Abstract
To accurately evaluate the dynamic elastic modulus (Ed) of wood in ancient timberwork buildings, the new materials of larch were used as the research object, and the stress wave nondestructive testing method was used to determine it. Based on nondestructive testing data, this paper proposed a method for predicting the Ed of larch using the principle of information diffusion. It selected the distance (D) from the bark to the pith in the cross-section of the wood and the height (H) from the base to the top in the radial section of the wood. The fuzzy diffusion relationships between the two evaluation indexes and the Ed were established using the information diffusion principle and the first- and second-order fuzzy approximate inferences in the fuzzy information optimization process. The calculation results showed that the dynamic elastic modulus model constructed by the information diffusion method can better predict the Ed of larch. The coefficient of determination between the measured value and the predicted value of the Ed was 0.861, they were in good agreement. The weights of the two influencing factors were 0.7 and 0.3, respectively, the average relative error of the fitted sample data was the minimum, which was 8.55%. This prediction model provided a strong basis for field inspection.
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Establishing a Dynamic Elastic Modulus Prediction Model of Larch Based on Nondestructive Testing Data
Liting Cheng,a Wei Wang b,c,d,* Zhiguo Yang,e and Jian Dai,b,c,d,*
To accurately evaluate the dynamic elastic modulus (Ed) of wood in ancient timberwork buildings, the new materials of larch were used as the research object, and the stress wave nondestructive testing method was used to determine it. Based on nondestructive testing data, this paper proposed a method for predicting the Ed of larch using the principle of information diffusion. It selected the distance (D) from the bark to the pith in the cross-section of the wood and the height (H) from the base to the top in the radial section of the wood. The fuzzy diffusion relationships between the two evaluation indexes and the Ed were established using the information diffusion principle and the first- and second-order fuzzy approximate inferences in the fuzzy information optimization process. The calculation results showed that the dynamic elastic modulus model constructed by the information diffusion method can better predict the Ed of larch. The coefficient of determination between the measured value and the predicted value of the Ed was 0.861, they were in good agreement. The weights of the two influencing factors were 0.7 and 0.3, respectively, the average relative error of the fitted sample data was the minimum, which was 8.55%. This prediction model provided a strong basis for field inspection.
Keywords: Stress wave; Distance base-top; Fuzzy matrix; Information diffusion
Contact information: a: College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China; b: College of Architecture and Urban Planning, Beijing University of Technology, Beijing 100124, China; c: Beijing Engineering Technology Research Center for Historic Building Protection, Beijing University of Technology, Beijing 100124, China; d: Key Science Research Base of Safety Assessment and Disaster Mitigation for Traditional Timber Structure (Beijing University of Technology), State Administration for Cultural Heritage, Beijing 100124,China; e: College of Petroleum Engineering, China University of Petroleum (Beijing), Beijing 102249, China;
*Corresponding author: chenglting@126.com; ieeww@bjut.edu.cn
INTRODUCTION
Wood is an important renewable natural resource and is widely used in construction, furniture, and bridges (Hou 2019). Because wood occupies an important position in ancient timberwork buildings, it has become one of the important goals of wood research to effectively, quickly, and accurately test the timber property of the wooden structure on site without damaging the material itself and the original structure of the ancient wooden structure members (Li 2015). Among the wood property indexes, elastic modulus is an important wood property index that can reflect the wood’s ability to change under load (Tian et al. 2017; Cavalheiro et al. 2018). At the same time, it is an important basis for nondestructive testing of wood and automatic classification of wood strength (Wang and Zhang 2006). Therefore, accurately predicting the elastic modulus of wood has an important practical significance. The traditional elastic modulus detection method is through mechanical experiments, which are destructive and not suitable for ancient buildings. In recent years, nondestructive testing technologies have been widely used in the field of wood property testing (Biechele et al. 2011; Ming et al. 2013; Chang 2017). The nondestructive testing methods currently applied to the determination of the elastic modulus of wooden components of ancient buildings are: ultrasonic method (Moreno-Chan et al. 2011), stress wave technology (Guan et al. 2013; Menezzi et al. 2014), SilviScan (Xu et al. 2012; Dahlen et al. 2018), and micro-drilling resistance instrument technology (Zhu et al. 2011; Sun 2012; Sun et al. 2012). These techniques have been widely used in nondestructive testing of wood properties (Zhu 2012; Dackermann et al. 2016; Wang et al. 2016; Haseli et al. 2020). It is of great practical significance to predict the elastic modulus index of the whole wood by using a reasonable calculation model, based on nondestructive testing technology to detect theof wood.
