Nondestructive Testing on Ancient Wooden Components Based on Shapley Value

College of Urban and Rural Development, Beijing University of Agriculture, Beijing 102206, China Business School, Beijing WuZi University, Beijing 101149, China School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048, China College of Architecture and Urban Planning, Beijing University of Technology, Beijing 100124, China College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China


Introduction
Due to thousands of years of history, many ancient buildings in China have been registered on the World Cultural Heritage list for their unique structures and cultural, historic, and artistic value.Most principal load-bearing components of these ancient buildings are made of wood, which, in combination with tenon and mortise connectors and bracket systems, has the advantage of long service periods, earthquake resistance, and hazard mitigation.However, as a biological material, wood is anisotropic [1], and its damage features and variations differ from those of other building materials.Wooden components are affected by temperature, humidity, the environment, and service life; thus, they are vulnerable to internal deterioration such as cavities, xylophage insects, and decay [2]. is will affect their mechanical properties of strength and stiffness and even lead to collapse [3], resulting in irrecoverable losses for nonrenewable architectural heritage.Currently, internal defects in wooden components are differentiated by qualitative visual identification and simple percussion methods, which, despite being simple and practical, largely depend on experience, although testing equipment is sometimes used for assistance [4].However, visual inspection is not effective for locating internal defects as it only involves the surface of elements.For instance, after dismantling wooden components, major errors are always found between predicted and actual internal defects.Additionally, blind dismantling of components destroys their authenticity, which not only has a serious impact on building appearance but also leads to incalculable losses.Furthermore, this method is problematic for internal defect estimation and scientific renovation programs.
In contrast, nondestructive testing technology aims to have no effect on the appearance, internal structure, or usability of ancient wooden components; thus, it represents a significant improvement from conventional timber and wooden material testing methods [5,6].In recent years, by employing methods such as X-ray scanning, Pilodyn [7], vibration methods [8], stress waves, drilling resistance [9][10][11][12][13], and ultrasonic waves [14][15][16] for the nondestructive testing of internal defects, researchers have determined that stress wave and impedance meter testing methods are the most suitable for defect detection because of their convenience, security, and effective visualization of results.For example, stress wave nondestructive testing and laboratory tests of the USS Constitution ship [17] showed that when the internal material is free of defects, the propagation path between the sensors is a straight line and the wave velocity is relatively small.Conversely, the propagation path between the sensors is disturbed by defects, resulting in a curved path and clearly increased stress wave propagation velocity in rotten wooden components; thus, internal decay and degradation can be better detected by stress wave testing.In addition, tests on the wooden components of ceilings and other parts of Papa Baroque conducted by Hungarian scholars, e.g.[18], demonstrated that stress wave technology can forecast the flexural strength of a single wooden component.Chinese researchers conducted impedance meter tests on buildings including the Forbidden City and Shanhaiguan Temple and found that resistance testing values are closely related to the air-dried density; therefore, internal defects can be directly determined from the resistance curve.Nonetheless, differences exist between testing results due to the respective advantages and limitations of these two methods.Moreover, no previous research has attempted error reduction or quantified the prediction outcomes when applying these nondestructive testing methods to the internal defects of wooden components.
erefore, in order to protect original wooden structures, combined predictions should be made of the internal defects and test results should be quantified to reduce error values.Reinforcement or replacement of wooden components is a problem which needs urgent solution in ancient architecture examination, design, and renovation [19][20][21].
In this study, we employ stress wave and drilling resistance tests to conduct nondestructive detection testing on the wooden components of an ancient building.We then match equivalent tree species to perform reverse laboratory tests, where we simulate common internal cavity defects in ancient architecture and then establish a nondestructive combined forecasting model to improve testing accuracy.By quantifying the internal defects, we develop a novel and practical combined prediction method for the nondestructive testing of internal defects in wooden components.erefore, this research provides strong data support for future renovation and reinforcement of wooden components, thereby promoting the conservation and effective utilization of architectural heritage.

