A risk index for the structural diagnosis of rockeries in Chinese classical gardens based on the MIVES model

Abstract

As an outstanding cultural heritage with both esthetic and historical significance, as well as a key component of Chinese classical gardens, rockeries are an extremely important cultural heritage that could be affected by a variety of man-made and natural factors; thus, a comprehensive and detailed assessment of their structural conditions is necessary. Structural risk, as both the most significant and most influential type of risk facing rockeries, could have a devastating impact on rockeries. However, previous studies have not considered the development of systemic structural diagnostic tools for rockeries. Therefore, to diagnose structural conditions by obtaining objective indicators that enable an efficient comparison between different rockeries, this study proposes a structural risk assessment tool to calculate the structural risk index for rockeries in Chinese classical gardens. This method was developed based on the MIVES model, which is a multi-criteria decision-making process founded on the concept of value function. The proposed method emphasizes a detailed diagnosis of the structural conditions of rockeries based on objective data and scientific force analysis. Compared to the health risk assessment method developed for rockeries in Chinese classical gardens in previous studies, this method represents a more detailed step toward assessing the structural condition of rockeries. Once the initial health risk of a rockery has been determined, a structural risk assessment should be carried out for those with a high-risk rating so that their structural condition can be evaluated and a more targeted response strategy can be adopted. The application of this method to four case studies demonstrated its usefulness.

Introduction

As an important part of Chinese classical gardens, rockeries have rich historical and cultural connotations and artistic value^1,2,3. One of the most fascinating aspects of Chinese culture that sets it apart from other great cultures worldwide is its enthusiasm for rocks⁴. Rockeries reflect the exquisite skills and wisdom of ancient craftsmen and carry rich historical and cultural connotations. They are tangible representations of Chinese gardening and mountain-stacking techniques, which are irreplaceable for the protection and inheritance of Chinese garden culture⁵. Therefore, the conservation and maintenance of rockeries in Chinese classical gardens are necessary.

Recently, several studies have focused on the conservation of rockeries, with some discussing the quantitative analysis method of rockeries, methods and technologies employed in monitoring rockeries^6,7,8, and the use of digital and information technologies to digitally survey rockeries^2,7,9,10,11. However, only a few studies have considered the diagnosis of rockeries. At present, the protection of rockeries is mostly based on the experience of rockery artisans, which is subjective and unscientific, and the lagging protection measures often miss the time of protection, giving way to irreversible damage to the rockery. Scientifically diagnosing the health risks of rockeries and preventing their damage is the key to timely and effective protection of rockery heritage.

According to previous research¹², structural risk is the biggest factor affecting the health condition of rockeries. Because most of the rockeries have undergone hundreds of years of wind and sun exposure, the weathering and deterioration of the surface of rockeries have been stable, do not have a significant impact on rockeries. However, as rockeries are made of irregular stones and heterogeneous structures combined with unique stacking techniques, their structures are very complex and, therefore, relatively fragile. Our research team conducted on-site surveys of nine classical Chinese gardens listed as World Heritage Sites. The results indicated that structural cracks of varying degrees appeared on the rockeries in seven of the nine gardens. The results also indicated the importance of the structural diagnosis of rockeries. However, the structural diagnosis of rockeries in Chinese classical gardens remains challenging. How to scientifically and effectively collect and analyze the relevant data, and determine the risk level of the rockeries has become the current issue that need to be resolved urgently. Previous studies have not yet established a methodology that is applicable to rock structure risk assessment. Therefore, it is necessary to develop a methodology that could be applied to the risk assessment of rockery structures.

With the progress of science and technology, heritage conservation no longer relies solely on traditional experience and intuition but increasingly incorporates scientific and technological means. Cultural heritage conservation has also begun to move towards digitalization. Finite element analysis has long been applied to other immovable cultural relics such as cave temples, ancient buildings, and ancient city walls to provide decision-making references for heritage preventive conservation work^13,14,15. To understand rockery structures more accurately, applying finite element analysis technology to the risk assessment of rockery structures could help develop rockery heritage protection in a scientific and precise direction¹⁶.