Information diffusion (Huang 2005; Hao et al. 2010) is a fuzzy mathematical processing method for set-valued samples that considers the optimal use of sample fuzzy information to make up for insufficient information. It turns a sample point of distinct values into a fuzzy set (Wang et al. 2019). It is built and developed on the basis of information distribution methods, the basic idea is to directly transfer the original information to the fuzzy relationship in a certain way, thereby avoiding the calculation of the membership function, so as to retain the original information carried by the original data to the greatest extent possible (Huang and Wang 1995). Information diffusion can be divided into two ways: one is to assign single-value samples to different discrete points to achieve data fuzziness, and the other is to find the information matrix determined by the discrete points of the two universes to get a fuzzy relationship between the two (He et al. 2008). This article adopted the second way to build a prediction model.
It is key to accurately detect the properties of wooden members for correctly evaluating the bearing capacity and service life of ancient wooden structure members. This has been proved in many on-site nondestructive testing practices. Based on the principle of information diffusion and the original data obtained by nondestructive testing technology, this paper established the distance (D) between the larch wood from the bark to the pulp pith, the height (H) from the base to the top of the tree, and the dynamic elastic modulus forecasting model. During on-site operation, determining the actual location of the measurement point on the cross-section and longitudinal section firstly, then, checking the Ed, collecting and summarizing all the results, removing the error value points, and selecting the proper weight value, then the Ed of the component is obtained. This study hopes to provide a basis and reference value for the rapid and accurate detection of the dynamic elastic modulus of ancient wooden structure members.
EXPERIMENTAL
Materials
The experimental materials were selected from larch (Larix gmelinii) materials that were above the fiber saturation point (FSP). Materials were purchased from Qingdongling Timber Factory (Tangshan City, Hebei Province, China). The diameter of the base was 50 cm, the diameter of the top was 40 cm, and the height was 400 cm. According to the number of annual rings and the information provided by the factory, the wood was approximately 300 to 400 years old. The wood was cut from the base to the top every 50 cm and was divided into 8 sections. The sections were labeled A, B, C, D, E, F, G, H and the shaded and light sides were marked (Figs. 1a, b, c). According to the standard GB 1929 (2009) for selecting test specimens, each section was sawn into 2 cm × 2 cm × 45 cm specimens. A total of 559 flawless specimens were selected. The test specimens were divided into 10 groups according to the actual depth (Tables 1a, b). Among these, group 1 was the pith, and group 10 was the bark. The number of specimens in each group is shown in Table 2. Group A was the base, and group H was the top.
Fig. 1. Schematic diagram of cross-section and radial sections: (a) schematic diagram of cross-section, (b) schematic diagram of cross-section interception, and (c) schematic diagram of radial section
Table 1(a). Groups Listed by Depth in Cross-section
Table 1(b). Groups Listed by Height in Radial Section
Table 2. Number of Specimens in Each Group
Equipment
The apparatuses used were a Lichen Technology blast dryer box 101-3BS (Shanghai Lichen Electronic Technology Co., Ltd., Shanghai, China), Lichen Technology electronic precision balance JA1003 (Shanghai Lichen Electronic Technology Co., Ltd., Shanghai, China), Vernier calipers (Guilin Guanglu Measuring Instrument Co., Ltd., Guilin, China), and FAKOPP microsecond timber (FAKOPP Enterprise Bt., Ágfalva, Hungary)( Figs. 2a, b, c, d).
Fig. 2. Experiment equipment– (a) dryer box; (b) electronic precision balance; (c) Vernier calipers; and (d) FAKOPP microsecond timber
Methodology Adopted
The propagation time of stress wave in wood was measured with a stress wave tester FAKOPP (FAKOPP Enterprise Bt., Ágfalva, Hungary). The two probes of the instrument were inserted into both ends of the test specimens and the propagation time was measured along the longitudinal direction of the test specimens. The angle between the two probes and the length of the test specimens was not less than 45°, and the distance between the two measurement points was measured, as shown in Fig. 3. During the measurement, the reading of the propagation time of the first tap was invalid. Starting from the second time, the average value of the propagation time obtained by continuously measuring three times was used as the final test result. According to the application principle of the stress wave tester (Zhang and Wang 2014), the propagation velocity and dynamic modulus of elasticity of the wood (Ed) were calculated using Eqs. 1 and 2. According to the actual position of specimens and the division method of the group, the average in the group was taken as the value of the stress wave propagation velocity and the Ed, see Table 3,
where L is the distance between the two sensors of the stress wave tester (m), T is the time recorded between the two sensors of the stress wave tester (μs), and Cc is the stress wave propagation velocity in wood (m/s). Equation 2 is as follows,
where Ed is the dynamic modulus of elasticity of wood (MPa), ρ was the density of wood (g/cm3).