Ancient Architecture Case Study
Located in the southern part of Beijing, China, the pavilion building analyzed in this study was built during the Ming Dynasty and is known as the altar.
ere are not many records regarding the renovation of this ancient building, and it was not renovated in recent years.In order to combat aging and damage at various levels, this building requires testing that does not damage its authenticity and involves minimal intervention.
According to the "Technical Code for Maintenance and Strengthening of Ancient Timber Buildings" (national standard of the People's Republic of China GB 50165-92), the ratio of the area of decay and aging metamorphism to the area of the whole section is p.Moreover, p > 1/5 is regarded as damage through decay and aging of the surface layer of the bearing timber pillar, p > 1/7 is considered as damage due to decay of the core material, and p > 1/8 is considered as residual damage through decay and aging of the surface layer of the beam or both ends of the beam or core.
Currently, aging and damage exists in the wooden components of the pavilion at various levels, the most prominent of which are surface damage and pollution and damage to the building materials.
2.1.Surface Damages.At the joints of the upper eaves and around the domes, the paint decorations on the rafters and sheathings have fallen off and the sheathings have rotten due to rain erosion (Figures 1(a) and 1(b)).ere is also cracking and bulging and peeling of paint decorations on the partially wooden pillar plaster (Figure 1(c)), wooden components of the fence in the west of the building are seriously aged and even cracked (Figure 1(d)), and the end of the Ang, a type of wooden component, is rotten (Figure 1(e)).

Pollution and Building Material Damage.
e color of the sharp wooden components at the wooden member top has changed due to mold from microorganisms in the airless atmosphere (Figure 2(a)).Additionally, during a site investigation, birds' nests were found and most wooden components were covered by their excrement and decayed (Figure 2(b)).On the upper side of the ring beams at the east side of the pavilion, the wooden components are incomplete and some are moldy (Figure 2(c)).After the site investigation, we found different degrees of damage outside the wooden components of the altar; this along with mold will lead to the degradation of components and the formation of internal damage.erefore, the external damage affects the internal damage.In order to determine the bearing capacity and safety of wooden members, a quantification of their internal defects is required.

Materials and Methods
Nondestructive testing was conducted on the main wooden components of beams and pillars, and internal defect data were collected.Some of the beams and pillars were then chosen as samples for detecting the tree species of the 2 Advances in Materials Science and Engineering components.We then matched equivalent tree species for an internal defect reverse laboratory test and analyzed the error values of each nondestructive method as well as their combined predicted values using the Shapley value algorithm.us, both nondestructive methods and prediction models were used to evaluate internal defect prediction of the wooden components.
3.1.Samples for Data Collection.Before that approximately ancient wooden buildings in the south and north of China were investigated (for example Shanxi Guanyin hall, Anhui Diao Xue Tang), revealing that the beam and pillar components are often damaged at both ends.Wooden pillars are eroded by rain and snow all year round, and the accumulated humidity at the bottom of the pillar, together with the normal load, encourages internal defects.e beams subject to the load of the upper wooden components all year round are easy to bend, and the overlap between beams is complex.erefore, we focused on the internal defects of these two major wooden components.On the premise of "minimum intervention," nondestructive testing was conducted on the load-bearing beams and pillars.

Stress Wave Measurement.
Internal nondestructive testing of the main load-bearing pillars and beams was conducted using the FAKOPP ten-probe stress wave measuring instrument.Stress wave sensors were installed in the top part (Section A), middle part (Section B), bottom part (Section C), beam end, and midspan (Figure 3), and testing sections were added if serious internal defects or uncertain factors were observed.Due to plaster on the pillar surfaces, the stress wave probes should be inserted to a depth of much Advances in Materials Science and Engineering greater than 50 mm for effective data collection.Sensor no. 1 was placed in the north direction, and the other sensors were distributed counterclockwise according to the perimeters of the components.e sensors were hit three consecutive times, and the average value of the resulting stress wave that propagated through the interior of the wooden components was taken as the propagation velocity.By matrix calculation and reconstruction of the propagation time of internal sections in different paths, the data were transformed into two-dimensional plots, in which the axes represent the specimen size and the left-hand color bar represents the condition of the wooden component.Dark green represents a healthy interior, which changes to yellow then red and finally blue as its condition deteriorates, representing a severe cavity.Internal defects in the wood were then analyzed using ArborSonic 3D.