Owing to the intricacy of heritage monuments, a thorough risk assessment should take data from multiple diagnostic procedures into account¹⁷. Therefore, multi-criteria decision-making methods based on the analysis of different sets of parameters have been successfully applied in risk assessment projects for cultural heritage^12,17,18,19. Among them, MIVES (Spanish acronym: Modelo Integrado de Valor para una Evaluación Sostenible, in English: Integrated Value Model for Sustainability Assessment) has been successfully applied to obtain a specific risk index for defining buildings^17,19,20. MIVES can solve predefined generic problems using a value function or index^19,21. It can assist in handling both homogeneous and heterogeneous alternatives and analyze intricate interactions between numerous divergent criteria^22,23, enabling decision-makers to rank various alternatives in order of importance. Therefore, after the structural analysis of rockeries, multi-criteria decision-making methods based on the MIVES model could combine different indicators to calculate the structural risk index of rockeries.

Considering the distinctive attributes and exceptional importance of rockeries in Chinese classical gardens, this study aims to develop a scientific structural risk assessment method based on the MIVES model. This method provides objective and quantitative evaluation indicators to aid decision-makers in diagnosing the structural integrity of rockeries. The suggested methodology was created utilizing objective, quantitative data gathered from technical surveys to provide a consistent method for diagnosing rock structures. To demonstrate its applicability, the method was applied to four case studies.

Structural risk assessment methodology for rockeries

Assessing the structural risks of rockeries is an intricate and thorough process. To achieve better results, it is essential to establish a scientific and standardized assessment method, and key indicators must be generated to measure the risk level of the structural conditions.

Figure 1 illustrates the proposed methodology, which comprises the following steps. First, historical information and documentation must be collected to obtain reliable information on the structural history of the rockeries. The disease status of rockeries should then be marked and documented by an in-situ investigation. Three-dimensional (3D) scanning and modeling of rockeries must be performed to provide data for force analysis based on ABAQUS¹³. Structural and force analyses of rockeries can provide an initial scientific understanding of their structure.

Fig. 1

Flowchart illustrating the proposed method for assessing structural risks.

Second, it is necessary to introduce key indicators that measure the structural risk level of rockeries, after which a comprehensive structural risk-assessment indicator system could be developed. The weights of the indicators are then determined^24,25.

Third, using the weights of each indicator, the general value function in the MIVES model is used to calculate the structural risk index of the rockeries. Once the structural risk indices are calculated, decision-makers can decide on the most effective action to improve the structure condition of rockeries. To determine if the protective measures are effective, the risk assessment procedure must be repeated at least once a year after decisions on potential courses of action have been made.

Structural stability analysis based on ABAQUS

ABAQUS, a commonly used finite element engineering simulation software, has unique advantages for analyzing complex mechanical structures^26,27. This software has various built-in material models and databases, which can simulate the basic properties and structures of common materials. It is particularly suitable for dealing with complex structural problems and nonlinear problems in various material models^28,29. It has been widely used to analyze building structures and concrete material^30,31,32.

As a typical complex structural nonlinear problem, the geometric model, material properties, load distribution, and boundary conditions of the rockery structure could be obtained by combining different option blocks in ABAQUS software. This software comprehensively simulated and analyzed the force state of the rock. Therefore, ABAQUS was selected as the finite element simulation and analysis carrier for subsequent finite element analysis.

A complete ABAQUS analysis process typically consists of three steps: a pre-processing stage using ABAQUS/CAE or other pre-processing software to graphically generate *. in-format model, the simulation stage using ABAQUS/Standard or ABAQUS/Explicit numerical model solving, and the solution process. The time required depends on the complexity of the analyzed model and the computing power of the computer. The postprocessing stage could be evaluated using the visualization module of ABAQUS/CAE or other postprocessing software to evaluate the calculation results, which could be displayed using cloud diagrams, animations, curve diagrams, and other methods. According to the characteristics of rockeries, the specific workflow of the finite element simulation and analysis could be divided into five parts: model construction, material property settings, mesh delineation, load application, and solution calculation.

• (1) Model construction

  Model construction is the basis of the finite element simulation and analysis. Based on the point cloud model obtained from the 3D digital mapping of the rock, reverse engineering modeling could establish a 3D solid model of the overall rock to clarify the overall structural stress points and weak points of the rock. After the overall and local modeling of the rockery, create a new ABAQUS/Standard or ABAQUS/Explicit working file and set the working directory, import the 3D model into ABAQUS software in the form of “parts,” and pay attention to the follow-up process to maintain the unity of the unit so that the overall structural stress and weak points could be clarified. Set the units of length, pressure, and mass to mm, MPa, and t, respectively.

• (2) Material properties

  According to previous research on rockery materials and experimental data from previous researchers³³, the general material parameters of rockeries could be obtained, and new material properties could be created in the properties module in ABAQUS. The material parameters were set according to this study’s requirements: the density was set to 2.7 × 10-9, the modulus of elasticity was 50,000, the Poisson’s ratio was 0.25, the cross-section was defined for the resulting material, and the cross-section properties were assigned to specific components.

