**Use of catastrophe theory to establish safety assessment model for timber-framed heritage buildings**,”

*BioResources*19(3), 6690-6710.

#### Abstract

Catastrophe theory was used to establish a safety assessment model to reduce the reliance on subjective judgments in evaluation of timber-framed heritage buildings. This study was conducted in three phases. Initially, a comprehensive evaluation index system was established from the perspective of foundation. It consisted of eight aspects and 25 safety evaluation indicators using superstructure load-bearing elements, maintenance structures, and their interconnections in timber-framed heritage buildings. The 25 safety evaluation indicators included foundation, base, stone piers, columns, beams, lintels (beams, pads, and other bending components), bracket sets, arches, maintenance walls, beam-brace connections, and roof structures. The bottom-level indicators in the index system were dimensionless. The second phase employed typical catastrophe models (cusp, swallowtail, and butterfly) for normalization, resulting in calculated catastrophe scales and evaluation levels. The case study of the Buddha Hall of Zhihua Temple, Beijing, was applied in the final phase. It was found that the catastrophe scales method solved the subjectivity issues in determining weights. Additionally, the calculations were found to be concise and reliable, providing accurate results. The model can be used as a theoretical reference for the future safety assessment of timber-framed heritage buildings.

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**Use of Catastrophe Theory to Establish Safety Assessment Model for Timber-Framed Heritage Buildings**

Wei Qian,^{a,b,c} ShuaiBing Li,^{a} and Wei Wang^{ a,b,c,}*

Catastrophe theory was used to establish a safety assessment model to reduce the reliance on subjective judgments in evaluation of timber-framed heritage buildings. This study was conducted in three phases. Initially, a comprehensive evaluation index system was established from the perspective of foundation. It consisted of eight aspects and 25 safety evaluation indicators using superstructure load-bearing elements, maintenance structures, and their interconnections in timber-framed heritage buildings. The 25 safety evaluation indicators included foundation, base, stone piers, columns, beams, lintels (beams, pads, and other bending components), bracket sets, arches, maintenance walls, beam-brace connections, and roof structures. The bottom-level indicators in the index system were dimensionless. The second phase employed typical catastrophe models (cusp, swallowtail, and butterfly) for normalization, resulting in calculated catastrophe scales and evaluation levels. The case study of the Buddha Hall of Zhihua Temple, Beijing, was applied in the final phase. It was found that the catastrophe scales method solved the subjectivity issues in determining weights. Additionally, the calculations were found to be concise and reliable, providing accurate results. The model can be used as a theoretical reference for the future safety assessment of timber-framed heritage buildings.

*DOI: 10.15376/biores.19.3.6690-6710*

*Keywords: Catastrophe theory; Timber structure; Heritage buildings; Safety assessment*

*Contact information: a: Faculty of Architecture, Civil and Transportation Engineering, Beijing University of Technology, Beijing 100124, China; b: Beijing Engineering Technology Research Center for Historic Building Protection, Beijing 100124, China; c: Key Science Research Base of Safety Assessment and Disaster Mitigation for Traditional Timber Structure (Beijing University of Technology), State Administration for Cultural Heritage, Beijing 100124, China; *Corresponding author: ieeww@bjut.edu.cn*

**INTRODUCTION**

Chinese architecture has an independent structural system. It has a long history and covers vast areas and regions of China. For thousands of years, although China has often been in contact with other countries in military and political aspects, the basic structure of architecture has not been affected by other influences and it still maintains its wooden structure. Chinese wooden structure building is not only the embodiment of the material form, but also carries the spiritual character and cultural gene characteristics of the Chinese nation, which has a very high historical, cultural and social value (Liang 2007). However, the safety protection of ancient buildings in China remains a concern. Many cultural heritage units are gradually deteriorating, and the damage to timber structures often goes unnoticed. Therefore, the safety identification and assessment of ancient buildings are crucial for preventive protection procedures.

