2.244 2.338 2.454 2.596 2.795 3.098 3.756 40 1.889 1.939 1.992 2.040 2.101 2.163 2.233 2.322 2.435 2.612 2.958 60 1.870 1.909 1.951 1.995 2.042 2.095 2.156 2.225 2.320 2.470 2.752 80 1.875 1.908 1.943 1.980 2.018 2.061 2.110 2.167 2.243 2.356 2.596 100 1.869 1.900 1.930 1.964 2.000 2.039 2.081 2.132 2.199 2.303 2.509 120 1.866 1.895 1.922 1.951 1.983 2.019 2.057 2.101 2.159 2.257 2.430 140 1.863 1.888 1.916 1.942 1.972 2.005 2.042 2.087 2.141 2.232 2.403 160 1.865 1.888 1.912 1.938 1.964 1.997 2.029 2.070 2.125 2.207 2.339 180 1.865 1.888 1.911 1.935 1.961 1.988 2.020 2.060 2.108 2.182 2.319 200 1.859 1.882 1.905 1.926 1.950 1.978 2.008 2.046 2.089 2.156 2.284 300 1.860 1.877 1.895 1.914 1.934 1.956 1.980 2.008 2.042 2.099 2.198 400 1.857 1.872 1.887 1.903 1.921 1.938 1.959 1.984 2.015 2.066 2.155 500 1.857 1.870 1.885 1.898 1.913 1.929 1.947 1.970 1.996 2.037 2.121 Analysis of Influencing Factors of Standard Values under Nonparametric Method The influencing factors of confidence Figures 12 through 14 show the relationship between S and Cv and r under the Normal distribution model, Lognormal distribution model, and 2-P-Weibull distribution model when nonparametric method is adopted through MATLAB analysis. Fig. 12. Influence of S and Cv on r under Normal distribution model According to Figs. 12 through 14, taking Cv = 0.2 and S = 4 as an example, the confidence r of Normal distribution model, Lognormal distribution model, and 2-P- PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8386 Weibull distribution model are 0.2330, 0.2355, and 0.2488, respectively; that is, in 10,000 samples, there are 2330, 2355, and 2488 mechanical properties values of original bamboo, respectively, which are less than 5% of their true value P. When Cv is constant, r increases with S. When S is constant, r increases with the increase of Cv. When Cv and S are the same, r under the three distribution models is roughly equal and the error is small. Fig. 13. Influence of S and Cv on r under Lognormal distribution model Fig. 14. Influence of S and Cv on r under 2-P-Weibull distribution model PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8387 Analysis of the values of S Figure 15 shows the relation surfaces of S, r, and n under each distribution model. It can be seen from the surface that the confidence degree r presents a nonlinear decreasing trend with the increase of the sampling sequence number S, and the slope of the surface decreases first and then increases. The r increases nonlinearly with the increase of the number of samples n, and the surface slope increases first and then decreases. The surfaces of different Cv almost coincide, which is consistent with the results in the literature. It can be seen that the influence of the coefficient of variation on the relationship between the confidence degree r and the sampling sequence number S is negligible. The minimum sample number n under different confidence and sampling sequence number requirements is shown in Tables 6 to 8. The minimum sample number n in the table is the value rounded up. Referring to Tables 6 to 8, the minimum number of samples under different probability distribution models, confidence, and sampling sequence numbers can be conveniently obtained. (a) (b) (c) Fig. 15. Relation between S and r and n: (a) Normal distribution model; (b) Lognormal distribution model; (c) 2-P-Weibull distribution model PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8388 Table 6. Relationship between Minimum Sample Number n, Confidence Degree r, and Order S (Normal Model) r S 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.99 1 15 16 19 21 24 28 32 37 45 58 86 2 34 37 41 44 48 53 58 67 76 93 124 3 54 58 62 66 72 78 85 93 105 125 166 4 74 78 83 88 94 102 109 120 131 152 200 5 94 99 104 110 117 125 134 145 159 181 227 6 113 120 126 132 139 147 156 167 184 210 252 7 134 140 146 154 162 170 179 192 210 233 285 8 154 160 166 175 185 192 205 216 233 259 315 9 173 181 188 196 206 215 226 240 256 282 345 10 192 201 208 218 228 237 249 263 282 310 374 11 214 222 230 239 249 260 270 284 305 335 397 12 233 242 250 260 270 280 293 308 330 361 423 13 254 262 272 281 292 303 315 333 352 386 450 14 273 282 292 303 313 325 339 354 378 409 479 15 293 304 313 324 335 346 362 376 397 434 500 Table 7. Relationship between Minimum Sample Number n, Confidence Degree r, and Order S (Lognormal Model) r S 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.99 1 15 16 18 21 24 27 32 37 46 59 88 2 34 37 40 44 49 54 58 67 77 94 126 3 53 58 62 67 72 78 85 93 105 124 160 4 73 79 81 89 94 102 109 