• (3) Mesh delineation

  Accurate model meshing is a key step in finite-element numerical simulations. The mesh quality significantly affects the numerical simulation results. A combination of automatic division and human control was adopted in mesh division based on the volume and external morphology of the stone block. When the model is simple, the meshing tool in ABAQUS software could perform hexahedral or tetrahedral meshes. When the model structure is more complex, the automatic meshing tool makes it difficult to meet the actual requirements, and the model must be manually divided with the help of other meshing tools. In this study, ABAQUS used a meshing tool to perform delineation for the geometric model of the combination of techniques. For the 3D model of the rockery, owing to its large volume, more surface details, a large number of meshes, and the original mesh data for the calculation will lead to a lack of software arithmetic; hence, a Hyper Mesh was used to carry out mesh delineation work and then export it in *. inp format back into ABAQUS for subsequent calculations.

• (4) Load applications

  Because it is difficult to investigate the underground foundation of the rockery accurately, the foundation part is not considered in the analysis process, and only the above-ground part of the rockery is considered as the solution object in the load application stage. The underground foundation of the rockery was ignored, and only the influence of gravity on the aboveground part of the rockery was considered. The external load was set as gravity, and the acceleration of gravity was set in the z-direction with a value of -9800. The bottom of the rockery was constrained and set as “completely fixed” while setting the boundary conditions. The constraints were set as “completely fixed.”

• (5) Calculate the solution

  Create a new job in the job module, check the finite element model in the job manager, and click Submit to solve the calculation after ensuring that the model is correct. The calculated results are presented in *. odb format in the preset working directory, and the results are viewed in the visualization module, which shows the simulation results of the displacement, stress, and strain of the rockery model through the stress cloud and strain cloud.

Risk assessment based on the MIVES model

MIVES is a Multi-criteria Decision-Making method based on Multi-attribute Utility Theory, incorporating the concept of value function to construct a model for decision-making. The MIVES model has been used in sustainability assessments of modern buildings^34,35,36, and increasingly in assessing the structural risks of built heritage¹⁷. The model operates in a straightforward manner, using a general value function formula to standardize indicator variables into a single evaluation unit. By combining these standardized variables with their respective weights, the risk index of the assessed object can be determined.

Therefore, in this study, the MIVES model was used to assess and calculate the structural risk indices of rockeries. The precise risk index enables landscape heritage managers to monitor changes in the rock’s structural risk index after each assessment cycle, allowing for appropriate action.

Table 1 Structural risk assessment indicator system of rockeries

Risk assessment indicator system

As rockeries are preserved in the natural environment while simultaneously being open to visitors, who are allowed to climb them, many risk factors affect their structural stability. Assessing the risk to structures is a complicated task, and the thoroughness of the indicators used is crucial for a methodical and comprehensive evaluation. Therefore, using the MIVES model, MCDM techniques were applied to develop a multi-level system, known as a decision tree, built around three factors to meet the objective of protecting the rockeries’ structure (Table 1).

• (1) Structural deterioration indicators

  Crack length (D1): As rockeries are composed of irregular monolithic stones, cracks of the same length have different severity for different stone sizes, and it is difficult to standardize judging standards. Therefore, in this study, the crack length index was set as the ratio of the current crack length to the extreme values of crack development length. The extreme values of crack development length are normally the stone length. This index was calculated using Eq.1:

  $$D1=\frac{{L}_{C}}{{L}_{S}}$$

  (1)

  where L_C is the current length of the crack, and L_S is the stone length.

  Crack width (D2): The width of a crack reflects its developmental stage. The wider the crack, the higher the structural risk of the rockery. It can be measured using a crack ruler. Crack widths are categorized into four classes: Class I(0–1 mm), Class II (1 mm–2 mm), Class III (2 mm–3 mm), and Class IV (greater than 3 mm) in studies related to masonry buildings and stone elements. When combined with rockery characteristics, this standard is too large and not directly applicable; the width of the cracks in the rockery was divided into five classes, Class I (0–0.5 mm), Class II (0.5 mm–1 mm), Class III (1 mm–2 mm), Class IV (more than 2 mm), and Class V (more than 5 mm).

  Crack density (D3): Typically, there is more than one crack in the rockeries. The higher the density of cracks in the rock, the worse the structural stability of the rock. This was calculated using Eq. 2:

  $$D3=\frac{L}{A}$$

  (2)

  where L is the total length of the cracks in the rock and A is the surface area of the rockery.