Drilling Resistance Tests Method.
e German-made IML-500 drill was used to conduct nondestructive drilling resistance testing on the wooden components.A microprobe with a length of 1.5 mm was driven into the interior of the components.e probe encounters different resistances as it moves forward, producing a relative resistance value of the path. is was plotted in two-dimensional plots of the internal defects where the x-axis represents the microprobe path length and the y-axis the relative resistance value (probe insertion resistance value and rotation resistance value).Based on the stress wave detection map, the defect locations of different sections were analyzed by the probe along three different paths at an angle of 60 °to collect impedance meter data.rough visual observation and smell, analysis of the macroscopic texture, color, and smell characteristics shows that heartwood (pillar B2) and sapwood (beam between B3 and B4) are clearly different and have no distinct taste.Analysis of vertical sections, cross sections, and tangential sections (Figures 4(c) and 4(d)), using microscope observations of specimens at 75x magnification, indicates that the tracheids range in tangential diameter from 25 to 56 µm (average 40 µm).e transverse section of earlywood tracheids is typically oblong and rarely polygonal, and dentate ray tracheids do not exhibit thickening and are sinuous and undular [22].Based on the macro and micro characteristics of the samples, we conclude that the samples between pillar B2 and B3-B4 are all Pinaceae Lindl, and the tree species are Larix of the Larix genus.e probe is inserted into the object to be measured by using the water content meter.After the special number is stabilized, the displayed number is the moisture value of the object to be measured.e moisture contents of the beam and pillar are less than 12%.4.2.Wooden Components' Internal Defects Analysis.Our analysis of the internal defects of the major wooden beam and pillar components shows no defects in the interior of beam components but serious defects in the interior of pillars B2 and D4.For example, Figure 5 shows Sections A, B, and C of pillar B2.Analysis by ArborSonic 3D revealed severe internal defects in Section C and no internal defects in Sections A and B (Figure 5(a)).To estimate whether the internal defects of Section C develop upwards or become more serious, all sensors were moved upwards by 50 cm (Section D) from Section C. e results show that a clear reduction of internal defects and of the affected area (Figure 5(a)), indicating that the bottom of the pillar is the original source of defects and Section C is the main location of internal defects (Table 1).By connecting each section of pillar B2 and analyzing the entire pillar, we verify that points 2-6 of Section C are the most serious and there are no defects in the interior of Sections A and B (Figure 5(b)).

Results and Discussion
As well as using the stress wave chart to determine internal defects in Section C, targeted drilling resistance test results also reveal the locations and sizes of internal defects along a single path (Figure 6).Different internal defect   Advances in Materials Science and Engineering lengths (x i , x i+1 , . . .x n ) can be formed by intersecting the three paths, and n triangles with diverse areas can be obtained by connecting neighboring defect points.e internal defect area of the entire testing section is obtained by where S 2 represents the total area of internal defects in the section, n is the number of defect edges, and x i is the defect length.According to previous research, when relative resistance values are less than 30% or below the average wave crest of the relative resistance values in the normal range, internal defects in this range are determined as cavities or cracks.In general, cavities are longer than cracks, so the internal defects of the wooden components can be defined as internal cavities.Specific testing data are shown in Table 1.