119 133 153 195 5 93 99 105 110 118 124 134 143 157 179 226 6 114 119 126 132 139 146 157 169 184 206 253 7 133 140 145 154 162 170 180 192 209 234 286 8 154 160 168 175 185 193 203 217 235 259 316 9 172 180 189 196 205 214 225 241 257 287 344 10 194 200 209 218 228 237 248 262 281 310 368 11 214 221 231 239 249 259 270 288 305 332 394 12 232 242 251 260 270 281 293 309 328 362 419 13 252 261 271 281 291 303 318 331 351 386 447 14 272 283 291 302 314 325 338 354 377 411 475 15 294 302 313 324 334 346 362 378 402 434 501 Determining the Standard Values of Mechanical Properties of Bamboo The parameter method is mainly based on the statistical mean μ and standard deviation δ of the sample to determine the calculation coefficient K, and the standard value of material strength is jointly determined by the above three. The non-parametric method is to arrange the strength in the sample in order from small to large, and compare the 5% quantile value with the strength data points corresponding to each order to determine the sampling sequence number. The corresponding strength value of the sequence number is the standard value of the bamboo. According to the above analysis, when the parameter method is adopted, the calculation coefficient K is not equal under different probability distribution models and coefficient of variation Cv. When the nonparametric method is adopted, the S value is the same under different probability distribution models and coefficient of variation Cv. When the sample data is not extensive enough, the use of PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8389 parameter method is easy to lead to the distortion of the results, especially when the number of samples is small. Therefore, it is suggested to use the non-parametric method to determine the standard values of mechanical properties of bamboo. Table 8. Relationship between Minimum Sample Number n, Confidence Degree r, and Order S (2-P-Weibull Model) r S 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.99 1 15 16 18 21 23 28 31 38 46 58 89 2 34 37 41 44 49 54 59 66 77 93 127 3 54 58 62 68 72 78 85 94 105 122 166 4 74 79 82 89 95 101 109 119 131 153 196 5 94 99 105 111 117 124 134 143 158 181 224 6 113 120 125 132 140 147 158 169 182 206 259 7 134 139 147 153 163 171 181 192 209 234 288 8 154 160 167 175 183 193 203 217 233 262 314 9 173 181 188 196 205 216 226 241 258 288 342 10 194 201 209 217 227 238 249 264 283 312 370 11 211 223 231 239 248 258 270 285 304 336 399 12 234 243 251 259 271 282 294 309 329 359 422 13 252 264 271 282 290 302 315 331 352 385 448 14 272 281 293 303 314 324 340 354 376 409 472 15 294 304 312 324 333 344 358 376 399 433 500 Linear function, Exponential function, and Power function are respectively used to fit the sampling sequence number S and minimum sampling number n, and the fitting results were obtained as shown in Table 9. As shown in Table 9, a linear function can be used to fit the relationship between the sampling sequence number S and the minimum sampling number n perfectly; that is, there is a linear correlation between S and n. The linear relation between S and n is defined as y = ax + b, where a and b are the calculation parameters. As observed in Table 9, values of a and b are different under different confidence levels. The value parameters a and b are fitted with the confidence r respectively, and the relation shown in Eq. 10 and Eq. 11, and the fitting curve shown in Fig. 16 is obtained. The decision coefficients R2 of Eq. 10 and Eq. 11 are all higher than 0.98, so it can be seen that Eq. 10 and Eq. 11 can perfectly fit the relationship between calculation parameters and r. Based on the above analysis, the relationship between the sampling sequence number, the minimum sample number, and the confidence of the mechanical properties of bamboo under the nonparametric method can be obtained, as shown in Eq. 12. The standard values of mechanical properties of bamboo obtained by the above methods are shown in Table 10. The values are determined by Eqs. 10 through 12: - . . 4r 4 64 0 053 a 1 e = + (R2 = 0.984) (10) .. - . 