  Crack direction (D4): Considering the rock’s structural characteristics, the crack direction affects the structural stability of the rock. Therefore, in this study, the horizontal direction cracks had the lowest development risk, whereas the vertical direction cracks had the highest risk. Combined with the preliminary site survey results of many rockeries, this indicator has been divided into five levels. To establish the grading criteria, horizontal direction cracks for Level I were assigned the value of 1, those with a horizontal direction angle of <30° for Level II were assigned 2, and those with a horizontal direction angle >30° but < 60° for Level III were assigned 3, those with a horizontal direction angle > 60° but < 90° for Level IV were assigned 4, and those with a vertical direction for Level V were assigned 5.

  Stone defect area (D5): This was calculated using the ratio of the rockery surface area to the deteriorated region. To avoid an insignificant value, D5 is calculated as follows:

  $$D5=\frac{{A}_{i}}{A}\times 100$$

  (3)

  where A is the surface area of the rockeries, and A_i is the area of the stone defect area.

• (2) Physical vulnerability indicators

  Force analysis indicator (D6): The ratio of the maximum stress to the destructive stress obtained from the finite-element simulation of the rockery could determine the possibility of rockery-producing cracks. Destructive stress originates from the general parameters of the rock and represents the possible damage to the stone in this state. D6 was calculated as follows:

  $$D6=\frac{{S}_{max }}{{S}_{0}}$$

  (4)

  where S_max is the maximum of rockeries, and S₀ is the destructive stress of rockeries.

  Velocity ratio of P wave (D7): This index uses rock inhomogeneities to cause changes in the ultrasonic signal to determine the degree of weathering of the rock surface. In stone heritage conservation projects, ultrasonic measuring instruments can be used to measure the wave velocity of longitudinal waves on the rock surface and calculate the wave velocity ratio of the rock on the rock face of the rockeries to determine the degree of the weathering of the rockeries. The D7 can be calculated as follows:

  $$D7=\sqrt{\frac{{V}_{{pf}}}{{V}_{{px}}}}$$

  (5)

  Where Vpx indicates the P-wave velocity of fresh rock, and Vpf indicates the P-wave velocity of weathered rock. Usually, it is necessary to collect samples through boreholes to measure the wave velocity of the fresh rock. However, considering the principle of minimum intervention in heritage conservation, obtaining fresh rock; therefore, it is impossible to strictly measure the wave velocity of fresh rock. However, to measure the weathering coefficient of the rock in this study, the wave velocity of fresh rockery rock was set to Vpx = 6130 m/s according to the material properties of rocks. Then, based on the wave velocity ratio, the weathering of the rock masses can be divided into five levels (Table 2).

  Table 2 Assessment table for weathering degree of rockery

  Uniaxial compressive strength (D8): This index refers to the maximum crushing stress that the rock can withstand and is an important index for measuring the mechanical state of the rock. Usually, compressive strength can be obtained through loading–damage experiments. However, considering the principle of non-destructive protection of cultural relics, the Leeb hardness value of the rockery surface could be measured using a Leeb hardness tester, according to which the curvilinear relationship between the Leeb hardness value of HL and the uniaxial compressive strength was calculated using the following equation:

  $${UCS}=8\times {10}^{-6}H{L}^{2.5}$$

  (6)

  where the uniaxial compressive strength is given in MPa.

  Capillary water absorption coefficient (D9): This index measures the water absorption energy of the rock surface and reflects its degree of weathering and porosity. High water absorption energy indicates low engineering-use-performance. According to the national standard “natural stone test method part III: water absorption, bulk density, true density, true porosity test (GB/T 9966.3-2020)” could be obtained from the water absorption rate of the rock, based on the water absorption rate was calculated to the rock's capillary water absorption coefficient. D9 was calculated as follows:

  $$D9=\frac{W}{\sqrt{t}}$$

  (7)

  where W is the water absorption per unit area (kg/m²) and t is the test time (h).

  Antiquity of the rockery (D10): This was used as a proxy for heritage value. Examining historical documents yields the value of D10, which can be expressed as the number of years after the completion of the rockery construction. To prevent the value of D10 from becoming too large, it was calculated as

  $$D10=\frac{A}{1000}$$

  (8)

  where A is the number of years after the construction of the rockeries.

  Past structural intervention (D11): This indicator describes how the original rockery’s structure was changed. Throughout the hundred years since they were built, the original construction of rockeries has been altered to suit the needs of various owners. These modifications might cause instability in the structures of the rockeries. By examining historical documents, one could determine the value of D11, which could be measured as the total number of structural interventions made in the history of rockeries.

