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An Experimental Study of the Hysteresis Model of the Kanchuang Frame Used in Chinese Traditional Timber Buildings of the Qing Dynasty

An Experimental Study of the Hysteresis Model of the Kanchuang Frame Used in Chinese Traditional... buildings Article An Experimental Study of the Hysteresis Model of the Kanchuang Frame Used in Chinese Traditional Timber Buildings of the Qing Dynasty 1 , 2 2 , 3 , 1 4 1 1 Junhong Huan , Xiaodong Guo * , Zhongzheng Guan , Teliang Yan , Tianyang Chu and Zemeng Sun School of Civil Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China; junhong_love@126.com (J.H.); guanzhongzheng@stdu.edu.cn (Z.G.); chuty1998@163.com (T.C.); sunzemeng19960326@163.com (Z.S.) Faculty of Architecture, Civil and Transportation Engineering, Beijing University of Technology, Beijing 100124, China Key Scientific Research Base of Safety Assessment and Disaster Mitigation for Traditional Timber Structure (Beijing University of Technology), State Administration for Cultural Heritage, Beijing 100124, China Beijing ZAJ Engineering Design Co., Ltd., No. 3 Chengxiangshiji Square, Beijing 100176, China; teliangyan@126.com * Correspondence: gxd@bjut.edu.cn Abstract: Kanchuang frames are important parts of traditional timber architecture in China. This paper used experimental and numerical methods to study the restoring force model of Kanchuang frames, which were used frequently in Chinese ancient timber structures, particularly in North China. The prototyped test model is a type of Chinese traditional timber architecture named Qilinyingshan. It was widely used in ancient timber buildings preserved from the Ming and Qing dynasties. This study analyzed the loading process and failure modes of the test model, and the skeleton curve and hysteretic curve data were collected. Moreover, a dimensionless skeleton curve model was developed Citation: Huan, J.; Guo, X.; Guan, Z.; based upon the findings. The hysteresis loops of the test model were also analyzed, and it was found Yan, T.; Chu, T.; Sun, Z. An that each hysteresis loop can be divided into several feature segments according to their stiffness Experimental Study of the Hysteresis at different loading stages. Regression analysis was also used to obtain the stiffness degradation Model of the Kanchuang Frame Used curvilinear equations of the feature segments. Finally, a hysteresis force model of a Kanchuang frame in Chinese Traditional Timber Buildings of the Qing Dynasty. was established. This study also found that the loading process can be divided into three stages: the Buildings 2022, 12, 887. https:// elastic stage, in which all of the components are in good condition; the elastic–plastic stage, in which doi.org/10.3390/buildings12070887 cracks gradually develop on the wall; and the new elastic–plastic stage, after which the wall collapses. It was found there was consistency between the restoring force model and the test results, indicating Academic Editors: Mahmud Ashraf that the model is valid and reliable. The skeleton curve model and hysteretic model provide reference and Wen-Shao Chang for the nonlinear seismic response of ancient timber architecture. Received: 19 April 2022 Accepted: 20 June 2022 Keywords: Kanchuang frame; restoring force model; ancient timber architecture; loading process Published: 23 June 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- 1. Introduction iations. Ancient timber buildings are considered cultural treasures of China. They represent Chinese history and culture. However, during the previous years, earthquakes have damaged or even destroyed a large number of traditional timber buildings. For example, nearly 10,000 traditional timber buildings were damaged in the Wenchuan earthquake in Copyright: © 2022 by the authors. Sichuan on 12 May 2008 [1]. The architecture of traditional Chinese timber buildings is Licensee MDPI, Basel, Switzerland. different from the architecture of modern buildings. In detail, in the structure system of This article is an open access article traditional Chinese timber buildings, columns are directly placed on plinths without any distributed under the terms and other fastening measures. Moreover, beams and columns are connected by mortise–tenon conditions of the Creative Commons joints without any nails and bracings, and all of the mass from the roof is loaded onto Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ the beams and columns. There are no weights on the infill walls [2,3]. Timber frames 4.0/). are usually the main-load bearing and force-resisting units of traditional Chinese timber Buildings 2022, 12, 887. https://doi.org/10.3390/buildings12070887 https://www.mdpi.com/journal/buildings Buildings 2022, 12, x FOR PEER REVIEW 2 of 23 Buildings 2022, 12, 887 2 of 22 no weights on the infill walls [2,3]. Timber frames are usually the main-load bearing and force- resisting units of traditional Chinese timber structures, which is different from modern struc- tures. It is thus of great importance to preserve these valuable cultural heritages by investigat- structures, which is different from modern structures. It is thus of great importance to ing the mechanical properties of the structure and components. preserve these valuable cultural heritages by investigating the mechanical properties of the structure and components. 1.1. History and Research of Chinese Traditional Structures 1.1. History and Research of Chinese Traditional Structures The previous historical literature on the earliest works of Chinese timber architecture The previous historical literature on the earliest works of Chinese timber architecture track its history to 5000 years ago. The most ancient known book about the construction track its history to 5000 years ago. The most ancient known book about the construction method of Chinese traditional timber buildings was published in the Song Dynasty [4]. method of Chinese traditional timber buildings was published in the Song Dynasty [4]. During recent modern times, many scholars have realized that this kind of cultural herit- During recent modern times, many scholars have realized that this kind of cultural heritage age is extremely valuable, and have begun to study and protect it. For example, in the is extremely valuable, and have begun to study and protect it. For example, in the 1930s, 1930s, Liang [5] took the lead in the study of the ancient timber buildings, and his studies Liang [5] took the lead in the study of the ancient timber buildings, and his studies were were focused on the architecture and anatomy of Chinese structures. Other scholars found focused on the architecture and anatomy of Chinese structures. Other scholars found that that the mortise–tenon joints are the weak points of timber structures, and thus studied the mortise–tenon joints are the weak points of timber structures, and thus studied their their mechanical properties. Eckelman et al. [6], Likos et al. [7], Huan et al. [8] and Chen mechanical properties. Eckelman et al. [6], Likos et al. [7], Huan et al. [8] and Chen et al. [9] et al. [9] studied the bending behaviour of different types of mortise–tenon joints in the studied the bending behaviour of different types of mortise–tenon joints in the timber timber structure. Other studies also focused on Dou-gong, a special part of traditional structure. Other studies also focused on Dou-gong, a special part of traditional timber timber structures (see Figure 1 [10]). These are commonly used in the main buildings of structures (see Figure 1 [10]). These are commonly used in the main buildings of traditional traditional palaces. Experimental studies [11–16] were carried out to investigate their fail- palaces. Experimental studies [11–16] were carried out to investigate their failure modes, ure modes, stiffness, yield load and other mechanical properties. In detail, Yeo et al. [17] stiffness, yield load and other mechanical properties. In detail, Yeo et al. [17] studied studied the seismic behaviour of Taiwanese timber brackets subjected to out-of-plane the seismic behaviour of Taiwanese timber brackets subjected to out-of-plane loading, loading, and the effect of their mechanical behaviour—such as failure modes—on the and the effect of their mechanical behaviour—such as failure modes—on the strength of strength of the mechanical model. Xie et al. [18] studied the static behaviour of a two- the mechanical model. Xie et al. [18] studied the static behaviour of a two-tiered Dou- tiered Dou-Gong system reinforced with super-elastic alloy, and found that the pre-strain Gong system reinforced with super-elastic alloy, and found that the pre-strain of the of the super-elastic alloy can significantly increase the damping ratio in the structure. Pre- super-elastic alloy can significantly increase the damping ratio in the structure. Previous vious studies [19–22] tried to evaluate the mechanical properties and safety state of an studies [19–22] tried to evaluate the mechanical properties and safety state of an entire entire timber structure through finite element modelling, mathematical methods, and a timber structure through finite element modelling, mathematical methods, and a shaking- shaking-table test. However, all of these studies focused on the timer components and table test. However, all of these studies focused on the timer components and timber timber frames without considering the infill walls. Recently, scholars have considered that frames without considering the infill walls. Recently, scholars have considered that infill infill walls might have a significant effect on the seismic performance of timber structures. walls might have a significant effect on the seismic performance of timber structures. For example, Vieux-Champagne et al. [23,24] studied the seismic performance of timber- For example, Vieux-Champagne et al. [23,24] studied the seismic performance of timber- framed structures filled with natural stones and earth mortar by introducing three scales framed structures filled with natural stones and earth mortar by introducing three scales of experiments. Ali et al. [25] studied the in-plane behaviour of full-scale Dhajji walls, a of experiments. Ali et al. [25] studied the in-plane behaviour of full-scale Dhajji walls, a wooden-braced frame with a stone infill system, and tested it under quasi-static loading. wooden-braced frame with a stone infill system, and tested it under quasi-static loading. Poletti and Vasconcelos [26] studied the seismic behaviour of the walls with masonry in- Poletti and Vasconcelos [26] studied the seismic behaviour of the walls with masonry filllath and plaster, and a timber frame with no infill. Dutu et al. [27,28] studied the seismic infilllath and plaster, and a timber frame with no infill. Dutu et al. [27,28] studied the behaviour of timber-framed masonry walls based on the static cyclic loading test, and seismic behaviour of timber-framed masonry walls based on the static cyclic loading test, proposed a simplified seismic evaluation method. All of these studies demonstrate that and proposed a simplified seismic evaluation method. All of these studies demonstrate the infill walls can significantly increase the stiffness, the ductility and the load-bearing that the infill walls can significantly increase the stiffness, the ductility and the load-bearing capacity of the timber frames, which consequently have a significant influence on the seis- capacity of the timber frames, which consequently have a significant influence on the mic behaviour of the entire building. seismic behaviour of the entire building. Figure 1. Structure of Dou-gong. Figure 1. Structure of Dou-gong. Buildings 2022, 12, x FOR PEER REVIEW 3 of 23 Buildings 2022, 12, 887 3 of 22 1.2. Research on Timber Frames with Infill Walls 1.2. Research on Timber Frames with Infill Walls Although the importance of infill walls has been studied in modern buildings, the infill walls of Chinese ancient timber architecture have not received attention in previous Although the importance of infill walls has been studied in modern buildings, the research, and they are considered to be nonstructural members. Moreover, limited re- infill walls of Chinese ancient timber architecture have not received attention in previous search has studied the seismic performance of timber frames with infill walls. Emile et al. research, and they are considered to be nonstructural members. Moreover, limited research [29] studied the failure modes, stiffness, strength (including the rate of degradation), and has studied the seismic performance of timber frames with infill walls. Emile et al. [29] energy dissipation capacity of Chinese traditional mortise–tenon jointed beam-column studied the failure modes, stiffness, strength (including the rate of degradation), and energy frames with wood panel infill. Xie et al. [30] also studied the influence of wood infill walls dissipation capacity of Chinese traditional mortise–tenon jointed beam-column frames on the se with wood ismipanel c performanc infill. Xie e of Chine et al. [30 s] e t also radit studied ional tim the ber str influence ucture of s throu woodg infill h shak walls ing t on ablthe e tests. It was found that the natural frequencies, damping ratio and acceleration responses seismic performance of Chinese traditional timber structures through shaking table tests. It of the model was found that with wood in the natural fill w frequencies, alls were grea damping ter than ratio those w and acceleration ithout. Chang e responses t al. [31of –33 the ] stmodel udied the mechanic with wood infill al walls behavio weru ers gr eater of trad than itiona those l ti without. mber she Chang ar wal etl and rei al. [31–33n ]forced studied planke the mechanical d timber she behaviours ar walls us of ing exper traditional iment timbs and c er shear alc wall ulatand ionsr . T einfor heir f ced indin planked gs showed timber shear walls using experiments and calculations. Their findings showed that the friction that the friction behaviour between board units and beams plays a major role in resisting behaviour between board units and beams plays a major role in resisting the lateral force the lateral force applied to the timber shear wall. The restoring force model is the founda- applied to the timber shear wall. The restoring force model is the foundation of the seismic tion of the seismic evaluation of structures. evaluation of structures. 1.3. Research and Application of the Restoring Force Model 1.3. Research and Application of the Restoring Force Model Many studies have investigated the restoring force model of connections and com- Many studies have investigated the restoring force model of connections and compo- ponents of structures [34–36], and they found that steel structures are rich in the restoring nents of structures [34–36], and they found that steel structures are rich in the restoring force model of reinforced concrete structures. However, only limited research has studied force model of reinforced concrete structures. However, only limited research has studied the restoring force model of ancient timber structures. Moreover, the restoring force the restoring force model of ancient timber structures. Moreover, the restoring force model model of reinforced concrete structures and steel structures cannot be used in timber of reinforced concrete structures and steel structures cannot be used in timber structures structures because of the huge difference in the machinal properties between timber and because of the huge difference in the machinal properties between timber and concrete. concrete. Therefore, it is essential to study the restoring force of Chinese timber frames Therefore, it is essential to study the restoring force of Chinese timber frames with infill with infill walls. In the present paper, a Chinese traditional timber frame with masonry walls. In the present paper, a Chinese traditional timber frame with masonry and wood and wood window infill—the Kanchuang frame (shown in Figure 2)—was tested under window infill—the Kanchuang frame (shown in Figure 2)—was tested under cyclic loading. cyclic loading. Kanchuang is a kind of wood window used in ancient timber buildings Kanchuang is a kind of wood window used in ancient timber buildings named “Kan”. named “Kan”. A Kanchuang frame is a timber frame with Kanchuang and a half masonry A Kanchuang frame is a timber frame with Kanchuang and a half masonry wall infilled. wall infilled. The loading process and failure modes of the test model were studied in The loading process and failure modes of the test model were studied in order to collect order to collect test data such as skeleton curves and hysteretic curves. The restoring force test data such as skeleton curves and hysteretic curves. The restoring force model for the model for the Kanchuang frame was developed to be used as the foundation for the seis- Kanchuang frame was developed to be used as the foundation for the seismic evaluation of mic evaluation of the Kanchuang frame. the Kanchuang frame. Figure 2. Dimensions of the experimental model (mm). Figure 2. Dimensions of the experimental model (mm). Buildings 2022, 12, 887 4 of 22 Finite element analysis has been frequently used to analyse the seismic performance of timber structures. However, a timber structure is quite different from the reinforced concrete structure. Moreover, most ancient timber buildings use various kinds of materials. Consequently, there are a lot of contact areas and small gaps between different compo- nents. This made them difficult to simulate using finite element analysis software, and the calculation results cannot be proven valid or reliable with the consideration of contacts, mechanical properties and material properties. Moreover, the calculation process is very complicated, and it takes a very long time to calculate the results. Nevertheless, it was found the building structures can be divided into sub-units. When the units’ hysteretic model, rigidity and strength are known, an analysis model can be used to calculate the sub-units. In this way, the contact analysis in the finite element calculation can be avoided, and we can improve the computational efficiency. Moreover, a correct hysteretic model is also crucial to nonlinear seismic analysis. Thus, there is a need to develop a simplified calculation model that can present the mechanical properties of the structure for nonlinear seismic analysis. The sub-units can be used to simplify the hysteretic model of the structure as mass elements or shell elements. Thus, the calculation model could be simpler than before, and the nonlinear seismic analysis methods—such as Newmark- —can be applied to calculate the seismic responses of the structure. 2. Experimental Studies 2.1. Specimen Fabrication In this study, a 1:2 scale Kanchuang frame model was used as the test specimen. This specimen of Chinese traditional timber architecture was collected from Qilinyingshan (see Reference [5]). The Kanchuang frame is widely used in ancient timber buildings of the Qing dynasty. Kanchuang frames consist of columns, fangs, wood windows, masonry walls, Shangkan, Fengkan and Baokuang. All of the wood specimens are connected with mortise–tenon joints. The structural properties of wood members can vary with the member size, or the size effect. According to the Buckingham theorem [37] and similitude theory, the structural properties of the prototype Kanchuang frame can be calculated by dividing the corresponding properties of the scaled specimens by the scale factors of the dimensions and mechanical properties of the materials [38]. The dimensional scale factor of the test model was 1/2. Table 1 shows the scale factor of the physical parameters. Table 1. Scale factors of the physical parameters. Length Area Displacement Elastic Modulus Force Drift Angle Moment Density Mass 1/2 1/4 1/2 1 1/4 1 1/8 2 1/4 The span of the frame was 1600 mm and the total height was 1800 mm. The dimensions of the masonry wall were 1450 mm  500 mm  250 mm. Figure 2 shows the layout and main dimensions of the model. The dimensions of the components are shown in Figures 3 and 4. 