The wooden structures are the main load-bearing system of timber-framed heritage buildings. Compared to modern buildings, heritage buildings have complex material properties and force systems, posing challenges and difficulties in safety assessment. Currently, evaluating approaches of the timber structures typically involve a combination of quantitative and qualitative methods. For instance, Cointe *et al.* (2007) used a method to assess the health of ancient wooden-framed structures based on field measurements and numerical coupling simulations. Garziera *et al*. (2025) utilized interferometric radar-based non-destructive testing to detect and evaluate the health conditions of ancient buildings. Lima *et al*. (2018) applied fiber optic grating sensors to monitor and assess the structural health of the Aviroda Church. In China, researchers have employed methods such as fuzzy comprehensive evaluation, grey system theory, neural network evaluation, and matter-element extension methods to investigate the safety assessment of timber-framed heritage buildings. However, these methods require the subjective determination of relevant indicator weights, influencing the objectivity of the assessment results. For example, Gu (2009) proposed a fuzzy comprehensive judgment theory for reliability assessment of brick and stone pagodas. Xu *et al*. (2017) applied analytic hierarchy process, grey fuzzy analysis, and grey whitening weight function to the safety assessment of wooden-framed heritage buildings. Additionally, fuzzy hierarchical analysis models and BP neural network models for the assessment of timber-framed heritage buildings have been used (Qin *et al*. 2017; Luo *et* *al*. 2020). Wang *et al*. (2022) applied the matter-element model to diagnose health and assess safety of ancient wooden structures, and Zhang *et al*. (2017) used an improved Elman neural network to predict the lifespan of ancient buildings. These methods mentioned above use subjective calculation of the weights.

Both Chinese and overseas researchers have applied catastrophe theory to research in various fields. For instance, Abrahamyan *et al.* (2023) applied catastrophe theory to voice quality research. Stamovlasis *et al.* (2022) utilized catastrophe theory in neuro-psychological studies, exploring the nonlinear impact of depression on financial capacity in individuals with mild cognitive impairment and dementia. Another study employed catastrophe theory to establish a risk assessment model for sudden water and mud in karst tunnels (Zhu *et al.* 2020). The catastrophe theory was applied for predicting construction risks in subway stations (Jiang *et al.* 2020), to assess the danger of gas pipelines in collapsed mining areas (Shu *et al.* 2017), to assess the fire risk of high-rise civil buildings (Zeng 2021), to assess the safety risk of highway bridge construction (Li 2023), and to evaluate the fire hazard of ancient wooden structures (Gao *et al.* 2023). However, in the field of safety assessment of wood-framed ancient buildings, the application of catastrophe theory is not yet common. Changes in the safety of ancient buildings can be understood based on the changes in the various components within the building caused by the qualitative change of the system. The transition of the system from a safe state into a hazardous state can be regarded as a mutation phenomenon. Thus, the formation of the damage pathway in a wood-framed ancient building is also in line with the laws of catastrophe theory. Therefore, the safety assessment of wood-framed ancient buildings has certain compatibility with the catastrophe theory. In addition, the catastrophe theory also considers the relative importance of each evaluation index, and combines qualitative and quantitative, and mainly quantitative, which effectively reduces the interference of human factors on the results and makes the final results more objective. This study integrated catastrophe theory into comprehensive evaluation and established an index system for the safety assessment of timber-framed heritage buildings. Quantitative recursive operations were performed based on normalization formulas, calculating the final catastrophe scale values for the safety level of timber-framed heritage buildings.

**EXPERIMENTAL**

The catastrophe theory was developed by René Thom, a French mathematician in the last century (Zhou 1989). It is a mathematical theory that investigates the discontinuous and sudden changes occurring in dynamic systems during continuous developmental processes, and the interweaving relationships with continuous factors of change. Many study subjects do not exhibit a continuous state but rather manifest a particular state abruptly at a critical point. Using the concepts from topological dynamics and singularity theory, catastrophe theory uses a potential function and characterizes the changing states of study subjects by establishing a potential function. This characterization process distinguishes the critical points at which the study subject undergoes a change. The theory then analyzes the discontinuous changes on either side of these critical points. Ultimately, the elementary catastrophe models are developed (Ling 1987).