• (3) Environmental stress indicators

  Rockery inspection cycle (D12): The rocker inspection cycle reflects the protection level. Based on previous studies, this indicator could be divided into five levels: all-weather monitoring and daily, weekly, monthly, and yearly monitoring, with assigned values of one, two, three, four, and five, respectively. The value of D12 could be obtained by exploring the management profiles of the rockeries.

Maximum Visitor Capacity (D13): Because of the characteristics of rockeries, they could climb and be viewed. Therefore, tourists' climbing behavior could also affect the structural stability of rockeries. The D13 value was obtained by consulting with the rockery management department. To prevent the value of D13 from becoming too large, it was calculated as follows:

$$D13=\frac{{N}_{v}}{10000}$$

(9)

where N_v is the maximum visitor capacity.

Number of trees in the rockery area (D14): Typically, rockeries are planted with trees, and the tree growth continues to exert pressure on them. Therefore, the greater the number of trees on the rockery, the higher the risk of compromising its structural stability.

Weights assigned to each indicator

Following the establishment of the indicator system, the AHP was used to determine the weight value of each indicator³⁷. The widely used decision analysis method enables an analyst to systematically evaluate the importance of different criteria by creating a matrix for pairwise comparisons. Ten experts were surveyed using a questionnaire to create the judgment matrix. Experts were chosen based on two requirements. First, their research interests had to be related to rockeries in Chinese classical gardens; and second, they had to conduct field investigations on rockeries. In this study, there are multiple judgment matrices for expert scores, and the geometric mean needs to be computed by multiplying the corresponding values and then taking the 1/10th power, with ten denoting the number of experts, to get a summary matrix. Then, the weight value of each indicator can be calculated. The specific steps are as follows:

• (1) Construct the hierarchical structure model, which is divided into the target, criterion, and indicator layers according to the difference of attribute division.

• (2) Construct a judgment matrix, comparing the importance of each indicator at the same Level. In this study, Saaty’s 1–9 scale has been used to construct a judgment matrix.

• (3) Conduct a consistency test. If the consistency ratio is less than 0.1, it means that the judgment matrix passes the test. Otherwise, it is necessary to reconstruct it. In this study, the consistency ratio is 0.0516, which means that the judgment matrix passed the consistency test.

As Table 3 shows, compared to the other two measures, the indicators of structural deterioration are of great importance, especially because cracks are frequently responsible for the significant degradation of rockeries, resulting in structural issues. The second most important group of indicators was the physical vulnerability indicator, which involved a force-analysis indicator. Such an analysis could provide key information regarding the structural conditions of the rockeries. Finally, the data from the environmental stress indicators had the lowest weight. This does not mean that environmental stress could be ignored as it is important in long-term structural assessments. However, the decision to adopt a lesser weight was due to the lag in the impact of environmental stress on the rockery structure.

Table 3 The weight of each indicator obtained by AHP

Risk index calculated based on the MIVES model

After determining the weights of the assessment indicators, the rockery’s structural risk index must be calculated using an appropriate method. The mathematical functions established in the MIVES convert variables with different units and scales into a set of variables with the same units and scales³⁶. However, a comprehensive value function for full spectrum assessment has been developed by creating positive, negative, and piecewise functions³⁸. Since the risk index in this study has been defined between 0 and 5, this novel and comprehensive value function has not been used in this study. The general value function used to calculate the risk index for each assessment index was as follows:

$${V}_{{index}}={K}_{i}\times \left\{1-\exp \left[-{m}_{i}{\left(\frac{\left|{x}_{i}-{x}_{min }\right|}{{n}_{i}}\right)}^{{A}_{i}}\right]\right\}+{S}_{min }$$

(10)

$${K}_{i}=\frac{{S}_{max }-{S}_{min }}{1-\exp \left[-{m}_{i}{\left(\frac{\left|{x}_{max }-{x}_{min }\right|}{{n}_{i}}\right)}^{{A}_{i}}\right]}$$

(11)

Where V_index represents the risk value of each indicator after uniform treatment, which was combined to calculate the HRI indicators of the rockery health risk index. The range of V_index values in this study was controlled to be between 0 (S_min) and 5 (S_max), where 0 represents the lowest risk and 5 represents the highest risk. xⁱ represents the quantitative value of each assessment indicator. K_i is a coefficient used to adjust the range of the V_index values. x_max and x_min represent the values on the x-axis corresponding to the highest and lowest values of the evaluation indicator function, respectively. m_i and n_i are constants used to adjust the geometry of the mathematical equations, that is, the values of the vertical and horizontal coordinates of the approximate inflection points, respectively. A_i denotes the shape factor of the function that defines the line shape of the curve. If A_i<1, the function curve is concave; if A_i=1, the function curve is straight; if A_i>1, the function curve is convex. The application of this function was divided into three steps.