2.2. Material Properties Ancient timber buildings are the cultural relics of a nation. It is prohibited for anyone to take materials from the standing ancient buildings. However, the material properties matter to the test results. In order to use materials similar to those from the ancient timber buildings, the researchers used Pinus sylvestris var. mongolica, lime and Xiaotingni brick to fabricate the specimen. Pinus sylvestris var. mongolica grows in Northeast China, and is a commonly used tree species in the repair and construction replacement of traditional timber structures. Lime was commonly used to make the mortar in ancient times. Xiaotingni brick is also widely used in the restoration of traditional buildings. Buildings 2022, 12, x FOR PEER REVIEW 5 of 23 Buildings 2022, 12, x FOR PEER REVIEW 5 of 23 Buildings 2022, 12, 887 5 of 22 Figure 3. Dimensions of the specimen (mm). Figure 3. Dimensions of the specimen (mm). Figure 3. Dimensions of the specimen (mm). (a) (a) (b) (b) Figure 4. Dimensions of the mortise and tenon joints (unit: mm): (a) mortise of the Bianting and Figure 4. Dimensions of the mortise and tenon joints (unit: mm): (a) mortise of the Bianting and Figure 4. Dimensions of the mortise and tenon joints (unit: mm): (a) mortise of the Bianting and tenon of the Zhongmo; (b) mortise of the Bianting and tenon of the Shangmo/Xiamo. tenon of the Zhongmo; (b) mortise of the Bianting and tenon of the Shangmo/Xiamo. tenon of the Zhongmo; (b) mortise of the Bianting and tenon of the Shangmo/Xiamo. Buildings 2022, 12, x FOR PEER REVIEW 6 of 23 2.2. Material Properties Ancient timber buildings are the cultural relics of a nation. It is prohibited for anyone to take materials from the standing ancient buildings. However, the material properties matter to the test results. In order to use materials similar to those from the ancient timber buildings, the researchers used Pinus sylvestris var. mongolica, lime and Xiaotingni brick to fabricate the specimen. Pinus sylvestris var. mongolica grows in Northeast China, and is a commonly used tree species in the repair and construction replacement of traditional timber structures. Lime was commonly used to make the mortar in ancient times. Xiaotingni brick is also widely used in the restoration of traditional buildings. Compressive strength tests were performed on nine cubic samples of traditional mor- tar, according to the standard test method for the performance of building mortar (JGJ/T Buildings 2022, 12, 887 70-2009 2009) [39]. The dimensions of the mortar sample were 70 mm × 70 mm × 70 6 mm of 22 (length × width × height). The mortar was made of lime and water. The mortar was poured into the molds and left to cure for 28 days at a temperature of 20 ± 2 °C and a relative air humidity of 95%. A universal testing machine was used to test the compressive strength Compressive strength tests were performed on nine cubic samples of traditional mortar, of the mortar samples. The average compressive strength of the mortar was 2.0 Mpa. according to the standard test method for the performance of building mortar (JGJ/T 70- Compressive strength tests were also carried out on the brick and masonry samples 2009 2009) [39]. The dimensions of the mortar sample were 70 mm  70 mm  70 mm according to the test method for wall bricks (GB/T 2542-2012) [40] and the standard test (length  width  height). The mortar was made of lime and water. The mortar was method for the basic mechanical properties of masonry (GB/T 50129-2011) [41]. Ten brick poured into the molds and left to cure for 28 days at a temperature of 20  2 C and a samples were made. The dimensions of the bricks were 75 mm × 75 mm × 60 mm (length relative air humidity of 95%. A universal testing machine was used to test the compressive × width × height). A universal testing machine was used to test the compressive strength strength of the mortar samples. The average compressive strength of the mortar was of the bricks. The average compressive strength of the brick samples was 9.3 Mpa. The 2.0 Mpa. dimensions of the prism in the compression tests including 12 layers of brick and 11 layers Compressive strength tests were also carried out on the brick and masonry samples accor of mort ding ar were 130 to the test mm method × 200 for mm wall × 415 mm ( bricks (GB/T length 2542-2012) × width × height [40] and ) (see the standar Figure 5) d. The test method averagefor com the pressiv basic e mechanical strength ofpr the operties masonry of s masonry amples w (GB/T as 3.4 50129-2011) Mpa. The av [er 41ag ]. T e Y enobrick ung’s samples modulus of the mas were made.onry sa The dimensions mples was 12 of 59 Mpa the bricks . Shwer ear e st75 ress mm test s were perf 75 mm  ormed on 60 mm (length nine tes t s width amples. The  height te). st A sample universal was c testing ompos machine ed of three layer was used s o tof brick test the and compr two layer essives str ofength morta of r. The di the bricks. mensi The ons aver of ta hg e e te comp st sampl ressive e were 130 strength mm of the × 200 brick mmsamples × 230 mm. was The whole 9.3 Mpa. The dimensions of the prism in the compression tests including 12 layers of brick and operation process was strictly controlled according to the Chinese National Standard 11 layers of mortar were 130 mm  200 mm  415 mm (length  width  height) (see (GB/T 50129-2011) [41] (see Figure 6). All of the samples were cured indoors at a temper- Figure 5). The average compressive strength of the masonry samples was 3.4 Mpa. The ature of 15–20 °C for 28 days. The test samples were then positioned on the center of the average Young’s modulus of the masonry samples was 1259 Mpa. Shear stress tests were test machine. Steel plates and sands were used to level up the bottom and top of the test performed on nine test samples. The test sample was composed of three layers of brick and sample. Vertical loads were applied by the testing machine, and the strength was calcu- two layers of mortar. The dimensions of the test sample were 130 mm  200 mm  230 mm. lated according to f = F/A (F is failure load and A is contact area). The average shear stress The whole operation process was strictly controlled according to the Chinese National of nine test samples was 0.041 Mpa (see Table 2). Standard (GB/T 50129-2011) [41] (see Figure 6). All of the samples were cured indoors at a temperature of 15–20 C for 28 days. The test samples were then positioned on the center Table 2. Mechanical properties of the mortar, brick and masonry (unit: Mpa). of the test machine. Steel plates and sands were used to level up the bottom and top of Compressive Strength Shear Strength of Young’s Modulus of the test sample. Vertical loads were applied by the testing machine, and the strength was Masonry Masonry Mortar Brick Masonry calculated according to f = F/A (F is failure load and A is contact area). The average shear 2.0 9.3 3.4 0.041 1259 stress of nine test samples was 0.041 Mpa (see Table 2). Buildings 2022, 12, x FOR PEER REVIEW 7 of 23 Figure 5. Compressive strength test of the masonry. Figure 5. Compressive strength test of the masonry. Figure 6. Shear stress test of the masonry. Figure 6. Shear stress test of the masonry. The material properties of the wood—including its modulus of elasticity, compres- sion strength density and moisture content—were tested, using the physical and mechan- ical methods for wood (GB 1927-1943-91) [42]. The dimensions of the wood specimens for the compression test and elastic modulus test were 30 mm × 20 mm × 20 mm and 300 mm × 20 mm × 20 mm. The mechanical properties of the wood are shown in Table 3. The load– deformation curves of the test samples are shown in Figure 7. Table 3. Mechanical properties of the wood. Parallel to Grain (MPa) Perpendicular to Grain (MPa) Moisture Density Compressive Compressive Elastic Compressive Elastic Content Elastic Modulus Strength Strength (T) Modulus (T) Strength (R) Modulus (R) 0.369 g/m 10.09% 46.21 8907 3.39 771 5.49 1620 T is the tangential direction and R is the radial direction. 120 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 0.0 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Deformation(mm) Deformation(mm) (a) (b) 2.0 20 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0123 4 0.00 0.25 0.50 0.75 1.00 1.25 1.50 Deformation(mm) Deformation(mm) (c) (d) Load(kN) Load(KN) Load(kN) Load(kN) Buildings 2022, 12, 887 7 of 22 Table 2. Mechanical properties of the mortar, brick and masonry (unit: Mpa). Compressive Strength Shear Strength of Young’s Modulus of Masonry Masonry Mortar Brick Masonry 2.0 9.3 3.4 0.041 1259 The material properties of the wood—including its modulus of elasticity, compression strength density and moisture content—were tested, using the physical and mechani- cal methods for wood (GB 1927-1943-91) [42]. The dimensions of the wood specimens for the compression test and elastic modulus test were 30 mm  20 mm  20 mm and 300 mm  20 mm  20 mm. The mechanical properties of the wood are shown in Table 3. The load–deformation curves of the test samples are shown in Figure 7. Table 3. Mechanical properties of the wood. Parallel to Grain (MPa) Perpendicular to Grain (MPa) Moisture Density Compressive Elastic Compressive Elastic Compressive Elastic Content Strength Modulus Strength (T) Modulus (T) Strength (R) Modulus (R) 0.369 g/m 10.09% 46.21 8907 3.39 771 5.49 1620 T is the tangential direction and R is the radial direction. Figure 7. Load–deformation curves of the test samples: (a) compressive strength test of the masonry samples; (b) shear stress test of the masonry samples; (c) compressive strength test of the wood samples (parallel to the grain); (d) compressive strength test of the wood samples (perpendicular to the grain, T); (e) compressive strength test of the wood samples (perpendicular to the grain, R). Buildings 2022, 12, x FOR PEER REVIEW 8 of 23 3.0 2.5 2.0 1.5 1.0 0.5 0.0 01 2345 Deformation(mm) (e) Figure 7. Load–deformation curves of the test samples: (a) compressive strength test of the masonry samples; (b) shear stress test of the masonry samples; (c) compressive strength test of the wood samples (parallel to the grain); (d) compressive strength test of the wood samples (perpendicular to Buildings 2022, 12, 887 8 of 22 the grain, T); (e) compressive strength test of the wood samples (perpendicular to the grain, R). 2.3. Testing and Measuring Schemes 2.3. Testing and Measuring Schemes Steel caps were positioned on the top of the columns. A distribution beam was then Steel caps were positioned on the top of the columns. A distribution beam was then positioned on the steel caps. Vertical loads of weight were hung at the two sides of the dis- positioned on the steel caps. Vertical loads of weight were hung at the two sides of the tribution beam, to simulate the dead-weight of the upper components and roof. The frame distribution beam, to simulate the dead-weight of the upper components and roof. The was installed in a vertical position with the bottom of the columns in hinge supports (see frame was installed in a vertical position with the bottom of the columns in hinge supports Figure 8). The vertical loads were calculated according to the relevant references [43–45]. (see Figure 8). The vertical loads were calculated according to the relevant references [43–45]. Using the similarity theory [46,47], the vertical loads placed on the 1/2 scale model were 12 Using the similarity theory [46,47], the vertical loads placed on the 1/2 scale model were kN. The displacement data were recorded using Displacement meter 3 in Figure 8. 12 kN. The displacement data were recorded using Displacement meter 3 in Figure 8. Figure 8. Schematic diagram of the loading equipment. Figure 8. Schematic diagram of the loading equipment. Cyclic loadings were applied using a hydraulic actuator with a displacement range of Cyclic loadings were applied using a hydraulic actuator with a displacement range 250 mm. The actuator was positioned at a height of 1.6 m, with a transverse cyclic load of ±250 mm. The actuator was positioned at a height of 1.6 m, with a transverse cyclic load with a programmed load cycle. A displacement meter (Displacement meter 3) and a force with a programmed load cycle. A displacement meter (Displacement meter 3) and a force sensor were used on the actuator. The force versus displacement curves were measured Buildings 2022, 12, x FOR PEER REVIEW 9 of 23 sensor were used on the actuator. The force versus displacement curves were measured directly. The horizontal cyclic loads were applied under displacement control according directly. The horizontal cyclic loads were applied under displacement control according to the ISO/FDIS 21581:2010(E) standards [48]. The frame was loaded with three initiation to the ISO/FDIS 21581:2010(E) standards [48]. The frame was loaded with three initiation cycles with an amplitude of 3 mm. The amplitude of the second three cycles was 5 mm. cycles with an amplitude of 3 mm. The amplitude of the second three cycles was 5 mm. The amplitude of the cycles gradually increased by a step of 5 mm until the amplitude The amplitude of the cycles gradually increased by a step of 5 mm until the amplitude was 70 mm. It then increased by a step of 10 mm until the amplitude reached 150 mm. was 70 mm. It then increased by a step of 10 mm until the amplitude reached 150 mm. Figure 9 presents the shape of the loading history. An overview of the test setup is shown Figure 9 presents the shape of the loading history. An overview of the test setup is shown in Figure 10. in Figure 10. -50 -100 -150 Figure 9. Loading scheme. Figure 9. Loading scheme. Figure 10. Test setup. 2.4. General Observation The Kanchuang frame was loaded until the displacement reached 150 mm (the drift angle was 3/32). Before the 10 mm (the drift angle was 1/160) cycles, no obvious damage was observed. Cracks primarily appeared in the side area of the masonry wall when the displacement reached 10 mm (see Figure 11). Then, the cracks developed further at an angle of 45° as the displacement increased. X-shaped cracks were formed when the dis- placement reached 50 mm (the drift angle was 1/32). Finally, cracks covered the whole wall and collapsed until the displacement reached 150 mm (the drift angle was 3/32). The tenons of the wood windows started to pull out of the mortise when the displacement reached 15 mm (drift angle is 3/320). Then, the gap gradually increased as the displace- ment increased. Finally, the gap reached 3 mm. Cracks appeared on the Lingtiao and Zaibian when the displacement reached 150 mm (the drift angle was 3/32). Figure 11 shows the final failure pattern. Δ/mm Load(kN) Buildings 2022, 12, x FOR PEER REVIEW 9 of 23 The amplitude of the cycles gradually increased by a step of 5 mm until the amplitude was 70 mm. It then increased by a step of 10 mm until the amplitude reached 150 mm. Figure 9 presents the shape of the loading history. An overview of the test setup is shown in Figure 10. -50 -100 -150 Buildings 2022, 12, 887 9 of 22 Figure 9. Loading scheme. Figure 10. Test setup. Figure 10. Test setup. 2.4. General Observation 2.4. General Observation The Kanchuang frame was loaded until the displacement reached 150 mm (the drift The Kanchuang frame was loaded until the displacement reached 150 mm (the drift angle was 3/32). Before the 10 mm (the drift angle was 1/160) cycles, no obvious damage angle was 3/32). Before the 10 mm (the drift angle was 1/160) cycles, no obvious damage was observed. Cracks primarily appeared in the side area of the masonry wall when was observed. Cracks primarily appeared in the side area of the masonry wall when the the displacement reached 10 mm (see Figure 11). Then, the cracks developed further at displacement reached 10 mm (see Figure 11). Then, the cracks developed further at an an angle of 45 as the displacement increased. X-shaped cracks were formed when the angle of 45° as the displacement increased. X-shaped cracks were formed when the dis- displacement reached 50 mm (the drift angle was 1/32). Finally, cracks covered the whole placement reached 50 mm (the drift angle was 1/32). Finally, cracks covered the whole wall and collapsed until the displacement reached 150 mm (the drift angle was 3/32). The wall and collapsed until the displacement reached 150 mm (the drift angle was 3/32). The tenons of the wood windows started to pull out of the mortise when the displacement tenons of the wood windows started to pull out of the mortise when the displacement reached 15 mm (drift angle is 3/320). Then, the gap gradually increased as the displacement reached 15 mm (drift angle is 3/320). Then, the gap gradually increased as the displace- increased. Finally, the gap reached 3 mm. Cracks appeared on the Lingtiao and Zaibian ment increased. Finally, the gap reached 3 mm. Cracks appeared on the Lingtiao and Buildings 2022, 12, x FOR PEER REVIEW 10 of 23 when the displacement reached 150 mm (the drift angle was 3/32). Figure 11 shows the Zaibian when the displacement reached 150 mm (the drift angle was 3/32). Figure 11 final failure pattern. shows the final failure pattern. (a) (b) (c) (d) (e) Figure 11. Cont. Δ/mm Buildings 2022, 12, 887 10 of 22 Buildings 2022, 12, x FOR PEER REVIEW 11 of 23 (f) Figure 11. Failure modes of the specimens: (a) crack on the left side of masonry wall at a displace- Figure 11. Failure modes of the specimens: (a) crack on the left side of masonry wall at a displacement ment of 10 mm (the drift angle was 1/160); (b) penetrating cracks on the right side of the masonry of 10 mm (the drift angle was 1/160); (b) penetrating cracks on the right side of the masonry wall at a wall at a displacement of 35 mm (the drift angle was 7/320); (c) the tenons pull out of mortise at a displacement of 35 mm (the drift angle was 7/320); (c) the tenons pull out of mortise at a displacement displacement of 15 mm (the drift angle was 3/320); (d) the tenons pull out of the mortise at a dis- of 15 mm (the drift angle was 3/320); (d) the tenons pull out of the mortise at a displacement of placement of 30 mm (the drift angle was 7/320); (e) cracks on the masonry wall at a displacement of 50 mm (the drift angle was 1/32); (f) failure modes of the specimens at a displacement of 150 mm 30 mm (the drift angle was 7/320); (e) cracks on the masonry wall at a displacement of 50 mm (the (the drift angle was 3/32). drift angle was 1/32); (f) failure modes of the specimens at a displacement of 150 mm (the drift angle was 3/32). 