Reverse Laboratory Test.
Based on the internal defect results of the wood components in the ancient buildings, we matched equivalent tree species and then processed samples by manual digging and cutting them to size.To reduce the error value between the experimental results and the data collected from the ancient buildings, we simulated all defect types of the samples as internal holes.According to the defect locations in the wood components, the defect distribution in the specimen was simulated in the core materials and in the sapwood.e specific design parameters of specimens are shown in Table 2.
After production, the specimens were kept in the laboratory at a temperature of 20 °and relative air humidity of 65% for 3 months.Tests were conducted when the air-dried moisture content was below 12%.Defect expansion over six different stages (when the defect area occupies 0, 1/9, 1/7, 1/5, 1/3, and 1/2 of the section area) demonstrates the initial, intermediate, and terminal stages of the defects.e average value of three hits on each sensor was taken as the propagation velocity.In order to avoid the discreteness of testing data due to specimens displaying diverse defect areas, stress wave data were collected three times.We analyzed 36 groups of nondestructive testing data and the average values of sections with the same defect areas.
Internal cavity simulation experiments of specimens D-1 and D-2 by the FAKOPP stress wave timer and IML impedance meter indicate that the larger the internal defect area, the more accurate the size and location of internal defect detection by the stress waves (Table 3).e boundaries of the cavities are blurred in the stress wave diagrams (Figures 7(a) and 7(b)).Moreover, the stress wave testing area, S 1 , is larger than the specimen's actual internal defect area, S 0 ; the absolute error of the stress wave, R 1 , is calculated as follows: R 1 and the relative error value, Q 1 , vary irregularly (Table 3). is illustrates that because of the severe damage to wooden components' internal fibers, the stress wave travel path changes and the propagation time increases, which leads to errors in internal defect area detection.
e crest (maximum drilling resistance) and trough (lowest rate drilling resistance) of the relative resistance values are revealed by inserting needles during impedance meter testing; therefore, the internal defects can be defined as cavities.
ese cavities can be used to determine the internal defect boundary and length; however, an impedance meter only reflects the internal defects in single path, so entire section defects cannot be detected.e impedance meter testing values, S 2 , of specimens are listed in Table 3.
e 36 groups of experiments prove that when the internal defect is a cavity with a diameter of more than 6 cm, the probe path will change direction due to a drop in the resistance value.Moreover, unstable swinging can easily cause the probe to break off or deviate from the anticipated testing path; thus, the absolute error value of drilling resistance, R 2 , is calculated as e data in Table 3 show that the R 2 value increases with defect expansion.A comparison of the above two testing methods indicates that the stress wave nondestructive testing method can make rapid and direct judgments of internal defect locations and lengths, but defect types cannot be determined clearly and defect boundaries are vague.Conversely, the impedance meter nondestructive testing method can make a precise judgment on internal defect length in a single path and quantify defect types; however, it cannot effectively determine the entire testing section, and the absolute error increases with the size of the internal cavity.

Numerical Analysis and Prediction Model
As shown in Table 3, errors exist in the stress wave and impedance meter nondestructive testing results, in order to reduce the area of internal defects of wood components;  therefore, we performed a combined prediction and evaluation for these two methods.

Combined Models Construction.
Stress wave and impedance meter methods have both advantages and disadvantages for testing internal defect locations and areas, so their results are evaluated comprehensively in order to construct a combined prediction model.On the assumption that the same wooden component internal defects are predicted by N testing methods, the combined forecasting model of the internal defects of N testing results is where f t is the predicted value of wooden component internal defects according to the combined forecasting model, f it is the predicted value of i wooden component internal defects, where i � 1,2. . . ., n, K is the weighting of internal defects, and K i is the weight of i wooden component internal defects, where i � 1,2. . . ., n and  N i�1 K i � 1.