• (1) Define the trend properties of each indicator

  There are two types of trend attributes in the function: positive and negative. Depending on the unit of measurement indicator, an increase will lead to either an increase or a decrease in the risk of the rockery structure. To determine if an increase will lead to an increase (as opposed to a decrease) in the risk of the rockery structure, the incremental function should be used if the indicator is positive. Otherwise, the decremental function should be used.

• (2) Define the shape of the value function

  The value functions had four shapes: convex, concave, linear, and S-shaped. Concave functions are more appropriate for numerical changes around the value of the lowest risk indicator with little or no increase in the value at risk. Convex curve functions could be used when the value of the risk corresponding to an indicator starts at the lowest value and increases rapidly with respect to the index. This type of function curve could also be used when the values of most of the indices under study are close to the lowest risk value. The linear function reflects a steady increase in risk generated by the indicator. A certain proportionality exists throughout the range of the indicator values. In the absence of defined special requirements, this function is the default choice. The concave and convex functions are combined to form an S-shaped function. The value of the risk increased significantly at the central value, whereas it did not change significantly near the minimum and maximum values. This type of relationship is selected when most indicator values are concentrated in the intermediate interval between minimum and maximum risk.

• (3) Determine the correlation coefficient of the value function

  After determining the increasing and decreasing nature and shape of the function, the next step is to determine the values of m_i, n_i, and A_i in the calculation formula for each indicator function. According to relevant research, these values were calculated using the formulae listed in Table 4³⁹.

• (4) Risk index calculation

  After calculating the score of each indicator in the rockery structure risk assessment indicator system, the final rockery structure risk index was obtained by multiplying the score of each index by the combined weight of each indicator. The calculation formula is as follows:

  $${V}_{{HR}}=\sum {W}_{{index}}\times {V}_{{index}}$$

  (12)

  Where V_HR is the final calculation of the structural risk index of the rockery, and its structural risk index takes the value range of zero to five. Table 6 shows the corresponding risk level of each range. W_index is the combination weight of each indicator, V_index is the value of each indicator calculated by the value function.

Table 4 References for values of m_i and n_i, A_i

Based on the data presented in Table 4, the correlation coefficients for the general value of the MIVES model are shown in Table 5. After obtaining the correlation coefficients, MATLAB software was used to generate a function graph of the rockery structural risk indicators. Figure 2 shows the resulting graph.

Table 5 Correlation coefficients in the formulae for each indicator function

Fig. 2

Plot of structural risk functions for 14 indicators.

Table 6 Risk index corresponds to risk level

Risk mitigation measures

After the structural risk index is obtained, specific mitigation measures must be implemented to reduce the structural risks faced by the rockeries. The proposed method demonstrates that the structural risk index of rockeries could be utilized to select the optimal and effective mitigation measures. Figure 3 shows the corresponding measurements.

Fig. 3

Risk mitigation measures.

In response to a higher risk level, immediate action should be taken, including the initiation of a repair project, thorough investigation and design, and the implementation of salvage protection measures to urgently address issues with the rockery. Techniques such as grouting reinforcement and anchor reinforcement can be used to repair cracks. Restoration strategies should strictly adhere to relevant regulations and guidelines, following the principle of minimal intervention while prioritizing the conservation of the rockery’s authenticity and integrity. Meanwhile, a comprehensive monitoring system should be established to track key cracks in the landscape heritage with high precision.

For medium- and low-risk levels, long-term restoration efforts are necessary, including a detailed maintenance plan for rockeries and regular repair and reinforcement work. Special attention should be given to areas prone to issues like corrosion and cracks to ensure r structural soundness and safety. A regular diagnostic mechanism should be put in place to record the effectiveness of each maintenance task. This system will allow for timely identification of problems, adjustments to the maintenance schedule, and the creation of a long-term maintenance profile to guide future decisions.

The risk assessment must be systematically updated after the corresponding risk-mitigation action is taken. This was performed to evaluate the effectiveness of the mitigation action. More steps must be taken if the assessment results remain unchanged.