3. Analysis of the Characteristic Curves 3. Analysis of the Characteristic Curves This study developed a hysteresis model of Kanchuang frames. Moreover, this paper This study developed a hysteresis model of Kanchuang frames. Moreover, this pa- mainly aimed to build the hysteretic model of the Kanchuang frame. The structural per- per form mainly ance oaimed f the Kanch to build uanthe g fram hyster e, such etic a model s its stif offnes thes de Kanchuang gradationframe. and ene The rgystr diss uctural ipa- performance tion, were discusse of the Kanchuang d in reference [49]. F frame, such igure as s its 12 and stiffness 13 show degradation that the c and urves ener argy e sldissipa- ightly tion, unsymme were discussed trical, and in the r refer easons ence inc [49 lud ]. Figur e the es fol12 low and ing:13 show that the curves are slightly unsymmetrical, and the reasons include the following: (1) Although the frame is symmetrical theoretically, in practice, the frame is not exactly Buildings 2022, 12, 887 11 of 22 Buildings 2022, 12, x FOR PEER REVIEW 12 of 23 Buildings 2022, 12, x FOR PEER REVIEW 12 of 23 (1) Although the frame is symmetrical theoretically, in practice, the frame is not exactly symmetrical. The timber frame of ancient timber buildings is not symmetrical, as the symmetrical. The timber frame of ancient timber buildings is not symmetrical, as the symmetrical. The timber frame of ancient timber buildings is not symmetrical, as specimens were handmade by the timber building restoration people. There are dif- specimens were handmade by the timber building restoration people. There are dif- the specimens were handmade by the timber building restoration people. There are ferent sizes of small gaps between the components, especially between the mortise ferent sizes of small gaps between the components, especially between the mortise different sizes of small gaps between the components, especially between the mortise and tenon. and tenon. and tenon. (2) The cracks and damage of the components did not appear in an exactly symmetrical (2) The cracks and damage of the components did not appear in an exactly symmetrical (2) The cracks and damage of the components did not appear in an exactly symmetrical way, and the curves are unsymmetrical in different loading directions. way, and the curves are unsymmetrical in different loading directions. way, and the curves are unsymmetrical in different loading directions. (3) The horizontal cyclic loads were applied using a hydraulic actuator. The actuator was (3) The horizontal cyclic loads were applied using a hydraulic actuator. The actuator was (3) The horizontal cyclic loads were applied using a hydraulic actuator. The actuator was positioned at one side of the frame. It wasn’t positioned in the middle of the frame, positioned at one side of the frame. It wasn’t positioned in the middle of the frame, positioned at one side of the frame. It wasn’t positioned in the middle of the frame, which also made the curves unsymmetrical. which also made the curves unsymmetrical. which also made the curves unsymmetrical. Drift angle Drift angle -0.0938-0.0625-0.0313 0.0000 0.0313 0.0625 0.0938 -0.0938-0.0625-0.0313 0.0000 0.0313 0.0625 0.0938 6 9.6 6 9.6 4 6.4 4 6.4 2 3.2 2 3.2 0 0.0 0 0.0 -2 -3.2 -2 -3.2 -4 -6.4 -4 -6.4 -6 -9.6 -6 -9.6 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 Δ(mm) Δ(mm) Figure 12. Load–displacement hysteresis curves. Figure 12. Load–displacement hysteresis curves. Figure 12. Load–displacement hysteresis curves. Drift angle Drift angle -0.0938 -0.0625 -0.0313 0.0000 0.0313 0.0625 0.0938 -0.0938 -0.0625 -0.0313 0.0000 0.0313 0.0625 0.0938 6 10 6 10 The first cycle The first cycle The second cycle 4 The second cycle 6 4 6 The third cycle The third cycle 2 3 2 3 0 0 0 0 -2 -3 -2 -3 -4 -6 -4 -6 -6 -10 -6 -10 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 Δ (mm) Δ (mm) Figure 13. Load–displacement envelope curves. Figure 13. Load–displacement envelope curves. Figure 13. Load–displacement envelope curves. 3.1. Load–Displacement Hysteretic Curve 3.1. Load–Displacement Hysteretic Curve 3.1. Load–Displacement Hysteretic Curve Figure 12 shows the load–displacement hysteretic curves of the Kanchuang frame. Figure 12 shows the load–displacement hysteretic curves of the Kanchuang frame. Figure 12 shows the load–displacement hysteretic curves of the Kanchuang frame. The curves were Z-shaped, particularly at the ultimate state. The pinch phenomenon The curves were Z-shaped, particularly at the ultimate state. The pinch phenomenon was The curves were Z-shaped, particularly at the ultimate state. The pinch phenomenon was was also shown in the curve. The pinch effect was clear as the displacement and loading also shown in the curve. The pinch effect was clear as the displacement and loading cycle also shown in the curve. The pinch effect was clear as the displacement and loading cycle cycle increased; it was caused by the plastic deformation of the wood, the pull-out of the increased; it was caused by the plastic deformation of the wood, the pull-out of the mor- increased; it was caused by the plastic deformation of the wood, the pull-out of the mor- mortise–tenon joints, and the crack development of the masonry wall. The peak values tise–tenon joints, and the crack development of the masonry wall. The peak values of the tise–tenon joints, and the crack development of the masonry wall. The peak values of the of the hysteretic loops increased as the loading displacements increased. The bending hysteretic loops increased as the loading displacements increased. The bending capacities hysteretic loops increased as the loading displacements increased. The bending capacities capacities of each loading displacement in the first cycle were higher than those in the of each loading displacement in the first cycle were higher than those in the second cycle. of each loading displacement in the first cycle were higher than those in the second cycle. second cycle. Moreover, the bending capacity of the second cycle was higher than that of the Moreover, the bending capacity of the second cycle was higher than that of the third cycle. Moreover, the bending capacity of the second cycle was higher than that of the third cycle. third cycle. This was caused by the accumulation of the damage and strength degradation This was caused by the accumulation of the damage and strength degradation of the mor- This was caused by the accumulation of the damage and strength degradation of the mor- of the mortise–tenon joints and the masonry wall. tise–tenon joints and the masonry wall. tise–tenon joints and the masonry wall. P(kN) P(kN) P(kN) P(kN) M(kN⋅m) M(kN⋅m) M(kN⋅m) M(kN⋅m) Buildings 2022, 12, x FOR PEER REVIEW 13 of 23 Buildings 2022, 12, 887 12 of 22 3.2. Load–Displacement Envelope Curve Figure 13 shows the load–displacement envelope curve of the Kanchuang frame: 3.2. Load–Displacement Envelope Curve (1) In different displacement cycles, the strength of first cycle was found to be stronger Figure 13 shows the load–displacement envelope curve of the Kanchuang frame: than the second and third cycles. The strength degradation was caused by the crack (1) In different displacement cycles, the strength of first cycle was found to be stronger development and the plastic cumulative deformation of the wood components. than the second and third cycles. The strength degradation was caused by the crack (2) Both the positive and negative curves followed the same trend up to 25 mm (the drift development and the plastic cumulative deformation of the wood components. angle was 1/64) displacement. Before the 25 mm (the drift angle was 1/64) cycles, no (2) Both the positive and negative curves followed the same trend up to 25 mm (the drift obvious damage occurred in the timber components and no penetrating cracks angle was 1/64) displacement. Before the 25 mm (the drift angle was 1/64) cycles, formed on the masonry wall. The whole frame was still in the elastic stage. no obvious damage occurred in the timber components and no penetrating cracks (3) During the 30–65 mm (drift angle during 3/160 and 13/320) cycles, strength degrada- formed on the masonry wall. The whole frame was still in the elastic stage. tion began in both the left (negative) and right (positive) loading directions while the (3) During the 30–65 mm (drift angle during 3/160 and 13/320) cycles, strength degrada- load was increasing. This happened because of the failure of the masonry wall, and tion began in both the left (negative) and right (positive) loading directions while the the tenons were pulled out of the mortises. The penetrating cracks began to appear load was increasing. This happened because of the failure of the masonry wall, and the in the masonry wall, and the crannies became larger as the loading displacement in- tenons were pulled out of the mortises. The penetrating cracks began to appear in the creased. The gaps between mortise and tenon joints of the frame and wood windows masonry wall, and the crannies became larger as the loading displacement increased. increased as loading displacement increased. In the 65 mm (the drift angle was The gaps between mortise and tenon joints of the frame and wood windows increased 13/320) cycles, the left and right side of the masonry wall began to collapse. During as loading displacement increased. In the 65 mm (the drift angle was 13/320) cycles, the loading process, the cracks on the right side (the positive direction) of the ma- the left and right side of the masonry wall began to collapse. During the loading sonry wall grew faster than those on the left side (the negative direction) (see Figure process, the cracks on the right side (the positive direction) of the masonry wall grew 11e). Therefore, the strength degradation of the frame in the positive loading was faster than those on the left side (the negative direction) (see Figure 11e). Therefore, weaker than that in the negative direction. The whole frame entered the elastoplastic the strength degradation of the frame in the positive loading was weaker than that in deformation stage. the negative direction. The whole frame entered the elastoplastic deformation stage. (4) After the 65 mm (the drift angle was 13/320) cycles, the stiffness increased in both the (4) After the 65 mm (the drift angle was 13/320) cycles, the stiffness increased in both right (positive) and left (negative) directions. The stiffness was stronger than that of the right (positive) and left (negative) directions. The stiffness was stronger than that the 30–65 mm (the drift angle during 3/160 and 13/320) cycles but weaker than that of the 30–65 mm (the drift angle during 3/160 and 13/320) cycles but weaker than before the 30 mm (the drift angle was 3/160) cycles. This is because the masonry wall that before the 30 mm (the drift angle was 3/160) cycles. This is because the masonry failed after the 65 mm (the drift angle was 13/320) cycles, and the components of the wall failed after the 65 mm (the drift angle was 13/320) cycles, and the components timber frames were compressed with each other. The structure reached a new elastic– of the timber frames were compressed with each other. The structure reached a new plastic stage characterized by the increase of the stiffness and the residual defor- elastic–plastic stage characterized by the increase of the stiffness and the residual mation of the timber components. deformation of the timber components. Research [29] has been conducted on traditional timber frames with wood plane in- Research [29] has been conducted on traditional timber frames with wood plane fill. This frame has been used in ancient timber buildings. The test model was made of infill. This frame has been used in ancient timber buildings. The test model was made of Pinus sylvestris var. mongolica. Figure 14 presents the test model and hysteretic curves of Pinus sylvestris var. mongolica. Figure 14 presents the test model and hysteretic curves of the test model. The peak values of two hysteretic curves at different displacement (drift the test model. The peak values of two hysteretic curves at different displacement (drift angle) cycles are different between Figures 12 and 14. This is because the dimensions, angle) cycles are different between Figures 12 and 14. This is because the dimensions, structures, wood spices and loading positions are different. The peak values (in Figure structures, wood spices and loading positions are different. The peak values (in Figure 14b) 14b) decreased while the loading displacement exceeded 200 mm (the drift angle was decreased while the loading displacement exceeded 200 mm (the drift angle was 1/8). 6/85). However, the peak values of the Kanchuang frame’s hysteretic curves did not de- However, the peak values of the Kanchuang frame’s hysteretic curves did not decrease. crease. Both hysteretic curves turned into Z shapes a despite the different dimensions, Both hysteretic curves turned into Z shapes a despite the different dimensions, structures, structures, wood spices and loading processes. Both curves showed a degrading trend wood spices and loading processes. Both curves showed a degrading trend while the while the loading cycles increased. loading cycles increased. (a) Figure 14. Cont. Buildings 2022, 12, x FOR PEER REVIEW 14 of 23 Buildings 2022, 12, x FOR PEER REVIEW 14 of 23 Buildings 2022, 12, 887 13 of 22 (b) Figure 14. Figures from reference [29]: (a) Test model; (b) hysteretic curves. (b) 4. Hysteresis Model Figure 14. Figures from reference [29]: (a) Test model; (b) hysteretic curves. Figure 14. Figures from reference [29]: (a) Test model; (b) hysteretic curves. 4.1. Envelope Curve 4. Hysteresis Model 4. Hysteresis Model The envelope curve is an essential part of the hysteresis model. In order to build the 4.1. Envelope Curve hysteresis model, the data consisting of the envelope curves should be nondimensional- 4.1. Envelope Curve The envelope curve is an essential part of the hysteresis model. In order to build the Δ P ised. The - relationship curves were thus obtained through the dimensionless pro- The envelope curve is hyster an essent esis model, ial part the of data the h consisting ysteresis mode of thel. In envelope order curves to build th should e be nondimensionalised. cessing of the test data of the test specimen, as shown in Equations (1) and (2). The D-P relationship curves were thus obtained through the dimensionless processing of hysteresis model, the data consisting of the envelope curves should be nondimensional- the test data of the test specimen, as shown in Equations (1) and (2). Δ P +− ised. The - relationship curves were thus obtained through the dimensionless pro-  ΔΔ =/ Δ + Δ /2 (1) () mm  cessing of the test data of the test specimen, as shown in Equations (1) and (2). D = D/ D + D /2 (1) m m +−  +− PP =/ P + P /2 (2)  () mm ΔΔ =/ Δ + Δ /2 (1) () mm  +  P = P/ P + P /2 (2) m m ± ± Δ P Δ P m m D and P are the+− loading displacement and the corresponding force. D and P are m m and  are the loading displacement and the corresponding force. and PP =/ P + P /2 (2) () mm  displacements in both the right (positive) direction and the left (negative) direction of the are displacements in both the right (positive) direction and the left (negative) direction of 65 mm cycles and the corresponding forces. Points O1, O2, A, B, D and E are vital points, ± ± the 65 mm cycles and the corresponding forces. Points O1, O2, A, B, D and E are vital Δ P Δ P m m and are the loading displacement and the corresponding force. and and points C and F are ultimate loading points (see Figure 15). It was found that the shapes points, and points C and F are ultimate loading points (see Figure 15). It was found that are displacements in both the right (positive) direction and the left (negative) direction of of the dimensionless envelope curves are close to the experimental data points. Table 4 the shapes of the dimensionless envelope curves are close to the experimental data points. the 65 mm cycles and the corresponding forces. Points O1, O2, A, B, D and E are vital presents the coordinates of the critical points and ultimate loading points; Table 5 shows Table 4 presents the coordinates of the critical points and ultimate loading points; Table 5 points, and points C and F are ultimate loading points (see Figure 15). It was found that the regression equations of the fitting envelope curves. The fitted envelope curves (see shows the regression equations of the fitting envelope curves. The fitted envelope curves the shapes of the dimensionless envelope curves are close to the experimental data points. Figure 15d) indicated a strength degradation trend in the different cycle processes, as is (see Figure 15d) indicated a strength degradation trend in the different cycle processes, as Table 4 presents the coord consistent inates of the cr with it the icatest l por in esults. ts and ultimate loading points; Table 5 is consistent with the test results. shows the regression equations of the fitting envelope curves. The fitted envelope curves (see Figure 15d) indicated a strength degradation trend in the different cycle processes, as is consistent with the test results. Experimental value Experimental value Theoretical value D 2 Theoretical value E Experimental value -3 -2 -1 0 1 2 3 Experimental value Theoretical value -3 -2 -1 0 1 2 3 Theoretical value O E 2 B -1 -1 A O O O A -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 2 -2 O -2 2 B -1 C A -1 -3 -3 -2 (a) (b) -2 Figure 15. Cont. -3 -3 (a) (b) P Buildings 2022, 12, 887 14 of 22 Buildings 2022, 12, x FOR PEER REVIEW 15 of 23 2 2 The first cycle Experimental value 1 1 The second cycle Theoretical value DD The third cycle 0 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -1 -1 -2 -2 -3 -3 (c) (d) Figure 15. Fitting curves of the skeleton curves of the Kanchuang frame: (a) the first cycle; (b) the Figure 15. Fitting curves of the skeleton curves of the Kanchuang frame: (a) the first cycle; (b) the second cycle; (c) the third cycle; (d) fitting envelope curves of three cycles. second cycle; (c) the third cycle; (d) fitting envelope curves of three cycles. Table 4. Coordinate critical points and ultimate loading points of the Kanchuang frame. Table 4. Coordinate critical points and ultimate loading points of the Kanchuang frame. Points O2 A B C O1 D E F Points O A B C O D E F 2 −0.046 −0.464 −0.998 1−2.301 0.046 0.462 1.002 2.311 The first cycle −0.187 −0.913 −1.007 −2.511 0.334 0.878 1.011 1.906 D 0.046 0.464 P 0.998 2.301 0.046 0.462 1.002 2.311 The first cycle 0.187 0.913 1.007 2.511 0.334 0.878 1.011 1.906 P −0.046 −0.461 −1.001 −2.309 0.046 0.459 1.005 2.304 The second cycle D 0.046 0.461 1.001 2.309 0.046 0.459 1.005 2.304 −0.173 −0.856 −1.051 −2.337 0.332 0.778 0.972 1.933 The second cycle P 0.173 0.856 1.051 2.337 0.332 0.778 0.972 1.933 −0.046 −0.461 −1.001 −2.309 0.046 0.459 1.005 2.304 The third 0.046 cycle 0.461 1.001 2.309 0.046 0.459 1.005 2.304 The third cycle −0.176 −0.856 −1.051 −2.337 0.303 0.778 0.972 1.933 P 0.176 0.856 1.051 2.337 0.303 0.778 0.972 1.933 Table 5. Equations of the envelope curves of the Kanchuang frame. Table 5. Equations of the envelope curves of the Kanchuang frame. Regression Equations Line The First Cycle The Se Regression cond CycleEquations The Third Cycle Line P =1.642Δ− 0.099 P =1.546Δ− 0.105 The First Cycle The Second Cycle The Third Cycle OA P =1.741Δ− 0.107 P =1.642Δ− 0.099 P =1.546Δ− 0.105 OA P= 1.741D 0.107 P= 1.642D 0.099 P= 1.546D 0.105 AB P=Δ 0.214 −P 0.814 = 0.214D 0.814 P=Δ 0.361 P=−0.361 0.690D 0.690 P=Δ 0.397 P= 0.397−D 0.6 30.636 6 AB BC P= 1.007D 0.555 P= 0.983D 0.066 P= 0.957D 0.076 P =0.214Δ− 0.814 P =0.361Δ− 0.690 P =0.397Δ− 0.636 OD P= 1.309D + 0.273 P= 1.080D + 0.282 P= 1.069D + 0.253 P=Δ 1.007 + 0.555 P=Δ 0.983 − 0.066 P=Δ 0.957 − 0.076 DE P= 0.246D + 0.765 P= 0.354D + 0.615 P= 0.333D + 0.595 BC EF P =1.007Δ− 0.555 P =0.983Δ− 0.066 P =0.957Δ− 0.076 P= 0.786D + 0.198 P= 0.740D + 0.228 P= 0.704D + 0.225 P=Δ 1.309 + 0.273 P=Δ 1.080 + 0.282 P=Δ 1.069 + 0.253 OD 4.2. Stiffness of the Hysteretic Loops P =1.309Δ+0.273 P =1.080Δ+0.282 P =1.069Δ+0.253 The stepsPof=Δ the 0.246 investigation + 0.765 of the stif P=Δ fness 0.354 of the+ 0.61 hyster 5 etic loops P=Δ wer 0.333 e followed. + 0.595 First, DE the characters of the hysteretic loops were analyzed in order to investigate the changing P =0.246Δ+0.765 P =0.354Δ+0.615 P =0.333Δ+0.595 rules of the stiffness. The findings show that the hysteretic loops of the Kanchuang frame P=Δ 0.786 + 0.198 P=Δ 0.740 + 0.228 P=Δ 0.704 + 0.225 EF were in the elastic stage