Typically, the variables in the potential function are divided into two categories based on the different states by which they are characterized. The first category is the state variables, primarily representing the behavioral states of the study subject itself. The second category is the control variables, used to characterize the factors influencing the changes in the variables on either side. Assuming the potential function is denoted as *f*(*x*), taking its derivative, and setting the 1^{st} derivative *f’*(*x*) = 0, yields its equilibrium surface. Working for the odd points set of the equilibrium surface with the 2^{nd} derivative *f”*(*x*) = 0 gives the bifurcation equation with only the control variables. When the control variables change to align with the equation, it signifies the occurrence of a catastrophe in the study subject. It allows the identification of critical points for each control variable causing the catastrophe (Li *et al*. 2011). Table 1 presents four typical elementary catastrophe models with their corresponding formulaic expressions.

**Table 1. **Four Common Catastrophe Models (Kang 2014)

Table 1 presents commonly used catastrophe models through meticulous calculations. While the computational process is intricate, the processes are relatively straightforward. Taking the swallowtail catastrophe model as an example, the general expression for its potential function is *f*(*x*) = *x*^{5}+*ax*^{3}+*bx*^{2}+*cx*. Calculating its first and second derivatives results in *df*(*x*) = 5*x*^{4}+3*ax*^{2}+2*bx*+*c*, *d ^{2}f*(

*x*) = 20

*x*

^{3}+6

*ax*+2

*b*, respectively. The bifurcation equation is obtained by eliminating the equilibrium surface

*df*(

*x*) = 0 and the odd points set

*d*(

^{2}f*x*) = 0.

From the bifurcation equations of the catastrophe models (see Table 1), the normalization equations for each model are developed. Given the various states of variables within the system, it is convenient to normalize the values of control and state variables to the range [0, 1]. This normalization process aligns with the principles of fuzzy membership functions, facilitating the direct calculation of the overall catastrophe membership function values using the provided formulas. Equations 1 to 4 below correspond to the normalization equations for the four common catastrophe models presented in Table 1.

For the fold catastrophe model:

Figure 1 shows schematic diagrams of the four common catastrophe models.

**Fig. 1.** Schematic diagram of four common mutation Model system

**The Process of Safety Assessment of Wood-framed Ancient Buildings**

The flowchart of this study on the safety assessment of ancient wood-framed buildings is shown below (Fig. 2):

**Fig. 2. **The process of safety assessment of wood-framed ancient buildings

**Safety Assessment of Timber-Framed Heritage Buildings (Based on Catastrophe Theory)**

*Safety assessment index system for timber-framed heritage buildings*

The factors influencing the safety of wooden components in ancient buildings are numerous, encompassing both quantitative and qualitative aspects. These factors are not entirely independent, and they continuously interact with each other. Therefore, when selecting indicators, priority should be given to those that can reflect the maximum amount of information with the least number of measurements. The structural characteristics of wooden structures in ancient buildings are considered. Standards such as “Reliability appraisal standards for civil buildings,” (GB 50292 2015), “Seismic appraisal standards for buildings,” (GB 50023 2009), “Technical specifications for maintenance and strengthening of wooden structures in ancient buildings,” (GB 50165-92 1993), and relevant literature are incorporated (Ma 2007; Gu 2009, Pan *et al*. 2016; Huan *et al*. 2019; Wang 2020; Wang *et al*. 2022). Then, the evaluation is categorized into eight aspects: foundation, base and stone activities, columns, beams (lintels, pads, and other flexural members), bracket sets, maintenance walls, beam-frame associations, and roof structures. A total of 25 assessment indicators are selected to establish an evaluation index system for wooden structures in ancient buildings, as illustrated in Fig. 3.

*Quantitative factor evaluation index grading standards*

Based on the varying levels of impact on safety, wooden structures in ancient buildings are classified into four categories (see Table 2), and the definitions of symbols in and Table 2 are explained (see Table 2.1). The standard for the evaluation indicators of safety factors in wooden structures of ancient buildings is used in this classification process.

**Fig. 3. **Wood structure safety evaluation index system

**Table 2. **The Assessment Index Classification Standard for Qualitative Factors