Application and results

Application to rockery at Ge Garden, Yangzhou, China

The rockery in the Ge Garden was chosen to demonstrate the effectiveness of the suggested structural risk assessment method. The Ge Garden is a masterpiece of the classical Chinese Garden located in Yangzhou, Jiangsu Province (Fig. 4). Ge Garden was listed as a nationally protected heritage site by the Chinese State Council in 1988. This rockery is in the southeast direction of holding the mountain building in Ge Garden (Fig. 4b), about 5.9m high, with an east-west length of 13.5 m, north-south length of 11.9 m; the stacking technique is relatively rich, with a certain representative of the rockeries in Chinese classical gardens⁴⁰.

Fig. 4: The Ge Garden and its environs.

a Case study location; b Ge Garden overview orthoimage; c Ge Garden rockery photographs.

To evaluate the structural dangers associated with stone artifacts, the field data must be systematically observed and analyzed. This makes it possible for experts to gather data from various domains and carry out a more thorough analysis to comprehend the structural circumstances⁴¹. Several preliminary in-situ investigations were conducted to assess the structural risk of the Ge Garden rockery and gain a thorough insight into its structural conditions.

To begin with, degradation data and a 3D model of the Ge Garden rockery were obtained by visual inspections and a 3D scan of the rockery (Fig. 5). A 3D scan can capture the point cloud data of the rockery. These data are then processed using Trimble Realworks software, sampled at a 50% rate, and encapsulated into a polygonal mesh with the encapsulation tool. Geomagic software is used to automatically check and repair the mesh. However, due to uneven point cloud data from the scanning process, there may be localized missing conditions, leading to holes and redundant surfaces in the mesh. These imperfections need to be manually patched to remove unnecessary parts and repair the holes. Once repaired, the model is rechecked in Geomagic, resulting in the final complete 3D model of the rockery. Then the 3D model could calculate the surface area of a rockery and could be inputted into ABAQUS to conduct a force analysis of the rockery.

Fig. 5

Reality-based 3D models of rockery in Ge Garden.

While 3D scanning technology accurately captures the external outline of the rockeries, it cannot detect the internal contacts and gaps between stones. To address this, the 3D model was adjusted in consultation with traditional rockery craftsman to estimate the internal structure. Although this method introduces some uncertainties, it is the most suitable approach given the current circumstances.

The 3D model was imported into ABAQUS software in *. inp format, material parameters were set, and loads were applied. Finally, the stresses in each rock direction under gravity loading were obtained via numerical simulations with a maximum stress concentration of 0.333 MPa and a destructive stress of 1.253 MPa (Fig. 6).

Fig. 6: Stress cloud diagram of the rockery in Ge Garden.

a South side view of stress cloud diagram. b North side view of stress cloud diagram.

Second, several nondestructive testing methods were used to determine the mechanical properties of the Ge Garden rockery. This includes ultrasound testing, which could obtain the value of D7; surface hardness testing, which could obtain the value of D8; and testing the water absorption coefficient by capillarity, which could obtain the value of D9. Ultrasonic testing was conducted using a portable ultrasonic detector (Proceq Pundit Lab+) to perform in-situ ultrasonic measurements on a rockery at an ultrasonic frequency of 54 kHz (Fig. 7c, d)⁴². Surface hardness testing was performed using a high-precision hardness tester to test the Leeb hardness value on the surface of the rockery; the high-precision hardness tester (COY TH110) was used in this study (Fig. 7a, b). The water absorption test of the rockstone was performed according to the Chinese National Standard (GB/T 9966.3-2020).

Fig. 7: Photographs of the site survey.

a Leeb hardness tester (COY TH110); b Surface hardness testing of rockery; c Portable ultrasonic detector (Proceq Pundit Lab+); d The ultrasonic testing of rockery.

Using the formulas and explanations from the preceding section, the values of each index could be determined once all in-situ examinations and studies of the Ge Garden rockery had been completed (Table 7).

Table 7 Indicator value of rockery at Ge Garden

Then, according to Eqs. (10) and (11), the risk value of each indicator was calculated and combined with the combined weight of each indicator (Table 3). Table 8 lists the indicator indexes and the final risk index.

Table 8 Indicator indexes and Risk Index

Application to other case studies

In addition to its application to the Ge Garden rockery, applications to three other classical Chinese rockeries in Suzhou, China—specifically, rockeries in the Master-of-Nets Garden, Lingering Garden, and Humble Administrator's Garden—are illustrated to demonstrate the application of the structural risk assessment method (Fig. 8).

Fig. 8: Location of the case studies.

a Rockery at the Humble Administrator’s Garden, b Rockery at the Lingering Garden, c Master-of-Nets Garden.