when the loading displacements were under 10 mm cycles. If P =0.786Δ+0.198 P =0.740Δ+0.228 P =0.704Δ+0.225 the loading displacements reached and exceeded 15 mm (the drift angle was 3/320), each hysteretic loop was divided into eight feature segments, and M, L, N, O, P and Q were the 4.2. Stiffness of the Hysteretic Loops intersection points of each segment (see Figure 16). K was used to represent the stiffness The steps of the investigation of the stiffness of the hysteretic loops were followed. of each feature segment (see Table 6). Secondly, dimensionless processing was conducted First, the characters of the hysteretic loops were analyzed in order to investigate the on the experimental data of the hysteretic loops based on Equations (1) and (2). Third, the changing rules of the stiffness. The findings show that the hysteretic loops of the Kan- regression analysis of dimensionless data points was obtained from the fitting equations chuang frame were in the elastic stage when the loading displacements were under 10 of K (see Table 7). Because of the discreteness of the data of K and K , no appropriate i 3 7 mm cycles. If the loading displacements reached and exceeded 15 mm (the drift angle was fitting equations were found. Average values were used and adopted to build the hysteresis 3/320), each hysteretic loop was divided into eight feature segments, and M, L, N, O, P model. Figure 17 shows the fitted curves, and it was found that the fitting curves simulated the experimental data. i and Q were the intersection points of each segment (see Figure 16). was used to P Buildings 2022, 12, x FOR PEER REVIEW 16 of 23 represent the stiffness of each feature segment (see Table 6). Secondly, dimensionless pro- cessing was conducted on the experimental data of the hysteretic loops based on Equa- tions (1) and (2). Third, the regression analysis of dimensionless data points was obtained from the fitting equations of (see Table 7). Because of the discreteness of the data of K K 3 7 and , no appropriate fitting equations were found. Average values were used and adopted to build the hysteresis model. Figure 17 shows the fitted curves, and it was found Buildings 2022, 12, 887 15 of 22 that the fitting curves simulated the experimental data. Figure 16. Feature segments of hysteresis curves. Figure 16.Feature segments of hysteresis curves. Table 6. Parameters of each feature segment. Table 6. Parameters of each feature segment. Loading Loading Rapid Unloading Loading Loading Rapid Unloading Rapid Rapid Feat Feature ure Loading Loading Strengthen Unloading Loading Loading Strengthen Unloading Elastic Strengthen Un Unloading loading Elastic Elastic Strengthen Unloading Unloading Elastic Segme Segment nt Elastic Stage 1 Stage 1 Elastic Stage 1 Elastic Stage 2 Stage 2 Elastic Stage 2 Stage 1 Stage 1 SStage tage 1 1 Stage 1 Stage 2 Stage 2 Stage Stage 2 2 Stage 2 Represent Represent K K K K K K K K 1 2 3 4 5 6 7 8 K K K K K K K K parameter 1 2 3 4 5 6 7 8 parameter Table 7. Fitting equations of the stiffness. Table 7. Fitting equations of the stiffness. Feature Segment Fitting Equations Feature Segment Fitting Equations −0.476 Loading elastic stage 1 K =0.424Δ+ 0.103 0.476 Loading elastic stage 1 K = 0.424D + 0.103 1 −3.186 K =0.021Δ+1.302 Loading strengthen stage 1 3.186 Loading strengthen stage 1 K = 0.021D + 1.302 K =12.52 Rapid unloading stage 1 Rapid unloading stage 1 K = 12.52 −0.319 K =1.192Δ− 0.975 Unloading elastic stage 1 0.319 Unloading elastic stage 1 K = 1.192D 0.975 −0.186 Loading elastic stage 2 K =2.07Δ−1.511 0.186 Loading elastic stage 2 K = 2.07D 1.511 −− 4 4.905 K =8.183×Δ 10 + 1.585 Loading strengthen stage 2 4.905 6 4 Loading strengthen stage 2 K = 8.183 10 D + 1.585 K =12.92 Rapid unloading stage 2 Rapid unloading stage 2 K = 12.92 −0.020 0.020 K =27.140Δ− 26.98 Unloading elastic stage 2 Unloading elastic stage 2 8 K = 27.140D 26.98 Based upon the analysis on the characteristics of the hysteretic loops, Equation (2) calculated the fitting curves between the displacement cycles and horizontal ordinates of the intersection points. 4M, 4L, 4N, 4O, 4P and 4Q are the dimensionless horizontal ordinates of the intersection points. Linear regression analysis was conducted on the points, and the fitting equations are shown in Table 8. The fitted curves are shown in Figure 18, and the data are close to the fitting line. Table 8. Fitting equations of points M, L, N, O, Pand Q. Points Fitting Equations D = 0.619D + 0.136 P D = 0.031D + 0.360 Q D = 0.950D 0.023 M D = 0.894D 0.017 L D = 0.048D 0.330 D = 0.618D 0.013 N Buildings 2022, 12, 887 16 of 22 Buildings 2022, 12, x FOR PEER REVIEW 17 of 23 4.0 3.5 3.0 −3.186 −0.476 2.5 K =Δ 0.021 + 1.302 K=Δ 0.424 + 0.103 2 3 R = 0.848 2.0 R = 0.953 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Δ Δ (a) (b) −0.319 K =Δ 1.192 − 0.975 K = 12.52 R = 0.994 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 (c) (d) 2.5 4.0 3.5 2.0 −0.186 3.0 −− 44.905 K =Δ 2.07 − 1.511 K =× 8.183 10 Δ + 1.585 1.5 R = 0.955 R = 0.598 2.5 1.0 2.0 0.5 1.5 0.0 1.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 (e) (f) 20 3.0 18 2.5 −0.020 K=Δ 27.140 − 26.98 16 2.0 R = 0.934 14 K = 12.92 1.5 12 1.0 10 0.5 0.0 6 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Δ Δ (g) (h) Figure 17. Stiffness degradation curves and fitting equations of the Kanchuang frame at different loading states: (a) K ; (b) K ; (c) K ; (d) K ; (e) K ; (f) K ; (g) K ; (h) K . 1 2 3 4 5 6 7 8 K K 7 5 K 1 6 K 4 K 2 Buildings 2022, 12, x FOR PEER REVIEW 18 of 23 Figure 17. Stiffness degradation curves and fitting equations of the Kanchuang frame at different K K K K K K K K 1 2 3 4 5 6 7 8 loading states: (a) ; (b) ; (c) ; (d) ; (e) ; (f) ; (g) ; (h) . Based upon the analysis on the characteristics of the hysteretic loops, Equation (2) calculated the fitting curves between the displacement cycles and horizontal ordinates of the intersection points. △M, △L, △N, △O, △P and △Q are the dimensionless horizontal ordinates of the intersection points. Linear regression analysis was conducted on the points, and the fitting equations are shown in Table 8. The fitted curves are shown in Fig- ure 18, and the data are close to the fitting line. Table 8. Fitting equations of points M, L, N, O, Pand Q. Points Fitting Equations O Δ− = 0.619Δ+ 0.136 ΔΔ =0.031+ 0.360 ΔΔ =0.950− 0.023 M ΔΔ =-0.894− 0.017 Buildings 2022, 12, 887 Δ− = 0.048Δ− 0.330 17 of 22 N ΔΔ =0.618− 0.013 0.0 0.43 -0.2 Δ= 0.031Δ+ 0.360 0.42 R = 0.920 -0.4 -0.6 0.41 -0.8 0.40 -1.0 Δ= −0.619Δ+ 0.136 0.39 R = 0.955 -1.2 0.38 -1.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0.0 0.5 1.0 1.5 2.0 2.5 (a) (b) 0.0 2.0 -0.2 -0.4 Δ= 0.950Δ− 0.023 Δ= −0.894Δ− 0.017 Q M -0.6 R = 0.998 R = 0.992 1.5 -0.8 -1.0 1.0 -1.2 -1.4 -1.6 0.5 -1.8 -2.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Buildings 2022, 12, x FOR PEER REVIEW 19 of 23 Δ Δ (c) (d) 1.4 -0.35 -0.36 1.2 Δ= 0.618Δ− 0.013 -0.37 R = 0.955 Δ= −0.048Δ− 0.330 1.0 -0.38 R = 0.794 0.8 -0.39 0.6 -0.40 -0.41 0.4 -0.42 0.2 -0.43 0.0 -0.44 0.0 0.5 1.0 1.5 2.0 2.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 (e) (f) Figure 18. Fitting curves of the piecewise points: (a) Point O; (b) Point P; (c) Point Q; (d) Point M; Figure 18. Fitting curves of the piecewise points: (a) Point O; (b) Point P; (c) Point Q; (d) Point M; (e) Point L; (f) Point N. (e) Point L; (f) Point N. 4.3. Hysteretic Rule 4.3. Hysteretic Rule The hysteretic model plays a crucial role in the nonlinear seismic response analysis of The hysteretic model plays a crucial role in the nonlinear seismic response analysis structures. The seismic response analysis results may be diverse when different hysteretic of structures. The seismic response analysis results may be diverse when different hyster- models are adopted. Different kinds of hysteretic model have been widely used to simulate etic models are adopted. Different kinds of hysteretic model have been widely used to the hysteretic behaviours of different structures. However, it was found that these models simulate the hysteretic behaviours of different structures. However, it was found that cannot simulate the strength degradation and the pinch phenomenon of the hysteretic these models cannot simulate the strength degradation and the pinch phenomenon of the loops of the Kanchuang frame. A new hysteretic model was proposed to simulate the hysteretic loops of the Kanchuang frame. A new hysteretic model was proposed to simu- hysteretic loops of the Kanchuang frame (see Figure 19). late the hysteretic loops of the Kanchuang frame (see Figure 19). The hysteresis rule is as follows. The loading path follows the envelope curves to point H first. Then, unloading along the paths H–Q with stiffnesses of . And unload- ing along path Q–P . Following this, the curve enters a new stage, loading along paths K K 5 6 P–O and O–G with stiffnesses of and . G is the peak point of the hysteretic loop. The negative unloading is then following along paths G–M and M–L with stiffnesses of K K K 7 8 1 and . It then loads along path L-N with the stiffness of . When the displace- ment load exceeds point N, the loading stiffness changes to , loads to the positive peak value H, and the first loading cycle finishes. After the loading displacement exceeds the initial loading displacement, its loading along the envelope curve is from point H to H’, and enters another loading cycle—H’-Q’-P’-O’-G’-M’-L’-N’ -H’—with stiffnesses of , K K K K K K K 2 3 4 5 6 7 8 , , , , , and . This is similar to the previous cycle. H ′ N ′ L′ Q ′ L Q M P′ O′ G′ Figure 19. Hysteretic model. L Δ Q Δ Δ Δ N P M Buildings 2022, 12, x FOR PEER REVIEW 19 of 23 1.4 -0.35 -0.36 1.2 Δ= 0.618Δ− 0.013 R = 0.955 -0.37 Δ= −0.048Δ− 0.330 1.0 -0.38 R = 0.794 0.8 -0.39 -0.40 0.6 -0.41 0.4 -0.42 0.2 -0.43 0.0 -0.44 0.0 0.5 1.0 1.5 2.0 2.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 (e) (f) Figure 18. Fitting curves of the piecewise points: (a) Point O; (b) Point P; (c) Point Q; (d) Point M; (e) Point L; (f) Point N. 4.3. Hysteretic Rule The hysteretic model plays a crucial role in the nonlinear seismic response analysis of structures. The seismic response analysis results may be diverse when different hyster- etic models are adopted. Different kinds of hysteretic model have been widely used to simulate the hysteretic behaviours of different structures. However, it was found that these models cannot simulate the strength degradation and the pinch phenomenon of the hysteretic loops of the Kanchuang frame. A new hysteretic model was proposed to simu- late the hysteretic loops of the Kanchuang frame (see Figure 19). The hysteresis rule is as follows. The loading path follows the envelope curves to point H first. Then, unloading along the paths H–Q with stiffnesses of . And unload- ing along path Q–P . Following this, the curve enters a new stage, loading along paths K K 5 6 P–O and O–G with stiffnesses of and . G is the peak point of the hysteretic loop. The negative unloading is then following along paths G–M and M–L with stiffnesses of K K K 7 8 1 and . It then loads along path L-N with the stiffness of . When the displace- ment load exceeds point N, the loading stiffness changes to , loads to the positive peak value H, and the first loading cycle finishes. After the loading displacement exceeds the initial loading displacement, its loading along the envelope curve is from point H to H’, Buildings 2022, 12, 887 and enters another loading cycle—H’-Q’-P’-O’-G’-M’-L’-N’ -H’—with stiffnesses of 18 of 22, K K K K K K K 2 3 4 5 6 7 8 , , , , , and . This is similar to the previous cycle. L′ Q L Q′ M ′ M P′ Figure 19. Hysteretic model. Figure 19. Piecewise hysteresis curves. The hysteresis rule is as follows. The loading path follows the envelope curves to point H first. Then, unloading along the paths H–Q with stiffnesses of K . And unloading along path Q–P K . Following this, the curve enters a new stage, loading along paths P–O and O–G with stiffnesses of K and K . G is the peak point of the hysteretic loop. The negative 5 6 unloading is then following along paths G–M and M–L with stiffnesses of K and K . It then 7 8 loads along path L-N with the stiffness of K . When the displacement load exceeds point N, the loading stiffness changes to K , loads to the positive peak value H, and the first loading cycle finishes. After the loading displacement exceeds the initial loading displacement, its loading along the envelope curve is from point H to H’, and enters another loading cycle—H’-Q’-P’-O’-G’-M’-L’-N’ -H’—with stiffnesses of K ,K ,K ,K ,K ,K ,K and K . This 1 2 3 4 5 6 7 8 is similar to the previous cycle. 5. Comparison of the Hysteretic Curves between the Analysis and Test The restoring force model and the experimental loading process were used to obtain the calculated hysteresis curves. Figure 20 shows the comparations of the experimental results and the calculation results of the Kanchuang frame. The test curves are coloured in blue, and the calculation results are red. It was found that the calculation curves are similar Buildings 2022, 12, x FOR PEER REVIEW 21 of 24 to the experimental curves; thus, the hysteretic model simulates the strength degradation, stiffness degradation and pinching effect of the experimental curves. Based on the model, the nonlinear seismic analysis of the traditional timber structure can be used. (a) Figure 20. Cont. (b) (c) Figure 20. Comparison between the calculation results and the experimental results: (a) first dis- placement cycle; (b) second displacement cycle; (c) third displacement cycle. 6. Conclusions A 1/2-scale model of a timber frame with a timber window and infilled masonry wall was tested. The failure modes, stress and stiffness of the structure were investigated. Moreover, the characters of the envelope curves and hysteretic loops were studied. Based on the results of this research, the following conclusions can be drawn: (1) The test results show that the loading process can be divided into three stages. At the elastic stage, no apparent damage was found through observation except for slight cracks on the wall. During the elastoplastic stage, the cracks on the wall became N Buildings 2022, 12, x FOR PEER REVIEW 21 of 24 Buildings 2022, 12, 887 19 of 22 (a) (b) (c) Figure 20. Comparison between the calculation results and the experimental results: (a) first dis- Figure 20. Comparison between the calculation results and the experimental results: (a) first displace- placement cycle; (b) second displacement cycle; (c) third displacement cycle. ment cycle; (b) second displacement cycle; (c) third displacement cycle. 6. Conclusions 6. Conclusions A 1/2-scale model of a timber frame with a timber window and infilled masonry A 1/2-scale model of a timber frame with a timber window and infilled masonry wall wall was tested. The failure modes, stress and stiffness of the structure were investigated. was tested. The failure modes, stress and stiffness of the structure were investigated. Moreover, the characters of the envelope curves and hysteretic loops were studied. Based Moreover, the characters of the envelope curves and hysteretic loops were studied. Based on the results of this research, the following conclusions can be drawn: on the results of this research, the following conclusions can be drawn: (1) The test results show that the loading process can be divided into three stages. At the (1) The test results show that the loading process can be divided into three stages. At the elastic stage, no apparent damage was found through observation except for slight elastic stage, no apparent damage was found through observation except for slight cracks on the wall. During the elastoplastic stage, the cracks on the wall became longer, cracks on the wall. During the elastoplastic stage, the cracks on the wall became wider and deeper as the loading displacement increased, and the gaps between the mortises and tenons steadily increased as the loading displacement increased. During the final new elastoplastic stage, the stiffness of the whole structure increased after the masonry wall collapsed. Brittle shear failure was observed in the masonry infill wall. At the end of the test, the masonry wall collapsed, but the timber frames did not fall apart. Slight cracks also appeared on the surface of the Lingtiao. (2) The pinching effect was observed from the hysteretic loops of the Kanchuang frame, indicating an occurrence of a slip between the timber components. The bearing capacity and stiffness of the frame were decreased but not lost, showing that the timber frame has good bearing and deformation capacities. (3) A dimensionless hysteretic model for the Kanchuang frame was established based on test results and numerical analysis. This model simulates the experimental curves’ strength degradation, stiffness degradation and pinching effect. The calculation results were consistent with the experimental results. They provide references for dynamic analyses of the traditional timber structure under dynamic loads. Buildings 2022, 12, 887 20 of 22 (4) This study provided a useful reference for the seismic evaluation and preservation of cultural heritage. This study also conducted dynamic analyses of the traditional timber structure under dynamic loads. However, in this study, the experimental and analytical studies were carried out on a scaled specimen, not a full-scaled one. Thus, further studies will be conducted to investigate whether this hysteretic model could apply to other frames of different dimensions. Moreover, the seismic and hysteretic behaviours of more timber frames with different types of infilled walls from ancient timber buildings will be studied. Author Contributions: Conceptualization, J.H. and X.G.; methodology, J.H. and X.G.; formal anal- ysis, J.H., X.G. and Z.G.; investigation, J.H., X.G., Z.G., T.Y., T.C. and Z.S.; writing—original draft preparation, J.H.; writing—review and editing, X.G., Z.G., T.Y., T.C. and Z.S.; supervision, X.G.; project administration, X.G.; funding acquisition, X.G. All authors have read and agreed to the published version of the manuscript. Funding: This study was financially supported by the National Key R&D Program of China (Grant Number 2019YFC1520803) and the Beijing Municipal Commission of Education–Municipal Natural Science Joint Foundation: “Research on Seismic Performance Evaluation of Beijing Ancient Timber Buildings Based on Value and damage Characteristics” (No. KZ202010005012). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The support of the National Key R&D Program of China (Grant Number 2019YFC1520803) and the Beijing Municipal Commission of Education–Municipal Natural Science Joint Foundation (Grant number KZ202010005012) is highly appreciated. Conflicts of Interest: The authors declare no conflict of interest. References 1. Zhou, Q.; Yan, W.M.; Yang, X.S.; Bao, J.J. Damage of ancient Chinese architecture caused by the Wenchuan earthquake. Sci. Conserv. Archaeol. 2010, 22, 37–45. 2. 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Safety State evaluation method based on attribute recognition model for ancient timber buildings. Adv. Civ. Eng. 2019, 2019, 3612535. [CrossRef] 22. Xie, Q.; Wang, L.; Zhang, L.; Hu, W.; Zhou, T. Seismic behaviour of a traditional timber structure: Shaking table tests, energy dissipation mechanism and damage assessment model. Bull. Earthq. Eng. 2019, 17, 1689–1714. [CrossRef] 23. Vieux-Champagne, F.; Sieffert, Y.; Grange, S.; Polastri, A.; Ceccotti, A.; Daudeville, L. Experimental analysis of seismic resistance of timber-framed structures with stones and earth infill. Eng. Struct. 2014, 69, 102–165. [CrossRef] 24. Vieux-Champagne, F.; Sieffert, Y.; Grange, S.; Nko’Ol, C.B.; Bertrand, E.; Duccini, J.C.; Faye, C.; Daudeville, L. Experimental analysis of a shake table test of timber-framed structures with stones and earth infill. Earthq. Spectra 2017, 33, 1075–1100. [CrossRef] 25. Ali, Q.; Schacher, T.; Ashraf, M.; Naeem, A.; Ahmad, N.; Umar, M. 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China Standards Press: Beijing, China, 2010. 49. Huan, J.H.; Guo, X.D.; Ma, D.H.; Guan, Z.Z. Seismic performance and damage evaluation of Kanchuang frame of ancient architecture. J. Civ. Environ. Eng. 2022, 44, 129–137. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Buildings Multidisciplinary Digital Publishing Institute