Optimal Weighting Method Based on the Shapley Value
Algorithm.e combined predicted overall values of the two nondestructive testing methods are distributed by a weighting method based on the Shapley value algorithm to determine the weight of each testing value [23][24][25].On the assumption that there are n nondestructive testing methods in the combined prediction, I � {1,2, . . .n} and, for any subset p, q  Advances in Materials Science and Engineering (any combination of n methods) of I, E(p), E(q) represents the internal defect testing error.e definitions are as follows: (1) For any subset p, q (any combination of n methods) of I, E(p) + E(q) ≥ E(p ∪ q) (2) p ⊆ I, x i represents the error value shared when the combination of i method ends, x i ≤ E(i) (3) e overall value E(n) produced by the combined internal defect prediction including n nondestructive   On the assumption that the average method is E i , the overall error value of the combined prediction is E i , and where m represents the sample numbers, e ij represents the absolute value of j samples of i nondestructive methods and absolute error of nondestructive testing data.e weight allocation formula based on the Shapley value algorithm is where w|p| is the weighting factor, representing the combined marginal contribution that i should make in combined nondestructive testing; p-{i} represents that model i is removed from the combination; i represents a certain nondestructive testing forecasting model involved in the combined prediction; E i is the error shared by the predicted model i, namely, the Shapley value algorithm; p is any subset of i; and |p| is the number of combined forecasting models.According to equations ( 4) and ( 5), the weight calculation formula of the combined prediction method is as follows: According to equations ( 4) and ( 5), the Shapley value of specimen D-1 stress wave nondestructive testing is Similarly, the error shared by specimen D-1 impedance meter nondestructive testing is E 2 � 9.39 cm 2 ; this explains that the Shapley value algorithm separately provides weight to each parameter; E 1 + E 2 � 23.84 cm 2 , showing that the sum of the error shared by the individual stress wave and impedance meter testing methods equals the overall error amount E. e shared values reflect the accuracy of each testing method, whose ultimate weight in the combined model depends on equation ( 6): Similarly, the allocated weights of the nondestructive testing combined model for specimen D-2 are w b1 � 0.45 and w b2 � 0.55.According to the above two allocated weights and equation ( 2), the internal defects of the combined model for specimens D-1 and D-2 are All combined forecasting models are recombined based on their corresponding stress wave and impedance meter testing results to predict internal defect areas.Wooden component internal defect area S 3 is obtained on the basis of the Shapley value combined prediction; the absolute error value, R 3 , and relative error value, Q 3 , are also obtained (Table 3).
According to Table 3, R 3 is less than R 1 and R 2. To further verify the prediction effect of the combined models on the basis of the Shapley value algorithm, the average relative error values, Q 1 and Q 2 , of the stress wave and impedance meter results are calculated.In Table 3, the Q 3 value of specimens D-1 and D-2 combined predicted internal defects is less than that of the individual error values of stress wave or impedance meter results.Moreover, the combined predicted internal defects are more accurate, with an average error value of less than 6%.
ese results indicate that combined prediction is applicable for detecting internal defect and quantitative analysis of wooden components.

Combined Prediction and Evaluation of the Wooden
Components' Internal Defects.Using Section C and Section D of pillar B2 as examples, we locate internal defects in the sapwood using stress wave two-dimensional plots, which are further confirmed by IML impedance meter tests, indicating that the internal defects are cavities.Internal defects of Section C and Section D are then predicted by hardwood pine and sapwood combined models on the basis of the Shapley value algorithm obtained from reverse laboratory tests.
e results indicate that the internal cavity area of Section C is 285.94 cm 2 , accounting for 1/3 of Section C, and that of Section D is 249.91 cm 2 , accounting for 1/4 of Section D. According to the Technical Code for Maintenance and Strengthening of Ancient Timber Buildings [26], compared to the entire section area, the decayed or cavity area in the load-bearing wooden components, P, is greater than 1/5.Advances in Materials Science and Engineering us, according to our testing data, defects exist in the interior of Section C and Section D of pillar B2 that require renovation.