In 1997, these three gardens were included on the World Heritage List of UNESCO. Detailed in-situ investigations and force analyses were performed to assess the structural risks of these rockeries.

Table 9 summarizes the resulting structural risks for all four case studies. As shown, one case study was at a lower risk level, two were at a medium risk level, and one was at a higher risk level. This study’s results confirm the need to study the structures of rockeries.

Table 9 Risk index and final assessment results for all the case studies

Apart from the valuable information supplied by the structural risk index, the list of recommended activities that can be selected aligns with the ultimate risk level. Maintenance and preservation work is sufficient for the MNG rockeries. The GG and HAG rockeries require more detailed structural analysis and long-term restoration. The rockery at LG requires structural reinforcement measures, and nondamaged structural diagnosis activities are recommended.

Discussion

This study proposed a systematic methodology for structural risk assessment based on the MIVES model for rockeries. Indicators were developed based on data obtained from in-situ investigations and force analyses, and they were designed to improve the objectivity of the structural diagnosis process. By calculating the structural risk index of rockeries using this method, decision-makers could determine the necessary steps in order of priority, thereby encouraging the implementation of minimum interventions with a solid scientific foundation. Unlike the health risk assessment method for rockeries developed in previous research, this method provides a more detailed assessment and analysis based on the health risk assessment method. After determining the health risk level associated with rockeries, it could be used to obtain more details regarding its structural conditions. Combining the MIVES model and finite analysis can provide a scientific risk index for managers, which can offer more detailed information by comparing risk assessment results over time or with different rockeries.

Based on assessment results, the most decisive indicators in GG, HAG, LG, and MNG are D3 and D6, D3 and D8, D2 and D3, and D3 and D6, respectively. The results indicate that D11 and D14 have mostly the same values. Therefore, it can be inferred that D3 is the most decisive indicator, and D11 and D14 are the least decisive indicators. The highest weights of D3 highlight the significant impact of crack density on the structural condition of rockeries. A high crack density in these case studies should be brought to the attention of rockeries’ managers. Additionally, while the value functions of some indicators are identical in this study, due to the different values of those indicators, those indicators still have different impacts on the final outcomes.

Although the proposed structural risk assessment method was successfully applied in four case studies, it is vital to highlight the limitations of this method. First, the four case studies were selected from a region south of the Yangtze River in China. Due to variations in climate and material characteristics, the evaluation criteria for rockeries from various regions need to be adjusted. Before using this method in different areas, rockeries’ material properties need to be tested. Second, a finite element simulation of the rockeries was performed with a homogenization model and common material parameters at room temperature to simulate the structural characteristics of the rockeries in an ideal state and to reduce the influence of other factors on the index assessment. Therefore, the unevenness of the rock material was not included in this study.

To inform future research on structural risk assessment methods. A finite-element model under different deterioration conditions should be established for structural simulation and analysis to inform future developments in the structural diagnosis of rockeries. For example, earthquakes, surrounding urban construction activities, and other external activities that have an overall structural impact on rockeries are important for the preventive protection of rockeries. Through a finite element simulation of the overall structural state of the rockeries under different circumstances, the monitoring data of the rockeries could be linked with the finite element analysis data to understand the structural changes of the rockeries in key areas in real time, thereby improving the response efficiency of protection work.

Conclusions

Rockeries, an important part of Chinese classical gardens, contain a wealth of historical and cultural connotations, as well as artistic value. However, their structural statuses have rarely been assessed. This study proposed a structural risk assessment method based on the MIVES model. This method was designed to provide a more detailed level of risk assessment, which assesses the structural risk that has the greatest impact on rockeries. To indicate that the method presented in this study is feasible as a standardized structural risk assessment method for rockeries, it was effectively applied to four rockeries in traditional Chinese gardens. In this study, the MIVES model was used to calculate the structural risk indices of the rockeries. This method converts variables with different units and scales into a set of variables with the same units and scales to calculate a standardized risk index.

Compared to the health risk assessment method developed for rockeries in previous studies, this method could be viewed as a further step in assessing the health condition of rockeries in classical Chinese gardens. Once the initial health risk of a rockery has been determined, a structural risk assessment should be conducted for those with a high-risk rating so that a more targeted response strategy could be adopted. Moreover, this method could generate standard outputs based on a standard objective index, which can facilitate communication of essential elements of recommendations to stakeholders who are not technical. Therefore, it is advisable to use this method to determine the structural risk levels of rockeries in Classical Chinese gardens.