An Experimental Study of the Hysteresis Model of the Kanchuang Frame Used in Chinese Traditional Timber Buildings of the Qing Dynasty

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buildings Article An Experimental Study of the Hysteresis Model of the Kanchuang Frame Used in Chinese Traditional Timber Buildings of the Qing Dynasty 1 , 2 2 , 3 , 1 4 1 1 Junhong Huan , Xiaodong Guo * , Zhongzheng Guan , Teliang Yan , Tianyang Chu and Zemeng Sun School of Civil Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China; junhong_love@126.com (J.H.); guanzhongzheng@stdu.edu.cn (Z.G.); chuty1998@163.com (T.C.); sunzemeng19960326@163.com (Z.S.) Faculty of Architecture, Civil and Transportation Engineering, Beijing University of Technology, Beijing 100124, China Key Scientific Research Base of Safety Assessment and Disaster Mitigation for Traditional Timber Structure (Beijing University of Technology), State Administration for Cultural Heritage, Beijing 100124, China Beijing ZAJ Engineering Design Co., Ltd., No. 3 Chengxiangshiji Square, Beijing 100176, China; teliangyan@126.com * Correspondence: gxd@bjut.edu.cn Abstract: Kanchuang frames are important parts of traditional timber architecture in China. This paper used experimental and numerical methods to study the restoring force model of Kanchuang frames, which were used frequently in Chinese ancient timber structures, particularly in North China. The prototyped test model is a type of Chinese traditional timber architecture named Qilinyingshan. It was widely used in ancient timber buildings preserved from the Ming and Qing dynasties. This study analyzed the loading process and failure modes of the test model, and the skeleton curve and hysteretic curve data were collected. Moreover, a dimensionless skeleton curve model was developed Citation: Huan, J.; Guo, X.; Guan, Z.; based upon the findings. The hysteresis loops of the test model were also analyzed, and it was found Yan, T.; Chu, T.; Sun, Z. An that each hysteresis loop can be divided into several feature segments according to their stiffness Experimental Study of the Hysteresis at different loading stages. Regression analysis was also used to obtain the stiffness degradation Model of the Kanchuang Frame Used curvilinear equations of the feature segments. Finally, a hysteresis force model of a Kanchuang frame in Chinese Traditional Timber Buildings of the Qing Dynasty. was established. This study also found that the loading process can be divided into three stages: the Buildings 2022, 12, 887. https:// elastic stage, in which all of the components are in good condition; the elastic–plastic stage, in which doi.org/10.3390/buildings12070887 cracks gradually develop on the wall; and the new elastic–plastic stage, after which the wall collapses. It was found there was consistency between the restoring force model and the test results, indicating Academic Editors: Mahmud Ashraf that the model is valid and reliable. The skeleton curve model and hysteretic model provide reference and Wen-Shao Chang for the nonlinear seismic response of ancient timber architecture. Received: 19 April 2022 Accepted: 20 June 2022 Keywords: Kanchuang frame; restoring force model; ancient timber architecture; loading process Published: 23 June 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- 1. Introduction iations. Ancient timber buildings are considered cultural treasures of China. They represent Chinese history and culture. However, during the previous years, earthquakes have damaged or even destroyed a large number of traditional timber buildings. For example, nearly 10,000 traditional timber buildings were damaged in the Wenchuan earthquake in Copyright: © 2022 by the authors. Sichuan on 12 May 2008 [1]. The architecture of traditional Chinese timber buildings is Licensee MDPI, Basel, Switzerland. different from the architecture of modern buildings. In detail, in the structure system of This article is an open access article traditional Chinese timber buildings, columns are directly placed on plinths without any distributed under the terms and other fastening measures. Moreover, beams and columns are connected by mortise–tenon conditions of the Creative Commons joints without any nails and bracings, and all of the mass from the roof is loaded onto Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ the beams and columns. There are no weights on the infill walls [2,3]. Timber frames 4.0/). are usually the main-load bearing and force-resisting units of traditional Chinese timber Buildings 2022, 12, 887. https://doi.org/10.3390/buildings12070887 https://www.mdpi.com/journal/buildings Buildings 2022, 12, x FOR PEER REVIEW 2 of 23 Buildings 2022, 12, 887 2 of 22 no weights on the infill walls [2,3]. Timber frames are usually the main-load bearing and force- resisting units of traditional Chinese timber structures, which is different from modern struc- tures. It is thus of great importance to preserve these valuable cultural heritages by investigat- structures, which is different from modern structures. It is thus of great importance to ing the mechanical properties of the structure and components. preserve these valuable cultural heritages by investigating the mechanical properties of the structure and components. 1.1. History and Research of Chinese Traditional Structures 1.1. History and Research of Chinese Traditional Structures The previous historical literature on the earliest works of Chinese timber architecture The previous historical literature on the earliest works of Chinese timber architecture track its history to 5000 years ago. The most ancient known book about the construction track its history to 5000 years ago. The most ancient known book about the construction method of Chinese traditional timber buildings was published in the Song Dynasty [4]. method of Chinese traditional timber buildings was published in the Song Dynasty [4]. During recent modern times, many scholars have realized that this kind of cultural herit- During recent modern times, many scholars have realized that this kind of cultural heritage age is extremely valuable, and have begun to study and protect it. For example, in the is extremely valuable, and have begun to study and protect it. For example, in the 1930s, 1930s, Liang [5] took the lead in the study of the ancient timber buildings, and his studies Liang [5] took the lead in the study of the ancient timber buildings, and his studies were were focused on the architecture and anatomy of Chinese structures. Other scholars found focused on the architecture and anatomy of Chinese structures. Other scholars found that that the mortise–tenon joints are the weak points of timber structures, and thus studied the mortise–tenon joints are the weak points of timber structures, and thus studied their their mechanical properties. Eckelman et al. [6], Likos et al. [7], Huan et al. [8] and Chen mechanical properties. Eckelman et al. [6], Likos et al. [7], Huan et al. [8] and Chen et al. [9] et al. [9] studied the bending behaviour of different types of mortise–tenon joints in the studied the bending behaviour of different types of mortise–tenon joints in the timber timber structure. Other studies also focused on Dou-gong, a special part of traditional structure. Other studies also focused on Dou-gong, a special part of traditional timber timber structures (see Figure 1 [10]). These are commonly used in the main buildings of structures (see Figure 1 [10]). These are commonly used in the main buildings of traditional traditional palaces. Experimental studies [11–16] were carried out to investigate their fail- palaces. Experimental studies [11–16] were carried out to investigate their failure modes, ure modes, stiffness, yield load and other mechanical properties. In detail, Yeo et al. [17] stiffness, yield load and other mechanical properties. In detail, Yeo et al. [17] studied studied the seismic behaviour of Taiwanese timber brackets subjected to out-of-plane the seismic behaviour of Taiwanese timber brackets subjected to out-of-plane loading, loading, and the effect of their mechanical behaviour—such as failure modes—on the and the effect of their mechanical behaviour—such as failure modes—on the strength of strength of the mechanical model. Xie et al. [18] studied the static behaviour of a two- the mechanical model. Xie et al. [18] studied the static behaviour of a two-tiered Dou- tiered Dou-Gong system reinforced with super-elastic alloy, and found that the pre-strain Gong system reinforced with super-elastic alloy, and found that the pre-strain of the of the super-elastic alloy can significantly increase the damping ratio in the structure. Pre- super-elastic alloy can significantly increase the damping ratio in the structure. Previous vious studies [19–22] tried to evaluate the mechanical properties and safety state of an studies [19–22] tried to evaluate the mechanical properties and safety state of an entire entire timber structure through finite element modelling, mathematical methods, and a timber structure through finite element modelling, mathematical methods, and a shaking- shaking-table test. However, all of these studies focused on the timer components and table test. However, all of these studies focused on the timer components and timber timber frames without considering the infill walls. Recently, scholars have considered that frames without considering the infill walls. Recently, scholars have considered that infill infill walls might have a significant effect on the seismic performance of timber structures. walls might have a significant effect on the seismic performance of timber structures. For example, Vieux-Champagne et al. [23,24] studied the seismic performance of timber- For example, Vieux-Champagne et al. [23,24] studied the seismic performance of timber- framed structures filled with natural stones and earth mortar by introducing three scales framed structures filled with natural stones and earth mortar by introducing three scales of experiments. Ali et al. [25] studied the in-plane behaviour of full-scale Dhajji walls, a of experiments. Ali et al. [25] studied the in-plane behaviour of full-scale Dhajji walls, a wooden-braced frame with a stone infill system, and tested it under quasi-static loading. wooden-braced frame with a stone infill system, and tested it under quasi-static loading. Poletti and Vasconcelos [26] studied the seismic behaviour of the walls with masonry in- Poletti and Vasconcelos [26] studied the seismic behaviour of the walls with masonry filllath and plaster, and a timber frame with no infill. Dutu et al. [27,28] studied the seismic infilllath and plaster, and a timber frame with no infill. Dutu et al. [27,28] studied the behaviour of timber-framed masonry walls based on the static cyclic loading test, and seismic behaviour of timber-framed masonry walls based on the static cyclic loading test, proposed a simplified seismic evaluation method. All of these studies demonstrate that and proposed a simplified seismic evaluation method. All of these studies demonstrate the infill walls can significantly increase the stiffness, the ductility and the load-bearing that the infill walls can significantly increase the stiffness, the ductility and the load-bearing capacity of the timber frames, which consequently have a significant influence on the seis- capacity of the timber frames, which consequently have a significant influence on the mic behaviour of the entire building. seismic behaviour of the entire building. Figure 1. Structure of Dou-gong. Figure 1. Structure of Dou-gong. Buildings 2022, 12, x FOR PEER REVIEW 3 of 23 Buildings 2022, 12, 887 3 of 22 1.2. Research on Timber Frames with Infill Walls 1.2. Research on Timber Frames with Infill Walls Although the importance of infill walls has been studied in modern buildings, the infill walls of Chinese ancient timber architecture have not received attention in previous Although the importance of infill walls has been studied in modern buildings, the research, and they are considered to be nonstructural members. Moreover, limited re- infill walls of Chinese ancient timber architecture have not received attention in previous search has studied the seismic performance of timber frames with infill walls. Emile et al. research, and they are considered to be nonstructural members. Moreover, limited research [29] studied the failure modes, stiffness, strength (including the rate of degradation), and has studied the seismic performance of timber frames with infill walls. Emile et al. [29] energy dissipation capacity of Chinese traditional mortise–tenon jointed beam-column studied the failure modes, stiffness, strength (including the rate of degradation), and energy frames with wood panel infill. Xie et al. [30] also studied the influence of wood infill walls dissipation capacity of Chinese traditional mortise–tenon jointed beam-column frames on the se with wood ismipanel c performanc infill. Xie e of Chine et al. [30 s] e t also radit studied ional tim the ber str influence ucture of s throu woodg infill h shak walls ing t on ablthe e tests. It was found that the natural frequencies, damping ratio and acceleration responses seismic performance of Chinese traditional timber structures through shaking table tests. It of the model was found that with wood in the natural fill w frequencies, alls were grea damping ter than ratio those w and acceleration ithout. Chang e responses t al. [31of –33 the ] stmodel udied the mechanic with wood infill al walls behavio weru ers gr eater of trad than itiona those l ti without. mber she Chang ar wal etl and rei al. [31–33n ]forced studied planke the mechanical d timber she behaviours ar walls us of ing exper traditional iment timbs and c er shear alc wall ulatand ionsr . T einfor heir f ced indin planked gs showed timber shear walls using experiments and calculations. Their findings showed that the friction that the friction behaviour between board units and beams plays a major role in resisting behaviour between board units and beams plays a major role in resisting the lateral force the lateral force applied to the timber shear wall. The restoring force model is the founda- applied to the timber shear wall. The restoring force model is the foundation of the seismic tion of the seismic evaluation of structures. evaluation of structures. 1.3. Research and Application of the Restoring Force Model 1.3. Research and Application of the Restoring Force Model Many studies have investigated the restoring force model of connections and com- Many studies have investigated the restoring force model of connections and compo- ponents of structures [34–36], and they found that steel structures are rich in the restoring nents of structures [34–36], and they found that steel structures are rich in the restoring force model of reinforced concrete structures. However, only limited research has studied force model of reinforced concrete structures. However, only limited research has studied the restoring force model of ancient timber structures. Moreover, the restoring force the restoring force model of ancient timber structures. Moreover, the restoring force model model of reinforced concrete structures and steel structures cannot be used in timber of reinforced concrete structures and steel structures cannot be used in timber structures structures because of the huge difference in the machinal properties between timber and because of the huge difference in the machinal properties between timber and concrete. concrete. Therefore, it is essential to study the restoring force of Chinese timber frames Therefore, it is essential to study the restoring force of Chinese timber frames with infill with infill walls. In the present paper, a Chinese traditional timber frame with masonry walls. In the present paper, a Chinese traditional timber frame with masonry and wood and wood window infill—the Kanchuang frame (shown in Figure 2)—was tested under window infill—the Kanchuang frame (shown in Figure 2)—was tested under cyclic loading. cyclic loading. Kanchuang is a kind of wood window used in ancient timber buildings Kanchuang is a kind of wood window used in ancient timber buildings named “Kan”. named “Kan”. A Kanchuang frame is a timber frame with Kanchuang and a half masonry A Kanchuang frame is a timber frame with Kanchuang and a half masonry wall infilled. wall infilled. The loading process and failure modes of the test model were studied in The loading process and failure modes of the test model were studied in order to collect order to collect test data such as skeleton curves and hysteretic curves. The restoring force test data such as skeleton curves and hysteretic curves. The restoring force model for the model for the Kanchuang frame was developed to be used as the foundation for the seis- Kanchuang frame was developed to be used as the foundation for the seismic evaluation of mic evaluation of the Kanchuang frame. the Kanchuang frame. Figure 2. Dimensions of the experimental model (mm). Figure 2. Dimensions of the experimental model (mm). Buildings 2022, 12, 887 4 of 22 Finite element analysis has been frequently used to analyse the seismic performance of timber structures. However, a timber structure is quite different from the reinforced concrete structure. Moreover, most ancient timber buildings use various kinds of materials. Consequently, there are a lot of contact areas and small gaps between different compo- nents. This made them difficult to simulate using finite element analysis software, and the calculation results cannot be proven valid or reliable with the consideration of contacts, mechanical properties and material properties. Moreover, the calculation process is very complicated, and it takes a very long time to calculate the results. Nevertheless, it was found the building structures can be divided into sub-units. When the units’ hysteretic model, rigidity and strength are known, an analysis model can be used to calculate the sub-units. In this way, the contact analysis in the finite element calculation can be avoided, and we can improve the computational efficiency. Moreover, a correct hysteretic model is also crucial to nonlinear seismic analysis. Thus, there is a need to develop a simplified calculation model that can present the mechanical properties of the structure for nonlinear seismic analysis. The sub-units can be used to simplify the hysteretic model of the structure as mass elements or shell elements. Thus, the calculation model could be simpler than before, and the nonlinear seismic analysis methods—such as Newmark- —can be applied to calculate the seismic responses of the structure. 