Fig. 3. Measuring the stress wave propagation time
Table 3. Dynamic Elastic Modulus of Wood at Each Location Measured by Nondestructive Testing
Next, to determine the density and the moisture content (MC). According to Fig. 4 to cut off the specimens, the size was 2 cm × 2 cm × 2 cm. It was measured with a Vernier caliper (Guilin Guanglu Measuring Instrument Co., Ltd., Guilin, China), and the mass of the test specimens was recorded with a balance. The density calculation formula, ρ = M / V, was then used to calculate the density of all the test specimens. Where, M was the mass (g), V was the volume (cm3). According to the actual position of specimens and the division method of the group, the average in the group was taken as the value of the density.
Fig. 4. Experimental specimen partition (units are in mm)
The MC of the test specimens was then determined according to the GB 1931 (1991) standard (Fig. 5). After drying in a drying box, the mass of the wooden block was measured, and the MC was calculated. The measurement results showed that the MC was 4% ~7%.
Fig. 5. Determination of moisture content by drying test specimens
RESULTS AND DISCUSSION
Establishment of Matrix Model
Establishment of fuzzy matrix between Ed and cross-sectional depth and radial section height
First, a fuzzy relationship between the cross-section distance (D) from the bark to the pith and the Ed was established. The range of D was 1 to 19, and the range of Ed was 5 to 15. The domain of discourse of the two was taken as Eq. 3,
UD = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}; VE = {5, 7, 9, 11, 13, 15} (3)
where UD is the domain of the cross-section distance, with a step size Δ = 2; and VE is the domain of the Ed with a step size Δ = 2.
This study selected the two-dimensional normal diffusion drop formula,
(4)
where uʹ = (u – a1) / (b1 – a1); vʹ = (v – a2) / (b2 – a2); uʹj = (uj – a1) / (b1 – a1); vʹj = (vj – a2) / (b2 – a2); a1 = min(uj), 1 ≤ j ≤ m; b1 = max(uj), 1 ≤ j ≤ m; a2 = min(vj), 1 ≤ j ≤ m; b2 = max(vj), 1 ≤ j ≤ m; uj is the discrete point of U, and vj is the discrete point of U, where j = [1, 2, …, r]. The variable h can be found using the equation h = 1.4208 / (m – 1), where m is the number of samples.
Using Eq. 3, the original information distribution matrix QD,E was obtained, and the original information was completed by normalization processing to obtain the fuzzy relation matrix RD,E between the cross-section distance (D) from the bark to the pith and the Ed. Using the MATLAB software (MathWorks, Inc., Natick, MA, USA) to calculate, the results are shown in Table 4.
Table 4. Initial Information Distribution Matrix QD,E
Similarly, the fuzzy relation between the height (H) from the base to the top in the radial section and Ed could be obtained, and then the original information distribution matrix QH,E and the fuzzy relation matrix RH,E could be gained, as shown in Table 5.
Table 5. Initial Information Distribution Matrix QH,E
Fuzzy approximate reasoning
In the construction of the model, the approximate inference formula, Bi = Ai × R, was used for prediction, where Ai in the formula was calculated according to Eq. 5,
when a ≤ amin, amin∈ Ai, Ai = [1, 0, ⋯, 0]
when a ≥ amax, amax ∈ Ai, Ai = [0, 0,⋯, 1]
when amin ≤ a ≤ amax, Ai = [max(0, 1 – |a – ai| / ∆)] (5)
where Δ is the step size; ∆ = ai+1 – ai; (i = 1, 2, ⋯, m).
The above is the first-order fuzzy approximation inference process. The cross and radial sections have different degrees of influence on the Ed (Fig. 6). Therefore, the second-order fuzzy approximation inference should be completed based on the influence.
According to Eq. 6, the result of the second-order fuzzy approximation inference can be obtained by the combination operation of Weight Array Aʹ and Fuzzy Matrix Rʹ,
Bʹ = Aʹ × Rʹ (6)
where Aʹ is the influence weight of each variable.