Conclusions
Internal defects within wooden components tend to reduce the compressive area per unit area.Individual nodes or component parts cannot be subjected to normal forces, and then the components of the skew flash or displacement, thereby eventually leading to deformation of ancient building-wood components.In this study, reverse laboratory tests were conducted on major wooden components belonging to the same tree species through nondestructive testing methods.According to defect detection data and experimental results, the following conclusions are drawn: (1) Internal defects mainly occur in key joints such as pillar bases and girders according to detection tests on wooden components of ancient buildings.Knowledge and analysis of wooden component internal defect characteristics can be used to determine preventive conservation and improve defect detection in ancient architecture.(2) e sizes and locations of wooden component cavities can be quickly obtained through stress wave tests and effectively visualized using two-dimensional plane detection diagrams.However, stress wave testing area is greater than the actual testing area, errors exist, and internal defect boundaries are blurred.(3) Using the stress wave results of internal defects in the wood, drilling resistance was used to detect the presence, type, and location of internal defects, which can be accurately confirmed by the corresponding two-dimensional plane detection diagrams.However, the results represent a single path of the testing section and weaker direct detection of internal defects in the entire section.e smaller the internal defects, the more accurate the drilling resistance testing result.e relative error values are directly related to expansion of the defect area.Moreover, more drilling resistance paths may provide more precise information.(4) e stress wave test chart can determine the approximate position of the internal damage in a wooden component, and the drilling resistance test can determine the shape of the internal damage.e drill resistance test depends on the stress wave test chart, and the drill resistance test results are based on the stress wave test.erefore, we determined the weight of each prediction method (stress wave and drill resistance tests) using the Shapley value algorithm weight allocation method.is exploits the advantages of both types of nondestructive testing, and using this model, we build an internal defect area prediction model.e predicted result of the combined model based on the Shapley value algorithm is more accurate than that of the single testing method; the internal defect area only exhibits a small prediction error.
rough experimental analysis, we determine that the mean relative error of the defect area of the combined prediction model is less than 6%.However, the Shapley value algorithm separately provides weights to each parameter; thus, if one parameter depends on another, different conclusions could be drawn.erefore, this study presents a new practical method for the quantitative evaluation of internal defects in wooden components, which will be beneficial for architectural heritage conservation as well as provide room for further research.
Data Availability e data used to support the findings of this study are available from the corresponding author upon request.
Figure 4(a) and 4(b) shows pillar B2 pillar and the beam between pillars B3 and B4.

Figure 3 :
Figure 3: Stress wave sensor locations in (a) pillars and (b) beams.

Figure 4 :
Figure 4: Analysis of wooden component samples for tree species analysis: (a) pillar B2, (b) beam between B3 and B4, (c) and (d) microscopic images of three types of section of the Larix tree species.

Figure 5 :
Figure 5: Stress wave analysis of pillar B2 showing (a) different sections of the pillar and (b) a longitudinal sectional of the internal defects.

Figure 6 :
Figure 6: Two-dimensional plot of the drilling resistance results of Section C. (a) Path one.(b) Path two.(c) Path three.Note: horizontal coordinates indicate the depth of the probe path and vertical coordinates show that the density distribution and the early and late wood performance of the probe differ as the probe advances inside the pillar.Location of damaged section in red box.

Figure 7 :
Figure 7: Image of specimens and stress wave diagrams for different stages of defect expansion for (a) the core material cavity experiment and (b) the sapwood cavity experiment.

Table 1 :
Nondestructive testing data of pillar B2.

Table 2 :
Wooden component sample descriptions used for the reverse tests.

Table 3 :
Comparison of nondestructive testing methods.
) 5.3.Weight Allocation and Combined Prediction Models of Internal Defects.According to the results of stress wave and impedance meter tests on different defect locations and areas in Table3, the results of the same specimen show that S 1 > S 0 and S 2 < S 0 .For example, the overall combined prediction error value for specimen D-1 determined by formula (3) is E � (26.37 + 21.31)/2 � 23.84 cm 2 .