2. Experimental Studies 2.1. Specimen Fabrication In this study, a 1:2 scale Kanchuang frame model was used as the test specimen. This specimen of Chinese traditional timber architecture was collected from Qilinyingshan (see Reference [5]). The Kanchuang frame is widely used in ancient timber buildings of the Qing dynasty. Kanchuang frames consist of columns, fangs, wood windows, masonry walls, Shangkan, Fengkan and Baokuang. All of the wood specimens are connected with mortise–tenon joints. The structural properties of wood members can vary with the member size, or the size effect. According to the Buckingham theorem [37] and similitude theory, the structural properties of the prototype Kanchuang frame can be calculated by dividing the corresponding properties of the scaled specimens by the scale factors of the dimensions and mechanical properties of the materials [38]. The dimensional scale factor of the test model was 1/2. Table 1 shows the scale factor of the physical parameters. Table 1. Scale factors of the physical parameters. Length Area Displacement Elastic Modulus Force Drift Angle Moment Density Mass 1/2 1/4 1/2 1 1/4 1 1/8 2 1/4 The span of the frame was 1600 mm and the total height was 1800 mm. The dimensions of the masonry wall were 1450 mm  500 mm  250 mm. Figure 2 shows the layout and main dimensions of the model. The dimensions of the components are shown in Figures 3 and 4. 2.2. Material Properties Ancient timber buildings are the cultural relics of a nation. It is prohibited for anyone to take materials from the standing ancient buildings. However, the material properties matter to the test results. In order to use materials similar to those from the ancient timber buildings, the researchers used Pinus sylvestris var. mongolica, lime and Xiaotingni brick to fabricate the specimen. Pinus sylvestris var. mongolica grows in Northeast China, and is a commonly used tree species in the repair and construction replacement of traditional timber structures. Lime was commonly used to make the mortar in ancient times. Xiaotingni brick is also widely used in the restoration of traditional buildings. Buildings 2022, 12, x FOR PEER REVIEW 5 of 23 Buildings 2022, 12, x FOR PEER REVIEW 5 of 23 Buildings 2022, 12, 887 5 of 22 Figure 3. Dimensions of the specimen (mm). Figure 3. Dimensions of the specimen (mm). Figure 3. Dimensions of the specimen (mm). (a) (a) (b) (b) Figure 4. Dimensions of the mortise and tenon joints (unit: mm): (a) mortise of the Bianting and Figure 4. Dimensions of the mortise and tenon joints (unit: mm): (a) mortise of the Bianting and Figure 4. Dimensions of the mortise and tenon joints (unit: mm): (a) mortise of the Bianting and tenon of the Zhongmo; (b) mortise of the Bianting and tenon of the Shangmo/Xiamo. tenon of the Zhongmo; (b) mortise of the Bianting and tenon of the Shangmo/Xiamo. tenon of the Zhongmo; (b) mortise of the Bianting and tenon of the Shangmo/Xiamo. Buildings 2022, 12, x FOR PEER REVIEW 6 of 23 2.2. Material Properties Ancient timber buildings are the cultural relics of a nation. It is prohibited for anyone to take materials from the standing ancient buildings. However, the material properties matter to the test results. In order to use materials similar to those from the ancient timber buildings, the researchers used Pinus sylvestris var. mongolica, lime and Xiaotingni brick to fabricate the specimen. Pinus sylvestris var. mongolica grows in Northeast China, and is a commonly used tree species in the repair and construction replacement of traditional timber structures. Lime was commonly used to make the mortar in ancient times. Xiaotingni brick is also widely used in the restoration of traditional buildings. Compressive strength tests were performed on nine cubic samples of traditional mor- tar, according to the standard test method for the performance of building mortar (JGJ/T Buildings 2022, 12, 887 70-2009 2009) [39]. The dimensions of the mortar sample were 70 mm × 70 mm × 70 6 mm of 22 (length × width × height). The mortar was made of lime and water. The mortar was poured into the molds and left to cure for 28 days at a temperature of 20 ± 2 °C and a relative air humidity of 95%. A universal testing machine was used to test the compressive strength Compressive strength tests were performed on nine cubic samples of traditional mortar, of the mortar samples. The average compressive strength of the mortar was 2.0 Mpa. according to the standard test method for the performance of building mortar (JGJ/T 70- Compressive strength tests were also carried out on the brick and masonry samples 2009 2009) [39]. The dimensions of the mortar sample were 70 mm  70 mm  70 mm according to the test method for wall bricks (GB/T 2542-2012) [40] and the standard test (length  width  height). The mortar was made of lime and water. The mortar was method for the basic mechanical properties of masonry (GB/T 50129-2011) [41]. Ten brick poured into the molds and left to cure for 28 days at a temperature of 20  2 C and a samples were made. The dimensions of the bricks were 75 mm × 75 mm × 60 mm (length relative air humidity of 95%. A universal testing machine was used to test the compressive × width × height). A universal testing machine was used to test the compressive strength strength of the mortar samples. The average compressive strength of the mortar was of the bricks. The average compressive strength of the brick samples was 9.3 Mpa. The 2.0 Mpa. dimensions of the prism in the compression tests including 12 layers of brick and 11 layers Compressive strength tests were also carried out on the brick and masonry samples accor of mort ding ar were 130 to the test mm method × 200 for mm wall × 415 mm ( bricks (GB/T length 2542-2012) × width × height [40] and ) (see the standar Figure 5) d. The test method averagefor com the pressiv basic e mechanical strength ofpr the operties masonry of s masonry amples w (GB/T as 3.4 50129-2011) Mpa. The av [er 41ag ]. T e Y enobrick ung’s samples modulus of the mas were made.onry sa The dimensions mples was 12 of 59 Mpa the bricks . Shwer ear e st75 ress mm test s were perf 75 mm  ormed on 60 mm (length nine tes t s width amples. The  height te). st A sample universal was c testing ompos machine ed of three layer was used s o tof brick test the and compr two layer essives str ofength morta of r. The di the bricks. mensi The ons aver of ta hg e e te comp st sampl ressive e were 130 strength mm of the × 200 brick mmsamples × 230 mm. was The whole 9.3 Mpa. The dimensions of the prism in the compression tests including 12 layers of brick and operation process was strictly controlled according to the Chinese National Standard 11 layers of mortar were 130 mm  200 mm  415 mm (length  width  height) (see (GB/T 50129-2011) [41] (see Figure 6). All of the samples were cured indoors at a temper- Figure 5). The average compressive strength of the masonry samples was 3.4 Mpa. The ature of 15–20 °C for 28 days. The test samples were then positioned on the center of the average Young’s modulus of the masonry samples was 1259 Mpa. Shear stress tests were test machine. Steel plates and sands were used to level up the bottom and top of the test performed on nine test samples. The test sample was composed of three layers of brick and sample. Vertical loads were applied by the testing machine, and the strength was calcu- two layers of mortar. The dimensions of the test sample were 130 mm  200 mm  230 mm. lated according to f = F/A (F is failure load and A is contact area). The average shear stress The whole operation process was strictly controlled according to the Chinese National of nine test samples was 0.041 Mpa (see Table 2). Standard (GB/T 50129-2011) [41] (see Figure 6). All of the samples were cured indoors at a temperature of 15–20 C for 28 days. The test samples were then positioned on the center Table 2. Mechanical properties of the mortar, brick and masonry (unit: Mpa). of the test machine. Steel plates and sands were used to level up the bottom and top of Compressive Strength Shear Strength of Young’s Modulus of the test sample. Vertical loads were applied by the testing machine, and the strength was Masonry Masonry Mortar Brick Masonry calculated according to f = F/A (F is failure load and A is contact area). The average shear 2.0 9.3 3.4 0.041 1259 stress of nine test samples was 0.041 Mpa (see Table 2). Buildings 2022, 12, x FOR PEER REVIEW 7 of 23 Figure 5. Compressive strength test of the masonry. Figure 5. Compressive strength test of the masonry. Figure 6. Shear stress test of the masonry. Figure 6. Shear stress test of the masonry. The material properties of the wood—including its modulus of elasticity, compres- sion strength density and moisture content—were tested, using the physical and mechan- ical methods for wood (GB 1927-1943-91) [42]. The dimensions of the wood specimens for the compression test and elastic modulus test were 30 mm × 20 mm × 20 mm and 300 mm × 20 mm × 20 mm. The mechanical properties of the wood are shown in Table 3. The load– deformation curves of the test samples are shown in Figure 7. Table 3. Mechanical properties of the wood. Parallel to Grain (MPa) Perpendicular to Grain (MPa) Moisture Density Compressive Compressive Elastic Compressive Elastic Content Elastic Modulus Strength Strength (T) Modulus (T) Strength (R) Modulus (R) 0.369 g/m 10.09% 46.21 8907 3.39 771 5.49 1620 T is the tangential direction and R is the radial direction. 120 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 0.0 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Deformation(mm) Deformation(mm) (a) (b) 2.0 20 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0123 4 0.00 0.25 0.50 0.75 1.00 1.25 1.50 Deformation(mm) Deformation(mm) (c) (d) Load(kN) Load(KN) Load(kN) Load(kN) Buildings 2022, 12, 887 7 of 22 Table 2. Mechanical properties of the mortar, brick and masonry (unit: Mpa). Compressive Strength Shear Strength of Young’s Modulus of Masonry Masonry Mortar Brick Masonry 2.0 9.3 3.4 0.041 1259 The material properties of the wood—including its modulus of elasticity, compression strength density and moisture content—were tested, using the physical and mechani- cal methods for wood (GB 1927-1943-91) [42]. The dimensions of the wood specimens for the compression test and elastic modulus test were 30 mm  20 mm  20 mm and 300 mm  20 mm  20 mm. The mechanical properties of the wood are shown in Table 3. The load–deformation curves of the test samples are shown in Figure 7. Table 3. Mechanical properties of the wood. Parallel to Grain (MPa) Perpendicular to Grain (MPa) Moisture Density Compressive Elastic Compressive Elastic Compressive Elastic Content Strength Modulus Strength (T) Modulus (T) Strength (R) Modulus (R) 0.369 g/m 10.09% 46.21 8907 3.39 771 5.49 1620 T is the tangential direction and R is the radial direction. Figure 7. Load–deformation curves of the test samples: (a) compressive strength test of the masonry samples; (b) shear stress test of the masonry samples; (c) compressive strength test of the wood samples (parallel to the grain); (d) compressive strength test of the wood samples (perpendicular to the grain, T); (e) compressive strength test of the wood samples (perpendicular to the grain, R). Buildings 2022, 12, x FOR PEER REVIEW 8 of 23 3.0 2.5 2.0 1.5 1.0 0.5 0.0 01 2345 Deformation(mm) (e) Figure 7. Load–deformation curves of the test samples: (a) compressive strength test of the masonry samples; (b) shear stress test of the masonry samples; (c) compressive strength test of the wood samples (parallel to the grain); (d) compressive strength test of the wood samples (perpendicular to Buildings 2022, 12, 887 8 of 22 the grain, T); (e) compressive strength test of the wood samples (perpendicular to the grain, R). 2.3. Testing and Measuring Schemes 2.3. Testing and Measuring Schemes Steel caps were positioned on the top of the columns. A distribution beam was then Steel caps were positioned on the top of the columns. A distribution beam was then positioned on the steel caps. Vertical loads of weight were hung at the two sides of the dis- positioned on the steel caps. Vertical loads of weight were hung at the two sides of the tribution beam, to simulate the dead-weight of the upper components and roof. The frame distribution beam, to simulate the dead-weight of the upper components and roof. The was installed in a vertical position with the bottom of the columns in hinge supports (see frame was installed in a vertical position with the bottom of the columns in hinge supports Figure 8). The vertical loads were calculated according to the relevant references [43–45]. (see Figure 8). The vertical loads were calculated according to the relevant references [43–45]. Using the similarity theory [46,47], the vertical loads placed on the 1/2 scale model were 12 Using the similarity theory [46,47], the vertical loads placed on the 1/2 scale model were kN. The displacement data were recorded using Displacement meter 3 in Figure 8. 12 kN. The displacement data were recorded using Displacement meter 3 in Figure 8. Figure 8. Schematic diagram of the loading equipment. Figure 8. Schematic diagram of the loading equipment. Cyclic loadings were applied using a hydraulic actuator with a displacement range of Cyclic loadings were applied using a hydraulic actuator with a displacement range 250 mm. The actuator was positioned at a height of 1.6 m, with a transverse cyclic load of ±250 mm. The actuator was positioned at a height of 1.6 m, with a transverse cyclic load with a programmed load cycle. A displacement meter (Displacement meter 3) and a force with a programmed load cycle. A displacement meter (Displacement meter 3) and a force sensor were used on the actuator. The force versus displacement curves were measured Buildings 2022, 12, x FOR PEER REVIEW 9 of 23 sensor were used on the actuator. The force versus displacement curves were measured directly. The horizontal cyclic loads were applied under displacement control according directly. The horizontal cyclic loads were applied under displacement control according to the ISO/FDIS 21581:2010(E) standards [48]. The frame was loaded with three initiation to the ISO/FDIS 21581:2010(E) standards [48]. The frame was loaded with three initiation cycles with an amplitude of 3 mm. The amplitude of the second three cycles was 5 mm. cycles with an amplitude of 3 mm. The amplitude of the second three cycles was 5 mm. The amplitude of the cycles gradually increased by a step of 5 mm until the amplitude The amplitude of the cycles gradually increased by a step of 5 mm until the amplitude was 70 mm. It then increased by a step of 10 mm until the amplitude reached 150 mm. was 70 mm. It then increased by a step of 10 mm until the amplitude reached 150 mm. Figure 9 presents the shape of the loading history. An overview of the test setup is shown Figure 9 presents the shape of the loading history. An overview of the test setup is shown in Figure 10. in Figure 10. -50 -100 -150 Figure 9. Loading scheme. Figure 9. Loading scheme. Figure 10. Test setup. 2.4. General Observation The Kanchuang frame was loaded until the displacement reached 150 mm (the drift angle was 3/32). Before the 10 mm (the drift angle was 1/160) cycles, no obvious damage was observed. Cracks primarily appeared in the side area of the masonry wall when the displacement reached 10 mm (see Figure 11). Then, the cracks developed further at an angle of 45° as the displacement increased. X-shaped cracks were formed when the dis- placement reached 50 mm (the drift angle was 1/32). Finally, cracks covered the whole wall and collapsed until the displacement reached 150 mm (the drift angle was 3/32). The tenons of the wood windows started to pull out of the mortise when the displacement reached 15 mm (drift angle is 3/320). Then, the gap gradually increased as the displace- ment increased. Finally, the gap reached 3 mm. Cracks appeared on the Lingtiao and Zaibian when the displacement reached 150 mm (the drift angle was 3/32). Figure 11 shows the final failure pattern. Δ/mm Load(kN) Buildings 2022, 12, x FOR PEER REVIEW 9 of 23 The amplitude of the cycles gradually increased by a step of 5 mm until the amplitude was 70 mm. It then increased by a step of 10 mm until the amplitude reached 150 mm. Figure 9 presents the shape of the loading history. An overview of the test setup is shown in Figure 10. -50 -100 -150 Buildings 2022, 12, 887 9 of 22 Figure 9. Loading scheme. Figure 10. Test setup. Figure 10. Test setup. 2.4. General Observation 2.4. General Observation The Kanchuang frame was loaded until the displacement reached 150 mm (the drift The Kanchuang frame was loaded until the displacement reached 150 mm (the drift angle was 3/32). Before the 10 mm (the drift angle was 1/160) cycles, no obvious damage angle was 3/32). Before the 10 mm (the drift angle was 1/160) cycles, no obvious damage was observed. Cracks primarily appeared in the side area of the masonry wall when was observed. Cracks primarily appeared in the side area of the masonry wall when the the displacement reached 10 mm (see Figure 11). Then, the cracks developed further at displacement reached 10 mm (see Figure 11). Then, the cracks developed further at an an angle of 45 as the displacement increased. X-shaped cracks were formed when the angle of 45° as the displacement increased. X-shaped cracks were formed when the dis- displacement reached 50 mm (the drift angle was 1/32). Finally, cracks covered the whole placement reached 50 mm (the drift angle was 1/32). Finally, cracks covered the whole wall and collapsed until the displacement reached 150 mm (the drift angle was 3/32). The wall and collapsed until the displacement reached 150 mm (the drift angle was 3/32). The tenons of the wood windows started to pull out of the mortise when the displacement tenons of the wood windows started to pull out of the mortise when the displacement reached 15 mm (drift angle is 3/320). Then, the gap gradually increased as the displacement reached 15 mm (drift angle is 3/320). Then, the gap gradually increased as the displace- increased. Finally, the gap reached 3 mm. Cracks appeared on the Lingtiao and Zaibian ment increased. Finally, the gap reached 3 mm. Cracks appeared on the Lingtiao and Buildings 2022, 12, x FOR PEER REVIEW 10 of 23 when the displacement reached 150 mm (the drift angle was 3/32). Figure 11 shows the Zaibian when the displacement reached 150 mm (the drift angle was 3/32). Figure 11 final failure pattern. shows the final failure pattern. (a) (b) (c) (d) (e) Figure 11. Cont. Δ/mm Buildings 2022, 12, 887 10 of 22 Buildings 2022, 12, x FOR PEER REVIEW 11 of 23 (f) Figure 11. Failure modes of the specimens: (a) crack on the left side of masonry wall at a displace- Figure 11. Failure modes of the specimens: (a) crack on the left side of masonry wall at a displacement ment of 10 mm (the drift angle was 1/160); (b) penetrating cracks on the right side of the masonry of 10 mm (the drift angle was 1/160); (b) penetrating cracks on the right side of the masonry wall at a wall at a displacement of 35 mm (the drift angle was 7/320); (c) the tenons pull out of mortise at a displacement of 35 mm (the drift angle was 7/320); (c) the tenons pull out of mortise at a displacement displacement of 15 mm (the drift angle was 3/320); (d) the tenons pull out of the mortise at a dis- of 15 mm (the drift angle was 3/320); (d) the tenons pull out of the mortise at a displacement of placement of 30 mm (the drift angle was 7/320); (e) cracks on the masonry wall at a displacement of 50 mm (the drift angle was 1/32); (f) failure modes of the specimens at a displacement of 150 mm 30 mm (the drift angle was 7/320); (e) cracks on the masonry wall at a displacement of 50 mm (the (the drift angle was 3/32). drift angle was 1/32); (f) failure modes of the specimens at a displacement of 150 mm (the drift angle was 3/32). 