In this study, the Aʹ value was determined by a cross-combination operation of two measured variables under different influence weights. Upon calculation, when the influence weight of the D and H was Aʹ = [0.7, 0.3], the average relative error of the predicted value was minimal, which was 8.55% (Figs. 6a, b; Fig. 7). In Fig. 6a, 2 was closer to the bark, and 19 was closer to the pith. In Fig. 6b, 0 was the base, and 400 was the top.
Fig. 6. Variation of Ed in (a) cross-section and (b) radial section
Fig. 7. Cross-combined calendar optimization algorithm calculates weights of influencing factors
Data normalization
To obtain the best predicted value, Bʹ is replaced into Eq. 7, and the result is the final predicted value,
where D is the final predicted value of Ed; Di is the grade value of Ed; k is a constant as appropriate, and k = 2 in this article.
Table 6 compares the results of the measured and predicted values of the Ed of larch under the information diffusion method, and the relative errors of the two results. It can be seen from the data and the prediction map that the model based on the information diffusion method predicts the relationship between the distance (D) from the bark to the pith, the height (H) from the base to the top of the wood, and the Ed, which is effective and practical. The information diffusion model does not need to understand the distribution of samples and does not need to construct membership functions. It uses the information diffusion principle to diffuse the original data. It requires less data and solves nonlinear and local minimum problems well (Zhang and Wang 2010; Li et al. 2014). These advantages are reflected in the above prediction model for the Ed of larch.
Table 6. Contrast of Predicted Value and Actual Value of Ed
It can be seen from Fig. 8a that the value of the Ed is very different from the distance (D) from the bark to the pith in the cross-section and the height (H) from the base to the top in the radial section of the wood. The variation trend is obvious, and it demonstrates the accuracy of the prediction model. As shown in the comparison with the multiple linear regression model (Fig. 8b), the prediction results based on the information diffusion model show high accuracy and stability and can clearly reflect the difference in elastic modulus at each position. The model can also provide guarantee for improving detection accuracy. From the above calculations, it can be seen that under the influence of the two weights of the distance (D) from the bark to the pith and the height (H) from the base to the top of the wood, the minimum average relative error of the fitting of the sample data was 8.55%. The coefficient of determination between the measured value and the predicted value of the Ed was R2 = 0.861 (Fig. 9). Cavalheiro et al. (2018) reported that through static bending and longitudinal vibration methods to generate regression models to estimate the Ed, the R2 value was 0.6265. From the comparison, the predicted value and the actual value were in good agreement.
Fig. 8. Data fitting
Fig. 9. Correlation between measured and predicted values
There are many factors that affect the Edof wood in actual engineering practice, such as moisture content, temperature, density, the annual ring width, wood grain, texture, differences between heart and sapwood, and some growth defects (Liu et al. 2014). Due to the relationship between these influencing factors and Ed, the detection results will be greatly different (Fig. 6b).Therefore, these constraints often cannot obtain enough sample data to estimate, so it will affect the accuracy of the prediction model. From the overall analysis, it can be seen that the prediction model proposed in this paper is still effective, it can provide a basis for evaluating the Ed of wood and provide a basis for further in-depth research, and it can also provide reference value for other related study on wood properties.
CONCLUSIONS
- Based on the information diffusion method, a prediction model was established between the distance (D) from the bark to the pith, the height (H) from the base to the top, and the Ed. Through analysis, the model is effective and practical. The coefficient of determination between the measured value and the predicted value of the Ed was 0.861, they were in good agreement. In actual field testing, when a sufficient number of distances (D) and heights (H) cannot be obtained, the accuracy of detecting the dynamic elastic modulus (Ed) of wood can be improved by using the information diffusion method. It provides a basis for field work.
- The calculation results showed that when the weights of the two influencing factors (D and H) were 0.7 and 0.3, the average relative error of the fitted sample data was the minimum, which was 8.55%.
- Compared with the multiple linear regression method, the prediction model based on the information diffusion method avoids the calculation of membership functions, and the prediction accuracy and stability of the model are significantly improved.
ACKNOWLEDGMENTS
This study was supported by the National Natural Science Foundation of China (51678005), Beijing Natural Science Foundation (8182008), the National Key R&D Program of China (Grant No. 2018YFD1100902-1), Beijing Municipal Commission of Education-Municipal Natural Science Joint Foundation (Grant No. KZ202010005012), and Beijing Municipal Commission of Education-Municipal Natural Science Joint Foundation (KZ202010005012).
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Article submitted: February 10, 2020; Peer review completed: April 25, 2020; Revised version received: and accepted; April 30, 2020; Published: May 7, 2020.
DOI: 10.15376/biores.15.3.4835-4850