5 5b 0 224 2 755r= (R2 = 0.981) (11) . - . . - . .5 5 4r 4 64 0 053n S 2 755r 0 224 1 e = + + (12) PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8390 Table 9. Fitting Relation Results r Linear Exponential Power Relation R2 Relation R2 Relation R2 0.50 S = 0.0502n + 0.2831 1 S = 1.8102e0.0032n 0.881 S = 0.0783n0.9208 1 0.55 S = 0.0487n + 0.1807 1 S = 1.7752e0.008n 0.885 S = 0.0699n0.9351 0.999 0.60 S = 0.0477n + 0.0499 1 S = 1.7359e0.0079n 0.885 S = 0.0543n0.9763 1 0.65 S = 0.0464n - 0.0699 1 S = 1.6987e0.0076n 0.889 S = 0.0465n0.9973 1 0.70 S = 0.0452n - 0.2435 1 S = 1.6451e0.0075n 0.893 S = 0.0369n1.032 1 0.75 S = 0.0443n - 0.4401 1 S = 1.59e0.0073n 0.895 S = 0.0276n1.0775 1 0.80 S = 0.0429n - 0.6139 1 S = 1.5418e0.0071n 0.897 S = 0.0213n1.1149 1 0.85 S = 0.0417n - 0.8872 1 S = 1.4676e0.0069n 0.902 S = 0.0149n1.1683 1 0.90 S = 0.04n - 1.1847 1 S = 1.3921e0.0067n 0.905 S = 0.0096n1.2317 1 0.95 S = 0.0379n - 1.6648 1 S = 1.2818e0.0063n 0.908 S = 0.0047n1.3338 0.999 0.99 S = 0.0342n - 2.5421 1 S = 1.0954e0.0057n 0.917 S = 0.0014n1.5026 0.997 (a) (b) Fig. 16. Relationship between K, r, and n in Normal distribution model: (a) relationship between a and r; and (b) relationship between b and r Table 10. Standard Values of Mechanical Properties of Original Bamboo UCS UCE MOR MOE UTS UTE USS CCS CTS n 13 13 7 6 13 14 9 12 4 fk 49.79 MPa 11.66 GPa 120.89 MPa 15.64 GPa 122.11 MPa 16.44 GPa 13.60 MPa 25.88 MPa 2.84 MPa CONCLUSIONS In this paper, tests of longitudinal grain compression, bending, longitudinal grain tensile, longitudinal grain shear, transverse grain compressive, and transverse grain tensile of original bamboo were conducted to study the influence of parametric and non- parametric methods on the standard values of mechanical properties of original bamboo. The main conclusions are as follows: 1. Ductile failure occurs when bamboo is subjected to longitudinal and transverse compressive resistances, and brittle failure occurs when bamboo is subjected to longitudinal tensile, flexural, transverse shear, and transverse tensile resistances. PEER-REVIEWED ARTICLE bioresources.com Li et al. (2023). “Standard bamboo properties,” BioResources 18(4), 8374-8393. 8391 2. Using the parameter method, the relationship between the confidence r and the calculation coefficient K, and the coefficient of variation Cv under the Normal model, Lognormal model, and 2-P-Weibull model was studied. The results show that under each distribution model, when Cv is constant, the confidence r increases gradually with the increase of K. When K is constant, the confidence r increases with the increase of Cv. When calculation coefficient K and coefficient of variation Cv are constant, the confidence r of the three probability distribution models is sorted as Lognormal model > Normal Model > 2-P-Weibull model. The calculation coefficient K presents a nonlinear increase trend of gradually increasing slope with the increase of confidence r, and a nonlinear decrease trend of gradually decreasing slope with the increase of the number of samples n. When the number of samples exceeds a certain value, K tends to coincide. 3. Under the non-parametric method, when the coefficient of variation Cv is constant, the confidence degree r increases with the increase of sampling sequence number S. When S is constant, r increases with the increase of Cv. When Cv and S are the same, r under the three distribution models is roughly equal. With the increase of sampling S, r presents a nonlinear decreasing trend, which decreases first and then increases. With the increase of the number of samples n, r presents a nonlinear increasing trend of the curve oblique, first increasing and then decreasing. The effect of Cv on the relationship between r and S is negligible. 4. Using the parameter method to study the standard values of mechanical properties of bamboo, the results easily can become distorted, so this paper suggests using the non- parameter method for analysis. The linear relation can perfectly fit the relation between the sampling sequence number S and the minimum sample number n. Based on this, the relation between S, r, and n is proposed. The standard values of UCS, UCE, MOR, MOE, UTS, UTE, USS, CCS, and CTS are 49.79 MPa, 11.66 GPa, 120.89 MPa, 15.64 GPa, 122.11 MPa, 16.44 GPa, 13.60 MPa, 25.88 MPa, and 2.84 MPa, respectively. ACKNOWLEDGMENTS The authors are grateful for the support of the National Key R & D Programme of China: [Grant Number 2017YFC0703500]. REFERENCES CITED ASTM D2915-17 (2022). “Standard practice for sampling and data-analysis for structural wood and wood-based products,” ASTM International, West Conshohocken, PA. Chen, L., He, L., Xie, J., Qiu, Z., Qi, J., Jiang, Y., Xiao, H., Chen, Y., de Hoop, C. F., and Hse, C.-Y. 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DOI: 10.1080/15376494.2023.2192710 Article submitted: September 2, 2023; Peer review completed: October 7, 2023; Revised version received: October 11, 2023; Accepted: October 12, 2023; Published: October 22, 2023. DOI: 10.15376/biores.18.4.8374-8393