3. Analysis of the Characteristic Curves 3. Analysis of the Characteristic Curves This study developed a hysteresis model of Kanchuang frames. Moreover, this paper This study developed a hysteresis model of Kanchuang frames. Moreover, this pa- mainly aimed to build the hysteretic model of the Kanchuang frame. The structural per- per form mainly ance oaimed f the Kanch to build uanthe g fram hyster e, such etic a model s its stif offnes thes de Kanchuang gradationframe. and ene The rgystr diss uctural ipa- performance tion, were discusse of the Kanchuang d in reference [49]. F frame, such igure as s its 12 and stiffness 13 show degradation that the c and urves ener argy e sldissipa- ightly tion, unsymme were discussed trical, and in the r refer easons ence inc [49 lud ]. Figur e the es fol12 low and ing:13 show that the curves are slightly unsymmetrical, and the reasons include the following: (1) Although the frame is symmetrical theoretically, in practice, the frame is not exactly Buildings 2022, 12, 887 11 of 22 Buildings 2022, 12, x FOR PEER REVIEW 12 of 23 Buildings 2022, 12, x FOR PEER REVIEW 12 of 23 (1) Although the frame is symmetrical theoretically, in practice, the frame is not exactly symmetrical. The timber frame of ancient timber buildings is not symmetrical, as the symmetrical. The timber frame of ancient timber buildings is not symmetrical, as the symmetrical. The timber frame of ancient timber buildings is not symmetrical, as specimens were handmade by the timber building restoration people. There are dif- specimens were handmade by the timber building restoration people. There are dif- the specimens were handmade by the timber building restoration people. There are ferent sizes of small gaps between the components, especially between the mortise ferent sizes of small gaps between the components, especially between the mortise different sizes of small gaps between the components, especially between the mortise and tenon. and tenon. and tenon. (2) The cracks and damage of the components did not appear in an exactly symmetrical (2) The cracks and damage of the components did not appear in an exactly symmetrical (2) The cracks and damage of the components did not appear in an exactly symmetrical way, and the curves are unsymmetrical in different loading directions. way, and the curves are unsymmetrical in different loading directions. way, and the curves are unsymmetrical in different loading directions. (3) The horizontal cyclic loads were applied using a hydraulic actuator. The actuator was (3) The horizontal cyclic loads were applied using a hydraulic actuator. The actuator was (3) The horizontal cyclic loads were applied using a hydraulic actuator. The actuator was positioned at one side of the frame. It wasn’t positioned in the middle of the frame, positioned at one side of the frame. It wasn’t positioned in the middle of the frame, positioned at one side of the frame. It wasn’t positioned in the middle of the frame, which also made the curves unsymmetrical. which also made the curves unsymmetrical. which also made the curves unsymmetrical. Drift angle Drift angle -0.0938-0.0625-0.0313 0.0000 0.0313 0.0625 0.0938 -0.0938-0.0625-0.0313 0.0000 0.0313 0.0625 0.0938 6 9.6 6 9.6 4 6.4 4 6.4 2 3.2 2 3.2 0 0.0 0 0.0 -2 -3.2 -2 -3.2 -4 -6.4 -4 -6.4 -6 -9.6 -6 -9.6 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 Δ(mm) Δ(mm) Figure 12. Load–displacement hysteresis curves. Figure 12. Load–displacement hysteresis curves. Figure 12. Load–displacement hysteresis curves. Drift angle Drift angle -0.0938 -0.0625 -0.0313 0.0000 0.0313 0.0625 0.0938 -0.0938 -0.0625 -0.0313 0.0000 0.0313 0.0625 0.0938 6 10 6 10 The first cycle The first cycle The second cycle 4 The second cycle 6 4 6 The third cycle The third cycle 2 3 2 3 0 0 0 0 -2 -3 -2 -3 -4 -6 -4 -6 -6 -10 -6 -10 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 Δ (mm) Δ (mm) Figure 13. Load–displacement envelope curves. Figure 13. Load–displacement envelope curves. Figure 13. Load–displacement envelope curves. 3.1. Load–Displacement Hysteretic Curve 3.1. Load–Displacement Hysteretic Curve 3.1. Load–Displacement Hysteretic Curve Figure 12 shows the load–displacement hysteretic curves of the Kanchuang frame. Figure 12 shows the load–displacement hysteretic curves of the Kanchuang frame. Figure 12 shows the load–displacement hysteretic curves of the Kanchuang frame. The curves were Z-shaped, particularly at the ultimate state. The pinch phenomenon The curves were Z-shaped, particularly at the ultimate state. The pinch phenomenon was The curves were Z-shaped, particularly at the ultimate state. The pinch phenomenon was was also shown in the curve. The pinch effect was clear as the displacement and loading also shown in the curve. The pinch effect was clear as the displacement and loading cycle also shown in the curve. The pinch effect was clear as the displacement and loading cycle cycle increased; it was caused by the plastic deformation of the wood, the pull-out of the increased; it was caused by the plastic deformation of the wood, the pull-out of the mor- increased; it was caused by the plastic deformation of the wood, the pull-out of the mor- mortise–tenon joints, and the crack development of the masonry wall. The peak values tise–tenon joints, and the crack development of the masonry wall. The peak values of the tise–tenon joints, and the crack development of the masonry wall. The peak values of the of the hysteretic loops increased as the loading displacements increased. The bending hysteretic loops increased as the loading displacements increased. The bending capacities hysteretic loops increased as the loading displacements increased. The bending capacities capacities of each loading displacement in the first cycle were higher than those in the of each loading displacement in the first cycle were higher than those in the second cycle. of each loading displacement in the first cycle were higher than those in the second cycle. second cycle. Moreover, the bending capacity of the second cycle was higher than that of the Moreover, the bending capacity of the second cycle was higher than that of the third cycle. Moreover, the bending capacity of the second cycle was higher than that of the third cycle. third cycle. This was caused by the accumulation of the damage and strength degradation This was caused by the accumulation of the damage and strength degradation of the mor- This was caused by the accumulation of the damage and strength degradation of the mor- of the mortise–tenon joints and the masonry wall. tise–tenon joints and the masonry wall. tise–tenon joints and the masonry wall. P(kN) P(kN) P(kN) P(kN) M(kN⋅m) M(kN⋅m) M(kN⋅m) M(kN⋅m) Buildings 2022, 12, x FOR PEER REVIEW 13 of 23 Buildings 2022, 12, 887 12 of 22 3.2. Load–Displacement Envelope Curve Figure 13 shows the load–displacement envelope curve of the Kanchuang frame: 3.2. Load–Displacement Envelope Curve (1) In different displacement cycles, the strength of first cycle was found to be stronger Figure 13 shows the load–displacement envelope curve of the Kanchuang frame: than the second and third cycles. The strength degradation was caused by the crack (1) In different displacement cycles, the strength of first cycle was found to be stronger development and the plastic cumulative deformation of the wood components. than the second and third cycles. The strength degradation was caused by the crack (2) Both the positive and negative curves followed the same trend up to 25 mm (the drift development and the plastic cumulative deformation of the wood components. angle was 1/64) displacement. Before the 25 mm (the drift angle was 1/64) cycles, no (2) Both the positive and negative curves followed the same trend up to 25 mm (the drift obvious damage occurred in the timber components and no penetrating cracks angle was 1/64) displacement. Before the 25 mm (the drift angle was 1/64) cycles, formed on the masonry wall. The whole frame was still in the elastic stage. no obvious damage occurred in the timber components and no penetrating cracks (3) During the 30–65 mm (drift angle during 3/160 and 13/320) cycles, strength degrada- formed on the masonry wall. The whole frame was still in the elastic stage. tion began in both the left (negative) and right (positive) loading directions while the (3) During the 30–65 mm (drift angle during 3/160 and 13/320) cycles, strength degrada- load was increasing. This happened because of the failure of the masonry wall, and tion began in both the left (negative) and right (positive) loading directions while the the tenons were pulled out of the mortises. The penetrating cracks began to appear load was increasing. This happened because of the failure of the masonry wall, and the in the masonry wall, and the crannies became larger as the loading displacement in- tenons were pulled out of the mortises. The penetrating cracks began to appear in the creased. The gaps between mortise and tenon joints of the frame and wood windows masonry wall, and the crannies became larger as the loading displacement increased. increased as loading displacement increased. In the 65 mm (the drift angle was The gaps between mortise and tenon joints of the frame and wood windows increased 13/320) cycles, the left and right side of the masonry wall began to collapse. During as loading displacement increased. In the 65 mm (the drift angle was 13/320) cycles, the loading process, the cracks on the right side (the positive direction) of the ma- the left and right side of the masonry wall began to collapse. During the loading sonry wall grew faster than those on the left side (the negative direction) (see Figure process, the cracks on the right side (the positive direction) of the masonry wall grew 11e). Therefore, the strength degradation of the frame in the positive loading was faster than those on the left side (the negative direction) (see Figure 11e). Therefore, weaker than that in the negative direction. The whole frame entered the elastoplastic the strength degradation of the frame in the positive loading was weaker than that in deformation stage. the negative direction. The whole frame entered the elastoplastic deformation stage. (4) After the 65 mm (the drift angle was 13/320) cycles, the stiffness increased in both the (4) After the 65 mm (the drift angle was 13/320) cycles, the stiffness increased in both right (positive) and left (negative) directions. The stiffness was stronger than that of the right (positive) and left (negative) directions. The stiffness was stronger than that the 30–65 mm (the drift angle during 3/160 and 13/320) cycles but weaker than that of the 30–65 mm (the drift angle during 3/160 and 13/320) cycles but weaker than before the 30 mm (the drift angle was 3/160) cycles. This is because the masonry wall that before the 30 mm (the drift angle was 3/160) cycles. This is because the masonry failed after the 65 mm (the drift angle was 13/320) cycles, and the components of the wall failed after the 65 mm (the drift angle was 13/320) cycles, and the components timber frames were compressed with each other. The structure reached a new elastic– of the timber frames were compressed with each other. The structure reached a new plastic stage characterized by the increase of the stiffness and the residual defor- elastic–plastic stage characterized by the increase of the stiffness and the residual mation of the timber components. deformation of the timber components. Research [29] has been conducted on traditional timber frames with wood plane in- Research [29] has been conducted on traditional timber frames with wood plane fill. This frame has been used in ancient timber buildings. The test model was made of infill. This frame has been used in ancient timber buildings. The test model was made of Pinus sylvestris var. mongolica. Figure 14 presents the test model and hysteretic curves of Pinus sylvestris var. mongolica. Figure 14 presents the test model and hysteretic curves of the test model. The peak values of two hysteretic curves at different displacement (drift the test model. The peak values of two hysteretic curves at different displacement (drift angle) cycles are different between Figures 12 and 14. This is because the dimensions, angle) cycles are different between Figures 12 and 14. This is because the dimensions, structures, wood spices and loading positions are different. The peak values (in Figure structures, wood spices and loading positions are different. The peak values (in Figure 14b) 14b) decreased while the loading displacement exceeded 200 mm (the drift angle was decreased while the loading displacement exceeded 200 mm (the drift angle was 1/8). 6/85). However, the peak values of the Kanchuang frame’s hysteretic curves did not de- However, the peak values of the Kanchuang frame’s hysteretic curves did not decrease. crease. Both hysteretic curves turned into Z shapes a despite the different dimensions, Both hysteretic curves turned into Z shapes a despite the different dimensions, structures, structures, wood spices and loading processes. Both curves showed a degrading trend wood spices and loading processes. Both curves showed a degrading trend while the while the loading cycles increased. loading cycles increased. (a) Figure 14. Cont. Buildings 2022, 12, x FOR PEER REVIEW 14 of 23 Buildings 2022, 12, x FOR PEER REVIEW 14 of 23 Buildings 2022, 12, 887 13 of 22 (b) Figure 14. Figures from reference [29]: (a) Test model; (b) hysteretic curves. (b) 4. Hysteresis Model Figure 14. Figures from reference [29]: (a) Test model; (b) hysteretic curves. Figure 14. Figures from reference [29]: (a) Test model; (b) hysteretic curves. 4.1. Envelope Curve 4. Hysteresis Model 4. Hysteresis Model The envelope curve is an essential part of the hysteresis model. In order to build the 4.1. Envelope Curve hysteresis model, the data consisting of the envelope curves should be nondimensional- 4.1. Envelope Curve The envelope curve is an essential part of the hysteresis model. In order to build the Δ P ised. The - relationship curves were thus obtained through the dimensionless pro- The envelope curve is hyster an essent esis model, ial part the of data the h consisting ysteresis mode of thel. In envelope order curves to build th should e be nondimensionalised. cessing of the test data of the test specimen, as shown in Equations (1) and (2). The D-P relationship curves were thus obtained through the dimensionless processing of hysteresis model, the data consisting of the envelope curves should be nondimensional- the test data of the test specimen, as shown in Equations (1) and (2). Δ P +− ised. The - relationship curves were thus obtained through the dimensionless pro-  ΔΔ =/ Δ + Δ /2 (1) () mm  cessing of the test data of the test specimen, as shown in Equations (1) and (2). D = D/ D + D /2 (1) m m +−  +− PP =/ P + P /2 (2)  () mm ΔΔ =/ Δ + Δ /2 (1) () mm  +  P = P/ P + P /2 (2) m m ± ± Δ P Δ P m m D and P are the+− loading displacement and the corresponding force. D and P are m m and  are the loading displacement and the corresponding force. and PP =/ P + P /2 (2) () mm  displacements in both the right (positive) direction and the left (negative) direction of the are displacements in both the right (positive) direction and the left (negative) direction of 65 mm cycles and the corresponding forces. Points O1, O2, A, B, D and E are vital points, ± ± the 65 mm cycles and the corresponding forces. Points O1, O2, A, B, D and E are vital Δ P Δ P m m and are the loading displacement and the corresponding force. and and points C and F are ultimate loading points (see Figure 15). It was found that the shapes points, and points C and F are ultimate loading points (see Figure 15). It was found that are displacements in both the right (positive) direction and the left (negative) direction of of the dimensionless envelope curves are close to the experimental data points. Table 4 the shapes of the dimensionless envelope curves are close to the experimental data points. the 65 mm cycles and the corresponding forces. Points O1, O2, A, B, D and E are vital presents the coordinates of the critical points and ultimate loading points; Table 5 shows Table 4 presents the coordinates of the critical points and ultimate loading points; Table 5 points, and points C and F are ultimate loading points (see Figure 15). It was found that the regression equations of the fitting envelope curves. The fitted envelope curves (see shows the regression equations of the fitting envelope curves. The fitted envelope curves the shapes of the dimensionless envelope curves are close to the experimental data points. Figure 15d) indicated a strength degradation trend in the different cycle processes, as is (see Figure 15d) indicated a strength degradation trend in the different cycle processes, as Table 4 presents the coord consistent inates of the cr with it the icatest l por in esults. ts and ultimate loading points; Table 5 is consistent with the test results. shows the regression equations of the fitting envelope curves. The fitted envelope curves (see Figure 15d) indicated a strength degradation trend in the different cycle processes, as is consistent with the test results. Experimental value Experimental value Theoretical value D 2 Theoretical value E Experimental value -3 -2 -1 0 1 2 3 Experimental value Theoretical value -3 -2 -1 0 1 2 3 Theoretical value O E 2 B -1 -1 A O O O A -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 2 -2 O -2 2 B -1 C A -1 -3 -3 -2 (a) (b) -2 Figure 15. Cont. -3 -3 (a) (b) P Buildings 2022, 12, 887 14 of 22 Buildings 2022, 12, x FOR PEER REVIEW 15 of 23 2 2 The first cycle Experimental value 1 1 The second cycle Theoretical value DD The third cycle 0 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -1 -1 -2 -2 -3 -3 (c) (d) Figure 15. Fitting curves of the skeleton curves of the Kanchuang frame: (a) the first cycle; (b) the Figure 15. Fitting curves of the skeleton curves of the Kanchuang frame: (a) the first cycle; (b) the second cycle; (c) the third cycle; (d) fitting envelope curves of three cycles. second cycle; (c) the third cycle; (d) fitting envelope curves of three cycles. Table 4. Coordinate critical points and ultimate loading points of the Kanchuang frame. Table 4. Coordinate critical points and ultimate loading points of the Kanchuang frame. Points O2 A B C O1 D E F Points O A B C O D E F 2 −0.046 −0.464 −0.998 1−2.301 0.046 0.462 1.002 2.311 The first cycle −0.187 −0.913 −1.007 −2.511 0.334 0.878 1.011 1.906 D 0.046 0.464 P 0.998 2.301 0.046 0.462 1.002 2.311 The first cycle 0.187 0.913 1.007 2.511 0.334 0.878 1.011 1.906 P −0.046 −0.461 −1.001 −2.309 0.046 0.459 1.005 2.304 The second cycle D 0.046 0.461 1.001 2.309 0.046 0.459 1.005 2.304 −0.173 −0.856 −1.051 −2.337 0.332 0.778 0.972 1.933 The second cycle P 0.173 0.856 1.051 2.337 0.332 0.778 0.972 1.933 −0.046 −0.461 −1.001 −2.309 0.046 0.459 1.005 2.304 The third 0.046 cycle 0.461 1.001 2.309 0.046 0.459 1.005 2.304 The third cycle −0.176 −0.856 −1.051 −2.337 0.303 0.778 0.972 1.933 P 0.176 0.856 1.051 2.337 0.303 0.778 0.972 1.933 Table 5. Equations of the envelope curves of the Kanchuang frame. Table 5. Equations of the envelope curves of the Kanchuang frame. Regression Equations Line The First Cycle The Se Regression cond CycleEquations The Third Cycle Line P =1.642Δ− 0.099 P =1.546Δ− 0.105 The First Cycle The Second Cycle The Third Cycle OA P =1.741Δ− 0.107 P =1.642Δ− 0.099 P =1.546Δ− 0.105 OA P= 1.741D 0.107 P= 1.642D 0.099 P= 1.546D 0.105 AB P=Δ 0.214 −P 0.814 = 0.214D 0.814 P=Δ 0.361 P=−0.361 0.690D 0.690 P=Δ 0.397 P= 0.397−D 0.6 30.636 6 AB BC P= 1.007D 0.555 P= 0.983D 0.066 P= 0.957D 0.076 P =0.214Δ− 0.814 P =0.361Δ− 0.690 P =0.397Δ− 0.636 OD P= 1.309D + 0.273 P= 1.080D + 0.282 P= 1.069D + 0.253 P=Δ 1.007 + 0.555 P=Δ 0.983 − 0.066 P=Δ 0.957 − 0.076 DE P= 0.246D + 0.765 P= 0.354D + 0.615 P= 0.333D + 0.595 BC EF P =1.007Δ− 0.555 P =0.983Δ− 0.066 P =0.957Δ− 0.076 P= 0.786D + 0.198 P= 0.740D + 0.228 P= 0.704D + 0.225 P=Δ 1.309 + 0.273 P=Δ 1.080 + 0.282 P=Δ 1.069 + 0.253 OD 4.2. Stiffness of the Hysteretic Loops P =1.309Δ+0.273 P =1.080Δ+0.282 P =1.069Δ+0.253 The stepsPof=Δ the 0.246 investigation + 0.765 of the stif P=Δ fness 0.354 of the+ 0.61 hyster 5 etic loops P=Δ wer 0.333 e followed. + 0.595 First, DE the characters of the hysteretic loops were analyzed in order to investigate the changing P =0.246Δ+0.765 P =0.354Δ+0.615 P =0.333Δ+0.595 rules of the stiffness. The findings show that the hysteretic loops of the Kanchuang frame P=Δ 0.786 + 0.198 P=Δ 0.740 + 0.228 P=Δ 0.704 + 0.225 EF were in the elastic stage when the loading displacements were under 10 mm cycles. If P =0.786Δ+0.198 P =0.740Δ+0.228 P =0.704Δ+0.225 the loading displacements reached and exceeded 15 mm (the drift angle was 3/320), each hysteretic loop was divided into eight feature segments, and M, L, N, O, P and Q were the 4.2. Stiffness of the Hysteretic Loops intersection points of each segment (see Figure 16). K was used to represent the stiffness The steps of the investigation of the stiffness of the hysteretic loops were followed. of each feature segment (see Table 6). Secondly, dimensionless processing was conducted First, the characters of the hysteretic loops were analyzed in order to investigate the on the experimental data of the hysteretic loops based on Equations (1) and (2). Third, the changing rules of the stiffness. The findings show that the hysteretic loops of the Kan- regression analysis of dimensionless data points was obtained from the fitting equations chuang frame were in the elastic stage when the loading displacements were under 10 of K (see Table 7). Because of the discreteness of the data of K and K , no appropriate i 3 7 mm cycles. If the loading displacements reached and exceeded 15 mm (the drift angle was fitting equations were found. Average values were used and adopted to build the hysteresis 3/320), each hysteretic loop was divided into eight feature segments, and M, L, N, O, P model. Figure 17 shows the fitted curves, and it was found that the fitting curves simulated the experimental data. i and Q were the intersection points of each segment (see Figure 16). was used to P Buildings 2022, 12, x FOR PEER REVIEW 16 of 23 represent the stiffness of each feature segment (see Table 6). Secondly, dimensionless pro- cessing was conducted on the experimental data of the hysteretic loops based on Equa- tions (1) and (2). Third, the regression analysis of dimensionless data points was obtained from the fitting equations of (see Table 7). Because of the discreteness of the data of K K 3 7 and , no appropriate fitting equations were found. Average values were used and adopted to build the hysteresis model. Figure 17 shows the fitted curves, and it was found Buildings 2022, 12, 887 15 of 22 that the fitting curves simulated the experimental data. Figure 16. Feature segments of hysteresis curves. Figure 16.Feature segments of hysteresis curves. Table 6. Parameters of each feature segment. Table 6. Parameters of each feature segment. Loading Loading Rapid Unloading Loading Loading Rapid Unloading Rapid Rapid Feat Feature ure Loading Loading Strengthen Unloading Loading Loading Strengthen Unloading Elastic Strengthen Un Unloading loading Elastic Elastic Strengthen Unloading Unloading Elastic Segme Segment nt Elastic Stage 1 Stage 1 Elastic Stage 1 Elastic Stage 2 Stage 2 Elastic Stage 2 Stage 1 Stage 1 SStage tage 1 1 Stage 1 Stage 2 Stage 2 Stage Stage 2 2 Stage 2 Represent Represent K K K K K K K K 1 2 3 4 5 6 7 8 K K K K K K K K parameter 1 2 3 4 5 6 7 8 parameter Table 7. Fitting equations of the stiffness. Table 7. Fitting equations of the stiffness. Feature Segment Fitting Equations Feature Segment Fitting Equations −0.476 Loading elastic stage 1 K =0.424Δ+ 0.103 0.476 Loading elastic stage 1 K = 0.424D + 0.103 1 −3.186 K =0.021Δ+1.302 Loading strengthen stage 1 3.186 Loading strengthen stage 1 K = 0.021D + 1.302 K =12.52 Rapid unloading stage 1 Rapid unloading stage 1 K = 12.52 −0.319 K =1.192Δ− 0.975 Unloading elastic stage 1 0.319 Unloading elastic stage 1 K = 1.192D 0.975 −0.186 Loading elastic stage 2 K =2.07Δ−1.511 0.186 Loading elastic stage 2 K = 2.07D 1.511 −− 4 4.905 K =8.183×Δ 10 + 1.585 Loading strengthen stage 2 4.905 6 4 Loading strengthen stage 2 K = 8.183 10 D + 1.585 K =12.92 Rapid unloading stage 2 Rapid unloading stage 2 K = 12.92 −0.020 0.020 K =27.140Δ− 26.98 Unloading elastic stage 2 Unloading elastic stage 2 8 K = 27.140D 26.98 Based upon the analysis on the characteristics of the hysteretic loops, Equation (2) calculated the fitting curves between the displacement cycles and horizontal ordinates of the intersection points. 4M, 4L, 4N, 4O, 4P and 4Q are the dimensionless horizontal ordinates of the intersection points. Linear regression analysis was conducted on the points, and the fitting equations are shown in Table 8. The fitted curves are shown in Figure 18, and the data are close to the fitting line. Table 8. Fitting equations of points M, L, N, O, Pand Q. Points Fitting Equations D = 0.619D + 0.136 P D = 0.031D + 0.360 Q D = 0.950D 0.023 M D = 0.894D 0.017 L D = 0.048D 0.330 D = 0.618D 0.013 N Buildings 2022, 12, 887 16 of 22 Buildings 2022, 12, x FOR PEER REVIEW 17 of 23 4.0 3.5 3.0 −3.186 −0.476 2.5 K =Δ 0.021 + 1.302 K=Δ 0.424 + 0.103 2 3 R = 0.848 2.0 R = 0.953 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Δ Δ (a) (b) −0.319 K =Δ 1.192 − 0.975 K = 12.52 R = 0.994 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 (c) (d) 2.5 4.0 3.5 2.0 −0.186 3.0 −− 44.905 K =Δ 2.07 − 1.511 K =× 8.183 10 Δ + 1.585 1.5 R = 0.955 R = 0.598 2.5 1.0 2.0 0.5 1.5 0.0 1.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 (e) (f) 20 3.0 18 2.5 −0.020 K=Δ 27.140 − 26.98 16 2.0 R = 0.934 14 K = 12.92 1.5 12 1.0 10 0.5 0.0 6 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Δ Δ (g) (h) Figure 17. Stiffness degradation curves and fitting equations of the Kanchuang frame at different loading states: (a) K ; (b) K ; (c) K ; (d) K ; (e) K ; (f) K ; (g) K ; (h) K . 1 2 3 4 5 6 7 8 K K 7 5 K 1 6 K 4 K 2 Buildings 2022, 12, x FOR PEER REVIEW 18 of 23 Figure 17. Stiffness degradation curves and fitting equations of the Kanchuang frame at different K K K K K K K K 1 2 3 4 5 6 7 8 loading states: (a) ; (b) ; (c) ; (d) ; (e) ; (f) ; (g) ; (h) . Based upon the analysis on the characteristics of the hysteretic loops, Equation (2) calculated the fitting curves between the displacement cycles and horizontal ordinates of the intersection points. △M, △L, △N, △O, △P and △Q are the dimensionless horizontal ordinates of the intersection points. Linear regression analysis was conducted on the points, and the fitting equations are shown in Table 8. The fitted curves are shown in Fig- ure 18, and the data are close to the fitting line. Table 8. Fitting equations of points M, L, N, O, Pand Q. Points Fitting Equations O Δ− = 0.619Δ+ 0.136 ΔΔ =0.031+ 0.360 ΔΔ =0.950− 0.023 M ΔΔ =-0.894− 0.017 Buildings 2022, 12, 887 Δ− = 0.048Δ− 0.330 17 of 22 N ΔΔ =0.618− 0.013 0.0 0.43 -0.2 Δ= 0.031Δ+ 0.360 0.42 R = 0.920 -0.4 -0.6 0.41 -0.8 0.40 -1.0 Δ= −0.619Δ+ 0.136 0.39 R = 0.955 -1.2 0.38 -1.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0.0 0.5 1.0 1.5 2.0 2.5 (a) (b) 0.0 2.0 -0.2 -0.4 Δ= 0.950Δ− 0.023 Δ= −0.894Δ− 0.017 Q M -0.6 R = 0.998 R = 0.992 1.5 -0.8 -1.0 1.0 -1.2 -1.4 -1.6 0.5 -1.8 -2.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Buildings 2022, 12, x FOR PEER REVIEW 19 of 23 Δ Δ (c) (d) 1.4 -0.35 -0.36 1.2 Δ= 0.618Δ− 0.013 -0.37 R = 0.955 Δ= −0.048Δ− 0.330 1.0 -0.38 R = 0.794 0.8 -0.39 0.6 -0.40 -0.41 0.4 -0.42 0.2 -0.43 0.0 -0.44 0.0 0.5 1.0 1.5 2.0 2.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 (e) (f) Figure 18. Fitting curves of the piecewise points: (a) Point O; (b) Point P; (c) Point Q; (d) Point M; Figure 18. Fitting curves of the piecewise points: (a) Point O; (b) Point P; (c) Point Q; (d) Point M; (e) Point L; (f) Point N. (e) Point L; (f) Point N. 4.3. Hysteretic Rule 4.3. Hysteretic Rule The hysteretic model plays a crucial role in the nonlinear seismic response analysis of The hysteretic model plays a crucial role in the nonlinear seismic response analysis structures. The seismic response analysis results may be diverse when different hysteretic of structures. The seismic response analysis results may be diverse when different hyster- models are adopted. Different kinds of hysteretic model have been widely used to simulate etic models are adopted. Different kinds of hysteretic model have been widely used to the hysteretic behaviours of different structures. However, it was found that these models simulate the hysteretic behaviours of different structures. However, it was found that cannot simulate the strength degradation and the pinch phenomenon of the hysteretic these models cannot simulate the strength degradation and the pinch phenomenon of the loops of the Kanchuang frame. A new hysteretic model was proposed to simulate the hysteretic loops of the Kanchuang frame. A new hysteretic model was proposed to simu- hysteretic loops of the Kanchuang frame (see Figure 19). late the hysteretic loops of the Kanchuang frame (see Figure 19). The hysteresis rule is as follows. The loading path follows the envelope curves to point H first. Then, unloading along the paths H–Q with stiffnesses of . And unload- ing along path Q–P . Following this, the curve enters a new stage, loading along paths K K 5 6 P–O and O–G with stiffnesses of and . G is the peak point of the hysteretic loop. The negative unloading is then following along paths G–M and M–L with stiffnesses of K K K 7 8 1 and . It then loads along path L-N with the stiffness of . When the displace- ment load exceeds point N, the loading stiffness changes to , loads to the positive peak value H, and the first loading cycle finishes. After the loading displacement exceeds the initial loading displacement, its loading along the envelope curve is from point H to H’, and enters another loading cycle—H’-Q’-P’-O’-G’-M’-L’-N’ -H’—with stiffnesses of , K K K K K K K 2 3 4 5 6 7 8 , , , , , and . This is similar to the previous cycle. H ′ N ′ L′ Q ′ L Q M P′ O′ G′ Figure 19. Hysteretic model. L Δ Q Δ Δ Δ N P M Buildings 2022, 12, x FOR PEER REVIEW 19 of 23 1.4 -0.35 -0.36 1.2 Δ= 0.618Δ− 0.013 R = 0.955 -0.37 Δ= −0.048Δ− 0.330 1.0 -0.38 R = 0.794 0.8 -0.39 -0.40 0.6 -0.41 0.4 -0.42 0.2 -0.43 0.0 -0.44 0.0 0.5 1.0 1.5 2.0 2.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 (e) (f) Figure 18. Fitting curves of the piecewise points: (a) Point O; (b) Point P; (c) Point Q; (d) Point M; (e) Point L; (f) Point N. 4.3. Hysteretic Rule The hysteretic model plays a crucial role in the nonlinear seismic response analysis of structures. The seismic response analysis results may be diverse when different hyster- etic models are adopted. Different kinds of hysteretic model have been widely used to simulate the hysteretic behaviours of different structures. However, it was found that these models cannot simulate the strength degradation and the pinch phenomenon of the hysteretic loops of the Kanchuang frame. A new hysteretic model was proposed to simu- late the hysteretic loops of the Kanchuang frame (see Figure 19). The hysteresis rule is as follows. The loading path follows the envelope curves to point H first. Then, unloading along the paths H–Q with stiffnesses of . And unload- ing along path Q–P . Following this, the curve enters a new stage, loading along paths K K 5 6 P–O and O–G with stiffnesses of and . G is the peak point of the hysteretic loop. The negative unloading is then following along paths G–M and M–L with stiffnesses of K K K 7 8 1 and . It then loads along path L-N with the stiffness of . When the displace- ment load exceeds point N, the loading stiffness changes to , loads to the positive peak value H, and the first loading cycle finishes. After the loading displacement exceeds the initial loading displacement, its loading along the envelope curve is from point H to H’, Buildings 2022, 12, 887 and enters another loading cycle—H’-Q’-P’-O’-G’-M’-L’-N’ -H’—with stiffnesses of 18 of 22, K K K K K K K 2 3 4 5 6 7 8 , , , , , and . This is similar to the previous cycle. L′ Q L Q′ M ′ M P′ Figure 19. Hysteretic model. Figure 19. Piecewise hysteresis curves. The hysteresis rule is as follows. The loading path follows the envelope curves to point H first. Then, unloading along the paths H–Q with stiffnesses of K . And unloading along path Q–P K . Following this, the curve enters a new stage, loading along paths P–O and O–G with stiffnesses of K and K . G is the peak point of the hysteretic loop. The negative 5 6 unloading is then following along paths G–M and M–L with stiffnesses of K and K . It then 7 8 loads along path L-N with the stiffness of K . When the displacement load exceeds point N, the loading stiffness changes to K , loads to the positive peak value H, and the first loading cycle finishes. After the loading displacement exceeds the initial loading displacement, its loading along the envelope curve is from point H to H’, and enters another loading cycle—H’-Q’-P’-O’-G’-M’-L’-N’ -H’—with stiffnesses of K ,K ,K ,K ,K ,K ,K and K . This 1 2 3 4 5 6 7 8 is similar to the previous cycle. 5. Comparison of the Hysteretic Curves between the Analysis and Test The restoring force model and the experimental loading process were used to obtain the calculated hysteresis curves. Figure 20 shows the comparations of the experimental results and the calculation results of the Kanchuang frame. The test curves are coloured in blue, and the calculation results are red. It was found that the calculation curves are similar Buildings 2022, 12, x FOR PEER REVIEW 21 of 24 to the experimental curves; thus, the hysteretic model simulates the strength degradation, stiffness degradation and pinching effect of the experimental curves. Based on the model, the nonlinear seismic analysis of the traditional timber structure can be used. (a) Figure 20. Cont. (b) (c) Figure 20. Comparison between the calculation results and the experimental results: (a) first dis- placement cycle; (b) second displacement cycle; (c) third displacement cycle. 6. Conclusions A 1/2-scale model of a timber frame with a timber window and infilled masonry wall was tested. The failure modes, stress and stiffness of the structure were investigated. Moreover, the characters of the envelope curves and hysteretic loops were studied. Based on the results of this research, the following conclusions can be drawn: (1) The test results show that the loading process can be divided into three stages. At the elastic stage, no apparent damage was found through observation except for slight cracks on the wall. During the elastoplastic stage, the cracks on the wall became N Buildings 2022, 12, x FOR PEER REVIEW 21 of 24 Buildings 2022, 12, 887 19 of 22 (a) (b) (c) Figure 20. Comparison between the calculation results and the experimental results: (a) first dis- Figure 20. Comparison between the calculation results and the experimental results: (a) first displace- placement cycle; (b) second displacement cycle; (c) third displacement cycle. ment cycle; (b) second displacement cycle; (c) third displacement cycle. 6. Conclusions 6. Conclusions A 1/2-scale model of a timber frame with a timber window and infilled masonry A 1/2-scale model of a timber frame with a timber window and infilled masonry wall wall was tested. The failure modes, stress and stiffness of the structure were investigated. was tested. The failure modes, stress and stiffness of the structure were investigated. Moreover, the characters of the envelope curves and hysteretic loops were studied. Based Moreover, the characters of the envelope curves and hysteretic loops were studied. Based on the results of this research, the following conclusions can be drawn: on the results of this research, the following conclusions can be drawn: (1) The test results show that the loading process can be divided into three stages. At the (1) The test results show that the loading process can be divided into three stages. At the elastic stage, no apparent damage was found through observation except for slight elastic stage, no apparent damage was found through observation except for slight cracks on the wall. During the elastoplastic stage, the cracks on the wall became longer, cracks on the wall. During the elastoplastic stage, the cracks on the wall became wider and deeper as the loading displacement increased, and the gaps between the mortises and tenons steadily increased as the loading displacement increased. During the final new elastoplastic stage, the stiffness of the whole structure increased after the masonry wall collapsed. Brittle shear failure was observed in the masonry infill wall. At the end of the test, the masonry wall collapsed, but the timber frames did not fall apart. Slight cracks also appeared on the surface of the Lingtiao. (2) The pinching effect was observed from the hysteretic loops of the Kanchuang frame, indicating an occurrence of a slip between the timber components. The bearing capacity and stiffness of the frame were decreased but not lost, showing that the timber frame has good bearing and deformation capacities. (3) A dimensionless hysteretic model for the Kanchuang frame was established based on test results and numerical analysis. This model simulates the experimental curves’ strength degradation, stiffness degradation and pinching effect. The calculation results were consistent with the experimental results. They provide references for dynamic analyses of the traditional timber structure under dynamic loads. Buildings 2022, 12, 887 20 of 22 (4) This study provided a useful reference for the seismic evaluation and preservation of cultural heritage. This study also conducted dynamic analyses of the traditional timber structure under dynamic loads. However, in this study, the experimental and analytical studies were carried out on a scaled specimen, not a full-scaled one. Thus, further studies will be conducted to investigate whether this hysteretic model could apply to other frames of different dimensions. Moreover, the seismic and hysteretic behaviours of more timber frames with different types of infilled walls from ancient timber buildings will be studied. Author Contributions: Conceptualization, J.H. and X.G.; methodology, J.H. and X.G.; formal anal- ysis, J.H., X.G. and Z.G.; investigation, J.H., X.G., Z.G., T.Y., T.C. and Z.S.; writing—original draft preparation, J.H.; writing—review and editing, X.G., Z.G., T.Y., T.C. and Z.S.; supervision, X.G.; project administration, X.G.; funding acquisition, X.G. All authors have read and agreed to the published version of the manuscript. Funding: This study was financially supported by the National Key R&D Program of China (Grant Number 2019YFC1520803) and the Beijing Municipal Commission of Education–Municipal Natural Science Joint Foundation: “Research on Seismic Performance Evaluation of Beijing Ancient Timber Buildings Based on Value and damage Characteristics” (No. KZ202010005012). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The support of the National Key R&D Program of China (Grant Number 2019YFC1520803) and the Beijing Municipal Commission of Education–Municipal Natural Science Joint Foundation (Grant number KZ202010005012) is highly appreciated. Conflicts of Interest: The authors declare no conflict of interest. References 1. Zhou, Q.; Yan, W.M.; Yang, X.S.; Bao, J.J. Damage of ancient Chinese architecture caused by the Wenchuan earthquake. Sci. Conserv. Archaeol. 2010, 22, 37–45. 2. 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Journal

BuildingsMultidisciplinary Digital Publishing Institute

Published: Jun 23, 2022

Keywords: Kanchuang frame; restoring force model; ancient timber architecture; loading process

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